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14 Geomagnetic Storms
Yohsuke Kamide and Yuri P. Maltsev

The present solar-terrestrial research has its roots
in studies of geomagnetic storms, well before thediscovery of the solar wind and various plasma
regions in the Earth’s magnetosphere. Geomagnetic
storms are phenomena which originate in the solar
corona and occur in the entire Sun-Earth system,
including the Earth’s upper atmosphere. A geomagnetic storm is deﬁned by the existence of the main
phase during which the horizontal component of
the Earth’s magnetic ﬁeld decreases drastically. Thisdepression, strongest at low latitudes, is thought tobe caused by the enhanced ring current in the innermagnetosphere. Its growth and decay is typicallymonitored by the Dst index. During a geomagnetic
storm, intense substorms take place successively athigh latitudes. Beginning with a brief description ofobservational characteristics of geomagnetic storms,
this chapter discusses how solar wind energy isdeposited into and is dissipated in the constituent
elements that are critical to magnetospheric andionospheric processes during geomagnetic storms.
Although most of the Dst variance during magnetic
storms can be reproduced by changes in the electricﬁeld in the solar wind and the residuals are uncorrelated with substorms, recent satellite observations of
the ring current constituents show the importanceof ionospheric ions. This implies that ionosphericions, which are associated with intense substorms,
are accelerated upward and contribute to the energy
density of the storm-time ring current. It is thus important to identify the role of substorm occurrencein the enhancement of magnetospheric convection
driven by solar wind electric ﬁeld. In evaluating thecontribution of various current systems in the Dst
decreases during geomagnetic storms, we contend
that not only the ring current but also the tail current
is important.

Contents
14.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

14.2

What is a Geomagnetic Storm? . . . . . . . . . . . . . . . . . . . 356

14.3

Ring Current as a Dominant Signatureof Geomagnetic Storms . . . . . . . . . . . . . . . . . . . . . . . . . . 358

14.4

Solar Wind Causes of Geomagnetic Storms . . . . . . . . 359

14.5

Magnetospheric Geometry During Geomagnetic
Storms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Auroral Electrojets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Auroral Oval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Standoﬀ Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Stable Trapping Boundary . . . . . . . . . . . . . . . . . . . . . . . 363

14.5.1
14.5.214.5.314.5.414.6
14.6.1
14.6.214.6.314.7
14.7.1
14.7.214.7.314.7.414.8

Storm-Time Magnetic Fields and Electric Fields
in the Magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Spatial Distribution of the Electric Currents . . . . . . . 363
Contribution of Diﬀerent Current Systems to Dst . . 364
Storm-Time Electric Fields
in the Magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Discussion on Open Issues of Geomagnetic Storm
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Inﬂuence of the Solar Wind Parameters on Dst . . . . . 367
Inﬂuence of the Substorm Expansion Phase on Dst . 368
What Causes Storm-Time Equatorward Shiftof the Auroral Oval? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Why Does the IMF Southward Component
Aﬀect Dst? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Yohsuke Kamide and Yuri P. Maltsev, Geomagnetic Storms.
In: Y. Kamide/A. Chian, Handbook of the Solar-Terrestrial Environment. pp. – ()
DOI: ./_ © Springer-Verlag Berlin Heidelberg 

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14.1 Introduction
Studies of the solar-terrestrial environment or space
physics have their roots in inspections of geomagnetic
storms in the 19th century. The discipline of thisresearch was called as Geomagnetism, which aimed
at locating and estimating the intensities of electric
currents that generate world geomagnetic disturbances
during geomagnetic storms. Since the end of the 19th
century, it became clear that great magnetic perturbations observed on the Earth’s surface are caused bycurrents ﬂowing in the inner magnetosphere (according
to the present-day terminology) as well as in the regionof auroras. It was in the 1950–60s when the existenceof the solar wind and the magnetosphere was predicted
and subsequently discovered by means of satellite measurements: see Sect. 14.4. The average conﬁgurationof the magnetosphere was modeled: see Sect. 14.7.
A number of plasma regions within the magnetosphere
were discovered and the role of each plasma region in
geomagnetic storm processes was identiﬁed.
It was Alexander von Humboldt (1769–1859) who
ﬁrst used “magnetisches Ungewitter” (magnetic thunderstorms) to describe the variability of geomagnetic
needles, which was associated with the occurrence ofauroras in the polar sky. Humboldt thought that worldwide magnetic disturbances on the ground and aurorasin the polar sky are two manifestations of the same
phenomenon, that is geomagnetic storms. He maintained a lifelong interest in geomagnetic disturbances,establishing a number of magnetic stations around theworld, notably in Britain, Russia, and in lands thenunder British and Russian rule, e.g., at Bombay, Toronto,
and Sitka. It was found by Humboldt that the signiﬁcantdecrease in the daily mean value of the horizontal
component is one of the chracteristic features of thestorm-time disturbance. Scientists during the First
Polar Year (1882–1883) deﬁned “geomagnetic storms”
as intense, irregular variabilities of geomagnetic ﬁeld
which occur as a consequence of solar disturbances.
S. Chapman thought that worldwide geomagnetic
disturbances during geomagnetic storms are a result ofelectric currents encircling the Earth, called the ring
current. It was then proposed that charged particles
originated from the Sun would drift around Earth,causing the decrease in the geomagnetic horizontal
intensity at the Earth’s surface.

The ﬁnal goal of the study of the solar-terrestrial
environment would be to be able to predict the
chain of processes that occur in the entire Sun-Earthsystem (Gopalswamy et al., 2006). Research of geomagnetic storms is the result of a convergence ofmulti-disciplinary sciences which developed fromseveral traditional ﬁelds of research, such as solar
physics, geomagnetism, auroral physics, and aeronomy.
Geomagnetic storms are multi-faceted phenomenathat originate at the solar corona and occur in thesolar wind, the magnetosphere, the ionosphere, andthe thermosphere. What in the solar wind causes
geomagnetic storms? How is the ring current energized
during geomagnetic storms? How do changes in themagnetosphere-ionosphere system aﬀect the Earth’s
upper atmosphere throughout the chain of storm
processes? These are some of the major questions
relating to geomagnetic storms. Because geomagnetic
storms take place in a wide range of plasma regions,they must be understood as a chain of processes fromthe Sun to Earth. It may thus be more appropriate to use
the terminology space storms instead of geomagnetic
storms: see Daglis (2001).

14.2 What is a Geomagnetic Storm?
The characteristic signature of a geomagnetic storm, or
simply a magnetic storm, is a depression in the H component of the magnetic ﬁeld lasting normally over one toseveral days. This depression is caused by the ring current ﬂowing westward in the magnetosphere, and can bemonitored by the Dst index. The Dst index is calculatedby N H − Hq
Dst = (14.1)
N n= cos ϕwhere H is the H component of the magnetic ﬁeld disturbance at a given station, Hq is the same component
over the quietest days, N is the total number of the stations, and ϕ is the station latitude. The cos ϕ factor is tonormalize magnetic disturbances at various latitudes tothe values at the equator.
It is generally accepted that storms are time periods
with a minimum of Dst less than − nT. According tothis criterion,  –  storm events occur annually, depending on solar activity. It is now commonly assumed
that the magnitude of magnetic storms can be deﬁned

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Fig. 14.1. Schematic illustration of the Dst variation fora typical geomagnetic storm. For the deﬁnition of theinitial, main, and recovery phases, see text

by the minimum Dst value (Gonzalez et al., 1994). In
the early years of research, a general picture of a typical magnetic storm emerged which had the features
shown in Fig. 14.1. This picture of the storm involved
a sudden positive increase in the H component (storm
sudden commencement or SSC) followed by a periodof arbitrary length in which the elevated ﬁeld does notchange signiﬁcantly: the initial phase. This phase is followed by the development of a depressed H component
transpiring over a period from one to a few hours: themain phase. The storm concludes by a slow recovery
over hours to tens of hours: the recovery phase.
The SSC is understood as being the eﬀect of a compression of the front side of the magnetosphere byenhanced solar wind pressure. The depression of themagnetic ﬁeld during the main phase was explained bySinger as the eﬀect of a ring current carried primarily
by energetic ions. It decays due primarily to chargeexchange, Coulomb interaction and wave-particle
interaction processes associated with neutral particles
in the volume of space occupied by the ring current
particles.
It was found that the direction of the north-south
component of the interplanetary magnetic ﬁeld (IMF)
regulates the growth and decay of the ring current. In
fact it became possible to reproduce successfully thegrowth of the ring current, i.e., variations of Dst, using,as input, only the component of the interplanetary electric ﬁeld in the ecliptic plane normal to the Sun-Earthline: see Burton et al. (1975). It was further recognizedthat the initial phase simply represented a period of timeafter the onset of the SSC during which the IMF wasoriented northward, i.e., little energy was entering themagnetosphere regardless of the speed and number density of particles in the solar wind. More importantly, it
was suggested that an SSC is not a necessary condition
for a storm to occur and hence the initial phase is notan essential feature to be a geomagnetic storm (Joselyn
and Tsurutani, 1990). In fact, the only basic element ofa storm is the signiﬁcant development of a ring current,
that is the existence of the main phase.
The major question which then arises regarding
the nature of magnetic storms involves the physicalprocesses leading to the growth of the ring current.
This question was apparently answered by Akasofu
and Chapman (1961) who noted that during the main
phase, there was violent auroral electrojet activity in themidnight sector with the amplitudes of the disturbances
there far exceeding the magnetic perturbation associated with the ring current itself. These disturbances at
auroral latitudes, i.e., polar substorms, were in some
way thought to be responsible for the growth of thering current. It was later shown that the storm timering current was carried by energetic ions with energies
typically in excess of several tens of keV (see Williams,
1987). The question of how ring current particles
attain their energies and whether substorms play animportant role in that process are still open. In otherwords, the relationship between substorms and storms
is currently poorly understood, and therefore basic
questions remain unanswered regarding the hypothesisof whether a magnetic storm is simply a superpositionof intense substorms (Akasofu, 1968).
In recent times it has become clear that during geomagnetic storms, signiﬁcant amounts of atomic oxygen are transferred from the auroral ionosphere into theKamide and Maltsev, “Geomagnetic storms” (2007)

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plasma sheet and ultimately form a signiﬁcant component of ring current population. Since charge exchange
processes aﬀect oxygen ions and protons diﬀerently, theobserved decay of the ring current can reﬂect the diﬀerent behavior of the components of the ring current due
to the two ion species.
The description of storm manifestations can befound in the reviews by, for example, Feldstein (1992),
Gonzalez et al. (1994), Kamide et al. (1998), Daglis
et al. (2003), and Maltsev (2004) and proceedings of theconferences, such as Tsurutani et al. (1997; 2006) and
Sharma et al. (2004).

14.3 Ring Current as a Dominant Signatureof Geomagnetic Storms
The Dst decrease, which is the main manifestation ofgeomagnetic storms, is caused by the ring current encircling Earth. The growth of the ring current begins
with the so-called injection of particles into the innermagnetosphere from the magnetotail. The ring current
is carried primarily by energetic ( –  keV) ions in
L   – . The concept of trapping charged particles in
the magnetic ﬁeld by, e.g., Störmer and Singer, was understood well before the discovery of trapped radiation
by Van Allen. See textbooks for charged-particle motions under the inﬂuence of magnetic and electric ﬁelds
in the magnetosphere.
The principal property of a geomagnetic storm is thecreation of an enhanced ring current, producing a decrease in the horizontal component of the geomagnetic
ﬁeld on the Earth’s surface. The strength of this perturbation on the Earth’s surface is approximately givenby the so-called Dessler–Parker–Sckopke relationship(Dessler and Parker, 1959; Sckopke, 1966):
ΔBB = EEm(14.2)

where ΔB is the ﬁeld decrease at the center of the Earthcaused by the ring current, B (. gauss) is the average equatorial surface ﬁeld, E is the total energy of thering current particles, and Em (=  ergs) is the totalmagnetic energy of the geomagnetic ﬁeld outside Earth.
According to the above relationship, the Dst value is, ina ﬁrst approximation, linearly proportional to the totalenergy of the ring current particles. This is the reason

why the Dst index is being used practically as a measure of the magnitude of geomagnetic storms. Vasyliunas (2006) showed that the Dessler–Parker–Schopke
formula can be generalized to include the eﬀect of ionospheric currents.
Parker (1957) established a hydromagnetic formalism, relating the magnetospheric currents to particle
pressures both parallel and perpendicular to the magnetic ﬁeld. The total current j summing over motionsof individual particles is: j = jD + jc , where jD is the drift
current caused by the magnetic ﬁeld gradient and ﬁeldline curvature, and jc is the current driven by gyrationeﬀects within the particle distribution.
Following the discovery by IMP-1 that the Earth’s
magnetic ﬁeld is consistently conﬁned and distorted by
the solar wind, various plasma regions in the magnetosphere, including the ring current and the plasma sheet,were identiﬁed. As energetic particles are injected into
the inner magnetosphere on the night side, they are inﬂuenced by forces due to curvature and gradient of the
Earth’s magnetic ﬁeld. Because of these forces, protons
drift westward from nightside toward dusk and electrons drift eastward from nightside toward dawn, comprising the net eﬀects as a ring current encircling Earthwestward. A geomagnetic storm is nothing but an enhancement of this ring current.
The ring current is not symmetric in local time.
Conducting extensive particle measurements in themagnetosphere, Frank (1967) was the ﬁrst who discovered the asymmetric nature of the ring current.
Measurements of the diﬀerential energy spectrums ofprotons and electrons over the energy range extending
from  eV to  keV were used. The total energy ofparticles of this energy range was found to be suﬃcientto account for the depression of the geomagnetic ﬁeld
in terms of the Dst index. Figure 14.2 shows intensities
of protons as functions of L during the diﬀerent phasesof a magnetic storm. It is evident that a severe increasein proton intensities over  < L < . is apparent in themain phase observations, with a maximum located nearL = ., and that by the recovery phase, this distribution
has substantially decreased in intensities with a peak
positioned at L = ..
It should be noted that Dst includes the magnetic effects not only of the symmetric ring current but of othercurrents, such as ionospheric, ﬁeld-aligned, and tail currents (e.g., Campbell, 1996). Using a numerical model-

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Fig. 14.2. Directional intensities of protons ( < E <  keV)
as functions of L during the pre-storm, main phase, recovery phase, and post-storm periods of an intense geomagnetic
storm. After Frank (1967)

ing of various current systems, Alexeev et al. (1996) havedemonstrated that the ground eﬀect of the tail current
during the main phase of geomagnetic storms can be of
the same order as the ring current. Dremukhina et al.
(1999) have applied this model to major geomagnetic
storms, showing that during the main phase, the contribution of the tail current to Dst is roughly equal to thatof the symmetric ring current, although the ring current
becomes dominant during the recovery phase.
The role of ionospheric particles in the developmentof the ring current during geomagnetic storms became
evident after the CHEM (charge energy mass) spectrometer onboard the AMPTE/CCE mission, which
was the ﬁrst experiment to investigate the near-Earthmagnetotail with multi-species ion measurements
extending in the higher-energy (  keV) range.
The great storm of February 1986 was studied by
Hamilton et al. (1988), showing that 60–% of thering current density near the maximum of the storm
was of ionospheric origin. The next opportunity for
multi-species measurements in the inner magnetosphere was provided by the MICS (Magnetospheric Ion
Composition Spectrometer) experiment onboard the
CRRES mission. Daglis (1997) studied the importanceof the ionospheric ion component in the ring current
during several storms observed by CRRES, showingthat during the main phase of great storms, the abundance of ionospheric-origin ions (O+ in particular) in
the inner magnetosphere is extraordinarily high. The
outstanding feature was the concurrent developmentof Dst and of the O+ contribution to the total particle
energy density: see Fig. 14.3 (Daglis et al., 1999).
The increased relative abundance of ionospheric O+
ions in the ring current during storms, besides inﬂuencing the ring current enhancement, inﬂuences the decay rate of the ring current, since the charge-exchange
lifetime of O+ is considerably shorter than the H+ lifetime for ring current energies. This implies that O+ dominated ring current will decay faster. Such a fast initial ring current decay, associated with a large O+ component during the storm main phase, has indeed been
observed: although “decay” of the ring current can also
occur as a result of the escape of energetic ions throughthe dayside magnetopause. This O+ dominance coupledwith the observations that a signiﬁcant fraction of H+
is also ionospheric in origin, suggests that the cause of
the ring current during great storms lies in the upward
acceleration of ionospheric ions along ﬁeld lines associated with the occurrence of intense substorms. Note,however, that the relative importance of ionosphericorigin and solar wind-origin ions varies considerablyfrom storm to storm.

14.4 Solar Wind Causes of Geomagnetic Storms
The evidence is overwhelming that solar wind dawn-todusk electric ﬁelds drive intense geomagnetic storms:
see McPherron (1997). These electric ﬁelds are caused
by a combination of solar wind velocity and southward
IMF. Of these two parameters, the southward ﬁeld is
probably more important because of its far greater variability. At least, two primary mechanisms are known tobe the main sources of enhanced dawn-to-dusk electric ﬁelds of substantial duration in the interplanetarymedium. They are interplanetary coronal mass ejections(ICME) and corotating interaction regions (CIRs): see
Fig. 14.4a and b, respectively (Tsurutani and Gonzalez,1997). CMEs are impulsive solar/coronal ejecta that occur near the maximum sunspot phase of the solar cycle.
Most geoeﬀective ICMEs are magnetic clouds, a subset

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Fig. 14.3. Time proﬁle of the contribution of H+ and O+
to the total energy density of energetic ion population in
the outer ring current during the great geomagnetic stormof March 1991 (top two panels) and the Dst index (bottom
panel). Adapted from Daglis, I.A., G. Kasotakis, E.T. Sarris, Y. Kamide, S. Livi, and B. Wilken, Variations of the ioncomposition during a large magnetic storm and their consequences, Phys. Chem. Earth, 24, 229–232, 1999

of ejecta characterized by large north-south componentsof the IMF. During the declining phase of the solar cycle, on the other hand, coronal holes tend to dominate,
expanding from the polar regions to equatorial regions:they emit fast plasma continuously, resulting in CIRs.
CMEs are distinct particle and ﬁeld structures withﬁeld orientations which are most numerous near solar
maximum, causing most major geomagnetic storms atthat phase of the solar cycle and possibly at other phases
as well (e.g., Gosling et al., 1991). The ejections that are
most eﬀective in creating magnetic storms are known
to be the ones that are fast, with speeds exceeding theambient wind speed by the magnetosonic wave speed,so that a fast forward shock is formed. As a fast plasmaand ﬁeld structure propagates from the Sun throughinterplanetary space, it sweeps up and compresses theslower plasma and ﬁeld ahead, creating a “sheath” between the shock and the interplanetary manifestation of
the ejecta.

Figure 14.4a illustrates the schematic of the regionsof possible intense IMF for such solar ejecta, in which
two types of possible satellite crossings, T1 and T2, areshown. The sheath ﬁelds leading the fast ejecta often
contain substantial north-south ﬁeld components, possibly due to compression and draping of the ambient
IMF over the ICMEs. Both the remnant ejecta ﬁelds andplasma and those of the sheath can be geoeﬀective, depending on the ﬁeld orientations. Roughly % of fastejecta do not cause major storms, however, because of
the lack of large southward ﬁeld components persistingfor three hours or longer.
During the declining phase of the solar cycle,
another type of solar/coronal event dominates. During
this phase, the coronal holes have expanded frompolar locations and extend into, and sometimes across,the equatorial regions. Fast ( –  kms), tenuous
plasma is continuously emitted from these solar regions. Because these regions are long-lived and evolve

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Fig. 14.4. (a) Schematic of a magnetic cloud region of intense IMF seen typically during solar maximum. T1 and T2 are two types ofsatellite crossings in the interplanetary structure. (b) Schematic of the formation of corotating interaction regions (CIRs) during
the descending phase of the solar cycle. The interaction between a high-speed stream (B) and a slow-speed stream (A) are shown
together with the CIR (shaded). The forward shock (FS), interface surface (IF), and reverse shock (RS) are also indicated. After
Kamide et al. (1998)

relatively slowly, they appear to “corotate” with the Sun.
If a coronal hole is near the ecliptic plane, the Earth’s
magnetosphere will be “bathed” in this stream once
per solar rotation. Typically, a heliospheric neutral
sheet/plasma sheet lies ahead of the fast stream ininterplanetary space. The characteristics of the plasma
sheet wind include low speed ( kms) and highdensity (tens of particlescm ). The interaction of thefast stream with the slow stream ahead creates a CIR.
CIRs are bounded on the leading and trailing edges by
forward- and reverse-propagating compressional waves,respectively. Figure 14.4b shows schematically the formation of CIRs in which magnetic ﬁeld ﬂuctuations arepresent in the high speed stream proper.
In addition to ICMEs and CIRs, there are “modulators” that do not directly drive magnetic storms without
an ICME or CIR, but increase/decrease the geoeﬀectiveness of the ICMEs and CIRs eﬀects through the diﬀerent

phases of individual geomagnetic storms, the sunspot
cycle, and seasons. A 27-day modulation in geomagnetic
activity has been noted since the 19th century. This periodicity, attributed to solar regions called “M-regions”
by Bartels, was later discovered to arise from high-speed
solar wind streams originating in coronal holes. While
ICMEs often contain sustained southward ﬁelds accompanied by fast wind speeds, the high-speed wind fromcoronal holes generally has relatively low ﬁeld magnitude and a radial orientation not conducive to production of steady and substantial north-south ﬁelds. However, the interaction of this fast wind with slower, denser
streamer wind, forming a CIR, produces geoeﬀective
ﬁeld compressions and deﬂections. Geomagnetic storms
associated with CIR-like plasma signatures rarely haveminimum Dst < − nT, and generally lack the sudden commencements often occurring for ICME-driven
storms.

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The coronal hole wind often holds continuous
Alfvénic activity, consisting of large-amplitude quasiperiodic ﬂuctuations in the IMF orientation, in-phasewith similar ﬂuctuations in the ﬂow direction, withperiods from tens of minutes to a few hours. In theinterplanetary region following CIRs, the southward
ﬁeld components caused by these waves can cause
magnetic reconnection, small injections of plasma
into the magnetosphere, and prolonged recovery
phases of the storms. Events of this type are known as“high-intensity, long-duration, continuous AE activity”(HILDCAA) events (Tsurutani and Gonzalez, 1997).
Although the average Bz component in HILDCAAs
is zero, the half-wave induced reconnection in themagnetospheric response results in a continuous
occurrence of substorms or other disturbances in themagnetosphere.

14.5 Magnetospheric Geometry
During Geomagnetic Storms
The magnetic conﬁguration in the magnetospherechanges signiﬁcantly during geomagnetic storms, withthe magnetic ﬁeld lines at the nightside stretching,magnetic ﬂux in the tail increasing, and dayside polarcusps shifting to lower latitudes.

14.5.1 Auroral Electrojets

storms because of the equatorward shift of the auroralelectrojets from the standard AE observatories.

14.5.2 Auroral Oval
The latitude of the particle precipitation regions, e.g. of
the auroral oval, also decreases as the storm develops.
The equatorward edge of the auroral oval can be approximated by a circle centered at midnight at the latitude of  under quiet conditions and  under moderately and strongly disturbed conditions (Starkov andFeldstein, 1967). The radius of the circle exhibits a very
good correlation with Dst. Meng (1984) studied the behavior of the equatorward edge of the auroral oval in thecourse of intense storms, revealing that the latitude ofthis edge follows approximately the Dst variation.
The points in Fig. 14.5, taken from Starkov (1993),
indicate the observed latitudes of the equatorward edgeof the auroral oval in the dusk-to-midnight sector, thedashed line being the least-squares approximation:Λ = . − . log Dst .

(14.4)The poleward boundary of the auroral oval also
moves to lower latitudes with storm intensifying,although the correlation with Dst is lower than for
the equatorward boundary. There is a class of auroras
associated with energetic proton precipitation, which
also shifts equatorward.

The auroral electrojets are known to shift equatorward
drastically during geomagnetic storms. During themain phase of intense storms, the westward electrojet
can cover the latitudes from  to  on the nightside, its total intensity reaching often as intense as MA. The eastward electrojet ﬂows in the dusk sector
at latitudes lower than those of the westward electrojet.
With Dst varying from 0 to − nT, the minimum
latitude appeared to lower down from  to (L changed from 7 to .), the rate of the equatorward
shift subsiding with storm intensity increasing.
Feldstein et al. (1997) suggested the following approximation for the corrected geomagnetic latitude Λ ofthe westward electrojet center under Dst:
Λ = . + . Dst .

(14.3)

Note that the AE indices underestimate the intensityof the auroral electrojets during intense geomagnetic

Fig. 14.5. Latitude of the equatorward edge of the auroral oval in
the dusk-to-midnight sector versus storm intensity. The points
indicate observations, and the dashed line is the least-squares
approximation ﬁt. After Starkov (1993)

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14.5.3 Standoff Distance
Up to distances of RE downtail, the magnetopause
can be well approximated by an ellipsoid. The standoﬀ
distance rs (the distance to the subsolar point) is controlled mostly by the solar wind dynamic pressure p.
Under weakly disturbed conditions this distance can betaken as (Roelof and Sibeck, 1993)
rs = rsa (pa p)

(14.5)

where rsa  RE is the average standoﬀ distance, andpa  . nPa is the average solar wind dynamic pressure. With the growth of geomagnetic activity, rs mostcommonly decreases due to an increase in p and otherfactors. Dayside magnetospheric erosion, deﬁned as inward magnetopause motion under invariable solar winddynamic pressure, is a phenomenon closely connectedwith a storm time decrease of the cusp latitude. A decrease in rs under p being constant begins once the IMF
turns southward.
A minimum standoﬀ distance rs = .RE wasreached during the strongest storm of the 20th century
on March 13, 1989, when Dst was as low as − nT.
During a storm on February 8–9, 1986 the daysidemagnetopause was detected at .RE (Hamilton et al.,1988), Dst being − nT.
Rufenach et al. (1989) studied 64 magnetopausecrossings by a geosynchronous satellite (r = .RE ).
Although the exact location of the subsolar point was
not known because MLT of the crossing sites varied
from 06 to  MLT, it can be inferred that the average rs
was smaller than .RE , considering that the subsolar
point is nearer to Earth than any other magnetopause
point. The average Dst for these 64 crossings was− nT, and the average solar wind dynamic pressurep   nPa.

14.5.4 Stable Trapping Boundary
The main energy content of the ring current is provided
by 10 to  keV protons. The ring current is formedby trapped particles. Under stationary conditions, thedrift trajectory of the particles with  pitch-angle isdescribed by the equation qΦ + μB = constant, where q
is the charge of a particle, Φ is the electric potential,
μ = w B is the magnetic moment (the ﬁrst adiabatic
invariant), and w is the energy of a particle perpendicular to the magnetic ﬁeld B. For energetic particles,

which are weakly inﬂuenced by the electric ﬁeld, thedrift trajectories in the equatorial plane coincide nearlywith the contours B = constant. It should be pointed
out, however, that in the ring current region the electric ﬁeld is reduced due to the shielding eﬀect of theﬁeld-aligned currents. For energetic particles with 
pitch-angle, the stable trapping boundary is the contour B = Bs , Bs being the magnetic ﬁeld at the subsolar point. This boundary is somewhat diﬀerent for particles with other pitch-angles, but we can consider thatthe majority of the trapped particles are inside the contour B = Bs .
During intense magnetic storms this contour approaches Earth. On the nightside the contour B = Bs
is displaced from RE under quiet conditions up to RE
under Dst = − nT. A decrease in the stable trapping area leads to contraction of the radiation belts. Under quiet conditions maximum ﬂux of high-energy (E 
 MeV) electrons is observed at L  . With storm intensifying, however, the high-energy electron belt is approaching Earth.

14.6 Storm-Time Magnetic Fields and Electric Fields
in the Magnetosphere14.6.1 Spatial Distribution of the Electric Currents
The overall pattern of large-scale electric currents includes the magnetopause currents and the currents inside the magnetosphere, both transverse and aligned tothe magnetic ﬁeld. The magnetopause currents shieldmagnetic ﬁelds associated with the sources inside Earth.
There are two regions of the large-scale ﬁeld-aligned
currents (FACs): the Region 1 FACs, ﬂowing at the polar cap boundary, and the Region 2 FACs at the equatorward edge of the auroral oval.
The currents that are transverse to the magnetic ﬁeld, are conﬁned to the near-equatorial layer,− < z < RE . Maltsev and Ostapenko (2004) usedthe database of Fairﬁeld et al. (1994) to restore thespatial distribution of the currents under various levels
of Dst. The results are shown in Fig. 14.6. One cansee that the longitudinal distribution of the westward
electric current is quite asymmetric in the day-night
direction, the nightside current intensity being several
times greater than that at the dayside. The dawn-dusk
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Fig. 14.6. Isocontours of the eastward current ﬂowing in the layer − < zSM < R E . The currents are expressed in kARE . Negative
contour levels indicate that currents ﬂow westward. After Maltsev and Ostapenko (2004)very sensitive to either geomagnetic activity or solar
wind conditions, with the maximum current located at − RE . The following approximations were obtained
for the total westward current ﬂowing between radial
distances of 4 and RE at the nightside and dayside,
respectively
Inight (MA) = . − . Dst ,Inoon (MA) = . − . Dst .

(14.6)(14.7)

Note that the model of Tsyganenko (2002) predicts
nearly a consistent response of these currents to Dst:
dInight  dDst = −., dInoon dDst = −..

14.6.2 Contribution of Different Current Systems to Dst
There are basically ﬁve types of magnetosphericcurrents, as schematically shown in the left panel ofFig. 14.7: the magnetopause currents shielding themagnetic ﬁeld of the Earth’s dipole; symmetric ring
current; magnetotail current system including thecross-tail current and closure currents on the magnetopause; Region 1 FACs closed in the ionosphere andon the magnetosphere ﬂanks or in the solar wind; andRegion 2 FACs closed through the partial ring current.
Each of the ﬁve basic current systems is closed. One cansee from the right column that, except for the Region 1
FACs, all current systems contribute to Dst, i.e.

Dst = k(δBmp + δBRC + δBct + δBpr )(14.8)

where the four quantities on the right-hand side designate the variations of magnetic disturbances at the
Earth’s center produced by the divergence-free part ofthe magnetopause currents, symmetric ring current,
cross-tail current along with the closure currents on themagnetopause, and partial ring current, respectively;
and k ( .) is related to the induction currents within
Earth.
The magnetic ﬁeld of the shielding current on themagnetopause can be found from the Mead (1964)
mpmodel: Bz ()  .Bs , where Bs is the magnetic ﬁeld
at the subsolar point on the magnetopause. The ﬁeld Bs
can be obtained from the pressure balance condition as
Bs = μ  psw , in which μ  is the magnetic permeabilityof vacuum, and psw is the solar wind pressure at thestagnation point that is nearly equal to the dynamic
pressure of the solar wind protons.
Note that the noon-side current (14.7) presents
the purely symmetric part of the ring current, without
a contribution of either partial ring current or cross-tailcurrent. A symmetric ring current of  MA magnitude
ﬂowing at distance of RE produces the disturbanceBrcz ()  − nT at the Earth’s center. Multiplying (14.7)
by −, we haveBrcz () = −. + . Dst .

(14.9)

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Fig. 14.7a–e. Electric currents (the left column, equatorial plane)
and the corresponding magnetic ﬁelds (the right column,noon-midnight meridian plane) of the ﬁve basic current systems. From top to bottom: (a) the divergence-free part of themagnetopause currents, (b) symmetric ring current, (c) crosstail current (the closure currents on the magnetopause are notshown), (d) Region 1 FACs, and (e) Region 2 FACs with thepartial ring currentWith the induction currents (k  .) included, we obtain the ground magnetic eﬀect to be
kBrc
z () = −. + . Dst(14.10)indicating that the contribution of the symmetric ring
current to Dst turns out to be %.
Subtracting (14.7) from (14.6) we can obtain thesum of the partial ring current (PRC) and near-Earthcross-tail current (NCT)I PRC + I NCT = . − . Dst .

(14.11)

Assuming that this combined current is conﬁned within
18 to 06 local time sector in the nightside, at distance of
6 RE , each MA of the current produces a disturbance of− nT in the Earth’s center or −  . (= k) = . nT on
the Earth’s surface at low latitudes. The corresponding
relationship of the disturbance to Dst becomes the formk BPRC= − + . Dst .
+ BNCTzz(14.12)

Consequently, the total contribution of the partial ringand near-Earth cross-tail currents to Dst is about %.
Using the paraboloid magnetospheric model, Alexeev
et al. (1996) and Dremukhina et al. (1999) predicted
about % contribution of the magnetotail current to
Dst.
The contributions of various currents to Dst can beevaluated not only from magnetic data but also from energetic particle observations. In particular, the symmetric ring current eﬀect can be estimated from the wellknown Dessler–Parker–Sckopke formula (Dessler andParker, 1959; Sckopke, 1966)
Brcz () = −μ  ERC
π ME(14.13)

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Y. Kamide and Y.P. Maltsev

where w is the plasma energy density, and dV is the elementary volume.
The energy density is related to pressure as

the ratio between the dayside and nightside currents is
reduced, so that the symmetric ring current contribution to Dst dropped to %.
Presumably, the discrepancy between the results ofthe two extensive statistical studies can be attributed tothe diﬀerent types of satellite orbits. The AMPTE/CCE
orbit was near the equatorial plane, so that all thetrapped particles at a given L-shell were included.
POLAR, whose orbit is limited in the meridian plane,
however, could observe only part of the trapped
particles.

w = p + p

14.6.3 Storm-Time Electric Fields in the Magnetospherein which ERC is the total energy of the particles trappedin the dipolar magnetic ﬁeld, and ME is the Earth’s magnetic moment. This is equivalent to Eq. (14.1). The totalenergy is given by
ERC =  w dVVwith p and p being the perpendicular and parallel (tothe magnetic ﬁeld) pressure, respectively. Under quiet
conditions, these quantities are maximum near L = (p   nPa, p   nPa) both in daytime and nighttime (Lui and Hamilton, 1992). At L =  the pressure
turns out to be nearly isotropic (p  p   nPa).
A statistical test of relation (14.13) was performedby Greenspan and Hamilton (2000). During 80 storm
events, AMPTE/CCE registered the energy content ERC
at distances from L =  to L = . Figure 14.8 shows theresult of the Dst dependence on ERC . The left panel corresponds to the 40 storms during which the satellite was
located on the nightside, while the right panel is referredto the 40 storms with the measurements performed on
the dayside. One can see the lack of any correlation between the dayside energy content and Dst, but at thenightside there is a high correlation. We should keep inmind that it is solely the symmetric ring current thatﬂows on the dayside, while on the nightside, there arealso cross-tail and partial ring currents. The right panelof Fig. 14.8 suggests that the symmetric ring current ispractically not related to Dst and can hardly be considered as a principal cause of the geomagnetic depression.
The second statistical test of (14.13), performed byTurner et al. (2001), yields somewhat diﬀerent results.
The authors restore the total energy content ERC in four
MLT sectors from particle observations by the POLAR
satellite. Their statistics included more than 1000 satellite passes through the ring current region along the polar orbit at altitudes from  to RE . The time period
covered was two and a half years. For moderately disturbed conditions (Dst  − nT), Turner et al. (2001)
revealed nearly symmetric ring current, which provided
% contribution to Dst. For storms with Dst = − nT,

The spatial distribution of the electric potential consists primarily of two cells, with maximum/minimum
located at the dawn/dusk polar cap boundary. On the basis of Dynamics Explorer-2 (DE2) observations, Weimer(1995) constructed the ionospheric potential distributions for various IMF clock angles in the GSM system:
see Fig. 14.9. The necessary condition for the initiationof the storm main phase is a prolonged southward IMF
of large values, so that the electric potential distribution
in the bottom row of Fig. 14.9 is typical for storm conditions.
The potential drop U between the centers of thedawn and dusk cells grows with the southward IMF increasing. In the study of Doyle and Burke (1983) the following empirical relation was obtained:
U(kV) = . + Ey (mVm)(14.14)

where Ey = −V Bz is the duskward component of the interplanetary electric ﬁeld, and V is the solar wind speed.
According to Boyle et al. (1997), however, the approximation that ﬁts best has the form
U = − V  + .B sin (θ)(14.15)in which B is the IMF modulus, V is the solar wind speed
in km/s, and θ = arccos(Bz B)GSM .
During moderate storms the potential drop acrossthe polar cap varies in a certain range, with the averagevalue rarely exceeding  kV and only rarely reaching kV. At the main phase of a storm on July 8–9, 1991
(Dstmin = − nT), the value of U was greater than kV, according to DMSP observations in the innermagnetosphere (Wilson et al., 2001). Magnetospheric第 4 章 論文集

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Fig. 14.8. The Dst index versus the energy content (dots)
according to AMPTE observations: (Left) at the nightside, (Right) at the dayside. The solid lines present thedependence expected from the Dessler–Parker–Sckopke
formula with the amplifying coeﬃcient k = . due tothe induction currents within Earth. After Greenspan and
Hamilton (2000)

observations show at times a signiﬁcant enhancement of
the electric ﬁeld during superstorms. A very strong relative increase (nearly 60 times) in the electric ﬁeld wasobserved by CRRES during a storm on March 24, 1991
(Wygant et al., 1998).

14.7 Discussion on Open Issuesof Geomagnetic Storm Dynamics
14.7.1 Influence of the Solar Wind Parameters on Dst
In examining storm time behavior of Dst, we recognizethat this index is typically presented as a sum of therapidly and slowly varying components as
Dst = DCF + Dst (14.16)

where DCF includes the rapidly varying (on a time scaleof a few minutes) component, which is controlled by
the solar wind dynamic pressure; and Dst  is the component varying on the time scale of more than several
hours. It is generally accepted thatdyn
DCF = a psw(14.17)dyn

where a is a coeﬃcient and psw is the dynamic pressureof the solar wind protons. In the study of O’Brien and
McPherron (2000) using observations over the periodof 30 years, it is shown that the value of a amounts
to . nT(nPa) . During storms occurring undersouthward IMF, the value of a decreases by a factor of 2.
The behavior of the slowly varying component of
Dst is commonly described by the following diﬀerential
equation:dDst 
Dst =Q−(14.18)dt
τwhere τ is the relaxation time, and Q is the coupling
function (i.e., the rate of energy injections) resulting
from solar wind-magnetosphere coupling. Burton et al.
(1975) found that Q is linearly proportional to the
duskward component of the solar wind electric ﬁeld.
Later a number of studies were performed aiming torelate the coupling function to other solar wind parameters. Table 14.1 summarizes the principal features of Q
and τ for the main phase (τmp ) and recovery phase (τrp )of storms.
From Table 14.1 one can see that there is a deal ofcontroversy with regard to the functional forms of Q andτ. An attempt was made to minimize this misleading effect by Maltsev and Rezhenov (2003). For this purpose
one can treat the relationship between dDst   dt anda certain solar wind parameter with all the others varying in a narrow range. More than , hourly valuesof the solar wind observations from the OMNI databasewere processed. It was found that (1) τ is governed by
the electric ﬁeld component Eyr = −V Bs , not by Dst  ,(2) a dependence of dDst   dt on the solar wind speed
persists both for southward and northward IMF, (3) nonoticeable correlations with the solar wind density or
horizontal IMF component were revealed, and (4) the
dependence on the ε parameter under the electric ﬁeld
Eyr = −V Bs being ﬁxed appeared to be weaker than thaton the electric ﬁeld Eyr under constant ε. The results canbe presented as Q = .−Eyr (.+Dst  .) −V and τ = . hrs. Most of the results presented in Table 14.1 suggest nearly linear coupling of Dst with the solar wind parameters and strongly non-linear decay, dependent on both Dst and Q. However, the technique forcalculating Q and τ does not seem to yield a unique solution, and we may have to enforce non-linear coupling
and linear decay.

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Y. Kamide and Y.P. Maltsev

Fig. 14.9. The potential distribution in the high-latitude ionosphere as a function of the IMF clock angle. After Weimer (1995)

14.7.2 Influence of the Substorm Expansion Phase on Dst
A substorm is a magnetospheric process of energy accumulation in the magnetotail as a result of solar windmagnetosphere coupling, followed by a subsequent explosive release of energy onto the polar ionosphere andinto the inner magnetosphere (Rostoker et al., 1980).
A typical substorm lasts from 1 to 3 hours. The stageof the energy accumulation, i.e., the growth phase, is ac-

companied by stretching of magnetic ﬁeld lines tailward.
The expansion phase is signiﬁed by dipolarization of themagnetospheric magnetic ﬁeld, and by a sudden brightening of the auroral arcs and their rapid expansion poleward and westward and intensiﬁcations of the auroralelectrojet in the ionosphere. Field-aligned currents increase with the onset of substorm expansion phase. They
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Table 14.1. Coupling function Q, which relates the Dst index to solar wind parameters, and relaxation times of the storm-associatedcurrents τmp and τrp for the main and recovery phases, respectively, as summarized by diﬀerent studies
No ReferencePudovkin et al. ()Q

τmp

Q = −. + .V (. − Bz ) for V (. − Bz )  .  −τ = . + . e

τrp−Q.

τ = . + . Dst  

Q =  nThrfor   V (. − Bz )  .  
Q =  for   V (. − Bz ) < Feldstein ()Q = .  − V (Bz − .σ)− .  − (V − ) + .
for V (Bz − .σ) < −Weak and moderate storms(Dst   − nT):
τ = . +  e.Qτ = . +  e. Dst

Q = − .  − (V − )for V (Bz − .σ)  −Strong storms (Dst  < − nT):
τ = . +  e.Qτ =  + . e. Dst−ε

τ =  hr for Dst  − nT
τ = . hr for −  Dst  − nT
τ = . hr for Dst < − nT

Gonzalez et al. ()Q



Valdivia et al. ()Q = −.E yr



O’Brien and McPherron () Q = −. (E y − .) for V Bs  . mVm τ = . exp . . + E yr

τ = .( − . Dst )

Q =  for V Bs < . mVm
τ = .( + .E yr )
 Maltsev and Rezhenov () Q = . − .E yr − V Q is expressed in nT/hr; τ is in hrs, Bz and Bs as well as the IMF variability σ are in nT; the electric ﬁeld component E y = −V Bz
is in mVm, the solar wind velocity V is in kms; the quantity Eyr = −V Bs is the duskward electric ﬁeld component; Bs is thesouthward IMF component in the GSM coordinates, with adopting Bs = Bz under Bz <  and Bs =  under Bz  ; andε = V B  sin (θ) l o where B is the IMF modulus, θ = arctan(By Bz ), and l o  RE is the eﬀective transverse size of themagnetospherering current component in substorm periods (Daglis
et al., 1997).
Nearly all storms at the main phase are accompanied
by intense substorms. Among rare storm events not accompanied by substorms are those occurred, for example, on November 24, 1981 (Yahnin et al., 1994) and on
July 15, 1997 (Zhou et al., 2003).
A statistical study of storm-substorm relationships
was presented by Iyemori and Rao (1996). Using the superimposed epoch analysis of more than 100 substorms,
they built average SYM-H index that is a one-minute resolution analog of Dst, having taken the substorm expansion onset for a zero moment. The result is shown separately for substorms at the storm main phase and thoseat the recovery phase, but for both phases, SYM-H (orDst) was found to show a slight weakening after the expansion onset of substorms.

Huang et al. (2004) examined a number of samplesof magnetic ﬁeld dipolarization associated with substorm onsets in the nightside magnetosphere, showingthat all of them lead to a noticeable increase in Dst by –  nT, i.e. to storm subsiding.
These observations have led a number of researchers
to the conclusion that there is no direct relation between substorms and storms, although both phenomena
develop under the same condition, namely, the southward IMF. The average substorm initiates under IMF
Bz  − nT, while the main phase of the average storm
proceeds under IMF Bz  − nT. This implies that an
average storm is accompanied by substorms with theintensity higher than the average intensity. Although
the ring current must be strengthened by energized ion
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eﬀect of magnetotail current reduction at substorm expansion onsets. This can explain why the resulting geomagnetic depression, as measured by Dst, weakens after
substorm explosive onset.

14.7.3 What Causes Storm-Time Equatorward Shiftof the Auroral Oval?
It is generally accepted that the equatorward edge of theauroral oval at the nightside is coincident with the outerboundary of the trapping region of energetic electrons.
In Fig. 14.10, the region of the trapped energetic electrons is shown to be conﬁned to the inner magnetosphere, while the auroral oval is associated with moredistant magnetospheric domains. We should then explain why the boundary between the inner and outermagnetospheric regions moves earthward during magnetic storms. Within the framework in Fig. 14.10, sucha reconﬁguration of the magnetosphere is caused by thelarge-scale currents in the magnetosphere.
It is clear from Fig. 14.7 that the FACs produce
mostly a longitudinally asymmetric distortion of thetrapping region boundary, without changing significantly its radial size. In the model of Alexeev et al.
(2000) the Region 1 FAC, whose intensity is  MA,
displaces the auroral oval sunward by  on the daysideand by  at night. The shielding currents on themagnetopause are capable of decreasing the extent of
the inner magnetospheric region but cannot change

noticeably its ionospheric projection. Thus a signiﬁcant decrease in the auroral oval latitude can only beachieved by redistribution of the magnetic ﬂux in themagnetosphere, speciﬁcally by its enhancement in theouter magnetosphere.
The ﬁrst attempt to interpret such a magnetic ﬂux redistribution in terms of the ring current (Siscoe, 1979a)
indicated that such a redistribution could be accomplished only if the ring current ﬂew beyond the stabletrapping region. In other words, it is the cross-tail current that should be considered as an ultimate cause ofthis eﬀect. In Siscoe (1979b) the contribution of the ring
current, deﬁned as a current ﬂowing within the stabletrapping region, to the observed storm time enhancement of the magnetic ﬂux in the polar cap was estimatedas only %. Schulz (1997) suggested that every  nTof the magnetic disturbance in the Earth’s center associated with the ring current (or  nT with the induction currents inside the Earth included) can displace theauroral oval toward the equator by  . From Fig. 14.5,
however, it is seen that in reality this displacement is, atleast, two times greater.
The reason why the ring current can only slightly affect the magnetic ﬂux redistribution between the innerand outer magnetosphere is that the area of the magnetic depression region, associated with the ring current, is rather small, compared to that of the cross-tailcurrent.
The total eﬀect of all magnetospheric currents canbe expressed as follows (Maltsev et al., 1996):
Bext
z () =

Fμ  psw + Brc
z () −
S

(14.19)where psw is the solar wind dynamic pressure in thestagnation point on the magnetopause, and Brcz () isthe magnetic ﬁeld of the ring current; and the physicalmeaning of the magnetic ﬂux F and cross-section S is
illustrated in Fig. 14.10.
The high-latitude magnetic ﬂux can be written as
Fig. 14.10a,b. Magnetospheric model used in the calculationsof storm time geomagnetic depression: (a) equatorial plane,(b) view from the duskside. S is the equatorial cross-sectionof the inner magnetosphere conﬁned by the contour B = Bs ,Bs being the magnetic ﬁeld in the subsolar point on the magnetopause. F is the magnetic ﬂux in the outer magnetosphere.
The outer magnetosphere contains both the closed magnetic
ﬁeld lines of the plasma sheet and open ﬁeld lines of the magnetotail lobes

F = πRE (Be ) sin θ a(14.20)where Be = , nT is the dipole magnetic ﬁeld at
the Earth’s equator, θ a is the colatitude of the equatorward edge of the auroral oval averaged over longitude.
The area of the equatorial cross-section S is equal toS = πrs , where rs is the radius of the contour B = Bs .
In the case of dipolar magnetic ﬁeld, rs = RE  sin θ a .

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For strong storms, however, this expression transfers to
rs = ()RE  sin θ a (Maltsev et al., 1996). As a result,
we have for storm-time conditions
S= πRE
.
 sin θ a(14.21)The equatorward boundary of the auroral oval can beapproximated by a circle centered at latitude of  at
midnight during disturbed periods, i.e.,Λ =  − θ a .

where C is the electric ﬁeld circulation along the contour
B = Bs , which separates the inner and outer magnetospheric regions: see Fig. 14.10. The circulation C can bedivided into the dayside and nightside portions:(14.22)

By combining (14.19) through (14.22), one can obtain
the relation of Λ to Dst = kBext
z (). The solid lines inFig. 14.11 show the dependence Λ (Dst) for two levelsof the solar wind dynamic pressure: psw =  and  nPa.
The dashed curve shows the observed dependence fromempirical formula Eq. (14.3). It is seen that the observed
dependence is close to the calculated one, indicating thatthe cross-tail current, or the magnetic ﬂux in the tail,is as important as the ring current. A major role of themagnetotail currents in the storm time decrease of theauroral oval latitude has also been pointed out by otherobservations.

14.7.4 Why Does the IMF Southward Component
Affect Dst?
A theoretical framework for the coupling function Q
has been developed by Arykov and Maltsev (1996) who
concentrated on the magnetotail current eﬀect on Dst,by including the cross-tail current, closure currents on
the magnetopause, and the partial ring current. Duringstorms, the third term (−FS) on the right-hand sideof (14.19) dominates. Having diﬀerentiated (14.19)
with respect to t and keeping in mind that, according
to (14.20) and (14.21), S is proportional to F − , we
have
dBext
 dF
z ()
=−
.
(14.23)dt
S dt
It is well known that the magnetic ﬂux in the tail grows
when the IMF is southward. According to (14.23), thisresults in strengthening the geomagnetic depression
kBext
z (). The high latitude magnetic ﬂux satisﬁes theMaxwell equationdF=C(14.24)dt

C = Cday + Cnight .

(14.25)For the dayside portion we adopt
Cday = U(14.26)

where U is the convection-associated potential drop between the dawn and dusk boundaries of the inner magnetosphere. The nightside portion Cnight can be presented as
F − F
Cnight = −(14.27)
τF
where F is the undisturbed quantity of the ﬂux, τF is therelaxation time. With the induction currents inside theEarth included, the geomagnetic disturbance H is
H = kBext
z ()(14.28)where k  .. The potential U is equal toU = χUPC(14.29)where UPC is the convection potential drop between
the dawn and dusk ﬂanks of the whole magnetosphere, which map onto the polar cap boundary, andχ is a certain coeﬃcient (χ < ). After substituting(14.24)–(14.29) into (14.23), we havedH
HF
= Q − −k
dt
τF
SτF
where

(14.30)UPC(14.31)
S
is the coupling function, which relates the magnetotail
current to the solar wind conditions.
Numerous studies have explored a statistical relation between the UPC and solar wind parameters: see
Table 14.1. To compare this theoretical elaboration withobservations, we can use empirical formula (14.14)
by Doyle and Burke (1983). Substituting (14.14) into
(14.31), with k = ., χ = ., and S = .   m(corresponding to the circle with a radius of .RE ),we obtain Q(nThr) = −.Ey (mVm) − . One
can see that the thus calculated coupling function is
Q = −k χKamide and Maltsev, “Geomagnetic storms” (2007)



227

Y. Kamide and Y.P. Maltsev
Fig. 14.11. Latitude of the equatorward edgeof the auroral oval Λ versus magnetic storm
intensity. The solid lines indicate the calculations, and the dashed line refers to the observations by Feldstein et al. (1997). After Maltsev et al. (1996)

close to what O’Brien and McPherron (2000) foundempirically. Thus, it is quite possible that the relationof geomagnetic depression to solar wind parameters bealmost entirely interpreted in terms of the magnetotail
current system.
From comparing (14.18) with (14.30), we can seethat τ = τF . That is, the decay of the storm time geomagnetic depression is related to relaxation of the magnetic ﬂux in the tail. A theoretical framework for thisprocess has yet to be developed.

14.8 Summary
Space storms, or geospace storms, are multi-faceted
phenomena that occur in the entire solar-terrestrial
environment. In this sense, geomagnetic storms are
only one of the manifestations of space storms. Beginning with a brief description of the average characterof geomagnetic storms along with some historicalaccount, this chapter describes magnetospheric-wide
dynamical changes during geomagnetic storms,
which are caused directly and indirectly by thesolar wind. Several unsolved questions are also discussed.
Acknowledgement. The authors would like to thankY.I. Feldstein and I.A. Daglis for their constructive
comments on an earlier version of the manuscript. This
work was supported in part by “A Cooperative Research
Program Between Japanese and US Space Weather
Centers of Excellence” (grants AOARD-- and-) of the Asian Oﬃce of Aerospace Researchand Development (AOARD), and in part by the Grantin-Aid for Creative Scientiﬁc Research “The Basic
Study of Space Weather Prediction” (grant GS,PI K. Shibata) of the Ministry of Education, Culture,

228

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Sports, Science and Technology (MEXT) of Japan.
This work was also supported in part by the Russian
Basic Research Foundation (grants -- and---BNTS) and by the Division of PhysicalSciences of the Russian Academy of Sciences (program
DPS-).

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FROM IGY TO eGY: THE IMPORTANCE OF REAL-TIMEDATA IN SPACE PHYSICS*
Y. KAMIDE
Solar-Terrestrial Environment Laboratory, Nagoya University
Nagoya, Aichi 464-8601, Japankamide@stelab.nagoya-u.ac.jp
Reviewing the history of space physics, particularly of solar-terrestrial relationships, overthe last fifty years from its birth at the International Geophysical Year (IGY, 1957-1958),
this paper emphasizes the importance of collecting data from space and from around theworld and processing these data, all in real time. These real-time data are essential to run
computer simulations under realistic boundary conditions and thus to make space
weather predictions. The availability of real-time data combined with real-time
simulations not only changes the practical style of research but also changes the conceptof solar-terrestrial physics in such a way that it becomes increasingly imperative tounderstand the whole spectra of processes, especially coupling between different plasma
regions and coupling between large-scale and small-scale processes. It is quite timelythat the scientific community is planning to conduct the electronic Geophysical Year
(eGY), because this international enterprise is harmonized with the rapid growth inmodern networking technologies and high-quality observation techniques.

1. Introduction
When the author entered the graduate school of the University of Tokyo, one of
the first subjects he learned about the Earth’s magnetosphere and its dynamics
was through the Axford/Hines theoretical model of plasma convection. Figure 1
shows a schematic illustration of the convection model Axford and Hines1
published in 1961. They proposed that plasma convection in the magnetospheremust be driven through viscous-like interactions between the solar wind andmagnetospheric plasmas, resulting in two large-scale vortices centered in thedawn and dusk hours of the magnetosphere. It was suggested that the driving
force of large-scale convection motion is associated with the tangential stressexerted by the solar wind on the magnetospheric surface, which was calledviscous-like interactions between solar wind and magnetospheric plasmas.

*

Based on the First Axford Lecture, delivered on July 10, 2006 at the occasion of the Third Annual
Meeting of the Asia and Oceania Geosciences Society held in Singapore.
1

Kamide, “From IGY to eGY: The importance of real-time data in space physics” (2007)
2

Fig. 1. Streamlines of plasma convection in the equatorial plane of the magnetosphere, proposed by
Axford and Hines in 1961. After Axford (1964).

Figure 2a is a projection of the convection pattern onto the polar ionosphere,
consisting again of two vortices. Figure 2b is the convection pattern in theionosphere calculated primarily from ground-based magnetometer data using amagnetogram-inversion technique: see also Figure 9. In both plots the so-called
twin-vortex patterns consisting of high and low potential vortices in the morningand evening sectors, respectively, can clearly be identified, although somedeformations in Figure 2b represent individual complications. Some of thedeformations originate from inhomogeneities in the ionospheric conductivity,
indicating that the ionospheric convection is not a simple geometrical projectionof magnetospheric convection along magnetic field lines. In a sense, what many
scientists in the field have been doing over the last 40 years is chasing the
Axford/Hines convection model by different means, such as radar, satellite andground-based measurements.
The present author is one solar-cycle younger than Axford. That is, when
Axford began his doctoral studies in 1957 at the beginning of the International
Geophysical year (IGY), I was an elementary school boy in Sapporo, Japan.
Fifty years ago, as I was about to enter a junior high school, I was fascinated by231

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3

a

b

Fig. 2a. Projection of magnetospheric convection onto the polar ionosphere. Points A and B canbe found in Fig. 1. After Axford (1964).
Fig. 2b. Example of the electric potential distribution calculated from real-time groundmagnetometer data through the KRM algorithm.

spectacular natural phenomena and was trapped by a number of questions
surrounding us, such as why ocean waves get excited even without strong winds,
what is the cloud formation mechanism generating different shapes, how lightfrom the Sun travels the long distance to the Earth, and how auroras are formed.
During the course of my “elementary school and junior high school” years,
which coincided with the IGY, I faced many interesting articles in newspapers
and science magazines about space and Antarctic expeditions: TV was not
popular at all at that time. Exciting to a young boy, these did bring much newknowledge about solar-terrestrial relationships into my world. For example, I
was very curious about auroral displays. Why are auroras most active on thenightside when the Sun is on the dayside? It was about that time that I, more or
less, decided that I wanted to be an Earth and space research scientist when I
grew up.
Looking back over the history of space physics, this paper presents theauthor’s personal view on the importance of conducting well-organized spaceground observations and of handling the resulting data products in real time.
The author does not pretend that this paper is unbiased toward the author’s own
experience, but he tries to recount some of the major accomplishments whichhave led to our present ability of specifying, as well as forecasting, the state of
the magnetosphere-ionosphere system on the basis of solar wind observations.

Kamide, “From IGY to eGY: The importance of real-time data in space physics” (2007)
4

2. Before IGY“Before IGY” may include many different meanings, depending on how far wego back before the IGY. Of course the author has no intention of taking thereaders back to Aristotle’s discovery of auroral breakup or the finding ofsunspots in China. These occurred well prior to the birth of science in thepresent sense.
It may be quite interesting to seek what the beginning of solar-terrestrial
relationships studies was. In tracing the history of the field, it can be found thatstudies of geomagnetic records and solar and auroral observations wereconducted in parallel with discoveries of the important laws ofelectromagnetism: see Figure 3. In other words, the study of geomagnetic
disturbances was one of the areas which contributed significantly to thedevelopment of electromagnetism in general, leading to Maxwell’s equations,which were systematically organized in 1873.
It is well known that Graham in London, and Hiorter and Celsius in Uppsala,Sweden, noticed that large disturbances in the geomagnetic field occurred
simultaneously at these two locations2. Gauss and Weber later showed thesemagnetic disturbances to be a worldwide phenomenon, such that magnetic“weather” is much less local than ordinary weather on the Earth’s surface or in
the troposphere. It was in September 1859, following Carrington’s observationof a group of giant while light flares on the Sun, that a great geomagnetic storm

Fig. 3. Some of the early important findings and laws in electromagnetism in the 18th and 19th
centuries. These led to Maxwell’s equations of electromagnetism.

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began3. These are some of the examples which indicate that the root of studies
of solar-terrestrial relationships goes back to the 18-19th centuries.
Who introduced the terminology “geomagnetic storm,” and when, into thescientific community? It is known that Alexander von Humboldt (1769-1859)
used “magnetisches Ungewitter,” (magnetic thunderstorms in English) to
describe the variability of geomagnetic needles, which were associated with theoccurrence of “light meteor” (i.e., auroras). Humboldt realized that geomagnetic
disturbances and auroras are two manifestations of the same phenomenon. He
maintained a lifelong interest in geomagnetic disturbances, establishing magnetic
stations around the world through his diplomatic contacts at, for example,
Bombay, Toronto, and Sitka.
It was during the First Polar Year (1882-1883) that scientists defined“geomagnetic storms” as “intense, irregular variabilities of the geomagnetic field
which occur as a consequence of solar disturbances.” It was still unclear how
intense was intense and how irregular was irregular, but it was worth noting that
the Sun was identified as the cause of geomagnetic storms.
3. Research Style near and during IGY
To obtain more complete knowledge of planet Earth, the member countries of
IGY worked together to set up a number of stations on the Earth’s surface, andeven to establish World Data Centers, with the notion that data obtained from theobservations during IGY are precious assets common to humankind. All Earth
Science disciplines including space physics were involved with this worldwide
campaign. The IGY officially adopted the use of artificial satellites to explorer
the Earth’s upper atmosphere. In fact, the Soviet Union and the United States ofAmerica successfully launched Sputnik 1 and Explorer 1 into orbits around theEarth in 1957 and 1958, respectively, opening the so-called space era. The
younger generation may not believe, however, that at the time of IGY, and eventhroughout the author’s graduate studies, scientists used microfilmed data insteadof digital data: thus making copies of data and scientific papers in journals
meant for the scientists to take their photographs, of course not in color.
Computers were not available, so that numerical calculations had to be made byhand, even for quite complicated ones.
Comparing what scientists knew fifty years ago about how geomagnetic
disturbances were generated with what we know today about the dynamics of themagnetosphere within the entire Sun-Earth system, we find that there is a great
difference between the two sets of pictures. This difference results partly fromprogressive changes in methodology over the last decades. The old techniqueKamide, “From IGY to eGY: The importance of real-time data in space physics” (2007)
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scientists had long relied on was to measure magnetic disturbances on the
Earth’s surface, monitoring the state of the electromagnetic environment before
the concepts of the magnetosphere, solar wind-magnetosphere coupling, andmagnetosphere-ionosphere coupling were introduced. After the 1960’s, in-situ
measurements of the plasma structure in the near-Earth space environment
became available. Adding to our wealth of knowledge has also been the constanttrial-and-error theoretical analysis of the space between the Sun and the Earth by
our frontier scientists, such as Axford, Dungey and Parker.
To realize the dramatic advance in technology for data handling over thelast fifty years, two examples are shown in which it can easily be guessed how
much time and energy scientists at that time must have spent for what we would
now need only a few seconds. Figure 4, taken from Akasofu4 shows the
distribution of auroral displays, which was constructed by using records fromeleven all-sky cameras of the IGY network. For obtaining this type of “world”
map, it is required first that all or most of the all-sky camera stations be cloudless.
One all-sky camera is able to cover only a very small fraction of the entire
auroral distribution: see the small circles in Figure 4. Second, all or most of the
all-sky cameras must be operating properly under freezing Arctic conditions. It
was heard from engineers, who were active during IGY that polar bears at times
knocked out their instruments. Finally, each of the auroral forms was manually
mapped to the ionospheric level whose altitude was assumed to be 110 km. This

Fig. 4. Example of the distribution of auroras, using different symbols for different shapes ofauroral forms. Each of the small circles indicates the field-of-view from each of the all-sky
cameras. Three lines for three L values, 4, 6, and 8, are shown. The black dot at the top of the
diagram is the direction of the Sun. After Akasofu (1964)

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was not a trivial work at all without digital data and computers. In this way, theworld aurora maps were generated for a number of days by repeating theprocedure for data obtained every 1-minute, having led to the concept of auroral
substorms5. It is now possible for a satellite to picture the entire polar region
almost instantly, regardless of the surface temperature and weather6-8.
Figure 5, taken from Fukushima9, is another 50-year-old example where theso-called equivalent current system in the ionosphere is mapped. Ground-basedmagnetometer records were used to obtain current streamlines in the ionospherethat can account for the observed geomagnetic perturbations at all stations. Thisis a good example of “easy to say but difficult to do practically.” That is,through a trial-and-error method by using a pencil and a rubber eraser, it was
required to draw two-dimensional, divergence-free streamlines to satisfy thefollowing conditions: the density of the streamlines is proportional to themagnitude of the magnetic perturbation and they are perpendicular to theobserved magnetic vectors at all the points. This time-consuming process isequivalent to solving a second-order differential equation for magnetic potentials,
from which one can obtain the equivalent current system in the ionosphere. It is,
at present, a matter of seconds by a computer to complete the entire process, but
at least several hours were needed to construct just one world map of theequivalent current system before Bostrom10 and Kamide et al.11 devised
automated algorithms.

Fig. 5. Example of hand-drawn equivalent current system obtained for the maximum epoch of an
intense substorm, which was then called as an elementary storm. A circle indicates the location of
the Sun. After Fukushima (1959).

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Fig. 6. Two auroral image data from the Polar spacecraft. Dynamic features of auroras can bemonitored, including the sudden creation of bright spots and their latitudinal and longitudinal
expansion. Courtesy of M. J. Brittnacher and G. K. Parks.

Without in-situ data from satellites, scientists of fifty years ago reliedheavily on ground-based observations of magnetic fields and auroras, fromwhich they only inferred physical parameters in space. The two examples
clearly demonstrate that deducing useful data products or physical parametersfrom magnetic and auroral observations was extremely laborious and often
required special personal techniques or talent.
Presently, not only ground magnetometer data but also data from various
spacecraft are available in real time or near real time. The entire high latitudes
can now be viewed from above the poles in different wavelengths6-8, so that the
global distribution of auroras and its dynamical changes can be monitored in realtime. Figure 6 is one such example, in which two auroral photographs takenfrom the Polar spacecraft are shown. We can see progressive changes in the
global distribution of auroras, particularly how bright spots generated in themidnight sector soon become “seeds” for a sudden expansion of auroral activity
in ten minutes.
Not only data from near-Earth satellites but also those from the upstream
solar wind are available in real time. Figure 7 shows an example of 24-hr realtime data from the ACE satellite, located at the L1 point between the Sun and theEarth. Note that in these real-time data, the last point at each quantity is notexactly of real-time observations but represents observations of five seconds ago,
for the light speed to travel from the satellite to the Earth.

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9Fig. 7. Example of real-time data of the solar wind and the interplanetary magnetic field, from the
ACE satellite. Adapted from the NOAA/SEC homepage.

4. From Discoveries to Predictions
Progress in any fields of natural science begins with “discovery” and ends with“proof.” Space physics is not an exception. “Proof” in our research area,
however, may be reworded by “prediction or reproduction” of solar-terrestrial
phenomena under study. The entire process in research can further be classified
into several practical steps. For example, the research procedure for individual
solar-terrestrial phenomena, such as magnetospheric substorms, which are thebasic energy process in the solar wind-magnetosphere-ionosphere system,
normally consists of the following four sub-steps:
1.
2.
3.
4.

Ascertaining characteristics of a phenomenon through observations
Introducing a hypothesis to account for the phenomenon
Testing repeatedly the hypothesis (or model) against observations
Evaluating the model quantitatively

In the field of solar-terrestrial relationships, the predictability of solar-terrestrial
phenomena is one of the key factors for evaluating a hypothesis. It is possible toKamide, “From IGY to eGY: The importance of real-time data in space physics” (2007)10

predict, at present, the occurrence of an intense magnetospheric substorm withperhaps more than 80% accuracy if (a) the solar wind variations are relatively
simple and (b) accurate information about the solar wind variability is given inreal time. This does not mean at all, however, that we are currently 80%
successful in predicting the occurrence of magnetospheric substorms, because
the solar wind variation is almost always complicated.
The advance in research of the Earth’s magnetosphere thus far has also beenachieved through repeating these four sub-steps. The magnetosphere, in which
the Earth’s magnetic field dominates, was first discovered through spacecraft
observations. The magnetotail and then the plasma sheet and other plasma
domains were subsequently discovered, and the nature of these plasma regions
was ascertained. Concurrent with these discoveries, extensive efforts were madeto search for the mechanism of the entry of solar wind energy into themagnetosphere and to locate the energy reservoir for substorms. A number oftheoretical models about the generation mechanism of substorms, i.e., the sudden
release of accumulated energy, were proposed, and repeatedly tested against new
data from spacecraft and ground-based observations. Computer simulations
were also conducted to try to reproduce the main features of substorms.
Prediction codes were evaluated according to their efficiency of predicting theoccurrence and intensity of substorms.
Table 1 summarizes the development of magnetospheric physics over thelast fifty years in three phases. Major discoveries are identified for each phase.
Note that the list for each phase comes from the point of view of substorm
research, and that different lists would result if we deal with different researchdisciplines, such as ionosphere-thermosphere coupling. Note also that substorm
research is a result of a convergence of multi-disciplinary sciences, whichdeveloped from several traditional fields of research, such as geomagnetism,auroral physics, and aeronomy.
It was during the first phase when the solar wind and the magnetosphere
were observed. Explorer 10 explored the nightside magnetosphere, i.e., themagnetotail boundary, as early as 1961. Without repeated crossings over theboundary, it would be impossible to discern the entire magnetotail. It was the
IMP 1 spacecraft, which discovered that the Earth’s magnetic field isconsistently distorted by the solar wind. The bow shock, the magnetosheath, andthe magnetopause were later identified12, 13. Figure 8, from an early paper by N.
Ness12, shows the initial results of IMP 1, mapping the positions of themagnetopause. IMP 1 also surveyed the lobe structure and length of themagnetotail, as well as the response to substorm activity. This series of239

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Table 1. Advances in the studies of magnetospheric dynamics over the last fifty years interms of magnetospheric substorms

Phase 1: Explorer and discovery of the magnetosphere
Property of the solar wind
Average configuration of the magnetosphere
Magnetopause, bow shock, and magnetotail
Plasma regions in the magnetosphere
Plasma sheet, lobe, ring current, and acceleration region of auroral particles
Dynamics of the magnetosphere
Plasma convection, and substorm-related changes

Phase 2: Understanding of solar-terrestrial processes
Solar wind control of substorms
Intensity, location, and probability of substorms
Configuration changes of the magnetotail
Plasma sheet thinning/thickening
Ring current formation
Electrodynamics of the ionosphere
Computer simulations

Phase 3: Predictions - Space weather applications
Now-cast of the state of the magnetos phere
Modeling of the entire the solar wind-magnetosphere system

important discoveries about the nature of the magnetosphere occurred in the
midst of extensive theoretical discussions among theorists1, 14, 15.
Subsequently, the configurations of both the dayside and nightside
magnetosphere were modeled. The presence of solar wind particles within themagnetosphere was confirmed. Changes of each plasma region at the expansion
onsets of substorms were identified. Along with ground-based observations ofauroral dynamics and magnetic perturbations in the polar region, satellites began
to assemble data for changes in the magnetotail, such as the large-scale topology
changes in fields and plasma flow associated with substorms.
In the second phase, which may be called the phase of understanding ofsolar-terrestrial processes, integration of satellite and ground-based observations
became essential. It was found that nonlinear coupling occurs in each of theplasma regions. More and more, computer simulations became a powerful toolfor quantitative understanding of such complicated processes. Substorms
involving both suddenly initiating and rapidly growing auroral breakups are
typical examples of such nonlinear processes. Our dream is to input current
solar wind data into the computer, which will then tells us how, when, and where
substorms of what magnitude will take place.
We are now entering the third phase, where physical relationships among
various phenomena occurring in different regions, described in detail by basic

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Fig. 8. IMP 1 and Explorer observations of the magnetospheric boundaries, generated by solar
wind-magnetosphere interactions. After Ness (1965).

equations, must be established. This is quite important not only toward acomplete understanding of the entire solar-terrestrial system in all scale sizes, butalso for space weather predictions, which are presently societal needs.
5. Real-Time Observations and Calculations
Over the last fifty years, the scientific community has made drastic changes in
the amount of data that space physicists must deal with. The rate of the increasein the data still keeps increasing at this moment. Are we able to continue tohandle such an increase for years to come? The answer is: “Probably yes,” or“Yes, at least up to the near future.” Fortunately, the rapid increase in thequantity of data we must handle has paralleled the rapid growth in modern
computing technologies. The development in technologies has coincided withthe urgent need to integrate observations and computer simulations in solarterrestrial physics. Observations need solid backup from theories, while theories
need realistic boundary conditions resulting from observations to run models.
While observations from the Earth’s surface can be considered as a type of“remote sensing” for solar-terrestrial processes and thus “indirect,” they
nevertheless generate high spatial/temporal resolutions. On the other hand,

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satellite observations, being in-situ and thus “direct,” provide only “point”
measurements along satellite orbits. Clearly, these are complementary to eachother.
The Solar-Terrestrial Environment Laboratory (STEL), installed a computer
system, called GEDAS (standing for Geospace Environment Data Analysis
System), to promote integrated studies of ground-based and satellite-based
observations as well as modeling and simulation research. GEDAS is not a
simple display system for real time data. In the GEDAS system, real-time data
are used as initial and boundary conditions for computer simulations to executein real time, permitting us to specify the state of the near-Earth plasma
environment and to predict space weather events that may occur or how thesubstorm just begun grows in the next several minutes. The correct answer will
be known in several minutes when real time data flow into the system. If thistype of test of a particular simulation model is not satisfactory, i.e., failing topredict what is coming next, one must modify the existing code and/or change
the parameters within a realistic degree, such that the revised simulation provides
better predictions for the next space weather events. This revision cannot be ad
hoc to try to account for only the next real-time observations, but must be
universal, leading to deepening of our basic knowledge of the solar-terrestrial
processes.
As one of the active projects currently underway at GEDAS, we collectground magnetometer data on a near real-time basis in an attempt to compute theinstantaneous, two-dimensional distribution of ionospheric parameters, such as
ionospheric electric fields and currents and field-aligned currents. This joint
effort of STEL and NOAA’s National Geophysical Data Center (NGDC) uses
operationally updated versions of the KRM and AMIE programs16, 17, 18.
In this real-time project, the uneven distribution of ground magnetometers is
one of the inevitable problems. The AMIE code, along with solar windobservations, is first used to estimate the global distribution of the electric
potential, which represents a statistical pattern of the ionospheric potential,commensurate with the solar wind condition. Once the global pattern iscalculated, that pattern is used as the boundary condition to calculate moredetailed structures of ionospheric parameters in a limited region, where groundmagnetometer data are available from a number of observatories in real time, on
the basis of the KRM algorithm.
Figures 9a and b present examples of the distribution of the electric
potential and the corresponding currents in the ionosphere, calculated in nearreal-time. It is clearly seen that deformations in the usual twin-vortex potential
pattern, which Axford envisaged, and strong auroral electrojets are reproduced.

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Fig. 9. Example of real-time calculations of the distributions of (a) electric potential and (b) the
corresponding ionospheric current vectors for disturbed periods.

Comparing these real-time weather maps with Figure 5, it is noticeable that there
is an enormous change in our knowledge about the ionospheric quantities overthe last fifty years. All one could obtain fifty years ago was the equivalentionospheric current system from ground magnetometer data, not “true” currents.
We calculate presently these ionospheric parameters every 10 minutes: see
http://gedas22.stelab.nagoya-u.ac.jp/index_j.html.
GEDAS provides thescientific community with specification of the geospace environment wellbeyond what is available from conventional geomagnetic activity indices, suchas the Kp and AE indices. Further, since the data products this particular
GEDAS program provides are based on real-time recordings of magnetic
perturbations from a number of stations, the output should be more realistic thanwhat empirical potential patterns using “point” measurements of the solar windestimate.
A weak point of our system is that to run the KRM algorithm for local
regions, the ionospheric conductance must be plugged in as an input parameter.
As a near-future project, the ionospheric conductance can be normalized using
other sets of real-time observations of the global distribution of auroras by polarorbiting satellites. Radar data in real time are also quite useful to estimate theionospheric conductances although they are basically “point” measurements.
This is an advantage of real-time data, making mutual calibrations possible. It is
important to point out that our output from this project is valuable for

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Fig. 10. Front view of magnetic configuration just before (Left) and after (Right) of the shock
arrival during the intense interplanetary disturbance of October 24, 2003, calculated from a solar
wind-magnetosphere coupling MHD simulation. Green and blue colors are used to distinguish
open and closed field lines, and field lines, which are not connected to the earth, are shown in red.
Solar wind data from ACE were used. Courtesy of T. Ogino.

understanding the status of the auroral electrojets, which are important forforecasting the strength of induced currents19, 20.
Figure 10 shows another example of real-time calculations at GEDAS21.
This is a front-view snapshot of the magnetic field configuration and the related

Fig. 11. Variations in the neutral density at an altitude of 300 km calculated by a global
thermosphere-ionosphere model. The red color region indicates an increase in the neutral density,and the numbers for the color contour are in the unit of 0.01%. The input parameters of the model
are the convection electric field and conductivities, and are taken from an MHD model for thecondition of the steady solar wind with southward IMF. Courtesy of H. Shinagawa.

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plasma temperature distributions in the magnetosphere just before andimmediately after an interplanetary shock arrived at the front of themagnetosphere for a unique space weather event of October 24, 2003, during
which several serious satellite anomalies, including the Japanese environmental
satellite ADEOS-2, were reported. For this MHD model, 1-min data of IMF(interplanetary magnetic field) and solar wind parameters from the ACE
spacecraft were used. It is possible to illustrate such magnetosphericconfigurations from different view-angles as well. In a sense, this simulation
system is a virtual magnetosphere in the eGY concept.
Figure 11 shows an example of changes in the neutral density at an altitudeof 300 km as a result of ionosphere-thermosphere coupling calculations22. This
coupling model uses, as the input parameters, not only the convection electricfield and ionospheric conductivities, but also the output from a solar windmagnetospheric MHD model, shown in Figure 10 whose input is real time solar
wind data. Instead of the ionospheric electric field mapped down from themagnetosphere obtained from the MHD simulation, it is possible to use theoutput from the KRM magnetogram-inversion as well. We can treat this type ofreal-time computing system as a virtual thermosphere in eGY. It is clear thatmutual calibrations of different real-time simulations can be conducted.
6. Issues For Future Studies
The Earth’s magnetosphere is a vast laboratory for plasma physics in which
unique processes occur, monitoring one of the aspects of Sun-Earth relationships.
It is bounded by the solar wind on the outer side and by the ionosphere on the
inner side. These boundaries represent dynamic borders within which themagnetosphere is influenced. As is evident from Table 1, during the past fifty
years we have seen significant advances in studies of magnetospheric physics.
These changes are primarily because of advances in various techniques, such as
satellites and radars, as well as computer simulations dealing with real time data.
As demonstrated, studies of the solar-terrestrial environment began withinspecting geomagnetic disturbances. Following the discoveries of the secrets of
the interaction between the solar wind and the Earth’s magnetic field, researchers
have almost reached a consensus about the average configuration of themagnetosphere. In the following, several major areas in which further
clarifications are required are discussed:

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6.1. Global vs. Local
There are still many unknowns about details of magnetospheric processes thatare very different in different substorms, representing perhaps different
preconditions and local ionospheric conditions. This variability seems to bequite similar to local weather in the troposphere, which is driven not only bylarge-scale air circulation resulting from the global pressure/temperature
distribution, but also by local boundary conditions, such as three-dimensional
structures of near-by mountains. Just suppose that these local conditions, forexample the plasma pressure in the magnetosphere, vary constantly. It would
then be hopeless to find some consistent features in small-scale auroral motions.
It may not be completely unlikely that by now, we already know what we aresupposed to know about large-scale substorm phenomenology, although it is still
extremely important to separate large-scale and small-scale phenomena in realtime observations, which are basically local measurements.
6.2. Macroscopic vs. Microscopic
To fully solve the substorm problem, both microscopic and macroscopic
processes must be accounted for together. “Microscopic” processes do notnecessarily mean “local” processes in 6.1, but represent those of particle scales.
This is simply because the two types of processes, such as acceleration of auroral
particles, i.e., a microscopic mechanism, and the generation of field-aligned
currents, i.e., a macroscopic mechanism, are taking place in the same highlatitude region at the same time. Individual electrons, for instance do not aware
whether they are participating in macroscopic or microscopic processes. One of
the typical examples is an auroral breakup at substorm expansions. To
understand these processes properly for the magnetosphere-ionosphere system, akinetic approach is necessary. On the other hand, macroscopic processes occur
as a result of large-scale energy conversion in the nearly entire magnetosphere,
responding to changes in the solar wind.
6.3. Average Configuration vs. Non-Steady Process
It is comparatively easy to assume that disturbed times are substorm times, andthe current system grows and decays as a whole in the entire ionosphere, but weall know that individual substorms are much more complex than this assumption
implies. Substorm expansions that take place in local regions near midnight areshort-lived. Thus, by observing an increase in the current intensity at one point,
we can never be certain whether the increase is the result of a global

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enhancement of the electric fields generated from the solar wind-magnetosphere
coupling or of a local enhancement in the ionospheric conductivities. Further,
substorms are a multi-dimensional phenomenon consisting of the directly-driven
process and the unloading process. Their relative importance varies fromsubstorm to substorm, and it even depends on the phases within a single
substorm.
It is also important to realize that averaging individual data tends to smearout important local aspects. Therefore, by employing averaging methods, wecannot obtain more than the well-known twin-vortex pattern of the electrostatic
potential that Axford predicted 45 years ago. The average picture often misleads
us about how nonlinear processes occur within such a large-scale system as themagnetosphere, particularly when the solar wind is highly variable. We are notallowed to consider that disturbed times are equivalent to substorm expansion
times.
6.4. The Solar Wind and the Boundary Condition to the Magnetosphere
The Sun does not determine all details of the processes in the entire solarterrestrial system, particularly those inside the magnetosphere. Therefore, using
data from the solar wind and the IMF alone, we cannot determine all processes
that occur within the magnetosphere. In other words, the solar wind only gives
the boundary condition to the magnetosphere, under which various types ofdisturbances, such as substorms and convection enhancements, take placeinternally. It is not too difficult to realize this by simply considering that the Sun
itself cannot decide where (what latitude and local time) the next auroral breakupof what intensity will take place in the polar ionosphere. It is the time history for,at least, several hours, of the inner magnetosphere and the polar ionosphere that
seems to be critical in determining detailed processes in the magnetosphereionosphere system.
It would not be highly recommended, in this respect, to try to reproduce all
the fine-scale variations of geomagnetic indices by using only solar windparameters. If a paper is published in which correlation between the solar windand geomagnetic indices are close to 1.0, arguably something must be wrong
with its analysis technique.
7. Conclusions
The final goal of Sun-Earth relationships research is to become to predict very
accurately geoeffective storm/substorm events when the “present” condition of
the Sun is given. For this to happen in reality, however, we will have to fully

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understand all the basic processes in the Sun-Earth system, such that all physicaldetails can accurately be coded in computers. Also, in order to achieve thisdegree of accuracy in the prediction of space weather, a super computer ofextremely high speed and an infinite memory size would be necessary.
It is quite obvious that we are currently far from this final goal. Therefore,
we are trying to understand details of multi time/spatial scale processes in thesolar-terrestrial system, and to use large, instead of infinite, computer memories.
We also make use of as many different types of observations as possible at alarge number of points. Computer simulations and modeling are extremely
powerful in filling gaps where no observation points exist on the Earth’s surface
and in space. It is very useful to rely not only on physics-based theories of theprocesses but also on empirical formulas.
This paper has attempted to demonstrate the importance of real-time data instudies of solar-terrestrial relationships. The importance of real-time data can besummarized in terms of the following specific meanings:
Space weather predictions
This paper has not explicitly addressed the usefulness of real-time data in space
weather predictions, but there is no doubt that real-time data are a must to
improve predictions of space weather.
Finding yourself in the global framework
In conducting any type of observations, it is important to identify the precise
location of your instrument within the entire solar-terrestrial system. For
example, without identifying the location of what a particular satellite ismeasuring in the large-scale energy flow from the Sun to the Earth, one is notable to discuss the physics of these measurements self-consistently. It is not
sufficient at all to know only that your radar is located accurately at geomagnetic
latitude 64.9 degrees and MLT = 23.6, measuring the southward electric field in
the region of the westward electrojet. It is crucial to understand whether you are
currently inside the convection electrojet system or a substorm expansion, both23
giving the same westward electrojet .
Test of your idea in real time
Real-time data can be used in computer simulations as the initial and boundary
conditions to run a model calculation. For example, with solar wind data as
input, one can run an MHD simulation, providing an output which predicts the
structure of the magnetospheric plasma regions for the next moment. The most
exciting point in such a scheme of integration of observations and simulations inreal time is that any researcher in the world can be a project leader, using real-Kamide, “From IGY to eGY: The importance of real-time data in space physics” (2007)
20

time observations to drive models in his/her own research projects in his/her own
office. In this way fresh ideas can be tested almost instantly against real data.
Output becoming input to other studies
An example has been shown of real-time calculations of two-dimensional
distribution of ionospheric electrodynamic quantities at high latitudes. Once theionospheric parameters have been computed, the output can be sent toinstitutions around the world, where this output will possibly become input forother modeling studies. For example, our electric field distribution in theionosphere obtained from GEDAS can be mapped to the magnetosphere and is24
therefore useful for tracing particles in the magnetosphere . Joule heating fromour calculations can be used as input for calculating neutral winds in thethermosphere, which will modify the electric field pattern in several minutes.
Education and outreach
It is evident that real-time data system such as GEDAS can be utilized as a live
classroom, in which students can see data directly from satellites and groundstations in real time. They can share the excitement of pursuing real time
research and of predicting geomagnetic storms and substorms along withtremendous auroral displays.
It is important to rephrase that real-time data combined with real-time
simulations are not only capable of changing the practical style of research but
also change the concept of research in solar-terrestrial physics. While most of
the previous studies in this field have attempted to account for one cross-sectionof one phenomenon in one plasma region within the entire Sun-Earth system, it
will become increasingly crucial to understand the whole spectra of processes,
particularly coupling between different plasma regions and coupling between
large-scale and small-scale processes. This trend will be more accelerated by the25
new international program called the eGY that the scientific community isplanning to undertake: see http://www.egy.org. It is an ICSU-endorsed initiativeof the International Union of Geodesy and Geophysics (IUGG), driven by theInternational Association of Geomagnetism and Aeronomy (IAGA). The eGY
program is quite timely in the sense that it uses the advantage of modern
networking technologies that are harmonized with high-quality observation
technologies. Promoting the development of virtual observatories is a central
feature of eGY.
No one was predicting during the IGY that fifty years later, the scientificcommunity would discuss this new international enterprise, eGY, for collecting
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and using these in computer simulations, all in real time. The present paper has
discussed how important it is, for further progress of solar-terrestrial physics, toconduct well-coordinated observations, to handle various data and datasets, andto process data products on a real-time basis. Although the author admits thatexamples and the discussion subjects given in the present paper are strongly
biased toward his own experience, it is hoped that the general conclusions areapplicable to other areas in Earth Sciences as well.
Acknowledgments
It was a great honor for me to deliver the First Axford Lecture. When Professor
Ian Axford was a visiting professor at our institute, we discussed seriously theneed of a new scientific society in Geosciences in the Asia/Oceania region. I
valued his extensive experience in establishing the European GeophysicalSociety (EGS), now the European Geosciences Union (EGU). While he wasstaying at our institute, we started to write a memorandum of intent. It was six
years ago that Professors Ian Axford and Wing Ip and myself began to establish
formally AOGS.
References
1. W. I. Axford and C. O. Hines, Can. J. Phys. 39, 1433 (1961); W. I. Axford,
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The Astrophysical Journal, 684: L45–L49, 2008 September 1
䉷 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THREE SUCCESSIVE AND INTERACTING SHOCK WAVES GENERATED BY A SOLAR FLARE
Noriyuki Narukage,1 Takako T. Ishii,2 Shin’ichi Nagata,2 Satoru UeNo,2 Reizaburo Kitai,2 Hiroki Kurokawa,2Maki Akioka,3 and Kazunari Shibata2
Received 2008 February 16; accepted 2008 July 25; published 2008 August 8

ABSTRACT
We discovered three successive Moreton waves generated by a single solar ﬂare on 2005 August 3. Although
this ﬂare was not special in magnitude or conﬁguration, Moreton waves (shock waves) successively occurred
three times. Multiple shock waves generated during a single ﬂare have not been reported before. Furthermore,
the faster second-generated Moreton wave caught up and merged with the slower ﬁrst-generated one. This is theﬁrst report of shock-shock interaction associated with a solar ﬂare. The shock-plasma interaction was also detected.
When the third-generated Moreton wave passed through an erupting ﬁlament, the ﬁlament was accelerated by
the Moreton wave. In this event, ﬁlaments also erupted three times. On the basis of this observation, we consider
that ﬁlament eruption is indispensable to the generation of Moreton waves.
Subject headings: shock waves — Sun: chromosphere — Sun: corona — Sun: ﬂares
servatory of Kyoto University. SMART has the ability to takesolar full-disk images in Ha at high spatial resolution (∼0.5⬙,∼360 km on the solar disk). The observing wavelengths are
not only Ha center but also Ⳳ0.5 and Ⳳ0.8 Å, thus enabling
us to monitor the line-of-sight motion of the chromosphericplasma. It takes about 38 s to obtain one set of these ﬁvewavelengths.
A Moreton wave is the chromospheric part compressed andreleased by a shock wave propagating in the solar corona (Uchida 1968; Uchida et al. 1973). Hence, we need to detect chromospheric downward and upward motion to identify a Moretonwave. SMART can detect such Doppler motion based on ob˚ and Ha Ⳳ 0.8 A
˚ in high spatial
servations in Ha Ⳳ 0.5 A
and temporal resolution. In this event, part of the Moreton wave
front was observed in all ﬁve wavelengths. However, the threesuccessive Moreton waves were detected in only Ha Ⳳ 0.5
˚ , especially Ha ⫺ 0.5 A
˚ . In the other wavelengths, it is hardA
to distinguish between the ﬁrst and second Moreton waves.
The suitable wavelength for the detection of Moreton wave
depends on the chromospheric downward and upward motion
caused by the coronal shock wave. Because of the observationin various Ha wings, SMART has the advantage of detecting
Moreton waves.
A ﬂare started at 04:54 UT near the southeastern limb andgenerated three successive Moreton waves. Figure 1 (Plate 1)
shows the original images in Ha center (Figs. 1a, 1e, and 1l)
and “Dopplergrams” (Figs. 1b–1d, 1f–1k, and 1m–1s) observedwith SMART. In this Letter, “Dopplergrams” are created using
the equation of log (Iblue /Ired ), where Ired and Iblue are the intensity
images almost simultaneously observed in the red and blue wings(Ha Ⳳ 0.5 Å), respectively. This method makes the line-ofsight motion clear. In the Dopplergrams, we can identify wavelike disturbances called Moreton waves which successively occurred three times and propagated. The red, green, and blue plus
signs indicate the wave fronts of the ﬁrst, second, and third
Moreton waves, respectively. This is the ﬁrst report of successive
Moreton waves. Although there is an observation where Moreton
waves propagated both northeastward and southward in the Xclass (X10) ﬂare on 2003 October 29 (Liu et al. 2006; Balasubramaniam et al. 2007), these waves were generated at thesame time and were not generated successively as in our case.
Furthermore, we observed an unprecedented phenomenon: the1. INTRODUCTION

Solar ﬂares are the largest explosive events in the solar system, and it is easy to imagine that such explosions generate
shock waves that propagate in the solar corona. In 1960, acoronal shock wave was indirectly discovered in the Ha spectral line (Moreton 1960; Smith & Harvey 1971); i.e., the intersection of a coronal shock wave and the chromosphere wasobserved as an Ha wavelike disturbances (Uchida 1968; Uchida et al. 1973). This wave was named a “Moreton wave” after
the discoverer. In the 1990s, solar observing satellites werelaunched and directly detected coronal shock waves in the extreme ultraviolet (Thompson et al. 1998; Biesecker et al. 2002)
and soft X-rays (Khan & Aurass 2002; Narukage et al. 2002,
2004; Hudson et al. 2003). Coronal waves have also been directly imaged by ground-based radio observations (White &Thompson 2005; Vršnak et al. 2005). However, observationsof shock waves were rare, because the shock wave signatures
are much fainter than the ﬂare and propagate very fast. The
generation mechanism of a Moreton wave has not been madeclear yet.
Here we show the discovery of three successive Moreton
waves (shock waves) generated by a single solar ﬂare. Furthermore, we found that the faster second-generated Moreton
wave caught up and merged with the slower ﬁrst-generated
one. The merging was also detected in radio as the sudden
enhancement of the signal. This is the ﬁrst report of shockshock interaction associated with a solar ﬂare. In § 2, we describe the observational data. In § 3, we examine the interactionof shock waves in detail. In § 4, our summary and discussion
are presented.
2. OBSERVATIONS

We discovered three successive ﬂare-generated Moreton
waves on 2005 August 3 with the Solar Magnetic Activity
Research Telescope (SMART) (UeNo et al. 2004) at Hida Ob1
Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan; narukage@solar.isas.jaxa.jp.
2
Kwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto
607-8471, Japan.
3
Hiraiso Solar Observatory, National Institute of Information and Communications Technology, Hitachinaka, Ibaraki 311-1202, Japan.

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Fig. 2.—Sequential images of erupting ﬁlaments. The ﬁeld of view is shown by the square with dotted line in Fig. 1t. (a) Preﬂare stable ﬁlament observed in
Ha line center. (b–r) Three erupting ﬁlaments associated with the three successive Moreton waves. These images are created by summing the two images almost
simultaneously observed in the Ha blue wings (⫺0.5 and ⫺0.8 Å ) to make erupting (line of sight) motion clear. These three erupting ﬁlaments are named “f1,”“f2,” and “f3” in order of occurrence time. Their footpoints seem to be almost ﬁxed at the same point marked by black crosses. Both ends of the ﬁlaments areindicated by black lines. The colored arrows connect such black lines and show the vectors of erupting ﬁlaments. The color of arrows indicates the azimuth angle(see color bar), where we deﬁne that the southward is 0⬚ of azimuth angle and that the azimuth angle increases counterclockwise.

faster second-generated Moreton wave caught up and mergedwith the slower ﬁrst-generated one in 05:01:36–05:02:14.4
Figure 2 indicates the motion of ﬁlaments. A ﬁlament stably
existed before the ﬂare as shown in Figure 2a observed in Ha
line center. Figures 2b–2r show the series of Ha blue wing
images which are created by summing the two images almost
˚ and Ha ⫺ 0.8 A
˚.
simultaneously observed in Ha ⫺ 0.5 A
These images show the moving feature toward us at a line-ofsight speed of several tens km s⫺1, because the Doppler shifts
˚ and Ha ⫺ 0.8 A
˚ corresponding to the Dopplerof Ha ⫺ 0.5 A
speeds of 23 and 37 km s⫺1, respectively. According to thisﬁgure, the ﬁlaments erupted three times. These three erupting
ﬁlaments are named “f1,” “f2,” and “f3” in order of occurrence
time. The relation between the erupting ﬁlaments and Moreton
waves is described in § 4 in detail.
Figure 3 shows the time evolution of this ﬂare, the Moreton
waves, and associated phenomena. Figure 3a is the soft X-ray(3–25 keV) intensity, which is emitted from thermal ﬂareplasma with a temperature of 10–20 MK and is roughly ameasure of the total released energy by a ﬂare. According tothis light curve, this ﬂare was classiﬁed as M-class and continued for about 1 hr. The black line in Figure 3b is the 2 GHz
4These three successive Moreton waves are easy to identify in movies; see
http://www.kwasan.kyoto-u.ac.jp/∼naru/three_successive_moreton_wave/
index.html.

radio ﬂux observed with the Nobeyama Radio Polarimeters (Nakajima et al. 1985). Gigahertz radio waves are generated by the
gyro-synchrotron mechanism of nonthermal electrons. The grayline in Figure 3b shows the hard X-ray count rate in 50–100
keV detected by the Ramaty High Energy Solar Spectroscopic
Imager (RHESSI) (Lin et al. 2002). Radio in the GHz range andhard X-rays in 50–100 keV are sensitive to instantaneously released energy. In this ﬂare, three radio ﬂux enhancements weredetected around 04:58:30, 05:00:00, and 05:04:15. The hard Xray count rate was also enhanced in these three timings. We
suggest that these energy releases are closely related to thethree successive Moreton waves.
3. INTERACTION OF SHOCK WAVES

In this ﬂare, Moreton waves were generated three times andﬁlaments erupted three times. Figure 3c shows the position of
the Moreton waves with diamonds and erupting ﬁlaments withplus signs. Using this ﬁgure, we can easily understand the
propagation of the Moreton waves. Around 05:01:36, the second Moreton wave (green diamonds) caught up with the ﬁrst
wave (red diamonds) at a distance of 150,000–200,000 km
from the ﬂare site. After this interference, these two waves
merged together and continued to propagate. This is the ﬁrst
observation of merging Moreton waves, i.e., shock-shock interaction associated with a solar ﬂare. However, each Moreton

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THREE SUCCESSIVE AND INTERACTING SHOCK WAVES

Fig. 3.—Time evolution of the ﬂare, Moreton waves, and associated phenomena. (a) Soft X-ray (3–25 keV) intensity observed with the GOES 10
satellite. The white region shows the period of (b–d). (b) The black and graylines show 2 GHz radio ﬂux observed with the Nobeyama Radio Polarimetersand hard X-ray count rate in 50–100 keV detected by RHESSI, respectively.
(c) Propagation of the three successive Moreton waves and eruption of ﬁlaments. The red, green, and blue diamonds show the wave fronts of the ﬁrst,
second, and third Moreton waves, respectively. The dash-dotted lines show
the second-degree polynomial ﬁts of the wave fronts. We note that we did not
apply the polynomial ﬁts to the ﬁrst Moreton wave front observed at 05:01:
36, when the ﬁrst Moreton wave seems to be interfered with by the second
one. The colored plus signs indicate the length of erupting ﬁlaments. Theircolors mean the azimuth angle as shown in Fig. 2. The blue-green, yellow,
and red plus signs are f1, f2, and f3 in Fig. 2, respectively. (d) Metric radio
spectrogram (25–2500 MHz) observed with the Hiraiso Radio Spectrograph.
We note that there are no data from 05:00:00 to 05:00:15 because of the
instrument’s calibration.

wave itself was not special. The average speeds of the ﬁrst,
second, and third Moreton waves are 460, 740, and 690 kms⫺1, respectively. The dash-dotted lines in Figure 3c show thesecond-degree polynomial ﬁts. On the basis of these ﬁts, theinitial speeds of the Moreton waves are 580, 1200, and 840
km s⫺1, where initial means the time when each wave front
became visible. The decelerations are 1.0, 2.4, and 0.7 kms⫺2. All three Moreton waves began propagation at a speed ofnear Alfvén speed (∼300–1000 km s⫺1; see Narukage et al.
2002, 2004) and decelerated during propagation. On the basis

L47of the polynomial ﬁts, the Moreton waves were located several
tens of thousands of kilometers away from the ﬂare sites whenthe energy releases occurred. These properties correspond toprevious studies of Moreton waves reported by Warmuth et al.
(2004).
Figure 3d shows the metric radio spectrogram (25–2500
MHz) observed with the Hiraiso Radio Spectrograph (HiRAS)(Kondo et al. 1995). The excited coronal electrons emit metricradio waves at the fundamental and harmonic plasma frequencyof the coronal electrons themselves. This emission is called aradio burst. The drift pattern of the radio burst from high to
low frequency means that the site of radio emission moves
from the lower to upper corona (Kai 1970). Speciﬁcally, a radio
burst caused by a shock wave is called a type II burst (Harvey
et al. 1974). In this ﬂare, HiRAS detected faint signals driftingfrom around 500 MHz to low frequency around 04:58:30, 05:
00:30, and 05:04:30. These might be type II bursts and related
to the Moreton waves. Moreover, the radio signal was suddenly
enhanced from 05:01:15 and continued to 05:02:45 as shownby arrow A in Figure 3d, just when the second Moreton wave
caught up and merged with the ﬁrst Moreton wave. This drifting
pattern consisted of two layers which started from frequencies
of 100 and 200 MHz at 05:01:15. The layers of lower andupper frequency were emitted at the fundamental and second
plasma frequency harmonics, respectively. On the basis of thecoronal electron density model proposed by Newkirk (1961)
the radio wave at 100 MHz is emitted by excited coronal electrons at an altitude of 170,000 km. This altitude corresponds
to the distance between the ﬂare site and the place where theﬁrst and second Moreton waves merged together (150,000–
200,000 km; see Fig. 3c). Hence, we interpret this radio enhancement to have been caused by the interaction of the shockwaves, namely, two shock waves merged together and thepower of the merged shock became stronger than the premerged
shock waves. Our event is the ﬁrst report of solar-ﬂare-associated shock-shock interaction.
Gopalswamy et al. (2001) reported a similar phenomenon ininterplanetary space. They found merging coronal mass ejections(CMEs) and a related radio enhancement caused by the interaction between shock waves ahead of the ﬁrst CME and the coreof the slow CME, i.e., interaction between a shock wave and amass of plasma. In our event, not only the shock-shock interaction but also the shock-plasma interaction was detected. Whenthe third Moreton wave (blue diamonds in Fig. 3c) passed
through the position of ﬁrst erupting ﬁlament f1 (blue-green plus
signs), the ﬁlament was accelerated from 60 km s⫺1 (which isan average speed from 05:01:36 to 05:03:30 in the plane of thesky) to 230 km s⫺1 (05:03:30–05:06:02, which is marked by ablack thick line in Fig. 3c). We consider that this accelerationof the ﬁlament is caused by the third Moreton wave. And, in
this timing, the radio signal was enhanced at 250 MHz as shownby arrow B in Figure 3d. The 250 MHz radio wave is emitted
at an altitude of 8500 km. This value is consistent with thecommon altitude of ﬁlaments. Hence, we argue that this radioenhancement was caused by the interaction between a shock
wave and a mass of plasma, like the report of Gopalswamy etal. (2001) and ﬁlament oscillations caused by Moreton waves
reported in some papers (Eto et al. 2002; Okamoto et al. 2004).
4. SUMMARY AND DISCUSSION

We observed the behavior of shock waves in detail. Although
this ﬂare was not special in magnitude or conﬁguration, shockwaves successively occurred three times. This means that shockNarukage et al., “Three Successive and Interacting Shock Waves” (2008)
L48

NARUKAGE ET AL.

waves might be generated by a ﬂare more frequently and successively than we had thought based on previous observations.
Moreover, these shock waves interacted with each other. Thissuggests that the conﬁgurations of ﬂare-associated shock waves
are more complicated than we had thought. These three successive and interacting Moreton waves are unprecedented andthus worthy to report themselves. What is more, these homologous Moreton waves gave us a unique opportunity to understand the generation and propagation mechanism of Moreton
waves.
First, we discuss the propagation of Moreton waves. The positions of Moreton waves can be ﬁtted using the second-degree
polynomial ﬁts as shown in Figure 3b. This suggests that theMoreton waves freely propagated with decelerations. The generation process of Moreton waves, i.e., shock waves, might becompleted until the Moreton waves became visible in Ha.
Next, we pay attention to the relation between erupting ﬁlaments and Moreton waves. In this event, ﬁlaments also erupted
three times. The erupting directions of f1, f2, and f3 are southward, south-southwestward, and southwestward, respectively.
This shows that the directions of ﬁlament eruptions changed
counterclockwise in the plane of the sky. This tendency issimilar to the Moreton waves. The ﬁrst Moreton wave began
to propagate southward (see Fig. 1b), the second wave began
to propagate in a direction slightly to the counterclockwise sideof the ﬁrst wave (see Figs. 1b and 1g), and the third wave ina direction to the counterclockwise side of the second wave
(see Figs. 1g and 1m). We also compared the speed of ﬁlament
eruptions and Moreton waves. The initial speeds of f1, f2, andf3 were 20, 170, and 50 km s⫺1, respectively, where the initial
speeds were derived using the three initial sequential images
for each eruption, namely, Figures 2b–2d for f1, Figures 2h–
2j for f2, and Figures 2k–2m for f3. The initial speeds of theMoreton waves are 580, 1200, and 840 km s⫺1. The second
Moreton wave is the fastest, the third is the middle, and theﬁrst is the slowest. This is the same order as the speeds of theerupting ﬁlaments. Hence, we consider that there are certain
correlations between the erupting ﬁlaments and Moreton waves
in their directions and speeds.
On the basis of the above observational results, we set upa hypothesis that the erupting ﬁlaments generated the Moreton
waves (shock waves). The initial speeds of the ﬁlaments reported in this Letter were 20–170 km s⫺1 in the plane of thesky, and the line-of-sight speeds were about several tens kms⫺1 because the erupting ﬁlaments were observed in Ha Ⳳ 0.5
˚ and Ha Ⳳ 0.8 A
˚ . These erupting ﬁlaments can drive theA

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shock waves (Moreton waves), because Chen et al. (2005)
showed that ﬂux rope at a speed of 100 km s⫺1 generates a
shock wave in their numerical simulation. If once the ﬁlament
eruption generated the shock waves, the shock wave propagates
faster than the erupting ﬁlament. After the shock wave goesaway from the erupting ﬁlament enough, the shock wave canfreely propagate. This is consistent with our observations where
the Moreton waves are more than 50,000 km away from theﬁlaments and propagated freely (see Fig. 2c). In addition, thishypothesis might explain the gap where the Moreton waves
were located several tens of thousands of kilometers away fromthe ﬂare sites when the energy releases occurred (see Figs. 2band 2c). According to Ohyama & Shibata (1997, 2008) not
only an X-ray plasma ejection but also an Ha ﬁlament started
to slowly eject at several tens km s⫺1 about 10 minutes before
the impulsive phase, and were suddenly accelerated to several
hundreds km s⫺1 just before or at about the onset of the impulsive phase. So, the ﬁlament eruptions in our events mightalso start before the impulsive phase, and might reach at a
distance of several tens of thousands km (p several tens km
s⫺1 # about 10 minutes) when the ﬂare energy releases wereobserved in radio and hard X-rays.
As mentioned above, our hypothesis that the erupting ﬁlaments
generated the Moreton waves (shock waves) is consistent withthe observation. However, we have not obtained clear evidence
for our hypothesis. To understand the generation and propagation
mechanism of Moreton waves, more observations with high temporal and spatial resolution are required. Our results show thatnew telescopes make it possible to observe the shock waves indetail. In addition to chromospheric observations with SMART,
coronal observations with the state-of-the-art X-Ray Telescope(XRT) and Extreme-Ultraviolet Imaging Spectrometer (EIS) on
board the Hinode satellite will give us more information about
ﬂare-associated shock waves (Narukage 2007), e.g., not only
plane-of-sky velocity but also line-of-sight velocity, temperature,and density of coronal plasma.
The authors thank especially M. Kadota and all other members of the Kwasan and Hida Observatories for useful comments. We also acknowledge an anonymous referee for his/her
useful comments and suggestions. This work was supportedby the Grant-in-Aid for the Global COE Program ‘‘The Next
Generation of Physics, Spun from Universality and Emergence”
from the Ministry of Education, Culture, Sports, Science, andTechnology (MEXT) of Japan.

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PLATE 1
Fig. 1.—Three successive Moreton waves and associated ﬂare observed with SMART. (a, e, l) Original Ha center images. The black crosses indicate thebrightest points in each image, where we suggest that energy was released by the ﬂare and the Moreton waves were generated. (b–d, f–k, m–s) “Dopplergrams”
˚ ),derived by the equation of log (Iblue /Ired ), where Ired and Iblue are the intensity images almost simultaneously observed in the red and blue wings (Ha Ⳳ 0.5 A
respectively. The red, green, and blue plus signs indicate the wave fronts of the ﬁrst, second, and third Moreton waves, respectively. (t) The square and circle
with solid line are the ﬁeld of view of (a–s) and the solar limb, respectively. The square with dotted line indicates the ﬁeld of view of Fig. 2. We note that thedark conﬁgurations located at the center part of (j) and the right-bottom part of (n) are clouds.

Asai et al., “Anemone Structure and Flares/CMEs” (2009)
Click
Here

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, A00A21, doi:10.1029/2008JA013291, 2009

for

Full
ArticleEvolution of the anemone AR NOAA 10798 and the related
geo-effective flares and CMEs
Ayumi Asai,1,2,3 Kazunari Shibata,4 Takako T. Ishii,4 Mitsuo Oka,4,5 Ryuho Kataoka,6Ken’ichi Fujiki,7 and Nat Gopalswamy8
Received 9 April 2008; revised 10 November 2008; accepted 17 December 2008; published 21 February 2009.

[1] We present a detailed examination of the features of the active region (AR) NOAA
10798. This AR generated coronal mass ejections (CMEs) that caused a large geomagnetic
storm on 24 August 2005 with the minimum Dst index of 216 nT. We examined theevolution of the AR and the features on/near the solar surface and in the interplanetary
space. The AR emerged in the middle of a small coronal hole, and formed a sea anemone
like configuration. Ha filaments were formed in the AR, which have southward axial
field. Three M class flares were generated, and the first two that occurred on 22 August
2005 were followed by Halo-type CMEs. The speeds of the CMEs were fast, and recorded
about 1200 and 2400 km s1, respectively. The second CME was especially fast, andcaught up and interacted with the first (slower) CME during their travelings toward Earth.
These acted synergically to generate an interplanetary disturbance with strong southwardmagnetic field of about 50 nT, which was followed by the large geomagnetic storm.
Citation: Asai, A., K. Shibata, T. T. Ishii, M. Oka, R. Kataoka, K. Fujiki, and N. Gopalswamy (2009), Evolution of the anemone AR
NOAA 10798 and the related geo-effective flares and CMEs, J. Geophys. Res., 114, A00A21, doi:10.1029/2008JA013291.

1. Introduction[2] Space weather has attracted a lot of attention in recent
times. Space weather research involves various related
fields, such as the solar surface, solar wind, interplanetary
space, geomagnetosphere, ionosphere, and atmosphere,
since a comprehensive understandings from active phenomena on the solar surface to the propagation of the disturbances toward Earth are crucially required for the studies.
[3] Vast plasma ejected from the solar corona in the formof coronal mass ejections (CMEs) leads to a geomagnetic
storm, and therefore, CMEs have been actively discussed.
Moreover, large geomagnetic storms are associated withlarge flares (that are emitting strong X ray), long duration
events (LDEs), fast CMEs, and so on [e.g., Gopalswamy etal., 2007]. Flare locations are another factor for a major
geomagnetic storm, since the flare location close to the disk
center indicates that the related CME is heading towards
Earth and is likely to cause a large geomagnetic storm[Manoharan et al., 2004]. Ejections that cause strong
disturbances with southward magnetic field in the interplan1
Nobeyama Solar Radio Observatory, National Astronomical Observatoryof Japan, Minamisaku, Japan.
2
National Astronomical Observatory of Japan, Mitaka, Japan.
3
Graduate University for Advanced Studies, Hayama, Japan.
4
Kwasan and Hida Observatories, Kyoto University, Yamashina, Japan.
5Center for Space Plasma and Aeronomic Research, University of
Alabama in Huntsville, Huntsville, Alabama, USA.
6
RIKEN, Wako, Japan.
7
Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya,Japan.
8
NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.

Copyright 2009 by the American Geophysical Union.
0148-0227/09/2008JA013291$09.00

etary space are also important. Coronal holes (CHs) are, on
the other hand, related with fast solar wind because of theiropen magnetic field, and therefore, themselves have been
another important factor for space weather studies. Whilelarge geomagnetic storms are caused by Earth-directed
CMEs [see, e.g., Gosling et al., 1990], weaker storms areassociated with high-speed streams from CHs [see, e.g.,
Sheeley et al., 1976]. However, storms related to high-speed
streams from CHs cause larger flux enhancement of MeV
electrons of the Earth’s Van Allen belt than the CMEassociated storms do on average [Kataoka and Miyoshi,
2006]. It has been, furthermore, reported that many fast
Halo CMEs are associated with CHs [Verma, 1998; Liu andHayashi, 2006]. The recent work done by Liu [2007]
showed that the speeds are faster even statistically thanthose of CMEs under the heliospheric current sheet. Therefore, in order to understand what kind of events cangenerate a large geomagnetic storm, it is necessary to study
active phenomena on the solar surface and the propagationin the interplanetary space, in relation to the surrounding
magnetic structure. In this paper we examine in detail theevolution of an active region (AR) that emerged in a CH
and the related flares, CMEs, and the geomagnetic storm to
elucidate how such a magnetic configuration works togenerate a geo-effective flares/CMEs.
[4] CMEs originating from AR NOAA 10798, generated
a large geomagnetic storm on 24 August 2005, which was
one of the 88 major geomagnetic storms reported by Zhang
et al. [2007]. This AR has been highly paid attention to,since it was one of the targets of the International CAWSES(Climate and Weather of the Sun-Earth System) Campaign(see http://www.bu.edu/cawses/secondcampaign.html), andat the related virtual conference (http://workshops.jhuapl.

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A00A21Figure 1. Overview of the geomagnetic storm that occurred on 24 August 2005 and the related solarterrestrial events. (top to bottom) SXR flux in the GOES 1.0 – 8.0 Å channel, proton fluxes in >10 MeV(black line) and >50 MeV (gray line) channels obtained with GOES, bulk velocity of solar wind Vsw
measured with ACE, total magnetic field strength jBj (black line) and Z component of the magnetic fieldBz (gray line) measured with ACE, and Dst index produced by the Kyoto University.
edu/s1/index.html), there were intensive discussions on theAR [e.g., Asai et al., 2006]. Figure 1 shows an overview ofthe geomagnetic storm and the related solar-terrestrial
events. The Soft X-ray (SXR) flux in the GOES 1.0– 8.0 Å
channel shows three M class flares that occurred on 22 and23 August 2005. The first two flares (marked with crosses)
were associated with the CMEs responsible for the geomagnetic storm in question. In the second panel we canrecognize the sufficient enhancements of the proton fluxesin >10 MeV (black line) and >50 MeV (gray line) channels
obtained with GOES. Both flares were followed byenhancements of solar energetic particles (SEPs), and thesecond flare’s was larger. The bulk velocity of solar wind
Vsw in the third panel and the total jBj (black line) and Z
component of the magnetic field Bz (gray line) in the fourth
panel were measured with the Advanced Composition
Explorer (ACE). The first shock was recorded at 05:35 UT
by ACE as shown by the dashed line. The same shock wasalso recorded by the Geotail satellite at 06:15 UT. Theinterplanetary magnetic field had a strong southward component of about 50 nT. The bottom panel shows the Dst
index produced by the Kyoto University. The decrease of
the Dst index was quite large, reaching 216 nT. In section
2 we describe the evolution of the AR, focusing on thephotospheric magnetic configuration, and the Ha filament
formed during the evolution of the AR and the coronalfeatures are also presented. In section 3 we discuss theflares/CMEs that occurred on 22 August 2005. In section 4

we shortly review the associated interplanetary disturbances. In section 5 we summarize our results.

2. Evolution and Structure of Active Region
NOAA 10798
2.1. Evolution[5] First, we examine the evolution of the photosphericmagnetic field. Figure 2 shows the continuum images (top),
the magnetograms (middle), and the extreme ultraviolet(EUV) images (bottom) of AR NOAA 10798. The continuum images and the magnetograms were obtained with theMichelson Doppler Imager (MDI [Scherrer et al., 1995])
aboard the Solar and Heliospheric Observatory (SOHO[Domingo et al., 1995]), while the EUV images are takenwith the Extreme-Ultraviolet Imaging Telescope (EIT[Delaboudiniére et al., 1995]) aboard SOHO. Each image
was taken at about 00:00 UT of the day. AR NOAA 10798
emerged on 18 August 2005 and rapidly evolved.
[6] Although the region showed a simple bipolar configuration, while it was in violation of the so-called ‘‘HaleNicholson’s magnetic polarity law’’ [Hale et al., 1919],
according to which the preceding spots in the southernhemisphere should have a negative polarity during solar
cycle 23. These ‘‘reverted polarity’’ ARs are statisticallymore likely to generate flares and CMEs [López Fuentes etal., 2003; Tian et al., 2005]. We checked all the ARs thatappeared in 2005, and found that only 7 ARs (5%),
including the AR NOAA 10798, were the ‘‘reverted polarity’’2 of 10

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A00A21Figure 2. Temporal evolution of the AR. The (top) continuum images and the (middle) magnetograms
observed by SOHO/MDI. (bottom) The EUV images obtained with SOHO/EIT. Each image was taken at
about 00:00 UT of the day. Solar north is up, and west is to the right.
ARs. Furthermore, 4 of the 7 ARs, including AR NOAA
10798, showed high solar activity. This implies that AR
NOAA 10798 had potentially a very complex structure. For
example, a highly twisted and kinked magnetic structure
may be embedded beneath the photosphere as Ishii et al.
[2000] and Kurokawa et al. [2002] reported. Indeed, we cansee the rotating motion of the pair of the sunspots counterclockwise during the disk passage. This AR further evolved
and generated an X17 flare on 7 September 2005 when it
returned as AR NOAA 10808 [Wang et al., 2006; Nagashima
et al., 2007].
[7] The top panel of Figure 3 shows the SXR lightcurves
obtained by GOES in the 1.0 – 8.0 Å (upper) and 0.5– 4.0 Å
(lower) channels. The bottom panel of Figure 3 shows thetime profiles of the magnetic flux of this AR. That for thenegative magnetic flux is multiplied by 1. The calculatedarea is 40000  40000, and is as wide as it covers the whole
AR. Following the emergence and evolution of the active
region from 18 August 2005, the magnetic fluxes as well as
the SXR intensity gradually increased, with three M classflares occurring on 22 and 23 August, before rotating
behind the west limb. In this paper we mainly discuss thefirst two flare that occurred on 22 August 2005, since thegeomagnetic storm on 24 August 2005 is attributed to
the related eruptions/CMEs.
2.2. Filament Formation[8] Second, we examine the filament formation in AR
10798, using the Ha images. A filament is just a visualizedpart of a helical flux rope, and it is only a fraction of thewhole of the flux rope. However, it is thought that afilament is located in the middle of the flux rope, and thatit even represents the whole structure [see, e.g., Low and
Hundhausen, 1995]. Therefore, an eruption of a filament isalways related with a large-scale disturbance of the coronalmagnetic field that often appears as an EIT dimming. As
Munro et al. [1979] suggested that more than 70% of CMEs

are associated with eruptive prominences or filament disappearances (with or without flares), and therefore, filament
eruptions are very important as a CME-associated phenomenon. Moreover, we often see an ejected filament observedin Has/microwaves or a plasmoid in SXRs at the center ofan ejecta, when it is accompanied by a flare. We can evenroughly extrapolate the magnetic configuration of a CMEfrom that of the ejected filament. For example, Rust [1994]
showed that the helicity of ejected filaments correspond tothe chirality of magnetic clouds passing Earth (about 4 days
after the eruptions).
[9] Figure 4 shows the temporal evolution of the AR in
the Ha images (top) and the magnetic field (bottom). The
Ha images in Figures 4a and 4b were taken with the Solar
Magnetic Activity Research Telescope (SMART) (http://
www.hida.kyoto-u.ac.jp/SMART/) at Hida Observatory,
Kyoto University. Figures 4c and 4d were obtained at theObservatoire de Paris, Section de Meudon, and the Big Bear
Solar Observatory, respectively. Both of these images wereobtained through the online data center of the Global HighResolution Ha Network (http://www.bbso.njit.edu/Research/
Halpha/). The magnetograms (Figues 4e – 4h) were taken by
SOHO/MDI.
[10] For three days after the emergence of the AR (until
20 August), a clear arch-filament system [Bruzek, 1967]
was seen (Figure 4a). The bipole-like systems bridged theneutral line and connected the spots of opposite polarity. On
the other hand, after 21 August 2005, these filamentary
structure was abruptly changed. In Figure 4b, some oblique
structure appeared, and showed prefilamental structure.
Comparing with the magnetograms (Figure 4, bottom), we
confirm that the magnetic field of these structure wasoriented from northwest to southeast, which means they
had southward magnetic field. About nine hours after this(Figure 4c), the sheared filamentary structure evolved to alarge Ha filament that lay on the magnetic neutral line
between the sunspots. The arrow in Figure 4c points to the

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Figure 3. Time profiles of SXR flux and magnetic fluxes. (top) SXR flux in the GOES 1.0 – 8.0 Å
(upper) and 0.5– 4.0 Å (lower) channels. (bottom) Magnetic flux of the AR observed with SOHO/MDI.
The calculated area is 40000  40000 centered on the middle of the AR, and is as wide as it covers thewhole AR. The time profile of the negative magnetic flux is multiplied by 1.

Figure 4. (top) Ha images. (a and b) Obtained with SMART at Hida Observatory, Kyoto University.
(c and d) Obtained at Observatoire de Paris, Section de Meudon, and Big Bear Solar Observatory,
respectively. (bottom) Magnetograms taken with SOHO/MDI.
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A00A21Figure 5. Coronal feature of AR NOAA 10798. (a) A SXR image obtained with GOES/SXI. (b) An EUV(195Å) image obtained with SOHO/EIT. The bright region near the center of the image is the AR. The
surrounding dark region is a CH. (c) A magnetogram taken with SOHO/MDI.
filament. The filament formation is consistent with what
Martin [1973] pointed out long ago: developed filaments
usually become apparent about the ‘‘fourth day’’ after the
initial formation of an active region. Unfortunately, there are
no data between Figures 4b and 4c, but we can see the newflux emergence around the magnetic neutral line, (compare
Figure 4f with Figure 4g). The formation of the filamentswith the southward magnetic field is probably related to theemerging flux.
[11] The first M class flare occurred at 00:44 UT on 22
August 2005, which is in the middle of the time between
Figures 4c and 4d. In Figure 4d, we can recognize thedisappearance of the Ha filament after the first flare, while a
new filament formed in the south part of the AR as pointed
by the arrow in the panel. Associated with the first flare, theHa filament formed in the northern part (Figure 4c) erupted,
and with the second flare, the southern one (Figure 4d)
erupted. The axial field of these filaments had southwardmagnetic field, which is easily inferred from the prefilamentary structure. Although we also checked the EUV datataken with EIT and the Transition Region and Coronal
Explorer, we could not find out any phenomena that canbe a source of the CMEs other than the filament eruptions.
We will discuss the flares and CMEs in more detail in
section 3.
2.3. Anemone Structure[12] Figures 5a and 5b show the coronal structure of theAR observed at about 00:00 UT on 20 August 2005, in SXR
and in EUV with Solar X-ray Imager (SXI) on board GOES
and SOHO/EIT, respectively. The bright structure near thecenter of the image is AR NOAA 10798. Figure 5c shows
the magnetogram taken by SOHO/MDI. The following
sunspot with the negative (that is black) polarity is thecenter of the EUV bright structure, and a radial array ofloops is formed. We also present schematic cartoons of
the magnetic structure of AR NOAA 10798 in Figures 6aand 6b.
[13] The appearance is like a sea anemone, and this typeof ARs is sometimes called ‘‘anemone structure’’ [Shibata
et al., 1994a, 1994b], or originally ‘‘fountain’’ [Tousey etal., 1973; Sheeley et al., 1975a] in the Skylab era. We callthese ARs ‘‘anemone ARs’’ in this paper. These anemone

ARs are often associated with the emerging fluxes within
unipolar regions [Sheeley et al., 1975b], and in most cases,
they appear in CHs [Asai et al., 2008]. Although characteristics of anemone ARs have been mainly discussed only in
SXRs, they are commonly seen under such a magnetic
configuration, even in a chromospheric line by the Solar
Optical Telescope on board Hinode [Shibata et al., 2007].
[14] As shown in Figure 5, AR NOAA 10798 is clearlysurrounded with a unipolar region with the positive magnetic polarity, and shows the anemone appearance both in
SXR and in EUV, and we can conclude that AR NOAA
10798 was a typical anemone AR. In emerging, the AR
interacted (reconnected) with the ambient coronal field, andmagnetic loops were arranged radially with the following
spot that has the negative magnetic polarity as the center of
the anemone structure. In Figure 5b the dark region surrounding the AR is a CH. On 22 August, when the flares/
CMEs in the matter occurred, the anemone appearance
somewhat changes as seen in Figure 2. This is caused byprojection like many anemone ARs, while some anemone
ARs keep the appearance even on the limb [Saito et al.,2000].

3. Flares and CMEs[15] Next, we focus on the two M class flares and theassociated CMEs. The first flare that was M2.6 on the
GOES scale, started at 00:44 UT, and peaked at 01:33 UT.
The second one was M5.6 on the GOES scale, and the startand the peak times were 16:46 UT and 17:27 UT, respectively. Both are LDEs, and showed clear arcade structure.
Figure 6c shows a schematic of the magnetic field during
the flares. The sites of the flares were (S11° W54°) and(S12° W60°), respectively.
[16] The two flares were associated with disappearancesof the Ha filaments, and Halo-type CMEs that wereobserved with the Large Angle Spectrometric Coronagraph(LASCO) aboard SOHO (see the SOHO/LASCO CME online
catalog, http://cdaw.gsfc.nasa.gov/CME_list/) [Yashiro et al.,2004]. LASCO images of the two CMEs (CME1 and CME2)
are shown in Figure 7. The left panels are the LASCO C2
running difference images overlaid with EUV images taken
by SOHO/EIT (195 Å), and the right panels are the LASCO

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A00A21Figure 6. Schematic cartoon of AR 10798 and related flares/CMEs. (a) A magnetic flux newly emerges
within a CH. (b) An anemone structure is generated, and an Ha filament is also formed above theemerged flux. (c) A magnetic reconnection occurs beneath the filament, which causes the filament
eruption. The ejected plasma is bent eastward by the surrounding magnetic field with positive magnetic
polarity. (d) The ejecta becomes a magnetic cloud (shown as a cylinder) that have a southward axial
magnetic field and is approaching to Earth. (e) Passage of a FR and the variation of the azimuthal angle ofthe magnetic field fB. When a FR passes the east of Earth, fB evolutes 90– 180 – 270 (90) degree.
C3 running difference images. CME1 was ejected mainly tothe northwest, and CME2 was to the southwest as indicatedby the arrows in Figure 7. The directions were roughly
consistent with the initial position of the Ha filaments (seesection 2.2).
[17] It is particularly notable that the CMEs were quite fast:
CME1 and CME2 had speeds of about 1200 and 2400 km s1,respectively. The speed of CME2 is ranked among the top
17 of all the 13,000 CMEs observed by SOHO/LASCO
until the end of 2007. Although the time interval between
the two flares/CMEs was about 16 hours, CME2 possibly
caught up with CME1 before reaching 1 AU [Gopalswamy
et al., 2001a]. Statistically, a CME ejected with the velocity
of VCME have an acceleration a m s2 = 2.193  0.0054 
VCME km s1 [Gopalswamy et al., 2001b], and the expected
accelerations for CME1 and CME2 are 4.3 and 10.8 m s1
for the current case. Therefore, we estimate that the interacting between CME1 and CME2 occurred at about 1 AU
(i.e., near Earth), by assuming constant accelerations for theCMEs.
[18] The interplanetary disturbance associated with the
Halo-type CMEs can be followed by using interplanetary
scintillation (IPS). When we see a radio source through ahighly turbulent plasma associated with a CME travelingfrom Sun, the radio source scintillates. Therefore, such
scintillations show us the electron density fluctuation
caused by the CME. As an effective indicator of the electron
density fluctuation, we often use g value (g) calculated from
IPS data (see Tokumaru et al. [2000, 2003, 2005] formore details). The g value represents the variation of theelectron density fluctuation in the solar wind (DNe), asR2
g2 / 10 DNe w(z)dz, where z is a distance along the line ofsight, and w(z) is the IPS weighting function given by Young[1971]. It is normalized to the mean level of density
fluctuations so that the quiet solar wind yields g value

Figure 7. White light CME images obtained with SOHO/LASCO. (left) C2 running difference images for CME1/CME2
overlaid with EUV images obtained with SOHO/EIT (195Å).
(right) C3 running difference images. The arrows roughly
point the main part of CMEs.

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Figure 8. Daily (Japanese daytime) sky projection mapsof g values obtained with IPS observations at STEL Nagoya
University. In each map, the center corresponds to Sun
center, and the dotted cocentric circles are constant radii
contours from Sun drawn at 0.3, 0.6, and 0.9 AU. Solidcircles indicate the points of the closest approach to Sun(P points) on the line of sight where g value data wereobtained (P point approximation). Dark gray and black
circles represent the locations where the g values are larger
than 1.5 and 2.0, respectively.
around unity, and the enhancement (g > 1) shows thepassing of a turbulent plasma.
[19] We examined the g values taken with IPS at SolarTerrestrial Environment Laboratory (STEL), Nagoya University [Kojima and Kakinuma, 1990; Asai et al., 1995;
Tokumaru et al., 2000]. Figure 8 shows the daily (Japanese
daytime) sky projection maps of the g values. In each map,the center corresponds to the location of Sun, and dotted
cocentric circles are constant radii contours from Sun drawn
at 0.3, 0.6, and 0.9 AU. The solid circles indicate the pointsof the closest approach to Sun (P points) on the line of sightwhere g value were obtained (P point approximation). The
locations of the stronger g values are emphasized by colors
and sizes of the circles. The dark gray and the black circles
represent the locations where the g values are larger than 1.5
and 2.0, respectively.
[20] In both panels of Figure 8 we can see density
fluctuations that were caused by the two Halo CMEs, while
we cannot distinguish the individual CMEs discretely due
the low spatial resolution of IPS. The front of the disturbance, which was caused by CME1, reached about 0.4 and
0.8 AU on August 23 and 24, respectively. CME1 is welldecelerated, and the speed is about 700 km s1. Thesefluctuations are distributed roughly in all direction.

4. Interplanetary Disturbance[21] Here, we investigate in more detail why a strong
disturbance with a magnetic field of about 50 nT arrived at
Earth. As we mentioned above (see section 4), CME2
probably caught up with CME1, and therefore, disturbanceis regarded as a merged product of CME1 and CME2,
although the interaction was not directly observed.
[22] Figure 9 shows a 7-hour interval corresponding to
the geo-effective part of the interplanetary disturbance from
Geotail. The top four panels present the magnetic field in
GSE coordinates obtained by the Magnetic Field Experiment(MGF [Kokubun et al., 1994]). The magnitude jBj and thex (Bx), y (By), and z components (Bz) of the magnetic field

269
A00A21are shown. The fifth panel shows the ion velocity Vxobserved with the Low Energy Particle Experiment (LEP
[Mukai et al., 1994]). The bottom panel shows the electron
density Ne observed by the Plasma Wave Instrument (PWI
[Matsumoto et al., 1994]). The density reached so high thatthe counts of the particle detectors onboard Geotail (andprobably ACE as well) were saturated, and therefore, it isunderestimated during the strong disturbance. To avoid theunderestimation of the density, we simply traced local
enhancements of the electrostatic noise that appears in the
dynamic spectra of the electric field as have been carried out
elsewhere (see, e.g., Figure 4 of Terasawa et al. [2005]).
Although the measurement also have a uncertainty, it ismore accurate than that by particle measurement experiments, since the counts were not saturated.
[23] A flux rope (FR) structure can be identified by thesmooth rotation of the magnetic field from 09:15 to 11:15 UT
as shown by the vertical dashed lines in Figure 9. It isnotable that the 2 hours duration of this FR was extremely
short compared to the typical duration of about 20 hours[Lepping et al., 2003; Gopalswamy, 2006]. The local
velocities of the FR is 650 km s1, as will be discussed
below. We estimated the radial size of the FR to be about
0.03 AU, by multiplying the local velocity by the 2 hours
duration. This value is also extremely small compared withtypical ones of 0.2– 0.3 AU [Forsyth et al., 2006]. This FR
showed the smooth rotation from positive to negative By, anegative Bz peak in the middle of the By rotation, andrelatively small Bx component. These can be roughly
explained by the passage of a right-handed flux rope withthe southward pointing axis field, which is consistent withthe magnetic field configuration of the associated Ha
filaments. The largest geomagnetic storm of cycle 23 that
occurred on 20 November 2003 was associated with asimilar FR [Gopalswamy et al., 2005].
[24] About 15 min before the front edge of the FR (i.e., atabout 09:00 UT), a solar wind discontinuity is identified by
the sudden increases in the magnetic field, solar wind speed,
and density as shown by the vertical solid line. From thevariation of velocity distribution function, we confirmed anabrupt increase of temperature (not shown) there, andconcluded that the discontinuity is a shock. We call the
discontinuity as the ‘‘second shock,’’ and the ‘‘first shock’’is for the one observed at the beginning of the event asshown with the dashed line in Figure 1 and the vertical solidline at about 06:15 UT in Figure 9.
[25] The extremely strong southward magnetic field, theunusual short duration of the FR (2 hours), and very small
separation between the second shock and the FR front(15 min) can be naturally explained, if we regard the
disturbance as the product of very fast shock wave associated with CME2 interacting with the slower body of CME1
in traveling to Earth. Therefore, CME2 suffered from astrong deceleration, which implies that there was a greatcompression of the interplanetary medium in front of CME2.
In this case the first and the second shocks are thought to be
associated with CME1 and CME2, respectively.
[26] The local velocities of the first and the second shocks
are measured by the positional relation between the ACE andGeotail satellites, and therefore, we can roughly estimate theiraccelerations (decelerations). ACE and Geotail were locatedat (223.7, 10.6, 4.5) and (12.9, 25.7, 1.9) RE (= 6378 km) at

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A00A21Figure 9. Geotail observation of the interplanetary disturbance on 24 August 2005. (top to bottom)
Magnitude and X, Y, and Z components in GSE coordinate of the magnetic field (MGF experiment), ion
velocity (LEP/SWI experiment), and electron density (PWI/SFA). The vertical solid lines show theshocks (the first and the second shocks). The vertical dashed lines show the flux rope.
09:00 UT on 24 August 2005 in GSE coordinate system,
respectively. As already mentioned, CME1 and CME2 wereejected with velocities of about 1200 and 2400 km s1, atintervals of 16 hours. On the other hand, the local velocitiesof the first and the second shocks are 650 and 710 km s1,respectively, and the time separation between them isreduced to only about 3 hours. Assuming the constantaccelerations, they are estimated to be 3.2 and 13 m s2.
As we calculated above, the accelerations for CME1 and
CME2 are statistically expected to be 4.3 and 10.8 m s1.
The additional deceleration of CME2 also indicates that itinteracted with slower CME1 and compressed the interplanetary medium there.

5. Summary and Discussions[27] In order to make clear the importance of an AR thatemerged in a CH to generate geo-effective flares/CMEs, we
examined the evolution of the AR NOAA 10798, the solar
events associated with a geomagnetic storm that occurred
on 24 August 2005, and the related interplanetary disturbances. The summary of the features of the AR and theevents is as follows: (1) Highly twisted and complex
magnetic flux emerged within a small CH on 18 August
2005, which was named NOAA 10798; (2) An anemone
type structure was generated, and Ha filaments that had
southward axial fields were formed on 21 August 2005;(3) Two halo CMEs associated with M class flares occurred
on 22 August 2005; (4) The CME speeds were fast,
especially the second one recorded 2400 km s1; (5) The
interplanetary disturbances with strong southward magnetic
field of about 50 nT and strong compression of plasma
were produced.
[28] The CMEs were particularly geo-effective, and theminimum Dst index was 216 nT. The reasons for the CMEs
to be so geo-effective were the high speeds of the two
CMEs and their interaction as well as the CMEs traveled
directly toward Earth. For the current case, the speed of
CME2 was faster and pushed the slower CME1, which led
to a unusual strong compression of the plasma at the front of
CME2.
[29] The high speeds of the CMEs are more notable. The
AR was large and very complex, and violated the HaleNicholson’s magnetic polarity law. These reverted polarity
ARs are statistically favorable to produce large flares.
However, it is suspicious whether just the violation of theHale-Nicholson’s law is responsible for high speed CMEsof about 2000 km s1, and it should be quantitatively andstatistically clarified in the future. In this paper we suggestthat the fast CMEs are probably a consequence of theeruption inside a CH from an anemone AR. This isconsistent with the association between fast Halo CMEs
and CHs as reported before [Verma, 1998; Liu and Hayashi,
2006; Liu, 2007].

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Asai et al., “Anemone Structure and Flares/CMEs” (2009)
A00A21ASAI ET AL.: ANEMONE STRUCTURE AND FLARES/CMES[30] Eruptive activities of anemone ARs are usually low[Asai et al., 2008], and often confined to small-scale
activities inside CHs that appears to be SXR bright points.
In some cases, anemone ARs can produce large SXR
coronal jets [Shibata et al., 1994b; Vourlidas et al., 1996;
Kundu et al., 1999; Alexander and Fletcher, 1999]. This isbecause the situation of an emergence of a magnetic flux
within a CH is suitable for magnetic reconnection with thesurrounding field to generate SXR coronal jets and/or Ha
surges [Yokoyama and Shibata, 1995, 1996]. Wang [1998]
indicates the possibility that even polar plumes are associatedwith jets from anemone ARs at high latitudes. Anemone
ARs are related to nonradial coronal streamers emanatingfrom magnetically high latitudes [Saito et al., 2000]. The
relation between anemone ARs and fast solar winds havealso been paid attentions to [Takahashi et al., 1994; Saito etal., 1994; Wang, 1998]. Saito et al. [2000] further discussed
the rotational reversing model and the triple dipole model toexplain the reversal of the solar surface magnetic field, andanemone ARs play an important role in this. This model
implies that anemone ARs are more conspicuous in the
decaying phase of a solar cycle as in the case of AR NOAA
10798.
[31] On the other hand, the deflection of CMEs eastward
by the interplanetary fields effectively worked in the current
case as shown in the Figure 6d. As Wang et al. [2006]
pointed out, the faster CMEs are deflected more eastward,
and therefore, the AR NOAA 10798 generated geo-effective
CMEs, although it was quite close to the southwest limb.
The azimuthal angle of the magnetic field measured fromthe x axis fB (= arctan(By/Bx)) changed 90°– 180°– 270°(90°) during the passage of the FR, which is consistent
with the guess that the deflection of the CMEs were sostrong that the axis of the FR passed through the east of
Earth. This is also consistent with the fact that the flares in
the next rotation (and renamed as NOAA 10808) did notaffect the magnetosphere so much [Wang et al., 2006;
Nagashima et al., 2007]. Furthermore, the extremely shortduration of FR and the missing of CME1 (see Figure 9) are
possibly explained by the skimming encounter with the
CMEs due to the strong deflection.
[32] The nature of the interplanetary disturbances andtheir impact on the magnetosphere strongly depend on thefeatures of emergence and evolution of an AR and therelation with the surrounding magnetic field. In this work
we succeeded to follow in detail the evolution of the AR
and the large geomagnetic storm resulting from eruptions in
the AR. The reconstruction of the proposed scenario using
numerical simulations will be attempted in the future.
[33] Acknowledgments. We first acknowledge anonymous referees
for their useful comments and suggestions. We wish to thank J. Kozyra,
K. Hayashi, N. V. Nitta, V. Yurchyshyn, and B. J. Thompson for fruitful
discussions and their helpful comments. This work was supported by the
Grant-in-Aid for the Global COE Program ‘‘The Next Generation of
Physics, Spun from Universality and Emergence’’ from the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of Japan.
This work was also supported by the Grant-in-Aid for Creative Scientific
Research ‘‘The Basic Study of Space Weather Prediction’’ (17GS0208,
Head Investigator: K. Shibata) from the Ministry of Education, Science,Sports, Technology, and Culture of Japan. This work was partially carriedout by the joint research program of the Solar-Terrestrial Environment
Laboratory, Nagoya University. We would like to acknowletge all themembers of the Geotail/PWI, LEP, and MGF for providing the data. We
would like to thank WDC for Geomagnetism, Kyoto Dst index service. Our

271
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thanks also go to the SMART teams of Hida Observatory, Kyoto University, Big Bear Solar Observatory, and Meudon Observatoire de Paris,
Section de Meudon, for letting us use the Ha data. We made extensive
use of SOHO, and ACE Data Center. M.O. was supported by the Grant-inAid for JSPS Postdoctoral Fellows for Research Abroad.
[34] Amitava Bhattacharjee thanks the reviewers for their assistance inevaluating this paper.

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A. Asai, Nobeyama Solar Radio Observatory, National Astronomical
Observatory of Japan, Minamimaki, Minamisaku, Nagano 384-1305,
Japan. (asai@nro.nao.ac.jp)
K. Fujiki, Solar-Terrestrial Environment Laboratory, Nagoya University,
Chikusa, Nagoya, Aichi 464-8601, Japan.
N. Gopalswamy, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA.
T. T. Ishii and K. Shibata, Kwasan and Hida Observatories, Kyoto
University, Yamashina, Kyoto 607-8471, Japan.
R. Kataoka, RIKEN, Wako, Saitama 351-0198, Japan.
M. Oka, Center for Space Plasma and Aeronomic Research, University of
Alabama in Huntsville, Huntsville, AL 35807, USA.

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Ueno et al., “CHAIN-project and installation of the flare monitoring telescopes” (2007)

Bull. Astr. Soc. India (2007) 35, 697–704

CHAIN-project and installation of the ﬂaremonitoring telescopes in developing countries
Satoru UeNo1∗, Kazunari Shibata2 , Goichi Kimura1 , Yoshikazu Nakatani1 ,Reizaburo Kitai1 and Shin ichi Nagata1
1

Hida Observatory, Kyoto University, Kurabashira, Kamitakara, Takayama city,
GIFU 506-1314, Japan
2
Kwasan Observatory, Kyoto University, Ohmine-cho, Yamashina-ku, Kyoto 607-8471, Japan

Abstract. The Flare Monitoring Telescope (FMT) was constructed in 1992
at the Hida Observatory in Japan to investigate the long-term variation of solar activity and explosive events, as a project of the international coordinatedobservations programme (STEP). The FMT consists of ﬁve solar imaging telescopes and one guide telescope. The ﬁve telescopes simultaneously observe thefull-disk Sun at diﬀerent wavelengths around H-alpha absorption line or in different modes. Therefore, the FMT can measure the three-dimensional velocity
ﬁeld of moving structures on the full solar disk without the atmospheric seeing
eﬀect. The science target of the FMT is to monitor solar ﬂares and erupting
ﬁlaments continuously all over the solar disk and as many events as possible
and to investigate the relationship between such phenomena and space weather.
Now we are planning to start a new worldwide project called as “Continuous
H-alpha Imaging Network (CHAIN)-project”. As part of this project, we are examining the possibility of installing telescopes similar to the FMT in developing
countries with cooperative help by the United Nations. We have selected Peru
as the candidate country where the ﬁrst oversea FMT will be installed, and are
beginning to study the natural environment, the seeing conditions, the proper
design of the telescope for Peru and the training and education programme ofoperating staﬀ, etc.
Keywords : Sun : ﬂares, shockwaves, ﬁlaments, prominences, coronal massejections, solar-terestrial relations∗ e-mail: ueno@kwasan.kyoto- u.ac.jp273

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1.

Introduction

The Hida Observatory has three solar telescopes, i.e., the Domeless Solar Telescope(DST), the Solar Magnetic Activity Research Telescope (SMART), the Flare Monitoring
Telescope (FMT) and the two nighttime telescopes, i.e., the 60 cm reﬂector and the 65 cm
refractor. Our main ﬁelds of research and education are solar physics, solar astro-plasma
physics, solar-system physics and solar-stellar physics. In recent years, studies of theenvironmental variation of the solar-terrestrial system has become quite important in theﬁeld of solar physics, because solar active phenomena have been recognized as the main
sources of space disturbances, and studies of solar active phenomena are indispensablebasis of space weather prediction. In our observatory, such studies have been done in twocomplementary ways. One way is to perform detailed studies of solar events with highspatial-resolution imaging, spectrum measurement with high wavelength- resolution andmulti-wavelength spectra measurement at the DST. However, its ﬁeld of view (FOV) is
limited to 6 arcmin, it is not enough for spatially large-scale events. The other way of ourstudies is the solar full-disk observations with the FMT and the SMART. With the FMT
we do simultaneous observations of the full-disk Sun at diﬀerent wavelengths around Halpha absorption line or in diﬀerent modes (Fig.1), and we do simultaneous observationof the full solar-disk vector magnetic ﬁeld on the photosphere and full-disk chromospherewith the SMART (UeNo et al. 2004). Therefore, the FMT and SMART particularly suitable for studying large-scale active phenomena on the whole solar-disk which are directly
connected with the space weather. In other words, observational studies with these twotelescopes can contribute well and to our understanding of space weather environment.
As space weather research can be advanced more eﬃciently with 24-hour continuous
observations of solar events, it is planned to create a world-wide observational networkwith FMT-type telescopes distributed all over the earth called as “Continuous H-alphaImaging Network (CHAIN)-project”. As part of the CHAIN-project, we are examining
the possibility of installing FMT-type telescopes in developing countries with cooperative
help by the United Nations. More precisely, we are considering Peru as the country wherethe ﬁrst oversea FMT will be installed.
In the following, we introduce characteristics of the FMT, some scientiﬁc results andour plans of installing the FMT in Peru.

2.

Characteristics of the FMT

The Flare Monitoring Telescope (FMT) was constructed in 1992 at the Hida Observatory
in Japan to investigate the long-term variation of solar activity and explosive events(Kurokawa et al. 1995). It has been one of the fruitful projects of the international
coordinated observations program (STEP) that started from 1991. The FMT consistsof ﬁve solar imaging telescopes and one guide telescope that have a simple and compact
design. We show the optical system and speciﬁcations of the telescopes in Fig. 2. OpticalUeno et al., “CHAIN-project and installation of the flare monitoring telescopes” (2007)

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699

Figure 1. Sample images obtained with ﬁve kinds of telescopes of the present FMT at Hida
observatory.

layout is almost common to the ﬁve telescopes. The images were obtained every two
seconds and recorded with time-lapsed video from 1992 to May 2006. From September
1996, we installed a digital data recording system of 1 min cadence. Moreover, in May
2006, we introduced a new digital CCD camera system (Takenaka System Co.LTD/
digital full frame shutter camera FC1500CL), so that pixel resolutions and digital bit
depths were improved (2.1 arcsec/pix, 10 bit) and the exposure-time was reduced (4ms). Normally, the time cadence for obtaining a series of digital images with this new
system is 20 seconds. Simultaneous exposures among ﬁve cameras is within 64 ns inprinciple. Therefore, the ﬁve telescopes can almost simultaneously observe the full-disk
Sun at diﬀerent wavelengths around H-alpha absorption line or in diﬀerent modes (Halpha line center, H-alpha -0.8 Å, H-alpha + 0.8 Å, continuum light, prominence modewith an occulting disk). In other words, the FMT can measure the three-dimensional
velocity ﬁeld of moving structures on the full solar disk without seeing errors.
The data that have been obtained with the FMT during 15 years are now publicly
opened on the web (http://www.kwasan.kyoto-u.ac.jp/general/facilities/fmt/database en.
html). At present, this data archive consists of the following four contents:
1. Event lists Monthly compiled lists: Physical characteristic and classiﬁcation are
given for every event. GIF-images are shown by clicking the record.
2. MPEG-movies of outstanding events: Classiﬁed into categories of ﬂares, ﬁlament
eruptions, surges and prominence eruptions.

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Figure 2. Optical system and speciﬁcations of the FMT.

3. Real-time images: Four kinds of the latest images (except continuum images).
4. All the digital raw images for download

3.

Representative scientiﬁc results with the FMT

In this section, we introduce two kinds of typical examples of the studies done with FMT
data. The ﬁrst theme is “investigation of the three- dimensional velocity ﬁeld of large-scaleﬁlament eruptions and their relationship with coronal mass ejections (CMEs)”. Making
the best use of the feature of the FMT, Morimoto & Kurokawa (2003a, 2003b) measured
3-D velocity ﬁelds of disappearing solar ﬁlaments. Then, they distinguished whether
each active ﬁlament really erupted or not by analyzing time-variations of their radial
upward velocities (Fig. 3) and investigated the relationship with coronal structures or
CMEs. They clearly showed that erupting ﬁlaments, which escaped from the sun, almost
perfectly corresponded to appearances of coronal arcade structures and CMEs. The future
subject in this theme will be the investigation of the correlation between “the velocity
strength and direction of the eruption” and “the strength of eﬀects of the corresponding
CMEs on the earth”.
The second theme of the typical study is “Moreton waves that accompany ﬂares”.
Eto et al. (2002) and Narukage et al. (2002) detected many Moreton waves in the chromosphere that accompanied solar ﬂares by investigating time-evolutions of the Doppler
velocity ﬁelds around ﬂares obtained with the FMT. The Moreton wave has theoretically
been considered as the cross section in the chromosphere of a shockwave generated bya strong solar ﬂare. Narukage et al. actually investigated the relationship between theMoreton wave and coronal wave observed in X-ray with Yohkoh-satellite (Fig. 4a). They
found the consistency among speeds, timings and directions (Fig. 4a,b). Moreover, they
conﬁrmed that their speeds indeed correspond to one of the expected MHD shockwaves.
By the way, more than half the number of Moreton waves that were found in the pastUeno et al., “CHAIN-project and installation of the flare monitoring telescopes” (2007)

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Figure 3. Transversal velocity ﬁeld distribution and time-variation of the Doppler velocity,
total velocity and upward velocity of the disappearing ﬁlament observed with the FMT (fromFigs 7 & 9 in Morimoto & Kurokawa 2003a).

have been discovered with the FMT. Therefore, we can say that the FMT is a very useful
instrument for inferring the characteristics of the shockwaves that are generated by solar
ﬂares and propagate to space.

4.

CHAIN-project and the plan of the FMT in Peru

Though the FMT is a simple and small-diameter telescope, it is a powerful instrument
for studying the large scale solar active phenomena that can aﬀect the space environment
as discussed above. Therefore, we want to monitor more solar ﬂares and erupting ﬁlaments continuously as many as possible by such characteristic telescopes. We are now
planning to execute ”Continuous H-alpha Imaging Network (CHAIN)-project” as part of
the CAWSES (Climate And Weather of the Sun-Earth System) project by distributing
several similar telescopes all over the world. Two groups of the telescopes are candidates
for network members. The ﬁrst group is made up of the existing H-alpha solar full-disk
telescopes. Staﬀ of our observatory have begun to contact ground-based observatories
in China, France etc. The second group is formed by newly installed H- alpha multiwavelength telescopes. As for this group, we are examining the possibility of installing
the telescopes in developing countries with aids from the United Nations. This is not only
to increase the number of ﬂare-monitoring locations, but also to encourage the educationand study of the solar- terrestrial physics in the developing countries. This project is also
being pushed by a new space weather project in Japan entitled “Basic Study of Space
Weather Predictions” under the CAWSES programme (Shibata and Kamide, 2007 (inpreparation)).
We have already selected Peru as the ﬁrst candidate where the oversea FMT will be第 4 章 論文集

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Figure 4. Relationship between the Moreton wave observed with the FMT and the coronal
wave detected with Yohkoh Soft X-ray telescope. (a) Time-evolution and direction of each wave.
(b) Timing and propagating speed of each wave. (From Figs 1 & 2 in Narukage et al. 2002).

installed, because the time diﬀerence between Japan and Peru is 14 hours and a telescopein Peru can almost cover Japanese nighttime. Moreover, Peru has kept deep connections
with Japan and especially with our university for 50 years. The relationship between
Japan and Peru is concretely shown in the paper of Dr. Ishitsuka in this book.
Currently, there are three candidate locations where a new FMT will be installedin Peru. Each position on the map of Peru is shown in Fig. 5. In general, there arequite a lot of ﬁne days in Ica city compared with that of Huancayo city. Moreover, theeducational and studying environment of the campus of Ica University is comparatively
excellent. For example, a coelostat for measuring the solar spectra will also be installedin this campus in 2007 with the support by National Astronomical Observatory of Japan,
and the campus ground will be developed and provided as the “Solar Station” by Ica
University (cf. Dr. Ishitsuka’s paper). Therefore, we are currently considering that thecampus of Ica University will be the ﬁrst candidate location. We will actually visit allof these sites in January 2007. After our actual inspection, we will ﬁnally decide thelocation for the FMT in Peru.
On the other hand, we still have a lot of items in various aspects that should be
investigated in advance, such as the seeing condition and the height of the turbulence
due to the heat haze at the candidate site, the best structure of the housing of thetelescope, the eﬃcient method of remodeling the telescope matching with the latitude ofPeru, the best combination of the observing wavelengths, the appropriate software fordata processing under the computer environment at Ica University, training and skill-upof the local staﬀs, etc. The biggest issue is a large amount of monetary budget. This
problem will also aﬀect future plans to install other FMTs in other developing countries.
We must select minimum necessary functions for oversea’s FMTs and study the way of
cutting down the expenses.

Ueno et al., “CHAIN-project and installation of the flare monitoring telescopes” (2007)

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Figure 5. The positions of candidate sites for installing the FMT in Peru besides Lima, thecapital city of Peru. ICA1: Ica University, ICA2: new educational astronomical observatory,
HUANCAYO: Huancayo Observatory. (Background map: from Google Earth).

5.

Conclusion and future subjects

In recent years, we recognize the scientiﬁc and social importance of the studies of theenvironmental variations of the solar-terrestrial system (space weather studies) in the ﬁeldof solar physics. Though the FMT at Hida observatory is simple and a small telescope,
it can measure the three-dimensional velocity ﬁeld of the full disk Sun without seeing
inﬂuence. Therefore, it is a powerful instrument for studying physical properties of largescale active phenomena on the whole solar-disk that are directly connected with thespace weather. In order to obtain more data of the solar ﬂare and erupting ﬁlaments
continuously, we are planning to install the ﬁrst oversea’s FMT in Peru as part of CHAINproject with cooperative help from the United Nations. The present ﬁrst candidate site
in Peru is the campus of Ica University. Currently, we are investigating various items,
aiming to start the operation of the FMT in Peru by the end of 2009, as follows:• The seeing conditions and the height of the turbulence due to the heat haze at thecandidate site
• The best structure to house the telescope (protecting it from sand storms)• The eﬃcient method of remodeling the telescope matching with the latitude of Peru
• The way of cutting down required expenses and ﬁnding the funds• The best combination of the observing wavelengths (improvement of the accuracyof the velocity estimation)• The appropriate software for data processing under the computer environment at
Ica Univ.

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• The training and skill of the local staﬀ (for daily operation, maintenance and management of the software)
We hope to install the FMT in Peru successfully and also to extend the CHAIN
network by promoting FMT’s in other countries in future.
Finally, we greatly appreciate the very kind support given for this project and forour talk in this meeting by staﬀ of IHY, United Nations and Dr. J.K. Ishitsuka, Dr. M.
Ishitsuka from Instituto Geoﬁsico del Peru (IGP).
This work was supported by a Grant-in-Aid for the 21st Century COE “Center for
Diversity and Universality in Physics” from the Ministry of Education Culture, Sports,
Science and Technology (MEXT), and by a Grant-in-Aid for Creative Scientiﬁc Researchof the MEXT “Basic Study of Space Weather Prediction” (17GS0208 PI: K. Shibata).

References
Eto, S., Isobe, H., Narukage, N., Morimoto, T., Thompson, B., Yashiro, S., Asai, A., Wang, T.,
Kitai, R., Kurokawa, H., & Shibata, K., 2002, PASJ 54, 481
Kurokawa, H., Ishiura, K., Kimura, G., Nakai, Y., Kitai, R., Funakoshi, Y., & Shinkawa, T.,1995, J. Geomag. Geoelectr., 47, 1043
Morimoto, T., & Kurokawa, H., 2003a, PASJ, 55, 1141
Morimoto, T., & Kurokawa, H., 2003b, PASJ, 55, 503
Narukage, N., Hudson, H.S., Morimoto, T., Akiyama, S., Kitai, R., Kurokawa, H., & Shibata,K., 2002, ApJ, 572 L109
UeNo, S., Nagata, S., Kitai, R., & Kurokawa, H., 2004, The Solar-B Mission and the Forefrontof Solar Physics, ASP Conference Series, Vol. 325 eds T. Sakurai & T. Sekii, 319

Zhao et al., “Subsurface Properties of Active Regions” (2010)
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C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

HIGH-RESOLUTION HELIOSEISMIC IMAGING OF SUBSURFACE STRUCTURES AND FLOWS OFA SOLAR ACTIVE REGION OBSERVED BY HINODE
Junwei Zhao1 , Alexander G. Kosovichev1 , and Takashi Sekii2

1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA
2 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

Received 2009 February 23; accepted 2009 November 10; published 2009 December 10

ABSTRACTWe analyze a solar active region observed by the Hinode Ca ii H line using the time–distance helioseismology
technique, and infer wave-speed perturbation structures and ﬂow ﬁelds beneath the active region with a high spatial
resolution. The general subsurface wave-speed structure is similar to the previous results obtained from Solar andHeliospheric Observatory/Michelson Doppler Imager observations. The general subsurface ﬂow structure is alsosimilar, and the downward ﬂows beneath the sunspot and the mass circulations around the sunspot are clearly resolved. Below the sunspot, some organized divergent ﬂow cells are observed, and these structures may indicate theexistence of mesoscale convective motions. Near the light bridge inside the sunspot, hotter plasma is found beneath,and ﬂows divergent from this area are observed. The Hinode data also allow us to investigate potential uncertainties
caused by the use of phase-speed ﬁlter for short travel distances. Comparing the measurements with and without thephase-speed ﬁltering, we ﬁnd out that inside the sunspot, mean acoustic travel times are in basic agreement, but thevalues are underestimated by a factor of 20%–40% inside the sunspot umbra for measurements with the ﬁltering.
The initial acoustic tomography results from Hinode show a great potential of using high-resolution observations
for probing the internal structure and dynamics of sunspots.
Key words: Sun: helioseismology – Sun: interior – sunspots

1. INTRODUCTION
Deriving subsurface structures and ﬂow ﬁelds of solar active
regions is one of the major topics for local helioseismology
studies. Using Ca ii K observations made at the geographic
South Pole in early 1990s, Duvall et al. (1996) found theﬁrst evidence of downdrafts below active regions throughmeasuring acoustic travel times by employing the time–distance
helioseismology technique. Later inversions (Kosovichev 1996)
using these travel time measurements found strong converging
downﬂows and areas of the excess of sound-speed beneathgrowing active regions. With the availability of Solar andHeliospheric Observatory/Michelson Doppler Imager (SOHO/MDI; Scherrer et al. 1995), Doppler observations that are seeingfree and more suitable for local helioseismology studies, moreinvestigations of the sunspot’s subsurface structures and ﬂow
ﬁelds have been carried out by use of various theoretical models:
ray-path approximation (e.g., Kosovichev et al. 2000; Zhao
et al. 2001), Fresnel-zone approximation (e.g., Jensen et al.
2001; Couvidat et al. 2004), and Born approximation (e.g.,
Couvidat et al. 2006). Despite the use of different approaches
of calculating travel time sensitivity kernels, the results remain
largely the same, i.e., for active region subsurface structures, anegative sound-speed perturbation was found up to 5 Mm below
the photosphere, and a positive perturbation was seen belowthat depth. For subsurface ﬂow ﬁelds, converging downﬂows
were inferred from the photosphere to about 5 Mm beneath it,
and divergent ﬂows were found below this layer. Another local
helioseismology technique, ring-diagram analysis, although it
could not provide subsurface ﬂow maps with such a high spatial
resolution, gave results that were in general agreement (Haber
et al. 2004; Komm et al. 2005). Some direct comparisons of these
two techniques were also performed (Hindman et al. 2004).
However, on the other hand, these results were not uncontroversial. By measuring acoustic phase shifts in sunspot penumbra
using MDI Doppler observations when sunspots were locatednear the solar disk limb, Schunker et al. (2005) demonstrated
that the measured acoustic travel times varied around sunspot
penumbra when the sunspot was located near the limb, and believed this effect was caused by the inclined magnetic ﬁeld.
While acknowledging the existence of such an effect in oscillation signals observed in Dopplergrams, Zhao & Kosovichev(2006) found out that such an effect did not exist in the oscillations observed in MDI continuum intensity. Additionally,
Rajaguru et al. (2006) showed that the phase-speed ﬁltering in
the time–distance acoustic travel time measurements would also
introduce errors in active regions where oscillation amplitudes
were suppressed. Using numerical simulations Parchevsky et al.
(2008) conﬁrmed this effect, and concluded that this effect led
to underestimation of the mean travel times at short distances.
Some recent attempts of using different ﬁltering procedures,e.g., ridge ﬁltering (Braun & Birch 2008), showed that different
ﬁltering might give different measurements. Thus, it is certainly
worthwhile examining how ﬁlters would change measurement
results. On the other hand, it is also arguable whether thosetime–distance observed acoustic travel time variations in solar magnetic areas could be interpreted as interior sound-speed
anomalies, or were caused by some surface magnetism effectsuch as the showerglass effect (Lindsey & Braun 2005a, 2005b).
Furthermore, it is also not clear how the interaction of acoustic
waves and magnetic ﬁelds would effect the phase of these waves,hence the travel time variations in the measured acoustic travel
times. It would be very difﬁcult to interpret the measurements
if acoustic waves experience some phase shifts at the boundaryof unmagnetized and magnetized areas (Cally 2009).
High spatial resolution observation of sunspots by the Solar
Optical Telescope (SOT; Tsuneta et al. 2008) onboard theJapanese solar spacecraft Hinode (Kosugi et al. 2007) not
only gives us another possibility of investigating subsurface
properties of solar active regions in addition to the alreadyexisting helioseismology instruments, but also provides anunprecedented high spatial resolution that has enabled some第 4 章 論文集

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studies that were improbable using these existing instruments.
In this paper, we analyze an active region observed by Hinodeutilizing the time–distance helioseismology technique, and infer
the wave-speed proﬁles and ﬂow ﬁelds beneath this region, andpresent these results in Section 3. We also analyze the acoustic
signals without applying any ﬁlters except ﬁltering out solar
convection and f-modes, and investigate how the phase-speedﬁltering effects our measurements. Such analyses can hardly
be carried out by use of MDI high-resolution observations, andthese results are presented in Section 4. In Section 5, we discussand summarize our results.

shown in Figure 1. We perform a 2×2 rebin before time–distanceanalysis, and the resultant resolution corresponds to 0.156 Mm
pixel−1 . During this 12+ hours observation, the pointing of SOT
was not sufﬁciently stable, and this is corrected by shifting
images to keep the sunspot center in a ﬁxed location during thewhole sequence.
As demonstrated by Sekii et al. (2007) and Nagashima et al.
(2009), the time–distance helioseismology technique works wellwith the Hinode Ca ii H data. We apply a similar measurement
and inversion procedure on this active region. Seven different
time–distance measurement annuli are used, and the annulus
ranges are 4.0–6.6, 6.5–9.7, 9.2–12.5, 12.0–15.9, 15.1–18.9,
18.5–22.3, and 21.6–25.4 Mm, respectively. Because of thelimited ﬁeld of view, longer annuli cannot be used in measuring
this data set, and this restricts us from reaching deeper interiors.
When we ﬁt the cross-covariance inside the sunspot umbra andpenumbra some misﬁttings occur, but the number of misﬁttings
is insigniﬁcant, and this should not alter our analysis and ﬁnal
results. The misﬁtted pixels are replaced by averaged values
from their neighboring pixels.
After the acoustic travel times are measured, inversions are
performed to infer the subsurface magnetoacoustic wave-speed
structures and ﬂow ﬁelds. The inversion technique, sensitivity
kernels, and the ﬁnal averaging kernels are essentially the sameas or similar to those presented by Zhao et al. (2007) except theslightly different annulus radii and depth coverage.

2. OBSERVATION AND DATA ANALYSIS

3. SUBSURFACE STRUCTURE AND FLOW FIELD

The sunspot inside active region NOAA AR10953 wasobserved by Hinode/SOT Ca ii H line continuously for a totalof 738 minutes, from 17:00UT 2007 May 2 through 05:58UT2007 May 3, with a cadence of approximately 1 minute. The
cadence was not uniform, but had a variation of a fraction of 1s, and this was corrected by interpolating data into a uniform
time grid of 1 minute. Each image, 2048 × 1024 pixels in size,
has a spatial resolution of 0. 108 pixel−1 , and a ﬁeld of view ofapproximately 160 × 80 Mm. An image from this observation is3.1. Subsurface Wave-speed Structure

Figure 1. Ca ii H image of the studied region from the Hinode/SOT.

A few selected images of the inverted wave-speed perturbations at different depths are displayed in Figure 2. At depths
shallower than ∼3 Mm, the wave-speed perturbation is dominantly negative inside the sunspot area, although small areas of
positive perturbations exist, in particular close to a light bridgein the upper umbra area. At the depth of 3.0–4.5 Mm, negative and positive wave-speed perturbations are mixed. Below

Figure 2. Wave-speed perturbation at different depths. The white contours in each panel indicate the boundaries of sunspot umbra and penumbra.

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Figure 3. Vertical view of wave-speed perturbation beneath the active region.

Figure 4. Averaging kernels, as functions of (a) depth and (b) horizontal
distance, for different targeted depths obtained from acoustic wave-speed
perturbation inversion.

approximately 6 Mm, the perturbations are predominantly positive inside the sunspot area. Please note that all depths here
are relative to where the Ca ii H line forms and the acoustic
signals are observed, i.e., approximately 250 km above the solar
photosphere (Carlsson et al. 2007).
A vertical cut through the center of the sunspot along theeast–west direction at Y = 39 Mm, as presented in Figure 3,
shows that the wave-speed anomalies extend about half of thesunspot size beyond the sunspot penumbra horizontally. In thevertical direction, the negative wave-speed perturbation extends
to a depth of ∼4 Mm. The positive perturbation is about 9 Mm
deep, but it is not clear whether it extends further, because ourinversion cannot reach deeper layers due to the limited ﬁeld ofview. The averaging kernels associated with this inversion are
shown in Figure 4.

In the quiet Sun, the measured mean acoustic travel times
often appear to be relatively constant with little correlation
with supergranulation structures. Hence, we can estimate ourtravel time measurement uncertainties by computing standard
deviations in a region of the quiet Sun. The errors in the inverted
wave-speed structures can then be estimated by convolving themeasurement uncertainties with the averaging kernels. Theseerror estimates are shown in Figure 5. It is found that the errors
decrease and the averaging kernels become broader with theincrease of depth. This is a general property of helioseismic
inversions, and the error magnitude and averaging kernel width
depend on the choice of regularization parameters. In this case,the parameters are chosen to provide a sufﬁcient smoothingof the regularized solution at different depths. This results inbroader averaging kernels and smaller error estimates in deeper
layers.
This picture of a sunspot’s subsurface wave-speed perturbation is made from intensity observations, rather than the morecommonly used Doppler velocity data. In spite of this, the subsurface wave-speed structure is remarkably similar to the previous results obtained using MDI Dopplergrams and various
types of acoustic sensitivity kernels (e.g., Kosovichev et al.
2000; Couvidat et al. 2006; Zharkov et al. 2007). It is also noticeable that no obvious effect caused by the inclined magnetic
ﬁeld is visible in the inverted sound-speed perturbation map,
although the sunspot is located quite a distance away from thesolar disk center. This is in agreement with the conclusion thatthe solar oscillation signals observed in intensity are not affectedby inclined magnetic ﬁelds (Zhao & Kosovichev 2006). Unfortunately, a direct comparison with MDI is not possible becauseof the lack of simultaneous uninterrupted observations for thissunspots or other sunspots.
3.2. Subsurface Flow Field
A few selected three-dimensional velocity maps are displayedin Figure 6 after a 4 × 4 rebinning in horizontal velocities inorder to reveal more clearly the larger scale patterns. It canbe found that at the depth of 0 to 3 Mm, the plasma ﬂows
toward the boundary of sunspot penumbra from outside. Inside
the sunspot penumbra, the horizontal ﬂow ﬁelds are not clearlyorganized in the rebinned images, but look well organized in theoriginal high-resolution ﬂow maps (see Section 3.3). Beneath
3 Mm, the horizontal ﬂow ﬁeld is largely divergent from thesunspot area, and extends quite far away. The vertical ﬂows are第 4 章 論文集

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Figure 5. (a) Measured mean travel times obtained in a piece of quiet Sun. Horizontal error bars indicate the measurement annulus ranges, and vertical error bars
indicate standard deviations from this area. (b) Error estimates from acoustic wave-speed perturbations at each targeted depth.

directed mostly downward until at least 4.5 Mm in depth inside
the sunspot area. Below 6 Mm or so, there seems to be a mixtureof the downward and upward ﬂows in this region.
A vertical view of the ﬂow ﬁeld (Figure 7) after linear
interpolation shows nicely the ﬂow structure beneath the active
region. Strong downdrafts are seen immediately below thesunspot’s surface, and extend up to 6 Mm into the deep.
The downdraft has a maximum speed larger than 500 m s−1 ,and is much stronger than the vertical ﬂow speed expected insupergranular structures. A little beyond the sunspot’s boundary,
one can ﬁnd both upward and inward ﬂows. Clearly, large-scalemass circulations form outside the sunspot, bringing plasma
down along the sunspot’s boundary, and back to the photosphere
within about twice the sunspot’s radius. It is remarkable thatsuch an apparent mass circulation is obtained directly fromthe helioseismic inversions without applying any additional
inversion constraints, such as forced mass conservation. The
mass circulation pattern in the previous MDI results (Zhao et al.
2001) was not that clear.
Based on analysis of artiﬁcial data, Zhao & Kosovichev(2003) demonstrated that a divergent (convergent) ﬂow structure
would result in a downward (upward) vertical ﬂow due to crosstalk effect, making the inversion of small vertical speed difﬁcult
or even impossible by our inversion technique. The same effect
was also discussed by Zhao et al. (2007) to explain why they
could not invert vertical ﬂows from the numerical simulations of
the quiet Sun. However, the downward ﬂows found in this study
is not due to the cross-talk effect, because the inﬂows aroundthe active region found in this study would have caused upward
speed due to this effect. This means that in our results the inferred
downward ﬂows are probably underestimated although it is notclear by how much. At the moment, we do not know how tooffset this effect in our inversions. This requires more work withsunspot numerical simulation data. However, it is important toemphasize that this effect does not affect the qualitative pictureof the ﬂow pattern below the sunspot obtained in our inversions.
3.3. High-resolution Subsurface Flow Field
Due to the very high spatial resolution of Hinode observation,
the time–distance analysis can also give high-resolution ﬂow
ﬁelds. One example, the horizontal ﬂow ﬁeld at 0–1.5 Mm
beneath the sunspot area, is shown in Figure 8. From thisﬂow map, one can easily identify small-scale divergent ﬂow
structures inside both the sunspot umbra and penumbra. The
scale of these ﬂow structures, 4–5 Mm, is larger than the
typical scale of solar granulations (1–2 Mm), but smaller thansupergranulations (20–30 Mm). The speed of these ﬂows isabout 300–500 m s−1 . They probably represent convective cells
in subsurface magnetized plasma of sunspots. It is intriguingthat small-scale vortex motions below sunspots were suggested
by Parker (1992) as a key element of the cluster sunspot model.
However, one should also be cautious in interpreting thesesmall-scale structures. The scale of these structures is similar to
the wavelength of acoustic waves used for the inferences at thisdepth, approximately 3 Mm. At this point, it is not clear how well
the structures of this scale are resolved in our measurements.
But the evidence of small-scale structures below sunspots isquite interesting, and certainly requires more observational andmodeling studies.
Another remarkable phenomenon in this high-resolution ﬂow
map is that in the light bridge area, located inside the upper partof the sunspot umbra (at X = 28–32 Mm and Y = 33–37 Mm),the ﬂows are divergent from the bridge with a speed of∼400 m s−1 . This shows a strong coupling between the sunspot
structure and subsurface dynamics. Thus, the high-resolution
subsurface ﬂow can be useful to monitor the dynamics andevolution of sunspots.
4. ANALYSIS WITHOUT PHASE-SPEED FILTERING
4.1. Averaged Travel Times
It has long been recognized from analysis of SOHO/MDI
data that for short distances, the time–distance cross-covariancefunctions from measurements may be corrupted by some signals that are not well understood (Duvall et al. 1997). In
Figure 7, we compare the time–distance diagrams calculated
for the Hinode observing run and for high-resolution MDI dataof the same duration and the same area size. As shown in theleft panel of Figure 9, for the high-resolution MDI observations, when the distance is smaller than ∼8 Mm, the acoustic
wave propagation signal is essentially lost in some horizontal
stripes, which block the measuring of acoustic travel times inthis distance range. It is not yet well understood how these horizontal stripes are generated, but they are thought to be related
to the instrument modulation transfer function (MTF) and instrumental distortions. Phase-speed ﬁltering was introduced toreduce the inﬂuence of these horizontal stripe signals at shortdistances (Duvall et al. 1997), but prior to the Hinode data there
was no way to investigate the effectiveness and accuracy of thisprocedure.
With the availability of Hinode high-resolution helioseismology observations, this difﬁculty can be resolved or partly resolved. In the Hinode time–distance diagram shown in the rightpanel of Figure 9, the acoustic signals are very nicely visible invery short distances as small as ∼2 Mm, although some shorter
and weaker horizontal stripes still exist. Judging from this time–
distance diagram averaged from the observations in the wholeﬁeld of view as displayed in Figure 1, it should be possible toZhao et al., “Subsurface Properties of Active Regions” (2010)
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Figure 6. Subsurface ﬂow ﬁelds at different depths. Background images indicate vertical ﬂow ﬁelds, with positive representing downﬂows and negative as upﬂows.
Arrows indicate horizontal ﬂows, with the longest arrow representing a velocity of approximately 500 m s−1 in each panel. White contours in each panel display theboundaries of sunspot umbra and penumbra.

make the travel time measurements for travel distances shorter
than 4 Mm even without using the phase-speed ﬁlter.
Without using the phase-speed ﬁlter (but with f-modes andsolar convection ﬁltered out), we obtain three different time–

distance diagrams when the central points of the annulus arelocated inside a quiet solar region, sunspot penumbra, andsunspot umbra, respectively. These time–distance diagrams are
obtained after averaging the outgoing and ingoing, i.e., both第 4 章 論文集

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Figure 7. Vertical view of ﬂow ﬁeld beneath the active region. The longest
arrow is approximately 500 m s−1 .

positive and negative lags of, cross-covariance functions. Fora better comparison, the quiet solar region is chosen at asize similar to the sunspot umbra. The Gabor wavelet ﬁtting(Kosovichev & Duvall 1996) is used to infer the acoustic travel
times for different distances and for all three diagrams, andthe results are shown in Figure 10. The time–distance diagram
looks quite noisy for the sunspot umbra, and less noisier forthe sunspot penumbra. However, the acoustic wave propagation
signals are strong and clear in both cases, and a robust ﬁtting is
not a problem for both diagrams.
It can be found that the measured acoustic travel times are
longer than the ray theory expectations at distances of about
6–7 Mm. This may be due to the fact that the second-skip
signals are not completely separated from the ﬁrst-skip signals,
and the ﬁtted values are thus slightly elevated. Since the travel309

time measurements inside the sunspot and in the quiet Sun
should be similarly affected in the ﬁtting, it is useful to use thequiet Sun measurements as a reference at this distance. To better
understand the travel time anomalies inside sunspots, we use thetravel time measured in the quiet Sun as a reference, and study
the relative travel times after the reference is subtracted fromthe travel times measured in the sunspot umbra and penumbra(see Figures 11(a) and (b)). It is clear that acoustic travel times
measured inside the sunspot umbra and penumbra are quite
similar, and both exhibit a positive anomaly of ∼40 s when the
distance is smaller than approximately 12 Mm, and a negative
anomaly of ∼35 s when the distance is between approximately
14 and 21 Mm.
For the correct interpretation of the travel time measurements, it is extremely interesting to compare these measurements with and without the phase-speed ﬁlter. For the measurements with the phase-speed ﬁltering, we average the mean travel
times, which are measured by use of seven different annulus
ranges using different phase-speed parameters as presented in
Section 3, inside the sunspot umbra and penumbra separately.
Figures 11(c) and (d) display such a comparison. It can be foundthat for both sunspot umbra and penumbra, the measurementswith and without the phase-speed ﬁltering are quite similar, i.e.,
the travel time anomalies are positive for travel distances shorter
than ∼13 Mm, but become negative for larger distances. However, it appears that for the sunspot umbra, the measurementswith the phase-speed ﬁltering underestimate the values by a factor of 20%–40%; and for the sunspot penumbra, the values arealso underestimated, but by a smaller factor. This underestimation is expected due to a combination of the smaller oscillation
amplitude in active regions and the use of phase-speed ﬁltering(Rajaguru et al. 2006; Parchevsky et al. 2008).

Figure 8. High-resolution horizontal ﬂow ﬁeld 0–1.5 Mm beneath the sunspot region. The background image is this sunspot observed at the photospheric level. The
longest arrow indicates a speed of 800 m s−1 .

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Figure 9. Time–distance diagrams obtained from MDI high-resolution Dopplergram (left panel) and Hinode (right panel) Ca ii H observations.

Figure 10. Time–distance diagrams obtained when the central points are located inside a quiet region, sunspot penumbra, and sunspot umbra, respectively. The dashedline in each panel is the travel times ﬁtted from the corresponding time–distance diagram.

Figure 11. (a) Comparison of travel times obtained in solar quiet region, sunspot penumbra, and sunspot umbra. Solid lines are the same as dashed lines in Figure 10;
and the dotted line is a theoretical expectation of travel times from ray theory. (b) Travel time differences relative to the quiet region for sunspot penumbra and umbra.
(c) Comparison of travel time differences with and without using phase-speed ﬁltering. Solid line is the same as the umbra line in (b); and the points with error bars
are mean travel times averaged from the travel time maps obtained with the use of phase-speed ﬁltering (i.e., measurements used in Section 3), but only points inside
sunspot umbra are used for average. Horizontal error bars indicate the annulus radius range; and the vertical error bars indicate the standard errors. (d) Same as panel(c), but solid lines and points with error bars are both obtained inside sunspot penumbra.

4.2. Travel Time Map
We have compared the averaged one-dimensional mean travel
time curves with and without the phase-speed ﬁltering, and it
is interesting to compare two-dimensional acoustic travel timemaps that are obtained from two sets of different measurements.
As it is well known that travel time measurements without
ﬁltering are noisier, we have succeeded in making clean travel
time maps for only three annuli among the seven that arepresented in Section 3. Figure 12 shows one such example
of comparing mean travel time maps with and without theﬁltering. There are many similarities as well as discrepancies
between these two travel time maps. It is clear that at thismeasurement distance, for both measurements, the negative
travel time anomalies are found outside the sunspot area,and mixed positive and negative time anomalies exist inside
the sunspot. However, it is also clear the detailed structuresof the map look different. Such comparisons are still at an

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Figure 12. Comparison of mean travel time maps obtained with (left panel) and without (right panel) the phase-speed ﬁltering. The annulus radius range is 12.0–
15.9 Mm, same for both measurements. White contours indicate the boundaries of sunspot umbra and penumbra.

early stage, and we believe a systematical study over suchcomparisons would lead us to a better understanding how
phase-speed ﬁltering, as well as other ﬁltering practices, affectmeasurements. Obviously, longer time series of high-resolution
sunspot observations will help to improve the results.
5. DISCUSSIONS AND SUMMARY
5.1. Subsurface Structure and Flow Field
By a time–distance analysis of unprecedented high spatial
resolution observations from the Hinode/SOT, we have investigated high-resolution wave-speed structures and mass ﬂows
beneath active region AR10953. For the subsurface wave-speed
structure, the inverted results are remarkably similar to previous
results based on MDI Dopplergrams with different inversion
techniques as well as different sensitivity kernels. For subsurface ﬂow ﬁelds, the general picture is similar to what has beenobtained from MDI observations, i.e., converging downward
ﬂows near the surface below the sunspot area. Despite the similarities, the current picture also has some differences comparedwith the earlier one. One difference is that the downward ﬂows
beneath the sunspot are more prominent in the ﬂow ﬁelds inferred from this study. Another remarkable result from this study
is the mass circulation outside the sunspot, which seems to keep
mass conservative and is much more clear than the previous
MDI inversions. It is widely believed in theory and in numericalsimulations that such downdraft and converging ﬂows near thesunspot surface play important roles in keeping sunspots stable(Parker 1979; Hurlburt & Rucklidge 2000; Hurlburt & DeRosa2008).
However, it is still not quite clear what the overall ﬂow
structures around the sunspot’s surface and interior look like.
It is already well known that there existed penumbral Evershed
outﬂows, outgoing moat ﬂows beyond the sunspot’s penumbra,
and inﬂows from the inner penumbra and umbra measured bytracking penumbra/umbra dots (e.g., Sobotka & Puschmann2009). For the sunspot’s interior, there were reports of outﬂows
from f-mode analysis (Gizon et al. 2000), inﬂows at the depthof 1.5–5 Mm (Zhao et al. 2001), and large-scale inﬂows around
active regions up to 10 Mm in depth from ring-diagram analysis(Haber et al. 2004). Based on all these observations, Gizon(2003) proposed a schematic ﬂow structure of the sunspot,and Hindman et al. (2009) recently proposed a similar one.
Both ﬂow structures show outﬂows near the surface, andinﬂows below that, with transition depth of these opposite ﬂows

Figure 13. Schematic plot of sunspot’s ﬂow structure near the surface and in
the interior.

undetermined. Here, based on all previous results, as well as on
the recent numerical simulation of Evershed ﬂows (Kitiashvili
et al. 2009), we also propose a schematic ﬂow structure of asunspot (Figure 13), slightly different from the plots of two
previous studies, but essentially similar. However, it is also
acknowledged that this ﬂow structure is not consistent with what
was recently found by Gizon et al. (2009) and recent numericalsimulations by Rempel et al. (2009). Thus, further observational
and theoretical studies are required to determine the subsurface
dynamics of sunspots.
The high-resolution subsurface wave-speed maps (Figure 2)
and ﬂow ﬁelds (Figure 8) present us an unprecedented opportunity to see the sunspot’s subsurface structures and dynamics
with many details. These results reveal the complexity of thesubsurface perturbations and dynamics and their relationshipwith the sunspot structure. The subsurface image at the depthof 0–1.5 Mm displays a larger wave-speed perturbation, presumably hotter in temperature, near the light bridge area thanother areas inside the sunspot, where the wave-speed perturbations are largely negative. This is in agreement with the surface
observations that light bridges are formed by upwelling of hot
plasma (Katsukawa et al. 2007). We also see clear divergent
ﬂows from the light bridge in a high-resolution subsurface ﬂow

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map at the same depth (Figure 8). This is consistent with theplasma upwelling that occurs in this area, but is not in good
agreement with the proper motion tracking results of this region(Louis et al. 2008).
It is also interesting to see quite a few divergent ﬂow cells
inside both the sunspot umbra and penumbra. The cells are bigger than granules and smaller than supergranules, and indicate
the existence of a mesoscale convection inside the sunspot. The
discovery of these motions is intriguing because they may be
a signature of the downdraft vortex rings around magnetic ﬂux
bundles, as suggested by Parker (1992). Parker’s conjecture wasthat the observed clustering of magnetic ﬂux bundles in sunspots
is at least in part a consequence of hydrodynamic attraction of
the downdraft vortex rings. Thus, it is very important to continue the investigations of the subsurface dynamics of sunspots,
both observationally by high-resolution helioseismology andtheoretically by realistic MHD simulations.
5.2. Analysis without Phase-speed Filtering
As introduced in Section 1, the phase-speed ﬁltering may
introduce some errors in active regions due to the smaller
oscillation amplitude in these areas (Rajaguru et al. 2006), andusing ridge ﬁltering may give different results than using phasespeed ﬁltering in active regions (Braun & Birch 2008). All ofthese prompt a re-examination of the use of phase-speed ﬁlteringand an evaluation of how phase-speed ﬁltering alters measured
acoustic travel times, in particular inside active regions.
The Hinode high-resolution observation has helped us toovercome an obstacle that acoustic travel times could hardly bemeasured in short distances inside active regions if phase-speedﬁltering was not applied. Our measured distance-dependent
acoustic travel time differences relative to the solar quiet region
obtained without using the phase-speed ﬁltering, as shown in
Figure 11, have evidently demonstrated that for short distances,
the acoustic travel time is longer inside both sunspot umbra
and penumbra than the quiet region, and for distances longer
than ∼13 Mm, the mean travel time is shorter. This is generally
in agreement with the measurements using phase-speed ﬁlters,
except that the ones with ﬁlters would underestimate thevalues by 20%–40% in sunspot umbra. It is believed thatthe underestimation is caused by a combination of oscillation
amplitude suppression inside active regions and the use ofphase-speed ﬁltering. Therefore, we believe that the phase-speedﬁltering does not change qualitatively the measured travel times,
and therefore the inferred interior properties of active regions,but would change the inferred values quantitatively.
The Hinode data also enable us to construct acoustic travel
time maps without using ﬁlters, and this gives us an opportunity
to assess the correctness and accuracy of various ﬁlters. Further
investigations are particularly important in this regard.
5.3. Summary
We summarize our results as follows.
1. By analyzing high-resolution Hinode Ca ii H observations using time–distance helioseismology, we have derived
wave-speed structures beneath solar active regions. The inferred structures are remarkably similar to results obtainedby various authors analyzing MDI Dopplergram observations, although the instrument, the spectrum line used toobserve, the type of observation (intensity and Doppler velocity), the inversion technique, and the sensitivity kernels
are mostly different.

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2. The subsurface ﬂow ﬁeld structure is generally in agreementwith previous MDI results, but it is clear that the downward
ﬂows near the sunspot surface is more prominent, and themass circulation around the sunspot is more clear and seems
self-consistent, although no mass conservation constraint isapplied in the inversion.
3. Our analysis without using phase-speed ﬁltering has convincingly demonstrated that the phase-speed ﬁltering does
not change travel time measurements qualitatively, but may
underestimate travel time anomalies inside active regions.
4. High spatial resolution subsurface ﬂow ﬁelds reveal quite
a few organized ﬂow structures inside sunspot umbra andpenumbra, which perhaps correspond to some convection
structures or downdraft vortex rings around magnetic ﬂux
bundles. In the light bridge area, subsurface hotter temperature is found, and plasma ﬂows divergent from thelight bridge are also seen. These initial results provide abasis for further development of high-resolution acoustic
tomography of sunspots, and comparison with numericalsimulations.
We thank Dr. I. Kitiashvili for making Figure 13 for us. We
also appreciate an anonymous referee for raising a few importantquestions that helped to improve the quality of this paper.
Hinode is a Japanese mission developed and launched by ISAS/
JAXA, collaborating with NAOJ as a domestic partner, NASAand STFC (UK) as international partners. Scientiﬁc operationof the Hinode mission is conducted by the Hinode scienceteam organized at ISAS/JAXA. This team mainly consists ofscientists from institutes in the partner countries. Support for the
post-launch operation is provided by JAXA and NAOJ (Japan),
STFC (UK), NASA, ESA, and NSC (Norway). This work was(partly) carried out at the NAOJ Hinode Science Center, which is
supported by the Grant-in-Aid for Creative Scientiﬁc Research“The Basic Study of Space Weather Prediction” from MEXT,
Japan (Head Investigator: K. Shibata), generous donations fromSun Microsystems, and NAOJ internal funding.
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PASJ: Publ. Astron. Soc. Japan 59, S637–S641, 2007 November 30
c 2007. Astronomical Society of Japan.


Initial Helioseismic Observations by Hinode/SOT
Takashi S EKII,1 Alexander G. KOSOVICHEV,2 Junwei Z HAO,2 Saku T SUNETA,1 Hiromoto S HIBAHASHI,3
Thomas E. B ERGER,4 Kiyoshi I CHIMOTO,1 Yukio K ATSUKAWA,1 Bruce L ITES,5 Shin’ichi NAGATA,6
Toshifumi S HIMIZU,7 Richard A. S HINE,7 Yoshinori S UEMATSU,1 Theodore D. TARBELL,7 and Alan M. T ITLE7
1 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588

sekii@solar.mtk.nao.ac.jp
2 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA
3 Department of Astronomy, School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033
4 Lockheed Martin Solar and Astrophysics Laboratory, B/252, 3251 Hanover St., Palo Alto, CA 94304, USA
5 High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA
6 Hida Observatories, Kyoto University, Kurabashira, Kamitakara-cho, Takayama, Gifu 506-1314
7 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,

3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510(Received 2007 July 27; accepted 2007 September 12)

Abstract
Results from initial helioseismic observations by the Solar Optical Telescope on-board Hinode are reported. It
has been demonstrated that intensity oscillation data from the Broadband Filter Imager can be used for various
helioseismic analyses. The k–! power spectra, as well as the corresponding time–distance cross-correlation
function, which promise high-resolution time–distance analysis below the 6-Mm travelling distance, were obtainedfor G-band and Ca II-H data. Subsurface supergranular patterns were observed from our ﬁrst time–distance analysis.
The results show that the solar oscillation spectrum is extended to much higher frequencies and wavenumbers, andthe time–distance diagram is extended to much shorter travel distances and times than were observed before, thus
revealing great potential for high-resolution helioseismic observations from Hinode.
Key words: Sun: helioseismology — Sun: interior — Sun: oscillations

1. Introduction
Local helioseismology has been a great success in revealing
subsurface structure of the Sun, particularly in detecting
three-dimensional ﬂow and sound-speed anomaly in active
regions (see, e.g., Zhao et al. 2001; Zhao & Kosovichev 2003).
Observations for local helioseismology require highresolution and wide coverage in both space and time. The
50-cm aperture Solar Optical Telescope (SOT: Tsuneta et al.
2007; Suematsu et al. 2007) on-board the Hinode satellite(Kosugi et al. 2007) provides 0:002 spatial resolution, thoughwith a limited ﬁeld of view [32800  16400 for the Narrowband
Filter Imager (NFI) and 21800  10900 for the Broadband Filter
Imager (BFI), nominally]. Except during the satellite eclipse
periods, Hinode can observe the Sun continuously with acadence of up to 15 s, or even less, with the actual limit
placed by the telemetry bandwidth and the capacity of the data
recorder, which are shared with other instruments as well asother SOT programs running at the same time frame.
The high resolution of SOT particularly offers a great
potential for local helioseismology. One of the main scientiﬁctargets of Hinode is to investigate the generation, evolution, andeventual dissipation of active regions on the Sun, in attempt toobservationally constrain theories on dynamo processes which
are thought to be responsible for the solar activity cycle. Local
helioseismology can contribute greatly in this area, not only via
active-region tomography, but also in measuring ﬂows at larger
scales, such as meridional ﬂow, which plays an importantrole in the current models of the solar activity cycle (Dikpati& Gilman 2006). Please note that even with the limitedﬁeld of view of SOT, and hence the limited depth coverage
for a tomographic study (see below), one can still measurenear-surface components of large-scale ﬂows in a small areaﬁrst, but then extend the measurement to a broader area simply
by repeating the observations at different locations.
It was planned to use Dopplergrams produced by NFI for
helioseismic observations on Hinode (Sekii et al. 2001; Sekii
2004). Not only oscillation signals in Dopplergrams are known
to exhibit a higher signal-to-noise ratio than an oscillation
signal in intensity, but also NFI has a larger ﬁeld of viewthan BFI. A larger ﬁeld of view translates into a better
depth coverage for tomography, because waves that penetrate
deeper into the Sun also travel further in horizontal directions.
Thus, the NFI Doppler observation was the preferred mode ofhelioseismic observation by Hinode.
However, a long series of Dopplergrams are still only being
tested at the time of writing. Instead, we examined BFIinstensity oscillation data from the H line (Ca II) and the G band
(CH).
The main aim of the present paper is to report that
helioseismic observations can indeed be made with BFI. As
an example, a result from our ﬁrst attempt at a time–distanceanalysis of subsurface ﬂow is presented.

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2. Data Analysis
A quiet region around the disk center was observed fromaround 2007 January 1, 16:00 UT to around 2007 January 2,
04:00 UT, for about 12 hours with a 12-minute interruption in
the middle of the run. Hinode was tracking the region during
the observation (except during the interruption mentioned
above); in addition, Correlation Tracker was used to maximize
the frame-to-frame correlation in the central region of the ﬁeldof view (Shimizu et al. 2007). No further tracking was done on
ground, unlike in the case of a sunspot study (Nagashima et al.
2007).
Both G-band images and Ca II-H images were acquired
at approximately 1-min cadence, Ca II-H images following
G-band images by 20 s. Sample images are shown in ﬁgure 1.
The CCD pixels were 22 summed onboard to reduce the dataamount, to double the effective pixel size to 0:00106. We applied
another 2  2 summing on the ground (the overall result wasa 4  4 summing) for the sake of faster analyses, though inthis case we no longer fully exploited the 0:002 resolution of
the telescope. The beneﬁt of using the full 2  2-summed data,
or observing with the full capacity of SOT (no summing), will
be investigated in future work.
The cadence was not exactly regular, due to the fact that
there are two clocks involved in controlling the equipment.
Not only are they on different cycles (0.5 s for Mission
Data Processor and 1.6 s by Polarization Modulator Unit),
but the clocks are slowly drifting away from each other.
Although programming tricks to correct this behavior are being
investigated and tested, they have not been applied to thecurrent datasets.
A simple test was run to check the effect of the slightly
irregular cadence, by putting through a sine wave on
this irregular temporal grid and then computing the power

Fig. 1. Sample images of the observed region in the G band (top) and
Ca II H line (bottom). Of the full ﬁeld of view, which is 21800  10900 ,only the central part of the left half (approximately 5400 2700 ) is shown.

[Vol. 59,

spectrum, while pretending that the cadence was exactly 1 min.
As we expected, the effect was insigniﬁcant, and therefore we
carried out all the following analyses assuming that the cadence
was exactly 1 min.
The projection effect was also neglected. Because the BFI
ﬁeld of view is so narrow it should not give rise to any
signiﬁcant inaccuracy around the disk center.
We used only the ﬁrst uninterrupted 6 hr of the data for basic
diagnostics products in the next section. For the time–distanceFig. 2. Power spectra (k–! diagrams) from the 6-hr G-band data (top)and Ca II-H data (middle), compared with the power spectrum froma 512-min MDI high-resolution Doppler data (bottom).

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Fig. 3. Time–distance diagrams from 6-hr Ca II-H data (left) and from 512-min MDI high-resolution Doppler data (right). The datasets are not co-eval.

analysis, the full 12-hr data were used. Before starting theanalyses that follow, ﬁrst we applied running difference ofimages; each image was subtracted from the following image,thus obviating the need for calibrating each image for dark,
ﬂat ﬁeld, and bad pixels. One might worry that employing
the running difference might increase the noise level, if thecadence is too short compared to the time scale over which
the signal changes, in which case the differences between two
successive frames are dominated by noise. This is not the casehere, when the 1-min cadence is not too high for the oscillation
signal that is dominated by 5-min oscillations.
3. Power Spectra and Time-Distance Diagram
In this section we present the most basic products ofhelioseismic observations: a so-called k–! diagram (or l–
diagram) and a time–distance diagram.
They are thencompared with results obtained from the Michelson Doppler
Imager (MDI) on-board SOlar and Heliospheric Observatory(Scherrer at al. 1995). The MDI data that we used wereacquired in the high-resolution mode. Although they too cover
the central region of the disk, they are not coeval with the SOT
data.
Figure 2 shows k–! diagrams obtained from the G-band
and Ca II-H data; the series of the differential intensity-ﬁeld
data (intensitygram) were Fourier transformed and converted
to power spectra, which were then azimuthally integrated
in wavenumber space.
Although no spherical-harmonic
decomposition was involved, degree l was assigned to eachcomponent by multiplying its wavenumber by the solar radius.
The power spectra were corrected for the ! 2 factor, where ! isthe angular frequency, because we took the running difference
before the Fourier transform.

In spite of only 6 hours of observation, the f-mode ridge isseen up to l  4000. We also see high-frequency interference
peaks (Jefferies et al. 1988; Duvall et al. 1991; Kumar et al.
1990) up to the Nyquist frequency. They seem disturbed
close to the Nyquist frequency, because the ridges fromboth the super- and sub-Nyquist ranges are running into theother regime. Compared to the k–! diagram obtained froma 512-min MDI high-resolution velocity dataset, also shown inﬁgure 2, the better performance of SOT at high wavenumbers,
where the high resolution matters most, is noticeable in how
the f-mode ridge extends to higher wavenumbers and, in thecase of the Ca II H data, the p1 ridge as well. However,as was expected, the SOT k–! diagrams show a higher
background-noise level compared to the Doppler measurement
by MDI.
It is apparent that in the G-band data the convective “noise”
is stronger, and for this reason, in the following analyses,
we used only the Ca II H data. A more detailed comparison
between the two, in particular the phase relations at various
frequencies, will be a subject of future work.
We then applied a Fourier–Bessel transform (see, e.g., Sekii& Shibahashi 2003) to the Ca II-H k–! power spectrum toobtain a time–distance cross-correlation function (ﬁgure 3).
The power spectrum uncorrected for ! 2 was used; this may beviewed as an application of a broad high-pass ﬁlter to the power
spectrum, the intention being ﬁltering out the convective noise.
The structure down to less than 0.1 heliographical degree
is seen with a high signal-to-noise ratio, which is very
encouraging. Once again, we can see the higher performanceof SOT, compared to MDI, also shown in ﬁgure 3, at small
spatial scales. The signal-to-noise ratio, however, is higher for
MDI at large distances, and thus both instruments complement
each other.

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4. Subsurface Flow
We analyzed the full 12-hr long series by a time–distancetechnique for local helioseismology. The gap between the two
6-hr stretches was ﬁlled with zeroes. Then, for measuring thetravel-time difference between the outward and inward wave
components, each pixel was cross-correlated with average
signals over annuli that covered, in radii, 4.00–6.62, 6.47–9.70,
9.24–12.47, 12.01–15.86, 15.09–18.94, 18.48–22.33, and21.56–25.41 (all in Mm). One such example as well as an MDI
counterpart (in high-resolution mode) is shown in ﬁgure 4.
The annuli were further divided into 4 sectors representing
north, south, east, and west, and opposing pairs of sectors weresimilarly cross-correlated to yield northward–southward andwestward–eastward travel-time differences.
All of the travel-time differences were then inverted by using
a ray-approximation kernel for acoustic wave propagating
through the solar interior. Figure 4 shows an outward–inward
travel-time difference map, as a proxy divergence of the ﬂow
ﬁeld. Inversions for horizontal ﬂow in the depth ranges of
0–1 Mm, 1–2 Mm, 2–3 Mm, and 3–4 Mm are shown in ﬁgure 5.
Inversions were done for the entire area in ﬁgure 4, but toavoid overcrowding the ﬁgure with arrows in ﬁgure 5, only
the central quarter is shown. The horizontal ﬂow patterns
are largely consistent with the outward–inward travel-timedifference, as was seen in a realistic numerical simulation by
Zhao et al. (2007). This indicates that the travel-time difference

Fig. 4. Outward–inward travel time difference maps from 512-min
MDI high-resolution Doppler data (top) and from the Ca II-H data(bottom), indicating a ﬂow divergence due to supergranulation. The
inner radius of the annulus used was 13.86 Mm and the outer radius
15.86 Mm. Dark patches are where the inward travel time is longer
than the outward travel time i.e., where the ﬂow is generally diverging,whereas brighter patches are where the ﬂow is generally converging.
Since the two datasets are not coeval, individual features are not to becompared directly.

[Vol. 59,

is a good proxy of ﬂow divergence, though to acquire
additional details we do need an inversion procedure. We cansee that supergranulation patterns are coherent vertically withina spatial scale of a few Mm.
5. Discussion
As we have demonstrated, SOT observation of intensity
oscillations can be used for high-resolution helioseismic
diagnosis of the Sun, in spite of its noise level being higher

Fig. 5. Subsurface ﬂow maps (indicated by arrows) obtained by atime–distance analysis of 12-hr Ca II-H data. The depth ranges are
0–1 Mm, 1–2 Mm, 2–3 Mm, and 3–4 Mm (top to bottom). The
longest arrow in the panels correspond to 0.51 km s1 , 0.52 km s1 ,0.30 km s1 , and the ﬁeld of view corresponds to the central quarterof the ﬁeld of view in ﬁgure 4.

Sekii et al., “Hinode/SOT Helioseismology” (2007)

295

No. S3]

S641

Hinode/SOT Helioseismology

than what can be expected from Doppler measurements. Thisis due to the fact that solar noise is higher in intensity than in
Doppler velocity.
From ﬁgure 2, it is apparent that SOT offers a good
opportunity to study high-wavenumber waves, particularly
the f modes. Ishikawa et al. (2007) have found ubiquitous
small-scale horizontal magnetic ﬁelds from a Hinode/SOT
observation of the photosphere. In addition to turbulence, thepresence of such magnetic ﬂuxes would affect the propagationof f-mode waves, which would be an important future work.
In the time–distance cross-correlation (ﬁgure 3), thestructure below 0:ı 5, which is about a distance of 6 Mm,
was observed to be much clearer than before. Sekii (2004)
used G-band data obtained from 50-cm La Palma SVST
to demonstrate a somewhat similar enhancement in spatial
resolution, but the noise level was much higher. It was
speculated that the noise came mainly from atmosphericseeing, and that spaceborne SOT would perform better.
Our current results show that SOT indeed does. Figure 4
demonstrates that a study of ﬁne structures in supergranulation
patterns beneﬁts from the high resolution of SOT; the brightand dark patches have a typical spatial scale of 20 Mm, buttransition from bright to dark takes place over a scale of lessthan 10 Mm, which MDI has difﬁculty to resolve. With SOT
results we can see smoother transitions.
To exploit this small-scale information would require moretuning and testing of the inversion, particularly the setup ofannuli. The trade-off between the resolution and the error

needs to be examined carefully. Once done, we will certainly
achieve a higher horizontal resolution in time–distanceinversions.
A similar improvement in performance is
expected in resolving, vertically, layers immediately below thesurface, since SOT can use a higher wavenumber, and hence
shallower-penetrating waves. This also applies to sound-speed
inversion, which has not yet been attempted.
Time–distance analysis in active regions, be it for ﬂow or forthermal (and even magnetic) structure, would need further care,particularly if we wish to go a step beyond phenomenologicalinversions; wave propagation in a magnetized atmosphere has
to be somehow taken into account. For the initial analysis ofsunspot oscillation data, see Nagashima et al. (2007).
Hinode is a Japanese mission developed and be launched by
ISAS/JAXA, collaborating with NAOJ as a domestic partner
as well as NASA and STFC (UK) as international partners.
Scientiﬁc operation of the Hinode mission is conducted by
the Hinode science team organized at ISAS/JAXA. This
team mainly consists of scientists from institutes in partnercountries. Support for the post-launch operation is provided
by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and
NSC (Norway). This work was carried out at the NAOJ
Hinode Science Center, which is supported by the Grant-in-Aid
for Creative Scientiﬁc Research “The Basic Study of Space
Weather Prediction” from MEXT, Japan (Head Investigator:
K. Shibata), generous donations from Sun Microsystems, and
NAOJ internal funding.

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Kosugi, T., et al. 2007, Sol. Phys., 243, 3
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Sekii, T., & Shibahashi, H. 2003, in Proc. SOHO 12/GONG+ 2002:
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Berger, T. E., Bush, R., & Scherrer, P. H. 2001, in Proc. SOHO
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Here

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, A10102, doi:10.1029/2009JA014167, 2009

for

Full
ArticleThree-dimensional MHD modeling of the solar wind structuresassociated with 13 December 2006 coronal mass ejection
R. Kataoka,1,2 T. Ebisuzaki,1 K. Kusano,3,4 D. Shiota,3 S. Inoue,3,5 T. T. Yamamoto,4and M. Tokumaru4
Received 17 February 2009; revised 10 June 2009; accepted 24 June 2009; published 6 October 2009.

[1] A 3-D magnetohydrodynamic (MHD) simulation is performed to reconstruct theinterplanetary propagation of a coronal mass ejection (CME) that occurred on 13 December
2006. A spheromak-type magnetic field is superposed on a realistic ambient solar wind toreproduce the large-scale interplanetary magnetic field (IMF) associated with the CME.
Here we show that a westward and southward directed spheromak CME with reasonablegeometric, dynamic, and magnetic parameters reproduces the magnetic cloud, interplanetaryshock, and sheath profiles as observed by in situ spacecraft. We suggest that the simple
solar wind model developed in this study is topologically complex enough to be consistent
with in situ observations, such as southward IMF associated with CMEs.
Citation: Kataoka, R., T. Ebisuzaki, K. Kusano, D. Shiota, S. Inoue, T. T. Yamamoto, and M. Tokumaru (2009), Three-dimensional
MHD modeling of the solar wind structures associated with 13 December 2006 coronal mass ejection, J. Geophys. Res., 114, A10102,
doi:10.1029/2009JA014167.

1. Introduction[2] It has been one of the most fundamental topics in thefield of Sun-Earth connection and space weather research
to understand the generation mechanism of a southward
pointing interplanetary magnetic field (IMF) at the Earth,since geomagnetic storms were found to be caused byprolonged, large-amplitude, and southward pointing IMF[Gonzalez and Tsurutani, 1987]. Coronal mass ejections(CMEs) and corotating interaction regions (CIRs) are thetwo major large-scale solar wind structures capable of producing such a geoeffective IMF [e.g., Kataoka and Miyoshi,
2006]. Comprehensive observations of the Sun and solar windduring solar cycle 23 enabled us to confirm that the major
cause of large geomagnetic storms is the large-amplitude IMF
associated with CMEs [Zhang et al., 2007] and relatively
small geomagnetic storms are caused by the southward IMF
associated with CIRs [Richardson et al., 2006].
[3] The magnetic cloud [Burlaga et al., 1981] is a subset of
CME-related IMF structures, characterized by low proton
temperature, enhanced IMF amplitude, and slow rotation of
IMF directions through a large angle [Klein and Burlaga,1982]. The southward pointing IMF within the magnetic
clouds usually satisfies the driving condition of geomagnetic
1
Advanced Science Institute, Institute of Physics and Chemical
Research, Wako, Japan.
2Interactive Research Center of Science, Tokyo Institute of Technology,
Meguro, Japan.
3
Institute for Research on Earth Evolution, Japan Agency for MarineEarth Science and Technology, Yokosuka, Japan.
4
Solar-Terrestrial Environment Laboratory, Nagoya University, Chikusa
City, Japan.
5
National Institute of Information and Communications Technology,
Koganei, Japan.

Copyright 2009 by the American Geophysical Union.
0148-0227/09/2009JA014167$09.00

storms. On the basis of the low b property, magnetic clouds
have been approximated by several types of force-free fields,such as a cylindrical flux rope [Goldstein, 1983; Marubashi,
1986], torus-shaped flux rope [Marubashi and Lepping,2007], or spheroidals [Vandas et al., 1993a, 1993b]. CMEdriven geomagnetic storms are caused not only by magnetic
clouds but also by many different types of CME-related IMF
structures, such as the IMF compressed by interplanetaryshock waves located ahead of high-speed CMEs [Tsurutani
and Gonzalez, 1997], and/or draped IMF created by the
propagation of CMEs in the ambient solar wind [Goslingand McComas, 1987; McComas et al., 1989; Watari et al.,
2004; Liu et al., 2008].
[4] Magnetohydrodynamics (MHD) [Alfvén, 1942] isthe fundamental physics for describing the global evolutionof the solar wind and IMF [see Parker, 2007]. Threedimensional MHD simulation is necessary to fully understand the dynamic variation of the complicated IMF structuresassociated with CMEs. A number of 3-D MHD simulationsof the solar wind have been performed during the last decade
to investigate the basic evolution and propagation of several
types of CMEs, such as the time-dependent pulse [Odstrcil
and Pizzo, 1999; Odstrcil et al., 2004; Hayashi et al., 2006;
Shen et al., 2007; Wu et al., 2007], the toroidal flux rope
[Vandas et al., 2002], the simulated flux rope [Manchester
et al., 2004a, 2004b], or the spheromak [Vandas et al., 1997,
1998]. The ambient solar wind used in their simulations areeither uniform [Vandas et al., 1997, 1998, 2002], structured[Odstrcil and Pizzo, 1999; Manchester et al., 2004a, 2004b],
or realistic [Odstrcil et al., 2004; Hayashi et al., 2006; Shenet al., 2007; Wu et al., 2007]. A self-consistent modeling of
the 3-D propagation of a magnetic cloud in a realistic ambient
solar wind remains a challenging problem.
[5] Integrated frameworks connecting the models of solar
corona, ejecta, and solar wind have been developed [e.g.,

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A10102KATAOKA ET AL.: MHD MODELING OF AN INTERPLANETARY CME

Figure 1. OMNI-2 solar wind and Dst profile for week of
the 13 December 2006 coronal mass ejection (CME) event.
From top to bottom, shown are (a) interplanetary magnetic
field (IMF) amplitude and the GSE southward component(solid line), (b) IMF longitude in GSE coordinates, (c) solar
wind speed, (d) number density, (e) proton temperature, and(f) Dst index.

Luhmann et al., 2004; Tóth et al., 2007]. However, it is still
difficult to simultaneously reproduce the in situ observationsof such basic solar wind structures as magnetic cloud, shock,and sheath. The purpose of this paper is to reproduce theinterplanetary propagation of a magnetic cloud in a realistic
ambient solar wind, using a newly developed 3-D MHD solar
wind model named ‘‘MHD cube,’’ in order to understand thebasic evolution of the magnetic cloud, interplanetary shock,and associated global IMF structures. We focus on a CME
event of 13 December 2006, the last CME-driven large storm
event during solar cycle 23. In section 2, we briefly review
the CME event to raise a puzzling issue showing thecoronagraph observation and in situ solar wind observation.
In sections 3 and 4, we describe the MHD cube model toexplain how to create the ambient solar wind and introduce atransient disturbance, respectively. In section 5, we compare
the in situ solar wind observation and our modeling results
to discuss the 3-D morphology of the CME evolution. In
section 6, we summarize the results and discuss the capabilities and limitations of the MHD cube model.

2. The 13 December 2006 CME Event[6] A high-speed halo-type CME was observed by Solarand Heliospheric Observatory (SOHO)/Large Angle andSpectrometric Coronagraph (LASCO) on 13 December

297
A10102

2006, associated with X3.4 flare at S06W23 (NOAA
AR10930). The CME speed is estimated as 1774 km s1
by Dr. S. Yashiro (http://cdaw.gsfc.nasa.gov/CME_list/). A
large isolated CME-driven geomagnetic storm was observed
on 14 – 15 December 2006. The provisional Dst index
reached 146 nT at the storm peak, as shown in Figure 1f.
This event is suitable for this study because HINODE data are
fully available for diagnosing the detailed vector magnetic
field of the active region.
[7] Figure 1 shows the solar wind profile at the Earthduring the week of the CME event. OMNI-2 hourly data is
obtained from the Web site (http://omniweb.gsfc.nasa.gov/).
After the sudden arrival of an interplanetary shock on 14
December 2006, typical magnetic cloud properties are identified by the enhanced IMF, slow IMF rotation, and low
temperature. The prolonged large-amplitude southward IMF
exists within the magnetic cloud that caused the large
geomagnetic storm. The magnetic cloud was embedded in
toward IMF polarity, and a sector boundary arrived at the endof 17 December 2006, followed by a typical corotatinginteraction region on 18– 19 December 2006. Another interplanetary shock associated with a different CME was observed on 16 December 2006.
[8] There are puzzling observations in the 13 December
2006 CME event. Using the Grad-Shafranov reconstruction
technique of the magnetic cloud [Hau and Sonnerup, 1999;
Hu and Sonnerup, 2002], Liu et al. [2008] suggested that the
Earth passed near the center of an inclined north – southaligned flux rope. The initial propagation direction of the
CME is, however, significantly westward and southward
as shown in Figure 2. Assuming that the flux rope CME
propagated significantly westward and southward, it is naturally expected that the Earth should pass the western edge ofthe flux rope, instead of the center of the flux rope. Understanding these observations in a simple scenario is therefore
not straightforward, and this is an interesting challenge toaddress the 3-D MHD modeling.

3. MHD Solar Wind Model[9] The time evolution of large-scale solar wind plasma
and IMF follows MHD (see Parker [2007]). The ideal MHD

Figure 2. Coronagraph difference images of the 13
December 2006 CME as seen by Solar and Heliospheric
Observatory (SOHO)/Large Angle and Spectrometric Coronagraph (LASCO) (left) C2 and (right) C3. White circles
indicate the position of the Sun. The images are adapted fromthe LASCO CME Catalog at NASA/Goddard Space Flight
Center.

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A10102

Figure 3. Inferred solar wind parameters at the inner boundary of 50 RS sphere. (a) Radial magnetic fieldand (b) radial speed are based on Wilcox Solar Observatory and interplanetary scintillation observations ofSolar-Terrestrial Environmental Laboratory, respectively. (c) Number density and (d) plasma temperatureare empirically calculated as functions of the radial speed. Solid curves show the current sheet location ofzero radial magnetic field.
equations to be solved in the conservation form are as
follows,
0

0
1
1 0 1
r
rv
0
B
B
C
C B0C
@ B rv C
I

BB
rvv
þ
p
t
C B C
þrB
@ ðe þ pt Þv  BðB  vÞ A ¼ @ 0 A;
@t @ e A
BvB  Bv0
pt ¼ p þ

B2
rv2
p
B2
þ ;
; e¼
þ
g1 2
2
2

ð1Þ

ð2Þ

where g = 5/3 is the ratio of specific heat, and the basic
quantities of r, v, p, and B represent normalized density,
velocity, pressure, and magnetic field, respectively. The cellcentered Total Variation Diminishing (TVD) scheme [Harten
and Hyman, 1983] with second-order MUSCL (Monotone
Upstream-centered Schemes for Conservation Laws) method[Van Leer, 1979] is applied using the Harten-Lax-van Leerdiscontinuity (HLLD) approximate Riemann solver [Miyoshi
and Kusano, 2005]. The HLLD method is excellent in termsof the positivity, simplicity, and efficiency [Stone et al.,2008], even comparing it to the most popular Roe method[Roe, 1981; Ryu and Jones, 1995]. The hyperbolic divergence cleaning technique of Dedner et al. [2002] is applied
to remove numerically finite values of div B.
[10] The numerical experiment is modeled as simply as
possible. The simulation box consists of 2563 uniform cubic
cells, with the spatial resolution of 2.0 solar radii (RS). The
Sun is located at the center of the cubic simulation box at theorigin (x, y, z) = (0, 0, 0) RS, while the Earth is fixed at a
position (x, y, z) = (215, 0, 0) RS, where the z axis is parallel to
the Sun’s rotation axis. Free boundary condition is applied at
the outer edges. Basic MHD quantities are inserted at a 50 RS
spherical surface as time-varying inner boundary to drive thesupersonic and super Alfvénic solar wind flowing outward inradial direction from the Sun.

[11] Figure 3 shows the assumed ambient solar windparameters at the inner boundary for the experiment of the
13 December 2006 CME event. The radial velocity map at asource surface of 2.5 RS was obtained from the interplanetary
scintillation (IPS) tomography method [Kojima et al., 1998],
based on two months data starting from 1 October 2006.
Solar-Terrestrial Environment Laboratory stopped the IPS
observation at 7 December 2006 for wintering of antennas at
the mountain areas. The two months period is the best timeinterval to obtain the whole velocity map without large data
gaps by the tomography method, and it is also a quiet period
during which the monthly velocity map does not change
significantly in each Carrington map. Constant speed isassumed to map from the source surface to the inner boundary at 50 RS. The magnetic field at the source surface of 2.5
RS is obtained from the Wilcox Solar Observatory’s potential
field model [Hoeksema et al., 1982], centering the
360°Carrington longitude of CR2050 on 14 November
2006. Centering at 14 November is consistent with the twomonth IPS observation. The result from the 2.5 RS radialmodel is selected because of its simplicity. The radial
magnetic field at the source surface Bs is scaled to the radial
IMF Br at 50 RS, assuming the decreasing strength with thesquare of distance. We further employ a simple power law
modification of the boundary magnetic field to push the flux
closer to the current sheet, creating a more uniform flux distribution away from the current sheet and a steeper gradientacross the current sheet as follows,
Br ¼ 10ðBs Þ0:5


2:5 2
;
50

ð3Þ

where we select the power index of 0.5 for simplicity, anda factor of 10 is multiplied to adjust it to the actual IMF
strength. A 14 deg eastward shift of the synoptic map is

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A10102

Figure 4. The ambient solar wind structure at 0400 UT on 12 December 2006. The amplitudes of MHD
parameters are shown at the equatorial plane: (a) interplanetary magnetic field strength, (b) solar windspeed, (c) number density, and (d) plasma temperature.
further applied, considering the approximate travel time fromthe solar surface to 50 Rs with a constant speed of 400 km s1.
[12] Following the method of Hayashi et al. [2003],
number density n and proton temperature T at 50 Rs distanceare calculated as empirical functions of radial speed V asfollows:

n ¼ 62:98 þ 866:4

3:402V; 1:549
100

ð4Þ

V
;
100

ð5Þ

the relation p = 2nkBT, where kB is the Boltzman constant.
The inner boundary map is rotated against the z axis at W =27.2753 day/rotation rate. The azimuth magnetic field B8
is added finally as a function of radial speed to reproduce theParker spiral:

T ¼ 0:455 þ 0:1943

where the units are n [/cc], T [MK], and V [km s1]. The
proton temperature is related to the MHD pressure using

Bf RW sin l
¼
;
Br
V

ð6Þ

where l is the colatitude from the Z axis, and R is 50 Rs. We
use a numerical relaxation technique to obtain the ambient
solar wind by rotating the inner boundary for a week inadvance, starting from the analytic solution of radial expansion flow with constant speed. Figure 4 shows the MHD

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Figure 5. Magnetic field lines of (left) the spheromak and (right) the closed loop morphology ofAR10930 computed using a nonlinear force-free (NLFF) model. The background contour shows positive(warm color) and negative (cold color) magnetic polarity of the photosphere. The red and white NLFF loops
correspond to the toroidal and poloidal fields of spheromak, respectively. A similar left-handed magnetic
field structure can be seen in both Figure 5 (left) and Figure 5 (right), where the toroidal fields direct roughly
westward and the poroidal fields surround over the toroidal fields from south to north.
parameters of the obtained ambient solar wind at the equatorial plane at 0400 UT on 12 December 2006. Basic structures such as the Parker spiral, heliospheric current sheet, andcorotating interaction regions are reproduced. The Earthrecently entered a toward IMF sector region, and will pass acurrent sheet and corotating interaction region in a week,
which is consistent with the observations of Figure 1.

a becomes negative for left-handed polarity. The sphericalBessel function and the derivative arej01 ð xÞ ¼

4. Spheromak CME Model[13] A spheromak-type magnetic field is introduced as thesimplest CME model. The spheroidal magnetic field may be anatural product of a plasmoid ejected from a solar flare via
reconnection [Gibson and Fan, 2008], although the actual
existence in the solar wind is under debate [e.g., Vandas et al.,1993a, 1993b, 2002; Farrugia et al., 1995]. In this paper, theinitial shape of the magnetic cloud is simply assumed to be asphere with the radius a. The inner magnetic field configuration is assumed to be the spheromak of Chandrasekhar andKendall [1957]:
Br ¼ 2B0

j1 ðarÞ
cosðqÞ;
ar

ð7Þ



Bq ¼ B0 j1 ðarÞ þ arj01 ðarÞ sinðqÞ;ð8Þ

Bf ¼ B0 j1 ðarÞ sinðqÞ;ð9Þ

where the polar coordinate system (r, q, 8) is defined by thepolar axis of the spheromak, a = 4.493409458a1 is theconstant derived from the force-free condition of r  B = aB
with the boundary condition of Br = 0 at r = a. The parametersin x  x cos x
;
x2

ð10Þ

2x cos x þ ðx2  2Þ sin x
:
x2

ð11Þ

j1 ð xÞ ¼[14] The magnetic helicity H in a spheromak is computedas
H ¼ a1

Z2p

Zp
df

0

Za
r2 dr sin qB2 ¼ 0:0454a4 B20 :

dq
0

ð12Þ

0

A realistic magnetic helicity can be estimated from HINODE
observations using the local correlation tracking method
coupled with induction equations [Kusano et al., 2002] toobtain the rough estimate of B0 = 1014 a2 T for AR10930.
We analyzed SOT/SP data obtained during December 9 – 13
during which the observed time intervals were 1 – 5 h. The
magnetic field B0 estimates the upper bounds. The model
constraints are B0 and the polar axis of spheromak, and theywould be useful for making predictions because B0 varies
event to event. In fact, there is a correlation between themagnetic parameters and flare activity [Yamamoto and Sakurai,
2009]. The nonlinear force-free method [Inoue et al., 2008]
based on the HINODE observations also gives anotherindependent estimation of H in the same order of magnitude.
An illustration of the 3-D magnetic field configuration of thespheromak is shown in Figure 5 (left), comparing it with thenonlinear force-free (NLFF) field (Figure 5, right) based on
the HINODE observation. A similar left-handed magnetic

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seems to be consistent with the HINODE observations(Figure 5). We discuss the need and the meaning of thisad hoc optimization process of arbitrary rotation in section 5.

5. Results and Discussions

Figure 6. Simulated solar wind profile at the Earth for the
week of the 13 December 2006 CME event. From top tobottom, shown are (a) IMF amplitude and the GSE southward
component (solid line), (b) IMF longitude in GSE coordinates,(c) solar wind speed, (d) number density, and (e) plasma
temperature. Blue lines show the solar wind profile computed
at a position 15 deg westward of the Earth.
field structure can be seen in both Figure 5 (left) and Figure 5(right), where the toroidal fields direct roughly westward andthe poroidal fields surround over the toroidal fields fromsouth to north.
[15] MHD quantities at the inner boundary are perturbed ina specified way as the spheroidal magnetic cloud moves
across the computational boundary. In this paper we deliberately choose the parameters to provide the best possible
results from this model. The magnetic field of the spheromak
is introduced by simply superposing it on the ambient field
to avoid any artificial div B problems. The radial speed Vr0
inside the cloud is assumed to be constant, where the transit
time t is directly related to the radius as t = 2a/Vr0, where theradius is assumed as a = 30 RS. The spheromak is coveredby a concentric spherical transition region where the speed
smoothly decreases to the ambient level at the radius of
40 RS using a trigonometric function of cos2. SOHO/LASCO
observed that the CME reached 6 RS at 0300 UT (Figure 2).
Assuming a constant deceleration of the CME speed from1800 km s1 at 6 RS to 1000 km s1 at 50 RS, the CME arrival
time at 50 RS is roughly estimated to be 0900 UT. The density
and temperature inside the magnetic cloud are assumed to be
twice that of the ambient solar wind. The ejection directionof the magnetic cloud is assumed to be 15 deg westward and
15 deg southward from the Sun-Earth line, roughly consistent
with the coronagraph observations in Figure 2. The polar axisof the ejected spheromak is assumed to be rotated 90 deg
toward the positive Y direction to find a better agreementwith the in situ IMF observations, although the original polar
axis at the lower corona mostly parallel to local north, which[16] Figure 6 shows a simulated solar wind profile at thevirtual Earth position for the same time interval of a week,
using the same vertical scale as Figure 1. To see a possible
range of variations at the 1 AU in situ sampling, the simulated
solar wind profile approximately 15 deg westward of the
Earth is also shown by blue lines. The most important point in
Figure 6 is the quantitative reproduction of the southward Bz
using the spheromak CME model. Many other basic structures are also reasonably reproduced as follows: The interplanetary shock arrived at 1900 UT on 14 December 2006,
which is 5 hours later than the actual arrival. The interplanetary shock and magnetic cloud arrived in the toward IMF
polarity before a current sheet arrival, followed by a corotating interaction region on 18– 19 December 2006 as observedin Figure 1, although the CIR structures is much fainter in themodel result. The rarefaction structure is also reproduced as
identified by the monotonic speed decrease with density
rarefaction, although significant perturbations coexist to
disturb the actual rarefaction profile in Figure 1. Predicting
the rarefaction structure is important for forecasting space
weather of terrestrial radiation belt. In fact, during the storm
recovery phase of this December 2006 storm, the radiation
belt electron flux was greatly enhanced during the rarefaction phase because of the ‘‘double rarefaction’’ mechanism[Kataoka and Miyoshi, 2008a, 2008b].
[17] The simulated solar wind profile in Figure 6 isobtained by rotating the polar axis of the ejected spheromak
by 90 deg anticlockwise, to see a better agreement with thein situ observation of the IMF. The in situ magnetic fieldrotation inside the magnetic cloud does not show a good
agreement if the polar axis of the spheromak is parallel tolocal north. This result may imply existence of a significant
anticlockwise writhe from the initial configuration occurring
below 50 RS, possibly below the coronal height where themagnetic field force dominates. The rotation of the magnetic
field vector at 1 AU associated with the spheromak CME issensitive to the initial magnetic field configuration at the
inner boundary and does not change significantly during
the propagation, because of relatively small magnetic force.
The amount of rotation of the field vector is also sensitive towhich part of the magnetic structure passes by the observer(thus the magnetic cloud trajectory), and a smaller magnetic
cloud is therefore more sensitive to this effect. The arrival
time of the interplanetary shock and/or magnetic cloud isearlier in case of high cloud density, because of the large total
momentum against the obstacle of slow ambient solar wind.
A systematic parameter survey using the MHD cube model tounderstand such sensitivities to various initial conditions isbeyond the scope of this paper and remains a future work.
[18] The basic structures such as magnetic cloud, shock,and sheath are investigated in more detail for the rotatedspheromak. Figure 7 shows the expanded view of velocity
and magnetic field vectors in GSE coordinates at the Earthposition for two days. The time resolutions for the ACE
magnetic field and velocity observations are 16 s and 64 s,

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A10102and Earth is implied within the magnetic cloud from electron
SEP observations [Liu et al., 2008]. During CME propagation, the overall morphology is elongated perpendicularly tothe radial direction, keeping the sense of the IMF rotations of
the spheromak. The meridional and azimuthal extent does notchange much during the propagation and is almost defined by
the initial size of the spheromak. Under the same conditionsof initial helicity and magnetic flux, the basic propagation
processes are not sensitive to the size of the spheromak. From
Figure 8, it is found that the Earth’s trajectory across aneastern shoulder of the deformed spheromak can explain both
the result of Liu et al. [2008] and the coronagraph observations described in section 2. The eastern shoulder trajectoryshould be observed as central passage of a flux rope at in situ
measurement, and the whole spheromak structure propagate
southward and westward as it appeared in the coronagraph.

6. Summary

Figure 7. Simulated solar wind profile at the Earth forthe time interval from 14 to 16 December 2006. We add 2 h tothe time axis of simulated profiles for a better comparison.
From top to bottom, shown are three components of (a, b, c)interplanetary magnetic field and (d, e, f) solar wind velocity
in GSE coordinates. The ACE observations are shown by
dots. Vertical solid lines show the interplanetary shock arrival,
and vertical dotted lines show the beginning of the magnetic
cloud.

respectively. We add 2 hours to the time axis of simulated
profiles just for a better comparison of the IMF rotation’s
turning from north to south (from positive ZGSE to negative
ZGSE). The flow direction inside the magnetic cloud iswestward (negative YGSE), as expected from the bulk motionof CME, while the flow deflection just after the shock arrival
is eastward (positive YGSE) and northward (positive ZGSE)
in both simulations and observations, as naturally expectedfrom the 3-D shock geometry [e.g., Kataoka et al., 2005].
[19] One of the most useful properties of MHD is theability to track the same magnetic field lines in order tounderstand the time evolution of the 3-D morphology.
Figure 8 shows the time evolution of the same magnetic fieldlines initially located inside of the spheromak. By definition,
spheromaks are magnetically closed structures fully detachedfrom the Sun. For example, Vandas et al. [1997, 1998] haveshown that a spheroid with its polar axis parallel to theambient evolves into a body similar to a toroid during
propagation in the solar wind. In our simulation, on thecontrary, most of the magnetic field lines of spheromak
CME are connected to the inner boundary from the beginning, as shown in Figure 8, since the magnetic field of thespheromak is introduced by simple superposition on theambient field. In fact, the IMF connection between the Sun[20] The MHD cube model was developed to reconstruct
the interplanetary propagation of CME on 13 December 2006
in a realistic ambient solar wind. The magnetic cloud andassociated structures, such as shock and sheath, are reasonably explained by the westward and southward propagationof a spheromak CME. Consequently, the spheromak CME
model is topologically complex enough to be consistent within situ observations, such as southward IMF associated with
CMEs. Introducing the 3-D spheromak-type CME magnetic
field is a new approach in this paper, while it is possible toimplement the spheromak CME in other MHD models, andthe expected results should not be significantly different
depending on different models.
[21] The MHD cube is the simplest model of the dynamic
solar wind, and free from a lot of problems in complicatedgrid systems. The uniform resolution makes the analysis
simple, and provides a good starting point for further development such as data assimilation and adaptive mesh refinement (AMR) technique. The uniform resolution of 2 RS iscoarse at the inner edge, and fine at 1 AU, which will benaturally improved by the AMR technique. The inner boundary can be replaced by some other solar wind models or
observations as desired in the future. The tunable free parameters of the spheromak CME model would be the initial
speed, width (size of spheromak), and direction, which are
related to observable quantities from a coronagraph. Further,detailed solar magnetograms, if available, can produce a
possible restriction in the polar axis and helicity of thespheromak.
[22] Limitations also exist. Odstrcil and Pizzo [1999]
suggested that initial CME position relative to the current
sheet is important for taking into account the interaction
between CMEs and the slow wind. A dynamic MHD coronalmodel is needed to address such interactions and the initial
evolution of CMEs and associated shock formations acrossthe critical point of solar wind acceleration. In fact, it has beensuggested that the initial CME evolution and shock formation
significantly depend on the coronal heating model [Jacobs
et al., 2005]. As members of a multi-institute collaborationof the space-weather-modeling taskforce in Japan [Shibata
and Kamide, 2007] that is working to solve such remaining
problems, we are developing an integrated framework to

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Figure 8. Time evolution of the same magnetic field lines inside spheromak CME at (left) 0400 UT on
13 December 2006 and (right) 0400 UT on 14 December 2006. (top) View from the north (XY plane).
(bottom) View from the east (YZ plane). The Sun is located at the origin, and the Earth is located at(X, Y, Z) = (215, 0, 0).
initiate the CME from the solar surface based on HINODE
observations for creating more realistic CME structures. The
initial results of our integrated framework will be shownelsewhere.
[23] Acknowledgments. The code development of the MHD cube by
R.K. owes thanks to ACT-JST Summer Seminar of Astrophysical Plasma
Simulation held by R. Matsumoto in 2003. R.K. thanks K. Tomisaka,
T. Hanawa, and T. Yokoyama for their teaching how to develop an MHD
code in the Summer Seminar. R.K. thanks T. Miyoshi of Hiroshima
University for his practical advice on the HLLD method. The CME catalog
is generated and maintained at the CDAW Data Center by NASA and the
Catholic University of America in cooperation with the Naval ResearchLaboratory. SOHO is a project of international cooperation between ESA

and NASA. The solar magnetic field data is obtained from the Wilcox Solar
Observatory. The OMNI-2 and ACE solar wind data are provided from
NASA/NSSDC. The Dst index is provided from Kyoto University. Hinodeis a Japanese mission developed and launched by ISAS/JAXA, with NAOJ
as domestic partner and NASA and STFC (UK) as international partners. It
is operated by these agencies in cooperation with ESA and NSC (Norway).
The work by R.K. was supported by a research fellowship of Special
Postdoctoral Research Program at RIKEN. R.K. thanks the RIKEN Super
Combined Cluster (RSCC) and Nagoya University in collaboration with
STEL for computational resources. This work was supported by the Grantin-Aid for Creative Scientific Research (20740286) and ‘‘The Basic Studyof Space Weather Prediction’’ (17GS0208, Head Investigator: K. Shibata)from the Ministry of Education, Science, Sports, Technology, and Cultureof Japan.
[24] Amitava Bhattacharjee thanks Sarah Gibson and another reviewer
for their assistance in evaluating this paper.

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T. Ebisuzaki, Advanced Science Institute, RIKEN, 2-1 Hirosawa, Wako,
Saitama 351-0198 Japan.
S. Inoue, National Institute of Information and Communications
Technology, 4-2-1 Nukui-Kitamachi, Koganei, Tokyo 184-8795, Japan.
R. Kataoka, Interactive Research Center of Science, Tokyo Institute ofTechnology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan.
D. Shiota, IFREE, JAMSTEC, 2-15 Natsushima-Cho, Yokosuka,
Kanagawa 237-0061, Japan.
K. Kusano, M. Tokumaru, and T. T. Yamamoto, Solar-Terrestrial
Environment Laboratory, Nagoya University, Furo-Cho, Chikusa-ku,
Nagoya 464-8601, Japan.

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New Solar Physics with Solar-B Mission
ASP Conference Series, Vol. 369, 2007
K. Shibata, S. Nagata, and T. Sakurai

Simulation Study of Transition Process from Long-Lived
Sigmoid to Flare Eruption
Kanya Kusano
The Earth Simulator Center, Japan Agency for Marine-Earth Scienceand Technology
Abstract. The formation mechanism of sigmoidal structure in the solar corona
and the transition process from long-lived sigmoid to ﬂare eruption are studied on
the basis of the Taylor’s minimum energy principle and the three-dimensional
magnetohydrodynamic numerical simulations. The simulation results indicatethat magnetic reconnection driven by the resistive tearing mode instability growing on current sheet, where magnetic shear is steeply reversed, may cause theformation of sigmoidal structure. The mechanism of that can be explained as
the spontaneous manifestation of the Taylor’s minimum energy state in magnetic arcade geometry. Furthermore, it is also numerically demonstrated thatthe formation of sigmoid can be followed by the eruption of magnetic ﬂux. In
the eruption, however, the sigmoidal ﬂux itself is not ejected upward, but it iscollapsed by the reconnection with overlaying magnetic ﬁeld. The results are
consistent with the reversed shear ﬂare model recently proposed by Kusano etal., which predicts that the coronal eruption is triggered by the collapsing ofmagnetic arcade due to the annihilation of positive and negative magnetic helicity.

1.

Introduction

Sigmoid is the characteristic morphology of forward-S or inverse-S shape (Rust& Kumar 1996), which was often observed by the Yohkoh soft X-ray telescope(SXT) (Tsuneta et al. 1991) in active regions. Because sigmoids tend to appear
associated with several eruptive events (Canﬁeld et al. 1999), they are widelybelieved to be some precursor phenomena for solar coronal eruption. Therefore, it is an important subject to reveal the formation process of sigmoids andthe transition dynamics from pre-ﬂare sigmoid to eruptive events, in order tounderstand the onset mechanism of the explosive activities in the solar corona.
Several papers have been published to explain the formation mechanismof sigmoids so far. The kink instability model, which was originally proposedby Rust & Kumar (1996), is most widely known, and it predicts that a ﬂux
tube twisted stronger than some criterion should evolve to sigmoid. Recently,
several authors (Magara & Longcope 2001; Fan & Gibson 2004; Kliem, Titov,& Török 2004; Inoue & Kusano 2006) performed the three-dimensional (3D)
magnetohydrodynamic (MHD) simulations of the kink mode instability of atwisted ﬂux rope, and they showed that a sigmoid-like current sheet can beformed as a consequence of the kink mode instability.
However, it is still a debatable issue whether the kink mode instability is
indeed responsible to the formation of sigmoids. For instance, Leamon et al.
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470

Kusano(2003) found that there is no evidence of eruption for values of large-scale total
twist approaching the threshold for the kink instability, though they sought 191
X-ray sigmoids. Yamamoto et al. (2005) analyzed the relationship between themagnetic helicity injection and the formation of sigmoid using the vector magnetograph data and the inversion technique of the induction equation (Kusano
et al. 2002), and found that the helicity injection rate normalized by the squareof magnetic ﬂux is only about 0.01 to 0.02 day−1 , although the sigmoidal loops
were observed to be formed within a few hours. This value of the helicity injection looks too small to explain the sigmoidal formation by the kink instability
model.
In this paper, we propose the another mechanism, which can explain thesigmoidal formation even though the ﬁeld line winding is too small to activate
the kink instability, on the basis of the numerical simulation and the Taylor’s
minimum energy principle (Taylor 1986). Also some issues of the new study that
we should do with the Solar-B observations to reveal the onset mechanism oferuption will be discussed in the ﬁnal section.

2.

Sigmoid as the Taylor’s Minimum Energy StateThe formation of long-lived sigmoid is one of the enigmas in the solar coronal
activity. However, the fact that sigmoids are spontaneously formed and can besustained for substantially longer period than the Alfvénic time-scale suggeststhat the structure is energetically preferential in the MHD system. According
to the Taylor’s energy principle (Taylor 1974), the MHD system, which is dominated by magnetic energy like the solar corona, has a tendency to relax toward
the linear force-free ﬁeld (LFFF), which is the solution of equation ∇×B = αB.
The LFFF may bifurcate into multiple states when the constant α is larger
than the lowest eigenvalue λ1 of the curl operator (Taylor 1986; Kusano et al.
1995; Kusano & Nishikawa 1996). One is called the coupled state, and anotheris the mixed state. When the magnetic helicity contained in the system is larger
than the criterion for the bifurcation, the minimum energy state is switched fromthe coupled state to the mixed state. Magnetic helicity in the coupled state is
totally generated by the coupling with the ﬁeld connecting to the boundary
condition, but that in the mixed state is given by the mixing of the eigenﬁeld
corresponding to λ1 and the ﬁeld connecting to the boundary condition. Since theeigenﬁeld in the magnetic arcade geometry forms helical ﬂux along the magnetic
neutral line, the ﬁeld lines of the mixed state makes sigmoidal structure. It
suggests the possibility that sigmoid is a manifestation of the Taylor’s minimum
energy state.
In fact, Nandy et al. (2003) observed that the distribution of the shear
parameter α in active regions tends to be ﬂattened indeed. Their observationsuggested that the Taylor-type relaxation could be operated in the timescale ofabout several days in the solar corona, although the direct evidence of the causal
relation between the Taylor-type relaxation and the sigmoidal formation has notbeen found yet.

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3.

471

Simulation Study of Sigmoidal Formation

Although the Taylor’s minimum energy principle can derive the equilibrium
state, which is most favorable energetically, it does not provide any information of the dynamics into the minimum energy state. Here, let us ask ourselves
what is the driver for the MHD energy relaxation, which is able to open the
dynamic route to access the minimum energy state in the solar corona. The resistive tearing mode instability is one candidate. Actually, in the toroidal pinch
experiments, the resistive tearing mode instability plays the role as the primary
driver for the energy relaxation (Taylor 1986; Kusano & Sato 1990).
Recently, Kusano (2005) carried out the numerical simulation in order todemonstrate the sigmoidal formation due to the tearing instability. The threedimensional (3D) simulation is initiated by adding a small perturbation to aquasi static arcade equilibrium, which was pre-calculated by two-dimensional(symmetric along magnetic neutral line) model. Magnetic shear of the initial
equilibrium state is reversed from right-handed to left-handed as one goes fromthe outside to the inside of some arcade surface. The shear reversal is so steep,that there is thin current sheet, which is unstable to the tearing mode, on theshear inversion layer.
Figure 1 shows the ﬁeld line evolution due to the growth of the resistive tearing instability. It clearly shows that the sheared ﬁeld evolves to the sigmoidal
ﬁeld, which forms an inverse-S shape. This result indicates that the sigmoidal
formation due to the tearing mode is possible indeed. Furthermore, it suggeststhat the morphology of sigmoid (S or inverse-S) is related to the sign of magnetic
shear near the magnetic neutral line. It would be consistent with the observation that more inverse/forward-S-shaped sigmoids were observed in the northern/southern hemisphere, in which negative/positive helicity corresponding to
the left/right-handed shear might be injected predominantly.
The shear parameter α of the sigmoidal ﬁeld is calculated, and is comparedto the theoretical prediction. If the mixed state is formed, α should be limitedby the eigenvalue λ1 , which is approximately given by the wavenumber 2π/h forcharacteristic scale h. Figure 2 represents the vertical distribution of ᾱ, which is α
axially averaged along a line parallel to magnetic neutral line, above the magnetic
neutral line of the sigmoidal region (see left panel of Fig.3), and shows that thevalue of ᾱ is almost matched to 2π/h, where h is the distance from the bottom
to the shear inversion layer. Note also that the distribution of α is ﬂattenedin the sigmoidal region, as predicted by the Taylor’s principle. The horizontal
wavelength of sigmoid along the magnetic neutral line is also compared to theeigenﬁeld, and it is conﬁrmed that the sigmoidal structure is well consistent
with the theoretical prediction of the helix in the minimum energy mixed state,
which is geometrically determined by the cross section of magnetic arcade. Refer
to Kusano (2005) for further detail.
The results above indicate that the Taylor-type relaxation driven by thetearing instability can work as the formation mechanism of sigmoidal structure.
The sigmoidal formation would be a natural process in magnetic arcade, andit can easily arise through the resistive instability of thin current sheet, eventhough the tightly twisted ﬂux tube does not preexist.

Kusano, “Transition from Long-Lived Sigmoid to Flare Eruption” (2007)

472

Kusanot=15.5t=33.2

Figure 1.
Top view of magnetic ﬁeld lines in the sigmoidal formation process in the simulation by Kusano (2005). The thickness of the ﬁeld line is
proportional to the intensity of electric current averaged along each ﬁeld line.

4.

Simulation Study of Eruption Mechanism

The numerical simulation by Kusano (2005) demonstrated not only the formation of sigmoid but also the eruption of that. When the calculation was simply
continued for much longer period than the Alfvén transit time after the sigmoid is formed, they found that magnetic cusp structure is created above thesigmoidal region and that the magnetic ﬂux is erupted upwards, as shown inright panel of Fig.3. The cusp structure, in which magnetic reconnection on thevertical current sheet drives the upward eruption, is well consistent with thestandard ﬂare model.
Here, we should emphasize that even after the eruption the weak sigmoidal
ﬁeld can survive near the photospheric boundary, because the sigmoidal ﬂux
itself is not ejected by the reconnection, but only the overlaying ﬂux is ejected.
It is quite diﬀerent from the previous model based on the kink instability, which
predicted that the ampliﬁed kinking deformation of sigmoid leads to the ejectionof sigmoid itself.

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Transition from Long-Lived Sigmoid to Flare Eruption40
t=30.1

20
0
α

-20
2π/h

-40
sigmoid
-60

t=0

-80
-100

h
0

0.05

0.1

0.15
z

0.2

0.25

0.3

Figure 2.
The vertical variation of the averaged shear parameter ᾱ for the
initial state (t = 0) and the sigmoidal state (t = 30.1) in the simulation by
Kusano (2005). The height of the shear inversion layer at t = 30.1 is denoted
by h, and the corresponding wavenumber (2π/h) is plotted by dot-dashedline.

In contrast with the previous model, the eruption in Kusano (2005) simulation arises as a consequence that the sheared ﬂux of the sigmoid is subject tosubstantial erosion by reconnection on the shear inversion layer. The physicalprocess of the eruption onset can be understood as a sort of loss-of-equilibrium,
which results through the annihilation of positive and negative magnetic helicity(Kusano et al. 2003, 2004). Therefore, the lifetime of sigmoid from the formation
to the eruption should be determined by the timescale of the tearing instability,
which causes the annihilation of magnetic helicity.
In the solar corona of temperature T = 106 [K], and plasma density n = 1015
−3[m ], the growth time of the most unstable tearing mode τtearing = (τA τη )0.5 isgiven by 4 × 10−5 (δ 3 /B)0.5 [sec] on the current layer of the width δ [m] and themagnetic ﬁeld B [T], where τA and τη indicate the Alfvén transit time and theresistive diﬀusion time, respectively. If B = 10−2 [T], and δ is about 500 [km],Kusano, “Transition from Long-Lived Sigmoid to Flare Eruption” (2007)

474

Kusanot=30.1t=58.7

Figure 3.
Three-dimensional structure of the magnetic ﬁeld lines in thesigmoidal region (t = 30.1) and that at the eruption phase (t = 58.7).

then τtearing is several days, and we can explain the sustainment of long-lived
sigmoids.
Finally, we mention that the ejected magnetic ﬂux forms a twisted ﬂux tube
in the plasmoid, and thus the kink instability may grow after the eruption. If
so, it is likely that the scenario of our tearing instability model leads to the kinkinstability dynamics, which may accelerate the ejected plasmoid into the coronalmass ejection (CME), as recently demonstrated by the simulation of Inoue &Kusano (2006).
5.

Discussion: What We Should Do with Solar-B

The numerical simulation clearly indicates the possibility that the tearing instability may produce the sigmoidal structure and can cause the eruptive dynamics.
However, there are still several limitation in the simulation model, and many
open questions remain to be solved. The most important problem is reality ofthe thin current sheet, which should exist relating to sigmoidal formation. The
vector magnetograph resolution of Solar-B/SOT is about 0.2” that corresponds
to 150 [km] on the solar surface. The systematic observation of current sheet
in sigmoidal regions might be the important task to reveal the mechanism of
sigmoids and the onset of eruption.
Another important task is the quantitative investigation of magnetic helicity
in sigmoidal regions. Several techniques have been developed to measure themagnetic helicity injection into the solar corona. Using the Solar-B/SOT data,
we can perform the non-interrupted observation of helicity injection activity,
which may greatly improve our understanding of the underlying mechanism
in sigmoidal formation. Also the reconstruction of three-dimensional magnetic
ﬁeld using the Solar-B/SOT data is much helpful to reveal the MHD propertyof sigmoidal ﬁeld. However, though many extrapolation techniques have been311

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475

proposed so far, they do not yet mature to be applied to practical data. A new
data assimilation method, which can directly connect observation and numericalsimulation, have to be developed to maximize the scientiﬁc information obtainedby the Solar-B.
Acknowledgments. The author acknowledges the Japan Society for the
Promotion of Science for the ﬁnancial supports. The author is grateful to Prof.
T.Sakurai, Dr.S.Inoue, Dr.T.Yokoyama and Mr.T.T.Yamamoto for useful discussions.
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Inoue, S. & Kusano, K. 2006, ApJ, 643 (in press)
Kusano, K. 2005, ApJ, 631, 1260
Kliem, B., Titov, V. S., & Török, T. 2004, A&A, 413, L23
Kusano, K., & Sato, T. 1990, Nuclear Fusion, 30, 2075
Kusano, K., Suzuki, Y., & Nishikawa, K. 1995, ApJ, 441, 942
Kusano, K., & Nishikawa, K. 1996, ApJ, 461, 415
Kusano, K., Maeshiro, T., Yokoyama, T., & Sakukrai T. 2002, ApJ, 577, 501
Kusano, K., Yokoyama, T., Maeshiro, T., & Sakukrai T. 2003b, Advances in Space
Research, 32, 1931
Kusano, K., Maeshiro, T., Yokoyama, T., & Sakukrai T. 2004, ApJ, 610, 537
Leamon, R. J., Canﬁeld, R. C., Blehm, Z.,& Pevtsov, A. A. 2003, ApJ, 596, L255
Magara, T. & Longcope, D. W. 2001, ApJ, 559, L55
Nandy, D., Hahn, M., Canﬁeld, R. C., & Longcope, D. W. 2003, ApJ, 597, L73
Rust, D.M., & Kumar, A. 1996, ApJ, 464, L199
Taylor J.B. 1974, Phys. Rev. Lett., 33, 1139
Taylor J.B. 1986, Revs. Modern Phys., 58, 741
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624, 1072

Tokumaru et al., “Non-Dipolar Solar Wind Structure” (2009)

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Here

313

GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L09101, doi:10.1029/2009GL037461, 2009

for

Full
ArticleNon-dipolar solar wind structure observed in the cycle 23/24 minimum
M. Tokumaru,1 M. Kojima,1 K. Fujiki,1 and K. Hayashi2
Received 14 January 2009; revised 25 March 2009; accepted 2 April 2009; published 5 May 2009.

[1] Interplanetary scintillation (IPS) observations made in
the Cycle 23/24 minimum using the Solar-Terrestrial
Environment Laboratory (STEL) multi-station system
indicated that during intervals the solar wind had asignificantly non-dipolar structure that consisted of fast windcomponents at the poles and the equator and slower windcomponents in between. The solar wind structure revealed
from the IPS observations was consistent with a marked
increase in the occurrence of fast winds observed in situ near
the earth. The poleward boundary of the slow wind region
observed during this minimum was ±30 north and south. In
addition, our IPS observations revealed that the organizationof the 3-dimensional solar wind was highly variable during
2007–2008. These features greatly differ from those observed
during the previous minima. This fact may be attributed to theweak magnetic field intensity at the poles during the Cycle 23/
24 minimum. Citation: Tokumaru, M., M. Kojima, K. Fujiki, andK. Hayashi (2009), Non-dipolar solar wind structure observed in thecycle 23/24 minimum, Geophys. Res. Lett., 36, L09101, doi:10.1029/
2009GL037461.

1. Introduction[2] Systematic changes in the 3-dimensional structure of
the solar wind, which are closely linked with the 11-year
sunspot cycle, have been disclosed from ground-based observations using interplanetary scintillation (IPS) for more than
three decades [e.g., Coles et al., 1980; Kojima and Kakinuma,
1990]. The solar wind observed in the previous two minimum
periods revealed a stable structure which was composed of alow-speed stream in a narrow latitude range around theequator and a high-speed stream over the poles. Here wedemonstrate that the solar wind structure determined frominterplanetary scintillation (IPS) observations of the SolarTerrestrial Environment Laboratory (STEL) of Nagoya University for the Cycle 23/24 minimum differs significantly
from this simple dipole-like feature. In section 2, we briefly
describe our IPS observations of the solar wind. In section 3,
we present the solar wind speed data obtained from our IPS
observations in the cycle 23/24 minimum. In section 4, weshow in situ observations which support our finding. In
section 5, we discuss the implications of our findings and insection 6 we summarize the results of this study.

2. IPS Observations[3] The IPS, which is a diffraction phenomenon caused
by density irregularities in the interplanetary medium, has
1
Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya,Japan.
2
W. W. Hansen Experimental Physics Laboratory, Stanford University,
Stanford, California, USA.

Copyright 2009 by the American Geophysical Union.
0094-8276/09/2009GL037461$05.00

been used as a useful ground-based method to observe thesolar wind out of the ecliptic [Dennison and Hewish, 1967].
As the diffraction pattern drifts owing to the solar windmotion, the flow speed can be determined by measuring thetime delay between two signals at geographically separated
sites. We have carried out IPS measurements of the solar
wind speed since the early 1980s, on a daily basis between
April and November/December, using the STEL 327-MHz
multi-station system [Kojima and Kakinuma, 1990]. Our
IPS observations enable us to produce a synoptic map of thesolar wind speed in a Carrington longitude-versus-latitudeformat for a given solar rotation, since approximately 1000of lines-of-sight data are usually available during thecorresponding 27-day period. The IPS data have been
deconvolved to retrieve the intrinsic distribution of solar
wind speed using the computer-assisted (CAT) tomographymethod [Kojima et al., 1998]. The high reliability of thesolar wind speed data deconvolved from our IPS observations has been confirmed by earlier works [e.g., Kojima etal., 1998, 2007]. For example, the solar wind data showed
excellent agreement with Ulysses latitude scan observations
during 2001 [Fujiki et al., 2003].

3. Solar Wind Speed Data
[4] Figures 1a–1d show the solar wind speed data obtained
from STEL IPS measurements for representative Carrington
rotations in the last three solar sunspot minima; i.e., CR 2070,
CR 2075, CR 1910, and CR 1790, which correspond to theperiods between May 13 to June 9, September 26 to October
10, 2008 (Cycle23/24), June 1 to 28, 1996 (Cycle 22/23), andJune 16 to July 13, 1987 (Cycle 21/22), respectively. The solidline in the synoptic map denotes the magnetic neutral line at
the source surface determined from WSO (Wilcox Solar
Observatory) magnetogram observations (http://wso.stanford.
edu/synsourcel.html) [Hoeksema et al., 1983]. Figures 1(left) and 1 (right) are the synoptic source surface velocity
map and the latitude profile of the solar wind speed, respectively. As shown here, the high-speed solar wind that emanates from the polar region is a common feature during
sunspot minimum periods. However, the solar wind observedin CR 2070 (Figure 1a) showed a clear difference in the flow
speed at the equator from those observed during the previous
two minimum periods (Figures 1c and 1d). The solar windspeed in CR 2070 significantly increased at the equator, while
its peak value (600 km/s) was somewhat lower than theflow speed of the fast polar streams. The mid-latitude regionsin CR 2070 were mostly dominated by the slow winds. Thus,
the solar wind observed for CR 2070 exhibited a significantly
non-dipolar structure composed of polar and equatorial faststreams with slow streams in between. Another important
point to note is that the non-dipolar structure found for CR
2070 was not discernible for CR 2075 (Figure 1b). This factsuggests that the solar wind structure evolved rapidly during

L09101

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L09101Figure 1. (left) Synoptic source surface maps and (right) latitude profiles of the solar wind speed. Maps were derived fromSTEL IPS observations for Carrington Rotations (CRs) (a) 2070, (b) 2075, (c) 1910, and (d) 1790. CRs 2070 and 2075
correspond to the Cycle 23/24 minimum, and CRs 1910 and 1790 correspond to 22/23 and 21/22 minima, respectively.
Synoptic map is drawn in the Carrington longitude-versus-latitude format, and red (blue) colours in the map represents slow
(fast) solar winds. Solid line in synoptic map denotes the magnetic neutral line at the source surface determined from WSO
magnetogram observations. Vertical bars in the latitude profile correspond to an rms value around the mean speed for agiven latitude.
2008. The latitude width of the slow wind region for CR 2075
is considerably larger than that for CR 1910. This fact may be
attributed to the complexity of the solar wind at each latitude
during 2008.
[5] A series of synoptic solar wind speed maps fromSTEL IPS observations are displayed in Figures 2a– 2c for

2008, 2007, and 1996, respectively. As revealed from
Figure 2, the solar wind speed structure evolved considerably during 2007 – 2008 despite quite a low level of solar
activity. The non-dipolar feature of the solar wind speed
was discerned for CRs 2070 – 2072 in 2008, and lessprominently for CR 2058 in 2007, and it appeared to be

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L09101Figure 2. A series of synoptic source surface maps of solar wind speed from STEL IPS observations for (a) 2008,(b) 2007, and (c) 1996. Color scale used here is the same as the one in Figure 1. White area in the map corresponds to
the region where no observational constraint is placed by IPS observations.
less distinct after CR 2074. Such a rapid evolution of the solar
wind structure during 2007 –2008 is in marked contrast tothat observed in the previous minima. The solar windobserved in 1996 (Figure 2c) showed a stable structure whichlasted over several solar rotations. In addition, Figure 2
reveals a significant difference in the latitude extent of theslow wind region between Cycle 23/24 and 22/23 minimum
periods. The slow wind region observed during 2007 – 2008
extended to 30° north and south, while the fast windoccasionally penetrated this latitude band. On the other hand,
the latitude extent of the slow wind observed in 1996 wasapproximately ±15°.

4. In Situ Observations at 1 AU
[6] The significant growth of the equatorial fast wind
revealed during 2007 – 2008 has been supported by in situ

observations conducted at earth orbit. Histograms of theoccurrence rate of the solar wind speed measured by spacecraft near the earth are indicated in Figures 3a– 3e for 2008,
2007, 2003, 2001, and 1996, respectively. Here, we used thesolar wind speed data collected by the Advanced Composition Explorer, ACE (http://www.srl.caltech.edu/ACE/ASC/)[McComas et al., 1998] or Wind (http://web.mit.edu/space/www/wind/wind_data.html) [Ogilvie et al., 1995] spacecraft.
Note that the histogram for 2008 (Figure 3a) produced fromthe ACE data from the beginning to July 22 of the year (thelatest date at which the Level 2 (verified) data were available
as of this writing). We have confirmed that the distribution
revealed by the histogram remains basically unchanged even
if the realtime ACE data covering a fuller span for 2008 areused. The histogram of 1996 data (Figure 3e) showed a sharp,
single peak profile with a maximum at 400 km/s, andthis fact is consistent with the solar wind speed distribution

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L09101Figure 3. Histograms of the occurrence rate of the solar wind speed measured at earth orbit in (a) 2008, (b) 2007, (c) 2003,(d) 2001, and (e) 1996. Here, in situ observations made by the ACE spacecraft are used for Figures 3a–3d, and Wind
spacecraft data are used for Figure 3e. Ndat, Vave, and Vmode shown in each plot correspond to number of data, average andmodal values of solar wind speed, respectively.
revealed from our IPS observations during the Cycle 22/23
minimum, since the earth’s orbit is located near the equator
where the slow speed component dominates (see Figure 2c).
The wind speed data in 2001 (Cycle 23 maximum) werebasically the same as those in 1996, as shown in Figure 3d,
since the slow speed stream became ubiquitous in the
heliosphere during the sunspot maximum. However, thehistogram profile revealed in 2007 and 2008 data was rather
different from the typical one. Although the primary peaks of2007 and 2008 data still occurred at 400 km/s, they weremuch less noticeable, and a secondary peak significantly
developed at 600 km/s (see Figures 3a and 3b). Thischange was consistent with the emergence of the equatorial
fast wind revealed by our IPS observations. We examined insitu data of the solar wind at earth orbit for other years
between 1995 and 2006, and confirmed that the pronounced
dominance of the slow-speed component was a common
feature for almost all years. One exception was in situ datataken in 2003, which corresponds to the declining phase of
the Cycle 23 (Figure 3c). The histogram for 2003 is

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L09101Figure 4. (left) Coronal magnetic fields within 2.5 solar radii and (right) synoptic coronal hole maps for CRs (a) 2070 and(b) 1910. The coronal magnetic fields were calculated from MDI/SOHO observations using the MHD code. The coronalhole map for CR 2070 was taken from the NSO/GONG magnetogram synoptic map archive (http://gong.nso.edu/data/magmap/archive.html). Green and red areas in this map denote positive and negative coronal holes, respectively. Coronalhole map for CR 1910 was from the NSO/Kit Peak synoptic coronal hole map archive (ftp://nsokp.nso.edu/kpvt/
coronal_holes/synoptic/).
somewhat similar to those for 2007 and 2008. Further
discussion on this similarity is beyond the scope of this paper.

5. Discussion[7] The peculiarity of the solar wind structure observed in
the Cycle 23/24 minimum can be attributed to that of the
global magnetic field structure of the corona. Figure 4
illustrates the MHD model calculations of the coronalmagnetic field within 2.5 Rs (the source surface) [Hayashi
et al., 2008] and the synoptic coronal hole maps for CR
2070 (Figure 4a) and CR 1910 (Figure 4b). The photospheric magnetic field data from the Michelson Doppler
Imager, MDI [Scherrer et al., 1995] onboard the Solar andHeliospheric Observatory (SOHO) spacecraft have beenused in these model calculations. The white and red lines
in Figure 4 represent the closed and open field lines, while
the blue and red colours on the sphere indicate the positive
and negative polarities of the large-scale photosphericmagnetic field. As for CR 1910, the low-to-mid-latitude
region was entirely occupied by the large-scale closed fieldlines, and the open magnetic fluxes originated exclusively
from the polar regions. In contrast, the open fluxes of CR

2070 emanated not only from the polar regions but also
from the low-latitude region, which is a source of theequatorial fast wind. Many mid- and low-latitude coronalholes were found for CR 2070, while the coronal holes wereconfined in the polar regions during CR 1910. The difference in coronal hole distribution between CRs 2070 and1910 is consistent with what was revealed from the model
calculations of the coronal magnetic field and also from our
IPS observations.
[8] The formation of an enhanced low-latitude open flux
was considered to be a manifestation of the solar magnetic
dynamo activity during the Cycle 23/24 minimum. The
solar magnetograph observations at the Wilcox Solar Observatory indicated that the polar field strength of the Cycle
23/24 minimum was significantly weaker than that of theprevious two minima (http://wso.stanford.edu/gifs/Polar.gif)[Hoeksema, 1995]. We consider that such a weak dipole
component is essential to account for the increase of the
low-latitude open flux and thus the non-dipolar solar windstructure in the Cycle 23/24 minimum. Furthermore, thesunspot counts in the Cycle 23/24 minimum were very low,
and the number of spotless days in 2008 was the highest in
the last 50 years. This was also a sign of the solar dynamo

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activity in this minimum [e.g., Schatten, 2005; Svalgaard etal., 2005]. Observations during Ulysses’s third orbit spacecraft showed that the polar fast wind was slightly slower,
significantly less dense, cooler, and had less mass andmomentum flux than during the previous solar minimum
orbit [McComas et al., 2008]. Thus, the Cycle 23/24
minimum exhibits several peculiar aspects from the viewpoint of solar wind and magnetism. We consider that itprovided an excellent opportunity to gain a better insight
into the solar wind formation and dynamo process.

6. Summary[9] Our IPS observations have disclosed that the solar
wind structure in the Cycle 23/24 minimum greatly differsfrom what was observed during the previous two minima.
Namely, the fast solar wind emanated not only from thepoles but also from the equatorial region, and the source ofthe slow solar wind split into two mid-latitude regions. The
poleward boundary of the slow wind during this minimum
extended to 30° north and south, and the solar windstructure evolved rapidly. These features significantly differ
from those observed during the previous minima. A distinct
increase of the occurrence rate of the high flow speed
measured in situ at 1 AU is consistent with our IPS
observations. Our finding is regarded as another manifestation to indicate the peculiarity of the solar magnetic activity
in this minimum.
[10] Acknowledgments. The IPS observations were carried out under
the solar wind program of the Solar-Terrestrial Environment Laboratory(STEL) of Nagoya University. This work was partly supported by the
Grant-in-Aid for Creative Scientific Research (17GS0208) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. We
wish to thank the ACE Science Center and the MIT space plasma group forproviding the solar wind plasma data collected by the ACE and Wind
spacecraft.

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doi:10.1029/2004GL021664.


K. Fujiki, M. Kojima, and M. Tokumaru, Solar-Terrestrial Environment
Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 4648601, Japan. (fujiki@stelab.nagoya-u.ac.jp; kojima@stelab.nagoya-u.ac.jp;
tokumaru@stelab.nagoya-u.ac.jp)
K. Hayashi, W. W. Hansen Experimental Physics Laboratory, Stanford
University, 491 South Service Road, Stanford, CA 94305-4085, USA.
(keiji@sun.stanford.edu)

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319

Solar Wind Forecast by Using Interplanetary Scintillation
Observations
Ken’ichi Fujiki, Hiroaki Ito and Munetoshi Tokumaru
Solar-Terrestrial Environment Laboratory (STELab), Nagoya University, Furo-cho, Chikusa-ku, Nagoya
464-8601, Japan
Abstract. Interplanetary scintillation (IPS) allows us to determine solar wind velocity and density structures over a relatively
short time by employing computer assisted tomography. This method can be applied to forecast solar wind changes for a
few days prior to its reaching Earth. We have been attempting solar wind forecasting by using IPS data observed at SolarTerrestrial Environment Laboratory (STELab), Japan. Here we introduce the solar wind forecast using STELab-IPS andevaluated dependences of the accuracy of the STELab-IPS forecasts on a number of line-of-sight observed with IPS and asolar activity by comparing comparing them with in situ measurements.
Keywords: solar wind, forecast, interplanetary scintillation
PACS: 90

INTRODUCTION

ing. In following section, we use forecast results from2001 and 2005 and explore the parameters that control
the accuracy of the forecast.

Radio waves from a compact radio source such as aquasar are scattered by irregularities of electron density
in the interplanetary medium [1]. The scattered waves
interfere with each other as they propagate toward theEarth producing amplitude scintillations which are thenseen as diffraction patterns at Earth. This phenomenon iscalled interplanetary scintillation (IPS). The IPS signal
contains the information of solar wind velocities anddensity fluctuations passing across a line of sight (LOS)
from an observer to a radio source.
We have carried out IPS observations at the SolarTerrestrial Environment Laboratory Nagoya University,
Japan (STELab-IPS) for over two decades [2]. The 327
MHz observing frequency is most sensitive to IPS signals from 0.2-1 AU distances from the Sun. By employing computer assisted tomography (CAT) to reduce the
LOS integration effect, solar wind velocity and density
structures can be determined with far more accuracy thanusing previous methods [3]. This technique can be applied to forecast the solar wind for a few days prior to its
reaching Earth (Fig.1(a)). The CAT algorithm we havedeveloped assumes that solar wind structure co-rotates
and requires a redundancy of LOS numbers. Because IPS
signal to noise increases during times of solar activity,
the accuracy of the solar wind forecast depends on solar
activity as well as the numbers of IPS LOS number forthe period of interest.
In this study, we simulated real-time analyses of aforecast procedure using IPS data obtained in 2001 (solar
maximum), 2002 (declining phase), and 2005 (solar minimum). In the next section, we describe the STELab-IPS
observations and our new method of solar wind forecast-

SOLAR WIND FORECAST USINGSTELAB-IPS
In this section we describe key details of the forecast
using STELab-IPS data and evaluate the forecast error.
The CAT produces a synoptic velocity distributionmap (V-map) using STELab-IPS data from a specific
Carrington rotation. The CAT analysis requires that thesolar wind structure is stable, i.e. co-rotational, over one
solar rotation because all of the IPS data in a given Carrington rotation are used to produce a V-map. For the purpose of solar wind forecast it is necessary to detect not
only a co-rotational structure but also a slowly/quickly
varying one. In this study we employed a modified version of the tomographic procedure (hereafter we simply
call it CAT2 ) that reduces the length of time averaging
which shows good coincidence between IPS and in situ
measurements even during solar maximum when solar
wind structure changes rapidly [4]. In the forecast procedure, a few tens of IPS observations are included in
the CAT2 algorithm to update the solar wind structure
inside of Earth’s orbit each day (Fig.1(b)). Then, forecast plots, that is, time profiles of the solar wind velocity
at the Earth position for several future days, are drawn.
These results are compared with in situ measurements toevaluate the accuracy of the solar wind forecast.
Here we describe the process of the forecast. The
LOSs observed in the last solar rotation are mapped on
the surface at a 1 AU distance from the Sun. After theCP1216, Twelfth International Solar Wind Conference,edited by M. Maksimovic, K. Issautier, N. Meyer-Vernet, M. Moncuquet, and F. Pantellini© 2010 American Institute of Physics 978-0-7354-0759-6/10/$30.00

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第 4 章 論文集

320

FIGURE 1. (a) Schematic view of the IPS forecast. Solid and dotted long arrows display the LOS. The LOS passing througheast Sun (solid lines) contain the information of solar wind being approaching the Earth. (b) IPS Tomographic map on October 13,
2005,deconvolved using data from the recent single rotation (grey map) and the LOSs obtained from the forecast date (solid line).
White area is data gap where the solar wind velocity cannot be deconvolved because of an insufficient LOS number. Day = 0 on thehorizontal axis corresponds to the earth’s longitude.

CAT analysis a tomographic V-map is determined for±180◦ centered at the Earth’s longitude (Fig.1(b)). The
east side (left hand side) of the V-map corresponds to the
future and a prediction length is defined as a longitudinal
width measured along the earth’s orbit where the solar
wind velocity is successfully deconvolved. In this case,the prediction length is about 7 days. Figure 2 shows avariation of the prediction length during October 2005.
Using this technique a forecast is always possible fivedays in the future, and it sometimes exceeds ten days. A
sampling of the solar wind velocity along the Earth’s orbit to the future side of the V-map produces the forecast
plot shown in figure 3. The STELab-IPS forecast velocity
increases up to 600km/s 2 days later and a similar velocity change is measured by the ACE satellite (Fig.3(a)).
Here we note that the spatial resolution of the V-map is
10◦ . Therefore it is impossible to use the CAT2 analysis
to reproduce the velocity jump detected by ACE less than
1 day in advance.
The CAT analysis is an iterative computing techniquethat minimizes the difference between a velocity distribution model and the velocity determined along each
LOS. Therefore the V-map always contains residual error. The residual error is suppressed where there areenough LOS crossing numbers, i.e. on the right-handside of the V-map. On the other hand, the residual error
appears to increase on the left-hand side which is used
for the forecast. A typical plot of the residual error isshown in figure 4. The residual error averaged over October 2005 is smaller than 20 km/s within a 4 day forecast
interval and then it rapidly increases as much as 80 km/s
beyond this. The increase of the residual error depends
only on the coverage of the LOS. The forecast is provided every four hours and forecast errors are defined as

FIGURE 2.

Prediction length during October 2005.

a standard deviation of the difference between forecasts
and ACE measurements by using data within 5 days as
follows,
2∑5Day=0 (VIPS −VACE )
σ=
,
(1)
N
here N is the number of times forecast. Figure 5 shows
the forecast errors and correlation coefficients between
the forecast and ACE measurements during October
2005. The difference and correlation coefficient are antiphased. Therefore we use the standard deviation as theerror in following evaluation the forecast accuracy.

RESULT
In this section, we evaluate the forecast errors during
three different periods of solar activity. Figure 6 shows

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Fujiki et al., “Solar Wind Forecast by using IPS Observations” (2010)

321

FIGURE 3. Good and bad examples of the IPS solar wind forecast. Black circles are the STELab-IPS forecast and the grey
circles are ACE measurements. Day=0 corresponds to the day the forecast is made.

IPS CAT2 is determined from 5 -10 days of observations
before the date the forecast is determined (Fig 1(b)).
Although an occurrence rate of CMEs during solar
maximum and the declining phase of the solar cycle are
one order of magnitude higher than that during solar minimum, a correlation between the forecast error and number of CMEs is not evident. On the other hand the error mildly depends on the number of LOS. A period of
decreasing LOS number corresponds to a period of thelarge forecast error. Roughly the forecast error tends toincrease when the number LOS summed past 7 days issmaller than 150 (case 1), i.e. around DOY 240 in 2005,
and when the number of LOS is decreasing (case 2). In
case 1, it is easy to explain that when the number of LOS
are smaller than 150 the data are not sufficient to deconvolve the solar wind structure. In case 2, the decreasing
number of LOS indicates that computed velocity distribution in the CAT2 analysis is deconvolved with a higher
weighting in the past and lower weighting in the future.
That the velocity in the future is strongly affected by thatof the past is understandable. In this case a varying transient structure blurs the co-rotational analysis.

FIGURE 5.

Error ( km / s )Correlation Coefficient

FIGURE 4. Residual error in the IPS tomographic procedure
averaged over October 2005.

SUMMARY
We have been challenging a forecast of the solar windby using STELab-IPS data applying the modified versionof computer assisted tomography. In this paper we evaluated the forecast errors which are attributed to the tomographic analysis and the solar activity. The real-time
forecast is simulated by using the data obtained in 2000
(solar maximum), 2002 (declining phase), and 2005 (solar maximum).
The prediction forecast determination is always atleast 5 days and sometimes exceeds 10 days during thesethree periods. However the residual errors of the CAT2
rapidly increases for forecasts longer than 4 days because

Correlation and difference during October 2005.

a variation of the forecast error from plots of two parameters, the LOS number and the CME number. The former is a proxy for the observation condition and latter a
proxy of solar transient activity. The velocity distribution
on the forecast side of the plot obtained by the STELab-665

第 4 章 論文集

Error ( km / s )Error ( km / s )Error ( km / s )322

FIGURE 6. Forecast error in different periods of solar activity (black circle); in 2000 (solar maximum; top panel), 2002 (thedeclining phase of the solar cycle; middle panel), and 2005 (at solar minimum; bottom panel). The error is smoothed over ± 1 day.
Red circles are a running sum (past 7 days) of the number of LOS whose P-point (the closest approach to the Sun) locates ±30◦
in latitude. Blue circles are a running sum (past 7 days) of number of CMEs as a proxy of solar transient activity. STELab-IPS
observations were stopped from August to September 2000 because of lightening damage.

ACKNOWLEDGMENTS

the redundancy of LOS is not sufficient to deconvolve thesolar wind structure in such regions (Fig 1(b)). Thus weconclude that the effective prediction length of the solar
wind forecast by IPS is 4 days. Additionally, the forecast error mildly depends on the number of LOS and its
variation. Successive observations over several days are
essentially important for the forecast.
We have evaluated the error from solar activity. According to figure 6, the forecast error does not depend
much on the number of CMEs which we use here asa proxy for solar transient phenomena. This result indicates that solar activity does not adversely affect the accuracy of the solar wind forecast. However, a verificationof the forecast during CME using an event study remains
to be done.
Finally we note that IPS observations with a newantenna (the Solar Wind Imaging Facility; SWIFT) has
began on September 2008, and this allows us to observeat least twice the number of radio sources per day. We
plan to upgrade the backends and feeds of the existing
old antennas to achieve simultaneous observations andstart these observations within two years.

This work was supported by the Grant-in-Aid for Creative Scientific Research "The Basic Study of Space
Weather Prediction" (17GS0208, Head Investigator: K.
Shibata) from the Ministry of Education, Culture, Sports,
Science, and Technology of Japan. We wish to thank the
ACE Science Center that provides the solar wind plasma
data collected by the ACE spacecraft. The CME catalog
which we used in this study is generated and maintainedat the CDAW Data Center by NASA and The Catholic
University of America in cooperation with the Naval Research Laboratory. SOHO is an international cooperative
project between ESA and NASA.

REFERENCES
1. A. Hewish, P. F. Scott, and D. Wills, Nature 203, 1214–1217(1964).
2. M. Kojima, and T. Kakinuma, J. Geophys. Res. 92,
7269–7279 (1987).
3. M. Kojima, M. Tokumaru, H. Watanabe, A. Yokobe,
K. Asai, B. V. Jackson, and P. L. Hick, J. Geophys. Res.
103, 1981–1990 (1998).
4. K. Fujiki, M. Kojima, M. Tokumaru, T. Ohmi, A. Yokobe,
K. Hayashi, D. J. McComas, and H. A. Elliott, Annales
Geophysicae 21, 1257–1261 (2003).

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Omura et al., “Generation Process of Chorus Emissions” (2008)
Click
Here

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A04223, doi:10.1029/2007JA012622, 2008

for

Full
ArticleTheory and simulation of the generation of whistler-mode chorus
Yoshiharu Omura,1 Yuto Katoh,1,2 and Danny Summers1,3
Received 2 July 2007; revised 5 November 2007; accepted 9 January 2008; published 25 April 2008.

[1] The generation process of whistler-mode chorus emissions is analyzed by both theory
and simulation. Driven by an assumed strong temperature anisotropy of energetic
electrons, the initial wave growth of chorus is linear. After the linear growth phase, thewave amplitude grows nonlinearly. It is found that the seeds of chorus emissions withrising frequency are generated near the magnetic equator as a result of a nonlinear
growth mechanism that depends on the wave amplitude. We derive the relativistic
second-order resonance condition for a whistler-mode wave with a varying frequency.
Wave trapping of resonant electrons near the equator results in the formation of anelectromagnetic electron hole in the wave phase space. For a specific wave phasevariation, corresponding to a rising frequency, the electron hole can form a resonantcurrent that causes growth of a wave with a rising frequency. Seeds of chorus elements
grow from the saturation level of the whistler-mode instability at the equator and thenpropagate away from the equator. In the frame of reference moving with the groupvelocity, the wave frequency is constant. The wave amplitude is amplified by the nonlinear
resonant current, which is sustained by the increasing inhomogeneity of the dipole
magnetic field over some distance from the equator. Chorus elements are generated
successively at the equator so long as a sufficient flux of energetic electrons with a strong
temperature anisotropy is present.
Citation: Omura, Y., Y. Katoh, and D. Summers (2008), Theory and simulation of the generation of whistler-mode chorus,
J. Geophys. Res., 113, A04223, doi:10.1029/2007JA012622.

1. Introduction[2] Whistler-mode chorus is a common form of very low
frequency (VLF) emission in the Earth’s magnetosphere andother planetary magnetospheres. Chorus typically comprises
repeated narrowband monochromatic signals of short duration (0.1 s) with rising frequency and occurs at frequencies
below the local electron gyrofrequency. There have been
extensive observations of magnetospheric chorus waves[e.g., see Helliwell, 1965; Tsurutani and Smith, 1974; Burtis
and Helliwell, 1976; Anderson and Kurth, 1989; Sazhin andHayakawa, 1992; Lauben et al., 1998; Meredith et al., 2001;
Santolik et al., 2003, 2004, 2005; Kasahara et al., 2005;
Chum et al., 2007, and references therein]. Gyroresonantinteractions of electrons with chorus play an important rolein the physics of the Earth’s radiation belts. In particular,energy diffusion due to electron gyroresonance with chorus
is a key mechanism for generating relativistic electrons in theregion 3 < L < 5 outside the plasmasphere during geomagnetic storms or periods of prolonged substorm activity[Summers et al., 1998, 2002, 2004a, 2004b, 2007a, 2007b;
1
Research Institute for Sustainable Humanosphere, Kyoto University,
Kyoto, Japan.
2
Planetary Plasma and Atmospheric Research Center, Graduate Schoolof Science, Tohoku University, Miyagi, Japan.
3
Department of Mathematics and Statistics, Memorial University of
Newfoundland, St. John’s, Newfoundland, Canada.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012622$09.00

Roth et al., 1999; Summers and Ma, 2000; Albert, 2002;
Miyoshi et al., 2003; Katoh and Omura, 2004; Summers,
2005; Horne et al., 2005; Varotsou et al., 2005; Shprits et al.,2006; Thorne et al., 2007]. Gyroresonant pitch-angle scattering of electrons by chorus can lead to significant precipitation into the atmosphere and net loss of energetic electrons
from the outer radiation belt [Lorentzen et al., 2001; O’Brienet al., 2004; Summers et al., 2005; Thorne et al., 2005].
[3] The processes governing the generation mechanism ofchorus emissions have been analyzed in a number of studies[Nunn, 1974; Omura and Matsumoto, 1982; Trakhtengerts,
1995, 1999; Nunn et al., 1997; Katoh and Omura, 2006a,
2006b, 2007a; Chum et al., 2007]. It is generally acceptedthat chorus waves can be excited by cyclotron resonance withanisotropic 10 – 100 keV electrons. Inhomogeneity in the
background magnetic field has been established as a controlling factor in the chorus generation mechanism. Nevertheless, the detailed generation process of chorus emissions
remains to be fully identified. Nunn [1974] and Nunn et al.
[1997] simulated the generation process of chorus emissions
by a Vlasov-hybrid simulation using a coherent wave as aseed of a chorus element. Omura and Matsumoto [1982]
performed simulations of whistler-mode wave-electroninteractions demonstrating the basic process of wave amplification via nonlinear resonant currents. Trakhtengerts[1995, 1999] proposed a generation mechanism of chorus
based on a model of a backward wave oscillator andnonlinear wave growth. In all of these previous studies, acoherent wave is assumed as a seed of a chorus element,

A04223

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A04223OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

while the whistler-mode instability driven by temperature
anisotropy gives a spectrum of waves that have positive
growth rates [Omura and Summers, 2004]. The essential
problem is how the coherent chorus elements with variablefrequency are excited in the equatorial region of the magnetosphere where the inhomogeneity of the magnetic field isvery small. As yet, it has not been clarified what determines
the frequency sweep rate of a chorus element. This is a key
issue in the present study. Omura and Summers [2006]
analyzed electron interactions with a coherent whistler-mode
wave generated in the magnetic equatorial plane and discussed the formation of resonant currents due to an electromagnetic electron hole in velocity phase space. Electromagnetic
electron holes play an essential role in our investigation.
[4] Recent spacecraft observations [e.g., Parrot et al.,
2003; Santolik et al., 2003, 2004; Breneman et al., 2007;
Chum et al., 2007] provide important clues for the generation process of chorus emissions. The generation region isfound to be in close proximity to the magnetic equator. In
situ observations of the wave amplitudes confirm that
amplitudes are sufficiently large for nonlinear trapping ofparticles to be significant. Recent analysis of chorus element
observations by Chum et al. [2007] shows that chorus
elements with rising frequencies are radiated from a nonmoving source near the geomagnetic equator. There is also amodel that assumes rapidly moving sources to interpretchorus emissions observed by the Custer satellites [Inan etal., 2004].
[5] In the simulation study by Katoh and Omura [2007a]using an electron hybrid code, it was demonstrated thatchorus emissions with a rising tone can be generated fromthermal noise in the region close to the equator. Nonlinear
wave trapping of resonant electrons becomes predominantnear the equator, and the velocity distribution of theelectrons is strongly modified, thereby giving rise to theresonant current. The trapping dynamics of resonant electrons are described by the first-order and second-orderresonance conditions. Further recent studies show that afraction of the resonant electrons are subject to acceleration
to relativistic energies [Omura et al., 2007; Katoh andOmura, 2007b]. Relativistic effects must therefore be incorporated in descriptions of the electron dynamics. Albert[2000] studied the interaction of relativistic resonant electrons with a monochromatic wave. We derive in the present
paper, for the first time, the relativistic second-order resonance condition for a chorus element with variable frequency.
We regard this condition as fundamental to the analysis of thechorus generation mechanism.
[6] The evolution of a whistler-mode wave is describedby wave equations that are based on Maxwell’s equations.
In past studies [Nunn, 1974; Omura et al., 1991], thedisplacement current has been neglected under the assumption that the electron plasma frequency is much greater thanthe electron cyclotron frequency. However, the electron
plasma frequency can be close to the electron cyclotron
frequency under disturbed magnetospheric conditions, especially in the region just outside the plasmapause [e.g.,
Sheeley et al., 2001]. Retaining the contribution of thedisplacement current, we derive a set of wave equations
describing the evolution of the wave amplitude and wave
phase, details of which are given in Appendix A. We study
the nonlinear dynamics of resonant electrons based on theA04223

exact second-order resonance condition which we derive insection 2. In section 3 we find the condition for the
maximum resonant current that modifies the wave amplitude. The relation between the frequency sweep rate andamplitude of chorus emissions in the generation region is
obtained in section 4. In section 5, we analyze the simulation data developed by Katoh and Omura [2007a, 2007b] todetermine the exact chorus wave amplitude in the source
region near the equator. We compare the simulation results
for the wave amplitude with our theoretical analysis.
Finally, in section 6 we discuss our results and present
the conclusions.

2. Resonance Conditions for Chorus-ElectronInteraction
2.1. Basic Formulae Based on the Cold Plasma
Dispersion Relation[7] As a preparation for our main calculation, we derivesome useful expressions for the phase velocity and groupvelocity of a whistler-mode wave propagating parallel tothe static magnetic field from a cold plasma dispersion
equation, namely, the relation between the wave frequency
w and the wave number k in a cold plasma [e.g., Stix,1992],
c2 k 2 ¼ w 2 þ

2
wwpe
:
We  w

ð1Þ

Here, c, wpe, and We are the speed of light, electron plasma
frequency, and electron cyclotron frequency, respectively.
For simplicity, the plasma frequency is assumed to beconstant, while the cyclotron frequency varies as a functionof position h, the distance along the magnetic field line of
the assumed dipole magnetic field measured from themagnetic equator.
[8] We introduce the dimensionless parameters d and x[Omura et al., 2007] defined by
w2
d2 ¼ 1  2 2 ;
c k

ð2Þ

wðWe  wÞ
:
w2pe

ð3Þ

and
x2 ¼

The whistler-mode dispersion equation (1) is rewritten as
d2 ¼

1
:
1 þ x2

ð4Þ

For typical conditions in the Earth’s outer radiation belt,
x  1 and d  1, which makes it possible to neglect thefirst term on the right-hand side of (1), although we
retain the term for completeness. The wave phasevelocity Vp is given by

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w¼ cdx:kð5ÞOmura et al., “Generation Process of Chorus Emissions” (2008)
OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

A04223Differentiating (1) with respect to w, we obtain2c 2 k2
We wpe@k:¼ 2w þ
@w
ðWe  wÞ2

ð6Þ

Solving (6) for the group velocity Vg = @w/@k, andmaking use of the parameters d and x, we find
Vg ¼


1
cx 2
We
:
x þ
d
2ðWe  wÞ

ð7Þ

[9] We assume a coherent whistler-mode wave withamplitude Bw and phase angle y. We further assume that
Bw is a slowly varying function of position h and time t andthat the phase y varies smoothly in space and time. The
frequency w and the wave number k are defined respectively
by w = @y/@t and k = @y/@h. In addition, we assume thatw and k are slowly varying functions of position h and timet. Since we can change the order of differentiation in thesecond-order derivative of y by h and t, i.e., @ 2y/@h@t, it
follows from the definitions of w and k that
@k
@w
¼
:
@t
@h

"
#
2
We wpe@k@w
2c k¼ 2w þ:@tðWe  wÞ2 @t
2

ð10Þ

ð11Þ

Using (7) and (8), we hence obtain from (11) an equation
for the wave number k in the form,
@k
@k
dw
@We
¼ Vg1:@h@t 2cx ðWe  wÞ @hUsing (10), we have the time variation of w as observed by
the particle,
dw @w
@w
¼
þ vk
¼
dt
@t
@h

ð12Þ

2.2. Resonance Velocity Seen by a Particle[12] We assume that an electron moves with a parallel
velocity vk and a perpendicular velocity v?. We define theresonance velocity VR that satisfies the Doppler cyclotron
resonance condition,


1
We
w
;
k
g


1


vk @w
:
Vg @t

ð15Þ

Noting that We is a simple function of h and that g is afunction of t as a property of the particle, we find 
vk @We We dg
d We
¼
 2
:
g @h
g dt
dt g

ð16Þ

Using (8), (10), and (12), we obtain the time variation of k
as seen by the the particle,
dk
¼
dt



vk @k
vk wd
@We
:
1

Vg @t 2cxðWe  wÞ @h
vk @w vk
dVR We dg 1
VR
1
¼ 2 þ
1

dt
kg dt kVg
Vg @t kg2d ðWe  gwÞ @We 1þ
;2ðWe  wÞ @h

ð17Þ

ð18Þ

where (5) and (13) have also been used.
2.3. Relativistic Second-Order Resonance Condition[13] The relativistic cyclotron resonance condition for awhistler-mode wave with a fixed frequency has recently
been analyzed by Omura et al. [2007]. We extend theiranalysis to the interaction with a chorus wave with varying
frequency. The variation of the kinetic energy K of aresonant particle is given by

Equations (8), (10), and (12) are key results that will be
used below.

VR ¼

ð14Þ

Substituting (15), (16), and (17) into (14), we derive theresult,[11] We then differentiate both sides of (1) with respect toposition h, to give at a specific time t the relation,"#
2
2
We wpe
wwpe
@k
@w
@We
:
¼ 2w þ2@hðWe  wÞ @h ðWe  wÞ2 @h  dVR 1 dw 1 d We
We 1 dk
¼
 w

:
dt
g k 2 dt
k dt k dt g

ð9Þ

From (7), (8), and (9), we find that
@w
@w
þ Vg
¼ 0:
@t
@h

A04223where g = [1  (v2k + v2?)/c2]1/2. If vk = VR, the electron
resonates with the wave and ‘‘sees’’ a constant phase of the
wave field. It is noted that we only have the single
cyclotron resonance for purely transverse waves propagating parallel to the static magnetic field. We now calculate
the time variation of VR as seen by the electron. We assume
that the wave fields are uniform in the transverse direction
and that the gyration of the electron does not cause any
variation of w or k or the cyclotron frequency We. From(13), we haveð8Þ[10] We differentiate both sides of (1) with respect to time
t at a fixed position h to obtain

2c2 k

325

dK
¼ eEw v? sin z;
dt

ð19Þ

where e, Ew, v?, and z are the electron charge, wave
electric field amplitude, perpendicular velocity, and angle
between the wave magnetic field Bw and the perpendicularvelocity v?, respectively. In the present case of field-aligned
propagation, the wave electric field Ew is perpendicular toBw, with Ew and Bw both lying in the plane transverse toð13Þ
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A04223OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

B0. From Maxwell’s induction equation, it follows that Ew /Bw = w/k. Noting also that K = m0c2(g  1), where m0 is
the electron rest mass, we find from (19) thatdg Ww wv? sin z
;
¼
dt
kc2

ð20Þ

where Ww = eBw/m0.
[14] The parallel component of the relativistic equation of
motion is given byd g vke v?  Bw þ B0r ;
¼
m0
dt

ð21Þ

where B0r is a radial component of the static magnetic field
B0, andr @B0
B0r ¼  c
:
2 @h

ð22Þ

Result (22) is derived by expanding B0 around the center ofcylotron motion in circular cylindrical coordinates. Notingthat the cyclotron radius rc is given by rc = gv?/We, we
obtain from (21) and (22) the equation [Dysthe, 1971],
 
d gvk
gv2 @We¼ Ww v? sin z  ?
:dt
2We @hð23Þ

The equation (23) describes the acceleration due to the wave
field and the mirror force. We then obtain the time
derivative of vk in the form
dvk Ww v? sin z vk dg
v 2 @We
¼
:

 ?
dt
g dt 2We @hgð24Þ[15] We assume that the first-order resonance condition,dz¼ k vk  VR ¼ 0;
dt

A04223

particles [Omura et al., 1991]. The parameter S is theinhomogeneity ratio given by( 
1
VR 2 @w
S ¼ 2 2 g 1
@t
Vg
wt d

 

2
kgv?2
d We  gw
@We
þ
 1þ
VR
:
2We
2 We  w@hð28Þ

It is noted that S, wt, and g are functions of v?. Expression(28) reduces to equation (11) of Omura et al. [2007] fora constant-frequency wave, and to equation (11) in thereview by Omura et al. [1991] for the nonrelativistic(g = 1) case. S is a ratio of a inhomogeneity term to theoscillatory term in the pendulum equation (S = 0). When
jSj > 1, there is no equilibrium state. When jSj < 1, there isa small amplitude solution that oscillates with the modified
trapping frequency wtr = wtdg 1/2. The factor dg 1/2 is dueto the relativistic effect.
[16] When a chorus element with a rising frequency(@w/@t > 0) propagates northward away from the magnetic
equator and interacts with resonant electrons moving in the
opposite direction, S takes a negative value. The fully
relativistic expression (28) for the inhomogeneity ratio fora parallel-propagating whistler-mode wave with variablefrequency is the single most important result in this paper.

3. Formation of the Nonlinear Resonant Current
at the Equator[17] The evolution of the wave amplitude of a chorus
element can be described by a wave equation derived fromMaxwell’s equations. Retaining the displacement current,
which has been neglected in previous studies of whistlermode waves, we obtain the wave equation describing
evolution of the wave amplitude BW
m Vg
@Bw
@Bw
þ Vg
¼  0 JE ;
@t
@h
2

ð25Þ

ð29Þ

and the dispersion relation with a nonlinear correction term,is satisfied, and then the second-order derivative of thephase z is given by

d2 z
d
¼k
vk  VR ;
2
dt
dt

ð26Þ

where we make use of the first-order resonance condition(26) to eliminate the derivative of k. Substituting (18),(20), and (24) into (26), we obtain the second-ordernonlinear ordinary differential equation for the phase anglez,
d2 z wt2 d 2ðsin z þ S Þ:
¼
dt 2
g

c2 k 2  w2 

ð30Þ

where m0 is the magnetic permittivity in vacuum. Details of
the derivation of (29) and (30) are provided in Appendix A.
The currents JE and JB are components of the resonantcurrent, parallel to the wave electric field Ew and wave
magnetic field Bw, respectively. The resonant current isformed by the interaction of resonant electrons with thewave. Expressions for JE and JB are

ð27Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Here, wt is the trapping frequency given by wt = kv? Ww ,and d 2 comes from the relativistic effect. If we use thenonrelativistic equations of motion, we set d = 1 and g = 1
in (27) to obtain a simplified equation for the resonant

2
wwpe
JB
;
¼ m0 c2 k
We  w
Bw

JE ¼

Z 1 Z 2p Z 1
0

0

1



½ev? sin z  f vk ; z; v? v? dvk dzdv? ;

ð31Þ

and
JB ¼

Z 1 Z 2p Z 1
0

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327

OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

A04223

A04223where f (vk, z, v?) is the distribution function of energetic
electrons. It is noted that the cold electrons supporting thewhistler-mode waves are not included in the distributionfunction. To evaluate the contribution of the term involving
JB/BW in (30), we assume that jJBj  jJEj. We justify thisassumption below. Equation (29) describes the growth of
the wave amplitude Bw. We can alternatively writedBw¼ wi Bw ;dt

ð33Þ

where wi is the wave growth rate. Equating the right-handsides of (29) and (33), we obtain
JE
2wi
¼
:
Bw
m0 Vg

ð34Þ

Setting jJEj = jJBj in (34), we determine the magnitude of
the right-hand side of (30) as"#
2
We wpeJB
m0 c k
wwi :
¼ 2þ
Bw
wðWe  wÞ22

ð35Þ

where wtr = wtdg 1/2, and we have assumed that vk  VR,
i.e., q  0. Eliminating the time variable t from (39) and(40) gives

Substituting (35) into (30), we obtain the result,c2 k 2 ¼ w2 1

2wi
w


þ

2 
wwpe
1
We  w
We wi:
We  w w

ð37Þ

We then have2
JE ¼ eV?0

Z 2p Z 10

1

g vk ; z sin zdvk dz:

q dq ¼ wtr2 ðsin z þ S Þ dz:

ð36Þ

When wi  w, equation (36) reduces to the cold plasma
dispersion relation for a whistler-mode wave (1). Thus, thederivation of the expression for the inhomogeneity ratio(28) is valid so long as the condition wi  w is satisfied.
[18] For simplicity of the analysis, we neglect the variation of the perpendicular velocity v?. We assume a delta
function for the velocity distribution of resonant electrons at
an average value V?0, i.e.,
f vk ; z; v? ¼ g vk ; z dðv?  V?0 Þ:

Figure 1. Trajectories of resonant electrons in the (q  z)phase space for the inhomogeneity ratio S = 0.4. The
phase angle z 0 is the center of trapping motion, while z 1 andz 2 indicate the boundaries of the trapping region at q = 0.

ð38ÞFrom (34) and (38) it follows that the phase angles z ofresonant electrons determine the rate of wave growth anddamping.
[19] Let us examine the dynamics of a resonant electron
as described by equation (27) [Dysthe, 1971; Vomvoridis
and Denavit, 1979; Matsumoto and Omura, 1981]. Introducing the variable q = k (vk  VR), we rewrite (27) in the
form,
dz
¼ q;
dt

ð39Þ

dq
¼ w2tr ðsin z þ S Þ;
dt

ð40Þ

and

ð41Þ

Integrating (41), we obtain the equation for the trajectories
of electrons in (q, z) phase space as
q2 þ 2wtr2 ðcos z  Sz Þ ¼ C;

ð42Þwhere C is a constant. In Figure 1, we plot the family ofelectron trajectories for S = 0.4. The first-order andsecond-order resonance conditions q = 0 and dq/dt = 0 are
satisfied at two different phases z 0 and z 1 that satisfy sin z +S = 0. At q = 0, the phase z 0 represents a stable equilibrium
point around which trapped resonant electrons rotate intime. The phase z 1 represents an unstable saddle equilibrium point. The separatrix of the trapping region originate
from this saddle point. Substituting the saddle point values(q = 0, z = z 1) into (42), we obtain the equation for the
separatrix,qs ðz Þ ¼

wtr

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2½cos z 1  cos z þ S ðz  z 1 Þ;ð43Þ

for z 1 < z < z 2, where z 2 is the phase of the separatrix
crossing q = 0. The separatrix boundary of the trapping
region plays an important role in the formation of theresonant current JE, as reported by Omura and Summers[2006]. Under the action of a slowly varying whistler-mode
wave packet, there develops an electromagnetic electron
hole in the phase space (vk, z). This is because the initiallytrapped resonant electrons inside the trapping region are
guided along the resonance velocity, while the untrappedelectrons outside the trapping region cannot cross theseparatrix so far as the structure of the trapping region
remains constant. Owing to the variation in Bw , v?, and thegradient of the static magnetic field, however, the values of
S and wtr change. When the absolute value of S decreases,

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A04223currents JE/J0 and JB/J0 as functions of S for 1 < S < 0.
The current JE/J0 takes the maximum value 0.975 at
S = 0.413. As we find in (29), a negative value of JE
contributes to wave growth, while JB only modifies thedispersion relation. Since the maximum value of JB is
only twice the maximum value of JE, the assumption
jJBj = jJEj we made in deriving (36) is justified as a crude
estimation of order of magnitude. We can conclude from(36) that the whistler-mode dispersion relation remains
linear so long as the growth rate wi is much smaller than
the frequency w.

4. Nonlinear Wave Growth

Figure 2. The magnitudes of the resonant currents JEand JB, normalized by J0, as functions of the inhomogeneity ratio S.
the range of z of the trapping region increases. When thefrequency wtr increases, the range of q of the trapping region
increases. Because of such an enlargement of the trapping
region, there occurs entrapping of resonant electrons near
the separatrix. Therefore, the electron hole can be filledwith some newly trapped electrons, although the number oftrapped electrons is usually smaller than the numberof untrapped resonant electrons. The difference in densitiesof the trapped and untrapped resonant electrons results inphase bunching of resonant electrons, which forms aresonant current.
[20] Let us assume that the total distribution function after
a certain period of resonant interaction is given by




 
g vk ; z ¼ g0 vk  gt vk ; z ;2
JE ¼ eV?0

0

1



SEQ ¼ 

gt vk ; z sin zdvk dz:

ð45Þ



@w
w 2 d2VR 2:
1
¼ 0:4 tg
Vg
@t

JE ¼ J0

1=2½cos z 1  cos z þ S ðz  z 1 Þ

sin zdz;

ð46Þ
@w 0:4d V?0 w
VR 2 Bw 2
1
W ;
¼
@t
g x c WeVg
B0 e

where J0 = (2e) d(m0kg)
have
JB ¼ J0

Z z2

1/2

V5/2
?0

GB1/2
w .

1=2



We
VR ¼ cdx 1 ;gw½cos z 1  cos z þ S ðz  z 1 Þcos zdz:

ð47Þ

We have evaluated the integrals in (46) and (47)numerically, and in Figure 2 we plot the dimensionless

ð50Þ

ð51Þ

and, from (7), we obtain a convenient formula for evaluating
the right-hand side of (50) in the form
1

Similarly, we

z1

ð49Þ

where, from (5) and (13),

z1

3/2

ð48Þ

Using w2t = kV?0Ww, we rewrite (49) as[21] Assuming that gt (vk, z) = G (= constant) inside thetrapping region and that gt (vk, z) = 0 outside it, we rewrite(45) asZ z2

g
VR 2 @w1:@tVg
wt2 d2

Setting SEQ = 0.4 in (48), we obtain a condition for the
maximum nonlinear growth of the wave asð44Þ

where g0(vk) is the unperturbed velocity distributionfunction, and gt (vk, z) is the distribution for the initiallytrapped electrons. Then, substituting (44) into (38), we haveZ 2p Z 1[22] As expressed by equation (29), a negative value of
JE produces wave growth. From Figure 2, we find that JE
maximizes at S = 0.41. Wave growth is due to thenonlinear dynamics of resonant electrons under the actionof a large-amplitude wave. According to linear theory, atthe equator there is a finite spectrum of whistler-mode
waves being generated by the temperature anisotropy of theresonant electrons. From these waves with different rates ofphase variation, a coherent wave can develop with a phasevariation @w/@t that maximizes the value of JE. Higher
wave amplitudes are thereby achieved. The development ofsuch coherent waves will be demonstrated by a selfconsistent simulation in section 5. By setting @We /@h = 0
in (28), the inhomogeneity ratio S at the equator is given by




VR
We
We
¼ 1 þ d2 x2 þ
1 :Vg
2ðWe  wÞ gw

ð52Þ

[23] As described by equation (29), the wave with a phasesatisfying the required frequency variation grows due to thenegative value of JE; SEQ then becomes smaller as the wave
grows. However, the second term on the right-hand side of(28) starts to contribute to the inhomogeneity ratio S
because of the increasing value of @We /@h as the wave

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A04223Figure 3. Dynamic spectra of the wave electric field (Ey , Ez) observed at different positions along thefield line. (a) Northern hemisphere: h = 4, 43, 82 cWe01. (b) Southern hemisphere: h = 5,  44, 83
cWe01.
packet propagates away from the equator. Thus, nonlinear
growth can continue over some distance from the equator.
[24] We now compare contributions to S of the spatial
inhomogeneity of the cyclotron frequency and the time
variation of the wave frequency. We rewrite (28) as


1
@w
@We
S ¼  2 2 s1
þ cs2
@h
@t
wt d

ð53Þ

and
s2 ¼


 1T2 cs2 @We
@w
¼
:
@t
T1
s1 @h
ð54Þ

ð55Þ

Comparing the first term (T1) and the second term (T2) of(28), we have

where


VR 2
s1 ¼ g 1 
Vg

 
 

g
w
V?0 2
d 2 ðWe  gwÞ VR
 1þ
:
2xd We2ðWe  wÞ cc

ð56ÞAssuming that T1/T2 = 1 at h = hc and that We(h) =We0(1 + ah2), we obtain the distance hc which represents the

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A04223transition at which the effects of the time variation of w andthe spatial variation of We are equal. We find that
hc ¼

 
1
s1 @w
;
2acWe0 s2 @t

ð57Þ

where a = 4.5/(LR E)2, corresponding to the Earth’s
approximate dipole magnetic field.
[25] If the seed of a chorus element is formed at theequator due to nonlinear growth, the inhomogeneity ratio S
takes the value 0.4 at the equator. The corresponding
frequency variation @w/@t, as expressed in (10), propagateswith the group velocity Vg. At the same time, the amplitude
Bw grows as a result of the negative nonlinear current JE, as
described by (29). In the frame of reference moving with thegroup velocity Vg, the frequency is unchanged while thewave amplitude grows. At the distance h = hc, the total
inhomogeneity, expressed by T1 + T2, doubles, while from(53) the inhomogeneity ratio S is inversely proportional to
the wave amplitude Bw. The maximum nonlinear wave
growth with S = 0.4 is sustained over h = 0  hc if the
wave growth makes the wave amplitude at h = hc twice aslarge as its equatorial value, namely, Bw(hc) = 2Bw(0).

5. Self-Consistent Particle Simulation[26] We have recently reproduced rising tones of whistlermode chorus emissions by a self-consistent particle simulation [Katoh and Omura, 2007a]. In the simulation we used
an electron hybrid code with a dipole magnetic field model.
We further assumed a spatially one-dimensional simulation
system aligned with the external magnetic field direction.
We performed detailed analyses of a simulation run that is
described by Katoh and Omura [2007a, 2007b]. Althoughdetailed parameters are found in the papers, we give somedescription of important parameters in the following. The
size of the simulation system, consisting of 16,384 grid4
points, and the simulation time are 983 cW1e0 and 4  10
W1
,
where
W
is
the
electrongyrofrequency
at
the
e0
e0
magnetic equator. These values respectively correspond to
the spatial scale 3350 km and time 455 ms for a backgroundmagnetic field intensity B0 at the magnetic equator of 500 nT
at L = 4. For the initial settings of the simulation, we loadedenergetic electrons, represented by 70 million superparticles, to form a highly anisotropic velocity distribution in
the equatorial region. The velocity distribution at theequator is assumed to be a Maxwellian in vk and a loss
cone in v?, and is assumed to have a high temperature
anisotropy. The plasma frequency of the cold electrons at
the equator is 4We0.
[27] A band of whistler-mode waves is generated in theearly stage of the simulation run at the equator through an
instability driven by the temperature anisotropy of theenergetic electrons. As shown in Figure 3, the electron
hybrid code produces successive formations of rising tone
elements propagating away from the equator. The panels in
Figure 3a and Figure 3b show the dynamic spectra atdifferent locations near the equator. Although the intensities
are relatively small at the equator (Figures 3a and 3b, top),chorus elements are formed at the equator and are amplified
as they propagate away from the equator. The averaged
frequency sweep rate of chorus emissions in the simulation

Figure 4. Spatial profiles of jJEj (solid line), jJBj(dashedline), and Bw (lower panel) at t = 20000 We01.
is estimated to be 5.0  105 W2e , corresponding to 62 kHz/s.
(In the work of Katoh and Omura [2007a] the frequency
sweep-rate was incorrectly given as 10.9 kHz/s.) On thebasis of the theoretical analyses given in the previous
sections, we discuss quantitatively the generation processof chorus emissions in the simulation results. Figure 4
shows the magnitude of the resonant currents JE and JB
1near the equator (h = 5  5 cW1e0 ) at t = 20,000 We0 . As
shown in Figure 3, the seeds of chorus emissions are
generated by the current JE, and the component JB takes
similar values to JE. Thus, the assumption jJEj = jJBj that we
made in deriving (35) is justified.
[28] Figure 5a shows the time history of the averaged
wave magnetic field in the equatorial region of the simulation for the whole simulation run. The solid line shows thewave magnetic field amplitude averaged over h = 10 
10 cW1e0 , and the dotted straight line indicates the maximum
linear growth rate. The dashed curve shows the amplitude
averaged over h = 100  100 cW1e0 . For the initial
parameters of the simulation assumed at the magnetic
equator, linear theory yields a positive growth rate >103
We0 within the frequency range w = 0.14  0.5 We0 and amaximum growth rate 2.67  103 We0 at the frequency
0.27We0. As shown in the enlarged plot in Figure 5b,whistler-mode waves grow with a maximum linear growthrate uniformly over h = 100  100 cW1
e0 in the early stage
1
up to the time t = 500 W1e0 . After t = 600 We0 ; however,there arises a discrepancy between the solid line and thedashed curve, thus indicating localized wave growth in theregion h = 10  10 cW1e0 . Stronger sporadic wave
growth, different from linear growth, takes place at latertime as we see in Figures 5a and 5b. We have plotted thespatial profile of the wave amplitude and its time evolution
in Figure 6. A series of discrete-time wave packets areformed near the equator and propagate away from theequator with growing wave amplitude. The spatial scale

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A04223OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

Figure 5. Time history of wave magnetic field amplitude
Bw averaged over h = 10  10 cWe01 (solid line) and h =100  100 cWe01 (dashed line). The dotted straight line
indicates the maximum linear growth rate.
of the wave packets is about a few tens of cW1e0approximately.
[29] Applying a discrete Fourier transform to the wave
field in the region h = 0– 30 cW1e0 , we obtained wave number
spectra of the wave amplitude BW and their time evolution.
These are shown in Figure 7a. The wave number spectra arecalculated from the transverse wave magnetic fields Bw1(hi)and Bw2(hi) (with hi = iDh, where i = 1  512), which are~ w1(ki) (i = 1  256),~ w1(ki) and B
Fourier-transformed to B
respectively; Dh is the grid spacing which is taken as~2~ 2 1/2 for the first
0.06 cW1e0 . We then calculated (Bw1 + Bw2)
32 modes (k1, k2, . . ..., k32), and these are plotted in
Figure 7a. We find that chorus elements shifting to higher
wave numbers are generated repeatedly. Seeds of chorus
elements are generated at wave modes around k  2 c1We0,
and the wave number progressively shifts to higher wave
numbers up to k  4 c1We0. We also find a band of strong
emissions in the wave number range k = 1  2 c1We0. The
original wave amplitude Bw = (B2w1 + B2w2)1/2, plotted in
Figure 7b, is associated with the long wavelength modes(k < 2.2 c1We0) and the chorus elements (k > 2.2 c1We0).

331
A04223The long wavelength modes grow continuously, and theirfrequencies are nearly constant or decrease slightly over thelong time evolution (Figure 3). This indicates that the lower
frequency emissions have linear growth rates determined by
the velocity distribution function of the energetic electrons
which are scattered in pitch angle and interact with lower
frequency waves at higher values of jVRj.
[30] To evaluate the wave amplitude of the chorus elements exactly, we have eliminated the lower wave number
modes (k < 2.2 c1We0) in the Fourier transformed fields~ w2, and reconstructed the wave amplitude by the~ w1 and B
B
inverse Fourier transform, as plotted in Figure 7c. In thesimulation run, we assumed V?0 = 0.70 c and wpe = 4 We0,
which determine the basic parameters as g = 1.4, x = 0.11,
and d = 0.99. We also assumed that w = 0.27 We0, whichcorresponds to the mode with maximum linear growth rate.
Inserting these parameters into (51) and (7), we obtain VR =
0.18 c and Vg = 0.16 c. Using these values in thesimulation, we rewrite (50) as Bw /B0 = 9.3 (@w/@t)W2
e0 .
Since we find @w/@t = 5.0  105 in the simulation, we canestimate the wave amplitude as Bw /B0 = 4.7  104, i.e.,log10(Bw /B0) = 3.3. This analytical estimate of the wave
amplitude of the chorus elements agrees very well withthose plotted in Figure 7c.
[31] For nonlinear wave growth, we need either a frequency variation or a spatial inhomogeneity of the static
magnetic field as expressed by (28). Around the equator,
where the flux of the resonant energetic electrons maximizes, the resonant current forms because of the electron
hole in the velocity phase space. Since the second term in(28) is negligible near the equator, the first term involving@w/@t dominates the inhomogeneity. Among the whistlermode waves with different phases, it is the modes thatsatisfy the second-order resonance condition with S  0.4
that undergo nonlinear wave growth to form chorus emissions. For computational efficiency in the present simulation, the spatial inhomogeneity of the dipole magnetic field
is enhanced to make the spatial scale smaller than that of thereal magnetosphere. The length-scale of the simulation
model has been reduced by a factor 3.5, which corresponds
to the assumption that a is replaced by as = 4.5 (3.5/L RE)2 =55/(LRE)2. At L = 4 in the Earth’s magnetosphere, we findFigure 6. Spatial profile and time evolution of the wave
amplitude Bw .

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A04223

A04223Figure 7. (a) Wave number spectrum of Bw over the spatial range h = 0 – 30 cWe01 and its time
evolution. The spectrum is calculated from the two orthogonal components of the transverse wave
magnetic field. (b) Original wave amplitude Bw(h, t) (h = 0  30 cWe01 and t = 20000 –30000 We1).
(c) Reconstructed wave amplitude Bw(h, t), corresponding to chorus emissions, after eliminating thelong-wavelength modes (k < 2.2 c1We0) from the wave number spectrum in Figure 7a.
as = 9.8  107 (c1We0)2. For the starting frequency ofchorus elements around w = 0.27 We0, we find s1 = 6.3, ands2 = 1.1 from (54) and (55). We then have hc = 150 cW1
e0
from (57). If the wave excited near the equator propagateswith a growing amplitude to the point h = hc, and if the
wave amplitude is greater than twice the equatorial amplitude, then wave growth can continue over a much longer
distance if there exists a sufficient flux of resonant particles
to form the resonant current.
[32] To find the assumed electron hole forming thenonlinear resonant current, we have calculated the perpendicular velocity flux distribution of the electrons. On thebasis of the velocity distribution function f (vk, v?) in thesimulation, we calculate
F? vk ; z ¼

Z 1
0



v2? f vk ; v? ; z dv?

ð58Þ

which we integrate over vk for a specific range of positionsh1  h  h2 to obtain the transverse flux distribution P(z, h).
We calculated P(z, h) for h = 0  15 cW1e0 , and plotted it in
Figure 8a. Since we find a depletion of flux in the range h =12– 14 cW1e0 , we provide the phase space plot of thetransverse flux F?(vk, z) for the h range in Figure 8b. We
also plotted a separatrix of the trapping region for the
averaged perpendicular velocity V?0 /c = 0.706 and the local
wave amplitude Bw /B0 = 0.001. We find some depletion of
the flux in the position of the electron hole, but the hole does
not appear clearly. This is because the resonance velocity VR

is a function of energy as we find in (13), and the location of
the separatrix depends on the energy range of resonant
electrons forming the flux. As we also have multiple waves
at different frequencies, a clear identification of the electron
hole is difficult. However, we find a depletion in the phaserange of the separatrix forming the transverse resonantcurrent causing the nonlinear wave growth.
[33] In Figure 9 we plot the wave amplitudes of thechorus elements (k > 2.2 c1We0). These have been obtainedby applying the method of Fourier transforms used in
producing Figure 7c to the whole of the northern hemisphere (h > 0). We find chorus elements are generated fromthe wave amplitude Bw /B0 = 5  104 and amplified to therange Bw /B0 = 1  2  103 at h = hc. Therefore, nonlinear
growth is sustained beyond the distance hc, as is confirmed
in Figure 9.
[34] The trapping frequency wt usually gives the oscillation frequency of the trapped particles. As is clear from(27), however, the relativistic electrons oscillate with frequency wtr (=wt dg 1/2). The time-scale Ttr of the oscillatory
motion of trapped resonant electrons is therefore given by
Ttr ¼


1=2
2p
cgx
¼ 2p
:
wtr
V?0 wWw dð59Þ

Substituting Ww = 5  104 We0 and other parameter values
into (59), we find Ttr = 250 W1e0 . For VR = 0.18 c, aresonant particle can move over one cycle Ttr a distance
45 cW1e0 which is much less than hc. Thus, the resonant

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A04223OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

Figure 8. Perpendicular velocity flux distributions ofelectrons (in arbitrary units) in (a) the phase space (h, z),and (b) the velocity phase space (vk, z) in the spatial range
h = 12– 14 cWe01 at t = 20,000 We01. The dashed line
indicates a separatrix of the trapping region for the average
perpendicular velocity V?0 = 0.706 c and the wave
amplitude Bw = 0.001 B0. A depletion of electron flux isfound in the phase range of the trapping region.

333
A04223

amplitude and the frequency. The wave amplitude varies in
the presence of the resonant current JE parallel to the wave
electric field, as expressed by (29). However, equation (10),which is valid when wi  w, shows that the wave frequency
does not vary in a frame of reference moving with the groupvelocity Vg. Therefore, the frequency variation should takeplace at the time of wave generation rather than during thewave propagation. At the equator coherent nonlinear growthoccurs for a wave with a specific phase variation @w/@t.
This produces a maximum growth rate corresponding to anegative value of JE. Our analysis predicts a frequency
sweep rate of a chorus element that is successfully confirmed by the computer simulation. The essential frequency
variation of a chorus element takes place as a result of
nonlinear wave growth due to the formation of an electromagnetic electron hole in the velocity space. By analyzing
the nonlinear dynamics of resonant electrons, we havefound that nonlinear wave growth maximizes at S = 0.4.
Waves with a specific phase variation @w/@t, corresponding
to S = 0.4 at the equator, should thus become dominant in
the system.
[36] We have also found the growth of lower frequency
waves at constant frequencies that corresponds to linear
growth in the simulation. To separate the chorus emissions
from the constant frequency waves at longer wavelengths,
we have applied a discrete Fourier transform to the wave
magnetic fields. We have separated the chorus elements
near the equator from the lower-frequency emissions andplotted the results in Figure 7c and Figure 9. We find thatthe wave amplitude and the frequency-sweep rate satisfy theformula for the frequency variation (50).
[37] The wave number spectra and their time variation near
the equator (h = 30  +30 cW1e0 ) are shown in Figure 7. The
selective nonlinear growth for waves with rising frequencies
takes place in the region close to the magnetic equator, andthe waves propagate away from the equator as shown in
Figure 9. When the waves grow, the spatial inhomogeneity
should decrease because the trapping frequency wt increases
as a function of the square root of the wave amplitude Bw. As
the waves propagate away from the equator, the second termin the formula for the inhomogeneity ratio S starts to increase.
This produces a finite negative S value that contributes to theelectrons should experience strong modification of theirorbits in the phase space and form an electromagnetic
electron hole as shown in Figure 1. However, the size of thehole depend on the value of v? of resonant electrons. In thepresent simulation we have a velocity distribution function
consisting of different value of v? the boundaries of thetrapping region does not appear clearly in Figure 8b. The
high-energy particles with different values of the Lorentz
factor also give different values of the resonance velocity
VR. This is another reason to obscure the existence of anelectron hole. Since the phase range of the electron hole isbasically the same, we do find depletion of resonant
electrons in the approximate range of z = 0.5p  1.5p in
Figure 8a, contributing to formation of the negative JE.

6. Discussion and Conclusions[35] We have derived the relativistic second-order resonance condition (28) for a wave with varying frequency. We
have also derived wave equations (29) and (30) for the wave

Figure 9. Wave amplitude Bw(h, t) of chorus emissions in
the northern hemisphere (h > 0). The long-wavelength
modes (k < 2.2 c1We0) have been eliminated.

11 of 14

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A04223OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

formation of the nonlinear resonant current JE. Therefore, thenonuniformity of the magnetic field near the equator plays a
key role in the amplification process of chorus emissions
formed at the equator.
[38] The inhomogeneity of the dipole magnetic fieldassumed in the present simulation has been enhanced tofacilitate the computation efficiency, i.e., we replace a by
as = 12.25 a. For a realistic magnetosphere, we obtainfrom (57) the relation: hc = 240 (s1/s2)(1/fce) (@f/@t) (km) at
L = 4. Since s1  6, s2  1, fce  14 kHz, and (@f /@t)
10 kHz/s for a typical chorus element, we then have hc
 1000 km. This is the size of the chorus wave generation
region within which the nonlinear wave growth takes place
primarily due to the frequency variation. The nonlinear
wave growth continues over the distance a few times
longer than hc because of the spatial inhomogeneity ofthe magnetic field.
[39] As shown by (50), the spatial inhomogeneity, whichis expressed by as, does not affect the relation between the
wave amplitude and the frequency sweep rate. Since thefrequency sweep rate @f /@t found in the simulation is
62 kHz/s, which corresponds to Bw /B0 = 4.7  104, thetypical frequency sweep rate 10 kHz/s in the magnetosphereshould correspond to Bw /B0 = 8  105, i.e., 40 pT. The
wave amplitude as a seed of chorus emissions is in areasonable range in comparison with the maximum wave
amplitude 150 pT observed by the Cluster spacecraft[Santolik et al., 2004].
[40] The frequency sweep rate is determined by thesaturation level of the whistler-mode instability driven by
the temperature anisotropy of energetic electrons at theequator. In the present simulation we have used a relatively
high-temperature anisotropy and have performed the simulation as an initial value problem without providing freshenergetic electrons into the simulation system. It is necessary to make a clear distinction between the saturation levelof the instability and the thermal fluctuation level that isenhanced in the particle simulation. In the real magnetosphere, the saturation level depends on the injection rate of
fresh electrons and their density and temperature anisotropy.
A smaller temperature anisotropy and a smaller flux of
injected energetic electrons would give a smaller saturationlevel and correspondingly a smaller frequency sweep rate.
[41] The concept of nonlinear wave growth based on thesecond-order resonance condition has been used in previous studies by Nunn [1974], Nunn et al. [1997], andTrakhtengerts [1995, 1999]. However, these authors havenot determined the relative roles of temporal frequency
variation and spatial variation of the gyrofrequency in contributing to the formation of chorus elements. The present
study concludes that the seed of the chorus element is formed
solely by the temporal frequency variation, or the secondorder phase variation in time, that maximizes the nonlinear
growth rate at the magnetic equator. This conclusion has been
confirmed by the electron hybrid simulation.
[42] We have assumed that chorus emissions propagate
purely parallel to the static magnetic field in the present
analysis. Recent observations by the Cluster spacecraft
indicate, however, that chorus emissions propagate obliquely to the observation points off the equator [Inan et al.,
2004; Chum et al., 2007]. The wave packet may undergo
Landau damping through propagation, and the electromag-

A04223

netic electron hole may not be maintained during theoblique propagation. Obliquely propagating whistler-mode
waves may also play a significant role in the precipitation ofenergetic electrons [Inan et al., 2003], while parallel propagating whistler-mode waves can accelerate electrons effectively to a few MeV energy, contributing to formation of theradiation belts [Omura et al., 2007]. The effect of obliquepropagation on nonlinear wave-particle interaction of chorus emissions is an important subject for future study.

Appendix A: Derivation of the Wave Equations(29) and (30)[43] We assume an electromagnetic wave propagating
parallel to a static magnetic field B0 directed along theh-axis. The wave fields are in the transverse plane containing x- and y-axes. We express the electric and magnetic field
vectors of the wave in the transverse plane by the complex~ w = Bw exp (iyB), respec~ w = Ew exp (iyE) and B
forms E
tively. We also assume that a transverse current J~c is formedby the cold electrons and that a transverse resonant current
J~R is formed by the resonant electrons. Maxwell’s equations
rE¼

@B
@t

ðA1Þ

and
1 @E
r  B ¼ m0 J þ 2
c @t

ðA2Þ

~w
~w
@E
@B
¼
@h
@t

ðA3Þ

~w
~w

 1 @E@B¼ m0 J~c þ J~R þ 2
:
c @t
@h

ðA4Þ

are expressed asiiHere, the cold current component J~c is given by Jc =
eneVe, where Ve is the flow velocity of the cold electron
fluid that supports the propagation of the whistler-mode
wave. The momentum equation of the cold electron fluid isgiven bymdVe
¼ eðE þ Ve  BÞ;
dt

ðA5Þ

which is rewritten as
w2pe
@ J~c
~ w þ iWe J~c :
E
¼
@t
m0 c2

ðA6Þ

~ w and J~c from (A3), (A4), and (A6), we obtain
Eliminating E

12 of 14

"

#


w2pe @
@
1 @2
@2
~w

 iWe
B
þ 2
c @t
@t
c2 @t 2 @h2


@
@ J~R
¼ im0
:
 iWe
@h
@t

ðA7Þ

Omura et al., “Generation Process of Chorus Emissions” (2008)
OMURA ET AL.: GENERATION PROCESS OF CHORUS EMISSIONS

A04223The left-hand side (LHS) of (A7) is expanded as
LHS ¼

Each term of LHS can be calculated as follows:
ðA9Þ

ðA10Þ

~w
We @ 2 B
w 2 We ~wWe @Bw¼i 2 Bexp ðiyB Þ
wþ2 2
2
2
c @t
c
c
@t

ðA11Þ

~w
@2B
~ w þ 2kWe @Bw exp ðiyB Þ¼ ik 2 We B
@h2
@h

ðA12Þ

~w
w2pe @ B
ww2pew2~ w þ pe @Bw exp ðiyB Þ;¼i 2 B
2
c @t
c
c2 @t

ðA13Þ

iWe

where the frequency w and wave number k are defined by
w = @yB/@t and k = @yB/@h for a whistler-mode wave
with right-hand circular polarization. We have neglected
the derivatives of the wave amplitude Bw and phase yB
that are higher than first-order.
[44] Decomposing J~R into components J~B parallel to thewave magnetic field and J~E parallel to the wave electricfield, we set
J~R ¼ ðJB  iJE Þ expðiyB Þ:

ðA14Þ

Substituting (A14) into the right-hand side (RHS) of (A7),and neglecting the derivatives of the currents JB and JE, we
find
RHS ¼ im0 k ðWe  wÞJB exp ðiyB Þ  m0 k ðWe  wÞJE exp ðiyB Þ:
ðA15ÞWe now express equation (A7) in the form of LHS = RHS.
We then multiply the right-hand and left-hand sides of (A7)
expanded above by exp (iyB), and separate the real andimaginary parts. We thereby obtain the respective equations!
w2pe þ 2wWe  3w2 @Bw@Bw
k þþ 2k ðWe  wÞ
c2
@t
@h
2

¼ m0 k ðWe  wÞJE ;

ðA16Þ

and
"
ðWe  wÞk 2 þ[45] Acknowledgments. Computation in the present study was performed with the KDK system of RISH and ACCMS (Academic Center for
Computing and Media Studies) at Kyoto University. This work waspartially supported by Grant-in-Aid 17340146 and 17GS0208 for Creative
Scientific Research ‘‘The Basic Study of Space Weather Prediction’’ of the
Ministry of Education, Science, Sports and Culture of Japan, and International
Communication Foundation. D. S. acknowledges support from the Natural
Sciences and Engineering Research Council of Canada under grant A-0621.
[46] Amitava Bhattacharjee thanks Richard Horne and another reviewer
for their assistance in evaluating this paper.

References



~w
@3B
@Bw
2~
2 @Bw
B
¼
iwk
þ
k

2kw
exp ðiyB Þ

w
@t@h2
@t
@h

i

A04223Using (5) and (7), we can rewrite (A16) and (A17) to obtainequations (29) and (30), respectively.

~w~w @3B~w~w~ w w2pe @ B
1 @3B
We @ 2 B
@2B

i 2
þ iWe
þ 2
:
2
3
2
2
2
c @t
@t@h
c @t
@h
c @t
ðA8Þ

~w
1 @3B
w3 ~
w 2 @Bw¼ i 2 B
exp ðiyB Þ
w3 2
2
3
c @t
c
c @t

335

#
w3  w2 We  ww2pe
Bw ¼ m0 k ðWe  wÞJB :
c2
ðA17Þ

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Y. Katoh, Planetary Plasma and Atmospheric Research Center, Graduate
School of Science, Tohoku University, Sendai, Miyagi 980-8578, Japan.
(yuto@pparc.geophys.tohoku.ac.jp)
Y. Omura, Research Institute for Sustainable Humanosphere, Kyoto
University, Uji, Kyoto 611-0011, Japan. (omura@rish.kyoto-u.ac.jp)
D. Summers, Department of Mathematics and Statistics, Memorial
University of Newfoundland, Elizabeth Avenue, St. John’s, NL, Canada
A1C 5S7. (dsummers@math.mun.ca)

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Omura et al., “Relativistic Turning Acceleration by Whisler Waves” (2007)
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Here

337

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A06236, doi:10.1029/2006JA012243, 2007

for

Full
ArticleRelativistic turning acceleration of resonant
electrons by coherent whistler mode waves in a
dipole magnetic fieldYoshiharu Omura,1 Naoki Furuya,1 and Danny Summers1,2
Received 28 December 2006; revised 14 February 2007; accepted 28 February 2007; published 20 June 2007.

[1] We report a very efficient process for accelerating high-energy electrons by coherent
whistler mode waves in the Earth’s dipole magnetic field, which we have found in ourrecent test particle simulations. The efficient acceleration process takes place forweakly relativistic seed electrons of a few hundred kiloelectronvolts. Under an assumptionthat the whistler mode wave packets are excited near the equatorial plane of the innermagnetosphere and propagate away from the equator, the acceleration process becomesirreversible. With a sufficiently long whistler mode wave packet of the order of 1 s, theenergetic electrons are accelerated to a relativistic energy range of a few megaelectronvoltsthrough a single resonant trapping process. We call this particular acceleration process
relativistic turning acceleration (RTA), which could be a viable mechanism for increasing
relativistic electron fluxes in the outer radiation belt. Necessary conditions for RTA area relatively large amplitude of whistler mode waves, in the range of 50 to a few hundred
picoteslas, and an initial kinetic energy of trapped electrons in the energy range of afew hundred kiloelectronvolts. The minimum energy of electrons accelerated by the
RTA process and the maximum energy attained by it are derived analytically and verified
by the test particle simulations.
Citation: Omura, Y., N. Furuya, and D. Summers (2007), Relativistic turning acceleration of resonant electrons by coherent whistler
mode waves in a dipole magnetic field, J. Geophys. Res., 112, A06236, doi:10.1029/2006JA012243.

1. Introduction[2] Over the course of a geomagnetic storm, the fluxof relativistic (>1 MeV) electrons in the outer radiation belt
(3 < L < 7) can undergo a net increase by several orders ofmagnitude over prestorm values, e.g., Baker et al. [1986,
1994, 1997], Li et al. [1997], Reeves et al. [1998, 2003],
Obara et al. [2001], Iles et al. [2002], Miyoshi et al.
[2003], Miyoshi and Kataoka [2005]. A physical mechanism that can account for the storm-time generation ofrelativistic electrons is cyclotron resonance with whistler
mode chorus waves outside the plasmasphere [Summers etal., 1998, 2002, 2004; Roth et al., 1999; Summers andMa, 2000; Meredith et al., 2002; Albert, 2002; Miyoshi
et al., 2003; Horne et al., 2005; Varotsou et al., 2005;
Summers, 2005; Shprits et al., 2006]. Whistler mode chorus
emissions in the inner magnetosphere are enhanced whensubstorm activity is enhanced [Meredith et al., 2001]. Consequently, the chorus diffusion mechanism is effective foraccelerating electrons during periods of prolonged substorm
activity, whether in the presence or absence of a magnetic
storm.
1
Research Institute for Sustainable Humanosphere, Kyoto University,
Uji, Kyoto, Japan.
2
Department of Mathematics and Statistics, Memorial University of
Newfoundland, St. John’s, Newfoundland, Canada.

[3] Quasi-linear diffusion theory can be effectively used
to describe electron cyclotron resonance with whistler modechorus waves [e.g., see Summers, 2005, Summers et al.,2007a, 2007b and references therein]. However, quasilinear theory is based on averaging processes, namely
ensemble averaging of the waves and gyrophase averagingof the particles, so it cannot take account of nonlinear
phenomena such as resonant wave trapping. Nonlinear
wave trapping has been demonstrated in self-consistent
simulation studies of whistler mode wave-particle interaction [Nunn et al., 1997; Omura and Summers, 2004; Katoh
and Omura, 2004, 2006a, 2006b]. Observations [e.g.,
Santolik et al., 2004; Kasahara et al., 2005] show thatwhistler mode chorus comprises discrete, coherent monochromatic elements rather than a broadband spectrum
assumed by quasi-linear theory. By performing test particle
simulations, Omura and Summers [2006] investigated electron interactions with a coherent whistler mode wave
generated in the magnetic equatorial plane. Omura andSummers found that electrons of energy of a few tens ofkiloelectronvolts can be accelerated to a few hundreds ofkiloelectronvolts by repeated trapping of resonant electrons
moving in the direction opposite to the direction of wave
propagation. It was found that a small fraction of electrons
are trapped by whistler mode waves propagating away fromthe equator and that electrons are accelerated by a few
percent of their kinetic energy, the particles being guided by

Copyright 2007 by the American Geophysical Union.
0148-0227/07/2006JA012243$09.00

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the varying resonant velocity that decreases as they movetoward the equator.
[4] In the present investigation, we extend the test particle
simulations of Omura and Summers [2006] and report a
particularly efficient form of particle acceleration. The
process involves a special form of resonant wave trappingwith the result that some electrons of energy of a few
hundred kiloelectronvolts can be accelerated to relativisticenergies of a few megaelectronvolts within a second. The
scenario for this special form of wave trapping, which
involves what we call relativistic turning acceleration(RTA), is as follows. Assuming that whistler mode wave
packets are excited at the equator and propagate away fromthe equator, then a small fraction of electrons approaching
the equator will be trapped. A fraction of these regularly
trapped electrons are found to turn around and move awayfrom the equator. These turning electrons are found toundergo the aforementioned significant energy increase,
and we refer to the energization process as RTA. We report
the results of our test particle simulations and show the RTA
phenomenon in section 2. In section 3, we carry out atheoretical analysis of the special trapping process and theacceleration of the turning electrons. We discuss our results
in section 4, and finally in section 5 we present ourconclusions.

2. Test Particle Simulation[5] We solve the relativistic equations of motion forelectrons assuming a dipole magnetic field and a coherent
whistler mode wave with a constant amplitude Bw andconstant frequency w. The numerical model and scheme
have already been described in detail in Omura andSummers [2006]. The whistler mode wave is assumed tobe generated in the equatorial plane and to propagate awayfrom the equator in the positive h direction, where h is the
distance along the dipole magnetic field line traced from theequatorial plane. The plasma density is assumed to beconstant along the magnetic field line.
[6] We suppose that B0 is the magnitude of the dipole
magnetic field in the equatorial plane at L = 4, and WEQ is theelectron (nonrelativistic) cyclotron frequency corresponding
to B0. We set w / WEQ = 0.4 and Bw / B0 = 2.4  104,corresponding to a wave frequency 5.6 kHz and wave
amplitude Bw = 120 pT. These are typical values for whistler
mode chorus waves observed by the CLUSTER spacecraft[Santolik et al., 2004]. Furthermore, we set wpe / WEQ = 2.0
where wpe is the electron plasma frequency and h WEQ/c =1600, with h = 5460 km, and c is the speed of light. We fix
the initial electron kinetic energy K0 = 424 keV, and we set
v||0 / c = 0.11, v?0 / c = 0.83, where v||0 and v?0 are initial
velocity components parallel and perpendicular to the external
dipole magnetic field, respectively; g 0 = [1  (v2||0 + v2?0) /c2]1/2 = 1.8 is the initial value of the Lorentz factor. We thentrace trajectories of 90 particles with different gyrophases z
equally spaced from 0 to 2 p.
[7] Figure 1 shows a trajectories of the 90 electrons as afunction of the kinetic energy in terms of megaelectronvolts.
Out of the 90 particles, 21 particles are trapped by the whistler
mode wave and undergo a dramatic acceleration. Others
remain at the energy level of about 400 keV, with some
perturbations depending on their phases. The trapped electrons

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Figure 1. Trajectories of resonant electrons interactingwith a coherent whistler mode wave. Trajectories of 90
particles are plotted.
are accelerated during their motion toward the equator. The
parallel velocity satisfies the cyclotron resonance condition,.
w  kvk ¼ We ðhÞ g

ð1Þ

where g = [1  (v2|| + v2?)/c2]1/2 and We(h) is the cyclotron
frequency at the distance h from the equator; g  1 is the
kinetic energy of an electron normalized by the rest
mass energy m0c2, as plotted on the right vertical axis of
Figure 1. From (1), the resonance velocity can be written
VR = [w  We(h)/g]/k. As the trapped electron approaches
the equator with a negative resonance velocity VR, theLorentz factor g increases, thereby making the magnitude
of VR progressively smaller. When g = We(h)/w, theresonance velocity becomes zero, and the trapped electronschange the direction of their motion to the direction of
propagation of the whistler mode wave. The dynamics andacceleration of the relativistic resonant electrons areanalyzed theoretically in the following section.

3. Theoretical Analysis of Relativistic Turning
Acceleration
3.1. Relativistic Resonance Condition and Variation ofKinetic Energy[8] The kinetic energy of a relativistic electron is given by
K ¼ m0 c2 ðg  1Þ;

ð2Þ

where m0 is the electron rest mass. The time variation of theelectron kinetic energy is given bydK
dðgvÞ
¼ m0 v ¼ eE  v:
dt
dt

ð3Þ

[9] We assume a purely transverse electromagnetic wave
propagating parallel to the static magnetic field B0 as
schematically illustrated in Figure 2. It follows that

2 of 11dK¼ eEw v? sin z;dt

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A06236frequency of the whistler mode wave is constant in space
and time, we obtain


dVR
1
VR d 2@We We dg¼ VR  þ
þ 2
;
dt
kg dt
kg 2ðWe  wÞ @hð9Þ

where d2 = 1  (w / ck)2. We calculate the right-hand side of(6) making use of (4), (7), and (9), and we obtain thefollowing second-order ordinary differential equation,
d2 z w2t d2¼
ðsin z þ S Þ;
g
dt 2

ð10Þ

where w2t = kv?Ww. We obtain the inhomogeneity ratio for a
relativistic electron in the form,
S¼

Figure 2. Whistler mode wave vector geometry in a dipole
magnetic field.
where z is an angle between the wave magnetic field Bw andthe perpendicular velocity v?. The wave electric field Ew isperpendicular to Bw in the plane transverse to B0 and thewave number vector k. The phase angle z determines the
dynamics of cyclotron resonant electrons, and its variation is
described by the differential equations [e.g., Matsumoto andOmura, 1981; Omura and Matsumoto, 1982; Omura et al.,
1991]


dz
¼ k vk  VR
dt

ð5Þ

d2 z
d
¼ k ðvk  VR Þ:
dt 2
dt

ð6Þ

[10] We have used the first-order resonance condition(1) in deriving (6). The resonance condition (1) is also
expressed by dz/dt = 0. The second-order resonancecondition for stable trapping of a resonant electron is
given by d2z/dt2 = 0. The parallel velocity v|| varies inaccordance with the Lorentz force of the wave magnetic
field and the mirror force of the static magnetic field. For
an electron interacting with a whistler mode wave with amagnetic field amplitude Bw, we have
 
d gvk
gv2 @We;¼ Ww v? sin z  ?
2We @hdt

c2 k 2 ¼ w2 þ

ww2pe
We  w;

2 þ d2
We  gw
kgv2? @We
VR 
:
We
@h
We  w

sin z þ S ¼ 0:

ð11Þ

ð12Þ

Combining (12) with (4), we find that the energy variationof a stably trapped resonant electron is given bydK¼ eEw v? S:
dt

ð13Þ

At the turning point where VR = 0, we derive from (11) the
result,S¼

gv? @We:
2d2 Ww We @hð14Þ

Noting that Ew / Bw = w / k, and g = We / w for VR = 0, we
obtain from (13) and (14) the relation,
dK m0 v2? @We:¼
dt
2kd2 @hð15Þ

Using (15) and (2) and setting VR = 0 in (9), we find thatdVR
1 wv? 2 @We¼:
2We kcddt@hð8Þ

where wpe is the electron plasma frequency, assumed
constant along the magnetic field line. Assuming that the

If S = 0, the differential equation has a solution that
oscillates around z = p with the relativistic nonlinear trapping
frequency wtdg 1/2. The phase angle z = 0 is a saddle point
for untrapped resonant electrons in the (v||, z) phase space.
In Figure 3, we plot the time histories of kinetic energies,
parallel velocities, perpendicular velocities, phase angles z,
and the inhomogeneity ratios S of trapped accelerated
electrons shown in Figure 1. The variations of z and S showthat the right-hand side of (10) is nearly equal to zero.
[11] The second-order resonance condition d2z/dt2 = 0[Nunn, 1974] gives the necessary condition for the resonant
trapping to take place, i.e.,

ð7Þ

where Ww = eBw / m0 and e is the charge of an electron.
We adopt the cold plasma dispersion relation for a whistler
mode wave [e.g., Stix, 1992].

1
2w2t d2

ð16Þ

For a trapped particle moving toward the equator with v|| < 0in the region where @We /@h > 0 to be reflected, theresonance velocity should satisfy dVR/dt > 0 at VR = 0.

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Figure 3. Trajectories of trapped resonant electrons, showing the kinetic energy K, parallel velocity v||,perpendicular velocity v?, the phase angle z between v? and Bw , and the inhomogeneity ratio S.
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Near the equatorial plane, the dipole magnetic field isapproximated byWe ðhÞ ¼ WEQ 1 þ ah2 ;ð20Þ

where a = 4.5 / (LRE)2 and RE is the Earth’s radius. It should
be noted that the quadratic approximation for the dipole field
is used only in the theoretical analysis, while the exact dipole
field is assumed in the test particle simulation. Using (18)–(20), we obtain
Ww
WEQ

Figure 4. Trajectories of 90 electrons in the (v||, v?) plane.

From (16), we find that turning is only possible before
reaching the equator. Once the particle passes through theequator, it is decelerated because the sign of @We /@hbecomes negative. The energy variation equation (15)implies that trapped electrons undergoing the turn continue
to be accelerated on one side of the equator. In the Northern
Hemisphere, where @We /@h > 0, a whistler mode wave,
generated at the equator, propagates with the wave number
k > 0. We find that dK/dt > 0 for trapped electrons. In theSouthern Hemisphere, where @We /@h < 0, we also find that
dK/dt > 0 because k < 0 for a wave propagating from theequator in the negative h direction. These characteristics ofthe trapped resonant electrons are displayed in the time
history plots in Figure 3. We also plot the trajectories of allparticles in (v||, v?) space in Figure 4. While untrappedelectrons interact with the wave only for a short time,essentially undergoing adiabatic mirror motion, the trappedresonant electrons near v|| = 0 can stay in resonance with thewave for a much longer time, being accelerated effectively
in the (v||, v?) phase space.

ajh0 jc
wd2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
w2
1
:
We ðh0 Þ2

ð21Þ

For the parameters L = 4, w / WEQ = 0.4, and wpe/WEQ = 2, theturning point positions h0 = (100, 1000, 10,000) km require
the inequalities Ww/WEQ > (0.057, 0.57, 6.5)  104, whichcorrespond to the respective wave amplitude restrictions Bw >(2.9, 29, 330) pT.

3.2. Wave Amplitude Necessary for Resonant
Acceleration at the Turning Point[12] We consider relativistic electrons trapped by a whistler mode wave packet moving away from the equator. At
the turning point h0 where VR = 0, the electrons will be
inside the trapping wave potential, where there exists a
stable equilibrium point satisfying the second-order resonance condition. The necessary condition for the trapping is
jSj < 1. Substituting (14) into this condition, we have
gv?
@We
< 1:
2d Ww We @h2

ð17Þ

We rewrite result (17) in terms of ratio of the wave amplitude
Ww to the static magnetic field at the equator WEQ as
Ww
v?
@We
>
;
WEQ 2d2 wWEQ @hð18Þ

where the resonance condition g = We / w with VR = 0 has beenused. The resonance condition also givessﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
w2
v? ¼ c 1  2 :
We

ð19Þ

Figure 5. The time history of the parallel velocities andkinetic energies K is plotted for eight particles whose
trajectories start at the mirror point where v|| = 0. The dashedcurves represent the analytical solutions for trappedresonant electrons given by (33) and (35).

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Figure 6. Trajectories of particles undergoing RTA for different wave frequencies. We set wpe / WEQ =2.0. The particles are trapped at h = 1000cWEQ1. The required wave amplitudes are 58, 75, 110, and
160 pT for w / WEQ = 0.5, 0.4, 0.3, and 0.2, respectively. The labels on the trajectories refer to the value of
w / WEQ.

3.3. Acceleration of Relativistic Electrons Near theTurning Point[13] Assume that an electron is trapped by a whistler
wave packet and guided to the turning point where @We /@h > 0.
If the wave packet terminates when the particle reaches
the turning point, the electron should start an adiabatic
motion with parallel velocity v|| = 0. The parallel velocity
increases to a negative value, and the electron moves
toward the equator. This means that the particle is at a
mirror point of the adiabatic motion in the dipole magnetic
field. We now assume a group of relativistic electrons (g > 1)
at the mirror point. Some electrons see the front of awhistler wave packet with a frequency w satisfying thecyclotron resonance condition kVR = w  We / g = 0. If the
wave amplitude Ww satisfies the inequality (21), some ofthe electrons become trapped by the wave packet and start
to be accelerated in the positive h direction, moving awayfrom the equator.
[14] The acceleration of the electrons trapped at theturning point is described by (16). Using (20), we rewrite(16) asdVR¼ h2 h;
dt

ð22Þ

pﬃﬃﬃ
where h = ws a / (kd) and s2 = 1  w2/W2e . Noting
dh /dt = VR, we haved2 h
¼ h2 h:
dt2

ð23Þ

pﬃﬃﬃ
We assume that h = h0 = ws0 a / (k0d0), a constant, near
the turning point h0, where the subscript ‘‘0’’ indicatesvalues at h = h0. Solving (23) with the initial value h = h0 at
t = 0, we obtainh ¼ h0 expðh0 t Þð24Þ

for time t > 0. We can also obtain the trajectory of a trappedelectron approaching the turning point with a negative
velocity (VR < 0). Assuming that the turning point occurs attime t = 0, we find the solution for t < 0 to be
h ¼ h0 expðh0 t Þ:

ð25ÞUsing (25) for t < 0, we integrate (22) to obtain
VR ¼ h0 ðh  h0 Þ:

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We rewrite (15), using (20) and (24), in the form,
dK m0 c2 s20 aWEQ h0expðh0 t Þ:
¼
dt
kd20

ð34Þ

[16] We assume that the particle is detrapped at a positionhD and at a time tD and integrate (34) over [0, tD] to obtain
the result,K ðtD Þ ¼ Kð0Þ þ

Figure 7. Relativistic cyclotron resonance condition andw  k dispersion relation; g E is the value of the Lorentz
factor at an entrapping point, and g D is the value at a
detrapping point.

Similarly from (24) for t > 0, we obtain the result,
VR ¼ h0 ðh  h0 Þ:

ð27Þ

The resonance velocity can also be expressed as a functionof g asVR ¼

w
We ðhÞ
1
:
k
wg

ð28Þ

The phase velocity w/k is given from the cold dispersion
relation (8) asw¼ cdx;kð29Þ

where x 2 = w (We(h)  w) / w2pe. From the definition of d2,
we obtain the relation
d2 ¼

1
:
1 þ x2

ð30Þ

By neglecting small variations in x, i.e., by setting x = x0,
we obtain from (27), (28), and (29) the following expressionthat is satisfied along a trajectory of a resonant electron:
ð31Þ

where the upper sign corresponds to t > 0 and the lower sign
to t < 0. Solving for g, we obtaing¼

We ðhÞ
1
w

1
pﬃﬃﬃ
s0 a
ðh  h0 Þ :d03.4. Relativistic Turning Acceleration[18] We estimate the maximum energy increase corresponding to the relativistic turning acceleration (RTA) process. We consider entrapped, resonant relativistic particles
turning at a very small distance from the equator andbecoming detrapped from the wave potential when moving
away from the equator. We denote the entrapping point as hE
and the Lorentz factor at that point as g E. Then, assuming h0 =0, we have from (31)hE ¼

VR ¼ h0 h0 ½expðh0 t Þ  1:
d0 We ðhE Þ
pﬃﬃﬃ
1 :
s0 a wg E

ð36Þ

As verified by Figure 4, we may assume VR  v?.
Neglecting the variation in the cyclotron frequency in (11),we set We  WEQ and find that the inhomogeneity ratio is
approximated by
S¼

ð32Þ

[15] We now examine the time variation of the kinetic
energy, starting from the turning point h0 at t = 0. From (22),
we obtain for t > 0 the relation

ð35Þ

Expression (35) shows that the kinetic energy K increases
exponentially in time.
[17] To confirm our analytical results, we performed afurther run of the test particle simulation, starting eightparticles with v|| = 0 at h0 = 30cW1
EQ = 100 km, withdifferent phase angles z. We assumed the initial perpendicular velocity v?0 = 0.92c, corresponding to g = 2.5. We
assumed wpe / WEQ = 2.0, and we set the wave amplitude
Bw = 1.0  104 B0 = 50 pT, which satisfies the condition(21). Six out of the eight electrons are trapped because theinhomogeneity ratio S is initially close to zero. The time
variation of the parallel velocities and the variation of the
kinetic energies are plotted in Figure 5. Since S increases
gradually, the trapped particles are detrapped one by one astime progresses. Trajectories of the trapped electrons show areasonable agreement with the dashed curves that represent
the analytical expressions (33) and (35). When the parallel
velocity of the trapped particles increases, the trajectories
deviate from the analytical expressions because the terms
involving VR on the right-hand side of (9), neglected near
the turning point, become significant.

h  h0 ¼

d0
We ðhÞ
pﬃﬃﬃ 1 
;
wg
s0 a

pﬃﬃﬃ
m0 c2 s0 WEQ ah0½expðh0 tD Þ  1:
wd 0

gcs0 ah:d 20 Ww

ð37Þ

As we see from Figure 3, the resonant wave trapping takes
place when jSj  1. Hence it follows thatð33Þ

7 of 11hE 

d20 Ww:
cs0 ag E

ð38Þ

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Substituting (36) into (38), we obtaingE

We ðhE Þ Ww d0
 pﬃﬃﬃ :c aw

g E;min ¼

ð39Þ

[19] After passing through the turning point (t > 0), thetrapped electrons continue to be accelerated as they moveaway from the equator. The timing of the detrapping depends
on the relative phase of the trapped electron with respect tothe stable equilibrium point of the second-order resonance.
The duration of the trapping maximizes if the particle isexactly at the equilibrium point. From (31), we find that the
maximum values of h = hD and the Lorentz factor g D at thetime of detrapping are related by the equation,
hD ¼



d0
We ðhD Þ
pﬃﬃﬃ 1 
:
wg D
s0 a

ð40Þ

From the trapping condition, jSj  1, we havehD 

d20 Ww:
cs0 ag D

ð41Þ

Substituting (40) into the inequality (41), we haveg D  g D;max ¼

We ðhD Þ Ww d0
þ pﬃﬃﬃ :
w
c a

ð42Þ

The gain in the kinetic energy (DK)max through therelativistic turning acceleration process is therefore 2m0 cWw d0
pﬃﬃﬃ;
ðDK Þmax ¼ m0 c2 g D;max  g E;min að43Þ

where we have assumed that We(hE)  We(hD). Noting thatd 0 is close to unity for small x20, we find that the maximum
energy gain does not depend appreciably on the wave
frequency, but does depend on the wave amplitude and thecurvature of the magnetic field. Assuming that the magnetic
field on the surface of the Earth is 32,000 nT, we rewrite(43) in the convenient form
ð DK Þmax ¼

5:6  104 Bw
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðMeVÞL2 1 þ x20 B0

ð44Þ

where x20 = w (WEQ  w) / w2pe.

4. Discussion[20] From (2), the kinetic energy of an electron is 510(g  1) keV. Relativistic electrons of a few hundred kiloelectronvolts to a few megaelectronvolts have a substantial
range of the resonance velocity VR = (w  We / g) / k. Wheng = We/w, we have VR = 0. Resonant electrons, satisfying thesecond-order resonance condition, are efficiently accelerated in the presence of coherent whistler mode waves
propagating away from the equator. Trapped particles, initially with a negative value of v||, are accelerated to a positive
value of v|| as VR changes as a function of the Lorentz factor g.
Throughout the turning process, the acceleration continues.
After the turning point, the acceleration rate increases, as

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described by (34), until the particles are detrapped because of
the increased value of jSj. The trajectories of the relativistic
turning acceleration (RTA) particles are described by (32). At
the turning point h = h0, we find g = We/w, which corresponds
to the kinetic energy K(0) = m0 c2 [We(h0) / w  1]. Entrappingof particles that pass through the turning point starts at theenergy KE = m0c2 (g E,min  1).
[21] The trapping can continue after the turning point up
to the distance hD where S = 1. Some particles are
detrapped at a shorter distance, depending on their locations
with respect to the stable equilibrium point given by thesecond-order resonance condition. Throughout the resonant
trapping process, a given resonant particle remains in one
particular hemisphere and is accelerated in the manner
described by (15). The maximum energy that can bereached is KD = m0 c2 (g D,max  1). The energy range
DK = KD  KE is proportional to the wave amplitude, while
the midpoint of the energy range is determined by the wave
frequency. For a small-amplitude wave, the trapping takes
place only at a short distance from the equator (h0  0), forwhich we find g 0 = WEQ / w. The electron plasma frequency
does not affect the nature of the acceleration because itbarely modifies the value of x as prescribed by the condition
wpe WEQ.
[22] At L = 4, whistler mode chorus waves typically occur
in the frequency range w = 0.2 – 0.5 We, which corresponds
to the turning point energy of 500 keV – 2 MeV. With awave amplitude of 100 pT (corresponding to Bw / B0 = 2 104), the energy range DK is then 700 keV from (44). For
this amplitude, the frequency w = 7 kHz and the gyrofrequency WEQ = 14 kHz, electrons of energy 150 keV can betrapped by the wave and accelerated to 850 keV.
[23] Figure 6 shows a parametric analysis of the RTA
process, by the test particle simulations, starting at a particle
position hE = 1000cW1
EQ = 3411 km. The required wave
amplitudes are 58, 75, 110, and 160 pT for w / WEQ = 0.5,
0.4, 0.3, and 0.2, respectively. The plasma frequency is
assumed to be constant: wpe / WEQ = 2.0. Waves with asmaller frequency can accelerate particles in a higher energy
range by a larger energy gain. This requires, however, alarger wave amplitude, as indicated by (43).
[24] Relativistic electrons in the Earth’s radiation belt canundergo adiabatic motions, mirroring in both hemispheres.
At the mirror points their parallel velocities become zeroand their pitch angles reach 90°. Let us consider particles
with Lorentz factors g in the ranges 2 – 5, and let us assume
that the particles encounter whistler mode wave packets
with frequencies w = 0.2– 0.5 WEQ. If the resonance condition g = We / w and the amplitude condition (21) aresatisfied, the particles will be trapped by the wave andexperience the second half of the RTA process (t > 0). The
process is demonstrated by the simulation presented in
Figure 5. In this case, most of the particles are trapped by
the incoming wave packet, unlike the previous case whereparticles see the preexisting wave packet at the time ofresonance. Therefore for a wave packet with an amplitude
well above the threshold given by (21), most of the resonantparticles are accelerated at the wavefront where the trapping
wave potential engulfs most of the particles because of thesmall value of S.
[25] The duration of each wave packet may be short,especially for chorus waves showing a rapid frequency

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A06236

Figure 8. Schematic illustration of relativistic turning acceleration (RTA); g E is the value of the Lorentz
factor at an entrapping point, and g D is the value at a detrapping point.
variation. However, RTA trajectories may well be attainedby many different wave packets at different times.
[26] In Figure 7 we illustrate the relativistic cyclotron
resonance condition and the w  k dispersion relation; weshow both the VR > 0 and VR < 0 cases. While we have only
presented our analysis for one hemisphere (with h > 0), the
RTA process also takes place for waves propagating in the
opposite direction. Thus, RTA takes place in both hemispheres
near the equator, as schematically illustrated in Figure 8.
[27] RTA is due to the nonlinear particle dynamics associated with resonant wave trapping. From (15), the particles
are always accelerated while they stay on one side of theequator. For a deceleration process to take place, the particles
have to encounter the waves propagating toward the equator.
Whistler mode chorus waves are generated at the equator
because of an injected population of electrons of energy

1).
10’s keV, with a strong temperature anisotropy (T? / T||
Santolik et al. [2004] find that the direction of whistler
mode wave propagation is predominantly away from theequator. A recent particle simulation also demonstrates thatwhistler mode wave packets corresponding to chorus
emissions grow progressively and propagate away fromthe equator [Katoh and Omura, 2007]. Therefore RTA iseventually an irreversible process. On average, the energy
gain of trapped resonant electrons by RTA is much higher
than the energy loss experienced by the untrapped resonant
electrons. We conclude that RTA is a viable mechanism forthe generation of relativistic electrons in the Earth’s outerradiation belt.
[28] The timescale of each RTA process is only of theorder of a second. If a sufficiently large temperature
anisotropy (T?/T||
1) is maintained by continuous

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injection of high-energy particles into the inner magnetosphere, successive chorus emissions are generated. Then,a series of effective RTA processes take place, resulting in a
rapid formation of a flux of relativistic electrons with atimescale much shorter than that typically predicted byquasi-linear diffusion theory. It can also be noted that
RTA becomes more effective for smaller L values, as shownby (44). This could explain a short timescale, less than
60 min, of flux enhancement of relativistic electrons in the
inner (L < 3) magnetosphere in association with large
substorms during the main phase of a magnetic storm
reported by Nagai et al. [2006].
[29] RTA is a unique feature of the dipole magnetic field
geometry. The process should also be operative in themagnetospheres of Saturn and Jupiter. We have presented
in expression (42) an estimate for the maximum energy
attainable by the RTA process. In a larger magnetospherewith a smaller gradient a such as the Jovian magnetosphere,
the maximum energy attainable by RTA becomes muchhigher. The wave amplitude is a critical value for controllingthe energy range of the accelerated electrons.

5. Conclusions[30] In performing test particle simulations of whistler
mode wave particle interaction in a dipole magnetic field,
we have found a very effective acceleration of relativistic
electrons to take place near the equatorial plane in the Earth’s
inner magnetosphere. We refer to this process as relativistic
turning acceleration (RTA), illustrated schematically in
Figure 8. Since the relativistic effect is essential for the
cyclotron resonance reported here, we have derived thesecond-order resonance condition in relativistic form. On
the basis of the first-order and second-order resonanceconditions, we have analyzed the acceleration process
theoretically and obtained the analytical trajectories ofresonant particles undergoing the RTA process. The analysis
yields the upper and lower thresholds of the kinetic energies
that can be affected and attained by RTA. The current
analysis assumes a wave packet with a duration of about
1 s and with constant frequency and constant amplitude.
Shorter pulses make the RTA process less effective becauseof earlier detrapping, but some electrons are retrapped bysubsequent pulses and further accelerated. Extension to aseries of shorter wave packets with variable frequency andamplitude such as chorus emissions is left as a future study.
[31] Acknowledgments. This work was partially supported by Kyoto
University Active Geosphere Investigations for the 21st Century Centers of
Excellence Program (KAGI21) and by grant-in-aid 17340146 and
17GS0208 for Creative Scientific Research ‘‘The Basic Study of Space
Weather Prediction’’ of the Ministry of Education, Science, Sports andCulture of Japan. D. S. acknowledges support from the Natural Sciencesand Engineering Research Council of Canada under grant A-0621.
[32] Zuyin Pu thanks Jay Albert and David Nunn for their assistance inevaluating this paper.

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N. Furuya and Y. Omura, Research Institute for Sustainable Humanosphere,
Kyoto University, Uji, Kyoto 611-0011, Japan. (nfuruya@rish.kyoto-u.ac.jp;
omura@rish.kyoto-u.ac.jp)
D. Summers, Department of Mathematics and Statistics, Memorial Universityof Newfoundland, St. John’s, Newfoundland, Canada. (dsummers@math.
mun.ca)11 of 11

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The Astrophysical Journal, 690:244–251, 2009 January 1
c 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.


doi:10.1088/0004-637X/690/1/244

ELECTRON SHOCK SURFING ACCELERATION IN MULTIDIMENSIONS: TWO-DIMENSIONALPARTICLE-IN-CELL SIMULATION OF COLLISIONLESS PERPENDICULAR SHOCK
Takanobu Amano1 and Masahiro Hoshino
Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan; amanot@stelab.nagoya-u.ac.jp
Received 2008 May 7; accepted 2008 August 31; published 2008 December 1

ABSTRACTElectron acceleration mechanisms in high-Mach-number collisionless shocks propagating in a weakly magnetized
medium is investigated using a self-consistent two-dimensional particle-in-cell simulation. Simulation results showthat strong electrostatic waves are excited via the electron–ion electrostatic two-stream instability at the leadingedge of the shock transition region as in the case of earlier one-dimensional simulations. We observe strong
electron acceleration that is associated with the turbulent electrostatic waves in the shock transition region. The
electron energy spectrum in the shock transition region exhibits a clear power-law distribution with spectral index
of 2.0−2.5. By analyzing the trajectories of accelerated electrons, we ﬁnd that the acceleration mechanism is very
similar to shock-surﬁng acceleration of ions. In contrast to the ion shock surﬁng, however, the energetic electrons
are reﬂected by electron-scale electrostatic ﬂuctuations in the shock transition region and not by the ion-scalecross-shock electrostatic potential. The reﬂected electrons are then accelerated by the convective electric ﬁeld infront of the shock. We conclude that the multidimensional effects as well as the self-consistent shock structure are
essential for the strong electron acceleration at high-Mach-number shocks.
Key words: acceleration of particles – cosmic rays – plasmas – shock waves

1. INTRODUCTION
It is generally believed that cosmic rays with energies up
to the knee (∼ 1015 eV) are produced at supernova remnant(SNR) shocks. There is indeed direct evidence for shock acceleration of cosmic-ray electrons to more than TeV energies(e.g., Koyama et al. 1995). Recently, TeV gamma rays fromsome shell-type SNRs have been detected by High Energy
Stereoscopic System (HESS), which implies the presence ofcosmic rays with ∼ 100 TeV energies (Aharonian et al. 2007).
Although it is still under active debate whether the primary particles emitting the gamma rays are either electrons or protons,
the morphological similarity between nonthermal X-ray emission and the gamma rays indicates that they are accelerated by
SNR shocks. Diffusive shock acceleration (DSA) is the mostwidely accepted theory for the shock acceleration of nonthermal particles (e.g., Blandford & Eichler 1987). The DSA theory
assumes the presence of magnetohydrodynamic (MHD) turbulence upstream of the shock. Energetic particles scattered by
MHD waves gain energy by diffusively crossing the shock frontback and forth many times. The central unresolved issue in DSA
theory is the well-known injection problem—since DSA is efﬁcient only for particles having enough energy to be scatteredby MHD waves, injection from a thermal pool to nonthermalenergies by some other mechanism is required. This requirement is very stringent, particularly for electrons because of theirsmall Larmor radii. Therefore, strong preacceleration mechanics are needed to explain the observed nonthermal emissions
from ultrarelativistic electrons in SNRs. Numerical studies using particle-in-cell (PIC) codes have been conducted to explore
the possibilities of direct electron energization at the vicinityof the shock that may provide a seed population for DSA (e.g.,
Dieckmann et al. 2000; Shimada & Hoshino 2000; McClements
et al. 2001; Hoshino & Shimada 2002; Schmitz et al. 2002).
Amano & Hoshino (2007) have recently shown that a fraction
1

Now at Solar-Terrestrial Environment Laboratory, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan.

of electrons may efﬁciently be injected into the DSA process
at high-Mach-number quasi-perpendicular shocks. Their onedimensional (1D) PIC simulations demonstrated that nonthermal electrons are generated by two successive different acceleration mechanisms, namely, shock-surﬁng acceleration (SSA)
and shock drift acceleration (SDA). They proposed an electron injection model based on the 1D simulation results, which
can account for the observed injection efﬁciencies (e.g., Bambaet al. 2003). However, the problem is that comprehensive theory of SSA does not exist at present. Therefore, the effects ofmultidimensionality on the injection efﬁciency were not taken
into account. Two- or three-dimensional self-consistent numerical simulations of high-Mach-number shocks are needed toevaluate the realistic injection efﬁciency.
It is well known that an important portion of dissipation at thecollisionless nonrelativistic shock is provided by the so-called
reﬂected ions. At the quasi-perpendicular shock with θBn > 45◦(θBn is an angle between the shock normal and the upstream
magnetic ﬁeld), the reﬂected ions gyrating in front of the shock
are accelerated by the convective electric ﬁeld in the upstream
region, and then transmitted to the downstream. Early hybrid
simulation (kinetic ions and massless electrons) studies showedthat the direct energization of the reﬂected ions contributes
importantly to the downstream thermalization (e.g., Leroyet al. 1982). However, it is generally considered that the energization of electrons at the collisionless shock is relatively weak.
Since Larmor radii of electrons are very small compared to thescale length of macroscopic electromagnetic ﬁelds, they are
considered to suffer only adiabatic heating by the compressedmagnetic ﬁeld at the shock. In contrast, the in situ observationsof the Earth’s bow shock demonstrated that this is not always
true (Gosling et al. 1989; Oka et al. 2006). Furthermore, radio
and X-ray observations strongly suggest that the nonthermal
electron acceleration is very efﬁcient in young SNRs. Microinstabilities in the shock transition region probably play an important role for the nonadiabatic energization of electrons. Recent
PIC simulations of quasi-perpendicular shocks have shown thata variety of instabilities can be excited in the shock transition

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ELECTRON SURFING ACCELERATION IN MULTIDIMENSIONS

region due to the presence of the reﬂected ions (e.g., Shimada &Hoshino 2000, 2004; Scholer et al. 2003; Matsukiyo & Scholer
2003, 2006; Muschietti & Lembège 2006). Among them, theBuneman instability (Buneman 1958), which is the electrostatic
two-stream instability between cold electrons and ions, is themost dominant mode at the high-Mach-number regime relevant
to SNRs. Hoshino & Shimada (2002) showed that the Buneman
instability plays a key role in the production of nonthermal electrons via SSA. Strong electrostatic potential produced by thenonlinear evolution of the instability can trap a fraction of electrons; the trapped electrons moving with the wave potential can
see an inductive electric ﬁeld arising from the relative velocity
between the wave and the background plasma. Therefore, they
can be accelerated in the transverse direction until they escapefrom the potential. Nonlinear 1D PIC simulations demonstrated
that SSA can quickly accelerate electrons to mildly relativisticenergies (McClements et al. 2001; Hoshino & Shimada 2002).
Therefore, it is believed that SSA plays an important role for an
efﬁcient electron injection. It is worth noting that the electronenergization by SSA relies on the assumption that the potential is uniform in the transverse direction, so that the electrontransport in the direction along the inductive electric ﬁeld isvery efﬁcient. However, it is well known that the Buneman instability at oblique propagation has growth rates comparable tothe parallel propagation (Lampe et al. 1974). The assumption of
1D wave potential is, therefore, not appropriate to evaluate therealistic efﬁciency of SSA. Ohira & Takahara (2007) recently
pointed out by performing two-dimensional (2D) electrostatic
PIC simulations that SSA may be inefﬁcient in multidimensions. Their conclusion was drawn from the observation thatthey did not observe nonthermal tails in the ﬁnal electron energy spectra. However, they used a homogeneous model of theshock transition region in which the plasma consisted of threecomponents—the upstream electrons and ions, and the reﬂectedion beam. We think the artifact introduced by their model should
be taken into account with great care.
Here, we report two-dimensional (2D) PIC simulation resultsof a high Mach number, perpendicular shock propagating in a
weakly magnetized plasma. Note that several numerical studies
of collisionless shocks using 2D PIC codes can be found in theliterature (e.g., Forslund et al. 1984; Lembege & Savoini 1992).
However, these studies considered only moderate Mach number shocks relevant to the Earth’s bow shock. At higher Mach
number regime, we ﬁnd strong electrostatic waves in the shocktransition region excited by the Buneman instability, as in thecase of 1D simulations. Efﬁcient electron acceleration associated with the large-amplitude electrostatic waves is observed.
It is shown that the nonthermal electrons are produced by amechanism similar to SSA of ions (e.g., Zank et al. 1996;
Lee et al. 1996). We argue that the effects of multidimensionality and the self-consistent shock structure are essential forthe production of nonthermal electrons at high-Mach-number
shocks.

2. SIMULATION
2.1. Simulation Setup
We use a 2D electromagnetic PIC simulation code, in which
both electrons and ions are treated as kinetic macroparticles, tostudy the dynamics of electrons and ions in a fully self-consistentshock structure. A shock wave is excited by the so-calledinjection method that is commonly used in 1D simulations.

Figure 1. Stacked proﬁles of compressional magnetic ﬁeld component Bz ,averaged over the y-direction. The vertical axis is normalized to the inverseof ion cyclotron frequency in the upstream.

A high-speed plasma consisting of electrons and ions is injectedfrom the boundary x = 0 of a 2D simulation box in the x−y plane
and travels toward the positive x-direction. The plasma carries
the uniform magnetic ﬁeld perpendicular to the simulation box(B0  ez ). At the opposite boundary, particles are specularly
reﬂected by the wall. Then, a perpendicular shock forms due tothe interactions between the incoming and the reﬂected particles,
and it propagates in the negative x-direction. Therefore, thesimulation is done in the downstream rest frame. The periodic
boundary condition is imposed in the y-direction.
We use the following plasma parameters in the upstream:
βe = βi = 0.5 (βj ≡ 8π nTj /B 2 ), where n, Tj , and B are thedensity, temperature, and magnetic ﬁeld strength, respectively.
The ratio of the plasma frequency to the electron cyclotron
frequency is ωpe /Ωce = 10. A reduced ion to electron mass
ratio of mi /me = 25 is used. These lead to an upstream Alfvénspeed of vA /c = 0.02. We use a plasma injection four-velocityof U0 /c = 0.2. The Alfvén Mach number of the resulting
shock wave is MA  14 in the shock rest frame. The gridsize of the simulation is taken to be equal to the electron
Debye length in the upstream. We use 4096 × 256 grid points
in the x- and y-direction, respectively. The physical size ofthe simulation box is Lx  204 c/ωpe and Ly  12.8 c/ωpe .
Initially, each cell contains 40 particles of each species in the
upstream. Note that the injection velocity of U0 /c = 0.2 is rather
high for simulations of realistic SNR shocks. We adopt thisvalue to reduce the computational costs. However, the dominant
instability in the present simulation is still the electrostatic modein contrast to an electromagnetic Weibel-like instability foundin relativistic shocks (e.g., Kato 2007). Hence, we think theessential physics does not change due to the use of an artiﬁciallyhigh shock speed.
We use the following units unless otherwise stated: time,
distance, velocity, and energy will be given in units of the−1
inverse of the electron plasma frequency in the upstream ωpe,
the electron inertial length c/ωpe , the injection velocity U0 , andthe upstream electron bulk energy 0 = (γ0 −1)me c2 where γ0 =
1 + (U0 /c)2 , respectively. The electric and magnetic ﬁelds arenormalized to the motional electric ﬁeld E0 = U0 B0 /γ0 c, andthe background magnetic ﬁeld B0 in the upstream, respectively.

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Figure 2. Snapshot of electromagnetic ﬁelds at ωpe t = 1000. From top to bottom, color images of Bz , Ey , and Ex are shown.

2.2. Shock Structure
We ﬁrst discuss an overall structure of the simulated highMach-number shock. Figure 1 shows the stacked magnetic
ﬁeld proﬁles Bz averaged over the y-direction. In this ﬁgure, we can clearly see a shock wave propagating in thenegative x-direction. The average shock-propagation speed isabout ∼0.4U0 , yielding a Mach number of MA ∼ 14 in theshock rest frame. Note that the vertical axis is normalized tothe inverse of ion cyclotron frequency in the upstream Ω−1ci(Ωci t = 250 ωpe t). The shock propagation is not stationary, butshows slight variation in the shock structure. It is known thatquasi-perpendicular shocks with high Mach numbers simulatedby 1D PIC codes typically show nonstationary behavior called
cyclic self-reformation, which occurs on a characteristic
timescale of 1–2 Ω−1
ci . However, the observed shock front shows
less time variability than usually observed in 1D, suggestingthat an efﬁcient plasma thermalization is suppressing the nonstationary behavior (Scholer & Matsukiyo 2004; Shimada &Hoshino 2005).
The snapshot of the electric and magnetic ﬁelds at ωpe t =1000 is shown in Figure 2. The leading edge of the shocktransition region is located at around x/c/ωpe  100. We can see
predominantly electrostatic ﬂuctuations at 100  x/c/ωpe 
110 in both Ex and Ey panels. These waves are excited via
the Buneman instability caused by the interactions between the
upstream electrons and the reﬂected ions. It should be notedthat the wavefronts of these electrostatic waves are oblique tothe shock normal, which is in sharp contrast to 1D. Furthermore,
the waves are not one-dimensional, having ﬁnite extent along the
wavefront. The excitation of multidimensional wave structureby the Buneman instability is consistent with the linear theory
and nonlinear 2D PIC simulations in a periodic simulation box(Lampe et al. 1974).
The reason why we observe the oblique wavefronts can beeasily understood by considering the Larmor motion of thereﬂected ions. Figure 3 displays the snapshot of the phase spaceof both electrons and ions, as well as the y-averaged magnetic
ﬁeld. The reﬂected ions can be easily identiﬁed in the top two
panels showing the phase-space plots of ions in (x, ui,x ) and(x, ui,y ). Since the reﬂected ions are accelerated in the positive

y-direction, they have a large bulk velocity not only in the x,
but also in the y-direction at the leading edge of the shockx/c/ωpe ∼ 100. The waves excited by the Buneman instability
propagate mostly parallel to the beam direction. Therefore,
it is not surprising that we observe the oblique wavefronts.
To be more precise, the instability excites a wide range ofoblique modes and the observed spatial proﬁle (wavefront) is asuperposition of waves with different wave vectors. However,
we observe the oblique wavefronts propagating parallel to thebeam probably because (1) the wave power peaks at the parallel
propagation, and (2) the wave propagation is symmetric withrespect to the beam. We have actually conﬁrmed that thesuperposed spatial proﬁle propagates almost perpendicular to
the wavefronts (parallel to the beam). It is worth noting that
this behavior agrees very well with that observed in periodic
simulations of the Buneman instability in 2D.
Looking at the electron phase-space plots (x, ue,x ), (x, ue,y )that are shown below the ions, we can ﬁnd strong electronenergization at the leading edge of the shock transition region
x/c/ωpe ∼ 100. It is clear that the energization of electrons
is associated with the strong electrostatic waves excited by
the Buneman instability as had been studied by 1D codes. As
we see below, however, the energization of electrons in 2D
occurs in a somewhat different manner, which is due to thedifferent properties of the strong electrostatic turbulence in thefoot region.
2.3. Energy Spectrum
The electron energy spectra shown in Figure 3 are integrated
over every 12.5c/ωpe interval to obtain Figure 4, showing the
averaged energy spectra around the shock transition region. One
can clearly ﬁnd power-law energy spectra within the shock transition region. The observed spectral slopes are ∼2.0−2.5. The
slope slightly steepens upon increasing the penetration into theshock. The downstream spectrum is essentially unchanged fromthe spectrum observed at the overshoot 125  x/c/ωpe  137.5(dash-dotted line). The steeper spectral indices observed in the
deeper shock transition region suggest that the nonthermal electrons are mostly produced at the leading edge of the shocktransition region. It is worth noting that a high-energy hump

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Figure 5. Time history of an accelerated electron: energy (left) and the ﬁrst
adiabatic invariant (right).

Figure 3. Snapshot of particle phase-space plots and compressional magnetic
ﬁeld proﬁle averaged over the y-direction. Color represents the logarithm of theparticle count in each bin. Note that the vertical scale of the electron energy
spectrum (the second panel from the bottom) is shown on a logarithmic scale.

Figure 6. Electron trajectory and electric ﬁelds Ex (left) and Ey (right),
respectively.

2.4. Particle Acceleration

Figure 4. Energy spectra of electrons around the shock transition region at
ωpe t = 1000. Each line shows a spectrum averaged over 75  x/c/ωpe  87.5
(thick solid), 87.5  x/c/ωpe  100 (solid), 100  x/c/ωpe  112.5(dashed), 112.5  x/c/ωpe  125 (dotted), and 125  x/c/ωpe  137.5(dash-dotted).

is observed in the distribution right before the shock transition
region 87.5  x/c/ωpe  100 (solid line). This hump corresponds to energetic electrons that are once reﬂected and are
gyrating in front of the shock.

In order to discuss particle acceleration mechanism in moredetail, individual trajectories of energetic electrons are analyzed.
Figure 5 shows the time history of energy and the ﬁrst adiabatic
invariant of a typical accelerated electron.
Here the ﬁrst adiabatic
2
invariant is deﬁned as μ ≡
u
2B,
and
is normalized to its⊥

upstream value μ0 = U02 2B0 . We use the downstream rest
frame as a reference frame to deﬁne the adiabatic invariant. The
particle’s position x is also plotted in Figure 6 as a functionof time. The color shows the components of the electric ﬁeld
Ex (left) and Ey (right). Note that the electric ﬁelds shownin the ﬁgure are measured at y = yp for each time step—Ex,y = Ex,y (x, yp , t), where yp represents the particle’s positiony. Thus, one may consider that the particle sees the time variationof a 1D shock structure. The particle trajectory in the x−yplane is shown in Figure 7. The symbols are plotted everyωpe Δt = 10 interval during the strong electron energization
ωpe t = 1000−1140.
The low-energy electron that is initially located in the far
upstream region begins to interact with the shock at ωpe t 
1040. It sees large-amplitude waves that have both Ex and第 4 章 論文集

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Figure 8. Schematic illustration of acceleration mechanism. Electrons areaccelerated in two steps: (1) they are accelerated by the stochastic electron
shock surﬁng in the shock transition region and preferentially transported to the
upstream region. (2) The accelerated electrons escaping into the upstream suffer
further acceleration by the constant motional electric ﬁeld.
Figure 7. Electron trajectory in the x−y plane. Symbols are plotted everyωpe Δt = 10 interval from ωpe t = 1000, during which the electron suffers thestrong energization.

Ey components during ωpe t  1040−1090, and is gradually
heated. Because of the compressed magnetic ﬁeld as well as
the change in the convective electric ﬁeld, the guiding center
velocity slows down in the shock transition region. Hence, theparticle trajectory is strongly deﬂected at ωpe t  1090. After
that, the particle is convected toward the negative x-direction
and is ﬁnally ejected into the upstream at ωpe t  1110. At
this time, the electron energy already increases by a factorof ∼ 20. During the stay in the upstream region, it sees theconstant upstream convective electric ﬁeld. Hence, the particle
is accelerated in the negative y-direction during its half gyration.
When it returns back to the shock ωpe t  1140, the energy isincreased by a factor of ∼ 40 from its initial value. During thisacceleration phase, the particle’s ﬁrst adiabatic invariant also
increases by a factor of ∼ 40. Note that, since the ﬁrst adiabatic
invariant is deﬁned in the downstream frame (not in the guiding
center frame), it oscillates with the electron cyclotron period. In
addition, its temporal average should also change as the particle
passes through the shock, even when the particle motion isstrictly adiabatic. However, this change is only of the order ofunity in the normalized unit, while the particle’s ﬁrst adiabatic
invariant increases by more than an order of magnitude. Thus,
it is obvious that the acceleration is a nonadiabatic process.
After ωpe t ∼ 1150, the particle energy further increases, but
only adiabatically, due to the compressed magnetic ﬁeld at theshock.
The particle acceleration process shown above is considered
to be a combination of two mechanisms: one is the energizationin the shock transition region, and the other is the accelerationin the upstream region (see Figure 8). We think that the former
acceleration mechanism may be understood as a stochastic acceleration by large-amplitude electrostatic turbulence. Consider
an electron that encounters a large-amplitude electrostatic wave.
If the electron encounters the wave at a certain gyrophase suchthat the particle velocity in the direction of the wave propagation is approximately equal to the phase velocity, it can travel (orresonate) with the wave during a certain time interval. Since thewave proﬁle propagates with the speed of the reﬂected ion beam
(which differs from the background plasma ﬂow speed), theresonant particle can see an inductive electric ﬁeld in the wave
rest frame. Therefore, electrons are accelerated in the transverse direction that is parallel to the wavefronts. The mechanismof particle acceleration is similar to SSA in 1D (McClements
et al. 2001; Hoshino & Shimada 2002); however, the difference is that the accelerated particles are not trapped in any
waves. Instead, they quickly move from one wave to anotherin a stochastic manner, and are accelerated when they are inresonance with the wave. Here we would emphasize two important characteristics of the electrostatic turbulence: (1) the
wavefronts are oblique to the shock normal, and (2) the turbulent region has a ﬁnite extent along the shock normal. Since thedirection of electron acceleration is approximately anti-parallelto the inductive electric ﬁeld, the accelerated electrons are preferentially transported in the upstream direction as schematicallyshown in Figure 8. Furthermore, since the turbulent region has
a ﬁnite extent, the accelerated electrons can eventually escape
into the upstream of the shock front. It should be noted that theelectron reﬂection is not induced by a macroscopic ﬁeld, suchas the magnetic ﬁeld gradient and the cross-shock electrostatic
potential. Since the Larmor radii of electrons are very small,deﬂection by magnetic ﬁeld alone cannot explain the observed
reﬂection. Similarly, the shock potential cannot reﬂect the negative charge. Indeed, we do not ﬁnd any reﬂected electronsin 1D PIC simulations of perpendicular shocks (e.g., Hoshino& Shimada 2002). The strong and multidimensional turbulent
electrostatic waves do play a role in the transport of the energetic
electrons. We also note that the electron reﬂection is not an artifact of the use of a small ion-to-electron mass ratio. Although
the scale length of the shock transition region is proportional
to the ion Larmor radius, the region of strong electrostatic turbulence always appears at the leading edge of the shock andthe scale length of the region depends only weakly on the mass
ratio ∝ (mi /me )1/3 (Papadopoulos 1988). Therefore, it is reasonable to expect that the electron reﬂection occurs in shocks
with realistic mass ratios.
The latter acceleration in the upstream can be easily understood by analogy with SSA of ions. The ion shock surﬁng is
caused by the shock potential, which reﬂects positively charged
particles. A fraction of ions reﬂected by the shock potential
can be accelerated by the constant motional electric ﬁeld during the Larmor motion in the upstream. On the other hand, theAmano and Hoshino, “Electron Surfing Acceleration in Multidimensions” (2009)353

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accelerated electrons observed here are reﬂected by the microscopic turbulent electrostatic waves. As a result, they suffer further acceleration by the motional electric ﬁeld in the upstream.
Because of this similarity, we consider the present electronacceleration process (including the former and the latter) as
SSA of electrons in multidimensions. The SSA in multidimensions is different from that discussed in 1D in the sense that thetrapping by the large-amplitude waves is no longer important.
The new mechanism is more like the ion shock surﬁng, whilein this case the turbulent electrostatic waves play the role of thereﬂecting wall. We think that the self-consistent shock structure
in multidimensions, that is, a ﬁnite extent of the turbulent region
along the shock normal as well as the oblique wavefronts, areimportant ingredients of the strong electron acceleration.
Let us compare the energy gain estimated from the above
argument with the simulation results. The energy gain ofelectrons from the motional electric ﬁeld E can be estimated
as
Δ
eEL
=
,
(1)
1/2me V02
1/2me V02
where L and V0 are the distance the particle travels along theelectric ﬁeld, and the upstream bulk velocity in the downstream
rest frame, respectively. Rewriting the electric ﬁeld by using therelative velocity difference between the background plasma andthe particle V as E = V B/c, we obtain  


Δ
V
c
Ωce
L
=
2
.
(2)
V0
V0
ωpe
c/ωpe
1/2me V02
For the estimate of the energy gain within the shock transition
region 1 , we use the drift velocity of the reﬂected ions in the
upstream frame V = Vr . Since the x and y components of
the reﬂected ions’ drift velocity measured in the rest frame of
the upstream electrons√ are Vr,x /V0 ∼ −2 and Vr,y /V0 ∼ 2,
we have Vr /V0 = 2 2. Substituting the measured penetration
distance of the particle L1 /c/ωpe ∼ 5, we obtain


Δ1
L1
=
2.8
×
∼ 14.
(3)
c/ωpe
1/2me V02
This estimate is smaller than the observed energy gain of ∼ 20
at ωpe t  1110, suggesting that the particle energy gain arises
not only from the motional electric ﬁeld but also from the wave
electric ﬁeld. The particle is actually accelerated by the large
positive Ex at ωpe t  1110 (see Figure 6). The sum of thisadditional energy from the wave electric ﬁeld and that estimatedfrom Equation (3) agrees well with the observed energy gain.
We note that the direct acceleration by the wave electric ﬁeld
should not be expected in periodic simulation models of theshock transition region that have commonly been used in theliterature. The difference obviously comes from the assumptionof the homogeneity made in the models: a spatial gradient of the
wave energy exists in a real shock transition region. Therefore,
the possibility for a particle to be accelerated by the wave electricﬁeld at the edge of the shock front remains ﬁnite.
The second step of the acceleration in the upstream region
can also be estimated by assuming V = V0 ,


Δ2
L2
=
∼ 20,
(4)
c/ωpe
1/2me V02
where a measured distance of L2 /c/ωpe ∼ 20 is used. This
energy gain is consistent with the simulation result.

We have seen that the energy gains of two acceleration phases
are comparable, thereby, both are important for nonthermal
particle acceleration. However, we think the former acceleration
within the shock transition region plays a more important role.
As a result of the ﬁrst step, energetic electrons are preferentiallytransported to the negative x-direction and eventually reﬂectedback to the upstream region, where they suffer a further
energization. Furthermore, the energy gain in the second step is
proportional to the Larmor radius of the preaccelerated electronin the upstream, which is determined by the energy gain in theﬁrst step.
3. CONCLUSIONS AND DISCUSSION
We have studied strong electron acceleration in a high-Machnumber, perpendicular shock by using a 2D PIC simulation
code. We demonstrate that nonthermal electrons with spectral
indices of 2.0−2.5 are generated in the shock. The efﬁcientelectron energization occurs at the leading edge of the shocktransition region through the interactions with large-amplitude
electrostatic waves produced by the Buneman instability. However, the electrostatic turbulence in 2D has considerably different characteristics compared to that in 1D: the growth of many
oblique modes produces multidimensional potential structures.
In addition, the wavefronts of the electric ﬁelds are oblique tothe shock normal and are almost perpendicular to the reﬂectedion beam. We show that these effects actually play a crucialrole in the electron acceleration. The electrostatic turbulencein the shock transition region enhances anomalous transport ofenergetic electrons toward the upstream and a fraction of electrons are reﬂected back from the shock front. These reﬂected
electrons suffer a further acceleration by the upstream convective electric ﬁeld. We call the acceleration mechanism as SSA;
however, the new mechanism is more like the classical shock
surﬁng of ions rather than that of electrons discussed previously
based on 1D simulations (McClements et al. 2001; Hoshino &Shimada 2002).
It is clear that periodic models of the shock transition
region, often used to investigate the nonlinear development ofbeam instabilities, are not appropriate to study the electron
acceleration process discussed here, because this requires a
spatial inhomogeneity inherent in the shock. We have also
performed 2D simulations by adopting a periodic simulation
model, which is similar to those used in the literature. We ﬁndthat the electron energization observed in the periodic modelis less efﬁcient than that shown in the present paper. Note that
our simulation results are basically consistent with those foundby Ohira & Takahara (2007). Since the spatial inhomogeneity
plays an essential role for the electron acceleration mechanism,
it is natural that we ﬁnd the signiﬁcant differences between
the numerical simulations of the self-consistent shock and theperiodic model. On the other hand, Dieckmann et al. (2008)
modeled perpendicular shocks by colliding two plasma clouds.
They found almost planar electrostatic waves, which contradicts
the results of our simulations. The use of a strong magnetic ﬁeld
(ωpe /Ωce = 5), or the short simulation time compared to theion cyclotron period in their simulations might be the reason for
this. In short, one should be careful in interpreting the results
obtained by adopting simplifying assumptions. We ﬁnd thatthe particle acceleration in the self-consistent shock structureis actually much more efﬁcient than in the periodic model. We
conclude that the SSA can play a role even in multidimensions
and will contribute importantly to the nonthermal production in
high-Mach-number shocks, although the mechanism is different

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from that previously discussed based on 1D PIC simulations.
We think that, however, the details of the mechanism are
not yet fully understood and several issues remain to beanswered.
An interesting question is: “What is the relative acceleration efﬁciency between 1D and 2D?” We have also performed
a 1D simulation with the same parameters. However, we do
not ﬁnd large-amplitude electrostatic waves at the leadingedge of the foot region—only less intense electrostatic waves,which cause a weaker electron heating, are observed in the
deeper, shock transition region. Although the condition of theBuneman instability is formally satisﬁed, the inhomogeneity
along the shock normal may prevent the wave growth because
the wavelength of the most unstable wave (∼ 2π V0 /ωpe ) is comparable to the convective Larmor radius of electrons (V0 /Ωce ).
We observe that the bulk of incoming electrons is merely decelerated at the leading edge of the foot region so as to cancel
the current produced by the reﬂected ions. This observation
may suggest that the threshold of the Buneman instability islower in 2D. The reason for this is that the y-component of
the relative drift can also contribute to the development of theinstability in 2D—since the system is homogeneous along thisdirection, the prediction of linear theory will hold. Comparisonswith higher Mach numbers and/or weaker magnetic ﬁelds, inwhich the Buneman instability is excited both in 1D and 2D, areanyway needed to discuss the relative efﬁciency. Nevertheless,if one invokes the 1D simulation results discussed in Amano &Hoshino (2007), the observed power-law index of the electronenergy spectrum is ∼3−4. The harder spectral index in 2D may
suggest that the electron acceleration is even more efﬁcient thanin 1D. A more detailed analysis of the differences between 1D
and 2D as well as the comparisons with the periodic model will
be reported elsewhere in future.
In the context of the electron injection into the DSA process,the maximum attainable energy is also important. The efﬁcientelectron injection in the quasi-perpendicular shock through
SSA followed by SDA requires that SSA should accelerate
electrons to energies of the order of the upstream bulk ion energy(Amano & Hoshino 2007). Although the present simulation
results satisfy the requirement, the mass ratio dependence ofthe maximum energy is not yet clear, and thus should be
investigated in more detail. More speciﬁcally, it is easy to
expect that the maximum velocity of accelerated electrons
depends on the phase velocity of the electrostatic waves,which does not depend on the mass ratio. Hence, one might
think that increasing the mass ratio leads to relatively lower
maximum energies. However, this argument may not apply
when multiple electron reﬂections occur. In the present study,
we have shown the trajectory of an accelerated particle thatis reﬂected by the shock only once. So far, we do not ﬁndany multiply reﬂected electrons. However, multiple reﬂections
may occur at shocks with different parameters. In the caseof the ion shock surﬁng, multiple reﬂections are believed toprovide an efﬁcient mechanism for injecting low-energy pickup ions into Fermi acceleration (Zank et al. 1996). We thinkthat the same can also happen for electron acceleration. The
property of the turbulent region will probably be important
for the multiple reﬂections; namely, the wave amplitude andthe width of the turbulent region. Hoshino & Shimada (2002)
showed that the energy gain by their SSA in 1D is proportional
to the amplitude of wave electric ﬁeld. Since the transport
of energetic electrons that plays a key role for the particle
acceleration is enhanced by electrostatic waves, the wave

Vol. 690

amplitude will also be important for SSA in multidimensions.
Since the saturation level of the Buneman instability increases
with increase in the Mach number, the electron acceleration
through multiple reﬂections may occur at higher-Mach-number
shocks. In addition, we think the width of the turbulent region, in
which the Buneman instability provides the dominant electronenergization, is also important. The Buneman instability rapidly
thermalizes electrons until the temperature approaches the
upstream bulk energy, which occurs on an extremely shortscale length of the order of (mi /me )1/3 V0 /ωpe (Papadopoulos
1988). Thus, the use of the real mass ratio increases the widthof the turbulent region (normalized to the wavelength of theinstability) by a factor of ∼4. The dependence of the acceleration
efﬁciency on these quantities will be another subject of future
investigation. Large-scale numerical simulation studies as wellas theoretical modeling are needed to improve our understandingof the strong electron acceleration process.
In the present study, we consider electron energization bylarge-amplitude electrostatic waves excited by the Buneman
instability. However, other instabilities may also contribute to
nonadiabatic heating and acceleration of particles in the shock.
Since 1D PIC simulations consider only instabilities having
wavevectors parallel to the shock normal, the effects of plasma
waves propagating in other directions are completely neglected.
It is well known that the cross-ﬁeld current ﬂowing transverseto the magnetosonic shock can be a source of free energy.
Such instabilities may play a dominant role at moderate-Machnumber shocks such as planetary bow shocks in the heliosphere,
in which the excitation of the Buneman instability is prohibited
due to large electron thermal velocities. It has been pointed outthat the enhanced dissipation by microinstabilities can modify
the nonstationary behavior of the macroscopic shock structure(Scholer & Matsukiyo 2004; Shimada & Hoshino 2005). We
also ﬁnd some differences in the shock structure between 1D and
2D. For instance, we observe a less nonstationary shock in 2D,
and the maximum compressed magnetic ﬁeld in the overshootregion of B/B0 ∼ 8 observed in 2D is signiﬁcantly reduced fromthat in 1D B/B0 ∼ 13. We think that microinstabilities do playa role in regulating the macroscopic shock structure. In addition,
we think the degree of freedom along the magnetic ﬁeld is also
another important subject. Recently, Umeda et al. (2008) have
performed 2D PIC simulations of the perpendicular shock, anddemonstrated that the electron acceleration efﬁciency is reduced
when the background magnetic ﬁeld lies in the simulation plane.
This is in clear contrast to the present results. We think the reason
for this discrepancy is that oblique wavefronts are produced inour simulations because the reﬂected ion beam rotates in theplane perpendicular to the magnetic ﬁeld. Another related workis the 2D simulations of the perpendicular shock by Hellinger
et al. (2007) using a somewhat stronger magnetic ﬁeld strength,
which evidences the emission of oblique whistlers in theshock transition region. According to the authors, these oblique
whistlers play the role of suppressing the self-reformation of theshock front. Burgess (2006) reported that the larger-scale shock
surface ﬂuctuations (rippling) enhance the efﬁciency of SDA
in the quasi-perpendicular shock. These effects should also betaken into account when one considers realistic electron heating
and acceleration efﬁciencies.
Finally, we would like to point out that an understanding of
the injection process is important for the nonlinear coupling
between energetic particles and the shock. There are observational indications that the magnetic ﬁelds at astrophysical shocks
are signiﬁcantly ampliﬁed by orders of magnitude from theAmano and Hoshino, “Electron Surfing Acceleration in Multidimensions” (2009)355

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ELECTRON SURFING ACCELERATION IN MULTIDIMENSIONS

typical interstellar value of a few μG (e.g., Bamba et al. 2005;
Uchiyama et al. 2007). It has been considered that the strong
ampliﬁcation is due to the action by the cosmic rays (Bell 2004).
Although the applicability of the simpliﬁed theory is still controversial, the nonlinear feedback due to the presence of cosmic
rays will be of great importance. It is indispensable to know thenumber of injected particles as well as their energy density forunderstanding the nonlinear interactions between the shock andenergetic particles. The microscopic dynamics in a thin, shocktransition region will have a nonnegligible impact on the global
shock evolution and the efﬁciency of particle acceleration tocosmic ray energies.
We thank N. Shimada, C. H. Jaroschek, and K. Nagata
for valuable discussions. T.A. is indebted to T. Terasawa,
M. Fujimoto, T. Yokoyama, M. Hirahara, and R. Matsumoto for
their helpful suggestions. This work was supported by JAXA
and the Solar-Terrestrial Environment Laboratory, Nagoya University. T.A. is supported by a JSPS Research Fellowship for
Young Scientists.
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356
PHYSICS OF PLASMAS 17, 032902 共2010兲

Structure of a strong supernova shock wave and rapid electron
acceleration conﬁned in its transition region
N. Shimada,1,a兲 M. Hoshino,1 and T. Amano2
1

Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan2
Department of Physics, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8602, Japan

共Received 5 October 2009; accepted 27 January 2010; published online 10 March 2010兲
A new rapid energization process within a supernova shock transition region 共STR兲 is reported byutilizing numerical simulation. Although the scale of a STR as a main dissipation region is only
several hundreds of thousands of kilometers, several interesting structures are found relating to thegeneration of a root of the energetic particles. The nonlinear evolution of plasma instabilities leads
to a dynamical change in the ion phase space distribution, which associates with change in the ﬁeld
properties. As a result, different types of large-amplitude ﬁeld structures appear. One is the leading
wave packet, and another is a series of magnetic solitary humps. Each ﬁeld structure has a
microscopic scale 共that is, the ion inertia length兲. Through the multiple nonlinear scattering between
these large-amplitude ﬁeld structures, electrons are accelerated directly. Within a STR, quick
thermalization realizes energy equipartition between the ion and electron; hot electrons play animportant role in keeping these large-amplitude ﬁeld structures on the ion-acoustic mode. The hot
electron shows non-Maxwellian distribution and could be the seed of further nonthermal population.
The “shock system,” where fresh incoming and reﬂected ions are supplied constantly, play anessential role in our result. With a perpendicular shock geometry, the maximum energy of theelectron is estimated by equating a width of the STR to a length of the Larmor radius of theenergetic electron. Under some realistic condition of M A = 170 and ␻ pe / ⍀ce = 120, maximum energy
is estimated to be ⬃10 MeV at an instant only within the STR. © 2010 American Institute ofPhysics. 关doi:10.1063/1.3322828兴
I. INTRODUCTION

Recently, the origin of the energetic electron related toastrophysical shocks has been investigated more and morefrom the point of view of the plasma kinetic processes.1–9
Within the theory of well-established classical diffusive
shock acceleration 共DSA兲 mechanism,10,11 a shock wave
tends to be treated as a simple discontinuity. At a shock
wave, however, we can ﬁnd that many kinds of plasma kinetics play a key role in the energy dissipation and generation of energetic particles. Correct treatment of the plasma
kinetic process around the shock may succeed to settle socalled “injection problem” to DSA process 共e.g., Refs.
12–14兲 as well as to capture a whole stage of DSA process
through nonlinear interaction between the shock structureand energetic particles.15–18 Furthermore, not only as an assistant role to DSA mechanism, investigation of the energy
release mechanism in terms of the plasma kinetics can also
be a powerful tool to search signiﬁcant electron energization
process in much smaller spatial/time scales than that DSA
mechanism may need.
In this paper, we focus on a nonrelativistic ion-electron
supernova shock with the perpendicular geometry where theshock normal is perpendicular to the direction of the background magnetic ﬁeld 共e.g., Ref. 19兲. Particle-in-cell 共PIC兲
simulation has been carried out to investigate coupling dynamics among the ion, electron, and ﬁeld in the shock trana兲

Electronic mail: nshimada@eps.s.u-tokyo.ac.jp.

1070-664X/2010/17共3兲/032902/10/$30.00

sition region 共STR兲. The STR is the front side of the shock
wave corresponding to the region ﬂow speed decreases
gradually from the upstream toward the shocked downstream
region. At the high Mach number shocks, part of the incoming ions is always reﬂected at the shock front and moves into
the upstream region. In detail, the STR means here the region from the leading edge of the reﬂected ion to around theend point of the ﬁrst macroscopic ion gyration in a shocked
region. The upstream electrons are immediately decelerated
just after they meet with the reﬂected ion. The velocity difference between the decelerated electron and the incoming
ion is large enough to excite a series of microscale potentials
through a strong two-stream instability, which grows into apacket of large-amplitude spatial-oscillating magnetic ﬁeld at
the leading edge of the reﬂected ion 共hereafter we call this“leading packet”兲. When the incoming ion begins to intermingle with the reﬂected ion, two ion components together
make complex vortices in the velocity phase space; the electric and magnetic ﬁeld structures change immediately. As aresult, just after this dynamical change, some components of
the leading packet are converted into magnetic solitary
humps almost standing in the electron bulk ﬂow frame. The
electrons bouncing around between the leading packet andthe standing magnetic humps get energy rapidly by the ﬁrstorder Fermi type process. We deﬁne here the ﬁrst-order
Fermi type acceleration as the acceleration of particles by ashock with an extended transition layer, while the originalﬁrst-order Fermi mechanism 共DSA兲 assumes an inﬁnitesi-

17, 032902-1© 2010 American Institute of Physics

Shimada et al., “Structure of a Strong Supernova Shock Wave...” (2010)
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Shimada, Hoshino, and Amano

mally thin transition layer. Among magnetic humps, the electrons are also bouncing around and sometimes getting energy
in a stochastic manner, which depends on mainly local motional electric ﬁeld caused by a combination of magnetic
ﬁeld of the humps and local bulk ﬂow variation due to themacroscopic ion gyromotion. Within the STR, strong thermalization occurs to attain energy equipartition between theelectron and ion. The hot electrons contribute to keep above
large-amplitude structures on the ion-acoustic mode. The
electron energization mechanisms stated above are quite dynamical process characteristic of a “shock system.”In the next section we introduce the simulation setup.
Section III presents simulation results: Sec. III A, macroscopic overview of shock wave properties; Sec. III B, theenergy redistribution process in the STR; and Sec. III C, thestructure of the STR and characteristic ﬁeld generation. In
Sec. III D we show electron energy spectra and discuss someexamples of electron trajectories in the nonlinear ﬁeld evolution. Section IV summarizes and discusses our results.
II. NUMERICAL SIMULATION SETUP

One-dimensional 共1D兲 electromagnetic relativistic PIC
simulation is adopted in the present paper. The shock wave iscollisionless and formed by so-called piston method. A highspeed plasma 共velocity u0兲 consisting of the electron and ion
共its absolute charge value is equal to the electron兲 is injectedfrom the left boundary 共x = 0兲 and travels toward x ⬎ 0 region. Initially, a uniform magnetic ﬁeld Bz 共strength B0兲 perpendicular to the ﬂow direction 共⬅z兲 is carried by theplasma. The y direction is deﬁned by the relation of each unit
vector, x ⫻ y = z. At the right boundary, the plasma ﬂow andall waves are reﬂected and accumulated to form a shock
wave propagating toward the left. The right boundary of thesimulation box is taken with enough distance from the left
boundary to allow free shock propagation without any
artiﬁcial downstream disturbances. The initial plasma parameters are as follows: u0 / c = 0.096, electron and ion beta␤e = ␤i = 0.1 共␤ j = 8␲T j / B20, where T j is the temperature for j
species, j = e for the electron, and j = i for the ion兲,
Alfvén Mach number 共M A兲 = 174, magnetosonic Mach number 共M s兲 = 159, frequency ratio of the electron plasma
oscillation to the electron cyclotron oscillation 共␻ pe / ⍀ce兲 is120, and mass ratio M / m = 100, where ⍀ce = eB0 / mc and␻ pe = 冑4␲n0e2 / m so that ␻ pe / ⍀ce ⬀ 冑n0 / B0. The quantities
n0, e, M, m, and c are respectively the upstream density, theelectric charge, the ion and electron mass, and the speed oflight. The electron plasma oscillation time 共2␲␻−1pe ⬅ T pe兲 isdivided into 115 numerical time steps. Each simulation cellhas 150 particles for each species of the electron and the ion.
We focus here to follow overall dynamics of shockpropagation as well as accurate electron dynamics undersome realistic plasma parameters, such as high Mach number
and the high frequency ratio of ␻ pe / ⍀ce, with statisticallyenough particles. Under realistic mass ratio, we require upstream incoming plasma ﬂow of ⬃6000 km/ s for above
setup condition of ␻ pe / ⍀ce = 120 and M A = 174, which is arather strong shock wave but not unrealistic 共for example, at
RCW 86 shock speed is estimated to 6000⫾ 2800 km/ s in

FIG. 1. 共Color online兲 The structure of a high Mach number shock wave.
From top, the electron momentum of x and y components 共Pex and Pey兲, theion momentum of x and y components 共Pix and Piy兲, the electric ﬁeld 共Ex兲,and the magnetic ﬁeld 共Bz兲. The contour shows ten logarithmic distributionsof the particle number. The electron and ion momenta are normalized by
mu0 and Mu0, respectively; the electric and magnetic ﬁelds are normalized
by the upstream motional electric ﬁeld E0 ⬅ u0B0 / c and upstream magnetic
ﬁeld B0, respectively. The horizontal axis is normalized by u0 / ⍀ci. Several
arrows 共a兲–共e兲 are referred to in Fig. 8. The normalized simulation time forthe ﬁgure is 7.1.

Ref. 20兲. Although, our simulation has been carried out under 1D and reduced mass ratio condition, results will be oneof the useful preparation steps toward complete elucidationof the real supernova shock dynamics. About some twodimensional 共2D兲 effect for the electron dynamics, although
in different parameter ranges, there are, for example, Amanoand Hoshino9 for nonrelativistic case and Kato21 andSpitkovsky22 for relativistic case.
III. SIMULATION RESULTS
A. Overview of shock wave properties

Figure 1 shows a snapshot of a part of the simulation
box in which a shock wave is propagating toward the left.
From top, the electron momentum phase space density of x
and y directions 共Pex and Pey兲, ion momentum phase spacedensity of x and y directions 共Pix and Piy兲, electric ﬁeld Ex,
and magnetic ﬁeld Bz. These quantities are normalized, respectively, by mu0 for the electron, Mu0 for the ion, by the
upstream motional electric ﬁeld E0 ⬅ u0B0 / c, and by the up-

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Structure of a strong supernova shock wave…

0.6

2

Ti / Mus2
Te / Mus

0.5
0.4
0.3
0.2
0.1
0

5
4
3
2
1
0
-1

30
15
0
-15
-30

Bz (low resolution)Bz (high resolution)

6

10
5
10
4
10
103
102
1
10
0
10

Ti /T0
Te /T0
0.5

1

1.5

2

2.5

x
FIG. 3. The effective temperature variation normalized by the shocked ﬂow
energy Mus2 共top兲 and normalized by the initial temperature T0 共bottom兲,where us 共=1.5u0兲 is an averaged injection ﬂow speed in the shock frame.
The dotted and solid lines correspond to the ion and electron temperature,
respectively. The arrow in the top panel at x ⬃ 0.92 shows a position ofabrupt rise in the Te. This position corresponds to the boundary between
regions I and II discussed in Sec. III C. The time in the ﬁgure is the same asthat in Fig. 1.

0.5

1

1.5
x

2

2.5

FIG. 2. Bz proﬁles with raw value 共left tics兲 and with smoothing value 共righttics兲 are drawn at an interval of ⬃0.1⍀−1
ci from bottom to top panel.
Smoothed proﬁles are obtained by spatial average over 5c / ␻ pi with eachdrawing point shifted at an interval of c / ␻ pi. The time of the middle panel
corresponds to the time in Fig. 1.

stream magnetic ﬁeld B0. The spatial unit for the horizontal
axis is u0 / ⍀ci 共⬅LSTR兲, which is a characteristic scale of the
STR, where ⍀ci = eB0 / Mc is the ion cyclotron frequency.
Some arrows labeled 共a兲–共e兲 will be referred to discuss theelectron energy spectra in Sec. III D and Fig. 8. The Ex proﬁle shows large and intense ﬂuctuation generated by microscopic plasma instabilities especially around x = 0.8⬃ 1.2
共see Sec. III C in detail兲.
Since magnetic ﬁeld proﬁle looks unfamiliar comparedwith a classical shock proﬁle, in Fig. 2, smoothing proﬁles
共use right tics兲 are drawn over raw-data proﬁles 共use left tics兲
with time interval ⬃0.1⍀−1ci . The middle panel is a snapshot
at the same time in Fig. 1. The time duration from bottom toa top panel corresponds to about 1.4TSTR 共⬅LSTR / us, characteristic STR passage time兲, where us = 1.5u0 is anaverage incoming ﬂow speed in the shock frame, i.e.,
shock speed. The normalization scales LSTR and TSTR corres-

pond to 115.2c / ␻ pi and 800␻−1
pi , respectively, where␻ pi = 冑4␲n0e2 / M is the ion plasma frequency. Smoothed proﬁles with low-resolution are obtained by spatial average over
5c / ␻ pi with each drawing point shifting at an interval ofc / ␻ pi. Smoothed proﬁles help to identify familiar shock
structure with ampliﬁed downstream ﬁeld about three times
as large as upstream magnetic ﬁeld as expected from
Rankine–Hugoniot 共RH兲 relation 共in our 1D case, compression ratio is three兲. As a result of the strong dissipation
共thermalization兲 process as shown in the next subsection, in a
low-resolution view, the ﬁeld proﬁles and particle distributions 共not shown兲 show little qualitative variation during
their propagation.23,24 We can discuss properties appear in
Fig. 1 共normalized simulation time is 7.1兲 as a typical stateof the shock wave. Figures 3–6, 8, and 9 are the same timewith Fig. 1, and Figs. 7, 10, and 11 include data fromt = 7.1– 7.7.
B. Rapid energy redistribution

Rapid and large velocity spread of the electron is seen atjust the entrance of the STR 共x = 0.8⬃ 1.0兲 in Fig. 1. In
Fig. 3 the quantity of the electron thermalization is shownas the effective temperature, Te, with solid lines comparedto the shock ﬂow energy Mu2s 共top: linear scale兲 and to
the initial temperature Te0 = Ti0 ⬅ T0 共bottom: log-scale兲.
The effective temperature is obtained by calculating兰f共p兲␥m共V − V兲2dp / 兰f共p兲dp, where f is a distribution function with p = ␥mV, ␥ is the Lorentz factor, V is the particle
velocity, and V is the averaged particle velocity. The ion
temperature 共Ti兲 calculated in the same way is also plottedShimada et al., “Structure of a Strong Supernova Shock Wave...” (2010)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6

Phys. Plasmas 17, 032902 共2010兲

Shimada, Hoshino, and Amano

102

ion
electrondiff. × 10

(a)
1

10

energy ratio (% in a logscale)potential

Exbulk flow

032902-4359

ionelectron

100
101
100

(b)

-1

10

-2

10

10-3
10-4

Bz

10-5
101
100

(c)

10-1
0.5

1

1.5
x

2

2.5

FIG. 4. Top: the bulk ﬂow speed of the ion 共solid兲, electron 共dashed兲, andthe ﬂow speed difference of these two components 共dotted, ampliﬁed tentimes兲 normalized by u0. Arrows in the panel shows the direction of theelectron component shooting out of the ion component. middle: Ex proﬁlenormalized by E0. Top and middle proﬁles are smoothed over 25c / ␻ pi witheach smoothing point at an interval of 2c / ␻ pi. Bottom: the electrostatic
potential corresponding to the above Ex proﬁle. The data are normalized by
Mus2 / 2e and smoothed over 5c / ␻ pi, with each smoothing point at an intervalof c / ␻ pi. The time in the ﬁgure is the same as that in Fig. 1.

with dotted lines in Fig. 3. The local enhancement in the Ti atx ⬃ 0.85 comes from non-Maxwellian behavior with superposition of the incoming and the reﬂected ion components.
When Ti enhancement appears due to the ion reﬂection, Te
rises up above Ti 共ⱖ104Te0兲 within a scale of 0.1LSTR 共leading edge of a classical foot region兲. In the past, a simulationof mildly relativistic low Mach number shocks 共M A ⬃ 2兲 under low frequency ratio condition 共␻ pe / ⍀ci ⬍ 1兲 shows quick
electron energization, which may realize energy equipartition
between ions and electrons 共e.g., Ref. 25兲. To our knowledge,
however, such strong electron thermalization observed in thepresent simulation 共Te ⱖ Ti兲 has not attained in any othernonrelativistic lower Mach number shock wave simulations.
The remarkable abrupt increase in Te indicated by an arrowat x ⬃ 0.92 corresponds to the position of an onset of the
dynamical change caused by the nonlinear saturation of theplasma instability 共see next subsection兲. This means that thenonlinear saturation process plays an important role in the
rapid energy equipartition between ions and electrons. The
local hump of Te ⬎ Ti around x = 1.5⬃ 1.9 is produced intermittently by a temporary variation in the macroscopic crossshock potential structure. In Fig. 4, proﬁles of Ex 共middle,
normalized by E0兲 and the cross-shock potential 共bottom,
normalized by Mus2 / 2e兲 are shown after smoothed over ion-

10-2
10-3
-4

10

10-5

Ex
0.5

1

1.5

2

2.5

x
FIG. 5. The variation in the energy ratio 共%兲 among 共a兲 the ions and electrons 共dotted and solid lines兲, 共b兲 the magnetic ﬁeld, and 共c兲 the electric ﬁeld
from the upstream to the downstream side of the STR 共x = 0.5– 2.5兲. Notethat the vertical axis is a log-scale. The time in the ﬁgure is the same as thatin Fig. 1.

scale 共25c / ␻ pi for Ex and 5c / ␻ pi for the potential兲. Overallvariation in Ex is brought by the polarization between thebulk electron and bulk gyrating ion. The bulk electron isovershooting the bulk ion motion. In the top panel of Fig. 4,
bulk ﬂow variations in the ion 共solid兲 and the electron
共dashed兲 are shown with the electron excess ampliﬁed tentimes 共dotted兲. Several arrows indicate directions overshooting electrons converge to or diverge to, which causes Exvariation and resultant cross-shock potential. The crossshock potential is one of the fundamental shock structures26
and also a critical quantity in terms of the electron injection
efﬁciency in DSA process.14,27 Time-averaged cross-shockpotential is about 0.3– 0.4Mus2 / 2e in our case and consistent
to the value used by Amano and Hoshino.14
Figure 5 illustrates a spatial variation in the energy density distribution 共percent in a log-scale兲 among the 共a兲 particles, 共b兲 magnetic ﬁeld Bz, and 共c兲 electric ﬁeld Ex. In panel
共a兲, the ion energy is drawn by dotted line, and the electronenergy is by solid line. The electron energy is calculated by
共␥ − 1兲mc2 共for ion energy, m is replaced by M兲. Each quantity is normalized by the total energy of all components.
Initially, the energy density ratio is about 99% for the ion and第 4 章 論文集

360032902-5Structure of a strong supernova shock wave…

FIG. 6. 共Color online兲 The enlargement picture of Fig. 1, an entrance of the
STR. From top, ion phase space density 共Pix兲, Ex, electric current Jy of theelectron 共the lower line兲 and total 共the upper line兲, and Bz. Smoothed proﬁles
共Jy and Ex兲 are obtained by averaged over c / ␻ pe, with each drawing point
shifted at an interval of c / 6␻ pe. The vertical line indicates boundary between the regions I and II. The time in the ﬁgure is the same as that inFig. 1.

about 1.0% for the electron. Almost half of the energy initially carried by the incoming ion is converted quite rapidly
into the electrons within a scale of ⬍0.1LSTR. The maximum
energies of Bz and Ex are only of the order of ⬃1% and⬃0.1%, respectively. Although, the energy carried by theﬁelds is much less than the particle energy, the ﬁelds areindispensable especially to the electron energization process
as shown below.
C. Structure of the STR

Figure 6 shows an enlargement picture of the entrance of
the STR 共x = 0.8⬃ 1.15兲 with Pix, Ex, the electric current ofy-component Jy 共in an online color version, the total current
is black line and the electron current is red line兲, and Bz,
from top to bottom. This region can be, roughly speaking,divided into two regions by the vertical line at x = 0.92. Following the change in the ion phase space distribution, otherphysical quantities also show clear qualitative change aroundthis line. The left region 共we call here “region I”兲 is where
we can easily distinguish the incoming ion 共positive velocity
component兲 from the reﬂected ion 共negative velocity component兲 in the Pix panel. In the region behind region I 共we callhere “region II”兲, the mixing between the incoming and thereﬂected ion population progresses rapidly in the phasespace. Let us point out some conspicuous structures seen inthese regions: the leading packet of the magnetic ﬁeld 共region I兲 and following magnetic solitary humps with largeamplitude electric ﬁeld oscillation 共region II兲. In region I, theamplitude of the magnetic ﬁeld oscillation reaches ten times

Phys. Plasmas 17, 032902 共2010兲

FIG. 7. 共Color online兲 Trajectories of large-amplitude electric ﬁeld 共Ex兲 inthe x-time phase space. The relative amplitude of Ex is shown by the contour
bar. Time is normalized by TSTR. The origin of the time in the ﬁgure is thesame as that in Fig. 1.

as large as the averaged value obtained from the RH relation,and in region II, the amplitude of the electric ﬁeld oscillation
has over 100 times as large as upstream motional electricﬁeld 共E0兲. In region I, the nonlinear evolution of the twostream instability between the decelerated hot electron andthe incoming ion28 causes some ﬂuctuation on the ionacoustic mode. As a result, a series of electrostatic potentials
is generated, which is seen as a modulation of the incoming
ion as well as the electric ﬁeld oscillation. Some electrons
are trapped around separatrix of these potentials and, consequently, dragged toward the +y direction due to motional
electric ﬁeld. Untrapped electrons in nonseparatrix region
tend to drift toward the −y direction under the inﬂuence of
local E ⫻ B drift. As a result, a series of clear electric current
ﬁlaments is formed by the trapped and untrapped electrons
共spiky variation from negative to positive in the Jy panel兲.
Since the partial derivatives along y and z vanish in our 1D
geometry and, in the Ampere’s circuital law, the Jy term ismore dominant here than the displacement current term⳵Ey / ⳵t, the relation reduces to −兩⳵Bz / ⳵x兩 ⬀ Jy, which facilitates the spatial modulation of Bz in the leading packet. A
series of resultant negative-positive structures of the magnetic ﬁeld indicates the possibility of the magnetic reconnection when we carry out 2D or three-dimensional 共3D兲 simulations. The leading packet propagates toward upstream witha velocity nearly equal to the shock speed, while each wavy
component of a large-amplitude magnetic ﬁeld drifts toward
the downstream in the shock frame.

Shimada et al., “Structure of a Strong Supernova Shock Wave...” (2010)
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Shimada, Hoshino, and Amano

5

6

number of electrons

361

10

10

105

10

104

10

4

1.5 < γ < 1.7

3

102

103

γ > 2.2

101

2

10

100

1

10

(A)

(B)

0

10

1

2

3

4

5

6

7

2

1

2

3

4

5

γ

Mu0 /210

corresponding
Maxwellian

510a. x=0.85 - 0.95
b. x=0.95 - 1.1
c. x=1.1 - 1.3
d. x=1.3 - 1.6
e. x=1.6 - 1.9

4

10

3

4

4×10
5

10

4

102

3

101

1

density

2

(C)

0

10

γ < 1.1

4

6×10

6

number of electrons

8×104

2

3

4

5

6

7

2

Mu0 /2
FIG. 8. 共Color online兲 The electron energy spectra. The energy is normalized by the ion injection ﬂow energy 共Mu20 / 2兲 in panels 共A兲 and 共C兲. Eachspectrum in panels 共A兲 and 共B兲 is sampled, from left, at just the entrance of
the STR 关共a兲: x = 0.85– 0.95兴, around the ramp region 关共b兲: x = 0.95– 1.1兴,around the magnetic overshoot region 关共c兲: x = 1.1– 1.3兴, around the magnetic undershoot region 关共d兲: x = 1.3– 1.6兴, and its downstream region 关共e兲:x = 1.6– 1.9兴. These regions correspond to the regions indicated by arrows
共a兲–共e兲 in Fig. 1. In panel 共C兲 the energy spectrum 共c兲 is drawn with acorresponding Maxwellian with the same effective temperature in this region. The time in the ﬁgure is the same as that in Fig. 1.

After the electron is heated rapidly and the ion components begin to be diffusive in region I, another instability
between the incoming and reﬂected ions 共ion-ion instability
hereafter兲 evolves. In region II, mixing between the incoming and the reﬂected ion components becomes almost complete. The trapped electrons in region I are detrapped here,
and clear Jy ﬁlaments and negative Bz disappear in region II.
During this process, some components in the leading packet
are modulated into the magnetic humps. A possible cause of
the magnetic hump formation is evolution and saturation of
the ion-ion instability, not general ion-ion “stream” instability 共e.g., Wu et al.29 in 1D and Ohira and Takahara30 in 2D兲.
This ion-ion instability looks nonlinear direct coupling between the incoming and reﬂected ion modulation, which is
supported by the hot electron on the ion-acoustic mode. The
linear analysis shows two-way unstable mode 共corresponding to the incoming and reﬂected ion兲 with the wavelength⬃2c / ␻ pi, which is a typical scale of the magnetic humps
with rather large growth rate of 20⍀ci – 30⍀ci 共see the
Appendix兲. Once the magnetic hump structures are formed,
they survive over a scale of ⬃LSTR in the downstream re-

1

0.8

1

1.2

1.4

1.6

1.8

x
FIG. 9. The electron number distribution with energy ␥ ⬎ 2.2 共top dashedline兲, 1.5⬍ ␥ ⬍ 1.7 共top solid line兲, and ␥ ⬍ 1.1 共middle solid line兲. The
electron density proﬁle normalized by the upstream density is shown forreference 共bottom兲. The density proﬁle is smoothed by averaged over c / ␻ pi.
Note that the top vertical axis is a log-scale. The time in the ﬁgure is thesame as that in Fig. 1.

gion. Some analytical study about the magnetic humps 共magnetosonic solitons兲, although under much smaller amplitudecondition, is reported by Pokhotelov et al.31
The Pix panel in Fig. 6 shows that a series of ion phaseholes is generated by the ion-ion instability in region II. In
Fig. 7, trajectories of the ion phase holes are shown as the
propagation of accompanying Ex in an x-time phase space.
The two-way complex propagation of the ion phase holes
affects electron energization process in a stochastic way.
These two-way streaming of the ion phase holes can survive
Landau damping because the phase speed of the ion
phase holes is much smaller than the thermal speed of the hot
electron.
In spite of such a high Mach number shock, Buneman
instability between the incoming electron and the reﬂected
ion previously reported 共e.g., Ref. 1兲 does not have a strong
effect on the electron energization here in comparison withother instabilities. The Buneman instability, however, plays
an important role to decelerate electrons and makes velocity
difference from the incoming ion population at the leadingedge of the STR.
When plasma ␤ is changed to a larger value, for example, ⬃1.0, we conﬁrm that a structure of the STR is almost identical to ␤ ⬃ 0.1 case here 共not shown兲. The STR

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Structure of a strong supernova shock wave…

FIG. 10. 共Color online兲 An example of the electron orbit around the entrance of the STR on the time-stacked proﬁles of the magnetic ﬁeld Bz
共right兲. The strength of Bz is shown by the contour. The left panel shows theelectron energy variation. The origin of the time in the ﬁgure is the same asthat in Fig. 1.

structure depends strongly on the separation in the velocity
phase space between the incoming and the reﬂected ion components and on their thermal width because that affects theway of the free energy release via evolution of the plasma
kinetic instabilities. At a high Mach number regime treated
here, the velocity separation is large enough to trigger strong
nonlinear evolution of the two-stream instability, whenever ␤
may be 0.1⬃ 1.
D. Electron energization process

As seen in Fig. 5共a兲, the electron energy density becomescomparable to the ion energy density instantaneously. How
much energy is attained by the most energetic electron at the
STR? Figure 8 shows spatial evolution of the electron energy
spectra around the STR: at just the entrance of the STR 关共a兲:
x = 0.85– 0.95兲, around the ramp region 关共b兲: x = 0.95– 1.1兴,around the magnetic overshoot region 关共c兲: x = 1.1– 1.3兴,around the magnetic undershoot region 关共d兲: x = 1.3– 1.6兴,and its downstream region 关共e兲: x = 1.6– 1.9兴. These regions
correspond to the arrows from 共a兲 to 共e兲 in the top of Fig. 1.
In Figs. 8共A兲 and 8共C兲 the energy is normalized by the injection ion ﬂow energy in the downstream frame 共Mu20 / 2兲.
The panel 共B兲 shows spectra in terms of ␥. The spectrum
becomes harder as it goes deeper in the STR. It is surprisingthat some electrons gain energy more than several times of
the incoming ion energy just within the entrance of the STR.

Phys. Plasmas 17, 032902 共2010兲

FIG. 11. 共Color online兲 An example of the electron orbit and energy variation in the downstream side. The format is the same as that in Fig. 10. Twoarrows in the right panel indicate the regions where the electron shows
unmagnetized behavior. The origin of the time in the ﬁgure is the same asthat in Fig. 1.

The most energetic electrons gain energy about 1.5 times
larger than the kinetic energy gain due to the ion specular
reﬂection at the shock front. Each spectrum shows strong
thermalization and a little enhancement in energetic electron
tails. In order to compare thermal population to the Maxwellian, we show the energy spectrum at 共d兲 with a corresponding Maxwellian of the same effective temperature of the region x = 1.3– 1.6 in panel 共C兲. Although we do not ﬁnd a clear
nonthermal tail, some excess is conﬁrmed in higher energy
region compared to the Maxwellian.
In Fig. 9, the electron number distributions of different
energy ranges are shown from x = 0.7 to x = 1.9. The top panel
shows a number of electrons with energy range of 1.5⬍ ␥⬍ 1.7 共solid兲 and of ␥ ⬎ 2.2 共dashed兲. The middle panel
shows a population of ␥ ⬍ 1.1. For reference, the smootheddensity proﬁle 共normalized by the upstream value兲 is drawn
in the bottom panel. Note that y-axis of the top panel is a
log-scale. The bulk electron distribution of ␥ ⬍ 1.1 does notentirely follow the shock-compressed density proﬁle. To
keep the shock-compressed density, more energetic electrons
are needed to appear behind x ⬃ 0.9. The lowest energy electrons 共␥ ⬍ 1.1兲 show a little but clear increase with a peak at
the leading packet. More energetic electrons in the top panel
show exponential rise and keep on their ﬂux in the downstream side. The width of the rising scale is comparable to
the Larmor radius 共rL兲 of the energetic electrons 共rL = 0.2 for␥ = 2.2 electrons and rL = 0.11⬃ 0.14 for ␥ = 1.5⬃ 1.7 elec-

Shimada et al., “Structure of a Strong Supernova Shock Wave...” (2010)
032902-8

Phys. Plasmas 17, 032902 共2010兲

Shimada, Hoshino, and Amano

trons兲. The energetic particles are conﬁned tightly in the STR
because the magnetic ﬁeld has the perpendicular geometry to
the shock normal direction. In our simulation, contrary to
expectation from DSA mechanism, the energetic electrons
are not hovering around over the upstream and downstream
regions.
Next, electron trajectories are investigated. Almost all
particles experience both of acceleration and deceleration ina long time duration. We focus here acceleration pattern andpick up two trajectories typical in the leading edge of the
STR 共Fig. 10兲 and its downstream region 共Fig. 11兲. In theright panel of Fig. 10, the electron trajectory is drawn overtime-stacked magnetic ﬁeld 共Bz兲 proﬁle. The strength of Bz isshowed by contour bar. A background thermal electron injected into the front of the STR gains energy due to thetrapping by the leading packet 共t = 0.0– 0.066兲 as well as due
to bouncing motion between the leading packet and magnetic
hump 共t ⱖ 0.066兲 up to ␥ ⬃ 3 共about 200 times as large as theinitial energy兲. The kinetic energy gain is brought nonadiabatically. The leading packet is propagating toward upstream
as a whole structure 共an array of stripes from the lower-rightto the upper-left in Fig. 10兲, but each large-amplitude wavy
structure is propagating toward downstream 共each stripe istoward the upper-right兲. On the other hand, a series of magnetic humps is almost standing in the electron bulk ﬂow. As
a result, the large-amplitude wavy structure is propagating
toward the humps. A electron is conﬁned and gains energy
through multiple nonlinear interaction with these converging
large-amplitude Bz structure similar to the ﬁrst-order Fermi
type acceleration mechanism.
Figure 11 shows another electron orbit among a series ofthe magnetic humps in the same format with Fig. 10. Theenergization process here is rather mild and stochastic. We
conﬁrm that the kinetic energy gain is also brought in a
nonadiabatic manner. A rather energetic electron has alarge Larmor radius; the effect of the ﬁelds cannot be symmetric for a round-trip of the electron gyromotion so thatstochastic energy change is brought about. In Fig. 11, theelectron shows unmagnetized behavior around the time
t = 0.05– 0.06 and t = 0.1– 0.15 共indicated by arrows in theright panel兲 corresponding to the turning point of the gyromotion. The gyro-orbit in a Bz ⬃ 0 ﬁeld has a quite small
curvature so that the electron is going along y-direction for awhile. The magnetic mirror force of the humps also keeps
the electron in the Bz ⬃ 0 region in this case. After detrappedfrom Bz ⬃ 0 region and magnetized, the electron gains someenergy due to the motional electric ﬁeld like so-called“pick-up ion” process. After that, the electron is bouncing
between humps and gains energy gradually up to ␥ ⬃ 4. The
nonlinear interaction with these standing magnetic humps
does not, however, always bring energy gain. It depends not
only on the local variation in the electric ﬁelds Ex 共see themiddle panel of Fig. 4兲 but also dominantly on the accumulation of the local Ey variation caused by the bulk ﬂow andBz variation. Around the region shown in Fig. 11, there is still
a positive and negative bulk ﬂow 共⬅u兲 variation due to themacroscopic ion gyromotion as shown in the top of Fig. 4.
The variation in Bz 共seen as amplitude variation of the magnetic humps兲 is connected with the bulk ﬂow variation. The

363

gyromotion of the energetic electrons can be synchronouswith the motional electric ﬁeld Ey = −e共⫾u兲Bz along they-direction. As a result, there is sometimes net energy gain
through a round-trip of the gyration.

IV. SUMMARY AND DISCUSSION

The numerical simulation of a collisionless shock wave
is reported under high Mach number and high frequency
ratio condition 共M A = 174 and ␻ pe / ⍀ce = 120兲 similar to shockwaves generated by the strong blast wave of supernovae. We
focus on the electron dynamics through the nonlinear
particle-ﬁeld coupling process. We found that the ﬁrst rapid
electron acceleration occurs in the thin shock front region.
Nonlinear evolution of the plasma instabilities between theions and electrons and following the ion-ion instability
causes strong particle energization. At the moment of thenonlinear saturation of the instability, strong mixing occurs
between the incoming and reﬂected ion component in thevelocity phase space as well as some considerable change inthe ﬁeld structures, for example, from the magnetic leading
packet to the magnetic solitary hump. The leading packet
consists of a series of large-amplitude negative-positive magnetic ﬁelds. 共When a 2D or 3D simulation is carried out, we
can expect an occurrence of the magnetic reconnection there.
It may bring about more interesting electron-ﬁeld dynamics.兲
Since each component of the leading packet and the magnetic hump are converging, some electrons scattered andbouncing between these two structures gain energy rapidly
by the ﬁrst-order Fermi type process. Some electrons aroundmagnetic humps also gain energy in a rather stochastic manner. In the magnetic hump region, there is still nonzero bulk
ﬂow variation due to the ion gyromotion. The electrons gain
energy not only due to the local Ex resulted from nonlinear
instability evolution but also due to the motional electricﬁeld Ey accompanied by the magnetic hump Bz and bulk
ﬂow. Depending on the gyrophase, an energetic electron
sometimes gains energy when its gyromotion couples with
the Ey variation of the ion-scale.
It is surprising that the electron energy density arises
rapidly at just the entrance of the STR and becomes comparable to the ion energy density. This energy equipartition ispushed by the nonlinear saturation process of the plasma
instability. The so-called injection problem in DSA mechanism may be settled if our result remains still true under therealistic dimension and mass ratio condition. We showed thatin our simulation, the energetic electrons are conﬁned within
a STR while they are interacting with the ﬁelds and gaining
energy so that we can estimate the maximum energy ␥max byequating the electron Larmor radius and characteristic scaleof the STR,
1 ⌫ 0u 0
mc2 冑 2
␥max − 1 ⬃
,
2 ⍀ci
eB

共1兲

where ⌫0 is incoming ﬂow Lorentz factor. Using u0= 2 / 3VAM A, the above equation results in第 4 章 論文集

364032902-9

␥max ⬃

Phys. Plasmas 17, 032902 共2010兲

Structure of a strong supernova shock wave…

冑

1+

冉 冊

1 M ␻ pe −2 2 2
M A⌫ 0 .
9 m ⍀ce

共2兲

When we adopt above equation to the shock wave case inthis paper, we have ␥max ⬃ 5, which is consistent to our resultof ␥max ⬃ 4. With realistic mass ratio M / m = 1836, ␥max canreach up to 21 共corresponding to ⬃10 MeV兲. Under a larger
mass ratio condition, the STR becomes wider in terms of thescale of electron dynamics, and the ion inertia becomes relatively larger. Since the ion inertia plays an important role toamplify the ﬁelds discussed in the current paper, we can
expect more electron energization. In fact, some periodic
simulations in Ref. 28 show that when the mass ratio islarger up to the realistic mass ratio, the electric ﬁeld supported by the ion inertia becomes more important in the electron energization process.
Recent observation shows shocked temperature Te ⬍ Ti
for strong shocks 共e.g., Refs. 32–36兲. Contrary to the observation, our simulation result shows Te ⱖ Ti. One reason forthat is a scale difference. Our simulation scale is only about
several LSTR much smaller than observation can resolve,
where LSTR ⬅ u0 / ⍀ci ⬃ 4.6⫻ 105 km 共for interstellar
density of 0.1 cm−3兲, ⬃1.5⫻ 105 km 共for 1.0 cm−3兲 using␻ pe / ⍀ci = 120 condition. The temperature far outside of the
STR is unfortunately beyond the scope of the paper. Anotherreason is a shock conﬁguration treated here, namely, the 1D
and perpendicular shock system, which is a rather strong
restriction. For example, 1D simulation tends to overheat
electrons compared to the 2D simulations.9,37 Althoughstrong thermalization indeed occurs in our simulation, thehot electrons shows non-Maxwellian distribution, and thenonthermal part grows as leaving off the STR. The hot electrons cannot only be a seed of the future nonthermal population but can also contribute to further acceleration by keeping the ﬁeld structures within a scale ⬃LSTR.
Unlike the classical DSA where the electrons are hovering around shocked and unshocked regions due to scattering
by large scale magnetohydrodynamic waves, the electrons in
our simulation experience multiple nonlinear interaction
even within one gyromotion with the large-amplitude ﬁeld
structures, which generated through the nonlinear evolutionof the plasma instabilities. These nonlinear transportations
may lead a breakout of the classical diffusion model. Recently, without the classical diffusion model, Malkov andDiamond38 also discussed particle acceleration process undermultiple nonlinear interactions among a train of magnetic
structure “shocklets” of c / ␻ pi scale in the conﬁned region of
the STR 共shock precursor兲. As a generator of the structure
shocklets, nonlinear evolution of the plasma instability isalso considered between cosmic ray ﬂow and shock incident
ﬂow. Not only for our rather lower energetic regime but also
for higher energetic regime, such microscopic coherent structures due to the nonlinear evolution of the plasma instabilities conﬁned in the STR are expected to play a key role in
the particle acceleration process.

ACKNOWLEDGMENTS

The numerical simulations have been supported by the
SX-6 system at ISAS/JAXA, Japan.
APPENDIX: LINEAR ANALYSIS OF THE PLASMAINSTABILITY

In order to ﬁnd linear growth modes of the time scalelarger than the electron gyration time, we examine the standard plasma dispersion relation under the magnetized electron and unmagnetized ion condition. As seen in the STR,
two ion components of the incoming ions and reﬂected ionsare adopted. We assume that 45% of the incoming ion is
reﬂected at the shock front 共the reﬂection ratio ␣r is 0.45兲.
The relative drift velocity between the two ion components is
assumed as 0.5u0共⬅u0d兲. The net electric current is canceledby the total ﬂows of the two ions and electron components.
As an initial temperature condition, we take 1 ⫻ 103T0 for theincoming ions and 3 ⫻ 103T0 for the reﬂected ions. The magnetic ﬁeld is assumed to be 3B0.
When considering waves perpendicular to the magnetic
ﬁeld, the plasma dielectric tensor ␧共k , ␻兲 is reduced to havethe forms

␻2pe n2
␻2pi0
␻
I
exp共−
␩
兲
+
␨i0Z共␨i0兲n2n⍀ce − ␻ ␻2n ␻ ␩␧xx = 1 + 兺+␻2pir␨irZ共␨ir兲,
␻2
␻2pe
␻
,
2 In exp共− ␩兲n⍀ce − ␻n ␻

␧yy = 1 + 兺

␻2pe
␻
,
2 n共In⬘ − In兲exp共− ␩兲␻n⍀ce − ␻
n␧xy = i 兺

where k and ␻ are the wave number and wave frequency
with the Doppler shift included. The wave frequency in therest frame 共␻r兲 is deﬁned by ␻r = ␻ + ku0d共1 − 2␣r兲. ␻ pe, ␻ pi0,
and ␻ pir are the plasma frequencies for the electron, incoming ion, and reﬂected ion, respectively. In and In⬘ are themodiﬁed Bessel function of nth order with the argument
␩ = v2e k2 / 2⍀2ce and its derivative. We take the summation for
n from ⫺10 to 10 here. Z is the plasma dispersion function
with the argument ␨i0 = 共␻ − ku0d兲 / kvi0 for the incoming ion
component and ␨ir = 共␻ + ku0d兲 / kvir for the reﬂected ion component, where vi0, vir, and ve are respectively the incoming,
reﬂected, and electron thermal velocities. Using the above
dielectric tensor, the dispersion relation is冋 冉 冊册

␧xx ␧yy −ck 2+ ␧2xy = 0.
␻

共A1兲

Figure 12 shows some of the solutions of Eq. 共A1兲. The
top panels include the real part of ␻, and the bottom panels
include the imaginary part of ␻ 共⬅␥, growth rate兲 versus
wavelength ␭. Panels 共a兲 and 共b兲 belong to the incoming ion
because their phase velocity is in the same direction as theincoming ion. Panels 共c兲 and 共d兲 belong to the reﬂected ion
with the same reason. Their harmonics in other wave number

Shimada et al., “Structure of a Strong Supernova Shock Wave...” (2010)
032902-10

-0.2

0.36

-0.21

0.35

-0.22

ω/ωpi

0.34

lation, especially in region II, we cannot separate clearly thelinear growth from other nonlinear plasma actions.
2

-0.25

0.3

-0.26

0.29

-0.27
1

1.5

2
2.5
λ/(c/ωpi)

3

(b)

0.035

-0.28

0.016

1.5

2

2.5
3
λ/(c/ωpi)

3.5

2

2.5
3
λ/(c/ωpi)

3.5

(d)

0.014

0.03

0.012

0.025

0.01

0.02

0.008
0.015

0.006

0.01

0.004

0.005
0

0.002
1

1.5

2
2.5
λ/(c/ωpi)

3

0

1.5

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1

-0.24

0.32
0.31

γ/ωpi

(c)
Te=10000T0
8000T04000T0
2000T0

-0.23

0.33

0.28

Phys. Plasmas 17, 032902 共2010兲

Shimada, Hoshino, and Amano

(a)

0.37

365

FIG. 12. The dispersion relation ␻ vs ␭ 共wave number兲 关共a兲 and 共c兲兴 andgrowth rate ␥ vs ␭ 关共b兲 and 共d兲兴 of the peculiar mode in the regime of theion-ion interaction on the incoming ion 关共a兲 and 共b兲兴 and on the reﬂected ion
关共c兲 and 共d兲兴. ␻ and ␭ are normalized by the plasma frequency of the upstream ion ␻ pi and its inertia c / ␻ pi, respectively.

ranges is not shown. ␻ and ␭ are normalized by the plasma
frequency of the upstream ion ␻ pi and its inertia c / ␻ pi,
respectively. The electron temperature Te is set up to2 ⫻ 103T0 共dotted兲, 4 ⫻ 103T0 共long-dotted兲, 8 ⫻ 103T0
共dashed兲, and 1 ⫻ 104T0 共solid兲. As the electron temperaturerises, the growth rate increases with its peak moving a little
toward a smaller ␭. The solution does not have a strong
dependence on the ion temperature, but smaller ion temperature results in larger growth rate. The wavelength ⬃2c / ␻ pi
共comparable to the normalized width ⌬x ⬃ 0.01兲 around thepeak of the growth rate corresponds to the width of the magnetic hump stated in Sec. III C. Coupling between such linear growth modes in the incoming and reﬂected ion component can be some cause of the magnetic hump generation.
However, since strong nonlinearity is observed in the simu-

第 4 章 論文集

366Electron acceleration via magnetic island coalescence
I. Shinohara, T. Yumura, K. G. Tanaka, and M. Fujimoto
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,
Sagamihara, Kanagawa 229-8510, Japan
ABSTRACTElectron acceleration via fast magnetic island coalescence that happens as quick magnetic reconnection triggering(QMRT) proceeds has been studied. We have carried out a three-dimensional full kinetic simulation of the Harriscurrent sheet with a large enough simulation run for two magnetic islands coalescence. Due to the strong inductive
electric field associated with the non-linear evolution of the lower-hybrid-drift instability and the magnetic island
coalescence process observed in the non-linear stage of the collisionless tearing mode, electrons are significantly
accelerated at around the neutral sheet and the subsequent X-line. The accelerated meandering electrons generated by
the non-linear evolution of the lower-hybrid-drift instability are resulted in QMRT, and QMRT leads to fastmagnetic island coalescence. As a whole, the reconnection triggering and its transition to large-scale structure work
as an effective electron accelerator.

INTRODUCTION
Numbers of observations have shown existence of energetic particles in space, so that particle acceleration
processes is an interesting topic in space plasma physics. In the Earth’s magnetotail case, emergence of MeV range
electrons associated with magnetospheric substorms has been known since 1970s, the production process of suchhigh energy electrons has been thought to be related to magnetotail reconnection (Terasawa and Nishida, 1976)
Recently, Øieroset et al. (2002) and Imada et al. (2007) reported detailed observations of energetic electrons in themagnetotail, and their results clearly show that energetic electrons appear around the magnetic reconnection sites.
Since typical electrostatic potential in the magnetotail is ~100 kV, direct acceleration through the cross-tailpotential is insufficient for the observed electron production, and more efficient acceleration process is necessary. In
the context of magnetic reconnection, several processes and their combinations are proposed to explain the observed
energetic electrons. There are basically four physical elements: (1) the meandering/Speiser motion around the X-line(Speiser, 1965), (2) pre-accelerated electrons around the X-line further accelerated by non-adiabatic acceleration in
the magnetic field pileup region (Büchner and Zelenyi, 1989), (3) Hoshino (2005) demonstrated that surfing
acceleration could generate high energy electrons under strongly driven reconnection, (4) Drake et al. (2006)
proposed that electrons can gain kinetic energy by reflecting from the ends of the contracting magnetic islands.
However, the reality of the electron acceleration process via magnetic reconnection still remains an open question.
In this paper, we point out another important physical element for electron acceleration associated withmagnetic reconnection. Carrying out a full kinetic simulation, we found that magnetic island coalescence can play animportant role for the electron acceleration. After precursor analytical studies of magnetic island coalescence(Zelenyi et al., 1988, 1990), magnetic island coalescence process as the tearing mode proceeds has been paid lessattention. Due to the strong induction electric field associated with tearing magnetic islands coalescence, electrons are
directly accelerated at around the X-line. Since the generated inductive electric field during the coalescence is quite
strong, the acceleration efficiency becomes high. We discuss the efficiency of this acceleration process.
SIMULATION RESULTSFirst of all, 3D simulation results are presented to show rapid magnetic coalescence subsequent to quick
magnetic reconnection triggering. We carried out a 3D electromagnetic full kinetic simulation of an ion-scale current
sheet. As the initial condition, we make use of the Harris current sheet for simplicity; Bx(z)=B0tanh(z/D),

15

Shinohara et al., “Electron acceleration via magnetic island coalescence” (2009)367

Fig. 1 (a) Temporal development of Bz (z=0) Fourier modes. Snapshots of the current density, Jy, contours
obtained at (b) T=4 (mode 2 growing), (c) T=10 (on the way of merging), and (d) T=16i-1.

nCS(z)=NCS/cosh2(z/D), Jy(z)=enCS(Ui+Ue), and Ui/Ue=-Ti,CS/Te,CS. D represents the thickness of the current sheet,
which we set D=0.5i (i=c/pi, the ion inertia length). The mass ration is taken to be mi/me=400, electron thermalvelocity is ve/c=1/3 (c is the light velocity), and pe/e=1.0. (pe and e are the plasma frequency and the electron
cyclotron frequency, respectively.) The simulation box
size in the x direction, Lx, is taken as Lx=24D that
corresponds to twice of the fastest growing tearing
wavelength and Ly almost corresponds to the lowerhybrid-drift wavelength. Periodic boundary conditions
are imposed in the x and y directions, while theconducting walls are set at the z boundaries, z=±4D.
The number of grids is Nx × Ny × Nz = 768 × 32 × 512(corresponding to Lx × Ly × Lz = 12I × 0.32I × 8i),and ~109 particles for both electrons and ions are usedin the calculation. It should be noted that no initial
field perturbation is added to the initial condition. In
the present case, the ion-electron temperature ratio istaken to be Ti,CS/Te,CS=8 so that the ions carry most of
the cross-field current. Stationary cold backgroundplasma is distributed outside the current sheet;tanh2(z/D),
NBK/NCS=0.1,
andnBK(z)=NBK
Te,BK=Ti,BK=Te,CS.
As reported in Shinohara and Fujimoto (2005),
the quick magnetic reconnection triggering is possible
in this initial setup. Figure 1a shows the temporal
Fig.2 Electron energy spectra16

第 4 章 論文集

368

Fig. 3 Positions of all the energetic electrons whose energy -1 > 2 at T=16 i-1 but -1 < 0.6 at T=12 i-1 areplotted. The electrons are sampled at (a) T=12, (b) 13, (c) 14, and (d) 16 i-1.

Fig. 4 Position of all the electrons of -1 > 2 at T=16 i-1. The format is the same as Fig 3.

development of Bz (z=0) Fourier modes. The fastest growing mode of the tearing instability, mode 2 (blue line),
starts to grow at T~1i-1 due to the strong modification of the current sheet structure by the non-linear effects of thelower-hybrid-drift instability. Mode 2 almost saturates at T~8i-1, and at this time, two magnetic islands areproduced. The characteristics of the system evolution are almost the same as the previous results of the Lx=12D case.
(Shinohara and Fujimoto, 2005) After the mode 2 saturation, magnetic island coalescence process can be seen during
T=8~16i-1. As shown in Figure 1a, mode 1 (red line) gradually grows and becomes the dominant mode at T~10i-1.
Figures 1b-d present snapshots of the current density, Jy, contours obtained at T=4 (mode 2 growing), 10 (on the wayof merging), and 16i-1. Figure 1d clearly shows that the magnetic island coalescence is completed at T~16i-1.
Tanaka et al. (2005) pointed out that the same spatial size of a magnetic island can be produced by another initialcondition of the D=1 case. However, the evolution time scale of the present case (T~16i-1) is much faster than thatof the D=1 case (T~45i-1). Moreover, the amount of magnetic flux contained in the large island is found to differ by~10. This difference of magnetic flux suggests the difference of the energy release efficiency.
Figure 2 shows the electron energy spectra integrated over the whole simulation box. During T=2~4i-1,
electron energization can be seen, and it is mostly due to the nonlinear evolution of the lower hybrid drift instability
and the quick magnetic reconnection triggering process. (Shinohara and Fujimoto, 2005) However, the mostprominent result here is that energetic electrons can be efficiently produced during the magnetic island coalescence.
Figure 2 shows that the electron energy spectrum becomes harder as the progress of the magnetic island coalescence,
T=8~16i-1.

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Shinohara et al., “Electron acceleration via magnetic island coalescence” (2009)369

Fig. 5 Accelerated electron trajectories obtained during T=12-16 i-1 (a) in the x-energy and (b) in the x-z
space. The red-colored lines indicate the interval T=13.00-13.33 i-1.

To clarify where and how electrons are accelerated during the magnetic island coalescence, electron trajectories
are examined. In Figure 3, positions of all the energetic electrons whose energy-1 > 2 at T=16 i-1 but -1 < 0.6 at
T=12 i-1 are plotted. ( is the relativistic factor.) Each electron position is colored by its energy. The electrons are
sampled at T=12, 13, 14, and 16 i-1 representing the acceleration associated with the magnetic island coalescence.
At T=12 i-1, all the plotted electrons have not been accelerated (-1 < 0.6), and Figure 3a clearly shows that all theelectrons are located in the upstream of the separatrix. As the magnetic island coalescence goes on, electrons movetoward the X-line and the accelerated electrons emerge in the downstream of the X-line. (Figure 3b, 3c) At final, as
shown in Figure 3d, the accelerated electrons are injected into the same magnetic flux tube. The result suggests thatelectrons are accelerated just around the X-line within the limited time period, T=12-16 i-1. Figure 4 shows the
position of all the electrons of -1 > 2 at T=16 i-1 (without the condition -1 < 0.6 at T=12 i-1). Comparing Figure
4 with Figure 3d, it suggests that the consequent energetic electrons are divided into two components: (1) Electrons
which are pre-energized by the tearing instability mediated by the quick magnetic reconnection triggering. This
component is distributed in the core part of the magnetic island. (2) Electrons which are accelerated during themagnetic island coalescence. This component exists in a magnetic flux tube of the outer part of the magnetic island.
Since the former part is well-energized before T=12 i-1, the acceleration of the latter part is much efficient.
To fully check the acceleration process, trajectories of typical higher energy electrons with -1 > 2 are plotted in
Figure 5. Figure 5 presents accelerated electron trajectories obtained during T=12-16 i-1. Since electron data aresampled intermittent timing, the trajectory lines are not continuous. Figure 5a shows the trajectories in the x andEnergy space. It clearly shows that all the electrons are quickly accelerated from -1 < 0.5 to -1 > 2 at aroundx/D=7.8 where is the X-line. The trajectories in the real x-z space are presented in Figure 5b. It confirms that theinitial acceleration occurs right at the X-line. To emphasize the time period, the trajectories for T=13.00~13.33 i-1
are red-colored. It suggests that the acceleration takes place within short period. Figure 5a also shows that, after
electrons are accelerated up to -1 > 2 at the X-line, they experience further acceleration in the magnetic field pileup
region similarly to previous results (e.g. Hoshino, 2005). The acceleration mechanism for this stage can be the nonadiabatic acceleration in the magnetic field pileup region for electrons with ~1, where 2 is the ratio of the magnetic(b)|B z (z =0)|max

E reconnection / B 0(a)Fig. 6 Time evolution of (a) the maximum |Bz| in the z=0 plane, and (b) the reconnection electric field.

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field curvature and the Larmor radius (Büchner and Zelenyi, 1989). As shown in Figure 3c and 3d, the magnetic flux
loop that contains energetic electrons is really contracted. Thus, the acceleration process proposed by Drake et al.
(2006) may contribute to the acceleration. These results suggest that the acceleration at the X-line works as theinjector for the acceleration in the magnetic field pileup region and in the magnetic flux loop.
Figure 6 shows the time evolution of the magnetic field and the reconnection electric field. At T~4i-1, a jumpof |B z (z=0)|max and enhancement of the reconnection electric field are seen, and they are the signatures of the quick
magnetic reconnection triggering. (Shinohara and Fujimoto, 2005) Corresponding to the magnetic island coalescence,
T=12~16i-1, a relatively large jump of |B z (z=0)|max and the emergence of the strong reconnection electric field areobserved. The observed reconnection rate during the magnetic island coalescence reaches (Erec/VAB0) ~0.4 which ismuch larger rate than those of the single island cases. It should be noted that such a high reconnection rate is possible
without any driven process. Although the magnetic island coalescence is the transient process, it can achieve highreconnection rate, and the enhanced reconnection electric field efficiently accelerate electrons right at the X-line.
DISCUSSIONAs described above, the key process of the electron acceleration associated with the magnetic island coalescenceis the acceleration at the X-line by the enhanced reconnection electric field. Since our simulation imposes the periodic
boundary condition in the y direction, electrons can continuously feel the reconnection electric field until they areejected from the X-line region. However, in reality, electrons can be accelerated within the limited space. Assumingthat electrons are accelerated along the straight orbit with the reconnection electric field, we can estimate how long
distance in the y direction is necessary to produce energetic electrons. The equation of the electron motion is

d
py
dt

e E y . (1)So, we get the following relation.

2 1

1/ 2

e 2 E y2p2
me2c 22 2emc

VA   mi   cE y  c   m  V B  iT  (2) e  A 0 t2

Substituting the simulation parameters and results, VA/c=0.05, mi/me=400, and cEy/VAB0~0.4, we get

2 1

1/ 2

~ 8 iT  (3)Thus, for the =2 electron, it takes T ~0.2 i-1 for the acceleration. This time scale is consistent with the simulation
result so that the assumption for the acceleration process looks good enough. The electron energy gain isE mc2  1 e E y y (4). Finally, the necessary distance in the y direction is expressed byy  i 

2 c   cE 

 1    y 
 VA  VA B0 

1~ 1000 (1) . (5)From (5), 1000 i is required for the production of the =2 electron. With the magnetotail parameter, 1000 i isequivalent to ~ 40 Re (i ~ 250 km). Therefore, even with the large reconnection rate ~0.4, the spatial scale of themagnetotail is smaller for the production of the =2 electron. However, this estimation is only focused on theacceleration right at the X-line. With the combination of the acceleration in the magnetic field pileup region, we can
expect the production of higher energy electrons through the magnetic island coalescence.
Since the present study is limited only for the two magnetic island merging case, how the electron acceleration
efficiency increases in the case of multiple magnetic island coalescence cases should be studied in the near future. If
more number of magnetic island coalescence is taken into account, the temporal and spatial scale length of theenhanced X-line region becomes longer as the size of the coalescence island size evolves larger. We expect that larger
multi-island coalescence can more effectively contribute to the production of the energetic electrons even if the timeinterval producing the energetic particles demonstrated in this paper is short.

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371

SUMMARY AND CONCLUSIONS
Performing a three-dimensional full kinetic simulation of the Harris current sheet, the magnetic island
coalescence has been studied. We found non-thermal electrons are efficiently generated during magnetic island
coalescence. The electron acceleration is mainly due to the meandering/Speiser motion right at the X-line and thegrad B/curvature drift acceleration in the magnetic field pileup region. Large inductive electric field (up to 40%
reconnection rate) generated during the magnetic island coalescence plays an essential role for the observed
acceleration.
ACKNOWLEDGEMENT
This work was supported by the Grant-in-Aid for Creative Scientific Research "The Basic Study of Space
Weather Prediction" (17GS0208, Head Investigator: K. Shibata) from the Ministry of Education, Science, Sports,
Technology, and Culture of Japan. Numerical simulations are performed on NEC SX6 at ISAS/JAXA.
REFERENCES
Büchner, J., and L. M. Zelenyi (1989), Regular and chaotic charged particle motion in magnetotail-like fieldreversals: 1. Basic theory, J. Geophys. Res., 94, 11,821– 11,842.
Drake, J. F., M. Swisdak, H. Che, and M. A. Shay (2006), Electron acceleration from contracting magnetic islands
during reconnection, Nature, 443, 7111, 553–556.
Hoshino, M. (2005), Electron surfing acceleration in magnetic reconnection, J. Geophys. Res., 110, A10, CiteID
A10215.
Imada, S., R. Nakamura, P. W. Daly, M. Hoshino, W. Baumjohann, S. Mühlbachler, A. Balogh, and H. Rème
(2007), Energetic electron acceleration in the downstream reconnection outflow region, J. Geophys. Res., 112,
A3, CiteID A03202.
Øieroset, M., R. P. Lin, T. D. Phan, D. E. Larson, and S. D. Bale (2002), Evidence for Electron Acceleration up to~300 keV in the Magnetic Reconnection Diffusion Region of Earth's Magnetotail, Phys. Rev. Lett., 89, 19, id.
195001.
Shinohara, I., and M. Fujimoto (2005), Formation of thin current layer associated with lower hybrid drift instability
and its relation to quick reconnection triggering, in Frontiers in Magnetospheric Plasma Physics, COSPAR
Colloquia Series Vol. 16, Eds: M. Hoshino. Y. Omura, and L. J. Lanzerotti, pp.123–129.
Speiser, T. W. (1965), Particle trajectories in model current sheet: 1. Analytical solution, J. Geophys. Res., 70,
4219–4226.
Tanaka, K. G., I. Shinohara, and M. Fujimoto (2005), Quick magnetic reconnection triggering in an ion-scale current
sheet: Transition from Type-I to Type-II, Geophys. Res. Lett., 32, 17, CiteID L17106.
Terasawa, T., and A. Nishida (1976), Simultaneous observations of relativistic electron bursts and neutral-line
signatures in the magnetotail, Planet. Space. Sci., 24, 9, 855–866.
Zelenyi, L. M., A. L. Taktakishvili, A Kinetic Theory of the Magnetic Islands Merging Instability, Plasma Physics
and Controlled Fusion, 30, 6, 663-679, 1988.
Zelenyi. L. M., J. G. Lominadze, A. L. Taktakishvili, Generation of energetic proton and electron bursts in planetary
magnetotails, J. Geophys. Res., 95, A4, 3883-3891, 1990.

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Magnetic Reconnection by a Self-Retreating X LineM. Oka,1,* M. Fujimoto,2 T. K. M. Nakamura,2 I. Shinohara,2 and K.-I. Nishikawa1
1

Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, Alabama 35805, USA
2
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan(Received 15 August 2008; published 14 November 2008)
Particle-in-cell simulations of collisionless magnetic reconnection are performed to study asymmetricreconnection in which an outﬂow is blocked by a hard wall while leaving sufﬁciently large room for theoutﬂow of the opposite direction. This condition leads to a slow, roughly constant motion of the diffusion
region away from the wall, the so-called ‘‘X-line retreat.’’ The typical retreat speed is 0:1 times the
Alfvén speed. At the diffusion region, ion ﬂow pattern shows strong asymmetry and the ion stagnation
point and the X line are not collocated. A surprise, however, is that the reconnection rate remains the same
unaffected by the retreat motion.
DOI: 10.1103/PhysRevLett.101.205004

PACS numbers: 52.35.Vd, 52.65.Rr, 94.30.cp

Magnetic reconnection triggers many explosive phenomena in laboratory and astrophysical plasma. Importance lies in the diffusion region where the MHD breaks
down and the kinetic scale comes in. It is a scientiﬁcchallenge to understand the structure of the diffusion region and its role in controlling the reconnection rate.
Owing to the growing power of computer capabilities,
full particle simulations has become popular these days.
It is now clear that the out-of-plane current remains localized and the reconnection rate remains fast while electrons
form a high-velocity jet that extends large distances downstream from the X line [1–4]. The region with the localized
current is termed ‘‘inner diffusion region’’ whereas the restof the diffusion region including the elongated electron
current layer is termed ‘‘outer diffusion region.’’ In general, simulations generate magnetic reconnection that issymmetric in both inﬂow and outﬂow directions.
Actual magnetic reconnection, however, occurs in amore complicated situation. During a magnetic reconnection in Earth’s magnetotail, an earthward outﬂow soon
collides with the dipole ﬁeld while the tailward outﬂow
is directed to the interplanetary space. As for the solar
ﬂares that take place above the area of emerging ﬂuxrope,
the upward outﬂow eventually merges into the solar windwhile the downward outﬂow collides with the magnetic
obstacle in the lower part of the corona. In these cases,
magnetic reconnection may not be symmetric about the Xline.
In this Letter, we investigate the consequences of blocking one side of the outﬂow. A hard wall is set up just aheadof an outﬂow while leaving a sufﬁciently large room for theoutﬂow of the opposite direction. As a result, reconnectedmagnetic ﬁelds are piled-up against the wall. By the time
the pile-up region reaches the diffusion region, the X line
starts to move away from the wall (‘‘X-line retreat’’).
Under such a strong inﬂuence of the boundary condition,
the structure of the diffusion region is modulated. In thepast, the X-line motion was considered by an analytical
0031-9007=08=101(20)=205004(4)treatment [5], MHD simulations [6], and a full particle
simulation [7], although these work did not shed light on
the modulated structure of the diffusion region. This Letter
will be the ﬁrst to report the structure of the diffusion
region in a self-consistent simulation of asymmetric outﬂow reconnection.
We utilized the two dimensional, particle-in-cell code[8,9]. The initial condition consists of a Harris current
sheet. The antiparallel magnetic ﬁeld and the density are
given by By ¼ B0 tanh½ðx  Lx =2Þ=D and Ncs ¼
N0 =cosh2 ½ðx  Lx =2Þ=D, respectively, where B0 is themagnetic ﬁeld at the inﬂow boundary, D is the halfthickness of the current sheet, N0 is the density at thecurrent sheet center, and Lx and Ly are the domain sizein x^ and y^ direction, respectively. D is chosen to be 1:5iwhere i is the ion inertial length. The inﬂow, backgroundplasma is represented by NB ¼ NB0 f1  1=cosh2 ½ðx 
Lx =2Þ=Dg where NB0 ¼ 0:2N0 . The pressure imbalanceby the nonuniform density is not important to the resultspresented below. The electron-to-ion temperature ratio is
set to be Te =Ti ¼ 1=5 for both the current sheet and thebackground. The frequency ratio !pe =ce ¼ 1:5 where!pe and ce are the electron plasma frequency and theelectron cyclotron frequency, respectively. The conducting
walls are used at x ¼ 0 and x ¼ Lx and symmetric boundary conditions were used at y ¼ 0 and y ¼ Ly . At thesymmetric boundary, particles are specularly reﬂected
whereas ﬁelds are given by @Bn =@n ¼ 0, Bt ¼ 0,
@Et =@n ¼ 0, and En ¼ 0, where E and B are the electricﬁeld and the magnetic ﬁeld vectors and the subscripts n andt denote the normal and tangential components, respectively. Reconnection is initiated at the distance H from thebottom (y ¼ 0) wall with a small magnetic island given by
the vector potential Az ¼ A0 expf½ðx  Lx =2Þ2 þ ðy 
HÞ2 =ð22i Þg where A0 ¼ 0:1B0 i . Ly is chosen to be large
enough to simulate effectively the free boundary conditionof the reality.

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Five simulation runs are performed to study the dependence on the initial distance H. The values of H along withthe other initial values are compiled in Table I. The initial
X line is at the center of the domain for Run 4, and thus, it
is a nonretreating case. Run 5 is an additional case intended
to study the dependence on the ion to electron mass ratio  mi =me . In all runs, we used average of 64 particles ineach grid cell. At the center of the current sheet, 276
particles per cell represents the unit density N0 .
Upper panels of Fig. 1 show snapshots of electron outof-plane current for Run 1. It is evident that as magnetic
reconnection proceeds, a downward ﬂow is blocked by thebottom wall and creates the pile-up region which expands
in time and eventually pushes the diffusion region upward.
The upward ﬂow, on the other hand, continues to grow inlength until its leading edge reaches the upper wall. The
choice of Ly is not sensitive to the results presented in thisLetter, and thus any effect from the upper wall is notimportant. The diffusion region is also elongated, and theﬁnal length of the electron current sheet becomes as long as
50i .
The black curve in Fig. 1(d) shows the X-line positiondeﬁned by the Bx reversal point, Bx ¼ 0. A distinct retreat
process starts at ci t  85. The X line moves away fromthe wall monotonically. Although a jump appears at ci t 
145 due to a magnetic island generated by a secondary
tearing instability, the X-line motion continues even after
the upward outﬂow reaches the upper wall. The approximate time of the outﬂow collision with the wall is ci t 
180.
Figure 1(e) shows various ﬂow speeds. While the upward ion ﬂow becomes super-Alfvénic and continues toincrease until the time of collision with the wall, the downward ion ﬂow reaches 0:3vA at ci t  80 when it isblocked by the bottom wall (marked by the arrow). The
downward ﬂow speed soon decreases and is followed by
the upward motion of the X line. In contrast, the retreat
speed and the ion ﬂow speed at the X line are roughly
constant at the same value 0:1vA , although disturbances
appear at ci t  145 due to a secondary tearing.
A striking feature can be found around the diffusion
region. Figure 2 shows an enlarged view of the diffusion
region for Run 1 at ci t  115 when the upward electron

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speed is saturated. In this ﬁgure, data are accumulated over
one gyro period. Coordinates are shifted so that the X line
comes to the origin. This simulation, along with otherswith different H, reveals that the ion and electron stagnation points are not collocated with the X line. As the pile-up
region expands, the lower half of the outer diffusion region
is modulated, and ion ﬂows are deﬂected. As a result, theion stagnation point is shifted downward about the distanceof i from the X line (marked by the blue horizontal
arrow). This separation between the stagnation point and
the X line is almost constant in time as shown in Fig. 1(d).
The shift of the ion stagnation point leads to an upward ion
ﬂow at the X line, resulting in a slow, rising motion of the X
line. While there is a high speed ion ﬂow in the upper halfof the diffusion region (the curve labeled ‘‘I’’ in the bottom
panel), the typical value of the upward ion ﬂow at the X
line was 0:1vA (‘‘II’’). The deﬂection can be recognized

TABLE I. Initial parameters and results of each run. Lengths
are normalized by the ion inertial length i , times are by the
inverse ion cyclotron frequency 1
ci and speeds are by the ion
Alfvén speed vA .  is the grid size. See text for details.

Run 1
Run 2
Run 3
Run 4
Run 5

Lx

Ly

H



i =

tcol

tret

vp

76.8
76.8
102.4
102.4
170.7

204.8
204.8
204.8
204.8
341.3

12.8
25.6
51.2
102.4
25.6

25
25
25
25
9

20
20
20
20
12

80
90
110
120
90

83
98
140

98

0.3
0.5
1.0

0.5

FIG. 1 (color online). Overview of the retreating X line for
Run 1: (a)–(c) normal electron current Jez . (d) positions of the X
line and the stagnation points. (e) the X line retreat speed,
upward and downward maximum ﬂow speeds for ions andelectrons, and the ﬂow speeds at the X line for ions and electrons.
A stagnation point is deﬁned by the ﬂow reversal point.

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FIG. 2 (color online). Enlarged view of the retreating X line for Run 1 obtained at ci t ¼ 115. The center panel shows the electron
ﬂow in y^ direction Vey (color code) as well as ion ﬂow directions in the x  y plane (vectors). The left panel shows quantities Bx ,pﬃﬃﬃﬃ
Vey = , and Viy along x ¼ 0 whereas the right panel shows the electric ﬁeld terms of the generalized Ohm’s law along x ¼ 0. The
bottom panel shows Viy cut along the horizontal lines in the color code. The purple and yellow arrows in the right panel show the outerand inner diffusion regions (DR), respectively.

as the double peak feature in the cut of the downward
outﬂow (‘‘III’’).
In contrast to ions, electrons do not show strong deﬂection and exhibit well-developed outﬂow jets both upwardand downward, although the downward jet is limited inlength. The typical length of the downward electron jet is5i and remains constant throughout the run. The maximum value of the electron ﬂow speed, however, is quite
fast and reaches 0:65vAe , as can be seen in the left panel.
This is almost the same as the maximum speed of the
upward electron jet. We have found that all retreating cases
show nearly the same maximum speed of the downward
electron jet, i.e., 0:7vAe . The length of the jet, on the other
hand, becomes slightly larger for larger H. For Run 3, thelength reached 10i .
We further examined the out-of-plane component of thegeneralized Ohm’s law along x ¼ 0 as shown in the rightpanel of Fig. 2 [4]. It is expressed asEz ¼ 

me vey @vez 1
1
þ vey Bx  r  
e
@y
c
ne

(1)

where ve is the electron bulk velocity,  ¼ pexz x^ þ peyz y^is the ﬂux of z-directed electron momentum in the reconnection plane (not including convection of momentum)with pe the electron pressure tensor. The observed electricﬁeld (solid dark curve) is balanced by the sum (dashedcurve) of the electron inertia, the Lorentz force, and thedivergence of the momentum ﬂux. Note that the time
variation term is negligible and is not considered here.
As is the case for nonmoving, symmetric reconnection
reported previously [4], the major contribution to the inner
diffusion comes from the pressure term. The upward outerdiffusion region is also similar to the symmetric case. On
the other hand, the downward outer diffusion region ishighly modiﬁed. Steeper acceleration of vey and piled-up
Bx make the Lorentz term proﬁle to be sharper. The inertia
term with the opposite sign is also enhanced because ofsteeper gradient of vez in the short jet. The striking featureis that both are enhanced in a balanced way such that Ez as
a whole shows the almost ﬂat proﬁle along the x-axis.
The above ﬁnding leads to the idea that the downward
outer diffusion region adjusts itself to buffer the effects of
the wall such that the reconnection electric ﬁeld is determined irrespective of the asymmetric boundary condition.
This idea is inspected in Fig. 3 which shows the reconnection rate, ER , of all runs. ER is determined by averaging theelectric ﬁeld over a small square region centered at the X
line. We have veriﬁed that taking the time derivative of thetotal magnetic ﬂux between the X line, and the bottom
center of the pile-up region yields identical proﬁle as ER .
For Runs 1–3 and 5, the upward outﬂow jet reaches theupper wall well after ci t  160. For Run 4, both upward

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FIG. 3 (color online). Dependencies of reconnection rate on
the initial distance from the wall (upper panel) and the mass ratio (lower panel). The horizontal arrows show periods of secondary island appearances. The straight vertical arrows show theapproximate times of the collision of the downward jet tcol which
is deﬁned by the peak in the proﬁles of the downward ion ﬂow
speed (see, for example, the blue arrow in Fig. 1). The dashedvertical arrows show the start times of the X-line retreat, tret . The
numbers annotated below each dashed arrow are the downward
ion ﬂow maximum vp =vA . There is no vertical arrow for Run 4
because it is the nonretreating case.

and downward jets collide at around ci t  120. It isevident that retreating cases (Runs 1–3) show almost thesame proﬁle as the nonretreating case (Run 4). Moreover,the proﬁle for Run 5 conﬁrms the earlier result that ER does
not depend on  [10]. Relatively large ﬂuctuations are due
to secondary islands whose appearance times are indicatedby the horizontal arrows. Also illustrated in Fig. 3 are thetimings of the jet collision with the wall as well as the
X-line retreat. It clearly shows that the rising motion of the
diffusion region is associated with the expansion of thepile-up region rather than the peak ER .
To summarize, the retreat motion of the X line is constant (0:1vA ). This speed does not depend on the initial
distance of the X line from the wall, H, except of course thenonretreating case. The downward outﬂow region islargely modulated by the asymmetry. The ion deﬂection
pattern depends on H and so is the length of the electron

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downward jet. The outer diffusion region is not elongatedas in the case of symmetric reconnection. In terms of thereconnection rate, however, it stays the same as the symmetric case thanks to the internal balance within the downward outer diffusion region.
In addition to the initial distance from the wall, a dependence on the mass ratio  should also be addressed.
What we have found here is the tendency for more magnetic islands to emerge with smaller . These islands mayhelp maintain fast reconnection [1]. In fact, as shown in
Fig. 3, the  ¼ 9 case (Run 5) generated two large islands,
yet keeping the same reconnection rate as the other runs.
Another important aspect of the  dependence is theseparation between the ion stagnation point and the X
line, but we could not measure xs for the  ¼ 9 casebecause of the successive generation of the secondaryislands. The mass dependence of the size of the separation
will be studied in our future paper. Note the fact, however,that the retreat speed was roughly equal to the ion ﬂow
speed so that the X line and the stagnation will be collocated in the X-line rest frame of reference.
Finally, it should be mentioned that the X-line speed canbe faster than the self-retreating speed of 0:1vA if, say, it isblown by another dominant X line. Recent two-ﬂuid simulations show that its reconnection rate is reduced andreconnection at the moving X line is eventually terminated.
Particle simulations for this kind of situations are alsoneeded.
This work was partially supported by the Grant-in-Aid
for Creative Scientiﬁc Research (17GS0208) from the
MEXT, Japan. M. O. was supported by JSPS.

*mitsuo.oka@uah.edu[1] W. Daughton et al., Phys. Plasmas 13, 072101 (2006).
[2] K. Fujimoto, Phys. Plasmas 13, 072904 (2006).
[3] H. Karimabadi, Geophys. Res. Lett. 34, L13104 (2007).
[4] M. A. Shay, Phys. Rev. Lett. 99, 155002 (2007).
[5] C. J. Owen and S. W. H. Cowley, Planet. Space Sci. 35,
451 (1987).
[6] E. R. Priest and T. G. Forbes, Astron. Astrophys. Rev. 10,313 (2002).
[7] M. Hesse and K. Schindler, Earth, Planets & Space 53,
645 (2001).
[8] M. Hoshino, J. Geophys. Res. 92, 7368 (1987).
[9] I. Shinohara et al., Phys. Rev. Lett. 87, 095001 (2001).
[10] M. Hesse, Phys. Plasmas 6, 1781 (1999).

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PHYSICS OF PLASMAS 15, 032902 共2008兲

Impulsive chromospheric heating of two-ribbon ﬂares
by the fast reconnection mechanism
M. Ugaia兲
Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama 790-8577, Japan

共Received 10 December 2007; accepted 29 January 2008; published online 10 March 2008兲
Chromospheric heating of two-ribbon ﬂares is quantitatively studied for different values of R0, theratio of the chromospheric plasma density to the coronal one, on the basis of the spontaneous fastreconnection model. In general, occurrence of impulsive chromospheric joule heating is delayed forthe larger R0 because of more Alfvén traveling time in the chromosphere. Once the chromosphericheating occurs, the temperature becomes more than 30 times its initial value for the case of R0= 400 in a pair of layers of deep chromosphere, and the region of high temperature shifts upward andbecomes broader with time, since the chromospheric thin layer of joule heating shifts upward
according to a pileup of reconnected ﬁeld lines in the ﬂare loop; then, chromospheric evaporation
grows and extends outward, and its velocity becomes comparable with the coronal downﬂow
velocity inside the loop boundary. The impulsive chromospheric heating is caused by drastic
evolution of the ﬂare current wedge, through which some part of the coronal sheet current suddenly
turns its direction to be concentrated into the chromospheric thin layer; simultaneously, amagnetohydrodynamic 共MHD兲 generator arises ahead of the ﬂare loop top to provide a new current
circuit inside the large-scale ﬂare current wedge. Hence, it is concluded that the powerful MHD
generator, sustained by the fast reconnection jet, drives the ﬂare current wedge to evolve, leading to
the impulsive chromospheric heating. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2884715兴
I. INTRODUCTION

It is well understood by observations that magnetic reconnection is essential for solar ﬂares.1 Theoretically, the fastreconnection mechanism involving standing slow shocks
should be most responsible for ﬂare phenomena, such as solar ﬂares and geomagnetic substorms. A possible fast reconnection conﬁguration was ﬁrst proposed by Petschek and extensively studied in two-dimensional 共2D兲 steadymagnetohydrodynamic 共MHD兲 ﬂows,2 and it has been believed that external boundary conditions control the fast reconnection conﬁguration.3 However, it was shown that thefast reconnection mechanism can be realized only when alocalized resistivity is applied.4 Hence, full agreement has
not yet been attained concerning the physical mechanism of
fast reconnection evolution, so that most of the previous theoretical studies on solar ﬂares were based on 2D cartoon
models surmised from the Petschek model.5 However, actual
ﬂares occur in three-dimensional 共3D兲 situations, so that it isessential to clarify such a 3D fast reconnection mechanismthat is applicable to complicated ﬂare phenomena.
Any effective reconnection process must provide a dissipation mechanism in the small X reconnection region ofelectron inertial length, where E · J ⬎ 0 and hence E = ␩J 共␩
⬎ 0 is the effective resistivity兲.6 The effective resistivity inspace 共collisionless兲 plasmas may be provided by currentdriven microturbulences, often called “anomalous
resistivity;”7 in fact, current-driven anomalous resistivities
are detected in laboratory plasmas.8 Therefore, in studyinga兲Electronic mail: ugai@cosmos.ehime-u.ac.jp.

1070-664X/2008/15共3兲/032902/9/$23.00

the fast reconnection mechanism theoretically, it is essential
to clarify the self-consistent interaction between the effective
resistivity in the small reconnection region and the ambient
global reconnection ﬂow.
In this respect, we have proposed the spontaneous fastreconnection model. First, the basic feature of a long current
sheet system was examined in detail, and it was demonstrated for 2D MHD ﬂows that, if there is no effective resistivity, the current sheet, initiated by a small disturbance, becomes drastically thinner with time by nonlinear 共pinch兲
effects, leading to extremely large current density 共in theframework of MHD, the current density becomes inﬁnitely
large兲.9 On the basis of this characteristic of the current sheet
system, we have demonstrated for a wide variety of anomalous resistivity models that once current-driven anomalous
resistivities are ignited, the fast reconnection mechanism canbe realized as an eventual solution of MHD by the nonlinear
instability due to positive feedback between anomalous resistivities and reconnection ﬂows.6,10 In general 3D situations, we ﬁnd that the fast reconnection mechanism can berealized only when the initial current sheet width is 3–4
times larger than its thickness, since the positive feedback
works only when this condition is satisﬁed.11,12
Recently, the 3D spontaneous fast reconnection model
has been successfully applied to geomagnetic substorms. For
instance, in situ satellite observations of the traveling compression regions associated with large-scale plasmoids traveling down the tail, known as deﬁnite signatures ofsubstorms,13 were ﬁrst explained exactly both qualitatively
and quantitatively.14 With respect to auroral ﬂares, the socalled substorm current wedge is known as one of the most

15, 032902-1© 2008 American Institute of Physics

Ugai, “Impulsive chromospheric heating of two-ribbon flares...” (2008)
032902-2

Phys. Plasmas 15, 032902 共2008兲

M. Ugaifundamental long-standing questions, since it has been believed by observations that drastic current wedge evolution
gives rise to impulsive auroral electrojets. For this question,
we ﬁrst demonstrated deﬁnitely that the magnetospheric current wedge drastically evolves when the fast reconnection jet
collides with the magnetic loop.12,15
In addition to substorms, the fast reconnection model has
been successfully applied to a variety of solar ﬂarephenomena.16,17 In our recent paper 共hereafter called Paper
1兲, the 3D spontaneous fast reconnection model was applied
to two-ribbon ﬂares, and the well-known morphological features are generally explained.18 The main theme of thepresent paper is to extend Paper 1 and study more quantitatively the physical mechanism of two-ribbon chromosphericheating for different parameter values. In addition, our concern is directed as clarifying the physical mechanism thatdrives the so-called ﬂare current wedge, through which a partof the coronal sheet current suddenly turns its direction to beconcentrated into the chromospheric thin layer, giving rise to
drastic current intensiﬁcation and to impulsive temperaturerise there.
II. SIMULATION MODELINGThe present simulation model is quite similar to that ofPaper 1, and computations will be done for different parameter values of the chromospheric plasma density.
A. Basic equations

The compressible MHD equations areD␳/Dt = − ␳ ⵜ · u,

␳De/Dt = − P ⵜ · u + ␩J2 ,␳共x,y兲 = 兵R0 exp关− 共x/0.4兲4兴 + 1其P共y兲/共1 + ␤0兲,

共3兲

where R0 is taken to be much larger than unity, so that thex = 0 plane corresponds to the deep 共lower兲 chromosphere,
and the middle chromosphere may be at x ⬃ 0.4. Here, Bz= 0 is assumed initially because of the symmetry boundary
conditions, although a nonzero Bz may have some effects on
the dynamics.19
The normalization of quantities, based on the initialquantities, is self-evident: Distances are normalized by thecurrent sheet half-thickness d0, B by Bx0, P by P0= B2x0 / 共2␮0兲; also, ␳ by ␳i = ␳共y = 0兲 in the corona, u by
VAx0共=Bx0 / 冑␮0␳i兲, time t by d0 / VAx0. The Alfvén velocity inthe ambient magnetic ﬁeld region 共1 ⬍ y ⬍ 4兲 in the corona isgiven by VAe = VAx0 / 冑␳e 关␳e = ␤0 / 共1 + ␤0兲兴. The normalized
temperature is given by T = P / ␳, so that T0 = 1 + ␤0 and T= T0 / 共R0 + 1兲 are the initial coronal and chromospheric temperatures, respectively.
Here, the conventional symmetry boundary conditions
are assumed on the z = 0, x = 0, and y = 0 planes. Hence, thecomputational region can be restricted to the ﬁrst quadrant
only and taken to be a rectangular box, 0 艋 x 艋 Lx, 0 艋 y艋 Ly, and 0 艋 z 艋 Lz; also, for simplicity, the symmetryboundary condition is assumed on the outer boundary plane
x = Lx, and on the other boundary planes 共y = Ly and z = Lz兲,the free boundary conditions are assumed.12,18
The effective resistivity for Ohm’s law 共normalized by
␮0d0VAx0兲 may be assumed to be18

=0
共1兲

for Vd ⬍ VC ,共4兲

where Vd共r , t兲 = 兩J共r , t兲 / ␳共r , t兲兩 is the relative electron-ion
drift velocity; or␩J共r,t兲 = k j关兩J共r,t兲兩 − JC兴2 for 兩J兩 ⬎ JC .

ⵜ · B = 0,

where D / Dt ⬅ ⳵ / ⳵t + u · ⵜ; the gas law, P = 共␥ − 1兲␳e, is assumed 关e is the internal energy per unit mass, and ␥ is the
speciﬁc heat ratio with ␥ = 5 / 3 assumed here 共an adiabaticcase兲兴, as is Ohm’s law, E + u ⫻ B = ␩J 共␩ may be an effective resistivity兲. The basic equations 共1兲 are transformed to aconservation-law form, and the modiﬁed Lax–Wendroff
scheme is used for the numerical computation.
B. Initial-boundary conditions

An antiparallel magnetic ﬁeld B = 关Bx共y兲 , 0 , 0兴 is initiallyassumed as Bx共y兲 = sin共␲y / 2兲 for 0 ⬍ y ⬍ 1; Bx = 1 for 1 ⬍ y⬍ 4; Bx = cos关共y − 4兲␲ / 1.2兴 for 4 ⬍ y ⬍ 4.6; Bx = 0 for y ⬎ 4.6;
also, Bx共y兲 = −Bx共−y兲 for y ⬍ 0. Fluid velocity u = 共0 , 0 , 0兲,and plasma pressure P共y兲 satisﬁes the pressure-balance
condition
P + B2x = 1 + ␤0 ,that P共y = 0兲 = 1 + ␤0 initially 共here, ␤0 = 0.15 is taken兲. Also,
plasma density ␳ initially satisﬁes

␩D共r,t兲 = kd关Vd共r,t兲 − VC兴 for Vd ⬎ VC ,

␳Du/Dt = − ⵜP + J ⫻ B,

⳵B/⳵t − ⵜ ⫻ 共u ⫻ B兲 = − ⵜ ⫻ 共␩J兲,J = ⵜ ⫻ B/␮0,

377

共2兲

where ␤0 is the ratio of the plasma pressure to the magnetic
pressure in the ambient magnetic ﬁeld region 1 ⬍ y ⬍ 4, so=0 for 兩J兩 ⬍ JC ,共5兲

where VC and JC are the thresholds. Here, we may take theresistivity model␩ = ␩S + ␩D + ␩J ,共6兲

where ␩S = ␩C共T / T0兲−1.5 is the Spitzer resistivity 共T0 = 1 + ␤0
is the initial coronal temperature兲. In what follows, we takekd = 0.003 and VC = 12 in Eq. 共4兲, k j = 0.0001 and JC = 3 in Eq.
共5兲, and ␩C = 0.0000001.18
In order to disturb the initial conﬁguration, a localized
resistivity is assumed at x = Lx in the 3D form

␩共r兲 = ␩0 exp兵− 关共x − Lx兲/kx兴2 − 共兩y兩/ky兲3 − 共兩z兩/kz兲3其,共7兲

which causes magnetic reconnection as a disturbance. Here,
we take kx = ky = 0.8 and kz = 5 with ␩0 = 0.02. This disturbanceequation 共7兲 is imposed only in the initial time range 0 ⬍ t⬍ 4, and the resistivity model equation 共6兲 is assumed fort ⬎ 4.

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FIG. 1. Proﬁles of the reconnection ouﬂow velocity ux along the x axis atdifferent times for the case of R0 = 400.

FIG. 2. Proﬁles of the reconnected ﬁeld component By along the x axis atdifferent times for the case of R0 = 400.

The parameters for the numerical computations are taken
to be Lx = 10, Ly = 9.6, and Lz = 9.8, with the mesh sizes, ⌬x= 0.04, ⌬y = 0.015, and ⌬z = 0.1.

region 共x = 10兲, and then it is quickly accelerated to the
Alfvén velocity VAe ⬃ 2.7, measured in the ambient magnetic
ﬁeld region, because of the strong motor 共J ⫻ B兲 force in thestanding slow shock layer associated with the fast reconnection mechanism. The chromospheric plasma in x ⬍ 0.4 canhardly move because of the large plasma density.
In the similar manner, Fig. 2 shows for R0 = 400 the proﬁles of the reconnected ﬁeld component By along the x axis.
The reconnected ﬁeld lines have been carried by the reconnection jet toward the chromosphere, giving rise to a largescale magnetic loop. The position, where the By ﬁeld has thepeak value, ﬁrst shifts to the negative x direction, and then it
shifts to the positive x direction for t ⬎ 44, since then themagnetic loop is expanding outward. In this case, the basic
structure of the magnetic loop and the fast reconnection jet,involving slow and fast shocks, is established at t = 44.
It may be instructive for later discussion to deﬁnitely
identify the characteristic regions associated with the resulting magnetic loop structure. Hence, Fig. 3 typically shows

III. RESULTS

Computations have been done for the cases of R0 = 0, 49,
100, and 400 in Eq. 共3兲 with the other conditions being quite
the same. The qualitative features of the resulting chromospheric heating are found to be generally similar, and thebasic morphological features were already reported for R0= 100 in Paper 1; also, the results for R0 = 49 are found to bequite similar to those for R0 = 100, except that the impulsive
chromospheric heating occurs at a somewhat earlier time for
R0 = 49. Hence, in what follows, the results for R0 = 400 will
mainly be shown, which may quantitatively be comparedwith those for R0 = 100, so that detailed structures of chromospheric heating will be clariﬁed.
A. General remarks

In the present ﬂare model, all the distinct plasma processes result from the fast reconnection mechanism proceeding in the corona, where magnetic reconnection grows at x
= Lx = 10, and the resulting fast reconnection jet collides withthe magnetic loop.18 Hence, in understanding the complicated physical processes, it is fundamental to show, ﬁrst ofall, how the fast reconnection mechanism in the corona
causes the Alfvénic jet and the large-scale magnetic loop
structure. We ﬁnd that the basic fast reconnection evolution
in the corona 共x ⬎ 1兲 is not signiﬁcantly inﬂuenced by thechromosphere, so that the resulting fast reconnection jet andmagnetic loop structure in the corona is found to be generally consistent with the one already reported in our previous
papers on the 2D and 3D dynamics of the magnetic loop for
R0 = 0,20 whereas the dynamics in the chromosphere 共0 ⬍ x⬍ 0.4兲 may strongly depend on R0.
Figure 1 shows for R0 = 400 the proﬁles of the fast reconnection jet ux along the x axis 共y = z = 0兲 at different times.
In the initial phase of the fast reconnection evolution 共t= 36兲, the jet begins to grow from near the X reconnection

FIG. 3. Typical proﬁles of the fast reconnection jet ux and the reconnectedﬁeld component By along the x axis at time T = 44 for the case of R0 = 100,
where X f and Xl indicate, respectively, the location of the fast shock and thatof the magnetic loop center.

Ugai, “Impulsive chromospheric heating of two-ribbon flares...” (2008)
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M. Ugai379
Phys. Plasmas 15, 032902 共2008兲

FIG. 5. Proﬁles in the x direction of the temperature T, measured at x
= 0共z = 0兲, and the downﬂow velocity ux ⬍ 0 measured at x = 0.4共z = 0兲 共left兲
for R0 = 100 and 共right兲 for R0 = 400.
FIG. 4. 共Color online兲 Magnetic ﬁeld conﬁgurations in the z = 0 plane andcontour lines of plasma pressure P in the x = 0 plane for R0 = 400 at different
times.

for R0 = 100 the proﬁles of ux and By along the x axis, whereX f indicates the location of the fast shock and Xl the locationof the magnetic loop center. As was previously shown indetail, the supersonic reconnection jet for x ⬎ X f in the high␤ region becomes subsonic across the shock by enhancing
the plasma temperature.20 In accordance with the magnetic
loop expansion, the locations of Xl and X f may move in thepositive x direction 共Figs. 1 and 2兲.
Figure 4 shows for R0 = 400 the magnetic ﬁeld conﬁgurations in the z = 0 plane and the plasma pressure distributions in the x = 0 plane at different times. At t = 42, the fastreconnection jet strongly compresses the magnetic loop top,
but the ﬁeld lines in the lower chromosphere 共x ⬍ 0.4兲 are notyet disturbed and are almost line-tied because of the large
plasma density, whereas for R0 = 100 the deep chromosphere
共x = 0兲 is already disturbed distinctly at t = 42;18 also, for R0= 0 it is disturbed at t = 40.5.15 In the case of R0 = 400, theplasma pressure P in the deep chromosphere 共x = 0兲 is impulsively enhanced at t = 44, and as time progresses, the chromospheric region of enhanced plasma pressure expands outward, leading to ﬂare loop expansion toward the corona.

B. Impulsive chromospheric heating

Here, let us examine quantitatively how the chromospheric temperature is enhanced according to the drastic evolution of the fast reconnection jet in the corona. First of all,Fig. 5 shows for the cases of R0 = 100 and 400 the
y-directional proﬁles of the temperature T in the deep chromosphere 共x = z = 0兲 as well as the plasma ﬂow velocity ux,
measured at x = 0.4共z = 0兲 at times t = 39 and 40, when thelower chromosphere is not yet disturbed drastically. In thistime range, the temperature is directly enhanced by the down
ﬂow 共ux ⬍ 0兲 of the coronal hot plasma heated by the reconnection process. Since the coronal plasma is of much lower

density, the resulting temperature rise is not so effective, despite that the down ﬂow velocity 共ux ⬍ 0兲 becomes considerably larger at t = 40 for R0 = 100.
The impulsive rise of chromospheric temperature may
occur when the chromospheric current density Jz, which initially ﬂows in the negative z direction, drastically increases.18
Hence, Fig. 6共a兲 shows for R0 = 400 the proﬁles of Jz along
the y axis 共z = x = 0兲 and indicates that Jz ⬍ 0 is remarkably
enhanced at t = 46 in the loop footpoint layer of reconnectedﬁeld lines 共Fig. 4兲, causing the anomalous resistivity ␩J,
whereas before t = 44 Jz ⬍ 0 does not signiﬁcantly increase in
the x = 0 plane. Figure 6共b兲 shows the y-directional proﬁles of
the resulting joule heating ␩J2 at x = 0.4 as well as at x
= 0共z = 0兲. At t = 44, joule heating does not yet occur effectively in the deep chromosphere 共x = 0兲, whereas it occurs in
the middle chromosphere 共x = 0.4兲. It may hence be statedthat for R0 = 400 the joule heating occurs effectively for t⬎ 44 in the chromosphere 共0 ⬍ x ⬍ 0.4兲; then, the thin layerof joule heating shifts upward with time.
Figure 7 shows for R0 = 100 and 400 the temperature
proﬁles along the y axis 共x = z = 0兲 at different times. At t
= 40 for R0 = 100 or at t = 42 for R0 = 400, the temperature
increases only by the downﬂow of coronal hot plasma 共Fig.
5兲. In both cases, as soon as the chromospheric joule heating
is impulsively triggered 共Fig. 6兲, the temperature is remarkably enhanced in the location of joule heating, so that thelayer of high temperature shifts upward and becomes broader
with time. These results demonstrate that the impulsive riseof chromospheric temperature is caused immediately by the
drastic chromospheric joule heating that occurs when thechromospheric current density Jz ⬍ 0 is notably enhanced.
In order to see the overall ﬂare loop structure, Fig. 8
shows the resulting temperature distribution at t = 50 for R0= 400. In the corona, the temperature is distinctly enhancedin the X reconnection region 共x ⬃ 10兲 and the fast reconnection jet region, where plasma density is remarkably
rareﬁed;20 also, it is notably enhanced along the magnetic
loop boundary and ahead of the loop top. In the deep chromosphere 共x = 0兲, where the initial temperature T = 1.15/ 401,
T increases in the shape of two-ribbons in view of the symmetry boundary conditions, and it becomes more than 30

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Impulsive chromospheric heating…

FIG. 6. Proﬁles in the y direction of 共a兲 the current density Jz, measured at
x = 0共z = 0兲, and 共b兲 the joule heating ␩J2, measured at x = 0 and x = 0.4共z= 0兲, for R0 = 400.

times its initial value. The resulting temperature rise of theoverall ﬂare loop is qualitatively consistent with that for R0= 100.18

Phys. Plasmas 15, 032902 共2008兲

FIG. 7. Proﬁles of the temperature T along the y axis at different times forthe cases R0 = 100 and R0 = 400.

共ux ⬃ 0兲. At t = 48, the chromospheric evaporation velocity
becomes comparable with the coronal downﬂow velocity,
and as time progresses the layer of chromospheric evaporation shifts in the positive y direction, which results from theﬂare loop expansion 共Fig. 9兲.

C. Chromospheric evaporation

The chromospheric temperature rise certainly leads to anincrease in plasma pressure, so that the chromosphericplasma of high density may erupt into the corona by the
resulting pressure-gradient force. Figure 9 shows for R0= 400 the plasma ﬂow vectors and the distributions of joule
heating ␩J2 at times t = 46 and 50. Inside the loop boundary,
there are two distinct plasma ﬂows: One is the coronal downﬂow toward the loop footpoint, which is mostly caused by
the slow shock standing along the loop boundary,20 and theother is the so-called chromospheric evaporation. As themagnetic loop expands outward, the chromospheric evaporation also extends outward into the corona.
In order to see quantitatively the chromospheric evaporation, Fig. 10 shows for R0 = 400 the y-directional proﬁles of
the ﬂow velocities across the loop boundary, measured at x
= 1.6共z = 0兲 just above the chromosphere at different times. In
this ﬁgure, ux ⬎ 0共uy ⬍ 0兲 indicates the chromospheric evaporation, and ux ⬍ 0共uy ⬎ 0兲 the coronal downﬂow 共Fig. 9兲,
whereas above the loop boundary there is no signiﬁcant ﬂow

FIG. 8. 共Color online兲 Temperature distribution for R0 = 400 at t = 50 withthe contour interval of 0.015 in the x = 0 plane and that of 0.8 for x ⬎ 0.

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Phys. Plasmas 15, 032902 共2008兲

FIG. 11. Proﬁles of Jz , Ez and E · J at times t = 44 and 46 along the x axis for
R0 = 400, where XG indicates the extent where E · J ⬍ 0.
FIG. 9. 共Color online兲 Plasma ﬂow vectors and distributions of joule heating
␩J2 for R0 = 400 in the x = 0, y = 0, and z = 0 planes.

D. MHD generator and current wedge evolution

In Paper 1, we demonstrated that the extreme increase in
the current density in the chromospheric two-ribbon layers iscaused by drastic evolution of the ﬂare current wedge.18
When the fast reconnection jet collides with the magnetic

FIG. 10. Proﬁles in the y direction of ux at times t = 46, 48, and 50, and uy
at t = 48, measured at x = 1.6共z = 0兲 for R0 = 400.

loop, the current wedge suddenly evolves, so that the coronal
sheet current, ﬂowing in the negative z direction, turns its
direction to be concentrated in the loop footpoint layer ofreconnected ﬁeld lines in the deep chromosphere, leading toa large-scale redistribution of the overall current system.12,15
In fact, the region, where the 3D fast reconnection processproceeds, is limited in a ﬁnite z-directional extent, in 兩z兩⬍ ZR say, so that the initial sheet current may drasticallychange its proﬁle in 兩z兩 ⬍ ZR.
An important question may be how such a vital current
wedge evolution can be realized by the fast reconnection
dynamics proceeding in 兩z兩 ⬍ ZR. In this respect, we suggested in Paper 1 that a MHD generator may work ahead ofthe magnetic loop, which could play a crucial role in driving
the current wedge to evolve. In order to examine this idea,
Fig. 11 shows the proﬁles of Ez , Jz and E · J共=EzJz兲 along thex axis across the magnetic loop top 共Fig. 2兲 for R0 = 400 andindicates that there is a region where Jz共⬃⳵By / ⳵x兲 becomes
positive and E · J is negative around x = 2 at t = 44 or aroundx = 2.5 at t = 46 关the extent of this region is indicated by XG,
which is located inside Xl ⬍ x ⬍ X f ahead of the loop top
共Figs. 2 and 3兲兴. In the region XG, E ⬃ −u ⫻ B, and E · J⬃ u · 共J ⫻ B兲, so that E · J ⬍ 0 indicates a MHD generator,
which is provided and sustained by the fast reconnection jet
ahead of the loop.
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Impulsive chromospheric heating…

FIG. 12. 共Color online兲 共a兲 Current ﬂow lines, denoted by C1, that passthrough the generator region XG at t = 46 for R0 = 400, and in 共b兲 the current
wedge 共CW兲 that is formed at t = 46 is also shown.

to the generator, Fig. 12共a兲 shows for R0 = 400 the current
ﬂow lines, deﬁned as the lines drawn along the current density vectors,15 that pass through the generator region XG at
t = 46. This ﬁgure indicates that a new current circuit, C1 say,is suddenly set up inside the magnetic loop to connect the
MHD generator ahead of the loop top to the loop boundary
above the chromospheric loop footpoint, where effective
joule dissipation occurs 共Fig. 9兲. Also, Fig. 12共b兲 shows theﬂare current wedge that has evolved simultaneously, so thatthe coronal sheet current turns its direction toward the loop
footpoint in the deep chromosphere to strongly intensify thecurrent density Jz ⬍ 0 in the x = 0 plane 共Fig. 6兲.
In order to see the current circuit C1 more generally, Fig.
13共a兲 shows for R0 = 100 that the current circuit C1 in factbuilds up, through which the MHD generator is connected to
the ﬂare loop boundary immediately above the chromospheric loop footpoint; also, Fig. 13共b兲 shows that the ﬂare
current wedge has simultaneously evolved. Hence, we again
see that the ﬂare current wedge evolves in accordance withthe current circuit C1 that evolves inside the large-scale ﬂare
current wedge. Since any distinct plasma process should becaused by the reconnection dynamics in 兩z兩 ⬍ ZR, we may

Phys. Plasmas 15, 032902 共2008兲

FIG. 13. 共Color online兲 共a兲 Current ﬂow lines, denoted by C1, that passthrough the generator region XG at t = 44 for R0 = 100, and in 共b兲 the current
wedge 共CW兲 that is formed at t = 44 is also shown.

conclude that the powerful MHD generator ahead of the loop
top plays a dominant role in driving 共or inducing兲 the ﬂare
current wedge to evolve drastically, leading to impulsive
heating in the deep chromosphere.
IV. SUMMARY AND DISCUSSIONIn Paper 1, morphological features of two-ribbon ﬂares
were examined. The present paper extends it and quantitatively studies the chromospheric two-ribbon heating for different values of the initial chromospheric density, R0. In general, for R0 Ⰷ 1, the chromospheric plasma cannot moveeasily, so that magnetic ﬁeld lines may be almost line-tied in
the deep chromosphere. For the case of larger R0, it takes
more time for the impulsive heating to take place in the deep
chromosphere 共x = 0兲. This may indicate that the drastic increase in the current density due to distinct changes inmagnetic ﬁelds in the lower chromosphere is delayed forlarger R0.
The chromospheric joule heating impulsively occurs
when the current density is suddenly enhanced in the thinlayer of the chromospheric loop footpoint around the separatrix of reconnected ﬁeld lines. Accordingly, the tempera-

Ugai, “Impulsive chromospheric heating of two-ribbon flares...” (2008)
032902-8

Phys. Plasmas 15, 032902 共2008兲

M. Ugaiture is simultaneously enhanced in the layer of deep chromosphere 共x = 0兲, and the chromospheric layer of enhanced
temperature shifts upward and becomes broader with time,since the separatrix moves upward in accordance with pileupof the reconnected ﬁeld lines inside the ﬂare loop. For thecase of R0 = 400, the enhanced temperature becomes morethan 30 times its initial value in the deep chromosphere,
whereas the temperature rise due to downﬂows of coronal
hot plasma is not so effective.
The impulsive chromospheric heating results in plasmapressure increase, so that the chromospheric plasma of highdensity eventually erupts outward into the corona, giving riseto the chromospheric evaporation. Hence, inside the ﬂareloop boundary, there are two distinct plasma ﬂows in the
inverse directions: One is the downﬂow of coronal hot
plasma toward the loop footpoint, and the other is the chromospheric evaporation. As the loop expands outward, thechromospheric evaporation also extends in the corona, andthe evaporation velocity becomes comparable with the coronal downﬂow velocity.
The impulsive intensiﬁcation of the chromospheric current density results immediately from the ﬂare current wedge
evolution, by which some part of the coronal sheet current,
ﬂowing into the fast reconnection region in 兩z兩 ⬍ ZR, suddenly
turns its direction to be concentrated into the chromosphericthin layer. At the same time, there appears a region, XG,
ahead of the ﬂare loop top, where a MHD generator, E · J⬃ u · 共J ⫻ B兲 ⬍ 0, works; accordingly, such a current circuit,
C1, is suddenly set up that connects the MHD generator to
the dissipation region of loop boundary above the chromospheric footpoint. Since any distinct plasma process should
be driven by the reconnection dynamics in 兩z兩 ⬍ ZR, we may
conclude that the powerful MHD generator ahead of the loop
top drives 共or induces兲 the ﬂare current wedge to evolve
drastically. This situation was already illustrated in Fig. 6 ofPaper 1, but that ﬁgure was not complete, so that it is redrawn in Fig. 14 on the basis of the present results 共Figs. 12
and 13兲.
The drastic current wedge evolution occurs when the fastreconnection jet collides with the ﬂare loop and the magnetic
ﬁelds in the loop are strongly sheared, leading to ﬁeldaligned currents due to the resulting sheared ﬁelds Bz inside
the ﬂare loop.15 In terms of MHD waves, large-amplitude
Alfvén waves may be triggered by the collision of the fastreconnection jet with the ﬂare loop top and propagate along
the loop toward the chromosphere. Hence, until the Alfvén
waves arrive at the footpoint of the deep chromosphere, distinct joule heating cannot be caused there. Since the Alfvén
velocity is proportional to ␳−1/2, the Alfvén traveling time inthe chromosphere becomes longer for the larger R0, so that
we understand the fact that the start of the impulsive chromospheric heating is delayed for the larger values of R0.
Finally, it may be instructive to discuss the previous theoretical models. The observed two ribbons have been considered to be consequences of heat ﬂow along the magnetic
ﬁeld line, which was examined by 2D MHD simulations on
the basis of the spontaneous fast reconnection model.21 However, it is difﬁcult to exactly obtain the heat conduction coefﬁcient along the complicated loop ﬁeld lines for coronal

383

FIG. 14. 共Color online兲 Schematic drawing of the current circuit C1, connected to the MHD generator ahead of the loop top where E · J ⬃ u · 共J
⫻ B兲 ⬍ 0; here, ER共⬃−u ⫻ B兲 is the reconnection electric ﬁeld, 2ZR is theextent where the fast reconnection proceeds, I0 is the initial coronal sheet
current, and some part of I0 becomes the ﬂare current wedge, CW.

plasmas, and the initial current sheet cannot be in static equilibrium in the presence of heat conduction. Hence, any heat
conduction is not assumed in the present model, whichshould be considered for a more realistic model. In view ofhelicity conservation, the present model is simple, and themagnetic helicity is conserved for the region extended to thenegative z direction because of the symmetry boundary conditions. In fact, an occurrence of fast reconnection mechanism may signiﬁcantly be limited in a local region of corona
where a current sheet of sufﬁciently large size is formed,
leading to drastic coronal heating 共Fig. 8兲. Hence, the fastreconnection mechanism could provide nanoﬂares and mayhence play an important role in effective coronal heating.22
ACKNOWLEDGMENTS

The author thanks T. Sugimoto for preparing the graphics. This work was supported by Grants-in-Aid for Creative
Scientiﬁc Research “The Basic Study of Space Weather Prediction” 共No. 17GS0208, Head Investigator, K. Shibata兲from the Ministry of Education in Japan, Mitsubishi Foundation, RISH of Kyoto University, and Solar-Terrestrial Environment Laboratory of Nagoya University.
K. Shibata, Adv. Space Res. 17, 9 共1996兲; Astrophys. Space Sci. 264, 129
共1999兲.
2
H. E. Petschek, NASA Special Publication 50 共National Aeronautics andSpace Administration, Washington, D.C., 1964兲, p. 425.
3
V. M. Vasyliunas, Rev. Geophys. Space Phys. 13, 303, DOI: 10.1029/
RG013i001p00303 共1975兲; E. R. Priest and T. G. Forbes, J. Geophys. Res.
91, 5579, DOI: 10.1029/JA091iA05p05579 共1986兲; E. R. Priest and L. C.
Lee, J. Plasma Phys. 44, 337 共1990兲.
4
M. Ugai and T. Tsuda, J. Plasma Phys. 17, 337 共1977兲; M. Yan, L. C. Lee,and E. R. Priest, J. Geophys. Res. 97, 8277, DOI: 10.1029/92JA00170
共1992兲.
5
E. R. Priest and T. G. Forbes, Astron. Astrophys. Rev. 10, 313 共2002兲.
6
M. Ugai and L. Zheng, Phys. Plasmas 12, 092312 共2005兲.
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A. T. Y. Lui, Space Sci. Rev. 95, 325 共2001兲; R. A. Treumann, Earth,Planets Space 53, 453 共2001兲.
8
H. Ji, M. Yamada, S. Hsu, and R. Kulsrud, Phys. Rev. Lett. 80, 3256
共1998兲; Y. Ono, M. Inomoto, Y. Ueda, T. Matsuyama, and Y. Murata,
Earth, Planets Space 53, 521 共2001兲.
9
M. Ugai, Phys. Fluids 29, 3659 共1986兲.
10
M. Ugai, Plasma Phys. Controlled Fusion 26, 1549 共1984兲; Phys. Fluids B
4, 2953 共1992兲; Phys. Plasmas 6, 1522 共1999兲.
11
M. Ugai and T. Shimizu, Phys. Plasmas 3, 853 共1996兲.
12
M. Ugai, Phys. Plasmas 14, 062902 共2007兲.
13
J. A. Slavin, M. F. Smith, E. L. Mazur, D. N. Baker, E. W. Hones, Jr., T.
Iyemori, and E. W. Greenstadt, J. Geophys. Res. 98, 15425, DOI:
10.1029/93JA01467 共1993兲.
14
M. Ugai and L. Zheng, Phys. Plasmas 13, 032901 共2006兲; 13, 062906
共2006兲.
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Impulsive chromospheric heating…

M. Ugai and K. Kondoh, Phys. Plasmas 13, 102903 共2006兲.
K. Shibata and S. Tanuma, Earth, Planets Space 53, 473 共2001兲.
17
H. Isobe, T. Miyagoshi, K. Shibata, and T. Yokoyama, Nature 共London兲
434, 478 共2005兲.
18
M. Ugai, Phys. Plasmas 14, 102904 共2007兲.
19
G. Hornig and E. R. Priest, Phys. Plasmas 10, 2712 共2003兲.
20
M. Ugai, Geophys. Res. Lett. 14, 103, DOI: 10.1029/GL014i002p00103
共1987兲; Phys. Plasmas 3, 4172 共1996兲; J. Geophys. Res. 104, 6929, DOI:
10.1029/1998JA900181 共1999兲; M. Ugai, K. Kondoh, and T. Shimizu,
Phys. Plasmas 10, 357 共2003兲.
21
T. Yokoyama and K. Shibata, Astrophys. J. Lett. 494, L113 共1998兲; Astrophys. J. 549, 1160 共2001兲.
22
E. N. Parker, Astrophys. J. 264, 642 共1983兲; J. A. Klimchuk, Sol. Phys.
234, 41 共2006兲.
15
16

Ugai, “The Evolution of Fast Reconnection in a Three-Dimensional...” (2008)

385

PHYSICS OF PLASMAS 15, 082306 共2008兲

The evolution of fast reconnection in a three-dimensional current
sheet system
M. Ugaia兲
Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama 790-8577, Japan

共Received 4 June 2008; accepted 22 July 2008; published online 13 August 2008兲
By showing the details of the numerical procedure, global dynamics of the current sheet system issystematically studied by three-dimensional magnetohydrodynamic simulations in the parameter
range where the numerical resistivity is much smaller than the physical resistivity. In the absence ofresistivity, initiated by a reconnection disturbance, current sheet thinning drastically occurs becauseof the sheet pinch, leading to extreme increase in current density around the X neutral point. For theuniform resistivity model, the drastic current increase is suppressed by the magnetic diffusion
共reconnection兲, but the reconnection jet cannot be accelerated effectively, so that any fastreconnection mechanism cannot evolve; for the smaller resistivity, the current density at the X point
becomes larger. Once current-driven anomalous resistivities build up, both the reconnection ﬂow
and the anomalous resistivity simultaneously grow to enhance each other, eventually giving rise to
the Alfvénic fast reconnection jet. However, if the current sheet width is smaller than three times its
thickness, the fast reconnection mechanism cannot be realized even in the presence of anomalous
resistivity. Hence, only when a thin current sheet of sufﬁciently large scale is formed in spaceplasmas, the fast reconnection mechanism is likely to evolve drastically, leading to distinct plasma
processes responsible for ﬂares and substorms. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2969737兴
I. INTRODUCTION

It is well understood by observations that magnetic reconnection is essential for solar ﬂares1 and geomagnetic
substorms.2 Theoretically, the fast reconnection mechanism
involving standing slow shocks should be most responsible
for ﬂare phenomena because of the resulting Alfvénic fastreconnection jet. A possible fast reconnection conﬁguration
was ﬁrst proposed by Petschek and extensively studied in
two-dimensional 共2D兲 steady magnetohydrodynamic 共MHD兲
ﬂows,3 and it has been believed that external boundary conditions control the fast reconnection conﬁguration, so thatmost of the theoretical studies have been directed to looking
for the fast reconnection mechanisms that can be realized
without any signiﬁcant inﬂuence of the effective resistivity.4
In general, any effective reconnection process must provide a dissipation mechanism in the small X reconnection
共diffusion兲 region, where E · J ⬎ 0 and hence E = ␩J 共␩ ⬎ 0 is
the effective resistivity兲.5 The effective resistivity in space
共collisionless兲 plasmas may be provided by current-driven
microturbulences, often called “anomalous resistivity;”6 infact, current-driven anomalous resistivities are detected inlaboratory plasmas.7 Therefore, in studying the fast reconnection mechanism theoretically, it is essential to clarify the
self-consistent interaction between the effective resistivity inthe small reconnection region and the ambient global reconnection ﬂow.
Recently, various reconnection modeling groups systematically examined the 2D reconnection evolutions in a current sheet system.8 In their simulations, artiﬁcial dissipation
terms were not shown, and for particle simulations it seemsa兲Electronic mail: ugai@cosmos.ehime-u.ac.jp.

1070-664X/2008/15共8兲/082306/10/$23.00

that the mesh numbers were too small to resolve exactly theelectron dynamics in the X reconnection region of electron
inertial length for a realistic mass ratio. They argued that theHall effects played a crucial role in attaining the fast reconnection rate almost independently of dissipation mechanisms. However, the importance of the dissipation mechanism around the X neutral point can be understood from thefollowing simple consideration. In the conventional 2D symmetric steady situation, employed by the theoreticalstudies,3,4 the generalized Ohm’s law becomes E = ␩J at the
X point, where the Hall term vanishes. For a fast reconnection rate E ⬎ 0.01VAB0, we obtain J ⬎ 0.01⫻ RmJ0, whereJ0 = B0 / 共␮0d兲 关d is the current sheet 共diffusion region兲
thickness兴. If only the classical resistivity is considered,Rm ⬎ 1010 and J ⬎ 108J0, so that if we take d ⬃ 107 m thediffusion region thickness becomes below 10 cm, which cannever be realized. Hence, the fast reconnection rate cannot besustained by classical resistivities; in other words, for any
fast reconnection mechanism responsible for substorms andﬂares, it is necessary to consider current-driven anomalous
resistivities.
It was shown for the ﬁrst time that the fast reconnection
mechanism can be realized as an eventual solution when alocalized resistivity is applied,9 so that we have proposed thespontaneous fast reconnection model. First, it was demonstrated for 2D MHD ﬂows that, if there is no effective resistivity, the current sheet, initiated by a small reconnection
disturbance, becomes drastically thinner by nonlinear 共pinch兲
effects, leading to extremely large current density in the X
region.10 On the basis of these results, we have demonstrated
for a wide variety of resistivity models and parameters thatonce current-driven anomalous resistivities are ignited, thefast reconnection mechanism can be realized as an eventual

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Phys. Plasmas 15, 082306 共2008兲

M. Ugaisolution of MHD equations by the nonlinear instability due
to positive feedback between anomalous resistivities and reconnection ﬂows.10,11 In general three-dimensional 共3D兲 situations, we argue that the fast reconnection mechanism can berealized when the current sheet width is sufﬁciently larger
than its thickness.12,13
The spontaneous fast reconnection model has been successfully applied to geomagnetic substorms14 and solar
ﬂares.15 For instance, traveling compression regions 共TCR兲,known as a clear signature of substorms,16 provide the mostdeﬁnite evidence that the fast reconnection mechanism isrealized according to substorm onset, and the 3D spontaneous fast reconnection model exactly explains the in situ TCR
signatures both qualitatively and quantitatively.17 Also, it isdemonstrated that the magnetospheric current wedge, known
as a long-standing question for substorms, can be drasticallyformed when the fast reconnection jet collides with the dipole ﬁeld 共magnetic loop兲.13,18 Recently, the current wedge
evolution is applied to two-ribbon ﬂares, and it is demonstrated that most of the morphological features of two-ribbon
ﬂares can be explained by this fast reconnection model; inparticular, the generator effect ahead of the loop top plays animportant role in the two-ribbon heating.19
However, it seems that the spontaneous fast reconnection
model is not fully understood; in particular, the physicalmeaning of the nonlinear instability 共positive feedback兲,which is the key to the spontaneous fast reconnection evolution in a current sheet system, seems to not be exactly recognized. This may be because it has been demonstrated bysimulations, so that it is difﬁcult for anyone to readily obtainthe same results; in fact, some details of the numerical procedure were often in the black box. Hence, in this paper weﬁrst show all the details of our numerical procedure, and the
major effects of numerical dissipation terms involved areclariﬁed. Then, we systematically examine the 3D global dynamics of long current sheet system, and the subsequent fastreconnection evolution is studied by 3D MHD simulations;in fact, the key physics of the fast reconnection dynamics
cannot be exactly understood without seeing the details of
the numerical procedure.
II. SIMULATION MODELINGThe simulation model is taken to be quite similar to thatof our previous papers; in particular, the initial magnetic ﬁeld
proﬁle and the boundary conditions are quite the same as ourrecent studies on the magnetic loop dynamics in a closed
system.18,19

B. Initial-boundary conditions

An antiparallel magnetic ﬁeld B = 关Bx共y兲 , 0 , 0兴 is initiallyassumed as Bx共y兲 = sin共␲y / 2兲 for 0 ⬍ y ⬍ 1; Bx = 1 for
1 ⬍ y ⬍ 4; Bx = cos关共y − 4兲␲ / 1.2兴 for 4 ⬍ y ⬍ 4.6; Bx = 0 fory ⬎ 4.6; also, Bx共y兲 = −Bx共−y兲 for y ⬍ 0. Fluid velocity
u = 共0 , 0 , 0兲 and plasma pressure P共y兲 satisﬁes the pressurebalance condition,
P + B2x = 1 + ␤0 ,

␳ = P/共1 + ␤0兲.

␳Du/Dt = − ⵜP + J ⫻ B,

Here, Bz = 0 is assumed initially because of the symmetryboundary conditions, although a nonzero Bz may have someeffects on the dynamics.20
The normalization of quantities, based on theinitial quantities, is self-evident: Distances are normalized
by the current sheet half-thickness d0, B by Bx0, P
by P0 = B2x0 / 共2␮0兲; also, ␳ by ␳i = ␳共y = 0兲, u by
VAx0共=Bx0 / 冑␮0␳i兲, time t by d0 / VAx0. The Alfvén velocity inthe ambient magnetic ﬁeld region 共1 ⬍ y ⬍ 4兲 is given by
VAe = VAx0 / 冑␳e ⬃ 2.7 关␳e = ␤0 / 共1 + ␤0兲兴, and the normalized
temperature is given by T = P / ␳ = 1 + ␤0.
Here, the conventional symmetry boundary conditions
are assumed on the z = 0, x = 0, and y = 0 planes. Hence, thecomputational region can be restricted to the ﬁrst quadrant
only and taken to be a rectangular box, 0 艋 x 艋 Lx,
0 艋 y 艋 Ly, and 0 艋 z 艋 Lz; also, for simplicity, the symmetryboundary condition is assumed on the outer boundary plane
x = Lx. On the other boundary planes 共y = Ly and z = Lz兲, thefree boundary conditions are assumed.9
The effective resistivity for the Ohm’s law 共normalizedby ␮0d0VAx0兲 may be provided by current-driven anomalous
resistivities in space plasmas. Here, we consider the following two anomalous resistivity models, ␩D and ␩J:5,11

␩D共r,t兲 = kd关Vd共r,t兲 − VC兴 for Vd ⬎ VC ,

␳De/Dt = − P ⵜ · u + ␩J ,
ⵜ · B = 0,for

Vd ⬍ VC ,共4兲

␩J共r,t兲 = k j关兩J共r,t兲兩 − JC兴 for 兩J兩 ⬎ JC ,

⳵B/⳵t − ⵜ ⫻ 共u ⫻ B兲 = − ⵜ ⫻ 共␩J兲,2

共3兲

where Vd共r , t兲 = 兩J共r , t兲 / ␳共r , t兲兩 is the relative electron-ion
drift velocity; also,

The compressible MHD equations areD␳/Dt = − ␳ ⵜ · u,

共2兲

where ␤0 is the ratio of the plasma pressure to the magnetic
pressure in the ambient magnetic ﬁeld region 1 ⬍ y ⬍ 4, sothat P共y = 0兲 = 1 + ␤0 initially 共here, ␤0 = 0.15 is taken兲. Also,
plasma density ␳ initially satisﬁes=0

A. Basic equations

J = ⵜ ⫻ B/␮0,

where D / Dt ⬅ ⳵ / ⳵t + u · ⵜ; the gas law, P = 共␥ − 1兲␳e, is assumed 关e is the internal energy per unit mass, and ␥ is the
speciﬁc heat ratio with ␥ = 5 / 3 assumed here 共an adiabaticcase兲兴, as is Ohm’s law, E + u ⫻ B = ␩J 共␩ may be an effective resistivity兲.

共1兲

=0

for

兩J兩 ⬍ JC ,共5兲

where VC and JC are the thresholds.
In order to disturb the initial conﬁguration, a localized
resistivity is assumed at x = Lx in the 3D form

Ugai, “The Evolution of Fast Reconnection in a Three-Dimensional...” (2008)
082306-3

Phys. Plasmas 15, 082306 共2008兲

The evolution of fast reconnection…

␩共r兲 = ␩0 exp兵− 关共x − Lx兲/kx兴2 − 共兩y兩/ky兲3 − 共兩z兩/kz兲3其,共6兲

which may provide a seed of magnetic reconnection. Here,
we take ␩0 = 0.02 and kx = ky = 0.8 with different values of kz.
This reconnection disturbance is imposed only in the initialtime range 0 ⬍ t ⬍ 4, and the resistivity models will be assumed for t ⬎ 4.

Here, let us show in some detail how the full MHD
equations 共1兲 is numerically solved. The numerical procedure
involved is rather simple and almost the same as the one
taken in our earliest papers.9–11 The basic equations 共1兲 aretransformed to the conservation form
共7兲

where U = 共␳ , ␳u , B , ⑀兲 共⑀ is the total energy density兲 describe
the MHD physical quantities, and F, G, and H are functionsof U.9 Hence, what MHD equations 共1兲 state is that each of
the physical quantities U be conserved.
ndenote U共x = i⌬x , y = j⌬y , z = k⌬z兲 at time
Let Ui,j,kt = n⌬t 共⌬x, ⌬y, and ⌬z are the mesh sizes, and ⌬t is the timestep兲; then, the two-step Lax–Wendroff scheme ﬁrst provides
the auxiliary values at time t = 共2n + 1兲⌬t by
2n+1
2n
2n
= Ui,j,k− 共⌬t/2⌬x兲共F2n
Ui,j,ki+1,j,k − Fi−1,j,k兲
2n2n− 共⌬t/2⌬y兲共Gi,j+1,k− Gi,j−1,k兲2n2n− Hi,j,k−1兲,− 共⌬t/2⌬z兲共Hi,j,k+1

共8兲

2n
2n
2n
2n
2n
Ui,j,k= 共U2nwhere
i+1,j,k + Ui−1,j,k + Ui,j+1,k + Ui,j−1,k + Ui,j,k+1
2n
2n2n+ Ui,j,k−1兲 / 6, and Fi,j,k means F共Ui,j,k兲, and so forth. Then, thephysical values at t = 共2n + 2兲⌬t are ﬁnally obtained by
2n+2
2n
2n+1
= Ui,j,k− 共⌬t/⌬x兲共F2n+1
Ui,j,ki+1,j,k − Fi−1,j,k兲 − 共⌬t/⌬y兲
2n+1
2n+1
2n+1
2n+1
⫻共Gi,j+1,k− Gi,j−1,k兲 − 共⌬t/⌬z兲共Hi,j,k+1− Hi,j,k−1兲.

共9兲These are the original two-step Lax–Wendroff scheme,
which assures that the physical quantities are precisely conserved numerically.21 Then, the numerical stability conditions are given by
共兩u兩 + c兲⌬t/⌬ ⬍ 1/冑2

共10兲

for the hyperbolic terms, where ⌬ is the minimum value ofthe spatial mesh sizes, and u and c are the characteristic ﬂow
velocity and wave speed, respectively; for the parabolic
terms

␩共2⌬t兲/共2⌬兲2 ⬍ 1/4,

2n
2n
2n
共1 − ␴兲Ui,j,k+ ␴/12共U2ni+1,j+1,k + Ui−1,j+1,k + Ui+1,j−1,k
2n
2n
2n
+ U2n
i−1,j−1,k + Ui+1,j,k+1 + Ui−1,j,k+1 + Ui+1,j,k−1
2n
2n
2n
2n
+ Ui−1,j,k−1+ Ui,j+1,k+1+ Ui,j−1,k+1+ Ui,j+1,k−1
2n+ Ui,j−1,k−1
兲,

共12兲

where ␴ is the smoothing coefﬁcient.23 Also, we take theartiﬁcial viscosity terms,21
共⌬t/2⌬x兲⌬0x共Qx⌬0xU兲 + 共⌬t/2⌬y兲⌬0y共Qy⌬0yU兲

III. NUMERICAL PROCEDURE

⳵U/⳵t + ⳵F/⳵x + ⳵G/⳵y + ⳵H/⳵z = 0,

387

共11兲

where ␩ is the 共maximum兲 resistivity.
For smooth ﬂows, the original Lax–Wendroff scheme isexcellent, but for shocks involved in the fast reconnection
mechanism it becomes numerically unstable. Hence, in orderto stabilize the numerical computation, the following two
artiﬁcial terms are added to the original scheme.22 First, the2nfollowing smoothing term is considered instead of Ui,j,kin
Eq. 共9兲:+ 共⌬t/2⌬z兲⌬0z共Qz⌬0zU兲,

共13兲

where for any f共x , y , z兲, 2⌬0x f = f共x + ⌬x , y , z兲 − f共x − ⌬x , y , z兲,and similarly for ⌬0y and ⌬0z. Also, Qx, Qy, and Qz are givenby
共Qx兲i+1,j,k = ␬0兩共ux兲i+2,j,k − 共ux兲i,j,k兩,
共Qy兲i,j+1,k = ␬0兩共uy兲i,j+2,k − 共uy兲i,j,k兩,

共14兲

共Qz兲i,j,k+1 = ␬0兩共uz兲i,j,k+2 − 共uz兲i,j,k兩,
where ␬0 is the viscosity coefﬁcient. This viscosity term,with U interpreted as U2n, is simply added to the right-handside of Eq. 共9兲 when the physical variables are computed at
the second step. Note that these numerical terms do not violate the conservation of quantities.
All of our 2D and 3D MHD reconnection simulations
have taken this simple numerical scheme with the uniform
mesh sizes. In what follows, the basic effects of the numerical 共artiﬁcial兲 terms 共␴ and ␬0兲 on the dynamics of the current sheet system will be shown deﬁnitely. For the present
fast reconnection process, the artiﬁcial viscosity 关Eq. 共13兲兴
must be effective only at slow shock layers, where the current density becomes distinctly large, so that the viscosity
coefﬁcient ␬0 may be modiﬁed as

␬0 = ␬ exp兵− 关共兩J兩 − JN兲/3兴2其=␬for兩J兩 ⬎ JN .

for

兩J兩 ⬍ JN ,共15兲

Here, we take the uniform mesh sizes ⌬x = 0.04,⌬y = 0.015, and ⌬z = 0.1 or ⌬z = 0.05; also, the time step⌬t = 0.001 is taken. These numerical parameters, which arequite similar to the ones taken in our previous simulations,are ﬁxed throughout the present study. Also, the computational domain is taken to be Lx = 10, Ly = 9.6, and Lz = 9.8 for⌬z = 0.1 or 4.9 for ⌬z = 0.05.

IV. RESULTS

In the 2D situation, initiated by the reconnection disturbance 共6兲, drastic current sheet thinning eventually occurs
indeﬁnitely if there is no effective dissipation mechanism
共␩ = 0兲.10 Here, the global dynamics of the current sheet system is systematically studied in the general 3D situation. In
general, the distinct current sheet thinning will be impededby some dissipation mechanism 共physical or numerical兲.
Hence, ﬁrst of all, in order to examine the effects of numerical dissipations, the current sheet dynamics with zero physi-

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FIG. 1. Temporal behaviors of Jz at x = 10共y = z = 0兲 and ux at x = 6共y = z = 0兲for different values of ␴ and ␬ for the case of ␩ = 0.

cal resistivity 共␩ = 0兲 is examined for kz = 5 in Eq. 共6兲; then,the self-consistent coupling with the physical resistivity isdemonstrated in detail.
A. Zero resistivity

When ␩ = 0, initiated by the reconnection disturbance
共6兲, the current sheet thinning occurs as follows.10 The initialreconnection causes the By ﬁeld component along the x axis,
so that the resulting motor 共JzBy兲 force tends to eject theplasma from near the X reconnection region, and the resulting plasma outﬂow leads to a decrease in the plasma pressure
there; accordingly, the ambient magnetized plasma is sucked
into the X reconnection region, and the resulting plasma inﬂow 共uy兲 causes the current sheet thinning locally, giving rise
to enhancement of the current density near the X region 共theeffective current sheet thickness dc may be dc ⬃ 1 / 兩Jz兩兲. The
increase in Jz in turn tends to enhance the JzBy force andfurther accelerate the outﬂow velocity ux, so that the current
sheet thinning drastically occur by the nonlinear sheet pinch
effect.10
Figure 1 shows for different numerical parameters
共␴ and ␬兲 the temporal behaviors of the current density Jz at
the X neutral point 共x = 10, y = 0, z = 0兲 and those of the outﬂow velocity ux measured at 共x = 6, y = 0, z = 0兲 with ␩ = 0 for
the case of kz = 5 in Eq. 共6兲. The current sheet thinning occurs
effectively for time t ⬎ 7, and for the cases of 共␴ = 0, ␬ = 0兲and 关␴ = 0, ␬ = 3共JN = 0兲兴, numerical ﬂuctuations suddenly occur when 兩Jz兩 ⬎ 12, leading to numerical instabilities, so thatfurther computations cannot be done; on the other hand, forthe cases of ␴ = 0.004 and ␴ = 0.01, the drastic increase in Jz
is eventually suppressed.
We ﬁnd that ␴ provides an effective “numerical resistivity,” whereas ␬ as well as the numerical dissipation involvedin the original scheme provides no signiﬁcant numerical resistivity. Hence, the growth of the outﬂow velocity ux for␴ = 0 results mainly from the initial reconnection disturbance
共6兲, whereas for ␴ = 0.004 and ␴ = 0.01 some numerical reconnection may further accelerate ux; however, the acceleration by the numerical resistivity is quite small, since the re-

FIG. 2. Temporal behaviors of Jz and Ez = ␩Jz at x = 10共y = z = 0兲 for different
magnitudes of the uniform resistivity ␩ 共␴ = ␬ = 0兲. In 共a兲 of the upper panel,
the temporal behavior of Jz for the case of ␩ = 0 and ␴ = 0.01共␬ = 0兲 is shownfor comparison.

sulting outﬂow velocity ux is not signiﬁcantly inﬂuenced by
the numerical terms 共Fig. 1兲. The larger ␴ gives the larger
numerical resistivity and leads to the slower growth of the
sheet pinch. Note that ␴ makes quantities be smoothed out
and is hence useful to stabilize the numerical computations,whereas it tends to exclude 共or mask兲 distinct processes involving signiﬁcant spatial changes; also, when 兩Jz兩 ⬎ 12, thenumerical resolution of the X reconnection region becomes
rather poor for the present mesh sizes.
B. Uniform resistivity model

Most of the conventional theoretical studies have considered that magnetic reconnection should not be inﬂuenced by
the effective resistivity form, so that the uniform resistivity
model has often been employed. Here, let us examine how
the local sheet pinch, initiated by the disturbance, is inﬂuenced by the uniform resistivity model. Here, a uniform resistivity is assumed after the disturbance with kz = 5 in Eq. 共6兲
is removed, so that magnetic reconnection proceeds immediately for time t ⬎ 4. Since the physical dissipation may suppress the drastic sheet pinch, we can take ␴ = ␬ = 0 here.
Figure 2 shows the temporal behaviors of Jz at the X

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FIG. 3. 共Color online兲 Magnetic ﬁeld and plasma ﬂow conﬁguration at
t = 30 for the uniform resistivity model with ␩ = 0.001. All the similar ﬁguresthat will appear will be shown in this manner with the same velocity scale
and the same view angle.

point 共x = Lx = 10, y = z = 0兲 as well as those of the electric ﬁeld
Ez = ␩Jz for different resistivity magnitudes. For each case,the local sheet pinch grows for t ⬎ 7, but the increase in Jz is
eventually impeded by the ﬁnite resistivity. The reconnection
rate, given by 兩Ez兩, indicates that any drastic reconnection
cannot occur. We ﬁnd that after Jz has the peak value resistive tearing occurs, but the tearing is very weak. Figure 3
shows the resulting magnetic ﬁeld and plasma ﬂow conﬁgurations for ␩ = 0.001, which indicates that the fast reconnection mechanism cannot evolve in any way; in particular, theoutﬂow velocity cannot be accelerated effectively without
any standing slow shock.
In general, the temporal changes in the current density Jz
at the X point are determined by the two physical factors.
One is the reconnection ﬂow associated with the initial disturbance in 0 ⬍ t ⬍ 4 and with the subsequent reconnection
occurring for 4 ⬍ t, which tends to enhance Jz as alreadystated. The other is the magnetic diffusion due to the ﬁnite
effective resistivity, which of course tends to reduce Jz.
When the convection effect due to the reconnection ﬂow isstronger than the magnetic diffusion 兩␩Jz兩, Jz should grow.
Hence, as shown in Fig. 2, Jz grows more rapidly for thesmaller resistivity ␩. Obviously, for the uniform resistivity
model, the reconnection rate 兩Ez兩 = 兩␩Jz兩 does not grow, since
the reconnection ﬂow velocities cannot be accelerated
effectively.11
In space plasmas, Spitzer resistivity is more realistic, but
as already shown, this resistivity becomes reduced at the X
point by the reconnection process because of the notabletemperature increase, so that the associated reconnection process becomes less effective than that of the uniform resistivity model.5,11 Hence, initiated by the reconnection disturbance, current density should be remarkably enhanced
around the X neutral point for the extremely small classicalresistivity. In such situations, current-driven anomalous resistivities are likely to be caused in space plasmas.6 Then, slow
shocks could build up, and the numerical dissipation terms
are required. Hence, before going further, let us estimate thenumerical resistivity involved, so that Fig. 2 indicates that

FIG. 4. Temporal behaviors of Jz and Ez = ␩JJzat x = 10共y = z = 0兲 for different
magnitudes of ␴ and ␬ for the resistivity model ␩J with JC = 4 and k j= 0.005.

the numerical resistivity for ␴ = 0.01 corresponds to the uniform physical resistivity with a magnitude between 0.0003
and 0.0005.
C. Anomalous resistivity model ␩J

Here, let us assume the current-driven anomalous resistivity model ␩J with k j = 0.005 in Eq. 共5兲 for the initial disturbance with kz = 5. In this case, the resistivity is zero until
the threshold JC is exceeded, so that the current density Jz
increases for t ⬎ 7 共Fig. 1兲. Once the anomalous resistivity ␩Jbuilds up, magnetic reconnection grows by the selfconsistent interaction between the anomalous resistivity andthe reconnection ﬂow. Here, let us consider the rather small
threshold value JC = 4, since the current density may be notably enhanced and precise numerical computations could
not be expected when 兩Jz兩 ⬎ 12 共Fig. 1兲.
Figure 4 shows the temporal variations of Ez = ␩JJz and
Jz at the neutral point 共x = 10, y = z = 0兲 for different values of␴ and ␬. In this case, the threshold JC = 4 is exceeded att ⬃ 10, when the reconnection rate 兩Ez兩 begins to increase
with increasing Jz 共hence ␩J兲; eventually, it has the peakvalue at t ⬃ 30. Figure 5 shows the temporal behaviors of theoutﬂow velocities ux, measured at 共x = 5, y = 0, z = 0兲 and共x = 6, y = 0, z = 0兲 on the x axis, and indicates that according
to the occurrence of magnetic reconnection at x = 10 the outﬂow velocity is drastically accelerated to the Alfvén velocity

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FIG. 6. Proﬁles of ␩JJz along the x axis at different times for the case of␴ = 0.01 and ␬ = 7共JN = 0兲 for the resistivity model ␩J with JC = 4 andk j = 0.005.

FIG. 5. Temporal behaviors of ux at x = 5共y = z = 0兲 and x = 6共y = z = 0兲 fordifferent magnitudes of ␴ and ␬ for the resistivity model ␩J with JC = 4 andk j = 0.005.

共VAe ⬃ 2.7兲 in the ambient magnetic ﬁeld region. The growthof the anomalous resistivity and that of the reconnection outﬂow velocity are inﬂuenced very little by ␴ or ␬, although
for the cases 共␴ = 0.006, ␬ = 0兲 and 共␴ = 0.01, ␬ = 0兲 numericalinstabilities occur for t ⬎ 30.
In 7 ⬍ t ⬍ 10, the increase in current density is consistent
with the sheet pinch with zero resistivity 共Fig. 1兲. After the
anomalous resistivity builds up at t ⬃ 10, the resistivity
grows locally around the X point 共at x = 10兲, so that magnetic
reconnection effectively occurs to accelerate the plasma outﬂow velocity from near the X point, which in turn allows theambient magnetized plasma to be sucked into the X point so
as to enhance the current density. In general, the resistivity
increase should reduce the current density; nevertheless, theanomalous resistivity can grow in the evolutionary stage
10⬍ t ⬍ 30, which is because the growth of the reconnection
ﬂows by the localized anomalous resistivity is so powerful.
An essential point is that both the anomalous resistivity andthe reconnection ﬂow grow simultaneously to enhance eachother as a positive feedback, so that this process can be considered to be a “nonlinear instability.”
During the time range 10⬍ t ⬍ 30 slow shocks build up
and extend outward from near the X point 共x = 10兲. At theslow shock layer, current density 共hence ␩J兲 becomes notably
large, so that the slow shock layer, attached to the reconnec-

tion 共diffusion兲 region, becomes wider and the diffusion region becomes longer; hence, resistive tearing is likely to occur in the long diffusion region. In fact, the tearing suddenly
occurs at t ⬃ 30, and the initial X point at x = 10 becomes an
O-type neutral point with the X point being shifted in the x
direction. Figure 6 shows the proﬁles of ␩JJz along the x axis
for the case of ␴ = 0.01 and ␬ = 7共JN = 0兲, which indicates thatthe anomalous resistivity ␩J, ﬁrst enhanced around x = 10 at
t = 30, moves in the x direction, and at t = 39 it is distinctly
localized around the new X point at x ⬃ 8.
Figure 7 shows the resulting magnetic ﬁeld and plasma
ﬂow conﬁgurations at t = 38, which indicates that the X point,
initially located at x = 10, has shifted to x ⬃ 8, where theanomalous resistivity ␩J is locally enhanced 共Fig. 6兲; on theother hand, an O-type neutral point is formed at x = 10 because of the tearing at t ⬃ 30. For this resistivity model, any
extremely thin slow shock layer cannot be sustained becauseof the enhanced resistivity there, and resistive tearing islikely to take place; hence, the fast reconnection process becomes strongly time-dependent or bursty.

FIG. 7. 共Color online兲 Magnetic ﬁeld and plasma ﬂow conﬁguration at
t = 38 for the case of ␴ = 0.01 and ␬ = 7共JN = 0兲 for the resistivity model ␩Jwith JC = 4 and k j = 0.005.

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FIG. 8. Temporal behaviors of Jz and Ez = ␩DJz at x = 10共y = z = 0兲 for different magnitudes of ␴ and ␬ for the resistivity model ␩D with VC = 12 andkd = 0.003.

FIG. 9. Temporal behaviors of ux at x = 5共y = z = 0兲 and x = 6共y = z = 0兲 fordifferent magnitudes of ␴ and ␬ for the resistivity model ␩D with VC = 12
and kd = 0.003.

D. Anomalous resistivity model ␩Dnection jet is little inﬂuenced by ␴ or ␬. Since the anomalous
resistivity and the reconnection ﬂow drastically grow simultaneously to enhance each other, the fast reconnection evolution has been considered to be a nonlinear instability.10,11
Figure 10 shows the resulting magnetic ﬁeld and plasma
ﬂow conﬁguration for ␴ = 0.01 and ␬ = 7共JN = 12兲, which isquite consistent with those already reported in our similar

Next, let us consider another anomalous resistivity
model ␩D with the threshold VC = 12 and kd = 0.003 in Eq. 共4兲.
Most of our previous studies have taken this resistivity
model to demonstrate the fast reconnection mechanism. As
before, kz = 5 is taken for the initial disturbance Eq. 共6兲, andFig. 8 shows the temporal behaviors of Ez = ␩DJz and Jz at
x = 10共y = z = 0兲 for different values of ␴ and ␬. Although thetime when the threshold VC = 12 is exceeded is somewhat
inﬂuenced by ␴ 共Fig. 1兲, the basic dynamics of the fast reconnection evolution is not signiﬁcantly inﬂuenced. When␩D builds up at t ⬃ 18, the self-consistent interaction between
the reconnection ﬂow and the anomalous resistivity gives
rise to the effective growth of the global reconnection ﬂow,
so that the reconnection rate Ez drastically grows at t ⬃ 30,
until the nonlinear saturation phase is attained at t ⬃ 35.
In this case, after ␩D is caused, the current density Jz
decreases unlike the case for the resistivity model ␩J; nevertheless, the reconnection rate 兩␩DJz兩 is remarkably enhanced.
This is because the plasma density ␳ is extremely rareﬁed at
the X point 共at x = 10兲 because of the rapid growth of thereconnection ﬂows. Figure 9 shows the temporal behaviorsof the reconnection outﬂow velocity ux, which indicates thatthe reconnection outﬂow velocity rapidly grows to attain the
Alfvén velocity VAe ⬃ 2.7 and that the resulting fast recon-

FIG. 10. 共Color online兲 Magnetic ﬁeld and plasma ﬂow conﬁguration at
t = 42 for the case of ␴ = 0.01 and ␬ = 7共JN = 12兲 for the resistivity model ␩Dwith VC = 12 and kd = 0.003.

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FIG. 11. Temporal behaviors of Ez = ␩DJz at x = 10共y = z = 0兲 for the cases ofkz = 2 and 3 for different values of ␴ and ␬ with the resistivity model ␩D with
VC = 4 and kd = 0.003.

studies. In fact, our previous studies for the resistivity model␩D have been shown mostly for ␴ = 0.01 and ␬ = 7; for instance, our latest papers on two-ribbon ﬂares took ␴ = 0.01
and ␬ = 7共JN = 6兲.19 As already demonstrated, it is the resulting Alfvénic fast reconnection jet that is most responsible for
distinct plasma processes observed in ﬂares and substorms.
Unlike the case of the resistivity model ␩J, the fast reconnection mechanism becomes almost quasisteady with verythin slow shock layers, although tearing is eventually causedin the diffusion region for this case, too.
E. Dependence on kz

Finally, let us show the reconnection evolutions forthe cases of kz = 2 and 3 for the initial disturbance 共6兲 withthe anomalous resistivity model ␩D. Here, we take⌬z = 0.05共Lz = 4.9兲 and the threshold VC = 4 and kd = 0.003
in Eq. 共4兲. Figure 11 shows the temporal behaviors of
Ez = ␩DJz at x = 10 for the cases kz = 2 and 3, where we assume
␴ = 0.01 and ␬ = 7 共JN = 0 or 6兲; also, Fig. 12 shows those of
the resulting reconnection ouﬂow velocity ux. As can beseen, the reconnection dynamics is little inﬂuenced by thenumerical viscosity term. In each case, the threshold VC = 4 isexceeded at t ⬃ 10, when the reconnection process, coupled
to the anomalous resistivity, builds up. For the case ofkz = 2, the reconnection rate or the reconnection ﬂow cannot
grow effectively. This is because the plasma inﬂow uz notably grows into the X point to prevent the effective growth of
the inﬂow uy of the ambient magnetized plasma.13 For thecase of kz = 3, both the reconnection rate and the outﬂow
velocity grows considerably, but the enhanced reconnection
rate is soon reduced; in particular, ux is decelerated before it
attains the Alfvén velocity VAe.
Figure 13 shows the resulting magnetic ﬁeld and plasma
ﬂow conﬁgurations for kz = 2 and 3 at time t = 35, when both
the reconnection rate and the outﬂow velocity have the peakvalues for kz = 3. For kz = 2, no signiﬁcant magnetic reconnection takes place, since the ambient magnetized plasma cannot
collapse into the X reconnection region. On the other hand,for kz = 3, the reconnection process becomes somewhat im-

FIG. 12. Temporal behaviors of ux at x = 5共y = z = 0兲 and x = 6共y = z = 0兲 for thecases of kz = 2 and 3 for different magnitudes of ␴ and ␬ with the resistivity
model ␩D with VC = 4 and kd = 0.003.

pulsive and causes the magnetic ﬁeld dipolarization 共or magnetic loop兲 in a very narrow channel, but the fast reconnection mechanism cannot be realized. Hence, it may be said
that when the z-directional width of the current sheet issmaller than three times its y-directional current sheet thickness, the Alfvénic fast reconnection jet cannot be caused.
V. SUMMARY AND DISCUSSION

In the present paper, all the details of the numerical procedure are shown, and the explicit numerical dissipation
terms involved in our MHD simulations are checked in detail. Then, we systematically examine the global dynamics oflong current sheet system by 3D MHD simulations in therange of the simulation parameters where the numerical resistivity involved, estimated to be at most 0.0005, is muchsmaller than the physical resistivity; hence, the reconnection
processes are little inﬂuenced by the numerical dissipation
terms in this parameter range. We then ﬁnd as in the 2D
situation that, initiated by a reconnection disturbance, thecurrent sheet becomes drastically thin by the pinch effect,leading to extreme increase in the current density around the
X reconnection region in the absence of effective resistivity.
For the uniform resistivity model, the local current increase
共sheet pinch兲 is suppressed by the magnetic diffusion, andUgai, “The Evolution of Fast Reconnection in a Three-Dimensional...” (2008)
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FIG. 13. 共Color online兲 Magnetic ﬁeld and plasma ﬂow conﬁguration at
t = 35 for the cases of kz = 2 and 3 for ␴ = 0.01 and ␬ = 7共JN = 6兲 with theresistivity model ␩D with VC = 4 and kd = 0.003.

the current density at the X point becomes larger for thesmaller resistivity, but the fast reconnection mechanism cannever be realized, since the reconnection jet cannot be accelerated effectively.
When the current density is extremely enhanced locallyin the X reconnection region, current-driven microturbulences are likely to be caused, leading to anomalous
resistivities.6 Once the anomalous resistivity ␩J 关Eq. 共5兲兴 is
ignited, both the reconnection ﬂow and the anomalous resistivity simultaneously grow to enhance each other, which ishence considered to be a nonlinear instability; accordingly,
the fast reconnection mechanism involving slow shocks iseventually set up, leading to the Alfvénic fast reconnection
jet. For this resistivity model, the shock layer, attached to the
X reconnection 共diffusion兲 region, cannot become thin because of the enhanced resistivity, so that the diffusion region
becomes longer and resistive tearing soon occurs. Hence, thefast reconnection process for this anomalous resistivity
model becomes strongly time-dependent or bursty.
For the anomalous resistivity model ␩D 关Eq. 共4兲兴 too,
once ␩D is ignited, the reconnection ﬂow drastically grows toenhance ␩D, since the plasma density is extremely rareﬁed in
the X reconnection 共diffusion兲 region; accordingly, the fastreconnection mechanism is established quasisteadily, although resistive tearing may eventually occur in the diffusion
region. The fast reconnection evolution can be considered to
be a nonlinear instability because of the simultaneous growthof the reconnection ﬂow and the anomalous resistivity. This

393

resistivity model has been taken by most of our previous
papers, and the 3D Alfvénic fast reconnection jet was applied
to distinct phenomena observed in ﬂares and substorms.
When the z-directional length kz 共normalized by d0兲 ofthe initial disturbance 共6兲 is small, the fast reconnection evolution becomes difﬁcult even for the resistivity model ␩D.
For the case of kz = 2, ␩D can be triggered, but any effective
reconnection cannot take place, since the ambient magnetized plasma cannot collapse into the X point. For kz = 3, thereconnection rate ﬁrst grows considerably but soon becomes
reduced before the reconnection jet is accelerated to the
Alfvén velocity, so that any fast reconnection mechanism
cannot be realized, although the reconnection process issomewhat impulsive. Since the large kz cannot be taken if thecurrent sheet width is small, it may be said that when thez-directional width of the initial current sheet is smaller than
three times its y-directional thickness, the fast reconnection
mechanism cannot be realized.
Most of simulation studies on magnetic reconnection
have assumed a large-scale long current sheet. As shown
here, if a seed of reconnection is imposed, notable current
sheet thinning occurs, and the drastic current density increase
can be suppressed by physical and/or numerical dissipation
mechanisms. Hence, it is essential to carefully check the effects of numerical dissipations on the reconnection process,since numerical resistivities, which may become distinct in
the regions of large current density, sometimes behave as
current-driven anomalous resistivities, leading to 共artiﬁcial兲
fast reconnection. For MHD simulations, physical dissipations are explicitly given, so that it is rather easy to estimate
the numerical dissipations, whereas for particle simulations it
may be quite difﬁcult to identify the physical dissipation
mechanism 共effective resistivity兲 in the small X reconnection
region of electron inertial length.
We have shown that if a thin current sheet of sufﬁciently
large scale is formed in space plasmas, the fast reconnection
mechanism may be likely to build up; in fact, the majorwell-known 共MHD兲 features of substorms and ﬂares can berelevantly explained by the spontaneous fast reconnection
model.13–15,17–19 However, MHD studies cannot treat the details of microphysics, so that we have simply assumed thatwhen the sheet pinch drastically grows so that the current
sheet thickness is of the ion 共or even electron兲 inertial length,
current-driven anomalous resistivities should be caused. Although we have demonstrated that such an effective resistivity that is driven by strong currents is most responsible forthe fast reconnection evolution, details of the dissipation
mechanism 共or the effective resistivity兲 at the X neutral pointshould be studied by precise particle simulations.

ACKNOWLEDGMENTS

This work was supported by Grant-in-Aids for Creative
Scientiﬁc Research “The Basic Study of Space Weather Prediction” 共17GS0208, Head Investigator: K. Shibata兲 from the
Ministry of Education in Japan, Mitsubishi Foundation,
RISH of Kyoto University, and Solar-Terrestrial Environment Laboratory of Nagoya University.

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J. A. Slavin, M. F. Smith, E. L. Mazur, D. N. Baker, E. W. Hones, Jr., T.
Iyemori, and E. W. Greenstadt, J. Geophys. Res. 98, 15425, DOI:
10.1029/93JA01467 共1993兲; J. A. Slavin, E. J. Smith, B. T. Tsurutani, D.
G. Sibeck, H. J. Singer, D. N. Baker, J. T. Gosling, E. W. Hones, and F. L.
Scarf, Geophys. Res. Lett. 11, 657, DOI: 10.1029/GL011i007p00657
共1984兲.
17
M. Ugai and L. Zheng, Phys. Plasmas 13, 032901 共2006兲; 13, 062906
共2006兲.
18
M. Ugai and K. Kondoh, Phys. Plasmas 13, 102903 共2006兲.
19
M. Ugai, Phys. Plasmas 14, 102904 共2007兲; 15, 032902 共2008兲.
20
G. Hornig and E. R. Priest, Phys. Plasmas 10, 2712 共2003兲; M. Ugai, ibid.
7, 867 共2000兲.
21
R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value
Problems, 2nd ed. 共Interscience, New York, 1967兲, p. 365.
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M. Ugai, Comput. Phys. Commun. 49, 185 共1988兲.
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T. Sato and T. Hayashi, Phys. Fluids 22, 1189 共1979兲.
13
14

Shimizu et al., “MHD Study of 3D Intermittent Fast Magnetic Reconnection” (2009)
The Astrophysical Journal, 707:420–427, 2009 December 10395
doi:10.1088/0004-637X/707/1/420


C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

MAGNETOHYDRODYNAMICS STUDY OF THREE-DIMENSIONAL FAST MAGNETIC RECONNECTION FOR
INTERMITTENT SNAKE-LIKE DOWNFLOWS IN SOLAR FLARES
T. Shimizu1,3 , K. Kondo1,4 , M. Ugai2,5 , and K. Shibata2,6

1 Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama City 790-8577, Japan; shimizu@cosmos.ehime-u.ac.jp
2 Kwasan Observatory, Graduate School of Science, Kyoto University, Kyoto City 607-8471, Japan

Received 2009 June 3; accepted 2009 October 1; published 2009 November 23

ABSTRACT
Three-dimensional instability of the spontaneous fast magnetic reconnection is studied with magnetohydrodynamics(MHD) simulation, where the two-dimensional model of the spontaneous fast magnetic reconnection is destabilized
in three dimensions. In two-dimensional models, every plasma condition is assumed to be uniform in the sheet
current direction. In that case, it is well known that the two-dimensional fast magnetic reconnection can be caused
by current-driven anomalous resistivity, when an initial resistive disturbance is locally put in a one-dimensional
current sheet. In this paper, it is studied whether the two-dimensional fast magnetic reconnection can be destabilized
or not when the initial resistive disturbance is three dimensional, i.e., that which has weak ﬂuctuations in the sheet
current direction. According to our study, the two-dimensional fast magnetic reconnection is developed to thethree-dimensional intermittent fast magnetic reconnection which is strongly localized in the sheet current direction.
The resulting fast magnetic reconnection repeats to randomly eject three-dimensional magnetic loops which arevery similar to the intermittent downﬂows observed in solar ﬂares. In fact, in some observations of solar ﬂares,
the current sheet seems to be approximately one dimensional, but the fast magnetic reconnection is strongly
localized in the sheet current direction, i.e., fully three dimensional. In addition, the observed plasma downﬂows
as snake-like curves. It is shown that those observed features are consistent with our numerical MHD study.
Key words: MHD – Sun: ﬂares
Online-only material: color ﬁguresassociated with the fast magnetic reconnection are often reported(Shibata et al. 1995; Masuda et al. 1994; Shibata 1996; Tsuneta
1996; Yokoyama et al. 2001; Longcope et al. 2007; Innes et al.
2003; Asai et al. 2004a, 2004b; Magara & Shibata 2008).
In some of those observation reports, the three-dimensional
structure of a fast magnetic reconnection has been discussed.
McKenzie & Hudson (1999), McKenzie (2000), Innes et al.
(2003), and Asai et al. (2004a) reported that the appearance of alarge-scale arcade structure and intermittent plasma downﬂows
are simultaneously observed in solar ﬂares and may be caused by
fast magnetic reconnection in the high altitude corona. According to their reports, the observed arcade structure is relatively
two dimensional but the plasma downﬂows observed on the topof the arcade structure are strongly localized in space, i.e., fully
three dimensional. In addition, the ejection of the plasma downﬂows is intermittent. The appearance of the two-dimensional
arcade structure suggests that the current sheet on the top of thearcade structure widely spreads in the sheet current direction
and the fast magnetic reconnection occurring there may be twodimensional in a macroscopic view. However, the appearance ofthe three-dimensional plasma downﬂows suggests that the fastmagnetic reconnection is strongly localized in the sheet current
direction, i.e., three dimensional. In fact, Innes et al. and Asaiet al. suggested that a fully three-dimensional fast magnetic
reconnection is intermittently repeated in a one-dimensional
current sheet. Also, Aschwanden (2002, Figure 18) proposeda similar theoretical model to explain bi-directional oscillativeelectron beams observed in solar ﬂares. However, theoreticallyand numerically, it is still unclear how the three-dimensional
fast magnetic reconnection occurs in a one-dimensional current
sheet.
There are some MHD studies for the large-scale arcade structure in solar ﬂares; for instance, Archontis et al. (2004, 2005)

1. INTRODUCTION
Fast magnetic reconnection provides a physical mechanism,
by which magnetic energy is explosively converted into plasma
kinetic and thermal energies. This mechanism has hence been
considered to play a crucial role in solar ﬂares and geomagnetic
substorms. When the magnetic reconnection process includes
slow shocks extending from the magnetic diffusion region,
the reconnection process becomes signiﬁcantly active, giving
rise to high-speed plasma jets and the ejection of large-scale
magnetic loops. Ugai (1991), Ugai & Shimizu (1996), and Ugai
et al. (2005) numerically revealed that the two-dimensional
fast magnetic reconnection can be established by a kind of
the current-driven anomalous resistivity embedded in a onedimensional current sheet, where a pair of slow shocks, highspeed plasma jets, and large-scale plasma loops, i.e., plasmoids,
have been observed. Recently, some similar two-dimensional
MHD studies were reported (Schumacher et al. 2000, Shibata &Tanuma 2001). In addition, with the recent drastic developmentof the computer performance, the MHD study of the fastmagnetic reconnection has been developed to three-dimensionalmodels (Linton & Priest 2003; Archontis et al. 2004, 2005;
Pontin & Craig 2005; Pontin et al. 2007).
On the other hand, according to the recent space satellite observations for solar ﬂares, the plasma jets and magnetic loops
3

Associate Professor, Research Center for Space and Cosmic Evolution,
Ehime University, Matsuyama City, Japan.
4 Researcher, Research Center for Space and Cosmic Evolution, Ehime
University, Matsuyama City, Japan.
5 Professor, Research Center for Space and Cosmic Evolution, Ehime
University, Matsuyama City, Japan.
6 Professor, Kwasan Observatory, Graduate School of Science, Kyoto
University, Japan.

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current sheet. After the initial resistive disturbance is removed,
the fast magnetic reconnection is spontaneously developed bya current-driven anomalous resistivity enhanced by a nonlinear
tearing instability. Hence, any externally driven mechanism is
not required to keep the reconnection process.
The compressible MHD equations are

numerically studied large-scale arcade structures observed insolar ﬂares. However, the three-dimensional localized plasma
downﬂows are not studied there. Also, Linton & Priest (2003),
Pontin & Craig (2005), and Pontin et al. (2007) reported threedimensional MHD studies for the fast magnetic reconnection.
In these studies, the initial conditions in the current sheet are
fully three dimensional in order to apply for the complicatedmagnetic ﬁeld conﬁgurations observed in solar ﬂares. In thiscase, the resulting fast magnetic reconnection directly becomesfully three dimensional. At this point, these three-dimensional
studies cannot explain why the three-dimensional fast magnetic
reconnection occurs in the one-dimensional current sheet. In
other words, these three-dimensional studies are not regarded
as a direct extension of the previous two-dimensional studies.
As for the MHD study for the both of the observed twodimensional large-scale arcade structure and three-dimensional
localized plasma downﬂows, there is a “patchy” reconnection
model based on a fully three-dimensional and non-steady fastmagnetic reconnection on ﬂux tubes, which was reported by
Linton & Longcope (2006). In this model, the one-dimensional
current sheet is destabilized by a fully three-dimensional resistive disturbance strongly localized in the sheet current direction. Hence, the resulting fast magnetic reconnection is strongly
localized in the sheet current direction and directly becomes
three dimensional. Since they focused on the dynamics of postreconnection jets, it is still unclear why the fast magnetic reconnection itself is three dimensional and intermittent, as observedin solar ﬂares.
Only focusing on the intermittent plasma ﬂows associatedwith the fast magnetic reconnection, the secondary or multiple
fast magnetic reconnection models proposed by Ugai (1984),
Shibata & Tanuma (2001), and Schmacher & Kliem (1996) may
be applicable. However, since these studies are two dimensional,
the models are not directly applied for the observations of thethree-dimensional plasma downﬂows in solar ﬂares.
In this paper, the two-dimensional fast magnetic reconnection
model proposed by Ugai is numerically developed to a threedimensional model. Note that the three-dimensional model
shown in this paper is based on current-driven anomalous
resistivity same as that of the three-dimensional model studied
by Ugai et al. (2005), but the initial resistive disturbanceis largely different. Because, in the studies done by Ugai,
the initial resistive disturbance to initiate the reconnection
process is strongly localized in the sheet current direction,
directly leading to the fully three-dimensional fast magnetic
reconnection. While the initial resistive disturbance employedin this paper is approximately two dimensional but only includes
a weak three-dimensional non-uniformity (i.e., ﬂuctuation) in
the sheet current direction. In this case, the resulting fastmagnetic reconnection is expected to be two dimensional,
but our MHD study shows that it results in a fully threedimensional fast magnetic reconnection. It means that the twodimensional fast magnetic reconnection is unstable for threedimensional resistive perturbations. In addition, the resulting
three-dimensional fast magnetic reconnection intermittently
occurs and the intermittent plasma downﬂows as snake-likecurves. In this paper, it is shown that the intermittency and snakelike outﬂows are very similar to the intermittent and snake-like
plasma downﬂows observed in solar ﬂares.

where β0 is the ratio of the plasma pressure to the magnetic
pressure in the magnetic ﬁeld region 1 < y < Y1 , where
β0 = 0.15 is taken. Fluid velocity u = (0, 0, 0) and plasma
density ρ = 1 are initially assumed. The normalization ofquantities is self-evident. Distances are normalized by d0 , B
2by Bx0 , and P√by Bx0/(2μ0 ); also, ρ is normalized by ρ0 , u by
VAx0 (= Bx0 / μ0 ρ0 ), time t by d0 /VAx0 , current density J by
Bx0 /(μ0 d0 ), and so forth.
For the boundary conditions of the computational region,
the symmetry boundary condition is assumed on the yz-planeat x = 0, xz-plane at y = 0, and xy-plane at z = 0 and
Lz . The other boundaries are assumed to be the open (free)boundary where all the quantities are determined by the statesof the inner region, so that the ﬁrst derivatives of the quantities
in the direction normal to the boundaries vanish, except thenormal component of B, which is determined by the solenoidal
condition. Accordingly, the computational region is restricted tothe ﬁrst quadrant only and taken to be a square box, 0  x  Lx ,0  y  Ly and 0  z  Lz . For the numerical conditions inthis paper, the computational region Lx = 44 and Ly = 9 aretaken with the mesh points Nx = 880 and Ny = 1000. In thispaper, the case of Lz = 20 and Nz = 100 is only studied. For
Lz > 10, the numerical results discussed below basically do notchange. While for Lz < 2, the three-dimensional fast magnetic
reconnection almost does not occur. In other words, in such acase, the two-dimensional fast magnetic reconnection seems to
be stable for three-dimensional resistive perturbation.
The MHD simulation procedures are basically the same asthat of Ugai et al. (2005), except the three-dimensional setup of
the initial resistive disturbance. The initial resistive disturbanceshown below is put to initiate a local tearing instability in the
one-dimensional current sheet during 0 < T < 4:

2. MHD SIMULATION SETUP

η0 = 0.04 exp(−α1 (x 2 + y 2 ))(1 + α2 cos(π z/Lz )),In the spontaneous fast reconnection model, the reconnection
process is initiated by a small resistive disturbance induced in thewhere α1 = α2 = 0.01. This disturbance works as a trigger for afast magnetic reconnection. For this initial resistive disturbance,

Dρ/Dt = −ρ · u, ρDu/Dt = − P + J × B,
∂B/∂t − × (u × B) = − ×(ηJ),ρDe/Dt = −P
· u + ηJ2 ,J = × B/μ0 ,· B = 0,(1)

where D/Dt ≡ ∂/∂t + u · ; the gas law, P = (γ − 1)ρe,
is assumed (e is the internal energy per unit mass and γ is the
speciﬁc heat ratio with γ = 5/3 assumed here), as is Ohm’s law,E + u × B = ηJ (η is the resistivity). The basic Equations (1)are transformed to a conservation-law form, and a two-step
Lax–Wendroff scheme is used for the numerical computation.
The initial current sheet system has a one-dimensional structure, i.e., the magnetic ﬁeld B = [Bx (y), 0, 0] is assumed as
Bx (y) = sin(πy/2) for 0 < y < 1; Bx = 1 for 1 < y < Y1 ;Bx = cos[(y − Y1 )π/1.2] for Y1 < y < Y1 + 0.6); Bx = 0 for
y > Y1 +0.6; also, Bx (y) = −Bx (−y) for y < 0. Here, Y1 = 5.0.
The plasma pressure P (y) initially satisﬁes the pressure-balance
condition
P + Bx2 = 1 + β0 ,
(2)

(3)

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reconnection is caused. This directly results from that the initialresistive disturbance is almost two dimensional. At the second
stage (50 < T < 80), the ﬁrst three-dimensional fast reconnection starts. Then the three-dimensional fast reconnection isintermittently caused at the third stage (80 < T ). It results inthe multiple three-dimensional fast reconnection which generates snake-like outﬂows similar to EUV observations in solar
ﬂares.
3.2. The First Stage (Two-dimensional Fast Reconnection)

(a)

(b)
Figure 1. (a) Magnetic ﬁeld lines and plasma ﬂow vector ﬁeld at T = 40.
(b) Proﬁles of ux , By , and uz along the x-axis.

the magnetic Reynolds number is 25 at the origin. Because of
α2 = 0.01, this resistive disturbance is slightly ﬂuctuated only
by 1%in the z-direction which is the sheet current direction. The
case of α2 = 0.0 exactly corresponds to the two-dimensional
fast magnetic reconnection (Ugai 1984). This z-directional
ﬂuctuation works as a perturbation to three-dimensionallydestabilize the two-dimensional fast magnetic reconnection. At
this point, note that the initial resistive disturbance employedby Ugai et al. (2005) used much larger α2 which was almost
equal to unity. Accordingly, in Ugai’s studies, the resulting
fast magnetic reconnection directly becomes three dimensional.
While the resulting fast magnetic reconnection shown in thispaper is expected to be approximately two dimensional, at least,
at the initial stage. In fact, the magnetic loop ﬁrstly ejected fromthe reconnection region is two dimensional, as shown later inFigure 1.
After T = 4, η0 is removed and, instead, the current-driven
anomalous resistivity shown below is set, which spontaneously
drives the fast magnetic reconnection:
η1 = 0.002(Vd − Vc )for

Vd > Vc(4)η1 = 0.0

for

Vd < Vc ,(5)

where Vd = |J |/ρ is the ion–electron drift velocity, and
Vc = 4.0 is set. The current density J is obtained from rot B
and is unity at the center of the initial current sheet, i.e., y = 0.
3. SIMULATION RESULTS

This simulation starts from the one-dimensional current sheet
which consists of x-directional anti-parallel magnetic ﬁeld lines.
There is no plasma ﬂow at time T = 0. Due to the initialresistive disturbance given by Equation (3), a weak magnetic
reconnection starts on the z-axis which becomes an initial
magnetic neutral line. After the initial resistive disturbance isremoved at T = 4, the current-driven anomalous resistivity
given by Equations (4) and (5) is assumed. The magnetic
reconnection process is temporally stopped by Equation (5)because the current sheet is thick at time T = 4. However, as theplasma frozen in the reconnected ﬁeld lines moves away fromthe z-axis, due to the magnetic tension, the current sheet aroundthe z-axis becomes thinner and Equation (4) starts to work
instead of Equation (5). Then the fast magnetic reconnection
spontaneously starts to develop.
Figure 1(a) shows some reconnected magnetic ﬁeld lines
traced from three x-directional straight lines on a plane of
y = 0 at time T = 40, which are respectively located on the xaxis(z = 0), z = 0.5Lz and z = Lz . Here, Lz = 20 is set. Notethat non-reconnected magnetic ﬁeld lines are not shown in thisﬁgure, which exist in the upper region of these reconnected ﬁeld
lines. Considering symmetry boundary conditions around the zaxis, a magnetic neutral line is formed along the z-axis. Since
every magnetic ﬁeld line is almost the same on each xy-plane of
the x-axis(z = 0), z = 0.5Lz and z = Lz , the reconnection
process is evidently two dimensional. It means that, at thistime, z-directional perturbation included in the initial resistive
disturbance, Equation (3), does not yet work. Considering thesymmetry boundary condition on the xz-plane at y = 0, a twodimensional magnetic loop is formed between x = 13 and 26.
Figure 1(b) shows proﬁles of By and ux along the x-axis, which
are respectively plotted as a solid line and a dash line. Also, uz
proﬁle is shown as a dotted line and is constantly zero, becauseof the symmetry boundary condition embedded on the xy-planeat z = 0. The plasma ﬂow speed ux plotted as a dash line
increases from zero at x = 0 and reaches the Alfvén speed VA
measured in the uniform magnetic ﬁeld region, which is initiallyunity. In this simulation, the Alfvén waves take T = 44 to pass
the distance of Lx = 44. Also, the reconnected ﬁeld intensity Byplotted as a solid line is zero at x = 0 and takes slightly positive
values in 0 < x < 10, i.e., the Alfvénic reconnection jet region.
Beyond about x = 10, By rapidly increases. It is the beginningof a two-dimensional magnetic loop. After By took a peak inthe magnetic loop, By slowly decreases to zero. These features
are the same as that of the two-dimensional studies reported in
Ugai & Shimizu (1996), Shimizu & Ugai (2003), and Ugai et al.
(2005).

3.1. Overview of Simulation

3.3. The Second Stage (Single Three-dimensional Fast
Reconnection)

As shown below, this simulation can be divided into threestages in time. At the ﬁrst stage (T < 50), two-dimensional fast

Figure 2(a) shows some magnetic ﬁeld lines at time T = 66
with the same format as Figure 1(a). In this ﬁgure, By contour

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(a)
Figure 3. Magnetic ﬁeld lines and By (>0) contour map at T = 138.

(b)

(c)
Figure 2. (a) Magnetic ﬁeld lines, contour map of By ﬁeld intensity and ﬂow
velocity at T = 66. Magnetic ﬁeld lines traced from three straight lines, i.e.,z = 0, z = 0.5Lz , and z = Lz for y = 0. By contour map is shown only
for positive intensity. (b) x-directional proﬁles of By and ux along the x-axis.
(c) x-directional proﬁles of By and ux along a line of z = Lz and y = 0.

map and plasma ﬂow vectors on the xz-plane at y = 0 are
also shown. The two-dimensional large magnetic loop observed
between x = 26 and x = 44(= Lx ) corresponds to that of
Figure 1(a) and has propagated in the positive x-direction. The
magnetic ﬁeld lines traced from the x-axis, i.e., z = 0, shows
another small magnetic loop between x = 3 and x = 8. Whilemagnetic ﬁeld lines traced from z = 0.5Lz and z = Lz do notshow such a loop conﬁguration between x = 3 and x = 8.
Also, a high By intensity region related to the small magnetic
loop can be observed between 3 < x < 8 and 0 < z < 10on the By contour map of the xz-plane at y = 0. Hence, thesmall magnetic loop is three dimensional, and it means that thethree-dimensional fast magnetic reconnection process started
there.
Figure 2(b) shows proﬁles along the x-axis with the sameformat as Figure 1(b). The dash line, i.e., ux proﬁle, takes aloose peak at about x = 22, and also, the solid line, i.e., By
proﬁle, takes a peak at x = 26. These peaks correspond to

the two-dimensional magnetic loop observed in Figure 1(b). In
addition, the dash line takes another positive peak around x = 4,
and the solid line takes a negative peak between x = 6 andx = 7. This By variation, including the change from positive tonegative, means that the small three-dimensional magnetic loop
has closed ﬁeld lines, as shown in Figure 2(a). In Figure 2(c),which shows similar proﬁles at z = Lz , there is no negative
By region in 2 < x < 6, in contrast to Figure 2(b). It meansthat the three-dimensional small magnetic loop is localized in
0 < z < 10 = 0.5Lz . In 0 < x < 3 in Figures 2(b) and(c), another negative By region is observed. This narrow regionis also a two-dimensional magnetic loop formed along the zaxis and consists of closed ﬁeld lines, because of the symmetryboundary condition around the z-axis. This small and long twodimensional magnetic loop rapidly grows after this time.
3.4. The Third Stage (Multiple Three-dimensional Fast
Reconnection)
Figure 3 shows some magnetic ﬁeld lines and By contour
map on the xz-plane at time T = 138, with a similar format
to those of Figures 1(a) and 2(a). In contrast to Figure 2,
some strong By intensity regions are locally observed on the
By contour map. The appearance of these By intensity regions
means that the three-dimensional fast magnetic reconnection
process has locally occurred at some points, resulting in themultiple formations of three-dimensional magnetic loops. The
two-dimensional magnetic loop observed in Figures 1 and 2 hasgone away out of x = Lx boundary. Another two-dimensional
magnetic loop has grown largely around the z-axis, which wasslightly observed in the vicinity of the z-axis in Figure 2(a).
In this section, it was shown that the initial z-directional
perturbation embedded in Equation (3) causes multiple threedimensional fast magnetic reconnection which is strongly localized in the z-direction, i.e., the sheet current direction. As shownin Figure 1, the ﬁrst stage of the fast magnetic reconnection isalmost two dimensional. It means that Equation (3) almost twodimensionally worked and the initial z-directional perturbation
was very weak. However, at the second and third stages, thefast magnetic reconnection was drastically developed in three
dimensions. It means that the two-dimensional fast magnetic
reconnection is unstable for a z-directional perturbation along
the magnetic neutral line. In fact, even when the z-directional
ﬂuctuation intensity α2 in Equation (3) is reduced from 0.01, theShimizu et al., “MHD Study of 3D Intermittent Fast Magnetic Reconnection” (2009)424

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Figure 4. By contour maps at time T = 130, 138, and, 161. Orange-colored regions indicate the positive By region. Solid contour lines indicate the negative By region.
Red lines are plotted by tracing test particles on the time-dependent ﬂow vector ﬁeld.
(A color version of this ﬁgure is available in the online journal.)

three-dimensional fast reconnection shown here occurs, as timeproceeds. In our previous study, it has been revealed that thisthree-dimensional instability is resulted from a nonlinear development of the three-dimensional fast magnetic reconnection(Shimizu et al. 2009).
According to incompressible Fourier MHD simulations executed by Dahlburg et al. (1992) and Dahlburg & Einaudi (2002),
it has been revealed that the one-dimensional current sheet canbe destabilized by the constant and uniform resistivity, leading to the three-dimensional tearing instability. With respect
to the three-dimensional instability of the reconnection process in the one-dimensional current sheet, these studies seemto be consistent with our studies shown here. However, in thefollowing section, it is shown that the three-dimensional fastreconnection driven by the current-driven anomalous resistivity employed here causes the drastically multiple formationsof three-dimensional reconnection jets which is similar to theintermittent and snake-like downﬂows observed in solar ﬂares.
3.5. Snake-like Outﬂow Jets
In this section, let us focus on the outﬂow jets generated in themultiple three-dimensional fast magnetic reconnection shown in
the previous section. Figures 4(a)–(c) show By contour maps on
the xz-plane, respectively, at time T = 130, 138, and 161.
In order to apply for a observation data of solar ﬂare, theseﬁgures have been rotated from Figures 1(a), 2(a), and 3(a); andhence, the x-axis in these ﬁgures is vertically drawn. The upper
abscissa is the z-axis which is the magnetic neutral line andthe reconnection jets ﬂow downward, i.e., toward the Sun. The
solid contour lines show the negative By regions, and the orangecolored regions show the positive By regions. A two-dimensional
magnetic loop observed around the z-axis in Figure 3 is drawn
as a non-colored (white) band-shape region in 0 < x < 4 and0 < z < 20, which is located in the upper region of theseﬁgures. It means that this two-dimensional magnetic loop has
the negative By region. On the other hand, for x > 5, somethree-dimensional magnetic loops are observed as negative andpositive By contour islands, which are non-uniform in the z-

direction. Closed magnetic loops which consist of closed ﬁeld
lines must have a pair of the negative and positive By contour
islands. While open magnetic loops only consist of a positive
contour island. Since these closed and open ﬁeld lines arecomplicatedly mixed, it is not so easy to distinguish them inthese ﬁgures. From Figures 4(a) to (c), most magnetic loops
tends to move in the positive x-direction, i.e., the downwardof these ﬁgures, with reconnection outﬂows. However, since
the multiple reconnection process simultaneously occurs on
more than two magnetic neutral lines, negative x-directional(upward) reconnection outﬂows may be caused between theseneutral lines. At the same time, some magnetic loops appear or
disappear, i.e., collapse, and, sometimes, are broaden in thex and z directions. In this simulation, initial magnetic ﬁeld
lines to be reconnected are limited only in 1 < y < Y1 ,and there is no magnetic ﬁeld in y > Y1 + 0.6 at time
T = 0. Accordingly, as time proceeds, the ﬁeld lines to bereconnected gradually reduce, and eventually, disappear. Then,the reconnection process terminates in T > 160.
In addition, in Figures 4(b) and (c), some snake-like red lines
are drawn. Also, in Figure 4(a), some of them are starting tobe drawn. These lines are plotted by tracing test particles on
the time-dependent plasma ﬂow ﬁeld (ux , uz ) on the xz-plane
for y = 0. Because of the symmetry boundary condition of thexz-plane at y = 0, since uy = 0 is always maintained on thexz-plane, the test particles started from the xz-plane stay on
the xz-plane. The start point for tracing these test particles isselected in strong positive By intensity regions, i.e., the orangecolored region. Since most magnetic loops tend to move withreconnected plasma downﬂows, these red lines of test particles
gradually extend in the positive x-direction, as observed from
Figures 4(a) to (c).
Figure 4(a) for T = 130 shows just after three test particles
started to be traced. These tracings of particles were started
at time T = 127, respectively, from (x, z) = (13.0, 13.0),(13.0, 13.5), and (13.0, 14.0), in which By intensity is positiveand relatively higher than that of the ambient regions. In
Figure 4(b), for T = 138, those test particle trajectories are第 4 章 論文集

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Figure 5. TRACE EUV image data (2002 April 21-XﬂareAR9906-01:49 and−01:50).
(A color version of this ﬁgure is available in the online journal.)

continued to be traced. After this, some additional test particles
start from some new points where By intensity is positive andrelatively higher. Finally, in Figure 4(c) for T = 161, every
trajectory exhibits snake-like curves and is elongated in thepositive x-direction with the plasma downﬂows.
4. INTERMITTENT DOWNFLOWS IN TRACE EUV DATA
Figures 5(a) and (b) show the EUV image data of solar
ﬂare which was obtained in TRACE satellite on 2002 April
21, respectively, at 01:48UT and 02:35UT. These observation
data have been reported by Innes et al. (2003). Also, Asaiet al. (2004a) reported similar observation data. As shown in
Figure 5(b), a large scale arcade structure appears, as the solar
ﬂare develops. According to their reports, the fast magnetic
reconnection is considered to be occurring at the upper sideof this image frame. In the upper region of Figure 5(a), somenarrow snake-like and dark downﬂows are observed, which aremarked with label “A.” Figures 6(a) and (b) show the details of
the snake-like downﬂows, which are, respectively, at 01:49UT
and 01:50UT. The lower side of these ﬁgures corresponds tosunward.
Some parallel downﬂows are observed in Figures 6(a) and (b),which are marked with labels “A” and “B”. Label A corresponds
to that of Figure 5(a). These two downﬂows seem to collide andFigure 6. TRACE EUV image data (2002 April 21-XﬂareAR9906-01:49 and−01:50).
(A color version of this ﬁgure is available in the online journal.)

tend to avoid each other, resulting in the snake-like curves.
However, these downﬂows do not seem to intersect. It suggeststhat the current sheet associated with the reconnection process isonly a ﬂat sheet, and those snake-like downﬂows coexist on thecurrent sheet. In addition, in contrast to the large-scale arcade
structure, these downﬂows are strongly localized on the current
sheet and intermittently generated (Innes et al. 2003; Asaiet al. 2004a, 2004b). It suggests that the three-dimensional andintermittent fast magnetic reconnection process is repeatedly
caused in a current sheet. These features of the snake-like andintermittent downﬂows are similar to the test particle trajectories
shown in Figure 4(c).
5. DISCUSSIONS
Figure 7 shows the By contour map on the xz-plane andplasma ﬂow vectors in a narrow region of 10 < x < 33 and10 < y < 15 at time T = 138. This narrow region corresponds
to a small square box in Figure 4(b). At this time, the snakelike red lines drawn in Figure 4(b) pass this narrow region in
Figure 7. Let us focus on this narrow region. Some reconnectedmagnetic ﬁeld lines drawn around z = 11 and 13 show that

Shimizu et al., “MHD Study of 3D Intermittent Fast Magnetic Reconnection” (2009)
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401

SHIMIZU ET AL.

Vol. 707YO-neutral line 1Oclosed field line 1
X-neutral line 1
open
field line 1

X
Z

O-neutral line 2
closed field line 2

open field lines 2

X-neutral line 2

Figure 7. By (> 0 and < 0) contour, some magnetic ﬁeld lines, and ﬂow vector
ﬁeld around snake-like reconnection jets. T = 138 corresponds to the third stageof this simulation, which is the intermittent and multiple three-dimensional
fast reconnection process. These traced magnetic ﬁeld lines show multiple
formations of magnetic loops. Observing the ﬂow vectors, these magnetic
loops have different propagation speeds. The x-directional ﬂow vectors aroundz = 12.5 and in x > 17 are slightly deﬂected to the negative z-direction. The
origin of the deﬂection seems that the fast speed ﬂows collide with the preceding
slow speed ﬂows. As a result, the outﬂow vector ﬁeld shows snake-like changes.

two closed magnetic loops are, respectively, formed around(x, z) = (17, 11) and (24, 13). The latter loop is much bigger
and thicker than the former loop.
More exactly, the inside of these magnetic loops has closed
magnetic ﬁeld lines, but the outside has open ﬁeld lines extended
to the positive x-direction. The open magnetic ﬁeld lines tend topush the magnetic loop in the positive x-direction, but the closed
ﬁeld lines will try to stagnate there. As a result, the plasma ﬂow
vectors for 12.5 < z < 15 disappear (decelerate) around x = 24
and again increase (accelerate) in x > 30. A similar ﬂow vector
change can also be observed in another closed magnetic loop
located around 10 < x < 24 and z = 11, where ux temporally
disappears and the plasma ﬂows stagnate.
Between these two magnetic loops, as x increases from 17
to 24, the z-directional component uz changes from slightly
positive values (uz > 0) to slightly negative values (uz < 0).
Accordingly, the x-directional ﬂow vectors snake-likely change
there. Since every initial magnetic ﬁeld line was exactly directedto the x-direction, these non-zero uz values are resulted fromthe interaction between those two three-dimensional magnetic
loops.
Figure 8 schematically shows the relations between the ﬂow
vector ﬁeld and ﬁeld line conﬁgurations, to explain what is
occurring in Figure 7. Along more than two x-type magnetic
neutral lines, the three-dimensional fast magnetic reconnection
simultaneously occurs, and a magnetic loop is generated fromeach neutral line. Then, two O-type magnetic neutral lines penetrate each magnetic loop. Accordingly, those regions between
O-type and x-type neutral lines are ﬁlled with closed ﬁeld lines.
Considering the magnetic tension, these closed ﬁeld lines will
try to stay there. While if the magnetic loop has open magnetic

Figure 8. Formation mechanism of snake-like reconnection jets in the third stageof this simulation. Because of the symmetry boundary condition on the z-axis,
the reconnected “open ﬁeld lines 1 and 2” in the x > 0 region mainly become
positive x-directional outﬂows. However, “X-neutral lines 1 and 2” separated
from the z-axis can create “closed ﬁeld lines 1 and 2” and, also, negative xdirectional outﬂows. Then, these positive and negative outﬂows collide around“O-neutral lines 1 and 2”. Since these magnetic loops are strongly localized in
the z-direction, the x-directional outﬂows show snake-like deﬂection in positive
and negative z-directions to avoid the closed magnetic loops.

ﬁeld lines, the magnetic tension will try to move the magnetic
loop in the positive or the negative x-direction. Since the magnetic loop is localized in the z-direction, such a magnetic tension
may avoid the plasma bulk sustained in the magnetic loop, resulting in the generation of non-zero uz components, leading to
the generation of snake-like plasma ﬂows.
This scenario to cause the snake-like ﬂows strongly supportsthat the intermittent downﬂows observed in solar ﬂares is causedby the fast magnetic reconnection. The most important thing isthat any distinct three-dimensional magnetic ﬁeld structure is notneeded in the initial current sheet to generate the intermittent andsnake-like downﬂows. Even in approximately one-dimensional
current sheet, small three-dimensional ﬂuctuation can generatesuch fully three-dimensional and intermittent downﬂows. Sincesuch three-dimensional and intermittent fast magnetic reconnections may be averaged in time and space, a two-dimensionallarge-scale magnetic arcade structure can be formed in downstream of the reconnection site, as observed in Figure 5(b).
6. SUMMARY
In this paper, it is demonstrated that two-dimensional fastmagnetic reconnection caused by a current-driven anomalous
resistivity is unstable in three dimensions and, even in a onedimensional current sheet, three-dimensional resistive perturbation embedded in Equation (3) can change the two-dimensional
fast magnetic reconnection to be fully three dimensional. In
addition, it was shown that the resulting three-dimensional fastmagnetic reconnection intermittently ejects magnetic loops, andthe dynamics of the magnetic loops can show snake-like trajectories of the plasma outﬂows. Such intermittent and snake-like
outﬂows are very similar to the EUV observation image data insolar ﬂares, which is reported by Innes et al. (2003) and Asaiet al. (2004a). The basic mechanism of the reconnection process
is also similar to the theoretical model of multiple fast magnetic
reconnection proposed by Aschwanden (2002, Figure 18), in第 4 章 論文集

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MHD STUDY OF 3D INTERMITTENT FAST MAGNETIC RECONNECTION

which many reconnection sites strongly localized in the sheet
current direction are widely distributed in the one-dimensional
current sheet.
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Shimizu et al., “Three-Dimensional Fast Magnetic Reconnection” (2009)403

PHYSICS OF PLASMAS 16, 052903 共2009兲

Magnetohydrodynamic study of three-dimensional instabilityof the spontaneous fast magnetic reconnection
T. Shimizu,1 K. Kondoh,1 K. Shibata,2 and M. Ugai1
1

Department of Computer Science, Ehime University, Matsuyama City 790-8577, Japan
Kwasan Observatory, Graduate School of Science, Kyoto University, Kyoto City 607-8471, Japan

2

共Received 25 April 2008; accepted 29 January 2009; published online 28 May 2009兲
Three-dimensional instability of the spontaneous fast magnetic reconnection is studied withmagnetohydrodynamic 共MHD兲 simulation, where the two-dimensional model of the spontaneous
fast magnetic reconnection is destabilized in three dimension. Generally, in two-dimensional
magnetic reconnection models, every plasma condition is assumed to be uniform in the sheet current
direction. In such two-dimensional MHD simulations, the current sheet destabilized by the initialresistive disturbance can be developed to fast magnetic reconnection by a current driven anomalous
resistivity. In this paper, the initial resistive disturbance includes a small amount of ﬂuctuations inthe sheet current direction, i.e., along the magnetic neutral line. The other conditions are the sameas that of previous two-dimensional MHD studies for fast magnetic reconnection. Accordingly, we
may expect that approximately two-dimensional fast magnetic reconnection occurs in the MHD
simulation. In fact, the fast magnetic reconnection activated on the ﬁrst stage of the simulation is
two dimensional. However, on the subsequent stages, it spontaneously becomes three dimensional
and is strongly localized in the sheet current direction. The resulting three-dimensional fast magnetic
reconnection intermittently ejects three-dimensional magnetic loops. Such intermittent ejections of
the three-dimensional loops are similar to the intermittent downﬂows observed in the solar ﬂares.
The ejection of the three-dimensional loops seems to be random but, numerically and theoretically,
it is shown that the aspect ratio of the ejected loops is limited under a criterion. © 2009 American
Institute of Physics. 关DOI: 10.1063/1.3095562兴
I. INTRODUCTION

Fast magnetic reconnection provides a physical mechanism, by which magnetic energy is explosively converted
into plasma kinetic and thermal energies. This mechanism
has, hence, been considered to play a crucial role in solar
ﬂares and geomagnetic substorms. Petschek1 predicted that,when the magnetic reconnection process includes slow
shocks extending from the magnetic diffusion region, thereconnection process becomes signiﬁcantly active, giving
rise to high-speed plasma jets and the ejection of large-scale
magnetic loops.
Ugai et al.2–5 numerically revealed that the twodimensional fast magnetic reconnection predicted by
Petschek can be established by a kind of the current driven
anomalous resistivity embedded in tearing instability, wherea pair of slow shocks, high speed plasma jets, and large-scale
plasma loops, i.e., plasmoids, have been observed. In themagnetic reconnection process, the resistivity is a key
mechanism to reconnect magnetic ﬁeld lines. In general, theanomalous resistivity means that the resistivity is nonuniform in space. Ugai assumed the current driven anomalous
resistivity, where the resistivity is nonlinearly enhanced
when the current sheet becomes thin. According to the numerical magnetohydrodynamic 共MHD兲 study, when the current driven anomalous resistivity is employed, fast magnetic
reconnection can be spontaneously established without any
external driven mechanism by which the current sheet iscompressed. In other words, when the uniform resistivity is
employed instead of the anomalous resistivity, fast magnetic
1070-664X/2009/16共5兲/052903/12/$25.00

reconnection is not established. However, even with the uniform resistivity, if an external driven mechanism is applied
for the current sheet, fast magnetic reconnection may beestablished.6 The relationship of the anomalous resistivity
and fast magnetic reconnection has become a controversial
topic for the MHD study.7,8 In this paper, we employ thecurrent driven anomalous resistivity model examined by
Ugai et al.2–5 and Shibata and Tanuma.9
With the recent drastic development of the computer performance, fast magnetic reconnection has been widely studied in many numerical simulations. Historically, the numerical MHD studies of fast magnetic reconnection started from
simple two-dimensional models, where the structure of theinitial current sheet is one dimensional. However, in recent,there are many three-dimensional MHD studies of fast magnetic reconnection. In recent numerical MHD studies related
to solar ﬂares,10–14 the initial current sheet or the other initialconditions are fully three dimensional, in order to apply forthe complicated magnetic ﬁeld conﬁgurations observed insolar ﬂares. Hence, the resulting fast magnetic reconnection
is directly three dimensional. At this point, those threedimensional studies may be not a direct extension of theprevious two-dimensional studies.
On the other hand, according to the recent space satellite
observations for solar ﬂares, the plasma jets and magnetic
loops associated with fast magnetic reconnection are often
reported.15–21 In some of those observation reports, the threedimensional structure of fast magnetic reconnection has been
discussed. Innes et al.20 and Asai et al.21 reported that the

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appearance of a large-scale arcade structure and intermittent
plasma downﬂows are simultaneously observed in solar
ﬂares and may be caused by fast magnetic reconnection in
the high altitude corona. According to their reports, the observed arcade structure is relatively large but the plasma
downﬂows observed on the top of the arcade structure arestrongly localized and intermittent. The appearance of thelarge-scale arcade structure suggests that the current sheet on
the top of the arcade structure widely spreads in the sheet
current direction and, from a point of a macroscopic view intime and space, the fast magnetic reconnection occurring
there may be two dimensional. While, the localized plasma
downﬂows suggest that the fast magnetic reconnection isstrongly localized in the sheet current direction, i.e., fully
three dimensional. In fact, Innes et al. and Asai et al. suggested that three-dimensional fast magnetic reconnection simultaneously occurs at some points along the magnetic neutral line. However, theoretically and numerically, it is still
unclear how the three-dimensional fast magnetic reconnection occurs there. For instance, the arcade structure studied insome MHD studies11,12 may be similar to the observed largescale arcade structure. However, the localized plasma downﬂows are not observed in those numerical studies.
As for the MHD study aimed to the both of the observed
large-scale arcade structure and localized plasma downﬂows,
there is “patchy” reconnection model based on a fully threedimensional and nonsteady fast magnetic reconnection on
ﬂux tubes, which was reported by Linton and Longcope.22 In
that model, the current sheet is destabilized by a fully threedimensional resistive disturbance strongly localized in the
sheet current direction. Hence, the resulting fast magnetic
reconnection is strongly localized in the sheet current direction and directly becomes three dimensional. Since they focused on the dynamics of postreconnection jets, it is still
unclear why the fast magnetic reconnection, itself, is three
dimensional and intermittent, as observed in solar ﬂares.
Only focusing on the intermittent plasma ﬂows associated with fast magnetic reconnection, the secondary or multiple fast magnetic reconnection models proposed by Ugai2,4
Shibata and Tanuma,9 Kitabata6 and Schumacher andKliem23 may be applicable. However, since those studies are
two dimensional, the models are not directly applied forthe observations of the three-dimensional and intermittent
plasma downﬂows strongly localized in the current sheet
direction.
In this paper, the two-dimensional fast magnetic reconnection model proposed by Ugai is numerically developed toa three-dimensional model. Note that the three-dimensionalmodel shown in this paper is different from that of the threedimensional model studied by Ugai et al.5,24 Because, in
those studies reported by Ugai, the initial resistive disturbance to initiate the reconnection process is strongly localized in the sheet current direction, directly leading to threedimensional fast magnetic reconnection. At this point, Ugai’s
previous model may be close to that of Linton andLongcope.22 While, the initial resistive disturbance employedin this paper is approximately two dimensional and includes
a “weak nonuniformity 共ﬂuctuation兲” in the sheet current direction. Accordingly, we can examine whether the two-

dimensional fast reconnection previously studied2–4,25,26 isunstable or not, for such a three-dimensional resistive perturbations. According to our numerical study shown in this paper, two-dimensional fast magnetic reconnection is unstable
and can be ﬁnally developed to fully three-dimensional andintermittent fast magnetic reconnection, if there is a weak
three-dimensional ﬂuctuation in the initial current sheet. Itmeans that, even in the approximately one-dimensional current sheet, three-dimensional magnetic loops 共plasmoids兲 canbe spontaneously and intermittently ejected from the reconnection regions, which are similar to the intermittent plasma
downﬂows observed in solar ﬂares.
II. MHD SIMULATIONS
A. Procedures of simulation

In the spontaneous fast reconnection model, the reconnection process is initiated by a small resistive disturbanceinduced in the current sheet. After the initial resistive disturbance is removed, the fast magnetic reconnection is spontaneously developed by a current driven anomalous resistivity
enhanced by a nonlinear tearing instability. Hence, any external driven mechanism is not required to keep the reconnection process.
The compressible MHD equations areD␳/Dt = − ␳ ⵜ · u,␳Du/Dt = − ⵜP + J ⫻ B,

⳵ B/⳵ t − ⵜ ⫻ 共u ⫻ B兲 = − ⵜ ⫻ 共␩J兲,␳De/Dt = − P ⵜ · u + ␩J2 ,J = ⵜ ⫻ B/␮0,

共1兲ⵜ · B = 0,

where D / Dt ⬅ ⳵ / ⳵t + u · ⵜ; the gas law, P = 共␥ − 1兲␳e, is assumed 关e is the internal energy per unit mass, and ␥ is the
speciﬁc heat ratio with ␥ = 5 / 3 assumed here, as is Ohm’slaw, E + u ⫻ B = ␩J 共␩ is a resistivity兲. The basic Eq. 共1兲 istransformed to a conservation-law form, and two-step Lax–
Wendroff scheme is used for the numerical computation.
The initial current sheet system has one-dimensional
structure, i.e., the magnetic ﬁeld B = 关Bx共y兲 , 0 , 0兴 is assumed
as Bx共y兲 = sin共␲y / 2兲 for 0 ⬍ y ⬍ 1; Bx = 1 for 1 ⬍ y ⬍ Y 1; Bx= cos关共y − Y 1兲␲ / 1.2兴 for 共Y 1 ⬍ y ⬍ Y 1 + 0.6兲; Bx = 0 for y ⬎ Y 1+ 0.6; also, Bx共y兲 = −Bx共−y兲 for y ⬍ 0. Here, Y 1 = 5.0. The
plasma pressure P共y兲 initially satisﬁes the pressure-balance
condition,
P + B2x = 1 + ␤0 ,共2兲

where ␤0 is the ratio of the plasma pressure to the magnetic
pressure in the magnetic ﬁeld region 1 ⬍ y ⬍ Y 1, where ␤0= 0.15 is taken. Fluid velocity u = 共0 , 0 , 0兲 and plasma density
␳ = 1 are initially assumed. The normalization of quantities isself-evident: Distances are normalized by d0, B by Bx0, and P
by B2x0 / 共2␮0兲; also, ␳ is normalized by ␳0, u by VAx0共=Bx0 /冑␮0␳0兲, time t by d0 / VAx0, current density J by Bx0 / 共␮0d0兲,and so forth.
For the boundary conditions of the computational region,
the symmetry boundary condition is assumed on the yz-planeat x = 0, xz-plane at y = 0, and xy-planes at z = 0 and Lz, so that

Shimizu et al., “Three-Dimensional Fast Magnetic Reconnection” (2009)
052903-3405
Phys. Plasmas 16, 052903 共2009兲MHD study for three-dimensional instability…

the computational region is restricted to the ﬁrst quadrant
only and taken to be a square box, 0 ⱕ x ⱕ Lx, 0 ⱕ y ⱕ Ly and0 ⱕ z ⱕ Lz. The other boundaries are assumed to be the open
共free兲 boundary where all the quantities are determined by
the states of the inner region, so that the ﬁrst derivatives of
the quantities in the direction normal to the boundaries vanish, excepting the normal component of B, which is determined by the solenoidal condition. For the numerical conditions in this paper, the computational region Lx = 30 and Ly= 9 are taken with the mesh points Nx = 880 and Ny = 1000.
Mainly, the case of Lz = 20 and Nz = 100 is studied but also the
Lz dependence is studied, keeping Nz = 100.
The MHD simulation procedures are basically the sameas that of Ugai et al.,5,24 excepting the three-dimensional
setup of the initial resistive disturbance. The initial resistive
disturbance shown below is put to initiate a tearing instability in the current sheet during 0 ⬍ T ⬍ 4.

␩0 = 0.04 exp关− ␣1共x2 + y 2兲兴 ⴱ 关1 + ␣2 cos共␲z/Lz兲兴,

共3兲

where ␣1 = ␣2 = 0.01. This disturbance works as a trigger forfast magnetic reconnection. For this initial resistive disturbance, the magnetic Reynolds number is 25 at the origin.
Because of ␣2 = 0.01, this resistive disturbance is slightly
ﬂuctuated only by 1% in the z-direction which is the
sheet current direction. The case of ␣2 = 0.0 corresponds tothe two-dimensional fast magnetic reconnection.2–4 The
z-directional ﬂuctuation works as a perturbation to threedimensionally destabilize the two-dimensional fast magnetic
reconnection. At this point, note that the initial resistive disturbance employed by Ugai et al.5,24 is largely different because ␣2 was much larger in those studies. In those studies,
the resulting fast magnetic reconnection directly became
three dimensional. While, the resulting fast magnetic reconnection shown in this paper is expected to be approximately
two dimensional, at least, on the initial stage. In fact, themagnetic loop ﬁrst ejected from the reconnection region is
two dimensional, as shown later in Fig. 2.
After T = 4, ␩0 is removed and, instead, the current
driven anomalous resistivity shown below is set, which spontaneously drives fast magnetic reconnection.

␩1 = 0.002共Vd − Vc兲,
␩1 = 0.0,

for

for

Vd ⬍ Vc ,Vd ⬎ Vc ,共4兲
共5兲

where Vd = 兩J兩 / ␳ is the ion-electron drift velocity and Vc= 4.0 is set. The current density J is obtained from rot B andis unity at the center of the initial current sheet, i.e., y = 0.
B. Simulation results
1. The ﬁrst stage „two-dimensional fast reconnection…

Figure 1 shows initial magnetic ﬁeld lines only for 0⬍ y ⬍ 2, which are directed in the x-direction. The initial current sheet is exactly one dimensional. For the space and timescales in this simulation, the Alfven speed VA in the uniform
magnetic ﬁeld region is set to be unity and the Alfven waves
take T = 30 to pass the distance of Lx = 30, which is thex-directional simulation box size. There is no plasma ﬂow at
T = 0. Due to the initial resistive disturbance given by Eq. 共3兲,FIG. 1. Initial conﬁguration. The xz-plane at y = 0 is the magnetic neutral
sheet. Some magnetic ﬁeld lines are traced on three xy-planes 共z = 0, 0.5Lz,
and Lz兲 only between 0 ⬍ y ⬍ 2. Here, Lz = 20. Every magnetic ﬁeld line isdirected to the x-direction. The symmetry boundary condition is assumed on
xz-plane at y = 0. Also, the xy-planes at z = 0 and Lz and yz-plane at x = 0 are
the symmetry boundary. The other boundaries are assumed to be the free
共open兲 boundary.

a weak magnetic reconnection starts on the z-axis. After the
initial resistive disturbance is removed at T = 4, the current
driven anomalous resistivity given by Eqs. 共4兲 and 共5兲 isassumed. The magnetic reconnection process is temporally
stopped by Eq. 共5兲 because the current sheet is thick at T
= 4. However, as the plasma frozen in the reconnected ﬁeld
lines moves away from the z-axis, the current sheet aroundz-axis becomes thinner and Eq. 共4兲 starts to work, instead of
Eq. 共5兲. Then, fast magnetic reconnection spontaneously
starts.
Figure 2共a兲 shows the magnetic ﬁeld lines on threexy-planes; z = 0 , 0.5Lz and Lz, and By contour map andplasma ﬂow vectors on xz-plane for y = 0, at T = 48. Considering the symmetry boundary condition on xz-plane at y = 0
and yz-plane at x = 0, the magnetic neutral line is formed
along the z-axis. Every magnetic ﬁeld line is almost the samefor these three xy-planes. Also, the By contour map andplasma ﬂow vectors are almost uniform in the z-direction.
Hence, at this time, the fast magnetic reconnection is twodimensional and a two-dimensional large-scale magnetic
loop is formed between 15⬍ x ⬍ 30. Figure 2共b兲 shows proﬁles of P , By and ux along the x-axis. The solid line ux rapidly increases from zero at x = 0 and reaches the Alfven speed
around x = 4. Then, it rapidly decreases at x = 12.5 indicatedby label “F” which is a fast shock. The dotted line shows thepressure proﬁle. Beyond the fast shock, a high pressure region in 15⬍ x ⬍ 26 corresponds to the magnetic loop observed in Fig. 2共a兲. The details of these features have been
already reported as two-dimensional studies of fast magnetic
reconnection in Shimizu and Ugai.25,26
Figure 3 shows the time variations of electric ﬁeld Ez atthree points; 共x , y , z兲 = 共0 , 0 , 0兲, 共0,0,4兲, and 共0,0,8兲, which are
regarded as the local magnetic reconnection rate on z-axis.
Until about T = 45, these three time variations of Ez almost

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Phys. Plasmas 16, 052903 共2009兲

FIG. 2. 共a兲 Magnetic ﬁeld lines, contour map of By ﬁeld intensity, and ﬂow
velocity at T = 48. Magnetic ﬁeld lines traced from 0 ⬍ y ⬍ 2 at x = 0 or 0
⬍ x ⬍ 30 at y = 0 are shown on three xy-planes. By contour map only for
positive intensity and ﬂow velocity are shown on the xz-plane. 共b兲
X-directional proﬁles of pressure p, By intensity and plasma ﬂow speed ux
along the x-axis. A fast shock 共label “F”兲 is formed at x = 12.5 which is in
front of the magnetic loop. The magnetic loop is observed as a high pressure
region in 15⬍ x ⬍ 26.

coincide. It means that the reconnection process is homogeneous with respect to the z-direction and, hence, two dimensional. For more details, until T = 4 indicated by “A,” theelectric ﬁeld injected by the initial resistive disturbance Eq.
共3兲 can be seen. At T = 4, Ez is reset by removing Eq. 共3兲 and,

FIG. 3. Time variations of electric ﬁeld Ez at three points; Z1共x , y , z兲
= 共0 , 0 , 0兲, Z2共0 , 0 , 4兲, and Z3共0 , 0 , 8兲 which are located on the magnetic
neutral line, i.e., z-axis. After T = 45, the z-dependence of these Ez proﬁlesappears, where three-dimensional fast reconnection starts.

FIG. 4. 共a兲 The ﬁgure format is the same as Fig. 2共a兲 but at T = 60. The
two-dimensional large magnetic loop shown in Fig. 2 has been a little
shifted in the positive x-direction. In addition, the secondary magnetic loop
indicated by A appears on the x-axis. This magnetic loop is localized in thez-direction and, hence, three-dimensional. 共b1兲 This ﬁgure format is thesame as Fig. 2共b兲 but at T = 60. The fast shock observed in Fig. 2共b兲 isindicated by F. Label B and C are the magnetic null points. In fact, the signof By intensity is reversed at those points. The high pressure region between
B and C corresponds to the magnetic loop indicated by A of 共a兲. 共b2兲 Thisﬁgure format is the same as 共b1兲 but at z = 0.5Lz. These proﬁles for x ⬎ 10 issimilar to those of 共b1兲 but the proﬁles for 5 ⬎ x is largely different.

then, Ez gradually increases, as fast magnetic reconnection
gradually starts with the enhancement of the current driven
anomalous resistivity, i.e., Equation 共4兲. At T = 20 indicatedby “B,” Ez reaches a loose peak which means the saturationof the ﬁrst stage of fast magnetic reconnection. At this time,a two-dimensional large-scale magnetic loop is generated, as
shown in Fig. 2. At about T = 55 indicated by “C,” the second
peak of Ez is observed for the solid line, i.e., z = 0. For theother lines, the peaks are looser than the peak of the solidline. It means that three-dimensional fast magnetic reconnection mainly occurs around z = 0, as shown in Fig. 4.

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2. The second stage „three-dimensional fastreconnection…

Figure 4共a兲 shows the numerical result similar to Fig.
2共a兲 but for T = 60. The two-dimensional large-scale magnetic loop in 20⬍ x ⬍ 30 corresponds to the one in 15⬍ x⬍ 25 of Fig. 2. In addition, another small magnetic loop is
observed in the vicinity of the origin, which is indicated bylabel A in Fig. 4共a兲. Since this magnetic loop is not observedin the magnetic ﬁeld lines of z = 0.5Lz and Lz, the loop structure is evidently three dimensional. In Fig. 4共a兲, in order toavoid confusions, the contour lines for By ⬍ 0 are not plotted,
which do not contribute to the positive x-directional magnetic tension to drive the fast magnetic reconnection jets inx ⬎ 0.
Figure 4共b1兲 shows the proﬁles of By, P, and ux. Twolabels B and C, which respectively indicate x = 2 and 4, aremagnetic null points, as shown in By proﬁle, i.e., dash line.
Also, there is another magnetic null point between B and C,
which almost coincides a peak of pressure proﬁle, i.e., thedotted line. Accordingly, the three-dimensional magnetic
loop seen in label A of Fig. 4共a兲 is located between B and C.
Since the interior of the magnetic loop is ﬁlled with highpressure plasma, the magnetic loop is a plasmoid. Such amagnetic loop cannot be observed in Fig. 4共b2兲 plotted forz = 0.5Lz and y = 0.
Figure 5共a兲 has the same format as Fig. 4共a兲 but for T= 72. The magnetic loop indicated by label “A” corresponds
to the one observed in Fig. 4, which has slightly moved in
the positive x-direction with the plasma ﬂow. In addition,
along the z-axis, a new two-dimensional magnetic loop starts
to grow as the third stage of fast magnetic reconnection. As
mentioned in Fig. 4共a兲, the contour lines for By ⬍ 0 are also
not drawn in Fig. 5共a兲. Accordingly, the contour lines for thetwo-dimensional magnetic loop along z-axis cannot be observed in Fig. 5共a兲.
In Fig. 5共b兲 which shows some proﬁles along x-axis, themagnetic loop between two magnetic null points C and “D”
is the same one between B and C in Fig. 4共b1兲. Anotherthree-dimensional magnetic loop observed in B and C, i.e.,
6 ⬍ x ⬍ 8, is generated by the forth stage fast magnetic reconnection, which is three dimensional but almost invisible inFig. 5共a兲.
Next, in order to see the ﬂow pattern near an active
magnetic diffusion region, Fig. 6共a兲 shows the local ﬂow
patterns on xz-plane for y = 0 and yz-plane for x = 0 at T
= 54. In addition, the By contour map is drawn on xz-plane
for y = 0. Also, some magnetic ﬁeld lines are shown on
xy-plane for z = 0. At this time, fast magnetic reconnection is
active at the origin because a magnetic loop is being formed
there, as shown in the By contour map and magnetic ﬁeld
lines. The bow-shape island observed on the By contour map,i.e., 0 ⬍ x ⬍ 1.5, presents the reconnected magnetic ﬁeld intensity, i.e., positive By intensity region of the magnetic loop.
Considering the symmetry boundary condition assumed on
xy-plane at z = 0, this magnetic loop is located between about−0.25Lz ⬍ z ⬍ 0.25Lz. Seeing the ﬂow pattern on yz-plane at
x = 0, the plasma inﬂow is concentrated to the origin. This
converging ﬂow on yz-plane is driven by fast-mode rarefaction waves generated in the magnetic diffusion region, i.e.,

FIG. 5. 共a兲 The ﬁgure format is the same as Fig. 2共a兲 but at T = 72. The
magnetic loop indicated by A in Fig. 4共a兲 can be also observed in A in thisﬁgure. Around z-axis, the third magnetic loop is formed and the structure isalmost two dimensional. 共b兲 The magnetic loop observed between C and D
corresponds to the secondary magnetic loop indicated by A of 共a兲, which has
a three-dimensional structure. In addition, the third magnetic loop is observed between D and x = 0, which has a two-dimensional structure. Another
共the forth兲 small magnetic loop is observed between B and C.

due to the pressure gradient effect. At this point, it is important that, as the spontaneous fast magnetic reconnection isdeveloped, the pressure and plasma density rapidly decreasein the magnetic diffusion region3 and fast-mode rarefaction
waves drive the reconnection inﬂow.
Figure 6共b兲 shows a cartoon of the ﬂow pattern on
yz-plane at x = 0. The plasma inﬂow concentrates to origin“o” where is the most active magnetic diffusion region. Considering the symmetry boundary condition assumed at z = Lz,a saddle point may appear at point “a” where the reconnection process is not so active and the plasma ﬂow is diverted
along the z-axis. Since the plasma ﬂow along z-axis, i.e., theneutral line, is unmagnetized, the converging plasma ﬂow
near origin o will prevent the reconnection process and thecurrent sheet may become thicker. While, the current sheetnear point a tends to become thinner because the diverted
unmagnetized ﬂow escapes from there. Accordingly, on thenext stage, the fast magnetic reconnection process may beenhanced at point a and, ﬁnally, the active reconnection region may switch between points a and o with the inversion of

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(a)

(b)

z=0

z=Lz

z-axisy-axismag. neutral lineaopoint "a"point "o"weak mag rec. region ==>>
strong mag. rec. region ==>>( slow drop of pressure <<==========>> strong drop of pressure )diverging flow ==>>
converging unmagnetized
thining of current sheet==>>inflow ==>>
enhancment of mag rec.
preventation of mag rec.

(b)

circulation(with random non-linear enhancement)

FIG. 6. The ﬁgure format is similar to Fig. 5共a兲 but at T = 54. In addition, an
active reconnection region 共0 ⬍ x ⬍ 10 and 0 ⬍ z ⬍ 0.5Lz兲 in the vicinity ofthe origin is only shown. The By contour map on the xz-plane 共only By ⬎ 0兲
and the magnetic ﬁeld lines on the xy-plane show the formation of a threedimensional magnetic loop around the origin. The ﬂow pattern on yz-plane
indicates the plasma inﬂow is concentrating toward the active magnetic
diffusion region, i.e., the origin. 共b兲 The ﬂow pattern on the yz-plane at x
= 0 is shown as a cartoon. Point o is the origin in 共a兲 and point a corresponds
to z = Lz on the z-axis. At these two points, the symmetry boundary conditionis assumed.

the unmagnetized ﬂow along the neutral line. As a result, thethree-dimensional reconnection process may be intermittently repeated, exchanging the active and inactive regions.
This oscillative switching mechanism is enhanced, as shownin Fig. 7.
Figures 7共a兲 and 7共b兲, respectively, show the uz and P
proﬁles along the z-axis and the time variations. The uz proﬁle at T = 48 共dashed line兲 is almost constantly zero. However, the uz proﬁle at T = 54 共dotted line兲 takes slight negative

FIG. 7. 共a兲 Uz proﬁles along the z-axis which is the magnetic neutral line.
The time variation is shown at T = 48, 54, and 63. Since the dotted uz proﬁlefor T = 54 takes negative values between 0 ⬍ z ⬍ 10, the plasma is ﬂowing tothe origin along the z-axis. This plasma ﬂow can be also observed in Fig.
6共a兲. However, the plasma ﬂow is not observed at T = 48. 共b兲 The pressure
proﬁle along the z-axis. At T = 54, the pressure gradient is slightly positive
which causes the negative plasma ﬂow along the z-axis, due to the fastmode rare-faction waves.

values in 0 ⬍ z ⬍ 10, which means the unmagnetized plasma
ﬂows are converging toward the origin, as observed along
the z-axis in Fig. 6共a兲. Then, at T = 63 共solid line兲, the ﬂow
direction is partially inverted and is largely ﬂuctuated, giving
rise to the oscillative enhancements of the reconnection process. At this time, the fast reconnection process becomesfully three dimensional. The time variations of the pressure
proﬁle in Fig. 7共b兲 show that the unmagnetized plasma ﬂow
uz toward the origin is driven by the pressure gradient alongz-axis, because the pressure proﬁle 共dashed line兲 at T = 48 isalmost constant but at T = 54 becomes a loose positive gradient slope in 0 ⬍ z ⬍ 10. As aforementioned, the positive gradient slope is originated in that the pressure in the active
magnetic diffusion region spontaneously decreases as the reconnection process proceeds.3
In this section, it was shown that the initial z-directional
perturbation embedded in Eq. 共3兲 results in threedimensional fast magnetic reconnection which is strongly localized in the z-direction, i.e., the sheet current direction. As
shown in Fig. 2, the ﬁrst stage of fast magnetic reconnectionis almost two dimensional. It means that Eq. 共3兲 almost twodimensionally worked and the initial z-directional perturbation was very weak. However, on the second stage, the fastmagnetic reconnection was drastically developed in three dimension. It means that the two-dimensional fast magnetic
reconnection is unstable for perturbations along the magnetic
neutral line. In fact, even when the z-directional ﬂuctuation
intensity ␣2 in Eq. 共3兲 is reduced from 0.01, the threedimensional fast reconnection occurs.
3. The other Lz values

In Secs. II B 1 and II B 2, we examined the case of Lz= 20 which is the z-directional size of the numerical simulation box. Because of the symmetry boundary condition assumed on the xy-planes at z = 0 and Lz, Lz value limits the
maximum wavelength for every z-directional MHD instabil-Shimizu et al., “Three-Dimensional Fast Magnetic Reconnection” (2009)
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6.0

ΔZ

4.0

2.0

0.0

(a)

(b)
FIG. 8. 共a兲 By contour map 共only By ⬎ 0兲 and ﬂow velocity on the xz-plane
at T = 78 for Lz = 20. Some magnetic loops are observed as high By intensity
regions. 共b兲 Similar to 共a兲 but for Lz = 2.5.

ity. Here, let us study how the generation of threedimensional magnetic loops depends on the Lz value. Figure
8共a兲 is again the case of Lz = 20. The By contour map and ﬂow
pattern are shown on xz-plane for y = 0 at T = 78. As mentioned in Fig. 4共a兲, in order to avoid confusions, the contour
map only shows the positive By intensity region, by which
the fast magnetic reconnection jets are driven in the positive
x direction. In general, since a magnetic loop which consistsof closed magnetic ﬁeld lines has a positive and negative By
intensity regions, it will be convenient to separate the positive and negative By intensity regions. In Fig. 8共a兲, three
three-dimensional magnetic loops are observed as high Byintensity regions on the contour map. The largest one is located from 共x , z兲 = 共5 , 0兲 to 共10,4兲, which is the bow-shape
region indicated by A and corresponds to the one observed in
Figs. 4 and 5. Note that, since the x-axis of Fig. 8 is assumed
as the symmetry boundary, the bow-shape region is symmetrically extended below the x-axis. In addition, two roundshape regions are, respectively, observed at 共x , z兲 = 共5 , 5兲 and共10,0兲. More rigorously, another weak bow-shape region may
be observed at 共x , z兲 = 共11, 20兲. Figure 8共b兲 is the case of Lz= 2.5 at T = 64 with the similar format as Fig. 8共a兲. The simulation conditions and procedures are the same as the case of
Lz = 20, excepting Lz value. Only a bow-shape region is observed around 共x , z兲 = 共3.0, 2.5兲. Comparing with Figs. 8共a兲and 8共b兲, we notice that the Lz variation drastically changes
the shape and location of the high By intensity regions on thexz-plane.
We examined more cases, changing Lz value. As shownin the comparison with Figs. 8共a兲 and 8共b兲, as Lz is larger, theformation of magnetic loops tends to be more complex. The
x and z directional sizes of those high By intensity regions

1.0

Δ X 2.0

3.0

FIG. 9. The aspect ratio map of magnetic loops detected in the cases of
Lz = 20.0, 10.0, 5.0, and 2.5. For cases of Lz ⬍ 2.5, no three-dimensional
magnetic loop was detected. The oblique dash line is drawn for 0.54⌬X= ⌬Z.

detected in various Lz cases are summarized in Fig. 9. In thisﬁgure, cases of Lz = 2.5, 5.0, 10.0, and 20.0 were examined.
⌬X and ⌬Z are measured on the By contour maps, such as
Figs. 8共a兲 and 8共b兲. More exactly, we ﬁrst deﬁned the shapeof the high By intensity region on xz-plane at y = 0 with thecontour line of half value of the peak By intensity, when thepeak By intensity in the region takes the maximum in time.
Second, we deﬁned ⌬X as the difference between the largest
and smallest values of x where the half value contour line ofhigh By intensity region reaches at a speciﬁed value of z
measured at the peak By intensity point. Third, we deﬁned⌬Z as the difference between the largest and smallest valuesof z where the half value contour line of high By intensity
region can reach at any value of x. In Appendix A, it isexplained how ⌬X and ⌬Z are measured, following thesedeﬁnitions of ⌬X and ⌬Z. These deﬁnitions are based on thefollowing theoretical discussions.
Figure 9 shows that the aspect ratio ⌬Z / ⌬X for every
observed magnetic loop is larger than 0.54. This feature is
supported by the following theoretical discussions. As wellknown, the reconnection jet’s speed is estimated as the Alfven speed VA measured in the upstream magnetic ﬁeld region. As shown in Fig. 3, when a magnetic loop is ejected,
the reconnection rate, i.e., electric ﬁeld at the magnetic neutral point, increases and then decreases. Hence, there is a
ﬁnite duration time ⌬T to generate a magnetic loop. In general, longer duration time will generate a larger magnetic
loop and vice versa. If we knew ⌬T, the x-directional size
⌬X of a magnetic loop may be estimated to be VA⌬T. Exactly speaking, since most magnetic loops tend to be compressed in the downstream, ⌬X will be less than VA⌬T. However, it will be hard to exactly predict the duration time ⌬T.
At this point, there is a simple criterion9 to limit ⌬T but it
gives only the maximum duration time on the basis of theclassical tearing instability. In this paper, we do not try to
predict ⌬T.
Alternately, let us focus on the limitation of ⌬Z. As
shown in Fig. 3, when a magnetic loop is ejected, the curveof the magnetic reconnection rate always takes a peak. Let us

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deﬁne ⌬T1 as the time spent for the growth phase in which
the reconnection rate increases. Then, let us deﬁne ⌬T2 as thetime for the decay phase in which the reconnection rate decreases. Evidently, the total duration time ⌬T is equal to⌬T1 + ⌬T2.
First, let us consider how ⌬T2 is predicted. As shown in
Figs. 6 and 7, when fast magnetic reconnection has been
activated under the pressure gradient along a magnetic neutral line, the unmagnetized plasma converging ﬂows driven
by the fast rarefaction wave, i.e., pressure gradient, along themagnetic neutral line prevents the reconnection process in aplace 关e.g., point o in Fig. 6共b兲兴. Because, the current sheet
tends to become thicker, there. On the other hand, the unmagnetized plasma diverging ﬂows may enhance in the otherplace 关e.g., point a in Fig. 6共b兲兴 because the current sheet
becomes thinner. Then, the pressure gradient will disappear
when the fast rarefaction wave swept over between points aand o. At this time, the three-dimensional fast magnetic reconnection will stop or becomes two dimensional. Then, thepressure gradient may be reversed. Finally, the active andinactive reconnection regions may be exchanged each other.
It suggests that the passing time of the fast rarefaction wave
between active and inactive reconnection regions will limit
the duration time ⌬T2. In other words, when ⌬T2 passed, thethree-dimensional fast magnetic reconnection cannot be
maintained.
For simplicity, let us assume that the active and inactive
reconnection regions are periodically located in thez-direction and ␭z is the wavelength, i.e., which is deﬁned as
the distance between a active neutral point and the next active neutral point. In this case, the distance between points aand o in Fig. 6共b兲 is regarded as 0.5␭z. Considering that thefast rarefaction wave speed is equal to the local sound speed
Cs on the magnetic neutral line, ␭z / 共2Cs兲 limits the maximum of ⌬T2. This maximum limitation implies that threedimensional fast magnetic reconnection may stop before thefast rarefaction waves have completely canceled the pressure
gradient. Since By, pressure and so on have a component ofz-directional wavelength ␭z in the high By intensity region,
we may approximately assume By = By0关1 + cos共2␲z / ␭z兲兴 as
the z-directional proﬁle, where 2By0 is the peak By value inthe region. In this case, 0.5␭z may be considered to be thez-directional size ⌬Z of high By intensity region.
Second, let us consider how ⌬T1 is predicted. If thegrowth rate of the three-dimensional fast magnetic reconnection were extremely slow, the pressure gradient along theneutral line will be canceled by the fast rarefaction wave
before the growth phase ends. Hence, ⌬T1 should be lessthan ␭z / 共2Cs兲 = ⌬Z / Cs.
Remembering ⌬X ⬍ VA⌬T and ⌬T = ⌬T1 + ⌬T2, we ﬁnally obtain the aspect ratio Cs⌬T / 2VA⌬T ⬍ ⌬Z / ⌬X as a criterion to generate three-dimensional magnetic loops. Here,let us consider how magnetic loops are collapsed by the lackof pressure balance. It may change values of ⌬X and ⌬Z.
Since the total pressure in the growing magnetic loop is nonequilibrium, the magnetic loop may start to collapse even in
the growth phase. More exactly, since the inside of the growing magnetic loop is higher pressure than the ambient regions, the high By intensity region tends to spread, and ac-

Phys. Plasmas 16, 052903 共2009兲

cordingly, ⌬X and ⌬Z will become larger. Let us take theminimum of ⌬Z which is equal to Cs⌬T derived above. This
corresponds to when the magnetic loop do not collapse in thez-direction. On the other hand, let us take the maximum of⌬X which is equal to VA⌬T, because, even if magnetic loop
collapses at VA, ⌬X cannot exceed VA⌬T, i.e., the Alfvenic
distance limit.
In the MHD simulations shown in this paper, Cs / VA
= 1.08 共=冑1.15兲 has been set in the initial condition and is
not largely changed during the simulation proceeds. The criterion of 0.54⌬X ⬍ ⌬Z is well satisﬁed in Fig. 9. In this theoretical discussion, we assumed that the neutral line, i.e.,
X-line, is parallel to the z-axis. In fact, every detectable
X-line was parallel or approximately parallel 共Appendix B兲.
In addition, we assumed that X-line almost do not propagate
in the positive and negative x-directions. In fact, thex-directional speed of X-line is much less than VA. At thesetwo points, the theoretical discussion shown in this section isconsistent with the numerical results studied in this paper.
III. SUMMARY AND DISCUSSION

In this paper, it was studied how the two-dimensional
fast magnetic reconnection driven by the current driven
anomalous resistivity responds for the three-dimensional resistive disturbance. Since the initial current sheet has a exactly one-dimensional structure and every setup of the simulation is exactly two dimensional, excepting the initialresistive disturbance Eq. 共3兲, a small amount of threedimensional ﬂuctuation embedded in Eq. 共3兲 can only change
the two-dimensional fast magnetic reconnection to be three
dimensional. At this point, MHD numerical studies reported
by Ugai et al.5,24 are basically different because the initialresistive disturbance employed in those papers includes alarge nonuniformity along the magnetic neutral line, i.e., ␣2value in Eq. 共3兲 is almost unity. The resulting fast magnetic
reconnection directly becomes three dimensional. While, ␣2value studied in this paper is 0.01. As shown in Fig. 2, theresulting fast magnetic reconnection is still two dimensionalon the ﬁrst stage. Then, on the second stage, threedimensional fast magnetic reconnection starts.
In general, two-dimensional fast magnetic reconnection
may be considered to be a nonlinear tearing instability. In
this paper, it was shown that, if the three-dimensional weakresistive perturbation is added in such a two-dimensionalmodel, the two-dimensional fast magnetic reconnection is
moreover destabilized and the ejected magnetic loop can bestrongly localized in the sheet current direction. In addition,
as shown in Figs. 5共b兲 and 8共a兲, the generation of such magnetic loops is intermittent and the generation pattern becomes random.
Also, it was revealed that the aspect ratio ⌬Z / ⌬X of theejected three-dimensional magnetic loops is always larger
than the ratio of the half value of the local sound speed in themagnetic diffusion region over the Alfven speed in the upstream magnetic ﬁeld region. In addition, it was shown that
this feature is supported by a simple theory of the threedimensional magnetic loop formation based on the competition of the z-directional local sound wave and theShimizu et al., “Three-Dimensional Fast Magnetic Reconnection” (2009)
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x-directional Alfven wave. The former wave can cancel thethree-dimensional fast magnetic reconnection at a time and aplace, and, otherwise, enhance at the other time and place.
The latter wave drives the fast magnetic reconnection, itself.
Note that the aspect ratio ⌬Z / ⌬X examined in this paper isonly for By ⬎ 0 regions on the current sheet. Hence, it is but
not directly for magnetic loops, themselves, which may consist of By ⬎ 0 and By ⬍ 0 regions. In other words, we do not
focus on the whole size of each magnetic loop in this paper.
These numerical results presented in this paper support
that the intermittent plasma downﬂows reported by Innes
et al.20 and Asai et al.21 is caused by three-dimensional fastmagnetic reconnection. Because, the fast magnetic reconnection shown in this paper is fully three dimensional on theﬁnal stage and intermittently ejects magnetic loops, like theintermittent downﬂows observed in the solar ﬂares. In addition, since the three-dimensional fast magnetic reconnection
simultaneously occurs at some points along the magnetic
neutral line, from a macroscopic view, the fast magnetic reconnection may seem to be two dimensional. Finally, thoseejected magnetic loops will pile up on the large-scale arcade
structure which is not simulated in this paper. Then, thelarge-scale arcade structure can gradually grow keeping thetwo dimensionality, as the pile up proceeds.
Here, it is important that, in order to cause threedimensional fast magnetic reconnection, the initial current
sheet do not necessarily need any distinctly threedimensional structure. In other words, even in the approximately one-dimensional current sheet, fully threedimensional and intermittent fast magnetic reconnection is
spontaneously established by a current driven anomalous resistivity. Once two-dimensional fast magnetic reconnection
occurs, any weak ﬂuctuation in the sheet current direction
can destabilize it to three dimensional and intermittent fastmagnetic reconnection.
In the next step of this study, the temporal dynamics ofejected magnetic loops must be studied for the collaboration
with the space observation data. In the temporal dynamics,
some of the intermittently ejected magnetic loops may ﬁnallymerge into a magnetic loop in the downstream region. The
magnetic loops detectable in space observations may be only
the relatively large-scale magnetic loops after merging. In
such a case, this three-dimensional intermittent fast reconnection model shown in this paper may be modiﬁed by thefractal and intermittent reconnection model which was proposed by Shibata and Tanuma,9 where they explained solar
ﬂare observation data as a time development from the microreconnection process to macroreconnection process. Since
their basic concept was a two-dimensional model, this threedimensional fast reconnection shown here should give a newview point to their model.

411

of Kyoto University and Solar-Terrestrial Environment
Laboratory 共STEL兲 of Nagoya University. The numericalcalculations were performed on the KDK computer system at
RISH and FUJITSU HP2500 parallel computer system at
Nagoya and Kyoto University Data Processing Centers.
APPENDIX A: MEASUREMENTS OF ⌬XAND ⌬ZIn this section, how to measure ⌬X and ⌬Z is explained,
which are respectively x and z-directional sizes of each magnetic loop. First, note that we focus on these sizes measured
at the end of the growth phase of a magnetic loop. Because,
we focus on the magnetic reconnection process, itself, rather
than the dynamics of magnetic loop. The shape of magnetic
loop may gradually change after the growth phase, due to thenonequilibrium of the pressure balance.
1. The ﬁrst step

Specify the end of the growth phase of each magnetic
loop, observing the time variation of By contour map on
xz-plane at y = 0. For this purpose, observe the By peak valueof each magnetic loop on By contour map, and then, ﬁndwhen the By peak value takes the maximum in time. Let us
deﬁne the timing as the end of the growth phase.
2. The second stepOn the By contour map at the end of growth phase, selecta contour line for the half value of the By peak value. Let us
deﬁne the shape of the magnetic loop with the selected contour line.
3. The third step

Measure the value of z at the By peak point. Then, measure ⌬X as the difference between the largest and smallest
values of x where the selected contour line passes at the
speciﬁed value of z. Then, measure ⌬Z as the difference
between the largest and smallest values of z where the selected contour line passes at any value of x.

ACKNOWLEDGMENTS

This work was supported by grants-in-aid from the Japanese Education Ministry; Ehime University 共Grant No.
15540475 to T.S.兲, and Kyoto University 共Grant No.
17GS0208 to K.S.兲. In addition, this work was supported by
the Research Institute for Sustainable Humanoshpere 共RISH兲

FIG. 10. By ⬎ 0 contour lines around origin. Case of Lz = 20 and at T = 65.

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Phys. Plasmas 16, 052903 共2009兲

FIG. 13. By ⬎ 0 contour lines around origin. Case of Lz = 20 and at T = 78.
FIG. 11. By ⬎ 0 contour lines around origin. Case of Lz = 20 and at T = 68.

As an example of the ⌬X and ⌬Z measurements, Figs.
10–13 are shown. These ﬁgures are obtained, respectively, at
T = 65, 68, 72, and 78 in the simulation run same as Fig. 8共a兲.
Figure 13 is a part of Fig. 8共a兲.
First, let us focus on the bow-shape high By intensity
region observed around 共x , z兲 = 共4 , 0兲 in Fig. 11. On the ﬁrst
step, the end of the growth phase is considered to be T = 68,
i.e., Figure 11. Because, the number of the contour lines ofwhich the bow-shape region consists is larger than those ofFigs. 10 共before兲 and 12 共after兲. In fact, the By peak value inthe region increases from 0.53 at T = 65 to 0.57 at T = 68.
Then, it decreases to 0.45 at T = 72. Here, the contour lines inthese ﬁgures are drawn only for By ⬎ 0 with the same interval⌬By = 0.05 from By = 0. Similarly, the end of the growthphase of the round-shape high By intensity region observed
around 共x , z兲 = 共4 , 6兲 is also considered to be T = 68. The endof the growth phase of another round-shape high By intensity
region observed around 共x , z兲 = 共9 , 0兲 is considered to be T
= 72. On the second step, the shape of each high By intensity
region is speciﬁed on the contour line for the half value ofFIG. 12. By ⬎ 0 contour lines around origin. Case of Lz = 20 and at T = 72.

each By peak value. On the third step, as shown in Fig. 11,⌬X1 = 0.8 and ⌬X2 = 1.2 are obtained at the By peak point ofeach By region. Then, ⌬Z1 = 6.5 and ⌬Z2 = 1.4 are obtained as
the z-directional whole size of the By region, where the halfvalues of ⌬Z1 is measured, because of the symmetry boundary condition along x-axis. Similarly, ⌬X3 = 0.8 and ⌬Z3= 1.9 are obtained in Fig. 12.
Figure 14 shows the case of Lz = 2.5, which is the same
simulation run as Fig. 8共b兲 but is obtained at T = 67. Clearly,
the number of the contour lines in Fig. 14 is much larger thanthat of Fig. 8共b兲. In addition, in contrast to Fig. 8共b兲, the peakof the high By intensity region in Fig. 14 is slightly separated
from the upper-boundary line, i.e., z = 2.5. Hence, ⌬Z4 should
not be taken as the half width. In Fig. 14, ⌬X4 = 0.8 and⌬Z4 = 0.8 are obtained. These obtained data 共⌬X , ⌬Z兲 aresummarized in Fig. 9.

APPENDIX B: SELF-FORMATIONS
OF MAGNETIC NEUTRAL

In this simulation, the magnetic neutral line, i.e., X-line,is spontaneously formed, which is not necessarily be parallelto the z-axis. However, in every case examined in this paper,
X-line tends to be parallel to the z-axis. In order to show thisfeature, in Figs. 10 and 13, the ﬂow vector ﬁeld is shown. In
this simulation, since the initial magnetic ﬁeld lines only
have x-directional component, the reconnection jet should be

FIG. 14. Case of Lz = 2.5 and at T = 67. This is the same simulation run as
Fig. 8共b兲.

Shimizu et al., “Three-Dimensional Fast Magnetic Reconnection” (2009)
052903-11

MHD study for three-dimensional instability…

413
Phys. Plasmas 16, 052903 共2009兲

FIG. 17. Contour lines of ␩J ⬍ −0.05. Case of Lz = 20 and at T = 75.

FIG. 15. 共a兲 Contour lines for Ux ⬎ 0 in the case of Lz = 20 and at T = 65. 共b兲
Contour lines for ␩J ⬍ −0.05 in the case of Lz = 20 and at T = 65.

FIG. 16. 共a兲 Contour lines of Ux ⬎ 0. Case of Lz = 20 and at T = 78. 共b兲
Contour lines of ␩J ⬍ −0.05. Case of Lz = 20 and at T = 78.

positive or negative x-directional. By observing these
x-directional diverging ﬂows, we can easily detect active
X-lines.
In Fig. 10, X-line1 associated with the magnetic loop
around 共x , z兲 = 共4 , 0兲 is indicated by a double dash line and isalmost parallel to the z-axis. More carefully observing thex-directionally diverging ﬂow vectors, the length of X-line1
is much shorter than ⌬Z1 of the bow-shape high By region.
Also in Fig. 13, X-line1 is still almost parallel and X-line 2
parallel to the z-axis appears. In addition, Fig. 6共a兲 shows the
beginning of the growth phase of the magnetic loop which isdeveloped to the bow-shape By region in Figs. 10–13. The
X-line is also along the z-axis.
Figures 15–18 more clearly show that X-line is almost
parallel to the z-axis. Figure 15共a兲 shows the ﬂow vector ﬁeld
and contour map of Ux ⬎ 0 at T = 65 in the case of Lz = 20.
Accordingly, it shows the details of Fig. 10. On X-line, thesign of Ux will be changed from positive to negative. Suchan X-line is indicated by label A, corresponding to X-line1 inFig. 10. The X-line seems to be not parallel to the z-axis, asvalue of z increases. However, we must focus on active
X-line that can produce a magnetic loop. In order to study theactivity of X-line, Fig. 15共b兲 shows the contour map of ␩J
less than ⫺0.05, where ␩J means the reconnection rate on
the X-line, and hence, the active magnetic diffusion region
can be observed as the region of ␩J ⬍ −0.05. The ﬂow vector
ﬁeld is same as that of Fig. 15共a兲, The active X-line indicatedby label A is being piecewisely separated but roughly parallelto the z-axis. Similarly, Figs. 16共a兲 and 16共b兲 show the case

FIG. 18. Contour lines of ␩J ⬍ −0.05. Case of Lz = 20 and at T = 82.

第 4 章 論文集

414052903-12

of T = 78. Also in these ﬁgures, the active X-line indicated bylabel A is approximately parallel to the z-axis.
Next, let us see the time development of the active
X-line in the series of Figs. 17, 16共b兲, and 18. The active
X-line indicated by label A in Fig. 17 共T = 75兲 is slightly
declined from the z-axis but, in Figs. 16共b兲 共T = 78兲 and 18
共T = 82兲, tends to gradually become parallel to the z-axis. In
Figs. 17 and 18, another active X-line is indicated by label B
around 3 ⬍ x ⬍ 7 and 4 ⬍ z ⬍ 6. From Figs. 17 and 16共b兲, theX-line is gradually growing and is slightly curved against thez-axis. Then, from Figs. 16共b兲 and 18, nonparallel part 共z⬍ 4.5兲 of the X-line is isolated from the main part 共4.0⬍ z兲 ofX-line. It suggests that, if the X-line largely declines from thez-axis, the X-line cannot be maintained to generate magnetic
loops, and hence, rapidly collapses.
1

Phys. Plasmas 16, 052903 共2009兲

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8

Okamotom et al., “Coronal Transverse MHD Waves in a Solar Prominence” (2007)
REPORT

Coronal Transverse Magnetohydrodynamic
Waves in a Solar Prominence
T. J. Okamoto,1,2* S. Tsuneta,1 T. E. Berger,3 K. Ichimoto,1 Y. Katsukawa,1 B. W. Lites,4S. Nagata,2 K. Shibata,2 T. Shimizu,5 R. A. Shine,3 Y. Suematsu,1 T. D. Tarbell,3 A. M. Title3
Solar prominences are cool 104 kelvin plasma clouds supported in the surrounding 106 kelvin coronal plasma
by as-yet-undetermined mechanisms. Observations from Hinode show fine-scale threadlike structures
oscillating in the plane of the sky with periods of several minutes. We suggest that these represent Alfvén
waves propagating on coronal magnetic field lines and that these may play a role in heating the corona.
olar prominences are classified as either
quiescent or active region (AR), the latter
referring to material suspended above sunspot magnetic regions. Quiescent prominences
often exist for many weeks at high solar latitudes,
whereas AR prominences can be dynamic andshort-lived. They are the most enigmatic of solar
structures supported by coronal magnetic field lines,
sometimes erupting as the source of coronal massejections, large-scale eruptions of plasma fromflaring solar active regions, that can have major
impacts on the terrestrial magnetic environment.
Recent ground-based observations have revealed

S

that AR prominences have numerous small
threadlike features (1), with continuous flow ofmaterial along the threads (2–9). Observations fromspace (10, 11) confirm these findings and show
additional dynamics related to coronal structure.
We report Hinode Solar Optical Telescope(SOT) (12, 13) observations of an AR prominence
in a 0.3-nm broadband region centered at 396.8
nm, the H-line spectral feature of singly ionized
calcium (Ca II). Radiation in this bandpass
typically has a temperature of less than 20,000 K.
We obtained over 1 hour of continuous SOT
images of NOAA AR 10921 on the west solar

www.sciencemag.org

SCIENCEVOL 318415

limb on 9 November 2006. The images show amultithreaded AR prominence suspended above
the main sunspot (Fig. 1). Although no simultaneous Ha images were taken, the Ca II H-line
prominence structures are consistent with the structures seen in lower-resolution Ha observations(14). The Ca II H-line movie (movie S1) shows
ubiquitous continuous horizontal motions along
the prominence threads. The origins of these flows
remain unknown. Some of the flows had constant
speeds of about 40 km s−1, whereas others accelerated monotonically in a more complicated fashion.
The Hinode SOT movies also reveal thatmany of the threads in the prominence underwent
vertical (i.e., in the plane of the sky) oscillatory
motions (Fig. 2) at periods of 130 to 250 s. The
1
National Astronomical Observatory (NAOJ), Mitaka, Tokyo
181-8588, Japan. 2Kwasan and Hida Observatories, Kyoto
University, Yamashina, Kyoto 607-8471, Japan. 3Lockheed
Martin Solar and Astrophysics Laboratory, B/252, 3251
Hanover Street, Palo Alto, CA 94304, USA. 4High Altitude
Observatory, National Center for Atmospheric Research,
Post Office Box 3000, Boulder CO 80307–3000, USA.
5
Institute of Space and Astronautical Science, Japan
Aerospace Exploration Agency (ISAS/JAXA), Sagamihara,
Kanagawa 229-8510, Japan.

*To whom correspondence should be addressed. E-mail:joten.okamoto@nao.ac.jp

7 DECEMBER 20071577

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416

Hinodevertical oscillatory motions were coherent overlengths as long as 16,000 km. One thread (Fig.
3A) had a vertical width of 660 km and an
oscillation period of 240 s. Comparison of theoscillation phase at various horizontal positions(Fig. 3, B to F) reveals that the thread oscillated inphase along its entire length. The vertical oscillation amplitudes of all threads we saw ranged from
400 to 1800 km, and thread widths were between
430 and 660 km (Table 1). Because we cannot
determine the angle between the plane ofoscillation and the line of sight to the prominence
threads, the oscillation amplitudes as well as thehorizontal velocities are minimum estimates.
The observed vertical oscillatory motion ismost likely due to propagating or standing Alfvén
waves on the horizontal magnetic field lines thatcompose the prominence. An alternate hypothesis
is longitudinal plasma motion along helical fieldlines. Helical field line structure has been observedin many previous prominence studies (15–17);
however, the helical field line hypothesis cannot
explain the several cases we found of synchronous
vertical oscillation of entire threads, for example,
Fig. 3. Alternately, if the observed threads are thinmagnetic flux tubes, the observed oscillations may
be fast magneto-acoustic kink modes propagating
along the tubes (18, 19). The observations shown
here lack line-of-sight Doppler velocity measurements, so we cannot determine the exact oscillation mode at this point.
These field lines connect to the source regionsin the photosphere (7), where they are excited by
the wide-spectrum p-mode oscillations originating in the convection zone. Each field line oscillated independently, as seen in the Hinodemovies. The synchronous oscillation along entire
threads shown in Fig. 3 implies that we cannot
distinguish the time difference at the minimum
and maximum amplitudes all along the thread.
The uncertainty in phase of the oscillation is no
more than 1/16 of the oscillation period, so weestimate that the minimum wavelength of theoscillation is 16 × 16,000 ≈ 250,000 km. The
wave speed is estimated to be >1050 km s−1 forthe average oscillation period of ~240 s. If weassume that the plasma density is 1010 cm−3 (20),the implied magnetic field strength is ~50 G for
the propagating Alfvén wave, in agreement withmeasurements and models of active region prominence magnetic fields (21, 22). The Poynting flux
carried by the observed waves is then estimated

20:21:04
20:21:49
20:22:34
20:23:20
20:24:05
20:24:50
20:25:34
20:26:19
20:27:05
20:27:50
20:28:35
5,000 kmFig. 2. Example of a vertical oscillation of a single prominence thread. A small field of view is extracted
from the larger field shown in Fig. 1 and shown in negative contrast. Tick marks have a spacing of 2000
km on the Sun, and UT time for each image is denoted on the left. The dashed line in each image indicatesan approximately constant height above the photosphere. This oscillating thread has a length of about
3600 km and width of 430 km. Steady flow at 39 km s−1 along the thread is evident. The verticalamplitude of the oscillation is about 900 km with a period of 174 s. The vertical speed is about 10 km s−1.

Fig. 1. High-resolution image on
the solar limb obtained with SOT
aboard Hinode. This observation
was performed with a cadence of
15 s from 19:33 to 20:44 UT on 9
November 2006. Tick marks havea spacing of 1000 km on the Sun.
A radial density filter is applied toshow the brighter photosphere andthe fainter coronal structures in thesame image. The main sunspot of
NOAA AR 10921 as well as thetrailing bright plage areas are visible on the disk. Above the limb,
ubiquitous vertical spicules are seen
below the horizontal threads of the
AR prominence. The cloudlike prom10,000 km
inence structure is located 10,000
to 20,000 km above the visible
limb and exhibits a very complex
fine structure with predominant horizontal threadlike features. The intensity of the prominence in Ca II H-line radiation is about 1% of the on-disk photosphere.

1578

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Okamotom et al., “Coronal Transverse MHD Waves in a Solar Prominence” (2007)

417

SPECIALSECTION
to be rv2VA ~ 2.0 × 106 erg s−1 cm−2 (where r is
the density, v is the velocity amplitude, and VA
is the Alfvén speed), a lower limit based on theminimum estimates of observed tangential velocities. Given a suitable dissipation mechanism,
this flux is sufficient to heat coronal loops withlengths longer than the estimated oscillation
wavelength (23).
In transiting from the photosphere to the chromosphere, the large density decrease with height
results in rapid increases in the acoustic and Alfvén
wave propagation speeds. This effective dis-continuity causes waves with periods longer thana cutoff period determined by the thermodynamic
conditions and magnetic field strength in the atmosphere to be reflected before reaching the coronal heights of active region prominences. Because
we find a typical oscillation period ~ 240 to 250 s,
we can infer that the Alfvén cutoff period is longer
than about 4 min for this prominence structure.
The limited field of view of our data prevents
us from determining the lengths of the field lines
threading the prominence. However, the frequency spectrum of the observed waves can be used toTable 1. Properties of moving threads with vertical oscillations. Asterisks mean averaged velocity.

1
2
3
4
5
6

Length
(km)

Width
(km)

Horizontal
velocity (km−1)

Vertical oscillation
period (s)Vertical oscillation
width (km)Height from thelimb (km)3600
16,000
6700
2200
3500
1700

430
660
580
360
430
510

39
15
39
46*
45*
25*

174 ± 25240 ± 30230 ± 87
180 ± 137
135 ± 21250 ± 17

904
1113
909
435
408
1771

18,30012,40014,70019,00014,30017,200

(km)
5,000
(km)

10,000AS1

S2 S3 S4

investigate the length of the field lines. Open fieldlines have a distinct wide spectrum above the
Alfvén cutoff frequency, whereas shorter closed
loops show multiple discrete resonances as afunction of loop length. The estimated Alfvén
speed and the mean observed period implies aminimum length of 250,000 km if this is thestanding wave in a closed-loop system.
Previous observations of waves in the solar
corona include Doppler velocity and periodic intensity oscillations in coronal loops as well as
flare-generated transversal displacements of active
region loops (24–28). Those oscillations are examples of magneto-acoustic waves propagating out
from photospheric source sites. Alfvén waves incoronal loops and prominences are also claimed toaccount for spectroscopic observations of nonthermal line widths in coronal emission lines (29, 30).
References and Notes
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Phys. 61, 39 (1979).

D (S3)

4,000

S5

3,000
2,000

0

0

10,000 20,000

30,00040,000 (km) 1,000
0
19:44

(km)
5,000

5,000

B (S1)

4,000

4,000

3,000

3,000

2,000

2,000

1,000

1,000

0
19:44
5,000

19:46

19:48

19:50

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19:52

5,000

C (S2)

4,000

4,000

3,000

3,000

2,000

2,000

1,000

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0

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E (S4)

19:44

19:46

F (S5)

0

19:52

19:44

19:46

(UT)

(UT)

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31. The authors thank H. Shibahashi, T. Sekii, R. Erdélyi, and
V. Nakariakov for comments. Hinode is a Japanese
mission developed and launched by ISAS/JAXA, with NAOJ
as domestic partner and NASA and Science andTechnology Facilities Council (STFC) (UK) as international
partners. It is operated by these agencies in cooperation
with European Space Agency and Norwegian Space
Centre (Norway). This work was carried out at the NAOJ
Hinode science center, which was supported by the
Grant-in-Aid for Creative Scientific Research, the Basic
Study of Space Weather Prediction (head investigator,
K.S.) from the Ministry of Education, Culture, Sports,
Science, and Technology, Japan, donation from Sun
Microsystems Incorporated, and NAOJ internal funding.
The National Center for Atmospheric Research issponsored by NSF. T.J.O. is supported by research
fellowships from the Japan Society for the Promotionof Science for Young Scientists.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/318/5856/1577/DC1
Movie S1
21 May 2007; accepted 6 November 2007
10.1126/science.1145447

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Tsuneta et al., “The Magnetic Landscape of the Sun’s Polar Region” (2008)

419

The Astrophysical Journal, 688:1374Y1381, 2008 December 1# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE MAGNETIC LANDSCAPE OF THE SUN’S POLAR REGION
1

S. Tsuneta, K. Ichimoto,1 Y. Katsukawa,1 B. W. Lites,2 K. Matsuzaki,3 S. Nagata,4 D. Orozco Suárez,5T. Shimizu,3 M. Shimojo,6 R. A. Shine,7 Y. Suematsu,1 T. K. Suzuki,8 T. D. Tarbell,7 and A. M. Title7
Received 2007 November 9; accepted 2008 August 1

ABSTRACT
We present observations of the magnetic landscape of the polar region of the Sun that are unprecedented in termsof spatial resolution, ﬁeld of view, and polarimetric precision. They were carried out with the Solar Optical Telescopeaboard Hinode. Using a Milne-Eddington inversion, we ﬁnd many vertically oriented magnetic ﬂux tubes with ﬁeld
strengths as strong as 1 kG scattered in latitude between 70 and 90 . They all have the same polarity, consistent withthe global polarity of the polar region. The ﬁeld vectors are observed to diverge from the centers of the ﬂux elements,
consistent with a view of magnetic ﬁelds that are expanding and fanning out with height. The polar region is also
found to have ubiquitous horizontal ﬁelds. The polar regions are the source of the fast solar wind, which is channeledalong unipolar coronal magnetic ﬁelds whose photospheric source is evidently rooted in the strong-ﬁeld, verticalpatches of ﬂux. We conjecture that vertical ﬂux tubes with large expansion around the photospheric-coronal boundary
serve as efﬁcient chimneys for Alfvén waves that accelerate the solar wind.
Subject headinggs: solar wind — Sun: faculae, plages — Sun: magnetic ﬁelds — Sun: photosphere
1. INTRODUCTION

Gregorian telescope with ﬁltergraph and spectropolarimeter aboard
the satellite Hinode (Kosugi et al. 2007). These observations are
unprecedented in terms of their very high spatial resolution, wide
ﬁeld of view, and high polarimetric sensitivity and accuracy inmeasurements of vector magnetic ﬁelds.

The Sun’s polar magnetic ﬁelds are thought to be a direct manifestation of the global poloidal ﬁelds in the interior, which serve
as seed ﬁelds for the global dynamo that produces the toroidal
ﬁelds responsible for active regions and sunspots. The polar regions are also the source of the fast solar wind. Although the polar
regions are of crucial importance to the dynamo process and theacceleration of the fast solar wind, their magnetic properties arepoorly known. Magnetic ﬁeld measurements in the solar polar
regions have long been a challenge: variable seeing, combinedwith a strong intensity gradient and the foreshortening effect at thesolar limb, greatly increases the systematic noise in ground-basedmagnetograms. Nevertheless, pioneering observations have been
carried out for the polar regions (Tang & Wang 1991; Lin et al.
1994; Homann et al. 1997; Okunev 2004; Okunev & Kneer 2004;
Blanco Rodriguez et al. 2007). These observations typically haveprovided measurements only of the line-of-sight magnetic component. Full Stokes polarimetry has also been carried out, but aswith most of the ground-based observations mentioned above, thespatial resolution of these measurements was limited by seeing(Lites 1996). Another limitation of past polar observations is thatthey have been restricted to individual polar faculae within a smallﬁeld of view and have not provided us with a global magnetic
landscape of the polar region, except for the Global Oscillation
Network Group and SOLIS (Harvey et al. 2007).
We have investigated the properties of the photospheric magnetic ﬁeld in the polar regions using the Solar Optical Telescope,
SOT (Tsuneta et al. 2008; Suematsu et al. 2008; Ichimoto et al.
2008; Shimizu et al. 2008). SOT is a diffraction-limited (0.200 Y 0.300 )2. STOKES MAPS OF THE POLAR REGION
Hinode observed the solar polar region on 2007 March 16,
when the south pole was located 7 inside the visible solar disk.
Stokes proﬁles of two Fe i lines (630.2 and 630.3 nm) were observed with the spectropolarimeter (SP) of SOT. The degrees oflinear (Stokes Q and U ) and circular (Stokes V ) polarization aredeﬁned as
R
R
R
Q(k)dkU (k)dkjV (k)jdk
R
R
R
;
;
;
Ic dk
Ic dk
Ic dk
respectively, where Ic is the continuum intensity and Q, U, and V
are the observed Stokes proﬁles. The SP wavelength sampling is2.15 pm. When the red lobe of the Stokes V proﬁle is positive, aminus sign is added, meaning that the line-of-sight ﬁeld is directed away from the observer. The integrations are performed
between 21.6 and 4.32 pm and between +4.32 and +21.6 pm
from the center of individual Stokes I proﬁles. These wavelength
ranges were determined by examining the actual data. If therange is too wide, the maps will be more susceptible to photon
noise. Maps of these degrees of polarization are hereafter called
the Stokes Q, U, and V maps and are shown in Figure 1. These
data were taken at 12:02:19Y14:55:48 UT on 2007 March 16.
Position angles for Stokes Q and U were determined as follows: 0 runs from east to west, and 90 is north-south. If thetransverse ﬁeld is directed to the west, it will be seen as white in
the Stokes Q map, and if it is directed north it will be black. If thetransverse ﬁeld is directed at 45 , it will be seen as white in the
Stokes U map, with black denoting 135 . Black (positive polarity)in the Stokes V map indicates that the magnetic ﬁeld is directed
toward the observer, and white (negative polarity) is directed awayfrom the observer.
We noticed scattered isolated patches in the Stokes Q map(Fig. 1b). They coherently have negative values (black), indicating

1

National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan.
High Altitude Observatory, National Center for Atmospheric Research, Boulder,
CO 80307-3000.
3
ISAS/JAXA, Sagamihara, Kanagawa 229-8510, Japan.
4
Kwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto 6078471, Japan.
5Instituto de Astrofı́sica de Andalucia, CSIC, E-18008 Granada, Spain.
6
Nobeyama Solar Radio Observatory, National Astronomical Observatoryof Japan, Nobeyama, Nagano 284-1305, Japan.
7
Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA 94304.
8
School of Arts and Sciences, University of Tokyo, Meguro, Tokyo 1538902, Japan.
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Fig. 2.— Polar view of the circular polarization (Stokes V ) signal, createdfrom Figs. 1d and 1e. The original observing ﬁeld of view ( Fig. 1) is 327.5200(east-west) by 163.8400 (north-south) and was converted to a map seen from above
the south pole. East is to the left, west is to the right, and the observation was carried out from the top down. Spatial resolution is lost near the extreme limb (i.e.,near the bottom of the ﬁgure). The ﬁeld of view is 327.5200 (east-west) by 472.9600(north-south along the line of sight). The ﬁeld of view for the line-of-sight direction (163.8400 ) expands to 472.9600 as a result of correction for foreshortening. The
pixel size is 0.1600 . Latitudinal lines for 85 , 80 , 75 , and 70 are shown as large
circles, while the plus sign marks the south pole. The image saturates at a degreeof polarization of 0.005. Black indicates the magnetic ﬁeld component toward
the observer, while white is that away from the observer. The small circle indicates the same magnetic patch highlighted in Fig. 1.

Fig. 1.— Polarization maps of the south polar region taken at 12:02:19Y
14:55:48 UT on 2007 March 16. (a) Continuum map; (b) Stokes Q map (transverse
magnetic ﬁeld); (c) Stokes U map (transverse magnetic ﬁeld); (d, e) Stokes V maps
(line-of-sight magnetic ﬁeld). East is to the left, and north is up. The gray scale indicates the wavelength-integrated total degree of polarization. The images saturate
at a degree of polarization of 0.003 for (b), (c), and (e), and at 0.02 for (d).
Black circles in (b), (c), and (d) indicate one fanning-out magnetic structure.

the presence of a magnetic component vertical (i.e., north-south)
to the local surface. These patches are associated with bipolarstructures in the Stokes U map (Fig. 1c). Figures 1d and 1e show
the Stokes V map. The same Stokes V image seen from just above
the south pole is shown in Figure 2, where the foreshortening effect has been corrected. All the pixels in the observing coordinate(as shown in Fig. 1) are mapped to the corresponding positions in
the polar maps (Figs. 2 and 3), and the gaps between the mapped
pixels are interpolated. Figures 1d and 2 show a bipolar structure
corresponding to the patches seen in the Stokes Q and U maps at
higher latitudes. One example is indicated by a small circle in Figures 1 and 2. The bipolar structure in the Stokes V map together
with the Q and U maps suggests the presence of fanning-out ﬂux
tubes vertical to the local surface.
Unipolar patches with positive polarity (black) are dominant
at lower latitudes in the Stokes V map. This is because the vertical ﬂux tubes located at lower latitudes have all their fanningout magnetic vectors directed toward the observer. The dominant
black polarity indicates a component of the ﬁeld oriented toward
the observer. This shows that all the vertical magnetic patches

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TSUNETA ET AL.

are coherently directed away from the Sun, consistent with thegeneral polar ﬁelds in 2007 (see Gopalswamy et al. 2003; Durrant
et al. 2004).
We also noticed in the Stokes U map that the positive polarity(white) in the bipolar structures is more dominant on the east side,
while negative polarity (black) dominates in the west. Since theﬂux tubes vertical to the local surface on the east side tilt toward
the plus axis of Stokes U, the bipolar structures on the east side are
more plus-biased (white), and vice versa. This explains the eastwest asymmetry of the bipolar structures in the Stokes U map.
3. POLAR LANDSCAPE
3.1. Milne-Eddington Fitting
We applied a least-squares ﬁt to the observed Fe i 630.15 nm
and Fe i 630.12 nm Stokes proﬁles, assuming a Milne-Eddington
atmosphere, with the MILOS code (Orozco Suárez et al. 2007a,
2007b). The 10 free parameters are the three components describing the vector magnetic ﬁeld (strength B, inclination angle , andazimuth angle ), the line-of-sight velocity, two parameters describing the source function, the ratio of line to continuum absorption coefﬁcients, the Doppler width, the damping parameter,
and the stray-light factor . To minimize the inﬂuence of noise,
we analyzed only pixels whose polarization-signal peaks exceed
a given threshold above the noise level . The noise level was determined in the continuum wavelength range of the proﬁles. The
ﬁtting was performed for pixels whose Q, U, or V signal is larger
than 5 . It turns out that 10.5% of the area meets this criterion.
There may be unresolved magnetic elements along the line ofsight, for example, rays passing obliquely through vertical ﬂux
tubes observed close to the limb. A nonzero stray-light factor 
may be interpreted as a parameter that includes both the ﬁlling factor of a nonmagnetized or weakly magnetized atmosphere along
the line of sight and the stray-light contamination factor. The
stray-light proﬁle is evaluated individually for each pixel as theaverage of the Stokes I proﬁles observed in a 100 wide box centered on the pixel (Orozco Suárez et al. 2007b). This arrangement also allows us to accurately estimate the stray-light proﬁlesof rapidly changing continuum intensity toward the limb. If thestray-light contamination is negligible, then the effective ﬁlling
factor of the magnetic atmosphere will be given by f = 1  .
The results of our inversions applied to the polar region showthat the distribution of the effective ﬁlling factor has a broad
peak at f = 0.15 with FWHM range 0.05 < f < 0.35 (see Fig. 5d
below). Orozco Suárez et al. (2007b) suggested that there is aconsiderable stray-light contribution and that the actual ﬁlling
factor may be larger than the nominal values derived above. As
an extreme case, we also estimate the magnetic ﬂux assuming
f = 1 in the subsequent sections.
3.2. Vertical Kilogauss Patches and Horizontal Field
Figure 3 is a map of the magnetic ﬁeld strength as seen fromjust above the south pole. Such a polar representation is needed
to correctly see the spatial extent and size distribution of the magnetic islands in the polar region. While many of them are isolated,and some have the form of a chain of islands, complex internal
structures are seen inside the individual patches. Many of thepatchy magnetic islands have very high ﬁeld strengths, reachingover 1 kG. These kilogauss patches coincide in position withthose seen in Figure 1: they are coherently unipolar, and likeplage and network ﬁelds at lower latitudes (Martı́nez Pillet et al.
1997), their magnetic ﬁelds are vertical to the local surface. The
fanning-out structure is conﬁrmed in the vector magnetic ﬁeld
map obtained from the least-squares ﬁtting.

We notice a clear tendency for patches to be larger in size withincreasing latitude, being as large as 500 ; 500 at higher latitudes
and 100 ; 100 at lower latitudes. Degradation in spatial resolution
due to the projection effect may contribute to the larger sizes athigh latitude. This, however, would imply that close to the solarlimb, we are observing ﬂux tubes higher in the atmosphere. The response function to temperature (del Toro Iniesta 2003) for the coreof the Stokes I proﬁle has a broad peak between 100 and 500 km
above a continuum optical depth of unity in Sun-center observations. The response function for a plane-parallel atmosphereviewed obliquely at an angle of 80 has a peak that is 50Y100 km
higher, implying that we are observing higher altitude atmosphereat 80 .
We are interested in the inclination angle i of the magnetic
ﬁeld vector with respect to the local normal. Close to the limb, it ispossible to determine i without the usual 180 ambiguity (del Toro
Iniesta 2003) in the transverse ﬁeld components: the inclination
angle is given by cos i = cos  cos  + sin  sin  cos (/2  ),
where  is the latitude,  is the inclination of the magnetic ﬁeld
vector, and  is the azimuth angle. Figure 4 shows the inclinationof the ﬁeld lines: red contours indicate regions where the local
inclination i is smaller than 25 (vertical), while blue contours
show regions with local inclination larger than 65 (horizontal).
All the large patches have ﬁelds that are vertical to the local surface (red), while the smaller patches tend to be horizontal (blue).
Most of the magnetic structures seen in Figure 3 thus have either
vertical or horizontal direction. These two types do not appear to
be spatially correlated.
Figure 4 also indicates that magnetic patches of larger spatial size coincide in position with polar faculae ( Lin et al. 1994;
Okunev 2004; Okunev & Kneer 2004). Figure 5c shows histograms of the continuum intensity relative to the local averageintensity for pixels with magnetic ﬁeld strength greater than 300
and 800 G. The distribution of the horizontal ﬁelds is essentiallysymmetric about the average intensity, while the vertical ﬁelds
tend to have higher continuum intensities. These bright points correspond to the polar faculae. The vertical kilogauss patches change
considerably in shape and distribution over 5Y10 hr, while thespatial distribution of the horizontal magnetic ﬁeld changes completely in 30 minutes or less. This is consistent with the observations of the horizontal ﬁelds made by SOLIS (Harvey et al. 2007)
and Hinode (Centeno et al. 2007; Ishikawa et al. 2008).
We manually selected 41 patches with vertical magnetic ﬁeld
to derive the distribution of magnetic ﬂux and patch size as a function of latitude.
P The total magnetic ﬂux of a magnetic patch isestimated as i Bi fi si, where Bi and fi are the intrinsic magnetic
ﬁeld strength and ﬁlling factor of the ith SOT pixel inside thepatch, respectively, and si is the common pixel size. The magnetic
ﬂux of the patches ranges from 1.8 ; 1018 to 1.0 ; 1020 Mx,
with mean ﬂux ofP2.7 ; 1019 Mx. We obtain the size of the magnetic patches as i si, where the summation is over pixels withper-pixel average ﬁeld strength Bi fi larger
Pthan 10 G: The size
increases by a factor of 1.92, and the ﬂux i Bi fi si decreases by
a factor of 1.33 between latitudes 70 and 85 . The total magnetic
ﬂux is preserved to within 40%.
Figure 5a shows a histogram (probability distribution function,PDF) of the intrinsic magnetic ﬁeld strength B for latitudes above
75 : vertical magnetic ﬁelds with inclination i < 25 dominate
the stronger ﬁeld regime, while horizontal ﬁelds with i > 65 aremuch more prevalent below 250 G. Of the pixels for which theinversion was performed, 41% contained vertical magnetic ﬁeld,
and 49% horizontal magnetic ﬁeld. Figure 5a can be comparedwith the PDF for the quiet Sun obtained with Hinode by OrozcoSuárez et al. (2007b, Fig. 7). A magnetic energy PDF, deﬁned by

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Fig. 3.— South polar view of the magnetic ﬁeld strength taken at 12:02:19 Y14:55:48 UT on 2007 March 16. The dimensions and orientation are the same as in Fig. 3.
The magnetic ﬁeld strength is obtained for pixels meeting a given threshold (see text).

Tsuneta et al., “The Magnetic Landscape of the Sun’s Polar Region” (2008)

423

Fig. 4.— Polar view in the continuum for latitudes above 80 , created from Fig. 1a. Colored contours indicate locations with average ﬁeld strength B f of 200 G. ( Theregion inside the contour has average ﬁeld strength larger than 200 G.) Red indicates regions where the local inclination i < 25 (vertical), while blue shows regions i > 65
( horizontal). East is to the left, and west is to the right. Latitudinal lines for 85 and 80 are shown, with the plus sign indicating the south pole. Near the extreme limb (to thebottom), spatial resolution is lost.

Fig. 5.— (a) Number of pixels as a function of magnetic ﬁeld strength (probability distribution function). Red lines indicate the vertical ﬁeld, blue the horizontal ﬁeld,and black the total. (b) Number of pixels times B2 as a function of magnetic ﬁeld strength. The bin size for the magnetic ﬁeld strength in (a) and (b) is 20 G. (c) Histograms ofcontinuum intensity for magnetic ﬁeld strengths above 300 G (solid lines) and 800 G (dashed lines). Since the continuum intensity rapidly decreases toward the limb, thehorizontal axis is the normalized excess continuum level with respect to the continuum averaged over a 6.400 box. (d) Filling factor as deﬁned in x 3.1. All panels are for
latitudes above 75 .

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MAGNETIC LANDSCAPE OF SUN’S POLAR REGION
the number of pixels multiplied by B2, as a function of ﬁeld strengthis shown in Figure 5b. This illustrates where the magnetic energy
is mainly located as a function of ﬁeld strength. The vertical ﬂux
tubes with higher ﬁeld strength are energetically dominant, while
weaker horizontal ﬂux tubes, by contrast, carry more energy.
3.3. Total Magnetic Flux in the Polar Region
The total vertical magnetic ﬂux in the SOT ﬁeld of view is
2.2 ; 1021 Mx, while the total horizontal ﬂux is 4.0 ; 1021 Mx.
The effective ﬁlling factor was taken into account in estimatingthese total ﬂuxes: the ﬂux of an individual pixel in Figure 3 is estimated as B f times the pixel size, with foreshortening correction.
Considering the stray-light contribution, the actual ﬁlling factor
may be larger than the nominal values derived with the leastsquares ﬁt. As an extreme case, we estimate the magnetic ﬂux
assuming f = 1: the total vertical magnetic ﬂux then becomes
9.9 ; 1021 Mx, and the total horizontal magnetic ﬂux becomes
2.0 ; 1022 Mx. The differences are factors of 0.22 and 0.2, whichroughly correspond to the average ﬁlling factor for the vertical andhorizontal ﬁelds, respectively. If the contribution from stray lightreaches 50% (Orozco Suárez et al. 2007b), the total vertical ﬂux
is estimated to be 7.2 ; 1021 Mx. Note that we have to be carefulin comparing the horizontal magnetic ﬂux with the vertical ﬂux
because of the different sensitivity in the degree of Stokes Q, U,
and V polarization to the transverse and line-of-sight magnetic
ﬁelds.
Since the inversion was performed for only the 10.5% of pixels
with high signal-to-noise ratio and the horizontal ﬁeld strength isgenerally smaller than the vertical, we also obtained the horizontal
magnetic ﬂux from the wavelength-integrated Stokes V signals
using a weak-ﬁeld approximation. This should be less sensitive
to error. We excluded pixels with ﬂux lower than 3  (7.2 Mx cm2),
performed foreshorteningpcorrectionin the pixel size, and added
ﬃﬃﬃ
a correction (a factor of 2) for the transverse horizontal ﬁelds
unseen in Stokes V. The total horizontal ﬂux thus obtained is4.35 ; 1021 Mx, which agrees well with the value above.
The total vertical magnetic ﬂux for the whole area at latitudes
above 70 is then estimated to be 5.6 ; 1021 Mx with the nominal ﬁlling factor considered and 2.5 ; 1022 Mx with f = 1, assuming that the unobserved polar region has the same magnetic
ﬂux as that observed with SOT. The estimate taking into accountthe effect of stray light is 1.8 ; 1022 Mx. Here we chose only theﬂux tubes vertical to the local surface, and the horizontal ﬂux is
not included. Since the surface area with latitude above 70 is
1.8 ; 1021 cm2, the average ﬂuxes are 3.1 G (with the nominal
ﬁlling factor f considered), 13.9 G (with f = 1), and 10.0 G (with
50% of stray light taken into account). Although these are themost accurate ﬂux estimates so far made for the polar regions,the numbers should be regarded as minimum values because of
the threshold in the selection of pixels for accurate inversion.
4. KILOGAUSS MAGNETIC PATCHESAND ACCELERATION OF THE FAST SOLAR WIND
4.1. Comparison between Photosphericand Interplanetary Magnetic Flux
The Hinode X-ray image taken on 2007 March 16 shows thatthe apparent polar coronal hole extends down to 60 Y70 in latitude. Thus, the entire region shown in Figure 3 is the photospheric base of the polar coronal hole. We next compare themagnetic ﬂux in the polar photosphere and that observed in interplanetary space during a different solar cycle. The mean magnetic ﬁeld strength observed by Ulysses in 1993Y1997 was 2.83 nT(2.83 ; 105 G) above 36 heliolatitude at 1 AU (McComas et al.

13792000), and the total magnetic ﬂux of the polar coronal hole isestimated to be 2 ; 1022 Mx, which is somewhat larger than the
total photospheric magnetic ﬂux obtained here with our effective
ﬁlling factor and close to the ﬂux with unity ﬁlling factor. We
consider these numbers to be consistent, since (1) the actual ﬁlling factor must be between 1 and the effective value derived
above, (2) there may be smaller undetected vertical ﬂux tubes,
as indicated by the presence of spicules, (3) the measurements
were made in different solar cycles, and (4) we obtained thephotospheric ﬂux at latitudes higher than 70 , whereas the polar
coronal hole extends below 70 .
4.2. Polar Flux Tubes and the Fast Solar Wind
The fast solar wind emanates from the polar regions ( Krieger
et al. 1973; Woch et al. 1997). The vertical ﬂux tubes should
undergo a large expansion between the photosphere and the lower
corona as a result of their high ﬁeld strength, the unipolarity, andtheir very limited number and size in the polar region. The total
area S of vertical ﬂux tubes with average ﬁeld strength B f > 200 G(Fig. 4, red contours) is 2.1 ; 1018 cm2, and the total surface
area of the photosphere corresponding to the SOT ﬁeld of view is
7.2 ; 10 20 cm 2. Thus, the areal expansion of individual ﬂux tubes
between the photosphere and the lower corona may reach a factorof 345.
The mean number density and velocity in the fast solar wind
as observed by Ulysses are 2.7 cm3 and 760 km s1, respectively, at 1 AU and heliolatitudes above 60 . These values exhibit little variation with heliolatitude (McComas et al. 2000).
On this basis, we estimate a total mass loss of 2.3 ; 108 kg s1
from one of the polar regions, assuming uniform-plasma parameters. The plasma density  at 5000 = 1 is 3 ; 107 g cm3, andthe upward speed associated with a fast solar wind is estimated to
be only 2 cm s1. The apparent Doppler velocity further decreases
if one takes into account the projection effect. Indeed, we do notsee any velocity feature at the locations of the vertical ﬂux tubes.
In a more mixed-polarity region, a larger fraction of the ﬁeld
lines will return at lower heights, allowing greater expansion for
those that are indeed open higher in the corona. But in the polar
regions, as soon as the vertical ﬁeld lines reach the chromosphereor the chromospheric-coronal boundary the ﬁelds will expand,since there is no obstacle to lateral expansion of the vertical ﬂux
tubes. Horizontal ﬁelds, although ubiquitous, would not reach thecorona. All the open ﬁeld lines forming the polar coronal holeessentially originate from such scattered, small, but intense magnetic patches, and the fast solar wind emanates from these vertical
ﬂux tubes seen in the photosphere (‘‘magnetic funnels’’; Tu et al.
2005).
Alfvén waves are believed to play a vital role in the acceleration and heating of the fast solar wind (Hollweg 1972; Suzuki& Inutsuka 2006). The Alfvén speed rapidly increases with height
as a result of the decrease in plasma density. A long-standing
problem is that Alfvén waves with wavelengths shorter than the
Alfvénic scale height tend to be reﬂected back (An et al. 1989;
Moore et al.1991). A rapid decrease in the magnetic ﬁeld strength
associated with rapidly expanding ﬂux tubes near the chromospheric boundary would reduce the vertical change in Alfvén speed,
resulting in a longer Alfvénic scale height. Therefore, the Alfvénic
cutoff frequency may be lower in the polar ﬂux tubes. We thus conjecture that Alfvén waves generated in the photosphere may be
more efﬁciently propagated to the corona through the fanningout ﬂux tubes. These ﬂux tubes may serve as the chimneys providing the entire coronal hole with Alfvén waves to accelerate
solar winds.

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TSUNETA ET AL.
5. DISCUSSION

We have discovered that the poloidal ﬁeld near the south solar
pole has the form of unipolar ﬂux tubes scattered throughout theregion rather than a weak extended ﬁeld. If a polar ﬁeld with thesame total magnetic ﬂux   BfS were uniformly distributed (Sbeing the total magnetic area), the estimated effective ﬁeld strength
would be about 10 G, as described above. The total magnetic
energy is then proportional to B2 fS = B. Thus, the surface poloidal magnetic energy is approximately 90 times larger than in
the case of a uniform magnetic ﬁeld if we take B  900 G, corresponding to the peak of the energy PDF in Figure 5b. The equipartition ﬁeld strength Be is that at which the magnetic energy
is equal to the kinetic energy of the surface granular motion:
Be = (4v 2 )1/2. The typical value of Be is about 400 G for granules with a velocity of v = 2 ; 105 cm s1 and the plasma density
 given in x 4.2. The magnetic ﬁeld strength for the majority ofthe patches is larger than the equipartition ﬁeld strength.
The observed unipolar, strong ﬂux tubes scattered about thepolar region are considered to represent seed poloidal ﬁelds fortoroidal ﬁelds ( Wang et al.1989a,1989b). Magnetic ﬂux is transported to the polar regions by meridional ﬂows and supergranular diffusion in the ﬂux-transport dynamo model ( Dikpati &Charbonneau 1999). Since the magnetic ﬁeld takes the form ofsuch isolated ﬂux tubes with super-equipartition strength insteadof the diffuse weak mean ﬁeld assumed in the ﬂux-transport dynamo (Dikpati & Charbonneau 1999), ﬂux transport on the Sun
would occur by means of an aerodynamic (drag) force against
the magnetic tension force and may be more difﬁcult than in themean ﬁeld case assumed in the models.
If the ﬂux tubes seen on the surface of the Sun are maintained
inside the Sun, this would affect a known difﬁculty in the mechanism (Elsasser 1956) to generate intense toroidal ﬁelds: asmaller ampliﬁcation factor is needed to generate the same toroidal ﬁeld from a poloidal ﬁeld with an intrinsic ﬁeld strength of
1 kG than from an average 10 G ﬁeld and thus may be achievable
within a solar cycle. We however recognize that there would
remain a serious energetics problem if the toroidal ﬁeld strength
indeed reached 100 kG (Schüssler 1996; Rempel 2006).
The total ﬂux of the vertical magnetic ﬁeld in the polar region
estimated here is at most 7.2 ; 1021 Mx at solar minimum, whilevarious measurements of the total magnetic ﬂux of single active
regions indicate 1022 Mx ( Longcope et al. 2007; Jeong & Chae
2007; Magara & Tsuneta 2008). Thus, the measured total polar
ﬂux scarcely corresponds to that of a single active region. The
total toroidal ﬂux would increase with time during the winding-up
process by virtue of differential rotation, and the concept of the-mechanism could be viable with the observations presented here.

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The transient horizontal magnetic ﬁeld discovered in the polar
region appears to have properties similar to those found in thequiet Sun and in active regions ( Lites et al. 2008; Centeno et al.
2007; Orozco Suárez et al. 2007a; Ishikawa et al. 2008; Ishikawa& Tsuneta 2009). In particular, the PDFs of the magnetic ﬁeld
strength for the polar region (Fig. 5a), quiet Sun, and active regions(Ishikawa & Tsuneta 2009) are remarkably similar, suggesting acommon local dynamo process (Cattaneo 1999) taking place allover the Sun.
The X-ray telescope and extreme-UV imaging spectrometer
aboard Hinode have observed remarkable activity in the polar
regions in the form of microﬂares and jets (Savcheva et al. 2007;
Cirtain et al. 2007). The lateral spreading of the vertical ﬂux
tubes to large areas may occur well above the formation heightof the two Fe lines, since there is no clear positional correlation
between the horizontal ﬁelds and the vertical ﬁelds, as seen in
Figure 4. These X-ray jets could be due to magnetic reconnection at the lateral magnetic contacts with the horizontal ﬁelds or
the transient emergence of separate bipolar ﬁeld lines (Shimojo
et al. 1998; Shibata et al. 1992).
In conclusion, the magnetic landscape of the polar region is characterized by vertical kilogauss patches with super-equipartition
ﬁeld strength, a coherence in polarity, lifetimes of 5Y15 hr, andubiquitous weaker transient horizontal ﬁelds. The lifetime of themagnetic concentrations in the quiet Sun has been estimated to
be 2500 s for 2.5 ; 1018 Mx, and 40 ks for 10 ; 1018 Mx, with
the SOHO Michelson Doppler Imager (Hagenaar et al. 1999). It
is important to clarify the similarities and differences between thepolar region and the quiet Sun with Hinode. We will discuss thisin a subsequent paper.

S. T. thanks T. Magara, M. Rempel, K. Fujiki, T. Rimmele, T. J.
Okamoto, N. Narukage, and R. Ishikawa for fruitful discussions.
The authors express sincere thanks to the ISAS/JAXA Solar-B
launch team, headed by Y. Morita, for their exceptional achievement. Hinode is a Japanese mission developed and launched by
ISAS/JAXA, collaborating with NAOJ as a domestic partner andNASA and STFC (UK) as international partners. This work was
carried out at the NAOJ Hinode Science Center, which is supported by the Grant-in-Aid for Creative Scientiﬁc Research ‘‘The
Basic Study of Space Weather Prediction’’ from MEXT, Japan(Head Investigator: K. Shibata), generous donations from Sun
Microsystems, and NAOJ internal funding. The SOT Focal Plane
Package of the Lockheed Martin Solar and Astrophysics Laboratory and the High Altitude Observatory was supported by NASA
contract NNM07AA01C.

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