A–1

付録 A

Space Weather Modeling: Status andProspects (Journal of GeophysicalResearch, Vol. 113, No. A3, 2008)“Space Weather Modeling: Status and Prospects” (Journal of Geophysical Research, Vol. 113, No. A3,
2008) は 2006 年 11 月 14 日–17 日に地球シミュレータセンター（横浜市）で開催された「CAWSESInternational Workshop on Space Weather Modeling」（本報告書 83 ページを参照）の集録として出版されたものである。

掲載論文• Introduction to special section on Space weather modeling: Status and prospects,
Kamide, Y. (2008),
J. Geophys. Res., 113, A03S01, doi:10.1029/2007JA012972• Magnetic flux emergence in the Sun,
Archontis, V. (2008),
J. Geophys. Res., 113, A03S04, doi:10.1029/2007JA012422• Radiation Belt Environment model: Application to space weather nowcasting,Fok, M., R. B. Horne, N. P. Meredith, and S. A. Glauert (2008),
J. Geophys. Res., 113, A03S08, doi:10.1029/2007JA012558• Nonlinear force-free coronal magnetic field extrapolation scheme based on the direct boundary
integral formulation,
He, H., and H. Wang (2008),
J. Geophys. Res., 113, A05S90, doi:10.1029/2007JA012441• Geomagnetically induced currents during intense storms driven by coronal mass ejections andcorotating interacting regions,Kataoka, R., and A. Pulkkinen (2008),
J. Geophys. Res., 113, A03S12, doi:10.1029/2007JA012487• Properties of AE indices derived from real-time global simulation and their implications for solarA–2
wind-magnetosphere coupling,
Kitamura, K., H. Shimazu, S. Fujita, S. Watari, M. Kunitake, H. Shinagawa, and T. Tanaka (2008),
J. Geophys. Res., 113, A03S10, doi:10.1029/2007JA012514• Numerical studies on three-dimensional earthward fast plasma flows in the near-Earth plasma
sheet by the spontaneous fast reconnection model,
Kondoh, K., and M. Ugai (2008),
J. Geophys. Res., 113, A03S07, doi:10.1029/2007JA012707• Flux enhancement of the outer radiation belt electrons after the arrival of stream interaction
regions,
Miyoshi, Y., and R. Kataoka (2008),
J. Geophys. Res., 113, A03S09, doi:10.1029/2007JA012506• A quantitative MHD study of the relation among arcade shearing, flux rope formation, anderuption due to the tearing instability,
Shiota, D., K. Kusano, T. Miyoshi, N. Nishikawa, and K. Shibata (2008),
J. Geophys. Res., 113, A03S05, doi:10.1029/2007JA012516• Parametric decay of circularly polarized Alfvén waves in the radially expanding solar wind,
Tanaka, S., T. Ogino, and T. Umeda (2007),
J. Geophys. Res., 112, A10110, doi:10.1029/2007JA012513• Nonlinear force-free modeling of the solar coronal magnetic field,
Wiegelmann, T. (2008),
J. Geophys. Res., 113, A03S02, doi:10.1029/2007JA012432• Key points model for polar region currents,
Xu, W., G. Chen, A. Du, Y. Wu, B. Chen, and X. Liu (2008),
J. Geophys. Res., 113, A03S11, doi:10.1029/2007JA012588• Some ubiquitous features of the mesospheric Fe and Na layer borders from simultaneous andcommon-volume Fe and Na lidar observations,
Yi, F., S. Zhang, X. Yue, Y. He, C. Yu, C. Huang, and W. Li (2008),
J. Geophys. Res., 113, A04S91, doi:10.1029/2007JA012632

ClickHere

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S01, doi:10.1029/2007JA012972, 2008

for

Full
ArticleIntroduction to special section on Space weather modeling: Status andprospects
Y. Kamide1
Received 4 December 2007; revised 29 December 2007; accepted 1 January 2008; published 5 February 2008.

Citation: Kamide, Y. (2008), Introduction to special section on Space weather modeling: Status and prospects, J. Geophys. Res., 113,
A03S01, doi:10.1029/2007JA012972.

[1] The importance of specifying and predicting the stateof the Earth’s space environment, i.e., space weather, hasrecently been increasingly realized, as dynamic activities of
the Sun, solar wind, magnetosphere, and ionosphere can
influence modern technology systems and even endanger
human life. It is also widely recognized that numericalsimulations driven by integrated ground-based and satellite
observations are a powerful tool for quantitatively understanding the complex Sun-Earth system, and they are a vitalmeans for predicting space weather.
[2] It is required, however, that a physical model of solarterrestrial phenomena be constructed as a basis of space
weather predictions. The model must deal with the key
processes spanning from flares and coronal mass ejections
on the Sun to geomagnetic storms and their effects in theupper atmosphere of Earth. To reach the goal effectively, it
is acute to ask a number of fundamental questions such as,
What is the triggering condition of solar flares and coronalmass ejections (CMEs)? What mechanism generates fastreconnection and leads to particle acceleration? Why particular solar flares cause superstorms on Earth? Working out
these basic questions is probably equivalent to understand-ing multiscale coupling of various physical processes occurring in the Sun-Earth system on different scales. It is
indispensable that the communications among different
fields, especially between the solar and geophysical communities, be enhanced more systematically.
[ 3 ] This special section results primarily from the
CAWSES International Workshop on Space Weather Modeling (CSWM), which was held on 14– 17 November 2006
in Yokohama, Japan, but includes several papers that were
not presented at the Workshop. It is hoped that this section
will provide a timely forum for discussing all aspects ofspace weather modeling research, aiming to summarize thecurrent status and future prospects of this rapidly growing
field. It contains original research papers as well as topicalreview papers on the advanced modeling of solar andheliospheric magnetic field; solar flares, CMEs, and filament eruption; solar wind; geomagnetic storms; and radiation belts.


Y. Kamide, Research Institute for Sustainable Humanosphere, Kyoto
University, Uji 611-0011, Japan. (kamide@rish.kyoto-u.ac.jp)

1
Research Institute for Sustainable Humanosphere, Kyoto University,
Uji, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012972$09.00

A03S01

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S04, doi:10.1029/2007JA012422, 2008

for

Full
ArticleMagnetic flux emergence in the Sun
V. Archontis1
Received 26 March 2007; revised 25 May 2007; accepted 12 July 2007; published 12 January 2008.

[1] Space weather research is closely connected with the study of the solar magnetic
activity. In past years, many solar missions (e.g., YOHKOH, SOHO, TRACE, and
RHESSI) have provided outstanding observations, which have been used to improve ourunderstanding of the structure and the dynamical evolution of solar magnetic fields. In
addition, the newly launched solar missions (e.g., Hinode and STEREO) will study theinteraction between the emerging magnetic field and the preexisting field in the corona(increasing our understanding of the causes of solar variability), and they will also observe
the three-dimensional evolution of solar eruptions as they leave the Sun and move into theinterplanetary space. One of the most important processes, responsible for many
dynamical phenomena observed in the Sun, is the emergence of magnetic flux from thesolar interior in active regions and the modification of the coronal magnetic field inresponse to the emergence. In fact, magnetic flux emergence might be responsible for theappearance of small-scale events (e.g., compact flares, plasmoids, and active-regionassociated X-ray brightenings) and large-scale events (e.g., X-class flares and CMEs),
which are major drivers of space weather. However, it is clear that the question of how
exactly the magnetic fields rise through the convection zone of the Sun and emerge
through the photosphere and chromosphere into the corona has still not been solved. It isbelieved that understanding the process of flux emergence is an important step toward theunderstanding of the initiation mechanism of eruptive events in the Sun, which isanother topic of great debate. This paper provides a brief review of the theory and thenumerical models, which have been used to study the process of magnetic flux emergence
into the outer atmosphere of the Sun. We underline the similarities and differences
between these models, and we compare the basic features of the numerical results withobservations. Finally, we review the recent progress and discuss what further
developments are required in the models to best describe the essential physics in theprocess of flux emergence.
Citation: Archontis, V. (2008), Magnetic flux emergence in the Sun, J. Geophys. Res., 113, A03S04, doi:10.1029/2007JA012422.

1. Introduction[2] Most of the solar activity is directly linked to theSun’s magnetic field. Indeed, observations have shown that
the Sun’s corona is highly structured, being threaded by acomplex network of magnetic fields. Understanding thestructure and the dynamic evolution of these fields isimportant because they are the building blocks of solar
activity. It is believed that solar eruptive phenomena result
from rapid changes in their structures and connections.
[3] The current state of the theory of the formation of theSun’s magnetic field suggests that it is produced by dynamo
action in the tachocline, an interface layer separating theconvection zone from the radiative zone. Then the dynamogenerated magnetic field is transported from the deep
interior of the Sun to the surface by magnetic buoyancy[e.g., Parker, 1955], which may well be coupled with the1

School of Mathematics and Statistics, University of St. Andrews, Fife,UK.
Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012422$09.00

convective motions [Parker, 1988]. Eventually, the buoyant
magnetic fields rise through the convection zone, intersect
the photosphere and create the observed sunspots andbipolar active regions [e.g., Zwaan, 1987]. The newly
emerged bipolar active regions are called emerging flux
regions (EFRs) [e.g., Zirin, 1970]. An extended review on
the structure and dynamics of magnetic fields in the solar
convection zone is given by Fan [2004]. The properties ofactive regions in terms of the dynamics of magnetic flux
tubes which emerge from the solar interior to the photosphere have been reviewed by Fisher et al. [2000].
[4] Shortly after the appearance of flux at the photosphere, a system of bright loops appears in the EUV and
X-ray detectors. Observations of large-scale magnetic fields
emerging into the corona have been identified by solar
satellites in the 1990s (e.g., TRACE and SOHO). The
coronal fields are normally outlined by plasma emitting in
EUV or X rays. Figure 1 is a high-resolution image of the
Sun taken by the X-ray telescope on Hinode satellite. This
image shows the detailed structure of active region loops
and X-ray bright points, which are seen as concentrations of

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Figure 1. First-light image from the X-ray telescope on
board the Hinode satellite. Credit: Japanese Aerospace
Exploration Agency (JAXA).

magnetic loops. The appearance of many brightenings
indicate that solar activity occurs all over the Sun.
[5] However, it is exceedingly difficult to measure coronal magnetic fields directly and, thus, we usually use theline-of-sight component of the magnetic field and vector
magnetograms from photospheric measurements, to extrapolate the magnetic field in the corona. For the photosphere,
direct measurements of the magnetic field, both by satellite
instruments (like MDI) as well as ground-based detectors,
yield a wealth of information. Vector magnetograms, inparticular, permit the reconstruction of all three componentsof the magnetic field.
[6] Schrijver et al. [2006] evaluated the performance of aseries of numerical models, which simulate nonlinear forcefree (NLFF) magnetic fields in the solar corona. The
numerical experiments were performed using vector magnetic field measurements of active regions. It was found thatthe solution depends strongly on the implementation of theboundary conditions. The ultimate goal of this comparison
between the numerical models is to find a robust method of
measuring coronal free energy in solar active regions.
[7] Another approach, which has been used to computethe large-scale magnetic field configuration of the solar
corona, is the potential field source surface (PFSS) models.
These models are simple to develop and implement andthey can also resolve larger-scale structures than the current
MHD models. On the other hand, they use idealized initialconditions and assumptions and they don’t treat important
processes (such as reconnection) properly. Riley et al.
[2000] showed that PFSS models are useful tools forcomputing the large-scale coronal field when timedependent changes in the photospheric flux can beneglected. Also, they found that PFSS models produce

A03S04similar results with the MHD models for configurations
based on untwisted coronal fields.
[8] Observations show that most of the coronal loops
seem to join the opposite polarities of the new active
regions, while others establish linkages to new active
regions, probably through reconnection of magnetic fieldlines. Space observations from YOHKOH, SOHO andTRACE have revealed evidence of magnetic reconnection[Martens, 2003] in emerging flux regions, in flares and it isbelieved that reconnection could power solar explosions
called Coronal Mass Ejections (CMEs). Antiochos et al.
[1999] presented the first model (known as breakout
model), which was consistent with many observational
properties and the energy requirements for CMEs. In thismodel, CMEs are triggered owing to reconnection between
a sheared arcade and neighboring flux systems.
[9] Eruptive phenomena, such as CMEs, flares andprominences can lead to solar and geomagnetic disturbances[Taylor et al., 1997] and disrupt terrestrial satellites andpower systems. Recent models have increased progress on
space weather prediction. The warning of an explosive
event from space weather on terrestrial and space systems
can be provided (a few hours ahead) on the basis ofobservations of such events from spacecrafts. The exact
impact, however, of the potential threat in the Sun-Earthenvironment requires further development of space weathermodels [Brun, 2007; Cole, 2003] and remains a greatchallenge.
[10] X-ray images from YOHKOH and vector magnetograms from ground-based observatories have shown the
topology of growing bipoles in areas of new magnetic
flux. The analysis of the proper motions of these bipoles
indicates that the flux bundles that they make up the
bipoles are twisted before they emerge [Leka et al.,1996]. Measurements of the twist of the emerging flux
systems have shown that the change of the twist is rather
small during the emergence [Wang and Abramenko,
1999]. In addition, photospheric measurements of thevector magnetic field have shown that the mean twist
in active regions is right-handed in the southern hemisphere and left-handed in the northern hemisphere[Pevtsov et al., 2001]. Soft X-ray data from observationsof active regions have repeatedly shown coronal loops
with a bright forward or an inverse S shape. The
structures with the forward S shape appear mostly inthe southern hemisphere while the inverse S shapestructures usually appear in the northern hemisphere.
Their shape appears to be helical because their magnetic
fields are twisted. Their central part is approximatelyaligned with the neutral line of the normal componentof the magnetic field in the photosphere. Filaments, witha forward or an inverse S shape, may also be visible
along the neutral line in sigmoid active regions [Rust andKumar, 1994].
[11] The above mentioned structures are called sigmoids.
Using YOHKOH solar X-ray images, it has been shown that
there is a relationship between structures with a sigmoidal
shape and eruptions of flux ropes in the corona [Sterling,
2000; Pevtsov, 2002]. For example, CMEs appear to originate preferentially in regions of the Sun’s corona thatexhibit sigmoidal structures [Canfield et al., 1999, 2000].
Although sigmoids play an important role in solar activity

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and their evolution is closely connected to space weather
forecasting [Sterling, 2000; Pevtsov, 2002], the origin andthe lifetime of sigmoids are still unknown.
[12] Another interesting issue, closely connected withthe study of flux emergence, is the actual structure of thesolar magnetic field. Observations have shown that magnetic fields on the photosphere are intermittent andmagnetic flux is predominantly concentrated in discrete
areas with kilogauss field strengths [Zwaan, 1987; Keller,
1992; Hagenaar, 2001; Socas Navarro and Sanchez
Almeida, 2003]. Resolution is a crucial factor on how
one interprets observations since the size of these areaschanges from sunspots down to very small scales, which
are difficult to resolve with the available observational
facilities. In the past few years, many observations haveshown that the observed field adopts the form of roundish
and discrete flux tubes while recent high-resolution
observations (i.e., with the Swedish Solar Telescope on
La Palma) show that the observed field has an intricate
topology. Theoretical arguments support also the intermittent morphology of the photospheric and the subsurface
magnetic fields, which are believed to be concentrated
into discrete flux tubes. Numerical MHD simulations ofmagnetoconvection [Galloway and Weiss, 1981; Nordlund
et al., 1992] have shown that magnetic flux is concentrated in the intergranular lanes by convective motions.
Eventually, the magnetic flux adopts the form of flux
tubes, which are intensified because of stretching andtwisting of their magnetic fieldlines by turbulent fluid
motions. It is worthwhile to mention that, magnetoconvection simulations on granule and mesogranule scales in
the upper part of the convection zone [Steiner et al.,
1998; Weiss et al., 2002; Stein and Nordlund, 2006] haveshown that the magnetic field lines of the flux tubes atthe surface of the Sun are connected to various regimes,
with a complex topology, below the surface and, thus, theconcept of discrete flux tubes for the weak magnetic
fields in the quiet Sun is not clear.
[13] Although it appears that we do not have yet acomplete picture of the topology of the magnetic field onor below the photosphere, the concept of isolated magnetic flux tubes has been used extensively over the past
decade or so in numerical experiments of flux emergencefrom the solar interior into the solar atmosphere. Most of
the numerical models use, as initial configuration, atwisted flux tube or a flux sheet below the photosphere.
Eventually, the initial flux system becomes unstable toperturbations or instabilities (i.e., the classical Parker
buoyancy instability [Parker, 1978]) and makes its way
up through the solar interior developing an W-loop shape.
As the buoyant flux system rises, the top of the W-loop
structure intersects the photosphere and creates sunspots
in bipolar regions. Finally, it emerges through the photosphere and chromosphere and expands into the corona.
However, even in the simplest configuration, the above
phenomenon is highly time-dependent, has a complex
three-dimensional geometry and the timescales of thevarious processes involved are remarkably different in
the subphotospheric layers and in the upper atmosphereof the Sun. Hence numerical experiments are necessary toprovide a first physical understanding of the flux emer-

A03S04gence process. This paper provides a review of the resultsof these numerical experiments.

2. Emergence Into a Field-Free Corona[14] The emergence of buoyant magnetic flux systems
from the convectively unstable solar interior into the higher
levels of the atmosphere is still a largely unexploredresearch domain and has been a subject of vigorous research
for the past three decades. In fact, the evolution of the risingflux systems occurs on the basis of the buoyancy instability
experienced by the plasma above the photosphere. There
has been much interest on the literature in the buoyant
instabilities in magnetized and in nonmagnetized plasmas(see the review by Hughes and Proctor [1998]). In thefollowing, we first report on magnetic buoyancy instabilities
in the framework of numerical MHD experiments of flux
emergence, then on the issue of twisted flux tubes andfinally on the dynamics of flux emergence.
2.1. Buoyant Instabilities[15] A first series of numerical experiments were twodimensional (2-D) and explored the excitation of magnetic
instabilities of a single flux system (tube or sheet) and its
subsequent emergence into a nonmagnetized corona.
Shibata et al. [1989a, 1989b] performed 2-D magnetohydrodynamic (MHD) experiments to study the nonlinear
evolution of the Parker instability, which is a kind of ideal
MHD instability driven by magnetic buoyancy. More precisely, they studied the undular mode k k B of the magnetic
buoyancy instability, where k and B are the wave number
and the initial magnetic field vector, in an isolated horizontal flux sheet. The background stratified atmosphere in theseexperiments consisted of two unmagnetized isothermal
layers as an simplified version of the Sun’s photosphere/
chromosphere and the ambient corona with a higher temperature. Also, small velocity perturbations were initially
imposed on the magnetic flux sheet to initiate the instability.
They found that as soon as the instability develops the flux
sheet rises as a result of enhanced magnetic buoyancy andeventually expands into the corona. The acceleration of therising loop shows a self-similar behavior in the low atmosphere. This self-similar solution reveals that the magnetic
loop is accelerated by the magnetic pressure gradient force,
which dominates the gravitational and the gas pressure
gradient force. The results of these simulations were consistent with the observed small rise velocity of magnetic
flux at the photospheric heights (v  1 Km sec1) and withstrong downdrafts (v  1 – 3 km sec1) at the footpoints ofthe expanding magnetic loop, which may correspond toobserved strong downdrafts near pores.
[16] Simulations by Kaisig et al. [1990] included aconvectively unstable layer below the photosphere. Thenvertical velocity fluctuations in the convection zone andhorizontal shear flows at photospheric heights were considered to study the evolution of the undular mode of magnetic
buoyancy of a horizontal flux sheet. The results indicatedthat the imposed velocity fluctuations can destabilize theinitial flux sheet, generating an upward-expanding magnetic
loop, as long as it is located within or just above theconvection zone but not if it is originally embedded in thehigher atmosphere. They also calculated eigenfunctions for

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the linear and nonlinear stability problem associated withthe particular initial condition that they were using.
[17] Kusano et al. [1998] investigated the Parker mode(k k B) and the interchange mode (k ? B) of magnetic
buoyancy instabilities with the aim to understand theemerging mechanism of magnetic loops in the solar corona.
They performed nonlinear, 2-D simulations in a weak twotemperature atmosphere, which consisted of the chromosphere and corona. A sheared magnetic flux was initiallyembedded in the bottom of the chromosphere and random
velocity perturbations were added into the equilibrium state
to initiate the experiment. They found that if the field issheared, a new instability could occur through the nonlinear
process of interchange instability, which leads to the formation of magnetic loops with a mushroom-like structure.
Current sheets are formed in the central lower part of themagnetic loops and reconnection occurs. As a result ofreconnection, magnetic bubbles are generated and eventually are detached from the original flux sheet and shortly
after they rise into the upper atmosphere.
[18] Matsumoto et al. [1993] used the same backgroundstratification and similar initial conditions with the previous
2-D numerical models, to study the nonlinear evolution of
EFRs [e.g., Zirin, 1970]. On the other hand, they considered
two different types of unperturbed magnetic flux system inthe lower atmosphere: a horizontal magnetic flux sheet anda horizontal magnetic flux tube. They performed 3-D MHD
simulations and they found that the expansion laws, derived
in the previous 2-D models, are modified because in three
dimensions the expansion of the rising magnetized volume
occurs also perpendicular to the fieldlines in two directions.
They also found that the evolution of the EFR depends on
the initial structure of the magnetic flux system. As an
example, the rise velocity of the expanding magnetic loops
obtained in these simulations was comparable to the observed rise velocity of arch filaments (10– 15 Km s1;[e.g., Chou and Zirin, 1988]) when the initial flux system isa flux sheet. On the other hand, the rise speed is too small
when a magnetic flux tube is considered as the unperturbed
magnetic flux system at the beginning of the simulation.
Another interesting feature presented in this paper is theevolution of an initial flux system, which consists of anumber of isolated flux tubes. In this model the flux tubes
interact with each other as they rise and they finally merge
into a bundle of expanding magnetic loops. Dense filaments
are formed in between the expanding magnetic loops, as
the plasma slides down along the outermost expanding
fieldlines, with a width about 800 Km and a rise speed
10 km s1.
[19] Finally, Archontis et al. [2004] showed that overdense flux can be transported into the atmosphere when thefollowing criterion is satisfied [Newcomb, 1961; Acheson,1979]:
!
~k 2
@
g
2
?
~
Hp ðlogBÞ >  bd þ k k 1 þ
:
~k 2
@z
2
z

ð1Þ

[20] In the above criterion, Hp is the pressure scale height,
z is the height, B is the magnetic field strength, g is the ratioof specific heats and the plasma-b is the ratio of the gaspressure over the magnetic pressure. There are also pertur-

A03S04k ? are thebations with wave vector k (where ~k k and ~horizontal components parallel and perpendicular to themagnetic field and ~k z is the vertical component). The
superadiabatic excess, d is given by d = r  rad, where r
is the actual logarithmic temperature gradient in theequilibrium stratification and rad is its adiabatic value.
[21] A crucial term in the above criterion is the bd term.
For an isothermal layer d = 0.4. Plasma-b, on the other
hand, becomes small as the magnetic pressure becomeslarger than the gas pressure when the uppermost layers of
the tube cross the photosphere. Thus the right-hand-side
term in the above criterion becomes smaller than the lefthand-side term and the instability is launched, carrying themagnetized plasma all the way up to the corona.
2.2. Twisted Flux Tubes[22] Observationally, there is evidence that the emerging
flux bundles, which rise through the solar interior and create
active regions at the photosphere, are twisted and have a
coherent configuration during their rise (see section 1). Alsoon theoretical grounds, numerical models in two dimensions
have shown that an initial twist is required for a tube toretain its coherent structure as it rises through the convection zone. A nontwisted tube splits into a pair of vortex
filaments rotating in opposite directions [e.g., Schuessler,
1979; Longcope et al., 1996]. The vortex filaments separate
horizontally from each other, owing to the buoyancy force
on the mass elements of the vortex filaments, and eventually
the rising motion of the buoyant tube turns into a horizontal
expanding motion. On the other hand, if the flux tube istwisted by a sufficient amount then the magnetic tension of
the twisted fieldlines can prevent the formation of vortex
filaments and the tube rises as a rigid body through theconvective envelope [e.g., Moreno-Insertis and Emonet,
1996; Emonet and Moreno-Insertis, 1998].
[23] The pitch angle (Y) of the twisted fieldlines around a
horizontal magnetic flux tube has to be above a threshold,for the transverse field to be dynamically important. If theflux tube is, both in pressure balance and thermal equilibrium with its surroundings, it will be less dense than theexternal plasma (and therefore will rise) by a value ofDr ðr  re Þ
1
¼
 ;
r
r
b

ð2Þ

where r is the density inside the tube and re is the density ofthe background atmosphere.
[24] The pitch angle of the twisted field lines of such atube has a threshold of ordertanY 

R
Hp

1=2
;

ð3Þ

where R is the radius of the tube and Hp is the local pressure
scale height.
[25] All the above models were two-dimensional. Dorch
and Nordlund [1998] performed 3-D simulations of buoyant
magnetic flux tubes ascending through a solar convection
zone model, and they showed that a weak random ortwisting component is sufficient to make the tube rise as acoherent structure. If the initial flux tube has a nontrivial

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Figure 2. (top) Motion of the top of the flux tube in time
for different field strengths, B0, and fixed twist, a = 0.4.
(bottom) Fixed initial field strength and different a. For
details, see Murray et al. [2006].
topology, the flux structure is held together for a long timeand is able to keep most of its buoyancy. Abbett et al. [2000]
found that the break-up of the flux tubes depends on thethree-dimensional geometry of the problem. If the curvatureof the upper part (apex) of an W-loop is large, the degree offragmentation of the loop as it reaches photospheric heights
is small. Also, Abbett et al. [2001] described how a buoyant
flux tube keeps its coherent structure under the action of the
Coriolis force. They also found that in the absence of forces
due to convective motions, a magnetic flux tube with strong
initial axial field strength will not be able to retain its
cohesion. Finally, Fan [2001] performed 3-D simulationsof arched flux tubes, which were formed from a horizontal
magnetic layer due to the nonlinear growth of the undular
instability. It was found that the rising arching tubes maintained their coherent structure as they moved through asignificant distance inside the computational volume. Thus,
although there was no net twist in the tubes, they emerged
through the convection zone without significant disruptionof their shape.
2.3. Dynamics of Flux Emergence[26] The dynamical emergence of magnetic flux from thesolar interior to the solar atmosphere is a big challenge in

A03S04numerical simulations. Many numerical experiments haveappeared in the literature during the past few years, whichyield insights into the dynamics of flux emergence and the
topology of the resulting structures. Most of the numericalmodels have used twisted flux tubes in the lower atmosphere as an initial unperturbed configuration for the magnetic field.
[27] Magara [2001] investigated the emergence and expansion processes of a twisted flux tube by means of 2.5D
MHD simulations. A highly stratified atmosphere was used,
including a layer with increasing temperature with depth forthe solar interior, an isothermal layer for the photosphere, atransition region and a high-temperature isothermal corona.
Initially, the tube rises through the convection zone bymagnetic buoyancy until it reaches the photosphere. However, the photosphere is strongly subadiabatic and, thus, theupper part of the tube slows down when it enters the low
atmosphere. The tube is flattened and a contact surface isformed in between the rising tube and the lower atmosphere. Eventually, the Rayleigh-Taylor instability acts on
this surface because dense plasma above the surface islocated on top of lighter plasma, which is more magnetized.
The tube emerges through the photosphere by the RayleighTaylor instability and starts to expand owing to the reduction of the background atmospheric gas pressure with heightand because the magnetic pressure of the emerging tube
becomes larger than the outside gas pressure. However,
although the tube expands into the outer atmosphere themain axis of the tube stays at the base or just below thephotosphere.
[28] Recently, 3-D MHD simulations have been carriedout by Murray et al. [2006] with the aim to understand therole of twist and magnetic field strength in shaping the
emergence process. It was found that when the value ofthe initial field strength, B0, and twist, a, is low the tube
cannot fully emerge into the corona, but it stays in the lower
atmosphere because the buoyancy instability criterion inequation (1) cannot be fulfilled. When the twist is fixed andthe field strength is varied the tube experiences different
magnitude of the buoyancy force, which is proportional to
B20. Thus, when the apex of the tube reaches the photosphere, it starts to rise into the upper atmosphere at different
times (see Figure 2, top). When the field strength is fixed
and the amount of twist is modified the tube rises withdifferent configuration for each a (see Figure 2, bottom).
More precisely, if a = 0.1 the tube flattens out at photospheric heights. For a = 0.2 the tube emerges at two side
locations because the draining of plasma from the upper partof the tube is more efficient at these locations. For larger
values of twist the tube rises and expands into the corona
adopting a dome-like structure. In addition, it has beenfound [e.g., Magara, 2007] that the expansion of an initiallyhigh twisted flux tube below the photosphere produces
coronal loops with sigmoidal structure while a tube with aweak twist produces expanded coronal loops with no clear
signature of sigmoidal structures.
[29] Fan [2001] used a density deficit profile to initiate
the rise of a flux tube from below the photosphere into thesolar atmosphere. The tube is more buoyant at the middle
than at the ends and evolves into an W-shaped loop as it
rises and expands in the atmosphere. At the beginning of theemergence a bipolar region is formed with a north-south

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Figure 3. (top) Outermost expanding fieldlines, which adopt a fan-like shape in the corona. (bottom)
Fieldlines at lower heights inside the rising flux system, which are less twisted. The innermost fieldlines
have an S-shape configuration.
orientation but later on the sunspots move toward an eastwest orientation, parallel to the axis of the emerging tube. In
fact, there is a shear horizontal velocity flow on the twosides of the neutral line, which agrees well with theobservations of emerging active regions by Strous et al.
[1996] and was diagnosed as a shear Alfvén wave in thesimulations by Manchester [2001].
[30] The expansion of an emerging twisted flux tube into
the higher levels of the atmosphere occurs in a runaway
fashion. This is because the distribution of the magnetic
pressure with height above the photosphere in the expanding volume of magnetized plasma is much larger than thegas pressure. The expansion takes place in all three directions: in fact, it expands faster in the horizontal directions
than in the vertical directions [e.g., Archontis et al., 2004].
Because of mass conservation, the fast rise and expansion of
the upcoming field cause strong downflows along theperiphery of the emerging fieldlines [e.g., Fan, 2001;
Magara and Longcope, 2003; Archontis et al., 2004].
[31] The outermost fieldlines of the rising flux system
adopts a fan-like shape as it expands into the higher levels of

the atmosphere [e.g., Fan, 2001; Magara, 2001; Archontis
et al., 2004]. The inner fieldlines do not follow the same
dynamical evolution and adopt shapes, which could appear
as sigmoids in the solar corona [e.g., Magara, 2004].
Figure 3 shows the shape of different sets of fieldlines
in a flux emergence experiment by Archontis et al. [2004].
It is worthwhile mentioning that, in most of the abovereferenced models, the fieldline on the main axis of theinitial flux tube rises, but very slowly and thus does notreally emerge into the corona. This is, probably, because
the curvature of this fieldline is very small and the dense
plasma does not drain sufficiently to make the buoyancy
effects to lift up the axis of the tube. However, fieldlinesthat are initially located just above the main axis of the
tube can emerge into the photosphere because their shapeis such that enables the magnetic pressure force to lift
them upward. In general, the fieldlines that emerge may beclassified into two categories: expanding fieldlines andundulating fieldlines [e.g., Magara and Longcope, 2003].
Also, the outermost fieldlines become more twisted, than
the inner fieldlines, as they expand and drive horizontal

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shearing motions in the low atmosphere that contribute to
the ejection of magnetic energy and helicity to the outer
atmosphere [e.g., Magara and Longcope, 2003].
[32] Abbett and Fisher [2003] coupled a subsurface
model of an emerging flux tube to a three dimensionalmodel corona. They first modeled the rise of a buoyant
magnetic flux system in a stable stratified convection zone.
Then, they used the time-dependent vector fields and scalarvariables in the upper subphotospheric boundary to drive a
3-D model corona. The simulations showed that the timedependent flows below the surface play a crucial role on the
dynamic evolution and subsequent morphology of an
emerging magnetic structure. At the beginning of thesimulation, the magnetic field that surrounds the emerging
system is not force-free. As the emergence proceeds and thevector fields at the photosphere evolve, the overlying
magnetic field relaxes to a more force-free state. The
development of models where high-resolution magnetograms of active regions will be used to drive dynamic
models of the solar corona is an interesting future task.
[33] Most of the MHD models described above do notinclude a realistic convection zone below the photosphere.
A first attempt to include the cell-like convection structure
in a flux emergence model was reported by Amari et al.
[2005]. In this model, a twisted flux tube is kinematicallyraised by a convection cell in the convection zone andevolves in a full MHD way after its emergence into thecorona. A photospheric layer is located in between theconvection zone and the corona. The effective resistivity
in this layer is larger than in the other two layers (resistivity
layer model (RLM)). It is shown that electric current andmagnetic flux emerge through the resistive photosphericlayer into the corona. The emergence of flux leads to an
arcade-like configuration at the beginning of simulation,
while later on the emerging flux evolves more rapidly
adopting the shape of a flux rope.
[34] Also, Cheung et al. [2007] modeled magnetic flux
emergence in granular convection performing radiative
MHD simulations. They showed that convection influences
the evolution of the emerging fields before and after theyreach photospheric heights. Subphotospheric upflows can
support the rising motion of some segments of the emerging
tube while downflows may suppress the emergence of othersegments. It was also found that flux tubes with small
longitudinal flux, are not highly buoyant and cannot risecoherently against the convective plasma motions. On theother hand, big flux tubes with large axial flux are able torise and eventually emerge at the surface disturbing the
granulation pattern owing to their dynamical horizontal
expansion.
[35] Leake and Arber [2006] performed 2.5D MHD
simulations to simulate the emergence of a twisted flux
tube into the solar atmosphere taking into account two
processes, which were not included in previous models:the partial ionization of certain regions of the solar atmosphere and the thermal conduction as a heat transfer
mechanism. It was found that when the dense plasma rises
and expands into the atmosphere is heated to its original
photospheric temperature rather than being cooled (adiabatic expansion). The inclusion of partially ionized plasma inthe chromosphere yields more rapid emergence and expansion of the rising field and a greater amount of flux into the

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corona. An important question then is if the resulting
coronal magnetic field is force-free. This is equivalent to
say that the current is parallel to the magnetic field and there
are no cross-field currents. It is found that when themagnetic field emerges through a partially ionized plasma,
the majority of the cross-field current is destroyed, and thus
the coronal magnetic field becomes force-free.
[36] Isobe et al. [2007] used 2-D simulations to study the
emergence of a very long flux sheet from a superadiabatically stratified layer that represents the convection zone into
the isothermal corona. A random velocity perturbation is
given in the initial flux sheet to excite the Parker buoyancy
instability. A number of loops are formed out of the current
sheet. They rise, expand and eventually reconnect at thelower atmosphere. Temperature in the reconnection outflows is enhanced compared to the background plasma. This
may account for the manifestation of Ellerman bombs.
During the rise of the loops, dense material is accumulated
at the valleys between neighboring loops. This dense
plasma is squeezed because of expansion of the rising loops
and creates elongated structures of cold and dense plasma,
which eventually is ejected to the outer atmosphere becauseof reconnection between the loops. At the end of theexperiment a very large loop of magnetic fieldlines isformed through successive reconnection of side-by-side
emerging loops at different heights of the atmosphere (seeFigure 4). This process is similar to those in the resistive
emergence model proposed by Pariat et al. [2004].

3. Emergence Into a Magnetized Corona[37] Detailed observations of magnetic fields in andaround active regions have shown that the emergence of
new magnetic flux causes noticeable changes in the topology and geometry of the magnetic flux systems in thecorona: first, by creating magnetic links to preexisting
magnetic fields [e.g., Longcope et al., 2005] and second,by triggering the ejection of collimated, high-velocity andhigh-temperature outflows of plasma observed in soft X
rays by the YOHKOH satellite [Shimojo et al., 1996;
Shimojo and Shibata, 2000]. These processes (change of
connectivity and emission of jets) can take place throughmagnetic fieldline reconnection whenever an upcoming anda preexisting magnetic flux system come into contact.
Reconnection is also responsible for significant localizedenergy release and the formation of a network of hot (with atemperature of few million degree Kelvin) plasma structures
observed by the YOHKOH soft X-ray telescope. Thus it isvery important to understand the dynamical interaction
between emerging flux and preexisting coronal magnetic
field and numerical simulations could provide a physicalunderstanding of the afore-mentioned process.
[38] The first models, which studied the presence of acoronal magnetic field and the interaction with the emerging
flux system were two-dimensional. Shibata et al. [1989a]
included a simple horizontal field in the corona and ahorizontal magnetic flux sheet below the photosphere tosimulate the emerging flux. Then, they performed a series ofexperiments changing the direction of the overlying field[Yokoyama and Shibata, 1995, 1996]. In the parallel-fieldcase (where the direction of the ambient field is parallel tothe direction of the uppermost fieldlines of the rising loop

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Figure 4. (a, b) Dense, cool material ejected together with (c) hot plasma from the reconnection sites ina flux emergence model by Isobe et al. [2007]. At the final stage of the experiment (Figure 4c),reconnected fieldlines join all the individual emerging loops and rise into the outer solar atmosphere.
upon contact) the rise velocity of the buoyant loop is
drastically suppressed, while in the antiparallel-field casereconnection occurs and the rising motion of the emerging
loop does not slow down. In the latter case, multiple
magnetic islands are created inside the current sheet, at
the interface between the two flux systems. These magnetic
islands are cool and dense structures, which were originallyformed in the lower atmosphere but are carried up within
the current sheet. Eventually, they are ejected sideways fromthe current sheet and they may be observed as Ha surges(cool jets). Also, the plasma behind the magnetic islands isheated by Joule dissipation to a few million degree Kelvinand is ejected along the reconnected fieldlines to both sidesof the current sheet, creating a pair of hot jets. In the case ofan oblique ambient field, one jet is ejected upward while thesecond one moves downward where it collides with smaller
emerging loops, which have been formed from the original
magnetic flux sheet below the photosphere, creating a fast
MHD shock. The collision compresses the plasma and thetemperature at the top of the loops increases. These hotloops may account for some observations of microflares orbright loops, which are found slightly shifted from the siteof the fast and hot outflows. In the case of a vertical coronalfield the jets are emitted in the vertical direction. However,
the general structure and topology around the emerging
region is similar to the oblique-field case. The above
numerical models reproduce some of the features presented
in the emerging flux model for the solar flare phenomenon
suggested by Heyvaerts et al. [1977], the standard CSHKP
model of solar flares (originally proposed by Carmichael[1964], Sturrock [1966], Hirayama [1974], and Kopp andPneuman [1976]) that explains their observable features on
the basis of magnetic reconnection and the revision of the
CSHKP model based on YOHKOH observations by
Shibata et al. [1995].
[39] The 3-D interaction between an emerging magnetic
flux tube and a large-scale horizontal coronal field wasstudied in a series of papers by Archontis et al. [2004],
Galsgaard et al. [2005], Archontis et al. [2005], Archontis
et al. [2006], and Galsgaard et al. [2007]. Experiments
were performed with the direction of the horizontal ambient
field being changed from parallel to antiparallel in steps of

45 degrees. In the antiparallel case, a dome-like current
concentration is formed at the interface when the twomagnetic flux systems come into contact. Eventually, thecurrent surface concentrates into a curved sheet, which iscontained in a vertical midplane that is rotated almost
5 degrees away from the initial axis of the flux tube. In
fact, the midplane of the current arch is rotated by an angle
that increases with the relative horizontal angle between thetwo flux systems into contact. At the initial stage of contactof the two systems the configuration of the magnetic field
vector across the current sheet is similar to the classical 2-D
X-type configuration. However later on, the orientation of
the field across the current sheet resembles a rotational
discontinuity such that the magnetic field never goes
through a null point. The above change in the profile ofthe magnetic field vector has important consequences forthe reconnection, which occurs in a full 3-D manner not
only at the top of the interface but all along the current
concentration. It was also found that magnetic fieldlines
reconnect in a continuous fashion while they are linked tothe diffusion region and that many fieldlines that belong
initially to the rising flux tube may reconnect more thanonce (multiple reconnection events). As a result of thereconnection process, the domain below the photosphereand the coronal domain, which were not joint at the
beginning of the experiments, become linked to each otherthrough the reconnected fieldlines. In fact, almost 75% of
the emerging flux becomes reconnected to the coronal fieldin the experiment with antiparallel magnetic flux systems.
[40] The three-dimensional current arch is the region
where the Joule dissipation has a significant impact on theheating of the plasma. It was found that the temperature in
the antiparallel case could be as high as 107 K but becomes
lower for the cases that are not so favorable for effective
reconnection. In any case, it is likely that current sheets,
formed between emerging and ambient magnetic fields,
may constitute a source of heating for the solar corona.
Similar to the 2-D models, the above 3-D experiments
showed that a pair of hot and high-velocity (with the peakvelocities typically reaching the local Alfvén velocity) jets
are emitted sideways from the current sheet (see Figure 5).
The jets do not look like horizontal thin layers (as in 2-D

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Figure 5. Curved current sheet (blue), the high-velocity outflows (green), and the reconnected fieldlines(yellow) in a 3-D flux emergence experiment [Archontis et al., 2005] where the emerging field reconnectswith an ambient field with antiparallel orientation.
models) but they are curved all along the sides of the current
sheet.
[41] Finally, the formation and evolution of 3-D plasmoids was investigated by Archontis et al. [2006]. The
formation of the plasmoids was possibly due to the tearing
mode instability. Two phases were apparent during theevolution of the system. In the first phase, the plasmoids
had the shape of solenoids lying along the current sheet with
their fieldlines connecting to the subphotospheric field orwith the coronal field. At this stage, they are cool (of theorder of 104 K) and dense and their velocity upon ejection is
as high as 15 Km s1. In the second phase, where themagnetic field across the current sheet undergoes through arotational discontinuity, the fieldlines in the plasmoid become much less tightly wounded. The plasmoids now
confine hot (of the order of 106 K) and less dense plasma
while they are ejected out of the current sheet with muchhigher speed (150 Km s1). The ejection of cool plasmoids may be compatible with observations of Ha or Hbsurges. It is also possible that the UV coil-like structures
observed in filament eruptions may account for the appearance of helical flux strands at the second phase of theexperiments. We should mention that the energy equation
used in these experiments is adiabatic. Also, like othersimulations, there is no radiative transfer or explicit coronalheating in the calculations. However, ohmic and viscous
dissipation are included in the above mentioned
experiments.
[42] Miyagoshi and Yokoyama [2004] studied the emergence of a flux sheet, by simulating the undular mode of themagnetic buoyancy instability, into an antiparallel ambient
field by means of 2-D MHD simulations including heat
conduction. They found that two different types of jets areformed around the emerging flux region: the classicalreconnection jets which are formed because of reconnection
between the two antiparallel flux systems and high-density
evaporation jets. The basic mechanism for the formation of
the evaporation jets is as follows: during reconnection, themagnetic energy is converted into thermal energy, which istransported to the chromosphere along the magnetic fieldlines by heat conduction. Chromospheric evaporation
occurs and dense plasma rises along the reconnected magnetic field creating a secondary pair of jets. However, thetemperature of these jets was found to be low because thecooling by conduction was more efficient than the heatingby magnetic reconnection.
[43] Isobe et al. [2005] performed 3-D MHD simulations
to study the interaction of an emerging flux sheet with an
oblique ambient field. They found that thin current sheets
and intermittent heating occurs at the interface between thetwo magnetic flux systems, as a result of the magnetic
Rayleigh-Taylor instability. Dissipation of the filamentary
current sheets leads to the heating of the plasma around
dense filaments, forming a system of hot and cold loops.
The Rayleigh-Taylor instability and the fast magnetic reconnection are coupled in a nonlinear way, leading tointermittent, patchy reconnection. These results may explain
the intermittent nature of coronal heating and the patchy
brightenings in solar flares.
[44] Archontis et al. [2007] performed 2.5D MHD simulations of a pair of twisted flux tubes as they emerge fromthe subphotospheric layers into the solar corona. A key
feature in this simulation is that the first flux tube emerges,
expands and creates a non uniform (in strength and direction) coronal field that the second tube emerges into. A
series of dynamical phenomena is produced by this modelin a self-consistent manner. A current sheet, with an archlike shape, is formed when the two systems start to interact.
Plasmoids are formed inside the sheet because of resistive
instabilities and when they are ejected out of the current
arch they carry cool and dense material with them. As
reconnection proceeds, the magnetic topology changes
dramatically: there are now four different flux systems
separated by a vertical current sheet, a configuration thathas been extensively used as an initial condition in manymodels for studying flares and eruptions of filaments.
Indeed, the general characteristics of the arcade structure
below the vertical current sheet may account for a compact

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Figure 6. Two flux systems come into contact. Eventually,a vertical current sheet is formed between them, a flux ropeis ejected into the outer atmosphere, and reconnection jets
heat the arcade-like structure at the base of the current sheet
and the top-loop of the emerging system on the right side of
the current sheet.
or an arcade flare associated with flux emergence. Also, hot
plasma (jet) is ejected upward from the current sheet and ismoving along the reconnected fieldlines at the top of thesecondary emerging system (see Figure 6). The temperatureenhancement along the upper part of the secondary emerging flux system may account for a loop brightening.

4. Flux Emergence and Eruptions[45] The emergence of magnetic flux from the solar
interior to the high atmosphere of the Sun may be connectedwith solar eruptions, such as flares, filaments and Coronal
Mass Ejections. In fact, it has been shown [Sterling andMoore, 2005] that even small-scale emergence of flux canchange the magnetic topology of a preexisting active region
and trigger an eruption.
[46] Eruptions and powerful explosions (e.g., fast CMEs)
is one of the basic topics of research in space weather
physics. However, details about how these eruptions are
triggered and propagate in the three-dimensional space arestill unknown. For details about the theory and the modelsof CMEs the reader is referred to the reviews by Klimchuck[2001], Forbes et al. [2006] and Mikic and Lee [2006]. In
the following, we outline results of numerical models thatincorporate flux emergence to study eruptions of magnetic
flux into the outer atmosphere of the Sun.
[47] Chen and Shibata [2000] proposed an emerging flux
trigger mechanism for CMEs using 2-D MHD simulations.

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Their model consists of a quadrapolar field in a twodimensional Cartesian plane. A detached flux rope islocated above the quadrapolar field of the CME source.
Two cases were studied. In the first case, magnetic flux
emerges within the filament channel with direction opposite
to the ambient coronal field, reconnection occurs below thedetached flux rope that leads to partial magnetic cancellation and the rise of the flux rope because of loss ofequilibrium. In the second case, the flux emerges at theright side outside of the filament channel. Reconnection firstoccurs between the emerging flux system and the outerfieldlines of the channel but it eventually proceeds in theinternal layers of the filament system and, thus, it leads tothe eruption of the flux rope similar to the first case. In bothcases, a vertical current sheet is formed below the flux rope
because of the reconnection between the emerging flux andthe ambient field. The upward reconnection jet inside thefilament channel pushes the flux rope toward the outer
atmosphere while below the current sheet, a cusp-shaped
structure with high temperature is formed. The fast ejectionof the flux rope may account for a CME ejection while thecusp-shaped structure has the characteristics of LDE (longduration events) flares.
[48] Shiota et al. [2005] used the same initial magnetic
field configuration as Chen and Shibata [2000] but theyalso included the effect of heat conduction and discussed thedifferences. They found that the dynamical properties (suchas velocity and magnetic fields) are very similar in the twomodels but the thermal properties (e.g., temperature, density) are different. For example, the temperature in thereconnection region and within the reconnection outflows
becomes lower. Also, the current density at the X-point in
the current sheet is larger and the width of the sheet isthinner. Also, they synthesized soft X-ray images from thedensity and temperature in the numerical results and compared them with YOHKOH observations. They showed the
Y-shape of the slow shocks associated with the reconnection, the cusp-shaped arcades below the current sheet andthe dimming above them, and a bright feature at the top of
the arcades that may correspond to the backbone of flarearcades observed by YOHKOH. Dubey et al. [2006] extended the model of Chen and Shibata [2000] by including
the effects of gravity, spherical geometry and a stratified
ambient medium in 2.5D simulations. They also studied
how the rate and the total amount of emerging flux affects
the velocity of the resulting CME-like structure (flux rope).
It was found that the latter factor plays a more crucial roleand the obtained flux rope velocities achieve higher values.
However, all these models cannot reproduce the ejection ofvery fast CMEs.
[49] The emergence of a twisted magnetic flux tube into apreexisting potential magnetic arcade in the corona has been
investigated in the work by Fan and Gibson [2003], Gibson
and Fan [2006] and Fan and Gibson [2006]. First, it was
found that a strong electric current concentration with aninverse-S shape is formed as the emerging tube develops
substantial writhing as a result of the kink instability. The
three-dimensional structure of the current is consistent withthe shape of X-ray sigmoids. Another interesting result isthat the emerging flux rope split into two parts during its
eruption and interaction with the ambient field. One part isbeing expelled into the outer atmosphere while the other10 of 13

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stays behind. Such a partial expulsion is consistent withobservations of CMEs. The full eruption of a kink-unstablecoronal magnetic flux rope, anchored in the photosphere,
was also studied by Török and Kliem [2005]. Their results
were in good agreement with the helical shape and the riseprofile of a very fast CME. They concluded that the helicalkink instability of a flux rope may be the mechanism thattriggers many solar eruptions.
[50] Manchester et al. [2004] investigated how a part of atwisted flux rope can erupt into the corona. They usedsimilar initial conditions to Fan [2001] but they decreased
the length of the buoyant section of the rising tube, so thatthe draining of the mass becomes more efficient. It was
found that the magnetic fieldlines inside the expanding
volume of the magnetized plasma are stretched as the tube
rises and eventually reconnect forming an internal current
sheet with an S-shape. Because of reconnection and to selfinduced shearing motions in the lower atmosphere, theupper half of the flux rope is detached from the lower halfand erupts into the corona. The general evolution of theabove system is consistent, at least qualitatively, withobservations of CMEs in which X-ray sigmoids may appear
after the onset of the eruption.

5. Discussion and Conclusions[51] Some of the most intense episodes of the Sun’s
activity are related to the dynamic process of magnetic flux
emergence from the solar interior into the low-density solar
corona. Over the past few years, numerical models of flux
emergence have been used to study and explain some of theobserved properties of solar active regions and eruptive
events. In fact, some of the experiments have been quite
successful in reproducing, mostly in a qualitative manner,
the appearance and evolution of bipolar regions in thephotosphere, the formation of sigmoidal structures strongly
reminiscent of the X-ray sigmoids observed in the Sun, themorphology of the magnetic fields in solar coronal loops,
the interaction between neighboring emerging flux systems,
the ejection of plasmoids and cool (Ha surges) and hot (Xray reconnection) outflows from sites of strong current
concentration, small-scale brightening events in the lower
atmosphere (e.g., Ellerman bombs), the formation of flares(arcade flares, compact flares, etc.) and finally the eruptionof magnetic flux ropes (e.g., CMEs) in the outer atmosphereof the Sun, which are very important components of theSpace Weather system. Thus the results of these numericalexperiments help us to construct a better picture of the solar
magnetic activity and the agreement between numericalexperiments and observations is very promising, but there
is much that remains to be done and there is still much tolearn about flux emergence and eruptions in the Sun.
[52] During the past few years, there has been a significant number of observations with increased resolution but
there is still no complete understanding of: first, the physics
behind the process of magnetic flux emergence from thedeep solar interior to the outer solar atmosphere and second,
how the emerging field interacts with preexisting magnetic
structures leading to dynamical ejections of plasma anderuptions of flux. The evolution of such systems constitute avery intricate problem (given the large range of length andtimescales, temperature, etc., involved in the process of flux

A03S04emergence) and, thus, observations are expected to provide
us with information of selected aspects, which then have to
fit together to build up a consistent physical picture of flux
emergence in a complete manner. Progress is achieved by
the recent solar mission of Hinode, which takes amazing
high-resolution observations providing detailed descriptionof the small-scale element and the large-scale magnetic field
distribution and their interactions.
[53] On the other hand, there are theoretical difficulties
since many numerical models are idealized. For example,
the choice of the initial conditions in the majority of theexperiments is not strictly in agreement with what has beenshown about the evolution of the magnetic fields in realistic
convection zone models. Also, there are problems if theinitial entropy distribution below the photosphere is not welldefined, as it influences the buoyancy and the emergence ofthe magnetic flux system to the photosphere. In addition,
the coupling between convection and small-scale flux
emergence into the corona has not been investigated so far.
[54] One of the major limitations of the 3-D simulationsof flux emergence into the atmosphere is the lack of anadequate description of the thermal behavior of the system.
Most of the existing simulations have assumed an adiabatic
energy equation (ohmic and viscous dissipation terms areusually included) and this means that the rapid expansion
into the corona is accompanied by rapid adiabatic cooling of
the emerging plasma. This is not observed on the Sun and,thus, additional physics must be included to model theenergy equation (e.g., radiative transfer in photosphereand chromosphere). In the corona, thermal conduction is adominant term in the energy balance and field-aligned
thermal conductivity must be also included along with
optically thin radiation.
[55] Also, emergence of magnetic flux within active
regions is often associated with flares and CMEs, but there
is no generally accepted explanation, so far, of how themagnetic field is stressed in the corona and what thetriggering processes are for the onset of the eruptions. Also,
there is no definitive modeling (including realistic geometry, kinetic effects and description of the thermodynamics ofthe plasma) of their three-dimensional magnetic structureand evolution into the heliosphere. Thus modeling a realistic CME and producing results that may be comparable withobservations has not yet been achieved.
[56] It is worthwhile to mention that a complete study offlux emergence in the solar atmosphere is a very fascinating
and computationally challenging problem. It is indisputablethat progress have gained through a series of numericalexperiments that have investigated the various physicalprocesses separately. We expect that the individual results
will be combined and will lead to important advances about
the nature and the dynamics of emerging flux systems in thenear future. Finally, detailed global 3-D numerical experiments using high-performance computing and simultaneous
high-resolution observations of the magnetic field structuresat different heights of the solar atmosphere, are needed toadvance our understanding further about the process ofmagnetic flux emergence and how it is associated withsolar eruptions.
[57] Acknowledgments. We thank A. W. Hood for many useful
discussions and constructive comments.

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ARCHONTIS: SOLAR FLUX EMERGENCE[58] Amitava Bhattacharjee thanks Mark Linton and another reviewer
for their assistance in evaluating the manuscript.

References
Abbett, W. P., and G. H. Fisher (2003), A coupled model for the emergenceof active region magnetic flux into the solar corona, Astrophys. J., 582,
475.
Abbett, W. P., G. H. Fisher, and Y. Fan (2000), The three-dimensional
evolution of rising, twisted magnetic flux tubes in a gravitationally stratified model convection zone, Astrophys. J., 540, 548.
Abbett, W. P., G. H. Fisher, and Y. Fan (2001), The effects of rotation on theevolution of rising omega loops in a stratified model convection zone,
Astrophys. J., 546, 1194.
Acheson, D. J. (1979), Instability by magnetic buoyancy, Sol. Phys., 62, 23.
Amari, T., J. F. Luciani, and J. J. Aly (2005), Non-current-free coronalclosure of subphotospheric MHD models, Astron. Astrophys., 426, 1047.
Antiochos, S. K., C. R. DeVore, and J. A. Klimchuk (1999), A model forsolar coronal mass ejections, Astrophys. J., 510, 485.
Archontis, V., F. Moreno-Insertis, K. Galsgaard, A. W. Hood, andE. O’Shea (2004), Emergence of magnetic flux from the convection zone
into the corona, Astron. Astrophys., 426, 1047.
Archontis, V., F. Moreno-Insertis, K. Galsgaard, and A. W. Hood (2005),
The three-dimensional interaction between emerging magnetic flux and alarge-scale coronal field: Reconnection, current sheets, and jets, Astrophys. J., 635, 1299.
Archontis, V., K. Galsgaard, K. Moreno-Insertis, and A. W. Hood (2006),
Three-dimensional plasmoid evolution in the solar atmosphere, Astrophys. J. Lett., 645, 161.
Archontis, V., A. W. Hood, and C. Brady (2007), Emergence and interaction of twisted flux tubes in the Sun, Astron. Astrophys., 466, 367.
Brun, A. S. (2007), Towards using modern data assimilation and weather
forecasting methods in solar physics, Astron. Nachr., 328, 329.
Canfield, R. C., H. S. Hudson, and D. E. McKenzie (1999), Sigmoidal
morphology and eruptive solar activity, Geophys. Res. Lett., 26(6), 627.
Canfield, R. C., H. S. Hudson, and A. A. Pevtsov (2000), Sigmoids asprecursors of solar eruptions, IEEE Trans. Plasma Sci., 28, 1786.
Carmichael, H. (1964), A process for flares, in Proceedings of AAS-NASA
Symposium on the Physics of Solar Flares, edited by W. N. Hess, p. 451,
NASA, Washington, D. C.
Chen, P. F., and K. Shibata (2000), An emerging flux trigger mechanism forcoronal mass ejections, Astrophys. J., 545, 524.
Cheung, M. C. M., and M. Schüssler (2007), Magnetic flux emergence ingranular convection: Radiative MHD simulations and observational signatures, Astron. Astrophys., 467, 703.
Chou, D., and H. Zirin (1988), The vertical structure of arch filament
systems in solar emerging flux regions, Astrophys. J., 333, 420.
Cole, D. G. (2003), Space weather: Its effects and predictability, Space Sci.
Rev., 107, 295.
Dorch, B., and A. Nordlund (1998), Numerical 3D simulations of buoyant
magnetic flux tubes, Astron. Astrophys., 338, 329.
Dubey, G. B., B. van der Holst, and S. Poedts (2006), The initiation ofcoronal mass ejections by magnetic flux emergence, Astron. Astrophys.,459, 927.
Emonet, T., and T. Moreno-Insertis (1998), The physics of twisted magnetic
tubes rising in a stratified medium: Two-dimensional results, Astrophys.
J., 492, 804.
Fan, Y. (2001), The emergence of a twisted W-tube into the solar atmosphere, Astrophys. J., 554, 1.
Fan, Y. (2004), Magnetic fields in the solar convection zone, Living Rev.
Solar Phys., 1, 1.
Fan, Y., and S. E. Gibson (2003), The emergence of a twisted magnetic flux
tube into a preexisting coronal arcade, Astrophys. J. Lett., 589, 105.
Fan, Y., and S. E. Gibson (2006), On the nature of the X-ray bright core in astable filament channel, Astrophys. J. Lett., 641, 149.
Fisher, G. H., Y. Fan, D. W. Longcope, and A. A. Pevtsov (2000), The solar
dynamo and emerging flux, Solar Phys., 192, 119.
Forbes, T. G., et al. (2006), CME theory and models, Space Sci. Rev., 123,
251.
Galloway, D. J., and N. O. Weiss (1981), Convection and magnetic field instars, Astrophys. J., 243, 945.
Galsgaard, K., F. Moreno-Insertis, V. Archontis, and A. Hood (2005), A
three-dimensional study of reconnection, current sheets, and jets resulting
from magnetic flux emergence in the Sun, Astrophys. J., 618, 153.
Galsgaard, K., V. Archontis, F. Moreno-Insertis, and A. Hood (2007), The
effect of the relative orientation between the coronal field and new emerging flux: I. Global properties, Astrophys. J., 666, 1.
Gibson, S. E., and Y. Fan (2006), The partial expulsion of a magnetic flux
rope, Astrophys. J. Lett., 637, 65.

A03S04

Hagenaar, H. J. (2001), Ephemeral regions on a sequence of full-disk
Michelson Doppler Imager magnetograms, Astrophys. J., 555, 448.
Heyvaerts, J., E. R. Priest, and D. M. Rust (1977), An emerging flux modelfor the solar flare phenomenon, Astrophys. J., 216, 123.
Hirayama, T. (1974), Solar quiescent prominence and the energy balance,
Solar Phys., 43, 323.
Hughes, D. W., and M. R. E. Proctor (1998), Magnetic fields in the solar
convection zone: Magnetoconvection and magnetic buoyancy, Annu. Rev.
Fluid Mech., 20, 187.
Isobe, H., T. Miyagoshi, K. Shibata, and T. Yokoyama (2005), Filamentary
structure on the Sun from the magnetic Rayleigh-Taylor instability, Nature, 434, 478.
Isobe, H., D. Tripathi, and V. Archontis (2007), Ellerman bombs and jets
associated with resistive flux emergence, Astrophys. J. Lett., 657, 53.
Kaisig, M., T. Tajima, K. Shibata, S. Nozawa, and R. Matsumoto (1990),
Nonlinear excitation of magnetic undular instability by convective motion, Astrophys. J., 358, 698.
Keller, C. U. (1992), Resolution of magnetic flux tubes on the Sun, Nature,359, 307.
Klimchuk, J. A. (2001), Theory of coronal mass ejections, in Space Weather,
Geophys. Monogr. Ser., vol. 125, edited by P. Song, H. J. Singer, and
G. Siscoe, pp. 143 – 157, AGU, Washington, D. C.
Kopp, R. A., and G. W. Pneuman (1976), Magnetic reconnection in thecorona and the loop prominence phenomenon, Solar Phys., 50, 85.
Kusano, K., K. Moriyama, and T. Miyoshi (1998), Nonlinear dynamics ofmagnetic buoyancy instabilities in a sheared magnetic field, Phys. Plasmas, 5, 7.
Leake, J. E., and T. D. Arber (2006), The emergence of magnetic flux
through a partially ionised solar atmosphere, Astron. Astrophys., 450, 2.
Leka, K. D., R. C. Canfield, A. N. McClymont, and L. van Driel-Gesztelyi(1996), Evidence for current-carrying emerging flux, Astrophys. J., 462,
547.
Longcope, D. W., G. H. Fisher, and S. Arendt (1996), The evolution andfragmentation of rising magnetic flux tubes, Astrophys. J., 464, 999.
Longcope, D. W., D. E. McKenzie, J. Cirtain, and J. Scott (2005), Observations of separator reconnection to an emerging active region, Astrophys.
J., 630, 596.
Magara, T. (2001), Dynamics of emerging flux tubes in the Sun, Astrophys.
J., 549, 608.
Magara, T. (2004), A model for dynamic evolution of emerging magnetic
fields in the Sun, Astrophys. J., 605, 480.
Magara, T. (2007), Dynamical and topological features of photospheric andcoronal activities produced by flux emergence in the Sun, Astrophys. J.,
653, 1499.
Magara, T., and D. W. Longcope (2003), Injection of magnetic energy andmagnetic helicity into the solar atmosphere by an emerging magnetic flux
tube, Astrophys. J., 586, 630.
Manchester, W., IV (2001), The role of nonlinear Alfvén waves in shear
formation during solar magnetic flux emergence, Astrophys. J., 547, 503.
Manchester, W., IV, T. Gombosi, D. DeZeeuw, and Y. Fan (2004), Eruptionof buoyantly emerging magnetic flux rope, Astrophys. J., 610, 588.
Martens, P. C. H. (2003), YOHKOH-SXT observations of reconnection,
Adv. Space Res., 32, 905.
Matsumoto, R., T. Tajima, K. Shibata, and M. Kaisig (1993), Three-dimensional magnetohydrodynamics of the emerging magnetic flux in the solar
atmosphere, Astrophys. J., 462, 547.
Mikic, Z., and M. A. Lee (2006), An introduction to theory and models of
CMEs, shocks and solar energetic particles, Space Sci. Rev., 123, 57.
Miyagoshi, T., and T. Yokoyama (2004), Magnetohydrodynamic simulationof solar coronal chromospheric evaporation jets caused by magnetic reconnection associated with magnetic flux emergence, Astrophys. J., 614,
1042.
Moreno-Insertis, F., and T. Emonet (1996), The rise of twisted magnetic
tubes in a stratified medium, Astrophys. J., 472, 53.
Murray, M. J., A. W. Hood, F. Moreno-Insertis, K. Galsgaard, andV. Archontis (2006), 3D simulations identifying the effects of varying thetwist and field strength of an emerging flux tube, Astron. Astrophys., 460,
909.
Newcomb, W. (1961), Convective instability induced by gravity in a plasma with a frozen-in magnetic field, Phys. Fluids, 4, 391.
Nordlund, A., A. Brandenburg, R. L. Jennings, M. Rieutord, J. Ruokolainen,
R. F. Stein, and I. Tuominen (1992), Dynamo action in stratified convection with overshoot, Astrophys. J., 392, 647.
Pariat, E., G. Aulanier, B. Schmieder, M. K. Georgoulis, D. M. Rust, and
P. N. Bernasconi (2004), Resistive emergence of undulatory flux tubes,
Astrophys. J., 614, 1099.
Parker, E. N. (1955), Hydromagnetic dynamo models, Astrophys. J., 122,
293.
Parker, E. N. (1978), Cosmical Magnetic Fields, Oxford Univ. Press, NewYork.

12 of 13

A03S04

ARCHONTIS: SOLAR FLUX EMERGENCE

Parker, E. N. (1988), Dynamical oscillation and propulsion of magnetic
fields in the convective zone of a star. VI: Small flux bundles, network
fields, and ephemeral active regions, Astrophys. J., 326, 407.
Pevtsov, A. (2002), Sinuous coronal loops at the Sun, in Multi-Wavelength
Observations of Coronal Structure and Dynamics, p. 125 – 134, Elsevier,
New York.
Pevtsov, A., C. Canfield, and S. M. Latushko (2001), Hemispheric helicity
trend for solar cycle 23, Astrophys. J., 549, L261 – L263.
Riley, P., J. A. Linker, Z. Mikic, R. Lionello, S. A. Ledvina, and J. G.
Luhmann (2000), A comparison between global solar magnetohydrodynamic and potential field source surface model results, Astrophys. J., 635,
1510.
Rust, D. M., and A. Kumar (1994), Helical magnetic fields in filaments,
Solar Phys., 155, 69.
Schrijver, C. J., M. L. Derosa, T. R. Metcalf, Y. Liu, J. McTiernan,
S. Rgnier, G. Valori, M. S. Wheatland, and T. Wiegelmann (2006), Nonlinear force-free modeling of coronal magnetic fields: Part I. A quantitative comparison of methods, Solar Phys., 235, 161 – 190.
Schuessler, M. (1979), Magnetic buoyancy revisited: Analytical and numerical results for rising flux tubes, Astron. Astrophys., 71, 79.
Shibata, K., T. Tajima, R. S. Steinolfson, and R. Matsumoto (1989a), Twodimensional magnetohydrodynamic model of emerging magnetic flux inthe solar atmosphere, Astrophys. J., 345, 584.
Shibata, K., T. Tajima, R. Matsumoto, T. Horiuchi, T. Hanawa, R. Rosner,and Y. Uchida (1989b), Nonlinear Parker instability of isolated magnetic
flux in a plasma, Astrophys. J., 338, 471.
Shibata, K., S. Masuda, M. Shimojo, H. Hara, T. Yokoyma, S. Tsuneta,
T. Kosugi, and Y. Ogawara (1995), Hot-plasma ejections associated withcompact-loop solar flares, Astrophys. J. Lett., 451, 83.
Shimojo, M., and K. Shibata (2000), Physical parameters of solar X-ray
jets, Astrophys. J., 542, 1100.
Shimojo, M., S. Hashimoto, K. Shibata, T. Hirayama, H. S. Hudson, andL. W. Acton (1996), Statistical study of solar X-ray jets observed with the
YOHKOH soft X-ray telescope, Publ. Astron. Soc. Jpn., 48, 123.
Shiota, D., H. Isobe, P. F. Chen, T. T. Yamamoto, T. Sakajiri, and K. Shibata(2005), Self-consistent magnetohydrodynamic modeling of a coronalmass ejection, coronal dimming, and a giant cusp-shaped arcade formation, Astrophys. J., 634, 663.
Socas-Navarro, H., and J. Sanchez Almeida (2003), Magnetic fields in thequiet Sun: Observational discrepancies and unresolved structure, Astrophys. J., 593, 581.

A03S04

Stein, R. F., and A. Nordlund (2006), Solar small-scale magnetoconvection,
Astrophys. J., 642, 1246.
Steiner, O., U. Grossmann-Doerth, M. Knoelker, and M. Schuessler (1998),
Dynamical interaction of solar magnetic elements and granular convection: Results of a numerical simulation, Astrophys. J., 495, 468.
Sterling, A. C. (2000), Sigmoid CME source regions at the Sun: Some
recent results, J. Atmos. Solar Terr. Phys., 62, 1427.
Sterling, A. C., and R. L. Moore (2005), Slow-rise and fast-rise phases of an
erupting solar filament, and flare emission onset, Astrophys. J., 630,
1148.
Strous, L. H., G. Scharmer, T. D. Tarbell, A. M. Title, and C. Zwaan (1996),
Phenomena in an emerging active region. I. Horizontal dynamics, Astron.
Astrophys., 306, 947.
Sturrock, P. A. (1966), Model of the high-energy phase of solar flares,
Nature, 211, 695.
Taylor, J. R., M. Lester, T. K. Yeoman, B. A. Emery, D. J. Knipp, D. Orr,S. I. Solovyev, T. J. Hughes, and H. Lühr (1997), The response of themagnetosphere to the passage of a coronal mass ejection on March 20 –21 1990, Ann. Geophys., 15, 671.
Török, T., and B. Kliem (2005), Confined and ejective eruptions of kinkunstable flux ropes, Astrophys. J., 630, 97.
Wang, T. J., and V. I. Abramenko (1999), Evolution of twist in an emerging
flux region NOAA 7321, in Magnetic Fields and Solar Processes: The
9th European Meeting on Solar Physics, ESA SP-448, p. 671, Eur. Space
Agency, Paris.
Weiss, N. O., M. R. E. Proctor, and D. P. Brownjohn (2002), Magnetic flux
separation in photospheric convection, Mon. Not. R. Astron. Soc., 337,
293.
Yokoyama, T., and K. Shibata (1995), Magnetic reconnection as the originof X-ray Jets and Ha surges on the Sun, Nature, 375, 42.
Yokoyama, T., and K. Shibata (1996), Numerical simulation of solar coronal X-ray jets based on the magnetic reconnection model, Publ. Astron.
Soc. Jpn., 48, 353.
Zirin, H. (1970), Active regions. I: The occurrence of solar flares and thedevelopment of active regions, Solar Phys., 14, 328.
Zwaan, C. (1987), Elements and patterns in the solar magnetic field, Annu.
Rev. Astron. Astrophys., 25, 83.


V. Archontis, School of Mathematics and Statistics, University of St.
Andrews, North Haugh, St. Andrews, Fife KY16 9SS, UK. (vasilis@
mcs.st-and.ac.uk)

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ClickHere

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S08, doi:10.1029/2007JA012558, 2008

for

Full
ArticleRadiation Belt Environment model: Application to space weather
nowcasting
Mei-Ching Fok,1 Richard B. Horne,2 Nigel P. Meredith,2 and Sarah A. Glauert2
Received 23 May 2007; revised 11 September 2007; accepted 9 October 2007; published 23 January 2008.

[1] A data-driven physical model of the energetic electrons in the Earth’s radiation belts,
called the Radiation Belt Environment (RBE) model, has been developed to understand
Earth’s radiation belt dynamics and to predict the radiation conditions found there. Thismodel calculates radiation belt electron fluxes from 10 keV to 6 MeV in the innermagnetosphere. It takes into account the realistic, time-varying magnetic field andconsiders effects of wave-particle interactions with whistler mode chorus waves. The
storm on 23–27 October 2002 is simulated and the temporal evolutions of the radial andpitch angle distributions of energetic electrons are examined. The calculated electron
fluxes agree very well with particle data from the low-orbit SAMPEX and LANL
geosynchronous satellites, when the wave-particle interactions are taken into accountduring storm recovery. Flux increases begin near the plasmapause and then diffuse
outward to higher L shells, consistent with previous findings from statistical studies. A
simplified version of the RBE model is now running in real time to provide nowcasting of
the radiation belt environment. With further improvements and refinements, this model
will have important value in both scientific and space weather applications.
Citation: Fok, M.-C., R. B. Horne, N. P. Meredith, and S. A. Glauert (2008), Radiation Belt Environment model: Application to
space weather nowcasting, J. Geophys. Res., 113, A03S08, doi:10.1029/2007JA012558.

1. Introduction[2] The Earth’s radiation belts consist of energetic electron (100 keV to several MeV) and ions (100 keV toseveral hundred MeV) trapped in the magnetosphere roughly from 1.2 < L < 8. The energetic electrons reside in two
distinct regions: the inner belt and the outer belt, which areusually separated by the slot region (1.8 < L < 3) of depleted
particle populations. Pitch angle diffusion loss of electrons
by interacting with whistler mode plasmaspheric hiss isbelieved to be the source of the slot region [Lyons et al.,
1972; Albert, 1994; Meredith et al., 2007]. The inner belt isrelatively stable while the outer belt is highly variable withgeomagnetic activity. The fluxes of energetic electrons in
the outer belt decrease during the main phase of a magnetic
storm due to adiabatic effect [Dessler and Karplus, 1961;
Kim and Chan, 1997]. Additional nonadiabatic processes
also contribute to the flux decrease in the storm main phase[Green et al., 2004]. During the recovery phase the flux ofenergetic electrons can change dramatically as well. Whileapproximately half of all moderate and intense storms cause
a net increase in the flux of energetic electrons by a factor of
2 or more, approximately a quarter of these storms result ina net decrease in the fluxes by more than a factor of 2

1
Geospace Physics Laboratory, NASA Goddard Space Flight Center,Greenbelt, Maryland, USA.
2
Physical Sciences Division, British Antarctic Survey, Cambridge, UK.

This paper is not subject to U.S. copyright.
Published in 2008 by the American Geophysical Union.

[Reeves et al., 2003]. This variability is caused by an imbalance between acceleration, transport, and loss processes allof which become enhanced during geomagnetic storms[Horne, 2002; Thorne et al., 2005; Horne et al., 2006;
Summers et al., 2007]. The ratio of poststorm to prestorm
flux is related to the solar wind speed [Paulikas and Blake,
1979; Reeves et al., 2003] and the direction of the IMF Bz
during the storm recovery phase [Iles et al., 2002]. In somemajor storms the flux enhancement takes place close to the
Earth and may even penetrate into the slot region [Baker etal., 2004; Zheng et al., 2006].
[3] The intensification of the radiation belts has significant impacts on the space environment. Moderate energy(10 to 100 keV) electrons can cause surface charging
effects and relativistic (0.1 to 5 MeV) electrons can cause
deep-dielectric charging on space systems [Baker, 2001].
Therefore understanding the physical processes that arecontrolling the development of the radiation belts during
active periods and being able to predict their variability
have both scientific and practical significance. Radial diffusion has traditionally been considered to be the leadingtransport and energization mechanism in the inner magnetosphere [Schulz and Lanzerotti, 1974]. However, it has
recently been suggested that electrons can be accelerated
efficiently by resonant wave particle interactions withwhistler mode chorus waves [Horne and Thorne, 1998;
Summers et al., 1998]. Several studies have shown anassociation between relativistic electron flux enhancements
and prolonged periods of enhanced chorus amplitudes
lasting for the order of several days [Meredith et al.,2002, 2003a; Miyoshi et al., 2003]. Furthermore, theoretical

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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHER

work suggests that the timescale for acceleration is typicallyof the order of 1 –2 d [Summers and Ma, 2000; Horne et al.,2005a, 2005b], consistent with observations. These studies
suggest that wave acceleration by whistler mode chorus
waves should be included in realistic physics-based modelsof the Earth’s radiation environment.
[4] A number of models have been established to simulate the radiation belt dynamics and to provide interpretationfor observable features. Two major approaches have beentaken in modeling the radiation belts: test particle andkinetic formulations. Li et al. [1993] and Hudson et al.
[1996] performed guiding center simulations of the storm
sudden commencement on 24 March 1991 that was induced
by an interplanetary shock. They used different models forthe compressional electric fields but both studies were able
to reproduce the rapid formation of a ‘‘new’’ electron belt at
L  2.5. Elkington et al. [2003] investigated the effects of
ULF waves on energetic electron dynamics by tracking theirguiding center motion in an asymmetric, compressed dipole
field. They found outer belt electrons are accelerated anddiffuse radially through drift resonance with Pc-5 ULF
waves. Using a test particle approach, Ukhorskiy et al.
[2006] simulated the evolution of outer belt electrons during
the magnetic storm of 7 September 2002. They found the
diamagnetic effect from the storm time ring current leads toexpansion of electron drift paths that intercept with themagnetopause, producing significant irreversible loss ofenergetic electrons at L > 5 during the storm main phase.
[5] Kinetic formulation is another commonly used technique to model the radiation belt. In a kinetic model theequation of the particle distribution function is solved
analytically or numerically. One simple approach is basedon a standard radial diffusion equation with diffusion
coefficients driven by the solar wind conditions or geomagnetic activity [Li et al., 2001; Albert et al., 2001]. Varotsou
et al. [2005] and Horne et al. [2006] combined radial
diffusion with acceleration and loss due to whistler modechorus waves and confirmed that wave acceleration bywhistler mode chorus is an important acceleration mechanism in the outer radiation belt. In particular, Horne et al.
[2006] modeled a moderate storm due to a high-speed solar
wind stream, typical of the declining phase of the solar
cycle. They found that the accelerated (MeV) electrons aretransported both inward and outward and increase in phasespace density by a factor of 10 between 3.5 < L < 7. For
models that cover a wide range of energy, drift motion
must be considered since convection is an important transport mechanism for lower-energy (<50 keV) particles[Bourdarie et al., 1997; Zheng et al., 2003; Miyoshi etal., 2006]. Using the relativistic RAM model with dipole
magnetic field, Miyoshi et al. [2006] simulated the dynamics
of energetic electrons during the October 2001 storm. They
reproduced the observed local time flux asymmetry for hot
electrons (30 keV). They also concluded that only convective transport and radial diffusion cannot explain theenhancement of relativistic electrons seen during storm
recovery.
[6] A convection-diffusion model, namely the Radiation
Belt Environment (RBE) model, has been developed tounderstand the radiation belt dynamics in order to predict
the flux variation during active times [Fok et al., 2001,
2005; Zheng et al., 2003]. The model was used to simulate a

A03S08substorm injection during a dipolarization of the magnetic
field [Fok et al., 2001]. Observable features during substorms, such as dispersionless injection and drift echoes, aresuccessfully reproduced. The electron flux enhancements
during two magnetic storms were also studied using theRBE model [Zheng et al., 2003; Fok et al., 2005]. They
found energization by the inductive electric field and bywhistler mode waves are crucial for the flux increase during
magnetic storms.
[7] The RBE model has been improved to include waveparticle interactions due to whistler mode chorus waves, tohave a higher-resolution in L shell, and extended to theinner belt. In modeling the wave-particle interactions,diffusion coefficients are taken from the Pitch Angle andEnergy Diffusion of Ions and Electrons (PADIE) code
[Glauert and Horne, 2005]. In this paper, we give a brief
description of the RBE formulation. We then present the
RBE simulation of the storm on 23– 27 October 2002. The
calculated radiation belt electron fluxes are compared withobservations from the Solar Anomalous and Magnetospheric Particle Explorer (SAMPEX) low-Earth orbit satelliteand Los Alamos National Laboratory (LANL) geosynchronous satellites. The dynamics of the energetic electrons are
discussed and the distinct roles of transport and wave
acceleration are quantitatively examined. With real-timesolar wind speed, density, interplanetary magnetic field(IMF), and Dst data as input, a simplified version of the
RBE model is running simultaneously to provide nowcasting of the radiation belt environment. Finally, we
discuss the potentials of the RBE model for space weather
applications.

2. Radiation Belt Environment Model[8] The Radiation Belt Environment (RBE) model is akinetic model that calculates the temporal variation of thephase space density of energetic electrons by solving thefollowing bounce-averaged Boltzmann transport equation[Fok et al., 2001; Zheng et al., 2003]:@fs  _  @fs@fs
1 @ pﬃﬃﬃﬃﬃ@fsþ h8_ i i
¼ pﬃﬃﬃﬃﬃ
þ li
M DMM
@t
@li
@8i
@M
M @M


1
@
@fs
T ð yÞ sin 2ao Dao aoþ@ao
T ð yÞ sin 2ao @ao


fs

loss
0:5t b cone

ð1Þ

where fs = fs(t, li, 8i, M, K), is the average distributionfunction on the field line between mirror points. Here li and
8i are the magnetic latitude and local time, respectively, atthe ionospheric foot point of the geomagnetic fieldpline.
Mﬃ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
is the relativistic magnetic moment and K = J/ 8mo M ,where J is the second adiabatic invariant. The motion of theparticles is described by their drifts across field lines which
are labeled by their ionospheric foot points. The innerboundary of li is at 11.8°, corresponding to L = 1.06. The
outer edge of the model is bounded by field lines with li at
70.2° and equatorial crossing at 10 Earth radius (RE),whichever is closer. The M range is chosen to well represent
the energy ranges of electrons from 10 keV to 6 MeV. The K
range is chosen to cover the loss cone so that particle

2 of 12FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08precipitations can be estimated as well. Here ao is theequatorial pitch angle and y = sinao. T(y) is given by:
T ð yÞ ¼

1
Ro

Z sm
0

ds
cos a

ð2Þ

where Ro is the radial distance in RE of the field line
equatorial crossing. The integration is along the field linefrom the equator to the mirror point. Here t b is the particle
bounce period.
[9] The left-hand side of (1) represents the drifts of theparticle population, and the terms on the right-hand side of(1) refer to diffusion and loss. The
 calculation of thebounce-averaged drift velocities, l_ i and h8_ i i, were described in detail by Fok and Moore [1997]. These drifts
include gradient and curvature drift and E  B drift fromconvection and corotation electric fields. The effects of
inductive electric field due to time-varying magnetic field
are also taken into account implicitly in the model. For thispurpose we have assumed that field lines are rooted at theionosphere so that the inductive electric field there is zero.
However, the shapes of field lines at higher altitudes vary as
a function of time according to the magnetic field model. If
field lines are perfect conductors, the field line motion athigh altitudes, for example, at the equator, will generate an
induction electric field of the form,
Eind ¼ vo  Bo

ð3Þ

where vo and Bo are the field line velocity and magnetic
field at the equator.
[10] The first term on the right-hand side of (1) represents
particle diffusion in M as a result of energy diffusion due tointeractions with plasma waves. The relation between theenergy diffusion coefficient (DEE) and the corresponding
coefficient in M (DMM) is given as
DMM ¼ DEE

@M
@E

2


¼ DEE

Eo þ E
Eo Bm

2
ð4Þ

where Eo is the electron rest energy and Bm is the magnetic
field at the mirror point. The second term on the right-handside of (1) represents pitch angle diffusion from interactions with plasma waves. For pure pitch angle diffusion(E unchanged) in the (M, K) coordinates, we first map theparticle phase space density from the (M, K) to (E, ao)coordinates, perform diffusion in ao, and then map the
updated distribution back to the (M, K) coordinates [Fok etal., 1996]. The diffusion terms are followed by the loss termof the loss cone, the boundary of which is assumed tocorrespond to mirror height of 120 km. Particles in the loss
cone are assumed to have a lifetime of one half bounce
period (0.5 t b).
[11] In this work, the bounce-averaged pitch angle andenergy diffusion coefficients, Dao ao and DEE, are given by
the PADIE code [Glauert and Horne, 2005]. Only resonance with lower-band whistler mode chorus (0.1 fce < f <0.5 fce) is considered. The presence of chorus waves isconfined between 15° and 15° magnetic latitude. The
exclusion of high-latitude chorus may cause underestimation of the loss of electrons at energies >1 MeV [Horne and

A03S08Thorne, 2003]. The diffusion coefficients are calculated as afunction of L shell, energy, pitch angle, and fpe/fce, the ratioof plasma frequency to the cyclotron frequency, and given
by
fpe 1
¼
fce B

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ne me
eo

ð5Þ

To calculate fpe/fce, which depends on the plasma density(ne), we have embedded inside the RBE model the coreplasma model of Ober and Gallagher [Ober et al., 1997].
The plasma model calculates the temporal evolution of theplasma flux tube contents and equatorial plasma density
distribution throughout the inner magnetosphere. The modelis driven by the same magnetic and electric fields as theRBE code. The PADIE diffusion coefficients are scaled witha chorus wave intensity of 104 pT2. To obtain the actual
diffusion coefficients, we estimate the chorus intensity at agiven location and time during the storm using the surveyof CRRES plasma wave data for lower-band chorus
presented by Meredith et al. [2001, 2003b]. For ourapplication the wave data were binned in L shell, magnetic
local time, and three levels of magnetic activity (Kp < 2; 2 
Kp < 4; Kp  4).
[12] Equation (1) includes multiple terms of different
processes. We use the method of fractional step to decompose the equation and solve only one term at a fractional
step [Fok et al., 1993]. To solve (1), we have to specify theelectric, magnetic fields, initial distribution, and particle
distribution on the nightside boundary, which is set at 10
RE or the last closed field line. In this work, we use the
Tsyganenko 2004 model [Tsyganenko and Sitnov, 2005] forthe magnetic field and Weimer model [Weimer, 2001] forelectric field. The electric field is updated every time step(3 s), since updating the electric field and the recalculationof the E  B drift are computationally fast. Updating themagnetic field, however, requires massive calculations offield line tracing and integration along field line in order torenew the particle energy and magnetic drift. The magnetic
field is thus updated every 5 min. The effect of radial
diffusion is integrated in these time-varying electric andmagnetic fields. The NASA trapped radiation model(AE8MAX) [Vette, 1991; Fung, 1996] is used for the initialcondition in the entire RBE spatial domain. The distribution
at the nightside boundary is assumed to be a kappa function
with density (Nps) and characteristic energy (Eps) modeledby linear relations with the upstream solar wind conditions[Zheng et al., 2003]:
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Nps ðtÞ ¼ ½0:02*Nsw ðt  2hrÞ þ 0:316* amu
Eps ðtÞ ¼ 0:016*Vsw ðt  2hrÞ  2:4

ð6Þ

where Nps is in cm3, Nsw is the solar wind density in thesame unit, amu is the atomic mass unit of the electron, Eps isin keV, and Vsw is the solar wind velocity in km/s. Note that
we assume a 2-h time lag between the plasma sheet
condition and solar wind condition at the daysidemagnetopause [Borovsky et al., 1998].
[13] Figure 1 summarizes the RBE model architecture
and logic. The only inputs to the model are the solar wind

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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08Figure 1. Model logic of the Radiation Belt Environment (RBE) model.

speed, density, and IMF and geomagnetic indices Dst (orsymH) and Kp. The magnetic field and electric field models
are driven by Dst or symH, upstream solar wind speed,density, and IMF. With the magnetic and electric fields
specified, the plasmasphere model solves the core plasma
density, which is used to calculate the plasma frequency
needed for the PADIE code. The diffusion coefficients arethen calculated for the chorus wave intensity determined by
the Kp value. The last piece is the plasma sheet distribution,
which is estimated from the solar wind density and speed(equation (6)). With all the auxiliary models in place, theRBE kinetic equation is solved to give the energetic electron

fluxes at all pitch angles in the inner and the outer radiation
belts.

3. Simulation of the Storm on 23–27 October
2002[14] The storm on 23– 27 October 2002 is a moderate
storm with minimum Dst reaching 98 nT. The storm istriggered by passing of solar wind pressure pulses andsouthward IMF. Figure 2 plots the Kp, symH, solar winddensity and speed, and IMF Bz, By measured from the ACE
satellite. The temporal resolution of the ACE data is 4 min

Figure 2. Kp, symH, solar wind density and speed, and IMF Bz, By measured from the ACE satellite on
23– 27 October 2002. The ACE data are 44 min shifted in time.
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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHER

Figure 3. Simulated 0.8 MeV electron flux at the magnetic
equator at quiet time, showing (left) the pitch angle
averaged flux and (right) the pitch angle anisotropy. The
white circles represent geosynchronous orbit.
and the data have been shifted by 44 min representing thetime for solar wind to travel from the satellite to the subsolar
point. The parameters plotted in Figure 2 are used to drive ourmagnetic (T04) and electric (Weimer 2000) field models.
[15] Figure 3 shows the simulated equatorial electron flux
at 0.8 MeV during the quiet time before the storm. Figure 3(left) is the pitch angle averaged flux and Figure 3 (right) is
the pitch angle anisotropy, defined as A = (j?  jk)/(j? + jk),
where j? and jk are perpendicular and parallel flux,
respectively. A = 0 represents perpendicular and parallel
fluxes are equal in magnitude, A > 0 perpendicular
distribution, and A < 0 parallel distribution. The whitecircles in Figure 3 represent geosynchronous orbit. It can beseen from Figure 3 (left) that the inner and the outer belts
are very well separated during quiet time. For this particularenergy, the peak flux in the outer belt is located at 4.5 RE.
In Figure 3 (right) a clear field-aligned pitch angle
distribution (PAD) is seen on the nightside extending fromdawn to dusk around the geosynchronous orbit. This fieldaligned feature of energetic electrons and ions on thenightside has been observed [Sibeck et al., 1987; Garcia,
1996; Friedel et al., 2006] and is an effect of drift shell
splitting in the asymmetric magnetic field. Particles with

A03S08different pitch angles injected from the nightside drift
differently to the dayside. To conserve the first adiabatic
invariant, perpendicular particles tend to drift farther awayfrom the Earth on the dayside where the magnetic field isstronger for a given radial distance. Perpendicular particles
originated from the nightside geosynchronous orbit may belost when their drift paths come across the daysidemagnetopause. The drift paths of parallel particles are
relatively circular and have closed drift paths around thegeosynchronous orbit. As a result, more particles with fieldaligned pitch angles are seen on the nightside thanperpendicular particles. The strong aligned region (bluepartial ring in Figure 3, right) represents the difference
between the last closed orbit of field-aligned electrons andthat of the perpendicular particles.
[16] The energetic electron distribution is simulated
throughout the storm on 23– 27 October 2002 using theRBE model. Figure 4 plots the 0.8 MeV pitch angle
averaged flux at 0200 UT, 25 October at the beginning of
the recovery phase, 50 h into the simulation. Figure 4 (leftand middle) shows the equatorial flux and pitch angle
anisotropy with the same formats as Figure 3. It can beseen that the electron flux in the inner part of the outer belt
has increased by an order of magnitude during the storm.
We will show later in the paper that this increase is mainly
coming from electrons interacting with chorus waves. The
field-aligned feature in the vicinity of the nightside geosynchronous orbit persists the whole time of the event (Figure 4,
middle). On the other hand, perpendicular PADs are seen on
the dayside near the magnetopause. These distinct PADs atnight and day can be envisaged from the meridian view inFigure 4 (right). The white curves are field lines withequatorial crossing points at 6.6 RE. Field-aligned distributions are characterized by lower flux at the equator than at
higher latitudes along the same field line. For the field
line highlighted at 0000 MLT, the equatorial flux is 3 
102 cm2 s1 keV1 sr1 (cyan) and the flux at 40°
magnetic latitude is 9  102 cm2 s1 keV1 sr1 (green),
three times of the value at the equator. The perpendicular
distribution near the dayside magnetopause is also seenfrom the relatively high flux around the equator. In the outerbelt where the electron flux is high, the enhancement isdominated by perpendicular particles.

Figure 4. Simulated 0.8 MeV electron flux at the magnetic equator 50 h into the simulation at 0200 on
25 October 2002 (left, middle) in the same format as Figure 3 and (right) showing the electron flux at thenoon-midnight meridian. Here, the white traces represent field lines with equatorial crossings at
geosynchronous orbit.
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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08Figure 5. (left) LANL SOPA geosynchronous electron fluxes compared with (right) RBE simulated
fluxes at three longitudes during the storm on 23– 27 October 2002.

[17] Next we examine the storm time development of theouter belt and assess the ability of the RBE model toreproduce the observed electron flux during this event.
Figure 5 (left) plots the observed electron fluxes at thegeosynchronous orbits from the LANL Synchronous Orbit
Particle Analyzer (SOPA) data and Figure 5 (right) plots the
RBE simulated fluxes, at three geographic longitudes:
7.7°E, 70.0°E, and 165.4°W. The SOPA fluxes decreaseduring the storm main phase on 24 October (Dst effect).
Electron fluxes start to recover toward the prestorm levels at
late main phase. The fluctuations in the SOPA fluxes on
26 October are results of substorm activity. The simulated
fluxes (Figure 5, right) exhibit similar temporal variation as
those of the SOPA data, and in general the magnitudes agree
very well with the data. However, the flux recovery starts
earlier than the observed data, and the model fluxes lacksubstorm-type fluctuations.
[18] The temporal evolution of the radial profile ofradiation belt particles during a storm provides important
clues to identify the energization mechanisms, such as radial
diffusion and wave acceleration. Figure 6 shows 2 –6 MeV
electron fluxes from the Proton/Electron Telescope (PET)on SAMPEX as a function of L shell and time for 23–
27 October 2002 (Figure 6, top). SAMPEX orbits the Earthat 520  670 km altitude and 82° inclination [Baker et al.,1993]. It samples magnetic field lines at the ionosphere thatthread the entire radiation belts every orbit period(90 min). PET is a zenith-pointing solid state telescopemeasuring electrons from 2 to 30 MeV. These electrons areeither trapped particles mirroring at SAMPEX altitudes orare precipitating into the atmosphere. The latter may be in
the drift loss cone and/or the local bounce loss cone
depending upon the longitudinal position of SAMPEX[Cook et al., 1993]. The SAMPEX fluxes shown in Figure
6 are with orbit resolution and smoothed over 15 orbits. As
shown in the figure, the outer belt, slot region, and inner belt
are clearly separated in this time period. The enhancementat L  3.4 on 23 October is a remnant from the activity
taking place earlier in the month. When the storm commences on 24 October, electron fluxes start to decrease. Near the
peak of the storm on late 24 October to early 25 October,electrons drift to higher L shells as a result of the ring
current buildup. In the recovery phase, electron fluxes
gradually increase. Flux enhancements are first seen at 3 <6 of 12

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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08especially during geomagnetic storms. The L values in the
RBE plots are calculated by:
L¼

Figure 6. (top) SAMPEX 2 – 6 MeV electron L-time
diagram during the storm on 23– 27 October 2002 andRBE simulated fluxes in T04 magnetic field (middle) withand (bottom) without inclusion of wave-particle interactions, respectively.

L < 4.5 and then expand in both inward and outward
direction.
[19] Figure 6 (middle and bottom) shows RBE simulated
fluxes displayed in the same layout as the SAMPEX L-time
plot. Note that the RBE data are equatorial fluxes rather thanfluxes at the ionosphere as the SAMPEX data. However, as
has been shown by Kanekal et al. [2001, 2005], thetemporal variability of low-altitude fluxes measured by
SAMPEX is nearly identical with the equatorial fluxes,ri
cos2 li

ð7Þ

where ri is the ionosphere distance in RE and li is themagnetic latitude of the ionospheric foot point of themagnetic field line. The color scales of the SAMPEX andRBE data are different. The RBE fluxes are higher
compared to SAMPEX since the latter observes a smaller
part of the equatorial pitch angle distribution. The RBE
fluxes with the inclusion of wave-particle interactions(Figure 6, middle) show similar features as the SAMPEX
data. One exception is the high intensity found in the
SAMPEX data on 23 October is not seen in the simulation.
This is due to the fact that the RBE model uses the statisticalmodel, AE8MAX, as initial condition and does not take into
account the previous activity. When the storm hit, the flux
decrease in the outer belt is also seen in the simulation. In
the storm recovery, the RBE model reproduces the observed
flux enhancements at about the same L shells.
[20] To identify the responsible mechanism for the flux
increase, a test run is performed without the chorus waveinduced diffusions (Figure 6, bottom). Two high-flux bands
are seen at around L = 3.2 and 4 during the storm recovery.
The enhancement at L  3.2 comes from inward radial
diffusion of electrons at higher L shells. The strong flux at
L  4 is result of particle injection during the storm main
phase. When wave diffusions are included (Figure 6,
middle), the enhancement at the inner edge of the outerbelt is diminished as a result of pitch angle diffusion loss.
On the other hand, with the seed population created fromparticle injection, energy diffusion at L  4 produces theflux increase in the outer belt. This enhancement cannot be
explained by radial diffusion and storm injection alone. Our
simulations confirm that local acceleration by wave particle
interactions with whistler mode chorus can account for theflux enhancement in the center of the outer belt during therecovery phase of this storm.
[21] It is well established that strong chorus accelerationof energetic electrons takes place just outside the plasmapause where fpe/fce is relatively low [Summers et al., 1998;
Horne et al., 2006]. To demonstrate our calculation isconsistent with this theory and to examine in detail how
the enhancement region evolves with time, simulated 2 – 6
MeV electron fluxes as a function of L and time are
replotted in 1-h temporal resolution (Figure 7, top). Figure
7 (middle and bottom) shows the Ober plasma density andfpe/fce, respectively. Here fpe/fce is plotted only for 2.5 < L <6.5, since this is the L range over which the wave particle
interactions are introduced into the model. It can be seenthat during storm recovery the values of fpe/fce are low
outside the plasmapause at 3 < L < 4. This is the region offavorable chorus-electron interactions and electron fluxes
start to increase at this location and time (Figure 7, top). As
wave-particle interactions continue, the high-flux region
extends to higher Ls and, to a less extent, to low L shells.
The expansion of the flux enhancement is a result of radial
diffusion since the active wave interaction region is shrinking in late recovery phase as indicated by the increasing

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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08belt at high energies. The RBE simulation is consistent withthese findings.

4. Space Weather Application

Figure 7. (top) RBE 2 – 6 MeV electron flux, (middle)
plasmasphere density, and (bottom) 8pe/8ce.

fpe/fce ratio. Green and Kivelson [2004] examined theenergetic electron data from POLAR/HIST instrument for
acceleration event between January 1997 and January 1999.
They conclude that the data is best explained by acceleration of an internal source and then outward radial diffusion
to geostationary orbit. Furthermore, Varotsou et al. [2005]
and Horne et al. [2006] found that local acceleration bywhistler mode chorus waves followed by inward andoutward radial diffusion can populate the entire radiation[22] As shown in Figure 1, the only inputs to the RBEmodel are solar wind speed, density, and the IMF, Dst, andKp data, which are all available near real time in public Web
sites. Even with increasing complexity, the RBE CPU
simulation/real time ratio is close to 1 when running on afast PC. The RBE model thus has great potential in space
weather applications in providing specification of the radiation environment in geospace. A simplified version of the
RBE model, using fewer grid points in M and K, is currently
running in real time to provide radiation belt nowcasting
updated every 15 min. The geosynchronous fluxes atlongitudes of GOES-11 and 12 are extracted from the
RBE real-time run and are plotted together with real-time
GOES electron (>0.6 MeV) data. The model-data comparison is continually posted at http://mcf.gsfc.nasa.gov/
RB_nowcast/. Figure 8 shows the RBE prediction and the
GOES data for the month of December 2006. The input data
to the RBE model are plotted in the bottom. The real-timeversion of the RBE model does not include wave-particle
interactions and thus Kp is not one of the input parameters.
As shown in Figure 8, the RBE model agrees well with the
GOES data. The simulation faithfully reproduces the diurnal
variation due to the offset of geographic and magnetic
equators. The RBE model also closely follows the observed
flux drop-off in response to decreasing Dst. The model
performance is less exemplary during quiet conditions,where the predicted flux is usually lower than observed.
Wave-induced diffusion is being implemented in the realtime version of the RBE model. We expect these additional
processes will improve the prediction accuracy not only at
the geosynchronous orbit but, more importantly, at the heartof the radiation belt. It should be emphasized that themeasurements presented in Figure 8 are all real-time data
without any selection or noise removal.
[23] The RBE model is an effective tool to predict the
response of the radiation belts to adverse solar wind conditions. However, the current design allows the model toperform near real-time nowcasting only. To enhance thespace weather applications of the model, it must provide
hours to days of lead time in forecasting. To accomplish thisgoal, the RBE code must be improved to yield higher
predictive accuracy. The model is then ready to beconnected to a model that is able to specify the upstream
solar wind conditions with sufficient lead time. Solar windmodels of this kind have been established or are underdevelopment [Fry et al., 2001; Detman et al., 2006]. The
merger of radiation belt and solar wind models represents avery worthwhile future effort in space weather applications[Akasofu, 2001].

5. Discussion and Conclusions[24] In equation (1) there is no explicit description ofradial diffusion transport in the RBE model. As we mentioned above, the effects of radial transport are represented
by the time-varying magnetic and electric fields. Since weonly update the magnetic field every 5 min, we are missing

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A03S08

FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08Figure 8. (top) Comparison of GOES electron fluxes (>0.6 MeV) with real-time RBE output at GOES
satellite locations and (bottom) inputs to the RBE model.
the diffusive effects from the high-frequency Pc4 – Pc5
fluctuations. In our previous studies [Fok et al., 2001;
Zheng et al., 2003], we have included a radial diffusion
term in the kinetic equation with diffusion coefficient given
by Cornwall [1972]. We found, during magnetic storms, thetransport due to time-varying magnetic field configuration ismuch larger than that from pure radial diffusion. The effectof radial diffusion may be stronger if the improved, realistic
diffusion coefficients are used [i.e., Elkington et al., 2003].
[25] The RBE model is sensitive to its auxiliary models:magnetic field model, electric field model, plasma sheet
model, plasmasphere model, and the wave-diffusion model.
Inaccuracy in these models will limit the RBE performance.
A good example is the selection of the magnetic fieldmodel. It is well known that the motions of energetic ionsand electrons are strongly controlled by the magnetic field
configuration. To illustrate the effects of magnetic field on the
RBE results, the October 2002 event is rerun using the
Tsyganenko 96 (T96) model [Tsyganenko, 1995; Tsyganenko
and Stern, 1996]. The T96 model does not consider thecontribution from the partial ring current and the history ofsolar wind parameters, as all these effects are included in theT04 model. Figure 9 shows the L-time plots of the simulated
electron flux in T96 magnetic field with (Figure 9, top) andwithout (Figure 9, bottom) chorus wave interactions. We firstcompare the electron fluxes calculated in the two Tsyganenko
models in the absence of VLF waves. It can be seen from
Figure 9 (bottom) that a flux enhancement is found at L  3.2
during the storm recovery on 25– 26 October, similar to butwith lower intensity than that using the T04 model (Figure 6,
bottom). Furthermore, in the T96 case, there is no noticeableenhancement at L  4 as in the T04 case. The lower flux in the
RBE-T96 run indicates that the T96 model underestimates
the storm time magnetic field disturbances and thus theefficiencies of radial diffusion and particle trapping. Since
the T96 model fails to provide a substantial seed populationin the outer belt, including chorus associated diffusion only
produces a weak enhancement there at late recovery(Figure 9, top).
[26] We have shown very good agreement between the
RBE simulation results and actual SAMPEX and geosynchronous particle data. Nevertheless, more work can bedone to improve the model further. Cross-diffusion inenergy and pitch angle, which has been found to be

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FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHERA03S08that the enhanced flux of energetic electrons that appearedin the slot region can be explained by wave acceleration bywhistler mode chorus waves [Horne et al., 2005; Shprits etal., 2006].
[28] As shown in section 3, simulations from the RBEmodel provide an explanation of the flux enhancements
seen by the SAMPEX satellite during a particular storm.
The RBE model will be even more relevant to the upcoming
Living With a Star Radiation Belt Storm Probe (RBSP)
mission, which has a low-inclination orbit and broad particle and wave measurements (NASA/TM-2002-211613).
The RBE or RBE-like models will be very useful in future
RBSP data analysis and interpretation. On the other hand,data from the RBSP mission can be used to probe theimportance of specific physical processes in the model.
[29] In summary, a physics-based Radiation Belt Environment (RBE) model has been developed to understand theradiation belt dynamics and provide real time predictions of
the radiation belt environment that can be compared withexisting and new data sets. To date the findings from thismodel development work include:[30] 1. In modeling the storm on 23– 27 October 2002,
acceleration by chorus waves is found to be responsible forelectron enhancement at the center of the outer belt.
[31] 2. The RBE model is running in real time to provide
nowcasting of the radiation environment. The RBE forecasting capability will be enhanced if it is connected to asolar wind forecast model.
[32] 3. Future improvements of the RBE model include
adding substorm effects, considering additional wave modes
and cross-diffusion.
Figure 9. L-time diagrams of RBE simulated electron flux
in T96 magnetic field with (top) and without (bottom)
wave-particle interactions.
important [Albert and Young, 2005], has not been includedin the model. Interactions with high-latitude whistler modechorus and other waves, such as electromagnetic ion cyclotron waves and plasmaspheric hiss, should be considered aswell. The diffusive effects caused by ULF waves withrealistic diffusion coefficients should be reconsidered. The
boundary condition of the RBE model (equation (5)) may
be oversimplified and does not currently include the effectsof substorm injection. All these processes will be introduced
into the simulation and explored in future studies.
[27] During quiet periods, the outer belt, the slot, and theinner belt are well-defined regions. However, during intensestorms, significant flux increases are seen in the slot region[e.g., Baker et al., 2004] and even deep into the inner belt.
The penetration distance has been found to be controlled by
the strength of the storm as indicated by the Dst index[Tverskaya, 1986, 1996; Tverskaya et al., 2003; Zheng etal., 2006]. On the other hand, the plasmapause displaces ina similar manner. Li et al. [2006] found a remarkablecorrelation between the inner edge of the outer belt electrons and the plasmapause location. Since strong chorus
acceleration of electrons takes place near the plasmapause[Horne et al., 2006], the storm time erosion of the plasmasphere may place the electron acceleration zone, and thus
flux enhancement, very close the Earth to the typical slot
and inner belt regions. This hypothesis was tested during theHalloween storm in 2003 [Baker et al., 2004]. It was shown[33] Acknowledgments. We like to thank Daniel Ober for providing
the core plasma model. The LANL SOPA data are contributed by Geoff
Reeves. SAMPEX data are provided by Shri Kanekal. Real-time GOES and
ACE data are available from the NOAA Space Environment Center. Real
time Dst data are provided by Kyoto University Geomagnetic Data Service.
Thanks to Daniel Spicer, Jay Albert, Dimitris Vassiliadis, and Shing Fungfor helpful suggestions and comments. The RBE real time runs were
performed by Henry Lo. This research is supported by NASA Science
Mission Directorate, Heliophysics Division, Living With a Star Targeted
Research and Technology program, under Work Breakdown Structure
936723.02.01.01.15 and 936723.02.01.01.27.
[34] Amitava Bhattacharjee thanks T. Paul O’Brien and another reviewer for their assistance in evaluating this paper.

References
Akasofu, S.-I. (2001), Predicting geomagnetic storms as a space weather
project, in Space Weather, Geophys. Monogr. Ser., vol. 125, edited by
P. Song, et al., pp. 329 – 337, AGU, Washington, D. C.
Albert, J. M. (1994), Quasi-linear pitch angle diffusion coefficients: Retaining
high harmonics, J. Geophys. Res., 99, 23,741 – 23,745.
Albert, J. M., and S. L. Young (2005), Multidimensional quasi-linear diffusion of radiation belt electrons, Geophys. Res. Lett., 32, L14110,
doi:10.1029/2005GL023191.
Albert, J. M., D. H. Brautigam, R. V. Hilmer, and G. P. Ginet (2001),
Dynamic radiation belt modeling at the Air Force Research Laboratory,in Space Weather, Geophys. Monogr. Ser., vol. 125, edited by P. Song, etal., pp. 281 – 287, AGU, Washington, D. C.
Baker, D. N. (2001), Satellite anomalies due to space storms, in Space
Storms and Space Weather Hazards, edited by I. A. Daglis, pp. 285 –
311, Kluwer Acad., Boston.
Baker, D. N., G. M. Mason, O. Figueroa, G. Colon, J. G. Watzin, and R. M.
Aleman (1993), An overview of the Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX) mission, IEEE Trans. Geosci. Remote Sens., 31, 531 – 541.
Baker, D. N., S. G. Kanekal, X. Li, S. P. Monk, J. Goldstein, andJ. L. Burch (2004), An extreme distortion of the Van Allen beltarising from the Halloween solar storm in 2003, Nature, 432, 878 –881.

10 of 12

A03S08

FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHER

Borovsky, J. E., M. F. Thomsen, and R. C. Elphic (1998), The driving of
the plasma sheet by the solar wind, J. Geophys. Res., 103, 17,617 –
17,639.
Bourdarie, S., D. Boscher, T. Beutier, J.-A. Sauvaud, and M. Blanc (1997),
Electron and proton radiation belt dynamic simulations during storm
periods: A new asymmetric convection-diffusion model, J. Geophys.
Res., 102, 17,541 – 17,552.
Cook, W. R., et al. (1993), PET: A proton/electron telescope for studies ofmagnetospheric, solar, and galactic particles, IEEE Trans. Geosci. Remote Sens., 31, 565 – 571.
Cornwall, J. M. (1972), Radial diffusion of ionized helium and protons: A
probe for magnetospheric dynamics, J. Geophys. Res., 77, 1756 – 1770.
Dessler, A. J., and R. Karplus (1961), Some effects of diamagnetic ring
currents on Van Allen radiation, J. Geophys. Res., 66, 2289 – 2295.
Detman, T., Z. Smith, M. Dryer, C. D. Fry, C. N. Arge, and V. Pizzo (2006),
A hybrid heliospheric modeling system; Background solar wind, J. Geophys. Res., 111, A07102, doi:10.1029/2005JA011430.
Elkington, S. R., M. K. Hudson, and A. A. Chan (2003), Resonant acceleration and diffusion of outer zone electrons in an asymmetric geomagnetic field, J. Geophys. Res., 108(A3), 1116, doi:10.1029/2001JA009202.
Fok, M.-C., and T. E. Moore (1997), Ring current modeling in a realistic
magnetic field configuration, Geophys. Res. Lett., 24, 1775 – 1778.
Fok, M.-C., J. U. Kozyra, A. F. Nagy, C. E. Rasmussen, and G. V. Khazanov(1993), Decay of equatorial ring current ions and associated aeronomicalconsequences, J. Geophys. Res., 98, 19,381 – 19,393.
Fok, M.-C., T. E. Moore, and M. E. Greenspan (1996), Ring current development during storm main phase, J. Geophys. Res., 101, 15,311 – 15,322.
Fok, M.-C., T. E. Moore, and W. N. Spjeldvik (2001), Rapid enhancementof radiation belt electron fluxes due to substorm dipolarization of thegeomagnetic field, J. Geophys. Res., 106, 3873 – 3881.
Fok, M.-C., Y. Ebihara, T. E. Moore, D. M. Ober, and K. A. Keller (2005),
Geospace storm processes coupling the ring current, radiation belt andplasmasphere, in Inner Magnetosphere Interactions: New Perspectives
from Imaging, Geophys. Monogr. Ser., vol. 159, edited by J. Burch etal., pp. 207 – 220, AGU, Washington, D. C.
Friedel, R. H., Y. Chen, G. D. Reeves, and T. Cayton (2006), Pitch angle
evolution of energetic electrons at geosynchronous orbit during disturbed
times, Eos Trans. AGU, 87(52), Fall Meet. Suppl., Abstract SM41C-08.
Fry, C. D., W. Sun, C. S. Deehr, M. Dryer, Z. Smith, S.-I. Akasofu,
M. Tokumaru, and M. Kojima (2001), Improvements to the HAF solar
wind model for space weather predictions, J. Geophys. Res., 106,
20,985 – 21,001.
Fung, S. F. (1996), Recent development in the NASA trapped radiation
model, in Radiation Belts: Models and Standards, Geophys. Monogr.
Ser., vol. 97, edited by J. F. Lemaire, D. Heynderickx, and D. N. Baker,
pp. 79 – 91, AGU, Washington, D. C.
Garcia, H. A. (1996), Energetic electron pitch angle distribution parametersat 6.6 Re, as deduced from GOES X-ray observations, Planet. Space Sci.,
44, 473 – 484.
Glauert, S. A., and R. B. Horne (2005), Calculation of pitch angle andenergy diffusion coefficients with the PADIE code, J. Geophys. Res.,
110, A04206, doi:10.1029/2004JA010851.
Green, J. C., and M. G. Kivelson (2004), Relativistic electrons in the outerradiation belt: Differentiating between acceleration mechanisms, J. Geophys. Res., 109, A03213, doi:10.1029/2003JA010153.
Green, J. C., T. G. Onsager, T. P. O’Brien, and D. N. Baker (2004), Testing
loss mechanisms capable of rapidly depleting relativistic electron flux inthe Earth’s outer radiation belt, J. Geophys. Res., 109, A12211,
doi:10.1029/2004JA010579.
Horne, R. B. (2002), The contribution of wave particle interactions toelectron loss and acceleration in the Earth’s radiation belts during geomagnetic storms, in Review of Radio Science 1999 – 2002, edited by W. R.
Stone, chap. 33, pp. 801 – 828, IEEE Press, Piscataway, N. J.
Horne, R. B., and R. M. Thorne (1998), Potential waves for relativistic
electron scattering and stochastic acceleration during magnetic storms,
Geophys. Res. Lett., 25, 3011 – 3014.
Horne, R. B., and R. M. Thorne (2003), Relativistic electron accelerationand precipitation during resonant interactions with whistler-mode chorus,
Geophys. Res. Lett., 30(10), 1527, doi:10.1029/2003GL016973.
Horne, R. B., R. M. Thorne, S. A. Glauert, J. M. Albert, N. P. Meredith, and
R. R. Anderson (2005a), Timescale for radiation belt electron acceleration by whistler mode chorus waves, J. Geophys. Res., 110, A03225,
doi:10.1029/2004JA010811.
Horne, R. B., et al. (2005b), Wave acceleration of electrons in the Van Allenradiation belts, Nature, 437, 227, doi:10.1038/nature03939.
Horne, R. B., N. P. Meredith, S. A. Glauert, A. Varotsou, D. Boscher, R. M.
Thorne, Y. Y. Shprits, and R. R. Anderson (2006), Mechanisms for theacceleration of radiation belt electrons, in Recurrent Magnetic Storms:
Corotating Solar Wind Streams, Geophys. Monogr. Ser., vol. 167, edited
by B. T. Tsurutani et al., pp. 151 – 173, AGU, Washington, D. C.

A03S08Hudson, M. K., S. R. Elkington, J. G. Lyon, V. A. Marchenko, I. Roth,
M. Temerin, and M. S. Gussenhoven (1996), MHD/particle simulationsof radiation belt formation during a storm sudden commencement, in
Radiation Belts: Models and Standards, Geophys. Monogr. Ser., vol.
97, edited by J. F. Lemaire, D. Heynderickx, and D. N. Baker, pp. 57 –62, AGU, Washington, D. C.
Iles, R. H. A., A. N. Fazakerley, A. D. Johnstone, N. P. Meredith, and
P. Buhler (2002), The relativistic electron response in the outer radiation belt during magnetic storms, Ann. Geophys., 20, 957 – 965.
Kanekal, S. G., D. N. Baker, and J. B. Blake (2001), Multisatellite measurements of relativistic electron: Global coherence, J. Geophys. Res.,
106, 29,721 – 29,732.
Kanekal, S. G., R. Friedel, G. D. Reeves, D. N. Baker, and J. B. Blake(2005), Relativistic electron events in 2002: Studies of pitch angle isotropization, J. Geophys. Res., 110, A12224, doi:10.1029/2004JA010974.
Kim, H.-J., and A. A. Chan (1997), Fully relativistic changes in storm time
relativistic electron fluxes, J. Geophys. Res., 102, 22,107 – 22,116.
Li, X., I. Roth, M. Temerin, J. Wygant, M. K. Hudson, and J. B. Blake(1993), Simulation of the prompt energization and transport of radiation
particles during the March 23, 1991, SSC, Geophys. Res. Lett., 20,2423 – 2426.
Li, X., M. Temerin, D. N. Baker, G. D. Reeves, and D. Larson (2001),
Quantitative prediction of radiation belt electrons at geostationary orbit
based on solar wind measurements, Geophys. Res. Lett., 28, 1887 – 1890.
Li, X., D. N. Baker, T. P. O’Brien, L. Xie, and Q. G. Zong (2006), Correlation between the inner edge of outer radiation belt electrons and theinnermost plasmapause location, Geophys. Res. Lett., 33, L14107,
doi:10.1029/2006GL026294.
Lyons, L. R., R. M. Thorne, and C. F. Kennel (1972), Pitch-angle diffusionof radiation belt electrons with the plasmasphere, J. Geophys. Res., 77,3455 – 3474.
Meredith, P. N., R. B. Horne, and R. R. Anderson (2001), Substorm dependence of chorus amplitudes: Implications for the acceleration of electrons to relativistic energy, J. Geophys. Res., 106, 13,165 – 13,178.
Meredith, N. P., R. B. Horne, R. H. A. Iles, R. M. Thorne, D. Heynderickx,
and R. R. Anderson (2002), Outer zone relativistic electron acceleration
associated with substorm-enhanced whistler-mode chorus, J. Geophys.
Res., 107(A7), 1144, doi:10.1029/2001JA900146.
Meredith, N. P., M. Cain, R. B. Horne, R. M. Thorne, D. Summers, and
R. R. Anderson (2003a), Evidence for chorus-driven electron acceleration
to relativistic energies from a survey of geomagnetically disturbed periods, J. Geophys. Res., 108(A6), 1248, doi:10.1029/2002JA009764.
Meredith, P. N., R. B. Horne, R. M. Thorne, and R. R. Anderson (2003b),
Favored regions for chorus-driven electron acceleration to relativisticenergies in the Earth’s outer radiation belt, Geophys. Res. Lett., 30(16),
1871, doi:10.1029/2003GL017698.
Meredith, N. P., R. B. Horne, S. A. Glauert, and R. R. Anderson (2007),
Slot region electron loss timescales due to plasmaspheric hiss and lightning generated whistlers, J. Geophys. Res., 112, A08214, doi:10.1029/
2007JA012413.
Miyoshi, Y., A. Morioka, T. Obara, H. Misawa, T. Nagai, and Y. Kasahara(2003), Rebuilding process of the outer radiation belt during the November 3, 1993, magnetic storm: NOAA and EXOS-D observations, J. Geophys. Res., 108(A1), 1004, doi:10.1029/2001JA007542.
Miyoshi, Y. S., V. K. Jordanova, A. Morioka, M. F. Thomsen, G. D.
Reeves, D. S. Evans, and J. C. Green (2006), Observations and modelingof energetic electron dynamics during the October 2001 storm, J. Geophys. Res., 111, A11S02, doi:10.1029/2005JA011351.
Ober, D. M., J. L. Horwitz, and D. L. Gallagher (1997), Formation ofdensity troughs embedded in the outer plasmasphere by subauroral ion
drift events, J. Geophys. Res., 102, 14,595 – 14,602.
Paulikas, G. A., and J. B. Blake (1979), Effects of the solar wind on
magnetospheric dynamics: Energetic electrons at the synchronous orbit,
in Quantitative Modeling of Magnetospheric Processes, Geophys.
Monogr. Ser., vol. 21, edited by W.-P. Olsen, pp. 180 – 202, AGU,
Washington, D. C.
Reeves, G. D., K. L. McAdams, and R. H. W. Friedel (2003), Accelerationand loss of relativistic electrons during geomagnetic storms, Geophys.
Res. Lett., 30(10), 1529, doi:10.1029/2002GL016513.
Schulz, M., and L. J. Lanzerotti (1974), Particle Diffusion in the Radiation
Belts, Phys. and Chem. in Space, vol. 7, Springer, New York.
Shprits, Y. Y., R. M. Thorne, R. B. Horne, S. A. Glauert, M. Cartwright,
C. T. Russel, D. N. Baker, and S. G. Kanekal (2006), Acceleration
mechanism responsible for the formation of the new radiation belt during
the 2003 Halloween solar storms, Geophys. Res. Lett., 33, L05104,
doi:10.1029/2005GL024256.
Sibeck, D. G., R. W. McEntire, A. T. Y. Lui, R. E. Lopez, and S. M.
Krimigis (1987), Magnetic field drift shell splitting: Cause of unusual
dayside particle pitch angle distributions during storms and substorms,
J. Geophys. Res., 92, 13,485 – 13,497.

11 of 12

A03S08

FOK ET AL.: RADIATION BELT MODEL AND SPACE WEATHER

Summers, D., and C. Ma (2000), A model for generating relativistic electrons in the Earth’s inner magnetosphere based on gyroresonant waveparticle interactions, J. Geophys. Res., 105, 2625 – 2640.
Summers, D., R. M. Thorne, and F. Xiao (1998), Relativistic theory ofwave-particle resonant diffusion with application to electron accelerationin the magnetosphere, J. Geophys. Res., 103, 20,487 – 20,500.
Summers, D., B. Ni, and N. P. Meredith (2007), Timescales for radiation
belt electron acceleration and loss due to resonant wave particle interactions: 2. Evaluation for VLF chorus, ELF hiss, and EMIC waves,
J. Geophys. Res., 112, A04207, doi:10.1029/2006JA011993.
Thorne, R. M., R. B. Horne, S. A. Glauert, N. P. Meredith, Y. Y. Shprits,
D. Summers, and R. R. Anderson (2005), The influence of wave-particle
interactions on relativistic electrons during storms, in Inner Magnetosphere Interactions: New Perspectives From Imaging, Geophys. Monogr.
Ser., vol. 159, edited by J. Burch, M. Schulz, and H. Spence, AGU,
Washington, D. C.
Tsyganenko, N. A. (1995), Modeling the Earth’s magnetospheric magnetic
field confined within a realistic magnetopause, J. Geophys. Res., 100,
5599 – 5612.
Tsyganenko, N. A., and M. I. Sitnov (2005), Modeling the dynamics of theinner magnetosphere during strong geomagnetic storms, J. Geophys.
Res., 110, A03208, doi:10.1029/2004JA010798.
Tsyganenko, N. A., and D. P. Stern (1996), Modeling the global magnetic
field of the large-scale Birkeland current systems, J. Geophys. Res., 101,
27,187 – 27,198.
Tverskaya, L. V. (1986), Electron-injection boundary in the Earth’s magnetosphere, Geomagn. Aeron., 26, 736 – 737.
Tverskaya, L. V. (1996), The latitude position dependence of the relativistic
electron maximum as a function of Dst, Adv. Space Res., 18, 135.

A03S08Tverskaya, L. V., N. N. Pavlov, J. B. Blake, R. S. Selesnick, and J. F.
Fennell (2003), Predicting the L-position of the storm-injected relativistic
electron belt, Adv. Space Res., 31, 1039 – 1044.
Ukhorskiy, A. Y., B. J. Anderson, P. C. Brandt, and N. A. Tsyganenko(2006), Storm time evolution of the outer radiation belt: Transport andlosses, J. Geophys. Res., 111, A11S03, doi:10.1029/2006JA011690.
Varotsou, A., D. Boscher, S. Bourdarie, R. B. Horne, S. A. Glauert, and
N. P. Meredith (2005), Simulation of the outer radiation belt electrons
near geosynchronous orbit including both radial diffusion and resonantinteraction with whistler mode chorus waves, Geophys. Res. Lett., 32,
L19106, doi:10.1029/2005GL023282.
Vette, J. I. (1991), The AE-8 trapped electron model environment, NSSDC/
WDC-A-R&S 91-24, NASA Goddard Space Flight Cent., Greenbelt, Md.
Weimer, D. R. (2001), An improved model of ionospheric electric potentials including substorm perturbations and applications to the Geospace
Environment Modeling November 24, 1996, event, J. Geophys. Res.,
106, 407 – 416.
Zheng, Y., M.-C. Fok, and G. V. Khazanov (2003), A radiation belt-ringcurrent forecasting model, Space Weather, 1(3), 1013, doi:10.1029/
2003SW000007.
Zheng, Y., A. T. Y. Lui, X. Li, and M.-C. Fok (2006), Characteristics of 2 –6 MeV electrons in the slot region and inner radiation belt, J. Geophys.
Res., 111, A10204, doi:10.1029/2006JA011748.


M.-C. Fok, Geospace Physics Laboratory, NASA Goddard Space Flight
Center, Code 673, Greenbelt, MD 20771, USA. (mei-ching.h.fok@nasa.gov)
S. A. Glauert, R. B. Horne, and N. P. Meredith, Physical Science
Division, British Antarctic Survey, High Cross, Madingley Road, Cambridge CB3 0ET, UK.

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A05S90, doi:10.1029/2007JA012441, 2008

for

Full
ArticleNonlinear force-free coronal magnetic field
extrapolation scheme based on the direct boundary
integral formulation
Han He1,2 and Huaning Wang1
Received 1 April 2007; revised 7 January 2008; accepted 5 March 2008; published 2 May 2008.

[1] The boundary integral equation (BIE) method was first proposed by Yan and Sakurai(2000) and used to extrapolate the nonlinear force-free magnetic field in the solar
atmosphere. Recently, Yan and Li (2006) improved the BIE method and proposed the
direct boundary integral equation (DBIE) formulation, which represents the nonlinear
force-free magnetic field by direct integration of the magnetic field on the bottom
boundary surface. On the basis of this new method, we devised a practical calculation
scheme for the nonlinear force-free field extrapolation above solar active regions. The
code of the scheme was tested by the analytical solutions of Low and Lou (1990) and wasapplied to the observed vector magnetogram of solar active region NOAA 9077. The
results of the calculations show that the improvement of the new computational scheme tothe scheme of Yan and Li (2006) is significant, and the force-free and divergence-freeconstraints are well satisfied in the extrapolated fields. The calculated field lines for
NOAA 9077 present the X-shaped structure and can be helpful for understanding themagnetic configuration of the filament channel as well as the magnetic reconnection
process during the Bastille Day flare on 14 July 2000.
Citation: He, H., and H. Wang (2008), Nonlinear force-free coronal magnetic field extrapolation scheme based on the directboundary integral formulation, J. Geophys. Res., 113, A05S90, doi:10.1029/2007JA012441.

1. Introduction[2] Both observations and theoretical analyses reveal thatthe magnetic field plays an important role in the activity
phenomena of the solar atmosphere: moving plasmas are
confined to magnetic field lines, and the magnetic field also
provides the energy for solar flares and other eruptive
phenomena [Tsuneta et al., 1992; Masuda et al., 1994;
Shibata et al., 1995; Wang et al., 1996; Tsuneta, 1996;
Priest and Forbes, 2002; Shibata, 2004; Lin et al., 2005;
Schwenn, 2006]. To understand the physical mechanisms ofthese activities in the solar atmosphere, an important step is
to find out the underlying structure of the magnetic field
above the related active region. Currently, the direct measurement of the magnetic field in the solar chromosphereand corona is not as sophisticated as observation in thephotosphere. The commonly used method for understanding
the configurations of the magnetic fields above the photosphere is extrapolation: the fields can be reconstructed froma physical model in which the observed photosphericmagnetic field is taken as a boundary condition. The
force-free field model is often adopted for this purpose,

1
National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, China.
2
Also at Kwasan and Hida Observatories, Graduate School of Science,
Kyoto University, Kyoto, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012441$09.00

since it is a reasonable approximation in the solar chromosphere and corona [Metcalf et al., 1995].
[3] The nonlinear force-free magnetic field, with the fieldlines being everywhere aligned parallel to the electric
current density, can be described by equations:ðr  BÞ  B ¼ 0;

ð1Þr B ¼ 0:ð2ÞEquation (2) is the divergence-free constraint of themagnetic field. The force-free constraint (1) can also be
written asr  B ¼ aðrÞB;ð3Þ

where a is a function of spatial location, usually called theforce-free parameter or force-free factor. It is constant alongeach field line, which can be determined from the bottom
boundary condition.
[4] Currently, several methods for the nonlinear forcefree field (NLFFF) extrapolation have been proposed[Sakurai, 1981; Yan and Sakurai, 2000; Wheatland et al.,
2000; Wiegelmann, 2004; Régnier and Amari, 2004; Valori
et al., 2005; Yan, 2005; Wiegelmann et al., 2006; Yan andLi, 2006; Amari et al., 2006; Wheatland, 2006; Song et al.,2006; Schrijver et al., 2006; Wiegelmann, 2007; Song et al.,2007; Valori et al., 2007; Fuhrmann et al., 2007]. As one of

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HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

A05S90

A05S90where r = [(x  xi)2 + (y  yi)2 + (z  zi)2]1/2 is the distance
between a variable point (x, y, z) and the given field point(xi, yi, zi), r0 = [(x  xi)2 + (y  yi)2 + (z + zi)2]1/2, as shownin Figure 1. The parameter l in equation (7) has the same
dimension (reciprocal of length) as the force-free factor a,which is defined by equation
Z
W

Figure 1. The geometry for application of the directboundary integral equation (DBIE).
the techniques for nonlinear force-free field modeling, theboundary integral equation (BIE) method was first proposedby Yan and Sakurai [2000]. Recently, Yan and Li [2006]
improved the BIE and proposed the direct boundary integral
equation (DBIE) formulation, which represents the forcefree magnetic field by direct integration of the magnetic
field on the bottom boundary surface. Figure 1 shows thegeometry for application of DBIE [Yan and Li, 2006]. On
the bottom boundary surface G (infinite plane), the boundary condition is
B ¼ B0 on G;

ð4Þ

where B0 denotes the known boundary values which can besupplied from vector magnetogram measurements. At
infinity, an asymptotic constraint condition is also introduced to ensure a finite energy content in the semispace W
above G, 
B ¼ O r2 when r ! 1;

ð5Þ

where r is the radial distance.
[5] According to the DBIE method, after a series ofderivations using the two constraint conditions (2) – (3)and the two boundary conditions (4) – (5), the magnetic
strength at the field point (xi, yi, zi) in W can be represented
by the equation [Yan and Li, 2006]:
Z
Bi ¼ @Y
B0 dG ¼
G @n

Z
G

@Y
B0 dG:@z

ð6Þ

The reference function Y in equation (6) is chosen as
Y¼

cosðlrÞ cosðlr0 Þ
;

4pr
4pr0

ð7Þ



Y l2 B  a2 B  ra  B dW ¼ 0:

ð8ÞIn the case of a nonlinear force-free field, corresponding toBix, Biy, and Biz in equation (6), there exist three componentsof the reference function, Yx, Yy, and Yz, together with lx, ly,
and lz defined locally at the given field point by equation (8).
The values of the three components of l are generally
different [Li et al., 2004; Yan and Li, 2006; He and Wang,2006].
[6] Once the parameter lx, ly, and lz at the field point (xi,yi, zi) are given, the magnetic field Bi can be calculated byintegration of the magnetic field on the bottom boundary
surface through the DBIE formulation (6). Because it iscostly to determine l directly from the volume integral
equation (8), Yan and Li [2006] devised an optimal
approach to find the suitable l values locally by using the
DBIE (6) together with the force-free constraint condition(1). At the field point (xi, yi, zi), they defined an evaluate
function as: jJ  Bj
fi lx ; ly ; lz ¼
; with J ¼ r  B;
jJjjBjð9Þwhich measures the absolute value of sine of the angle
between J and B. For any initial values of lx, ly, and lz, B
at the field point and its neighborhood can be calculated by
DBIE (6), then the value of fi(lx, ly, lz) can be obtained byequation (9). In an ideal situation, the suitable l values(denoted by l*x, l*y, and l*z) can be determined byfi lx* ; ly* ; l*z ¼ 0;

ð10Þwhich is equivalent to the force-free field equation (1). In
practical computing, one needs to find (l*x, l*y, l*z) thatsatisfies
fi lx* ; l*y ; lz* ¼ min fi lx ; ly ; lz :ð11ÞA downhill simplex method can be employed to fulfil thetask of equation (11) [Nelder and Mead, 1965; Yan and Li,
2006]. The computation of Yan and Li [2006] shows that if
equation (10) is satisfied, the divergence-free constraint (2)can also be satisfied at the neighborhood of the given field
point. This property indicates that the divergence-freeconstraint (2) has been involved during the derivation of
the DBIE formulation (6).
[7] We followed the main idea of the strategy proposedby Yan and Li [2006] and devised the upward boundary
integration scheme for the nonlinear force-free field extrapolation. That is, while we take into account the wholeboundary data information through the DBIE formulation(6), the parameter l at a given field point can be determined

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HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

A05S902. Upward Boundary Integration Scheme

every layer. In some circumstances, especially in the case of
jJj or jBj approaches to zero in equation (9), equation (11)may fail to achieve a convergent result. The problem
appears in the form of isolated points where singularities
appear in the raw data of Gn+1, which can be eliminated bysmoothing processes with the help of the nearest grid points
in the x- and y- directions.
[10] In the original procedure of Yan and Li [2006], thevalues of fi and J in equation (9) are calculated in theinfinitesimal neighborhood ±d of the field point in the x-, y-,and z-directions (small cubic volume surrounding the field
point). The computing of the integration in equation (6)should be done seven times (six sides of the small cube plus
the field point itself) to obtain the value of fi and J. To
reduce the loads of computing and fully utilize the boundary
data information at Gn in our scheme, we calculate fi and J
in a small square pyramid between the field point i and theneighboring grid points at the boundary Gn, as shown in
Figure 3. The height of the pyramid is l (distance between
Gn+1 and Gn) and the side length of the square base is 2l,
where l denotes the space between two consecutive grid
points at Gn (the scale of one pixel). Given an arbitrary l at
field point i, to obtain the values of fi(l) and J(l) in
equation (9) we only need to carry out the integration of
DBIE (6) one time at the field point i to calculate Bi(l), thus
saving computing time as compared to the original procedure of Yan and Li [2006]. Once Bi(l) is known, Jx, Jy, andJz can be calculated in two isosceles triangles and the squarebase of the pyramid, respectively, as indicated by different
colors in Figure 3. Bo at the center of the square base, as
shown in Figure 3, is employed to complete the calculationof fi(l). Then we can use equation (11) to find the suitablevalue of l*. Once l* is determined, Bi(l*) will be the final
result at the field point i for the nonlinear force-free fieldmodeling.
[11] We employed the same code of the downhill simplex
method as used by Yan and Li [2006] to perform multidimensional minimization of fi in equation (11). The initialstarting point of (lx, ly, lz) for the downhill simplex[8] In the original computational procedure proposed by
Yan and Li [2006], the equation (11) has better convergenceproperty at the field points with lower altitude (explained inAppendix A). To take advantage of this property and to
avoid the increasing errors at the field points with higher
altitude, we devised a new scheme for nonlinear force-freefield extrapolation based on the DBIE formulation (6),which we called the upward boundary integration scheme.
[9] In the new scheme, as shown in Figure 2, thecomputation is carried out upwardly, from G0 (photosphere)
to G1, then from G1 to G2, and so on. In the original
procedure of Yan and Li [2006], the bottom boundary was
fixed to the photosphere, while in our scheme the bottom
boundary for applying the DBIE formulation (6) is moved
upwardly layer by layer. That is, we always calculate thefield distribution at Gn+1 from the data of the new bottom
boundary Gn. The step distance between the two consecutive layers is the same as the space between the twoconsecutive grid points at Gn (the scale of one pixel). Since
Gn+1 is very close to Gn, the field points at Gn+1 are always atvery low altitude relative to the bottom boundary Gn. Thenequation (11) can be used to achieve good convergence at

Figure 3. The diagram to illustrate the calculation ofJ(r  B) in the small square pyramid between the field
point i and the neighboring grid points at Gn. The height ofthe square pyramid is l and the side length of the squarebase is 2l, where l denotes the space between two
consecutive grid points at Gn. Jx and Jy are calculated in
two isosceles triangles, respectively, and Jz is calculated in
the square base of the pyramid, as indicated by different
colors.

Figure 2. The diagram to illustrate the upward boundary
integration scheme for nonlinear force-free field extrapolation based on the DBIE formulation (6). The computation iscarried out layer by layer upwardly, the field distribution at
Gn+1 is calculated from the data of Gn.
locally through the force-free equation (1) or equation (10)with the help of only the neighboring boundary data
information. Unlike the original procedure of Yan and Li[2006], our computation is carried out upwardly layer bylayer. The procedure and techniques of the new scheme isdescribed in section 2. The code of the scheme was testedby the analytical solutions of Low and Lou [1990] and wasapplied to the observed vector magnetogram of solar active
region NOAA 9077. The results of test calculations using
the analytical solutions are presented in section 3, and theresults of the calculations by using the observed vector
magnetogram are shown in section 4. In section 5, we givethe summary and conclusion.

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HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

Figure 4. The diagram to illustrate the boundary conditions for application of the DBIE method to a solar active
region with concentrated magnetic flux. At the bottom
boundary G, the observed vector magnetogram is bounded
in a finite square area which covers the main magnetic fluxof the active region. The remained flux outside the square
area is relatively very weak, and can be considered to be
zero as an approximation approach. The cubic volume
above the square boundary data area represents the output
region of the code.

method is selected as (0, 0, 0) in our code. The characteristic
length scale of l in the downhill simplex method is
specified as the maximum absolute value of force-free
factor a at the boundary surface G0 because l and a havethe same dimension (reciprocal of length) and the sameorder of magnitude [Li et al., 2004; Yan and Li, 2006; He
and Wang, 2006]. For Case I and Case II in section 3(analytical solutions of Low and Lou [1990]), the maximum
absolute values of a are 11.3 and 11.1, respectively. If we
adopt one pixel as the length unit (with a 64  64 grid forboundary area x, y2[1, +1], see section 3) as in our code,
the two values become 0.353 and 0.347. For the data ofactive regions observed in the photosphere (see section 4),
we choose the characteristic length scale of l as 10 
108 m1 [Pevtsov et al., 1995]. Also in our code, thelength unit is one pixel, for the common active regions with afield of view 30000  30000 and pixel number 64  64, thecharacteristic length scale of l becomes 0.342, whichapproximates to the values used by Case I and Case II of
the analytical solutions.
[12] As described in section 1, the DBIE formulation (6)demands that the bottom boundary G be an infinite planesurface [Yan and Li, 2006]. Figures 4 and 5 illustrates how
to apply the DBIE to a solar active region with concentrated
magnetic flux. At the bottom boundary G, the observed
vector magnetogram is bounded in a finite square area
which covers the main magnetic flux of the active region
as shown in Figure 4. The remained flux outside the square
area is relatively very weak and can be considered to be zeroas an approximation approach. Then, in practical calculation, we only need to carry out the integration of DBIE (6)A05S90over the finite square area of the active region, while thebottom boundary G is still an infinite plane surface.
[13] Considering that the main magnetic flux region as
well as the field lines may expand at higher layers, thesquare area for the integration is enlarged gradually layer bylayer during the calculation as illustrated in Figure 5 (inpresent code, from Gn to Gn+1, each side of the square area
expanding by one pixel). Meanwhile, we keep the original
pixel number of the square area at all layers by resampling
the grid points to save computing time. Thus the space
between the two consecutive grid points at Gn as well as the
distance between the two consecutive layers also increases
gradually with height, as shown in Figure 5. After the field
distributions at a series of layers are obtained, the values ofthe magnetic field in the space between the layers can becalculated through the technique of interpolation.
[14] In practical calculation, the code reads the boundary
data (square area, N  N array in the code), calculates thenonlinear force-free field distributions at all layers, andreforms the grid structure to a regular form through interpolation. The output of the code is the field distribution in a
cubic volume (N  N  N array in the code) which is justabove the boundary data area, as illustrated in Figures 4
and 5. The current code is written in IDL programming
language. The time needed for a 64  64  64 output gridis about 13 h on an 1.86 GHz Intel processor.

3. Testing the Code by Using the Analytical
Solutions of Low and Lou [1990][15] First, we test the code described in section 2 by using
the analytical nonlinear force-free field solutions given by
Low and Lou [1990]. The fields of the analytical solutions
are basically axially symmetric. The point source of theaxisymmetric fields is located at the origin of the sphericalcoordinate system, with the axis of symmetry pointing to
the Z direction associated with the spherical coordinate
system. By arbitrarily positioning the plane surface boundary G of DBIE in the space of the analytical fields, weFigure 5. The diagram to illustrate how to fit expanding
main magnetic flux region as well as field lines at higher
layers in the new computational scheme for the DBIE
method. The area for the integration of the DBIE is enlarged
gradually layer by layer. Meanwhile, pixel number of theareas is fixed at all layers by resampling the grid points tosave the computing time. The square dashed line indicatesthe final output region of the code.

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A05S90point source (origin of the spherical coordinate system), andF is the angle between the normal direction of G and the Z
axis associated with the spherical coordinate system [Lowand Lou, 1990; Schrijver et al., 2006]. In both cases, the field
distributions in the modeling volume bounded by x, y 2 [1,+1] and z 2 [0, 2] (x, y, and z are Cartesian coordinates defined
locally on the boundary surface G) were calculated based on
the bottom boundary data in the area x, y 2 [1, +1], just as
described in section 2 and illustrated in Figures 2 –5. The
pixel numbers of the bottom boundary data are 64  64.
3.1. Direct Comparison[16] The extrapolated field lines for Case I and Case II arecompared with the analytical solutions in Figures 7 and 8,
respectively. Left columns are images of analytical solutions
and right columns are images of calculated fields. It can beseen from Figures 7 and 8 that the orientations of theextrapolated field lines basically coincide with the analyticalsolutions. Figures 7 and 8 only show the inner volume (x, y 2[0.5, +0.5] and z 2[0, 1], with 32  32  32 grid) of themodeling space as indicated in Figure 6 by square dashedlines, where the quality of agreement between the extrapolated field and the analytical solutions is better than in themargin region.
[17] A vector correlation metric Cvec is employed toquantify the degree of agreement between the analyticalFigure 6. The global field configurations of the twoanalytical NLFFF solutions given by Low and Lou [1990],
which are employed to test the validity and accuracy of the
NLFFF extrapolation code (see section 3). The parameters
for Case I are n = 1, m = 1, L = 0.3, F = p/4, the parameters
for Case II are n = 3, m = 1, L = 0.3, F = 4p/5, where n andm are the eigenvalues of the solutions, L is the distance
between the plane surface boundary G and the point source(origin of the spherical coordinate system), and F is theangle between the normal direction of G and the Z axis
associated with the spherical coordinate system [Low andLou, 1990; Schrijver et al., 2006]. The radius of the spheresis equal to L(=0.3). The long thick lines indicate theposition and scale of the bottom boundary data area (x, y 2[1, +1]). The square dashed lines represent the innervolume (x, y 2 [0.5, +0.5] and z 2 [0, 1]) of the modelingspace. x, y, and z are Cartesian coordinates defined locallyon the plane surface boundary G.
obtain a different kind of boundary conditions prepared forextrapolation. Two cases are selected to test the validity andaccuracy of our code as shown in Figure 6, which are thesame cases as used by Schrijver et al. [2006]. The parameters for Case I are n = 1, m = 1, L = 0.3,F = p/4, theparameters for Case II are n = 3, m = 1, L = 0.3,F = 4p/5,
where n and m are the eigenvalues of the solutions, L is the
distance between the plane surface boundary G and the

Figure 7. The extrapolated field lines in the inner volume(x, y 2 [0.5, +0.5] and z 2 [0, 1]) of the modeling spacefor Case I compared with the analytical solution of Low andLou [1990]. (a) and (c) Left column is images of theanalytical solution and (b) and (d) right column is images ofthe extrapolated field. Top row are images in top view,
bottom row are images in 3-D view. Closed field lines areplotted in blue, and field lines that leave the modeling box
are in red.

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A05S90

A05S90Lorentz force and Ld of divergence of the fields, as used by
Schrijver et al. [2006]. Lf and Ld are defined as:
Lf ¼

1
V

Z

Ld ¼

Figure 8. The extrapolated field lines in the inner volume(x, y 2 [0.5, +0.5] and z 2 [0, 1]) of the modeling spacefor Case II compared with the analytical solution of Low
and Lou [1990]. (a) and (c) Left column is images of theanalytical solution, and (b) and (d) right column is imagesof the extrapolated field. Top row are images in top view,and bottom row are images in 3-D view. Closed field lines
are plotted in blue, and field lines that leave the modelingbox are in red.
field B and the extrapolated field b as used by Schrijver et
al. [2006]. To define Cvec, the equationP
Cvec ¼

P

i Bi

bi

2P
2
i jBi ji jbi j

1=2

;

ð12Þ

with Bi and bi as the field vectors of the analytical field B
and the extrapolated field b at each grid point i. If B and b
are identical, Cvec = 1; if Bi ? bi, Cvec = 0. We calculated thevalues of Cvec at each layer in the central domain (x, y 2[0.5, +0.5] and z 2[0, 2], with 32  32  64 grid) of themodeling space, the results are shown in Figure 9. Since weonly use the finite boundary data in the area x, y 2 [1, +1]
with 64  64 grid, it can be seen that the extrapolated fields
b deviate from the analytical fields B gradually with theincreasing of height. At the lower layers and in the central
domain, B and b get the best agreement, as illustrated in
Figures 7 and 8.
3.2. Internal Consistency of the Extrapolated Field[18] The internal consistency of the calculated field ismeasured by the force-free constraint (1) and divergencefree constraint (2). To check the extent to which theextrapolated fields satisfy the force-free and divergence-freeconstraints, we introduced the integral measures Lf of theB2 jðr  BÞ  Bj2 dV ;

ð13Þ

V

1
V

Z
jr Bj2 dV :

ð14Þ

V

We calculated Lf and Ld of the extrapolated field at eachlayer in the central domain (x, y 2 [0.5, +0.5] and z 2 [0, 2],with 32  32  64 grid, the unit of length in the calculationis one pixel) and plotted the curves of Lf versus height andLd versus height. The results for Case I and Case II are
shown in Figures 10 and 11, respectively. The curves foranalytical solutions of Low and Lou [1990] are also
calculated and shown in Figures 10 and 11 for reference.
[19] The curves for the analytical solutions (Figures 10aand 10c, Figures 11a and 11c) show a typical profile of Lf
versus height and Ld versus height for the nonlinear forcefree field. That is, the measures Lf and Ld decrease to zero
rapidly with the increase of height. The relatively large
values of Lf and Ld at the lower layers are the effects of thediscrete grid points and the finite-difference method used in
the calculations, and the values of Lf and Ld at the bottom
layer (layer number 0) are in the same order of magnitude.
(Exact values of Lf and Ld at the bottom layer can be foundin Tables 1 and 2.)[20] The curves for the extrapolated field (Figures 10b
and 10d and Figures 11b and 11d) show similar propertiesof the profile as do the analytical solutions, which indicatesthat the force-free constraint (1) and divergence-free constraint (2) are well satisfied in the extrapolated field.
Comparing the analytical solutions, the larger values of Lf
and Ld for the extrapolated field are the results of the errors
existing in the calculated fields, which were introduced by
the numerical computation and are amplified by the jBj2
terms in equations (13) and (14).
[21] Besides the values at each layer, the integral measures Lf and Ld were also calculated over the entire volume(x, y 2 [1, +1] and z 2 [0, 2], with 64  64  64 grid) as
well as the inner volume (x, y 2 [0.5, +0.5] and z 2 [0, 1],with 32  32  32 grid). All the values of Lf and Ld forCase I and Case II are listed in Tables 1 and 2, respectively.
The maximum values of jBj at the bottom boundary are also
included in Tables 1 and 2 for reference.
3.3. Comparing With Original Computational Schemeof Yan and Li [2006][22] As discussed in section 2 and Appendix A, in theoriginal computational scheme of Yan and Li [2006], theequation (11) has better convergence property at the field
points with lower altitude (near the bottom boundary) than
the field points with higher altitude (away from the bottom
boundary). To take advantage of this property and avoid theincreasing errors at the field points with higher altitude, we
devised the new upward boundary integration scheme for
NLFFF extrapolation. To check the improvement of the newscheme to the original computational procedure of Yan andLi [2006], we produced the curves of Cvec versus height for
Case I by using the original computational scheme proposed

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Figure 9. The curves of vector correlation metric Cvec (defined in equation (12)) versus height for
Case I (dotted line) and Case II (dashed line). The extrapolated fields are calculated by the code of the
new computational scheme. Cvec is calculated at each layer in the central domain x, y 2 [0.5, +0.5]
and z 2 [0, 2], with 32  32  64 grid.

Figure 10. Lf versus height and Ld versus height curves of the extrapolated field by using the newscheme for Case I compared with the analytical solution of Low and Lou [1990]. (a) and (c) Left column
is curves of the analytical solution, and (b) and (d) right column is curves of the extrapolated field. Lf and
Ld are integral measures of the Lorentz force and divergence, as defined in equations (13) and (14). Lf and
Ld are calculated at each layer in the central domain (x, y 2 [0.5, +0.5] and z 2 [0, 2], with 32  32 
64 grid), the unit of length in the calculation is one pixel.
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A05S90Figure 11. Lf versus height and Ld versus height curves of the extrapolated field by using the
new scheme for Case II compared with the analytical solution of Low and Lou [1990]. (a) and (c) Leftcolumn is curves of the analytical solution, and (b) and (d) right column is curves of the extrapolated
field. Lf and Ld are integral measures of the Lorentz force and divergence, as defined in equations (13)and (14). Lf and Ld are calculated at each layer in the central domain (x, y 2 [0.5, +0.5] and z 2 [0, 2],with 32  32  64 grid), the unit of length in the calculation is one pixel.
by Yan and Li [2006] and by using the new upward
boundary integration scheme. The two curves are comparedin Figure 12. The curve of Cvec versus height measures theagreement between the extrapolated field and the analyticalsolution at each layer. Cvec = 1 represents the best agreement as explained in section 3.1.
[23] It can be seen from the curves in Figure 12 that theextrapolated field from the original scheme of Yan and Li[2006] basically agrees with the analytical solution at lower
layers, but at higher layers, the two fields are totallydifferent. The extrapolated field by the new scheme is still
similar to the analytical field at higher layers, as shown in
Figure 12. The extrapolated field lines of the new scheme(Figure 7d) compared with the field lines of the originalscheme (Figure A1) also illustrate this improvement.

Table 1. Integral Measures Lf and Ld of the Extrapolated Field forCase Ia

Table 2. Integral Measures Lf and Ld of the Extrapolated Field forCase IIa

Lf over the inner volumeb
Ld over the inner volume
Lf over the entire volumec
Ld over the entire volume
Lf at the bottom layer of inner volume
Ld at the bottom layer of inner volume
Lf at the bottom layer of entire volume
Ld at the bottom layer of entire volume
Maximum of jBj at the bottom boundaryaAnalytical
Solution

Extrapolated
Field

0.008500
0.008694
0.001015
0.001037
0.15450.15800.037440.03827
299.4

1.228
1.853
0.20450.2793
11.04
22.71
3.066
5.861

Lf over the inner volumeb
Ld over the inner volume
Lf over the entire volumec
Ld over the entire volume
Lf at the bottom layer of inner volume
Ld at the bottom layer of inner volume
Lf at the bottom layer of entire volume
Ld at the bottom layer of entire volume
Maximum of jBj at the bottom boundary

Lf and Ld are defined in equations (13) and (14), and the unit of length in
the calculations is one pixel. The nonzero values of Lf and Ld for the
analytical solution are due to the discrete grid points and the finitedifference method used in the calculations.
b
Inner volume is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 1], with 32 
32  32 grid.
cEntire volume is bounded by x, y 2 [1.0, +1.0] and z 2 [0, 2], with64  64  64 grid.

Analytical
Solution

Extrapolated
Field

3.561
3.134
0.42440.3734
80.76
69.46
19.56
16.82
2517.7

80.28
52.48
9.901
6.555
1050.4
646.7
255.1
160.8

a
Lf and Ld are defined in equations (13) and (14), and the unit of length in
the calculations is one pixel. The nonzero values of Lf and Ld for the
analytical solution are due to the discrete grid points and the finitedifference method used in the calculations.
b
Inner volume is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 1], with32  32  32 grid.
cEntire volume is bounded by x, y 2 [1.0, +1.0] and z 2 [0, 2], with64  64  64 grid.

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A05S90Figure 12. The curve of Cvec versus height for Case I by using the original computational scheme
proposed by Yan and Li [2006] (dashed line), compared with the curve produced by using the new
upward boundary integration scheme (dotted line). Cvec is defined in equation (12) and is calculated ateach layer in the inner volume (x, y 2 [0.5, +0.5] and z 2 [0, 1], with 32  32  32 grid).
[24] Several techniques are introduced in the code of the
new extrapolation scheme to save computing time as
described in section 2. To calculate the NLFFF in a volume
with a 64  64  64 grid for Case I, the current code of the
new scheme needs about 13 h on an 1.86 GHz Intelprocessor in IDL programming language, while the codeof the original scheme of Yan and Li [2006] (also in IDL
programming language) needs about 104 h on the same
computer.

of agreement between the analytical fields B and theextrapolated field b in the modeling volume. The first metric
Cves are defined in equation (12), the other four metrics aredefined as [Schrijver et al., 2006]:
CCS ¼

1 X Bi bi;
M i jBi jjbi jð15ÞP
i jbi  Bi j
P;
i jBi jð16Þ

En0 ¼ 1 

3.4. Comparing With Other Integration Scheme for
NLFFF Extrapolation[25] Five evaluation metrics Cves, CCS, E0n, E0m and e are
introduced by Schrijver et al. [2006] to quantify the degree

Table 3. Evaluation Metrics for Case Iab

New scheme of DBIE (entire volume )New scheme of DBIE (inner volumec)
New scheme of DBIE (lower central domaind)
Original scheme of DBIE (lower central domain)
Integral scheme implemented by Liu (entire volume)eIntegral scheme implemented by Liu (inner volume)
Integral scheme implemented by Liu (lower central domain)

Cves

CCS

E0 n

E0m



0.97
0.98
0.99
0.98
0.88
0.98
0.98

0.86
0.94
0.97
0.85
0.47
0.85
0.94

0.57
0.72
0.81
0.75
0.10
0.71
0.77

0.045
0.41
0.70
0.60
3.00
0.44
0.69

1.08
1.06
1.04
0.96
1.10
0.87
0.86

a
The five evaluation metrics are defined in equations (12) and (15) – (18). In the new and original computational schemes of DBIE, only bottom boundary
data (x, y 2 [1, +1] with 64  64 grid) were used for extrapolation, while in the scheme of Liu, data of the bottom boundary and the four side boundariesof the modeling volume were used.
b
Entire volume is bounded by x, y 2 [1, +1] and z 2 [0, 2], with 64  64  64 grid.
c
Inner volume is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 1], with 32  32  32 grid.
dLower central domain is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 0.5], with 32  32  16 grid.
e
The values of the integral scheme implemented by Liu are recalculated by courtesy of Yang Liu. The small differences between the values listed here
and the values in the paper of Schrijver et al. [2006] are due to the different accuracies of the numerical solutions for the analytical field.

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Table 4. Evaluation Metrics for Case IIab

New scheme of DBIE (entire volume )New scheme of DBIE (inner volumec )New scheme of DBIE (lower central domaind )Original scheme of DBIE (lower central domain)
Integral scheme implemented by Liu (entire volume)eIntegral scheme implemented by Liu (inner volume)
Integral scheme implemented by Liu (lower central domain)

Cves

CCS

E0n

E0m



0.97
0.97
0.98
0.97
0.97
0.97
0.98

0.65
0.81
0.90
0.63
0.41
0.54
0.74

0.077
0.48
0.65
0.60
0.04
0.47
0.62

12.4
2.5
0.044
0.075
14.1
2.2
0.043

1.06
1.05
1.04
0.99
1.04
1.03
1.03

a
The five evaluation metrics are defined in equations (12) and (15) – (18). In the new and original computational schemes of DBIE, data of the bottom
boundary area x, y 2 [1, +1] (64  64 grid) were used for extrapolation, while in the scheme of Liu, data of the bottom boundary area x, y 2 [3, +3](192  192 grid) were used.
b
Entire volume is bounded by x, y 2 [1, +1] and z 2 [0, 2], with 64  64  64 grid.
c
Inner volume is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 1], with 32  32  32 grid.
dLower central domain is bounded by x, y 2 [0.5, +0.5] and z 2 [0, 0.5], with 32  32  16 grid.
e
The values of the integral scheme implemented by Liu are recalculated by courtesy of Yang Liu. The small differences between the values listed here
and the values in the paper of Schrijver et al. [2006] are due to the different accuracies of the numerical solutions for the analytical field.

Figure 13. Results of the NLFFF extrapolation for active region NOAA 9077 by using the code of the
new scheme. (a) The vector magnetogram used by the code. The white contours represent the positive
polarity of Bz, the black contours represent the negative polarity of Bz, the contour levels are ±50, 100,
200, 500, 1000, 1500, 2000, 3000 G. Small arrows overlying the contours represent Bt (transverse
component). The data was observed at 0414 UT on 14 July 2000 by the Solar Magnetic Field Telescope(SMFT) at Huairou Solar Observing Station. The field of view is 26900  26900 with pixel number as 64 
64 (4.200/pixel). The square dashed line indicates the central domain (32  32 grid) of the magnetogram.
(b) The top view of the extrapolated field. (c) The top view of the extrapolated field in the central domain.
(d) The 3-D view of the extrapolated field in the central domain along the direction of the polarity
inversion line as indicated by an arrow in the right margin of Figure 13c. Closed field lines are plotted inblue color, field lines that leave the modeling box are in red color. All field lines are plotted from layer 1(see Figure 17) to avoid the influence of noises in the data of bottom boundary.
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A05S90Figure 14. (a) and (b) The whole 3-D views of the closed field lines and the U-shaped field lines in theextrapolated field of NOAA 9077 (see section 4 and Figure 14e). The closed field lines are plotted fromlayer 1 (see Figure 17) to avoid the influence of noises in the data of bottom boundary. The U-shapedlines are plotted in the volume between layer 1 and layer 19. (c) and (d) The side views of the closed fieldlines and the U-shaped field lines along the direction of Y-Axis. (e) Diagram to illustrate the X-shaped
structure of the field lines in the extrapolated field of NOAA 9077 as shown in Figure 13d. The dashedcurve represents the U-shaped field lines above the X-point, which are not plotted and thus are displayed
as a cavity in Figure 13d.
Em0 ¼ 1 

1 X jbi  Bi j
;
M i
jBi j
P

 ¼ Pijbi j2i jBi j

2

;

ð17Þ

ð18Þ

where Bi and bi are the field vectors of the analytical field B
and the extrapolated field b at each grid point i, M is thetotal number of vectors in the volume. If B and b are
identical, all of the five metrics equal one.
[26] To have a quantitative comparison to other integration schemes and other NLFFF extrapolation methods, we
calculated the five metrics for the extrapolated field of the
new computationalscheme inthe entirevolume(x,y 2[1,+1]
and z 2 [0, 2], with 64  64  64 grid), inner volume(x, y 2 [0.5, +0.5] and z 2 [0, 1], with 32  32  32 grid),
and the lower central domain (x, y 2 [0.5, +0.5] and z 2[0, 0.5], with 32  32  16 grid). The results for Case I and
Case II are listed in Tables 3 and 4, respectively. The valuesof the integral scheme implemented by Liu in the paper of
Schrijver et al. [2006] and the original computational
scheme proposed by Yan and Li [2006] (only for lower
central domain) are also included in Tables 3 and 4 forreference.
[27] It should be noted that the boundary conditions in
our calculations are somewhat different to the boundary
conditions used by Schrijver et al. [2006]. In the work of
Schrijver et al. [2006], the data on all six boundaries of themodeling volume are available for Case I, the data on thebottom boundary area x, y 2 [3, +3] (with 192  192 grid)are provided for Case II. In our calculations, we use thebottom boundary data in the area x, y 2 [1, +1] (with 64 
64 grid) for both Case I and Case II. Tables 3 and 4 showthat the degree of agreement of the new scheme is better
than the integral scheme implemented by Liu [Schrijver etal., 2006].

4. Applying the Code to the Observed Vector
Magnetogram of Solar Active Region[28] In this section, we check the validity and compatibility of the extrapolation code to deal with the observed
boundary data of solar active regions. The vector magnetogram employed for testing was observed by Solar Magnetic
Field Telescope (SMFT) [Ai, 1987], which is located at
Huairou Solar Observing Station of NAOC (National
Astronomical Observatories, Chinese Academy of Sciences)in Beijing. The active region associated with the magnetogramis NOAA 9077. The data were observed at 0414 UT on 14 July
2000, several hours before the Bastille Day event (X5.7 flare) at
1024 UT [Deng et al., 2001; Yan et al., 2001; Liu and Zhang,2001; Zhang et al., 2001; Zhang, 2002; Tian et al., 2002;
Somov et al., 2002; Wang et al., 2005].
[29] The original field of view of the magnetogram is
31300  21800 from a 512  512 pixel size CCD. We
reformed the magnetogram to a square area (26900 
26900) through cropping and interpolation and reduced the

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Figure 15. Comparison between the extrapolated field lines of NOAA 9077 and the TRACE 195Å
image. (a) The vector magnetogram of the selected region which covers the main polarity inversion lineof NOAA 9077. The data were observed at 0414 UT on 14 July 2000. (b) The TRACE 195Å image of thesame region. The data were observed at 0412 UT on 14 July 2000. (c) The U-shaped field lines overlying
the magnetogram of Bz. (d) The U-shaped field lines overlying the TRACE 195Å image. (e) The closed
field lines overlying the magnetogram of Bz. (f) The closed field lines overlying the TRACE 195Å image.
The U-shaped lines in Figures 15c and 15d are plotted in the volume between layer 0 and layer 19. The
closed field lines in Figures 15e and 15f are plotted from layer 0 (see Figure 17).

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Figure 16. Lf versus height and Ld versus height curves ofthe extrapolated fields by using the code of the new scheme
for NOAA 9077. Lf and Ld are defined in equations (13) and(14) and were calculated at each layer. The unit of magnetic
field is Gauss; the unit of length in the calculation is one
pixel.
pixel number to 64  64 (4.200/pixel). The final magnetogram which is ready for the NLFFF extrapolation is shownin Figure 13a. The extrapolated field lines are shown in
Figures 13b – 13d. The lines in blue color are closed fieldlines with both footpoints being anchored at the bottom
boundary, the red lines representing the field lines that leave
the modeling box. Figure 13b is the top view of the field
configuration in the modeling volume above the wholemagnetogram. Figures 13c and 13d show the detailed
structures above the central domain of the magnetogram(32  32 grid, indicated by a square dashed line in
Figure 13a). Figure 13c is the top view of the field lines,
Figure 13d is the 3-D view of the field lines along thedirection of the polarity inversion line as indicated by anarrow in the right margin of Figure 13c. All the field lines in
Figures 13b– 13d are plotted from layer 1 (see Figure 17) toavoid the influence of noises in the data of the bottom
boundary (the magnetogram observed in the photosphere).
[30] In Figures 13b and 13c, we can see the compact
loops with different orientations aligned over the polarity
inversion line of the magnetogram. The 3-D view of the

A05S90field lines in Figure 13d shows that the loops above the righthalf of the polarity inversion line in Figure 13c is lower thanthe loops above the left half of the polarity inversion line.
Figure 13d also shows that in the region with lower arcade(right half of Figure 13c), the open field lines together withthe underlying compact loops present an X-shaped structure. The diagram to illustrate this X-shaped structure issketched in Figure 14e. The dashed curve in Figure 14e
represents the U-shaped field lines above the X-point, which
are not plotted and thus are displayed as a cavity in
Figure 13d. The whole 3-D views of the closed field lines
and the U-shaped field lines above the main polarity
inversion line of the magnetogram are shown in Figures 14a
and 14b, the side views of the field lines along thedirection of Y-axis are shown in Figures 14c and 14d,
respectively.
[31] To check the extent to which the extrapolated field
can reflect the real distributions of the coronal magnetic
fields, We compared the extrapolated field lines of NOAA
9077 with the EUV images of solar atmosphere obtained by
the Transition Region and Coronal Explorer (TRACE)
satellite [Handy et al., 1999]. The top views of the closed
field lines and the U-shaped field lines are plotted in
Figure 15, and are compared with the TRACE 195Å imageof the same region at almost the same time (0412 UT on
14 July 2000). It can be seen from Figures 15e and 15f thatthe orientations of the closed field lines above the left halfof the polarity inversion line basically coincide with thecoronal loops observed in the TRACE 195Å image, whilein the region above the right half of the polarity inversion
line, Figures 15c and 15d show that the distribution of the
U-shaped field lines basically coincides with the configuration of the filament channel.
[32] To check the extent to which the extrapolated fieldsof the active region NOAA 9077 satisfy the force-freeconstraint (1) and divergence-free constraint (2), we calculated the integral measure Lf and Ld at each layer of thecalculated field, and produced the curves of Lf versus heightand Ld versus height, as we have done for the analyticalfield in section 3. The two curves are shown in Figure 16.
The profiles of the curves in Figure 16 show similar
properties to the analytical solutions in section 3. That is,
the measures Lf and Ld decrease to zero rapidly with theincrease in height, indicating that the force-free anddivergence-free constraints are well satisfied in the extrapolated field.
[33] Being suitable for dealing with the noisy vector
magnetogram observations is one of the great advantagesof the BIE/DBIE method [Yan and Sakurai, 2000; Yan,
2005; Yan and Li, 2006]. The integration over the wholeboundary in the DBIE formulation (6) can efficiently
suppress the influence of the noises in the boundary data
to the convergence property of equation (11). Thus theextrapolation code for the analytical solutions in section 3
can be applied to the real vector magnetograms directly.
Figure 17 shows the magnitude distribution of the extrapolated field for NOAA 9077 at four successive layers.
Figure 17a is the magnitude distribution of jBj at the bottom
boundary (layer 0), Figures 17b– 17d are the magnitude
distribution at layers 1 to 3. Since the field are calculated

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A05S90

HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

A05S90Figure 17. Magnitude distribution of the extrapolated field by using the code of the new scheme for
NOAA 9077 at four successive layers. (a) The magnitude distribution of jBj at the bottom boundary(layer 0); (b) – (d) The magnitude distributions of jBj at layers 1 to 3. The unit of magnetic field (Z-Axis)
is Gauss.
layer by layer, the noises in the boundary data are eliminated quickly with the increase in height.

5. Summary and Conclusion[34] On the basis of DBIE formulation (6), we devised the
upward boundary integration scheme for the nonlinear
force-free field extrapolation. In this new scheme, thebottom boundary for applying the DBIE formulation (6) is
moved upwardly layer by layer. That is, we always calculate
the field distribution at Gn+1 from the data of the newbottom boundary Gn, as shown in Figure 2. While we take
into account the whole bottom boundary data information at
Gn through the DBIE formulation (6), the suitable value ofparameter l at a given field point in Gn+1 can be determined
locally through the force-free constraint condition (11) withthe help of only neighboring boundary data information at
Gn, as shown in Figures 2 and 3.
[35] The main techniques employed in the new scheme
include (1) the bottom boundary for applying the DBIE is
moved upwardly layer by layer to achieve the best convergence property and accuracy, as shown in Figure 2; (2) the
parameter l at a given field point is calculated in a small
square pyramid (sketched in Figure 3) to fully utilize theboundary data information at Gn, and thus save the computing time; (3) the square area for computing the integration of DBIE (6) is enlarged gradually layer by layer to fit
the expanding field, at the same time, pixel number of thesquare areas is fixed at all layers by resampling the grid
points to save the computing time as illustrated in Figure 5.

[36] The code of the new computational scheme wastested by the analytical solutions of Low and Lou [1990]
and are applied to the observed vector magnetogram of solar
active region. In the direct comparison between theextrapolated fields with the analytical solutions of Low
and Lou [1990], the orientations of the extrapolated fieldlines are basically coincide with the analytical solutions.
The quantitative comparison shows that the extrapolated
fields deviate from the analytical fields gradually with theincrease in height. At the lower layers and in the central
domain, the best agreements are obtained. Since we only
use the finite bottom boundary data in the calculation, andthe analytical solutions of Low and Lou [1990] present
global configurations [Low and Lou, 1990; Wang and
Sakurai, 1998; Li et al., 2004; He and Wang, 2006], it isnatural that the extrapolated fields deviate from the analyticalfields at higher layers. The force-free constraint (1) anddivergence-free constraint (2) for the nonlinear force-freefield are well satisfied in the extrapolated field as discussed in
section 3.2.
[37] In the original computational scheme of Yan and Li[2006], the equation (11) has better convergence property at
the field points with lower altitude (near the bottom boundary) than the field points with higher altitude (away from thebottom boundary). The new upward boundary integration
scheme is proposed for taking advantage of this property andto avoid the increasing errors at higher altitude. The improvement of the new scheme to the original computational
procedure of Yan and Li [2006] is significant as shown insection 3.3. The accuracy of the new scheme is better than the

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Figure A1. The extrapolated field lines in the innervolume (x, y 2 [0.5, +0.5] and z 2 [0, 1]) of the modelingspace for Case I by using the original computational
procedure proposed by Yan and Li [2006]. (a) The image in
3-D view. (b) The image in side view along the direction of
Y-axis. Closed field lines are plotted in blue, and field linesthat leave the modeling box are in red.
integral scheme implemented by Liu [Schrijver et al., 2006]
as discussed in section 3.4. If we only care about the magnetic
field near the bottom boundary (at layers with very low
altitude), the integral scheme of Liu can also give reasonable
results as shown in Tables 3 and 4.
[38] The calculations using the observed vector magnetogram of solar active region NOAA 9077 in section 4
demonstrate that the DBIE formulation (6) and the upward
boundary integration scheme can be applied to solar active
regions for the nonlinear force-free field extrapolation. Only
the bottom boundary data in the photosphere are needed in
the calculation, and the DBIE method can suppress thenoises in the observed data through the integration over thewhole bottom boundary. The force-free and divergence-freeconstraints are well satisfied in the extrapolated field asshown in Figure 16.
[39] Since the computation is carried out point by point
and layer by layer as described in section 2, fine structures
can be preserved in the extrapolated field of NOAA 9077 as
illustrated in Figures 13– 15. In the region above the left
half of the polarity inversion line of the magnetogram, theorientations of the closed field lines basically coincide withthe coronal loops observed in the TRACE 195Å image asshown in Figures 15e and 15f. In the region with the dark
filament above the right half of the polarity inversion line,
the extrapolated field lines present the X-shaped structure as
demonstrated in Figures 13– 15. Beneath the X-point are thelow-lying compact loops along the polarity inversion line,
while above the X-point, the distribution of the U-shaped
field lines coincides with the configuration of the filament
channel as shown in Figures 15c and 15d, which can be
helpful for understanding the magnetic structure of the dark
filament as well as the magnetic reconnection processduring the Bastille Day flare [Tsuneta et al., 1992; Masuda
et al., 1994; Shibata et al., 1995; Tsuneta, 1996; Masuda etal., 2001; Fletcher and Hudson, 2001; Priest and Forbes,
2002; Somov et al., 2002; Shibata, 2004].
[40] The application of the new DBIE extrapolation
scheme to the real vector magnetograms is still in apreliminary stage. More efforts on the comparison between

A05S90the extrapolated field lines and the coronal loop observations are needed. The recent data obtained by Hinode(Solar-B) satellite are very valuable for this purpose [Kosugi
et al., 2007]. Topological methods and techniques will be
useful for quantitatively analyzing the topological propertiesof the extrapolated fields [Wang et al., 2000, 2001; Longcope,
2005; Zhao et al., 2005]. Moreover, the DBIE formulation
can be applied to the spherical boundary case [Aly andSeehafer, 1993; Li et al., 2004; Yan, 2005; He and Wang,2006]. The upward boundary integration scheme can also be
adapted to the case with a spherical boundary. By using thefull disk observations of vector magnetograms [UeNo et al.,
2004; Zhang et al., 2007; Su and Zhang, 2007], it ispossible to model the global or large-scale structures of acoronal magnetic field with a spherical boundary and study
their relationships to the coronal mass ejections (CMEs)[Chen and Shibata, 2000; Chen et al., 2002; Zhang andLow, 2005; Zhou et al., 2006a, 2006b].

Appendix A: Convergence Property of theOriginal Computational Procedure Proposed by
Yan and Li [2006][41] We use the analytical nonlinear force-free field
solutions of Low and Lou [1990] to investigate the convergence property of the original computational procedure
proposed by Yan and Li [2006]. The boundary conditions
and boundary data are the same as Case I described in
section 3. The field distribution in the space above thebottom boundary is calculated by using the original DBIE
extrapolation code provided by Yan and Li (http://
srg.bao.ac.cn/dbie-des.htm) [Yan and Li, 2006].
[42] The extrapolated field lines in the inner volume (x, y2 [0.5, +0.5] and z 2 [0, 1], with 32  32  32 grid) are
shown in Figure A1. The values of the extrapolated field aresmoothed with the help of nearest grid points before thefield lines are plotted. Figure A1 is the 3-D view, Figure A1
is the side view along the direction of Y-Axis.
[43] It can be seen from Figure A1 that the orientations of
the field lines at lower altitude basically coincide with theanalytical solution (Figure 7c). But the field lines at higher
altitude are contorted and lose regularity. This result indicates that in the original computational procedure of Yan
and Li [2006], the equation (11), employed to find thesuitable l values for DBIE (6), has a better convergenceproperty at the field points with lower altitude (near thebottom boundary) than the field points with higher altitude(away from the bottom boundary). To take advantage of thisproperty and avoid the increasing errors at the field points
with higher altitude, we devised our new scheme fornonlinear force-free field extrapolation based on the DBIE
formulation (6). In the new scheme, the values of magnetic
field are calculated layer by layer upwardly as described in
section 2.
[44] Acknowledgments. The authors thank the colleagues for valuablecomments and discussions during the CAWSES International Workshop on
Space Weather Modeling (CSWM) at the Earth Simulator Center in
Yokohama. We are grateful to the reviewers for valuable comments andsuggestions to improve the paper and Clia Goodwin for great help toimprove the readability of the manuscript. We thank Yihua Yan and
Zhuheng Li for helpful discussions on the principle of the BIE/DBIE
method and the techniques of force-free field extrapolation. H. He greatly
benefited from the deep and helpful discussions with the colleagues at

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HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

Kwasan and Hida Observatories of Kyoto University. We gratefully
acknowledge Huairou Solar Observing Station of NAOC and TRACE
team for providing the high-quality data of vector magnetograms and EUV
images. We also thank Yang Liu for kindly permitting us to use his resultsin this paper. This work is jointly supported by National Basic Research
Program of China (973 Program) through grant 2006CB806307, National
Natural Science Foundation of China (NSFC) through grants 10673017 and
10733020, and Projects KJCX2-YW-T04 and KGCX3-SYW-403-10 ofChinese Academy of Sciences (CAS).
[45] Amitava Bhattacharjee thanks Edward Deluca and another reviewer
for their assistance in evaluating this paper.

References
Ai, G. X. (1987), Solar magnetic field telescope, Publ. Beijing Astron. Obs.,
9, 27 – 36.
Aly, J. J., and N. Seehafer (1993), Coronal force-free magnetic field:
Source-surface model, Sol. Phys., 144, 243 – 254.
Amari, T., T. Z. Boulmezaoud, and J. J. Aly (2006), Well posed reconstruction of the solar coronal magnetic field, Astron. Astrophys., 446,
691 – 705.
Chen, P. F., and K. Shibata (2000), An emerging flux trigger mechanism forcoronal mass ejections, Astrophys. J., 545, 524 – 531.
Chen, P. F., C. Fang, K. Shibata, and Y. H. Tang (2002), Coronal massejections and emerging flux, Adv. Space Res., 30, 535 – 543.
Deng, Y., J. Wang, Y. Yan, and J. Zhang (2001), Evolution of magnetic
nonpotentiality in NOAA AR 9077, Sol. Phys., 204, 11 – 26.
Fletcher, L., and H. Hudson (2001), The magnetic structure and generationof EUV flare ribbons, Sol. Phys., 204, 69 – 89.
Fuhrmann, M., N. Seehafer, and G. Valori (2007), Preprocessing of solar
vector magnetograms for force-free magnetic field extrapolation, Astron.
Astrophys., 476, 349 – 357.
Handy, B. N., et al. (1999), The transition region and coronal explorer, Sol.
Phys., 187, 229 – 260.
He, H., and H. N. Wang (2006), The validity of the boundary integral
equation for magnetic field extrapolation in open space above a sphericalsurface, Mon. Not. R. Astron. Soc., 369, 207 – 215.
Kosugi, T., et al. (2007), The Hinode (Solar-B) mission: An overview, Sol.
Phys., 243, 3 – 17.
Li, Z., Y. Yan, and G. Song (2004), Properties of the boundary integral
equation for solar non-constant-a force-free magnetic fields, Mon. Not.
R. Astron. Soc., 347, 1255 – 1265.
Lin, J., Y.-K. Ko, L. Sui, J. C. Raymond, G. A. Stenborg, Y. Jiang, S. Zhao,
and S. Mancuso (2005), Direct observations of the magnetic reconnection
site of an eruption on 2003 November 18, Astrophys. J., 622, 1251 – 1264.
Liu, Y., and H. Zhang (2001), Relationship between magnetic field evolution and major flare event on July 14, 2000, Astron. Astrophys., 372,
1019 – 1029.
Longcope, D. W. (2005), Topological methods for the analysis of solar
magnetic fields, Living Rev. Sol. Phys., 2, 7. (Available at http://www.
livingreviews.org/lrsp-2005-7)
Low, B. C., and Y. Q. Lou (1990), Modeling solar force-free magnetic
fields, Astrophys. J., 352, 343 – 352.
Masuda, S., T. Kosugi, H. Hara, S. Tsuneta, and Y. Ogawara (1994), A
loop-top hard X-ray source in a compact solar flare as evidence formagnetic reconnection, Nature, 371, 495 – 497.
Masuda, S., T. Kosugi, and H. S. Hudson (2001), A hard X-ray two-ribbon
flare observed with Yohkoh/HXT, Sol. Phys., 204, 55 – 67.
Metcalf, T. R., L. Jiao, A. N. McClymont, R. C. Canfield, and H. Uitenbroek(1995), Is the solar chromospheric magnetic field force-free, Astrophys. J.,
439, 474 – 481.
Nelder, J. A., and R. Mead (1965), A simplex method for function minimization, Comput. J., 7, 308 – 313.
Pevtsov, A. A., R. C. Canfield, and T. R. Metcalf (1995), Latitudinalvariation of helicity of photospheric magnetic fields, Astrophys. J., 440,
L109 – L112.
Priest, E. R., and T. G. Forbes (2002), The magnetic nature of solar flares,
Astron. Astrophys. Rev., 10, 313 – 377.
Régnier, S., and T. Amari (2004), 3D magnetic configuration of the Ha
filament and X-ray sigmoid in NOAA AR 8151, Astron. Astrophys., 425,
345 – 352.
Sakurai, T. (1981), Calculation of force-free magnetic field with non-constanta, Sol. Phys., 69, 343 – 359.
Schrijver, C. J., M. L. Derosa, T. R. Metcalf, Y. Liu, J. McTiernan, S. Régnier,
G. Valori, M. S. Wheatland, and T. Wiegelmann (2006), Nonlinear forcefree modeling of coronal magnetic fields part I: A quantitative comparisonof methods, Sol. Phys., 235, 161 – 190.
Schwenn, R. (2006), Space weather: The solar perspective, Living Rev. Sol.
Phys., 3, 2. (Available at http://www.livingreviews.org/lrsp-2006-2)

A05S90

Shibata, K. (2004), Magnetic reconnection in solar and astrophysical plasmas, in Physics of Magnetic Reconnection in High-Temperature Plasmas,
edited by M. Ugai, pp. 193 – 237, Res. Signpost, Trivandrum, India.
Shibata, K., S. Masuda, M. Shimojo, H. Hara, T. Yokoyama, S. Tsuneta,
T. Kosugi, and Y. Ogawara (1995), Hot-plasma ejections associatedwith compact-loop solar flares, Astrophys. J., 451, L83 – L85.
Somov, B. V., T. Kosugi, H. S. Hudson, T. Sakao, and S. Masuda (2002),
Magnetic reconnection scenario of the Bastille Day 2000 flare, Astrophys.
J., 579, 863 – 873.
Song, M. T., C. Fang, Y. H. Tang, S. T. Wu, and Y. A. Zhang (2006), A newand fast way to reconstruct a nonlinear force-free field in the solar corona,
Astrophys. J., 649, 1084 – 1092.
Song, M. T., C. Fang, H. Q. Zhang, Y. H. Tang, S. T. Wu, and Y. A. Zhang(2007), Reconstructing spherical nonlinear force-free field in the solar
corona, Astrophys. J., 666, 491 – 500.
Su, J. T., and H. Q. Zhang (2007), The effects of polarization crosstalk andsolar rotation on measuring full-disk solar photospheric vector magnetic
fields, Astrophys. J., 666, 559 – 575.
Tian, L., J. Wang, and D. Wu (2002), Non-potentiality of the magnetic field
beneath the eruptive filament in the Bastille event, Sol. Phys., 209, 375 –389.
Tsuneta, S. (1996), Structure and dynamics of magnetic reconnection in asolar flare, Astrophys. J., 456, 840 – 849.
Tsuneta, S., H. Hara, T. Shimizu, L. W. Acton, K. T. Strong, H. S. Hudson,
and Y. Ogawara (1992), Observation of a solar flare at the limb with the
YOHKOH Soft X-ray Telescope, Publ. Astron. Soc. Jpn., 44, L63 – L69.
UeNo, S., S. Nagata, R. Kitai, H. Kurokawa, and K. Ichimoto (2004), The
development of filter vector magnetographs for the Solar Magnetic
Activity Research Telescope (SMART), in Ground-Based Instrumentation
for Astronomy, edited by A. F. M. Moorwood and M. Iye, Proc. SPIE,
5492, 958 – 969.
Valori, G., B. Kliem, and R. Keppens (2005), Extrapolation of a nonlinear
force-free field containing a highly twisted magnetic loop, Astron. Astrophys., 433, 335 – 347.
Valori, G., B. Kliem, and M. Fuhrmann (2007), Magnetofrictional extrapolations of Low and Lou’s force-free equilibria, Sol. Phys., 245, 263 –285.
Wang, H., C. Liu, Y. Deng, and H. Zhang (2005), Reevaluation of themagnetic structure and evolution associated with the Bastille day flareon 2000 July 14, Astrophys. J., 627, 1031 – 1039.
Wang, H. N., and T. Sakurai (1998), Cross-section variations of coronalmagnetic loops, Publ. Astron. Soc. Jpn., 50, 111 – 123.
Wang, H. N., Y. H. Yan, T. Sakurai, and M. Zhang (2000), Topology ofmagnetic field and coronal heating in solar active regions, II. the role ofquasi-separatrix layers, Sol. Phys., 197, 263 – 273.
Wang, H. N., Y. Yan, and T. Sakurai (2001), Topology of magnetic field andcoronal heating in solar active regions, Sol. Phys., 201, 323 – 336.
Wang, J., Z. Shi, H. Wang, and Y. Lü (1996), Flares and the magnetic
nonpotentiality, Astrophys. J., 456, 861 – 878.
Wheatland, M. S. (2006), A fast current-field iteration method for calculating nonlinear force-free fields, Sol. Phys., 238, 29 – 39.
Wheatland, M. S., P. A. Sturrock, and G. Roumeliotis (2000), An optimization approach to reconstructing force-free fields, Astrophys. J., 540,
1150 – 1155.
Wiegelmann, T. (2004), Optimization code with weighting function for thereconstruction of coronal magnetic fields, Sol. Phys., 219, 87 – 108.
Wiegelmann, T. (2007), Computing nonlinear force-free coronal magnetic
fields in spherical geometry, Sol. Phys., 240, 227 – 239.
Wiegelmann, T., B. Inhester, and T. Sakurai (2006), Preprocessing of vector
magnetograph data for a nonlinear force-free magnetic field reconstruction, Sol. Phys., 233, 215 – 232.
Yan, Y. (2005), On the coronal magnetic field configuration and solar flare/
CME process, Space Sci. Rev., 121, 213 – 221.
Yan, Y., and Z. Li (2006), Direct boundary integral formulation for solar
non-constant-a force-free magnetic fields, Astrophys. J., 638, 1162 –1168.
Yan, Y., and T. Sakurai (2000), New boundary integral equation representation for finite energy force-free magnetic fields in open space above thesun, Sol. Phys., 195, 89 – 109.
Yan, Y., M. J. Aschwanden, S. Wang, and Y. Deng (2001), Evolution ofmagnetic flux rope in the active region NOAA 9077 on 14 July 2000, Sol.
Phys., 204, 27 – 40.
Zhang, H. (2002), Magnetic field, helicity and the 2000 July 14 flare insolar active region 9077, Mon. Not. R. Astron. Soc., 332, 500 – 512.
Zhang, H. Q., et al. (2007), Solar magnetism and the activity telescope at
HSOS, Chin. J. Astron. Astrophys., 7, 281 – 288.
Zhang, J., J. Wang, Y. Deng, and D. Wu (2001), Magnetic flux cancellation
associated with the major solar event on 2000 July 14, Astrophys. J., 548,
L99 – L102.

16 of 17

A05S90

HE AND WANG: NLFFF EXTRAPOLATION SCHEME BASED ON DBIE

Zhang, M., and B. C. Low (2005), The hydromagnetic nature of solar
coronal mass ejections, Annu. Rev. Astron. Astrophys., 43, 103 – 137.
Zhao, H., J. X. Wang, J. Zhang, and C. J. Xiao (2005), A new method of
identifying 3D null points in solar vector magnetic fields, Chin. J. Astron.
Astrophys., 5, 443 – 447.
Zhou, G. P., J. X. Wang, and J. Zhang (2006a), Large-scale source regionsof earth-directed coronal mass ejections, Astron. Astrophys., 445, 1133 –1141.

A05S90Zhou, G. P., Y. M. Wang, and J. X. Wang (2006b), Coronal mass ejections
associated with polar crown filaments, Adv. Space Res., 38, 466 – 469.


H. He and H. Wang, National Astronomical Observatories, Chinese
Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing
100012, China. (hehan@bao.ac.cn; hnwang@bao.ac.cn)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S12, doi:10.1029/2007JA012487, 2008

for

Full
ArticleGeomagnetically induced currents during intense storms driven bycoronal mass ejections and corotating interacting regions
Ryuho Kataoka1,2 and Antti Pulkkinen3,4Received 23 April 2007; revised 12 June 2007; accepted 10 July 2007; published 19 January 2008.

[1] Geomagnetically induced currents (GIC) flowing in ground-based conductor systems
during large geomagnetic storms are one of the most significant space weather
phenomena that affect our ground-based technological systems. Here we show that GIC
activity in subauroral latitudes depends on the storm phase and on the interplanetary
drivers, such as coronal mass ejections (CMEs) and corotating interacting regions (CIRs).
For example, it is shown that GIC amplitudes are relatively small during CIR storms incomparison to CME storms. However, Pc3-5 pulsation activity during CIR storms
drives long-lasting GIC in the local prenoon sector. Despite of the differences between
CME and CIR storms, the relationship between GIC and the time derivative of thehorizontal ground magnetic field is always the same. A novel power law equation isderived to accurately characterize the relation.
Citation: Kataoka, R., and A. Pulkkinen (2008), Geomagnetically induced currents during intense storms driven by coronal massejections and corotating interacting regions, J. Geophys. Res., 113, A03S12, doi:10.1029/2007JA012487.

1. Introduction[2] Geomagnetically induced currents (GICs) flowing inlong technological conductor networks on the ground are
one of the manifestations of geomagnetic storms and can bethought as an end link of the chain of space weather
processes from the surface of the Sun to the surface of the
Earth. GIC poses a potential threat to the normal operationof technological systems such as power lines, pipelines, andrailway systems [e.g., Lanzerotti, 1979; Boteler et al.,1998].
[3] The fundamental challenge of GIC research is to
identify, understand, and model the different geophysicalprocesses associated with large GIC events. The present
understanding is that a number of different geophysicalprocesses are capable of driving large GICs; storm suddencommencements (SSC), geomagnetic pulsations, and auroral substorms have been identified as important causes forlarge GICs [Boteler, 2001; Lam et al., 2002; Kappenman,
2003; Pulkkinen et al., 2003, 2005].
[4] Most of the earlier studies, like those cited above, on
the ionospheric and magnetospheric drivers of GICs havebeen more or less event based and did not provide any directmeans for generalizations. Rigorous statistical analysis of
GIC and the time derivative of the horizontal groundmagnetic field (denoted hereafter dB/dt), a quantity closely
coupled to GIC via Faraday’s law of induction, have been
1
Computational Astrophysics Laboratory, Institute of Physics andChemical Research (RIKEN), Saitama, Japan.
2
Solar-Terrestrial Laboratory, Nagoya University, Nagoya, Japan.
3
Goddard Earth Sciences and Technology Center, University of
Maryland, Baltimore County, Baltimore, Maryland, USA.
4
NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012487$09.00

carried out by Viljanen [1997], Viljanen et al. [2001, 2006],
Weigel et al. [2002], Weigel and Baker [2003], Wintoft[2005], and Pulkkinen et al. [2006].
[5] However, basic characteristics of GIC during intensestorms are still not well known. Recently, Pulkkinen andKataoka [2006] performed a time-frequency analysis of
GICs during the 10 greatest superstorms to show a clear
local time and storm phase dependence of the GIC spectra.
Such superstorms are driven by fast coronal mass ejections(CMEs) [e.g., Kataoka et al., 2005]. Miyoshi and Kataoka[2005] showed evidence that there are significant differences in the magnetospheric response between intensestorms driven by CMEs and corotating interaction regions(CIRs), and therefore the GIC response is also expected tobe different for the CME and CIR storms. The purpose ofthis study is to investigate the possible differences of GIC
variations between CME and CIR storms.

2. Data and the Analysis Method[6] The main data to be analyzed is composed of GIC
measurements carried out since November 1998 in the
Finnish natural gas pipeline at the Mäntsälä pipeline section[Pulkkinen et al., 2001]. The magnetic latitude of themeasurement site is about 57 degrees, and the magnetic
local time is about UT plus 3 h. The GIC data are obtainedwith a 10 s temporal resolution. The IMAGE magnetometer
chain [Lühr et al., 1998] is used to provide more generalgeomagnetic context. Nurmijärvi Geophysical Observatory(NUR), a part of the IMAGE chain, is located about 30 km
southwest from the GIC measurement site. Also, the magnetometer data are obtained with a 10 s temporal resolution.
Table 1 lists the coordinates of the IMAGE magnetometerstations used in this study.
[7] Figure 1 shows a correlation between minimum Dst
and maximum GIC for the isolated intense storms (mini-

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A03S12

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Table 1. Coordinates of IMAGE Magnetometer Stations Used in This Study
AbbreviationName

GEO Latitude

GEO Longitude

CGM Latitude

CGM Longitude

NAL
LYR
HOR
SOR
KIL
MUO
PEL
OUJ
HAN
NUR
TAR

Ny Ålesund
Longyearbyen
HornsundSørøyaKilpisjärvi
MuonioPello
Oulujärvi
Hankasalmi
Nurmijärvi
Tartu

78.92
78.20
77.00
70.54
69.02
68.02
66.90
64.52
62.30
60.50
58.26

11.95
15.82
15.60
22.22
20.79
23.53
24.08
27.23
26.65
24.65
26.46

75.25
75.12
74.13
67.34
65.88
64.72
63.55
60.99
58.71
56.89
54.47

112.08113.00109.59106.17103.79105.22104.92106.14104.61102.18102.89mum Dst < 100 nT) during solar cycle 23 (years 1996 to2005) as identified by Kataoka and Miyoshi [2006]. Continuous GIC data are available for 32 CME storms and3 CIR storms, as indicated by open and solid circles,
respectively. The superstorms investigated by Pulkkinen
and Kataoka [2006] are included in the CME storms only
if the superstorms are isolated without multiple occurrencesof intense storms within 4 d. The maximum amplitude of
GIC is calculated for the time interval 1 d before and after
the Dst minima. The correlation coefficient is 0.67 for CME
storms, implying that large GIC events tend to be associatedwith large CME storms.
[8] As is seen from Figure 1, the three CIR storms are
smaller in amplitude both in terms of Dst and GIC. The
small number of CIR storms is due to the fact that CIRs
rarely produce intense storms. In fact, typical CIRs have alimited interplanetary magnetic field strength of up to about
20 nT [Richardson et al., 2006]. However, since intense
CIR storms can be a potential cause for large GIC events, itis worthwhile to include these events in our analysis.
[9] Generally, the waveform of GICs tends to be very
noisy and nonstationary [Pulkkinen and Kataoka, 2006].
The S-transform [Stockwell et al., 1996], capable of handing
the noisy nonstationary data, is a time-frequency analysis ofa time series h(t) and is defined as:S ðt; f Þ ¼

Z 1

j f j f 2 ðttÞ2hðt Þ pﬃﬃﬃﬃﬃﬃ e 2 ei2pft dt;
2p
1

lytic signal, and the amplitude jS(t, f)j is color coded todisplay the S-transform spectra.

3. Results[11] First, we show typical examples of the GIC spectra
during comparable nightside CME and CIR storm events.
The Dst minima for the selected CME and CIR storms are110 nT at 1930 UT (2230 MLT) on 11 May 2002 and109 nT at 1730 UT (2030 MLT) on 11 February 2004,
respectively. Using the nightside events, we can directlycompare the results with the results for superstorms, as
shown by Pulkkinen and Kataoka [2006].
[12] Figures 2 and 3 show GIC and the S-transform of
GIC for the selected events. It is seen that around the storm
peak taking place in the nightside, turbulent broadband
spectra are seen in both CME and CIR storms. After thestorm peak, the CIR storm has a stronger and longer-lasting
GIC activity than that of the CME storm even though the
Dst minima for the storms are similar to each other. Local
daytime GIC in Pc3-5 pulsation ranges (0.3– 10 min) areð1Þ

where t is time and f is frequency. The kernel of the Stransform is defined by a Gaussian with the window widthof 1/f to give the best resolution in time-frequency space.
The normalized Gaussian kernel also gives a direct
mathematical connection to the Fourier transform via theintegration over t.
[10] Before calculating the S-transform spectra of GIC for
all 24-h long segments centered around Dst minima ofstorm events, there are three preprocessing steps as follows:(1) We include the margins of 2 h time series just before andafter the 24-h time series to reduce an artificial edge effects;(2) Hanning window is applied on the 5% of the total 28 
360 points at the edges to further reduce the edge effect;(3) Hilbert transform is applied to obtain the analytic signal
in which the spectral power is localized into positivefrequencies. Finally, we apply the S-transform to the ana-

Figure 1. Scatterplot of maximum jDstj and jGICj foreach of the Dst < 100 nT storms used in this study.
Coronal mass ejection (CME) storms are indicated by open
circles and corotating interaction regions (CIR) storms bysolid circles.

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Figure 2. S-transform spectrum of a CME storm on 11 May 2002 showing (a) the waveform ofgeomagnetically induced currents (GIC), (b) the S-transform spectral power, and (c) the FFT power
spectrum obtained by integrating the S-transform over time. The left half of Figure 2b is the local
nightside (1800 – 0600 MLT), and the right half is the local dayside (0600 –1800 MLT). Red diamonds
indicate the Dst index. The storm peak is shown by a vertical dotted line.
especially enhanced during the recovery phase of the CIR
storm. Comparing the Figures 2c and 3c, the GIC enhancement in the Pc3-5 range during the CIR storm can also beseen from the FFT spectra. Strong monochromatic GIC in
the Pc5 range (2.5– 10 min), as was seen for superstorms by
Pulkkinen and Kataoka [2006], is not present during therecovery phase of the CME storm. Although not shown
here, the spectral features observed above are common also
for the other CME and CIR storms studied in this work.

[13] We then compare the time derivative of the horizontal magnetic field (dB/dt) observed at NUR with the GIC
using all of the CME and CIR storm events. The time
derivative is calculated by using the three-point central
difference formula. To compare the GIC and dB/dt, thedata are divided into 1 h intervals, each containing 360
sample points. The maximum amplitudes of jdB/dtj and
jGICj are calculated in each interval, and 1 h values are
obtained for four different time intervals: just after the storm

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Figure 3. S-transform spectrum of a CIR storm on 11 February 2004. The format is the same as in
Figure 2.
sudden commencement (SSC), main phase, recovery phase,and 24 h after the end of the recovery phase.
[14] In this paper, the beginning of the main phase isdetermined by the existence of minimum d(Dst)/dt or two
successive points of d(Dst)/dt < 20 nT/h, where the time
derivative is calculated using a two-point forward difference
formula. If it is not possible to detect the beginning by thesecriteria, a typical time period of 6 h before the Dst minimum
is taken as the main phase interval. The end of the recovery
phase is determined by the recovery of the Dst index to 50%of the Dst minimum. If it is not possible to detect the end bythese criteria, typical time period of 12 h after the Dst
minimum is taken as the recovery phase interval.

[15] From Figure 4 showing the scatterplot and histograms of maximum jdB/dtj and jGICj for CME storms, it isseen that from the main phase (red) to the recovery phase(blue) the amplitudes of jGICj and jdB/dtj decrease morethan a half an order of magnitude on average. Further, fromthe recovery phase (blue) to the 24 h after the end of therecovery phase (green), the amplitudes decrease again morethan a half an order of magnitude on average. However,independent of the storm phase, all of the data points
distribute around the same regression line. The correlation
coefficient is better for jdBy/dtj (CC = 0.97) than that of
jdBx/dtj (CC = 0.94), where the subscripts x and y denotethe geographic north-south and east-west components, re-

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Figure 4. Scatterplot and histograms of maximum jdB/dtj showing (left) the x-component and (right) ycomponent and maximum jGICj during the main phase (red), recovery phase (blue), 24 h after the end ofrecovery phase (green), and just after the storm sudden commencements (black) of CME storms.
spectively. The difference in correlation clearly reflects thegeographical orientation of the pipeline; GIC flow in thepipeline responds differently for east-west and north-south
geoelectric field in association with the temporal changes ofthe magnetic field [Pulkkinen et al., 2001]. The relationshipbetween jdBy/dtj and jGICj can be approximated by powerlaw equation,
jGICjðAÞ ¼ 100:67 jdBy =dtj0:84 ðnT=sÞ:ð2Þ

From Figure 5 showing the scatter plot and histograms of
maximum jdB/dtj and jGICj for CIR storms, it is seen that
although the correlation between the two quantities isslightly worse, the basic trend of the regression line is thesame as that for the CME storms. Again, the correlation
coefficient is better for jdBy/dtj (CC = 0.91) than that of
jdBx/dtj (CC = 0.78). The approximately log-normal
histograms in Figures 4 and 5 suggest that the GIC
amplitude rarely exceeds the 10 A level even in the main
phase of CIR storms. From Figure 5 it is also seen that thestorm phase dependence of the jdB/dtj and jGICj amplitudes

Figure 5. Same as Figure 4 but for CIR storms.
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Table 2. Statistical Parameters of |GIC| Amplitudes for Different Storm Phasesa
Interplanetary Driver

Storm Phase

Total Numberof Storms

Total Numberof Hours

Average (A)Maximum (A)

CME
CME
CME
CME
CIR
CIR
CIR

SSC
main
recovery
after recovery
main
recovery
after recovery

32
32
32
32
3
3
3

32
247
380
403
25
29
69

3.3
3.8
1.5
0.6
2.0
1.3
1.0

31.6
32.0
18.2
13.0
7.3
4.1
4.3

a
From left to right, columns show the storm driver, storm phase, the total number of the storm events, the total number ofsample hours, mean values, and maximum values.

is relatively small for CIR storms, ranging well below anorder of magnitude on average. Comparing the recovery
phase (blue) and late recovery phase (green) in Figures 4
and 5, it is apparent, in agreement with Figures 2 and 3, thatthe jdB/dtj and jGICj amplitudes during the recovery andlate recovery phases are larger for CIR storms on average.
Some statistical parameters of jGICj are summarized inTable 2. The mean values are calculated after taking thelogarithm. This is reasonable since the hourly values of both
jGICj and jdB/dtj follow the log-normal distribution asshow in the histograms in Figures 4 and 5.

4. Discussion[16] Figures 2 and 3 imply that partially different physicalmechanisms drive GIC during CME and CIR storms. The
most interesting observation is that, during CIR storms, GIC
is enhanced in the Pc3-5 pulsation range, especially in thelocal dayside during the recovery phase. The Pc3-5 pulsations are related to the high-speed Alfvénic solar windstreams originating from large coronal holes [seeRichardson et al., 2006]. Thus these coronal hole streams
are the origin for the long-lasting GIC activity during CIR
storms. On the other hand, for the February 2004 event
shown in Figure 3, the very quiet period of GIC lasting 8 h
just after the storm peak was associated with unusuallystable northward interplanetary magnetic fields embedded
within the CIR and is not necessarily a general feature of
CIR-associated storms.
[17] The well-known correlation of Pc3-5 pulsation wave
power with the solar wind speed [Singer et al., 1977;
Engebretson et al., 1998], suggests that the Kelvin-Helmholtz instabilities (KHI) at the magnetopause may act as apossible energy source for the pulsations. Actually, using
multiple in situ satellite observations, Rae et al. [2005]
investigated very strong narrowband monochromatic Pc5
pulsation, as seen in GIC during superstorms [Pulkkinen
and Kataoka, 2006] and concluded that the discrete Pc5
pulsation is the result from the excitation of a magnetospheric waveguide mode driven by KHI at the magnetopause. Other mechanisms, however, possibly contribute tolarge GIC in the Pc3-5 range. On the basis of the IMAGE
magnetometer data, Howard and Menk [2005] suggested
that the daytime Pc3-4 waves as observed in this paper are
generated by the upstream ion-cycrotron resonance and that
there is no evidence of generation by the KHI. Baker et al.
[2003] surveyed a 10-year data set of CANOPUS tostatistically investigate the Pc5 pulsation and suggested
from the property of field-line resonance that the energy

source of Pc5 pulsation is not only KHI but also that asignificant part of the Pc5 pulsations should have some
irregular or impulsive energy source such as travelingindentations on the magnetopause [Mathie and Mann,
2000] or solar wind buffeting. They also noted that transient
events such as magnetic impulse events [e.g., Kataoka etal., 2001] contribute to the largest power of Pc5 spectra in alimited dawn sector between 65 to 70 magnetic latitudes.
[18] Despite the spectral differences found between CME
and CIR storms, hourly dB/dt is found always to be an
excellent indicator of the hourly GIC activity, and therelationship between the GIC and the dB/dt is always thesame in a very large dynamic range of about three orders ofmagnitude, as shown in Figures 4 and 5. This finding is inagreement with earlier results [see, e.g., Viljanen et al.,2001, and references therein]; the novel result is that therelationship can be made very accurate by using thefunctional form given in equation (2). It should be noted
here that there are examples [Trichtchenko and Boteler,
2006] where GIC behavior follows the magnetic fieldstrength (B) rather than its time derivative. However, bymaking plots similar to Figures 4 and 5 for B, we haveconfirmed that dB/dt is a better indicator for GIC activity
than B at Mäntsälä.
[19] For additional insights to jdB/dtj behavior, as shownin Figure 6, we construct a polar map showing the global
distribution of average amplitude of the hourly maximum
jdB/dtj and root mean square of the hourly maximum jdBx/
dtj and jdBy/dtj. We use the eleven meridional magnetometers listed in Table 1. All six CIR events in the list of
Kataoka and Miyoshi [2006] and an additional recent CIR
event that occurred in the end of August 2005 are used inconstructing the map. However, even with the additional
CIR events the number of data points was, unfortunately,
too small to construct a map for the CIR main phase. A totalof 11  24 average points (11 points in latitude and 24
points in MLT) are linearly interpolated and color contoured. Note again that the average values are calculated
after taking the logarithm.
[20] Weigel et al. [2002, 2003] computed similar averagepolar maps of jdB/dtj, but they did not make a classificationof different storm phases. The results here indicate that it
may be beneficial to separate predictions of jdB/dtj bydifferent classes based on, for example, storm phase andinterplanetary driver. For example, the prenoon peak of CIR
storms (Figure 6c) is about twice as strong on average thanthat of CME storms (Figure 6b). Further, comparing the
Figures 6a and 6b, the local time sector having the greatest
activity may vary as a function of CME storm phase, for

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Figure 6. Polar map of the average amplitude of the hourly maximum jdB/dtj during (a) the main phaseand (b) the recovery phase of CME storms and (c) during and 24 h after the recovery phase of CIR
storms. Solid lines indicate the 80, 70, and 60 MLAT, and the dotted line indicates the latitude of the
NUR station. Open squares indicate the meridional locations of the IMAGE chain.
example, the greatest activity is appeared in the dawn andpostmidnight sector during the main and recovery phase,respectively.

5. Conclusions[21] It was shown that subauroral GIC activity depends
on the solar wind driver and phases of strong geomagnetic
storms. For example, GIC amplitudes are relatively small
during CIR storms in comparison to CME storms. However,Pc3-5 pulsation activity during CIR storms drives longlasting GIC in the local prenoon sector. Despite of thedifferences, the maximum hourly value of jdB/dtj is always
an excellent indicator for the maximum hourly amplitude ofjGICj in the Finnish pipeline for any local time and any
storm phase of CME/CIR storms. A novel power law
equation was derived to accurately characterize the relationship between jdB/dtj and jGICj.
[22] Acknowledgments. GIC measurements in the Finnish pipeline
owned by the Gasum Oy Company are carried out by the Finnish
Meteorological Institute. A. Viljanen of Finnish Meteorological Instituteis acknowledged for providing the GIC data used in the study. We thank theinstitutes who maintain the IMAGE magnetometer array. The work by RK
was supported by a research fellowship of the Japan Society for the
Promotion of Science for Young Scientists and a research fellowship ofSpecial Postdoctoral Research Program at RIKEN. 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.
[23] Amitava Bhattacharjee thanks Ian Richardson and Yusuke Ebihara
for their assistance in evaluating this paper.

References
Baker, G. J., E. F. Donovan, and B. J. Jackel (2003), A comprehensive
survey of auroral latitude Pc5 pulsation characteristics, J. Geophys. Res.,
108(A10), 1384, doi:10.1029/2002JA009801.

Boteler, D. (2001), Assessment of geomagnetic hazards to power systems in
Canada, Natural Hazards, 23, 101 – 120.
Boteler, D. H., R. J. Pirjola, and H. Nevanlinna (1998), The effects ofgeomagnetic disturbances on electrical systems at the Earth’s surface,Adv. Space Res., 22, 17 – 27.
Engebretson, M., K.-H. Glassmeier, M. Stellmacher, W. J. Hughes, and
H. Luhr (1998), The dependence of a high latitude Pc5 wave power onsolar wind velocity and on the phase of high speed solar wind streams,
J. Geophys. Res., 103, 26,271.
Howard, T. A., and F. W. Menk (2005), Ground observations of highlatitude Pc3-4 ULF waves, J. Geophys. Res., 110, A04205,
doi:10.1029/2004JA010417.
Kappenman, J. G. (2003), Storm sudden commencement events and theassociated geomagnetically induced current risks to ground-based systems at low-latitude and midlatitude locations, Space Weather, 1(3),1016, doi:10.1029/2003SW000009.
Kataoka, R., and Y. Miyoshi (2006), Flux enhancement of radiation belt
electrons during geomagnetic storms driven by coronal mass ejections
and corotating interaction regions, Space Weather, 4, S09004,
doi:10.1029/2005SW000211.
Kataoka, R., H. Fukunishi, L. J. Lanzerotti, C. G. Maclennan, H. U. Frey,
S. B. Mende, J. H. Doolittle, T. J. Rosenberg, and A. T. Weatherwax(2001), Magnetic Impulse Event: A detailed case study of extended
ground and space observations, J. Geophys. Res., 106, 25,873 – 25,889.
Kataoka, R., S. Watari, N. Shimada, H. Shimazu, and K. Marubashi (2005),
Downstream structures of interplanetary fast shocks associated with coronal mass ejections, Geophys. Res. Lett., 32, L12103, doi:10.1029/
2005GL022777.
Lam, H.-L., D. H. Boteler, and L. Trichtchenko (2002), Case studies ofspace weather events from their launching on the Sun to their impacts on
power systems on the Earth, Ann. Geophys., 20, 1073 – 1079.
Lanzerotti, L. J. (1979), Geomagnetic influences on man-made systems,
J. Atmos. Terr. Phys., 41, 787 – 796.
Lühr, H., A. Aylward, S. C. Bucher, A. Pajunpaa, K. Pajunpaa, T. Holmboe,
and S. M. Zalewski (1998), Westward moving dynamic substorm features
observed with the IMAGE magnetometer network and other groundbased instruments, Ann. Geophys., 16, 425 – 440.
Mathie, R. A., and I. R. Mann (2000), Observations of Pc5 field line
resonance azimuthal phase speeds: A diagnostic of their excitation mechanism, J. Geophys. Res., 105, 10,713 – 10,728.
Miyoshi, Y., and R. Kataoka (2005), Ring current ions and radiation belt
electrons during geomagnetic storms driven by coronal mass ejections
and corotating interaction regions, Geophys. Res. Lett., 32, L21105,
doi:10.1029/2005GL024590.

7 of 8

A03S12KATAOKA AND PULKKINEN: GIC DURING CME AND CIR STORMS

Pulkkinen, A., and R. Kataoka (2006), S-transform view of geomagnetically induced currents during geomagnetic superstorms, Geophys. Res.
Lett., 33, L12108, doi:10.1029/2006GL025822.
Pulkkinen, A., A. Viljanen, K. Pajunpää, and R. Pirjola (2001), Recordings
and occurrence of geomagnetically induced currents in the Finnish natural gas pipeline network, J. Appl. Geophys., 48, 219 – 231.
Pulkkinen, A., A. Thomson, E. Clarke, and A. McKay (2003), April 2000
storm: ionospheric drivers of large geomagnetically induced currents,
Ann. Geophys., 21, 709 – 717.
Pulkkinen, A., S. Lindahl, A. Viljanen, and R. Pirjola (2005), Geomagnetic
storm of 29 – 31 October 2003: Geomagnetically induced currents andtheir relation to problems in the Swedish high-voltage power transmission
system, Space Weather, 3, S08C03, doi:10.1029/2004SW000123.
Pulkkinen, A., A. Viljanen, and R. Pirjola (2006), Estimation of geomagnetically induced current levels from different input data, Space Weather,4, S08005, doi:10.1029/2006SW000229.
Rae, I. J., et al. (2005), Evolution and characteristics of global Pc5 ULF
waves during a high solar wind speed interval, J. Geophys. Res., 110,
A12211, doi:10.1029/2005JA011007.
Richardson, I. G., et al. (2006), Major geomagnetic storms (Dst < 100 nT)
generated by corotating interaction regions, J. Geophys. Res., 111,
A07S09, doi:10.1029/2005JA011476.
Singer, H. J., C. T. Russel, M. G. Kivelson, E. Q. Greenstadt, and J. V.
Olson (1977), Evidence for the control of Pc3-4 magnetic pulsations by
the solar wind velocity, Geophys. Res. Lett., 4, 377 – 381.
Stockwell, R. G., L. Manshinha, and R. P. Lowe (1996), Localization of thecomplex spectrum: The S-transform, IEEE Trans. Signal Proc., 44(4),998 – 1001.

A03S12Trichtchenko, L., and D. Boteler (2006), Response of power systems to thetemporal characteristics of geomagnetic storms, paper presented at Canadian Conference on Electrical and Computer Engineering, IEEE, Ottawa,
Ontario, Canada.
Viljanen, A. (1997), The relation between geomagnetic variations and theirtime derivatives and implications for estimation of induction risks, Geophys. Res. Lett., 24, 631 – 634.
Viljanen, A., H. Nevanlinna, K. Pajunpää, and A. Pulkkinen (2001), Time
derivative of the horizontal magnetic field as an activity indicator, Ann.
Geophys., 19, 1107 – 1118.
Viljanen, A., E. I. Taskanen, and A. Pulkkinen (2006), Relation between
substorm characteristics and rapid temporal variations of the groundmagnetic field, Ann. Geophys., 24, 725 – 733.
Weigel, R. S., and D. N. Baker (2003), Probability distribution invarianceof 1-minute auroral-zone geomagnetic field fluctuations, Geophys. Res.
Lett., 30(23), 2193, doi:10.1029/2003GL018470.
Weigel, R. S., D. Vassiliadis, and A. J. Klimas (2002), Coupling of the solar
wind to temporal fluctuations in ground magnetic fields, Geophys. Res.
Lett., 29(19), 1915, doi:10.1029/2002GL014740.
Weigel, R. S., A. J. Klimas, and D. Vassiliadis (2003), Solar wind coupling
to and predictability of ground magnetic fields and their time derivatives,
J. Geophys. Res., 108(A7), 1298, doi:10.1029/2002JA009627.
Wintoft, P. (2005), Study of the solar wind coupling to the time difference
horizontal geomagnetic field, Ann. Geophys., 23, 1949 – 1957.


R. Kataoka, Computational Astrophysics Laboratory, Institute of Physics
and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198,
Japan. (ryuho@riken.jp)
A. Pulkkinen, NASA Goddard Space Flight Center, Code 674, Greenbelt,
MD 20771, USA. (antti.pulkkinen@gsfc.nasa.gov)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S10, doi:10.1029/2007JA012514, 2008

for

Full
ArticleProperties of AE indices derived from real-time global simulation andtheir implications for solar wind-magnetosphere coupling
K. Kitamura,1,2 H. Shimazu,1,3 S. Fujita,3,4 S. Watari,1 M. Kunitake,1 H. Shinagawa,1,3
and T. Tanaka3,5Received 1 May 2007; revised 4 October 2007; accepted 15 November 2007; published 8 February 2008.

[1] Real-time magnetohydrodynamic (MHD) simulation of the solar windmagnetosphere-ionosphere (S-M-I) coupling system was used to calculate auroralelectrojet (AE) indices. This simulation reproduces the magnetic field configurations in themagnetosphere, magnetospheric convection, and field-aligned currents (FACs) using the
upstream boundary conditions with the interplanetary magnetic field (IMF), solar windspeed, temperature, and proton number density measured by the ACE spacecraft. The
electrical potential at 3 RE (Earth radii) from the center of the Earth is mapped on theionosphere. The ionospheric currents are deduced from Ohm’s law to match thedivergence of Pedersen and Hall currents from FACs. The AE indices are obtained fromthe magnetic field perturbation caused by the simulated ionospheric currents. We
compared the simulated AE indices for 247 d with the AE indices deduced from themagnetic variations at up to 12 stations located around the auroral latitude. The results
show that the simulated AE reproduces the observed AE indices well. Of the 247 d, 64%
had cross-correlation coefficients of more than 0.5. We also found that the simulated AE
indices do not correlate well with the observed AE indices when the standard deviations ofvariations in the observed AE indices are less than 100 nT. When variations in the AE
indices are small, some of the short-period perturbations of the electromagnetic energy
flowing from the solar wind into the magnetosphere is absorbed or filtered in the real S-MI coupling system by some mechanism that is not included in our MHD simulation andthat the resulting fluctuation in the AE indices is damped compared with the simulation.
Citation: Kitamura, K., H. Shimazu, S. Fujita, M. Kunitake, H. Shinagawa, and T. Tanaka (2008), Properties of AE indices derived
from real-time global simulation and their implications for solar wind-magnetosphere coupling, J. Geophys. Res., 113, A03S10,
doi:10.1029/2007JA012514.

1. Introduction[2] Recent improvements in computer calculation speeds
have made it possible for simulations of large-scale phenomena to be done. Global magnetohydrodynamics (MHD)
simulation has played a significant role in the field of theinteraction between the solar wind and the Earth’s magnetosphere. It has revealed the shape of the magnetosphere,
the structure of the distant magnetic tail, and how theclosure of the Region-1 field-aligned currents (FACs) are
reproduced [Walker et al., 1993; Fedder and Lyon, 1995;
Raeder et al., 1995; Spicer et al., 1996; Elsen and Winglee,

1
National Institute of Information and Communications Technology,
Tokyo, Japan.
2
Now at Tokuyama College of Technology, Yamaguchi, Japan.
3Core Research for Evolutional Science and Technology, Japan Scienceand Technology Agency, Saitama, Japan.
4
Meteorological College, Chiba, Japan.
5
Department of Earth and Planetary Sciences, Kyushu University,
Fukuoka, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012514$09.00

1997; Janhunen and Koskinen, 1997; White et al., 1998;
Song et al., 1999].
[3] The ACE spacecraft’s provision of real-time solar
wind data makes it possible to run a global MHD modelof the magnetosphere in real time. We have developed amodel that simulates real-time global MHD of the solar
wind interaction with the magnetosphere. The MHD codeused here was developed by Tanaka [1994] to enable highly
accurate calculations of the MHD simulation for systems
with a potential magnetic field (intrinsic geomagnetic field).
Tanaka’s [1995] MHD simulation reproduced the FACs in
the magnetosphere and indicated that the distribution of thesimulated FACs has signatures typical of the current patterns of regions 1 and 2, which depend on the Z-componentof the interplanetary magnetic field (IMF). Tanaka also
advanced our understanding of the structure and origin ofmagnetospheric convection [Tanaka, 1999].
[4] Another recent real-time simulation model of the
global magnetosphere has being implemented at the Community Coordinated Modeling Center (CCMC) (http://
ccmc.gsfc.nasa.gov) [Narock et al., 2004]. This model isbased on the space plasma simulation model of the Space
Weather Modeling Framework (SWMF) at the University of

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A03S10KITAMURA ET AL.: AE INDICES DERIVED FROM MHD SIMULATION

Michigan [Tóth et al., 2005]. These kinds of real-timemodels are useful both in predicting the space weather
environment and in studying physical processes in space.
Another successful global magnetospheric model that canalso be used to derive global geomagnetic indices is theOpen Global Geospace Circulation Model (openGGCM),
which was originally developed as an MHD model ofEarth’s magnetosphere at UCLA in the early 1990s by
J. Raeder.
[5] Simulation is based on physical laws, but approximations and numerical errors are included in it. Thus all
simulations must be verified against real (observed) data.
Moreover, simulations in which conditions can be set freely
are indispensable tools in understanding the essential elements of the observed results. Because of these characteristics, linking simulation results with observations is very
important.
[6] Real-time simulation data and observed data can becomplementary. With in situ observation by a satellite, even if
many satellites are used, it is difficult to get a global image ofthe magnetosphere. In contrast, global simulation can visualize the entire image and conditions of the magnetosphere.
[7] For a quantitative comparison of observed and simulated data to be valid, the two types of data must have thesame physical parameters. We use geomagnetic indices forthis comparison. The most typical indices deduced from theground magnetic variations are Dst and auroral electrojet(AE) indices. The Dst index represents the axially symmetric component of magnetic variations at low latitudes on the
ground and is generally used as a measure of the intensity ofmagnetic storms. However, MHD simulation cannot accurately reproduce the Dst index because lack of the particle
dynamics in MHD cannot reproduce the ring current that
contributes strongly to the Dst index.
[8] The AE index, on the other hand, is deduced frommagnetic variation at auroral latitudes on the ground, which
is produced mainly by an ionospheric current called theauroral electrojet. The auroral electrojet is generated by theelectric field imposed on the polar ionosphere and by
confined high electric conductivities, which are mainly
associated with substorms. The AE indices are derived fromdata generated by up to 12 geomagnetic observatories near
the aurora oval located at roughly longitudinally equidistantintervals [Takahashi et al., 2004]. Using 1-min sampling
magnetic data, the H- (horizontal) component magnetic
variation at each station whose base line is subtracted issuperposed on the same time axis. In this plot the upper andlower envelopes are defined as the AU and AL indices,
respectively. These provide the measure of the maximum
eastward and westward electrojet currents at any time
[Davis and Sugiura, 1966; Kamei et al., 1981]. The AE
and AO indices are derived from the AL and AU indices asfollows:
AE ¼ AU  AL
AO ¼ ð AU þ ALÞ=2

Comparing the result of simulation with these indices aswell as satellite observation is very valuable because theindices are derived by superposing data from a wide area.
[9] We aim to determine whether our real-time MHD
simulation is practical in actual space weather forecasting.

A03S10We introduce our real-time global MHD simulation model
first. Then, we derive the AE index from the simulation andcompare it with observed indices. Finally, we discuss theimplications for the characteristics of the model that concern
solar wind-magnetosphere-ionosphere (S-M-I) coupling.

2. Real-Time Global MHD Simulation Model[10] Tanaka [1994, 1995] developed a three-dimensional(3-D) MHD simulation model to investigate S-M-I coupling. He based his simulation code on an unstructured grid
system to cope with the differences between the scales forthe magnetosphere and the ionosphere and on a finite
volume total variation-diminishing (TVD) scheme to capture discontinuity. The inner boundary of the grid system isset at 3 Earth radii (RE) from the center of the Earth. The
ionospheric potential can be deduced from the electric
potential at 3 RE, which is mapped on the ionosphere along
the dipole field lines. The ionospheric current, which can be
decomposed into the Pedersen and Hall currents, is derived
from Ohm’s law in the ionosphere. The current closes in theionosphere as follows:r srFI ¼ Gm ðrot B1 nb Þ ¼ Jjj ; s ¼ sEUV þ k1 sDiff ð P; rÞ þ k2 sJ Jjj ; 
Fm ¼ FI  k3 f1 Jjj jJjj j;where s is the ionospheric conductivity tensor, FI is theionospheric potential, Fm is the magnetospheric potential,
B1 is the variable component of the magnetic field, nb is an
outward unit vector, Jk is the electric current parallel to themagnetic field, and Gm is a geometrical factor associatedwith the mapping along the field lines from r = 3 RE to r = 1
RE. The current in the ionosphere closes with the FAC.
Values sEUV, sDiff, and sJ are the conductivities due to thesolar EUV, the diffuse auroral precipitation, and the FAC,
respectively. The diffuse auroral precipitation sDiff isdefined as a function of the pressure P and density r. Here
f1 is a function being 1 at the upward FAC and 0 at the
downward FAC, and k1  k3 are scaling constants [Tanaka,
2000]. In this simulation the geographical north and thegeomagnetic north are defined on the same axis. The
resistivity (h) is defined in the present MHD simulation as:
h ¼ k4 f2 ð xÞjJ j2 =jBj2

where f2 (x) is a time-independent fixed function (x indicatespointing to the Sun from the Earth), k4 is a scaling constant.
Here f2 (x) is small in the near-Earth region at x > 20 RE,
linearly increases down tail, and saturates at x > 60 RE[Tanaka, 2000].
[11] Using the simulation code developed by Tanaka[1994, 1995], Den et al. [2006] started real-time global
MHD simulation using the supercomputer at the National
Institute of Information and Communications Technology(NICT). They used the density, velocity, temperature, and Zand Y- components (in the GSM coordinate system) of the
IMF of real-time solar wind data, which were recorded by
the Advanced Composition Explorer (ACE) spacecraft
located near the L1 point [Zwickl et al., 1998]. We use

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A03S10Figure 1. Types of graphics posted on Web site, showing (top left) magnetic field lines connected to the
Earth. Blue and red lines are connected to northern and southern hemispheres, respectively. Some fieldlines with ends connected to both hemispheres are blue and some red. Also shown are (top right) plasmapressure on meridian plane, where the circle in the center represents Earth, (bottom left) contour lines
represent electric potential and color code shows electrical conductivity of polar ionosphere viewed from
North Pole, and (bottom right) input parameters observed by ACE spacecraft: solar wind velocity,
density, and Z- and Y-components of IMF, respectively.
the modified spherical coordinates [Tanaka, 2000], and grid
numbers 44, 56, and 60 for the r, theta, and phi directions,
respectively.
[12] Processing the real-time solar wind data and calculating the MHD simulations takes only a few minutes,
enabling us to obtain the 3-D distribution of the magnetic
field and plasma parameters in the magnetosphere during
the travel time of the solar wind. The travel time of the solar
wind from the position of the ACE spacecraft to themagnetopause is approximately 1 h, though travel time
basically depends on the solar wind velocity and the distance
between the ACE spacecraft and the Earth. Thus real-time
simulation is used to forecast the space weather of themagnetosphere for the 1-h period following the simulation.
[13] We generally simulate a period of 24 h a day for365 d. The results are illustrated visually, and the resulting
images can be posted to our Web site in almost real time
(http://www2.nict.go.jp/y/y223/simulation/realtime/). The
four kinds of graphics we post on the Web site are shownin Figure 1. The upper left panel shows magnetic field lines

around the Earth, and the upper right one shows the plasmapressure on the meridian plane. The circle in the center ofthe latter represents the Earth. Figure 1 shows that oursimulation reproduces the bow shock and the tail plasma
sheet. In the lower left panel, the contour lines represent theelectric potential, and the color code shows the electric
conductivity of the polar ionosphere viewed from the North
Pole. The lower right panel shows input parameters observed by the ACE spacecraft: solar wind velocity, density,and the Z- and Y-components of the IMF. The images arearchived in a movie format, and current and archived
images can be accessed on the Web site.
[14] The example shown in Figure 1 is for a period after
an interplanetary shock impacted the magnetopause. A
pressure pulse propagates from the subsolar point to themagnetotail. When the Z-component of the IMF becomes
negative, the current sheet thinning and plasmoid ejection
can be reproduced intermittently. When the plasmoid isejected, the conductivity of the polar ionosphere is enhanced on the nightside. When the Y-component of the

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A03S10KITAMURA ET AL.: AE INDICES DERIVED FROM MHD SIMULATIONIMF is not oscillatory, a clear convection cell structure (around cell and a crescent cell) can be observed [Tanaka,
2001]. The shape of the cells depends on the direction of the
Y-component of the IMF.

3. Comparison of the Simulated and Observed
AE Indices[15] The geomagnetic variations on the ground are theconsequence of the energy propagation furthest downstream

A03S10in the S-M-I coupling system and contain integrated information about the magnetosphere. Therefore the derivationof the magnetic field disturbance on the ground is suitablefor verifying the availability of the actual space weather
forecast using our simulation. The magnetic fields generated
by the ionospheric current in the simulation are used toderive the AE index in real time. Although the actual AE
index is deduced from the H-component variations in themagnetic field at the selected geomagnetic stations, which
are located at the auroral latitudes, the simulated AE index
here is deduced from the H-component magnetic variations
at all grid points located at latitudes between 60° and 70°.
There are seven grid points in the latitudinal direction and
40 in the longitudinal direction used to derive the AE index[Den et al., 2006].
[16] We use digital data from the Quick-Look (QL) AE
indices, which are calculated and distributed by the World
Data Center (WDC) for Geomagnetism, Kyoto (http://swdcwww.kugi.kyoto-u.ac.jp/AEasy/index.html) as the observed AE indices. In general, the QL AE indices are basedon semi-real-time magnetic data, which may include spikenoises or data errors, so it is not always appropriate to use
the QL indices for precise analysis. It is also noted that datafrom all 12 stations was not always available. When thishappened, we used the magnetic variation data from theobservatories that were operating to derive the QL AE
indices. The real-time solar wind data measured by ACE
are not processed by accurate data cleaning. Here, the QL
AE index can be appropriately compared with the simulated
AE index because one purpose of this study is to evaluate
how well the AE index using real-time data predicts space
weather.
[17] The simulated AE indices are calculated using realtime solar wind data obtained by the ACE spacecraft at apoint more than 200 RE upstream from the Earth, so theobserved AE indices lag the simulated AE indices by aduration that depends on the speed and the structure of thesolar wind. To obtain the time lag, we calculate the crosscorrelation coefficients of the observed AE indices to thesimulated AE indices whose time series are shifted from 0 to
120 min in 1-min steps. The time lag is defined as theshifted time with the maximum correlation coefficient
between the observed AE index and time-shifted simulated
AE index. We obtain daily correlation coefficients and lags
using 1-min sampling data over 24 h. The AU, AL, and AO
indices are calculated in the same way as the AE indices. In

Figure 2. (a) Comparison of simulated and observed
indices for 28 July 2006. AE, AU, AL, and AO indices areshown in respective order from top to bottom. Blue and redlines represent observed and simulated indices, respectively,
calculated from all the grid points located at latitudes between
60° and 70°. Black dotted lines are simulated indicescalculated from 12 grid points closest to AE observatories.
Time lag between simulated and observed AE indices is
71 min. MaxCCs shown in upper right of each panel are
maximum correlation coefficients between observed andsimulated (all stations) indices. Also shown is the (b) timelag and cross-correlation coefficients between simulated andobserved AE indices for 28 July 2006.
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A03S10KITAMURA ET AL.: AE INDICES DERIVED FROM MHD SIMULATION

Figure 3. Comparison between time lag and 1/V (V: solar
wind speed) for events in which maximum correlation
coefficients (maxCCs) are more than 0.4. Solid line
represents least-square fitting. Broken line represents
expectation from propagation of solar wind from positionof ACE (220 RE) to Earth.
this procedure, if the solar wind speed changes drastically in
1 d, the results of the analysis for that day may containsignificant errors. To clarify this effect, we calculated thedaily mean solar wind speed (Vmean), the daily maximum
speed (V max ) and the minimum speed (V min) for 398 dfrom August 2005 to September 2006. The ratio Rsw =(V max  V min )/Vmean for 398 d shows that 78% of
Rsws are less than 0.3, that is, the errors caused by changes
in the solar wind speed within a day are generally small.
[18] The simulated and observed indices for the interval
between August 2005 and September 2006 were used in thecalculations. Because of an interruption in computer operations, data for some simulated indices are missing. We also
removed data that included obvious errors for either simulated or observed AE indices. As a result, to calculate theAE, AU, AL, and AO indices, we used data collected for
247, 238, 247, and 201 d, respectively.
[19] Figure 2a shows a comparison of simulated andobserved indices for 28 July 2006. The AE, AU, AL, and
AO indices are shown in respective order from the top to thebottom panel. The blue and red lines indicate observed andsimulated indices, respectively, calculated from all grid
points located at latitudes between 60° and 70°. The black
dotted lines represent the simulated index calculated fromthe 12 grid points closest to the 12 AE observatories, which
are overwritten in each panel. We used the geomagnetic
coordinates for the 12 stations to derive the index.
[20] The figure shows that, as expected, the absolutevalues of the simulated AL and AU indices derived fromthe all grid-point data are generally larger than those fromthe 12-grid-points. Since there are two stations whose
geomagnetic latitude is above 70°, the 12-grid-points index
exceeds the all-grid-points index in some places. If wecompare the AL and AU indices, the difference between
all-grid-points indices and the 12-grid-points indices is

A03S10rather large in the AU index. High-latitude eveningsidestations contribute to the AU index [Allen and Kroehl,
1975]. The UT variation in the AU index shows thistendency.
[21] However, the difference between the 12-grid-points
indices and the all-grid-points indices is small comparedwith the difference from the observed indices. This indicatesthat the derivation in the AE index from all grid points
between 60° and 70° does not cause serious discord for thestatistical analysis. Hereafter, we use as the simulated
indices those calculated from all the grid point data.
[22] We calculate the cross correlation coefficient between the simulated index and the observed one. The
correlation coefficients of the AE index for the time lags
are shown in Figure 2b. The correlation coefficients are
distributed according to a quadratic function of the timedelay. In this case, the maximum correlation coefficient(maxCC) between the simulated (red line) and observed AE
indices (blue line) is 0.77 at a 71-min lag. This value isroughly accurate when the solar wind speed for the day,
330 km/s, is taken into account.
[23] The observed AE index shows clear auroral activities
from 0200 UT to 0500 UT with a maximum amplitude of
1000 nT. The simulated AE index (red) also shows
activities from 0200 UT to 0530 UT with a peak amplitudeof 1700 nT, which is larger than that of the observed AE
index. In addition, a large fluctuation with an amplitude of
700 nT appears in the simulated AE index, whereas nosuch fluctuation appears in the observed AE index. The
amplitude of the fluctuation in the simulated AE index isgenerally larger than that in the observed AE index. In
contrast, the simulated AE index for the AE activities from
0600 UT to 0800 UT with an amplitude of 800 nT,
corresponds well to the observed AE index.
[24] The AE index is derived by subtracting the AL fromthe AU, so the difference between the simulated andobserved AE indices can be considered in terms of thedifference between the AU and AL indices. We calculate
the maxCC for the AE and for the AU, AL and AO indices.
The maxCCs of the AL and AO indices are 0.766 and 0.596,
respectively, while the maxCC of the AU index is 0.162,
which is much less than those of the other indices. The
envelopes of the simulated AL index correlate well withthe observed AL index during 0200 – 0500 UT and 0600–
0800 UT, though the large fluctuation in the simulated AL
index from 0200 UT to 0600 UT does not appear in theobserved AL index. The amplitudes of the simulated AL
activities during the above-indicated intervals are roughly
comparable to that of the observed AL index. In contrast,
the time variation of the simulated AU index is quite
different from that of the observed AU index. The intensityof the simulated AU index is larger than that of theobserved AU index almost all day long, especially during
the above-indicated intervals. The difference between theamplitudes of the simulated and observed AU indicesreaches 500 nT in some cases.
[25] This indicates that the westward electrojets are
reproduced well by the present simulation, whereas theeastward electrojets are sometimes not reproduced well. In
general, the intensity of the auroral electrojet is predominant
in the westward direction during the interval of the substorm. Therefore during the substorm, the simulated AE

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A03S10KITAMURA ET AL.: AE INDICES DERIVED FROM MHD SIMULATION

A03S10Figure 4. Occurrence frequency of maximum correlation coefficients of (a) AE, (b) AU, (c) AL, and(d) AO indices. Vertical and horizontal axes represent occurrence of days and maximum correlation
coefficient (maxCC), respectively. Range of horizontal axis is between 0 and 1, and negative maxCCs(less than 1% of total) are not shown here.
index can be reproduced well. We used one event as an
example of a typical substorm event and as one of the bestcorrelated events. Similar signatures are observed in othersubstorm events. To clarify what constitutes a factor limiting the reproducibility of the simulation, it is necessary tostatistically investigate the correlations between the simulated and observed indices.
[26] The time lags between the observed and simulated
AE indices are also important in evaluating the results of thesimulations. The time lag mainly represents the travel timeof the solar wind from the position of the ACE spacecraft tothe Earth. Thus the time lag should decrease as solar windspeed increases. Figure 3 shows the relationship between
the time lag and the solar wind speed observed by the ACE
spacecraft. Although the data points scatter, 1/V (V: the solar
wind speed) increases as the time lag increases, and thecorrelation coefficient between the two parameters is 0.53.
The distribution of the data points can fit the linear function
y = 1.26  105 x + 1.40  103 (x: time lag [min], y: 1/(solar wind speed) [1/km/s]). This indicates that the time lagbasically corresponds to the travel time of the solar windspeed. However, the y-intercept of 1.40  103 suggeststhat the time lag is not due to solar wind speed alone. The
broken line in Figure 3 represents the expected relationshipbetween 1/V and the time lag under the assumption that the
ACE spacecraft is located 220 RE upstream from the Earth.
The figure shows that most of the events have a longer-thanexpected time lag. This means that the time lag cannot be

explained by the propagation time of the solar wind from
ACE to the Earth alone. The time constant of the magnetospheric response to the solar wind may significantly affectthe time lag between the observed and simulated AE
indices.
[27] Figure 4 shows the occurrence frequencies of the
maxCCs for AE, AU, AL, and AO indices. In Figure 4a,
most of the maxCCs for the AE index are distributed in therange between 0.4 and 0.8. Of the total of 247 d, 158 (64%)
showed a maxCC of more than 0.5. This shows thatsimulation can accurately reproduce the AE activities formost days and is useful in predicting the AE index for the
next hour. However, there were also 89 d (36%) for which
the maxCCs were less than 0.5, which is not accurateenough to be used in prediction.
[28] The occurrence frequencies of the maxCCs for the
AU, AL, and AO indices are shown in Figures 4b, 4c, and 4d,
respectively. The distribution of the occurrence frequencyof the AL indices is similar to that of the AE indices; 54%of the days analyzed had maxCCs of over 0.5. In contrast,
the occurrence frequencies of the maxCCs for the AU and
AO indices have rather different distributions. As can beseen in Figure 4b, the occurrence of the maxCCs for the AU
indices was broadly distributed, with a formless peakaround 0.4, and only 37% of the days had maxCCs of morethan 0.5. Of the four types of indices, the AO indices
showed the worst correlation, with 96% of the days having
maxCCs of less than 0.5.

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Figure 5. Relationship between maximum cross-correlation coefficients (maxCC) and standard deviation ofobserved AE (STD(AEobs)). Dashed horizontal line represents maxCC of 0.5, and dashed vertical line represents
STD(AEobs) of 100 nT.
[29] A statistical analysis of the cross-correlations for the
247 d shown in Figures 3 and 4 indicates that the simulated
AE index corresponds fairly well to the observed AE index,
but that, for 36% of the days, the simulated AE indices show
very little correlation with the observed AE indices. The
factors in these lower maxCCs should be researched further.
[30] It is well known that the AE index is correlated withsolar wind parameters and the IMF. Variations in the AE
index and substorms occur as a consequence of the processes in the solar wind-magnetosphere coupling system.
However, the triggering mechanism of substorms is still
unknown. Lyons et al. [1997, 2005] found that a large
majority of the substorms with well-defined onset times are
triggered by northward turning of the IMF. In addition, theexternal cause of the triggering, that is, change in the
dynamic pressure under a strong southward IMF, also
causes auroral brightening of the substorm [Shi et al.,2005, 2006]. In contrast, Kitamura et al. [2003, 2005]
reported that quasi-periodic substorms occur during steadysouthward IMFs with no prominent northward turning. Thismeans that the electromagnetic energy that flows continuously into the magnetosphere during the southward IMF isreleased as quasi-periodic substorms not by an external
trigger but by an internal process of the magnetosphere.
Even if the internal process is related to the substorm
trigger, the AE index still correlates fairly well with the
IMF and solar wind parameters.
[31] To compare the solar wind conditions and maxCCsof the AE indices, we used the Z-component of the IMF,
solar wind speed, and proton number density, all of which
are measured by the ACE spacecraft, to calculate the crosscorrelation coefficients with the maxCC obtained as described above. The results showed no direct influence of thesolar wind parameters on the maxCCs. We also calculated
the cross-correlation coefficients between the maxCCs andthe standard deviations of these solar wind parameters to

A03S10investigate the influence of the fluctuations of these solar
wind parameters on the maxCCs. All the correlation coefficients calculated this way were less than 0.1 (data notshown here), which indicates that solar wind fluctuations
have no direct influence on the maxCCs.
[32] To predict activities of the AE index, we should pay
more attention to cases of intense AE activity rather than tomild AE activity. For this reason, we calculated the standard
deviations of the observed AE indices (STD(AEobs)) for
24 h and compared them with the maxCCs. Figure 5 shows
the results for this comparison. The distribution of the plots
seems to be scattered, and no obvious linear relationship canbe observed in Figure 5. There were 120 d (out of 247) for
which the STD(AEobs) was over 100 nT. The maxCCs for
74% of these 120 d were over 0.5. This means that thecorrelations between the simulated and observed AE indices
are better for days with observed AE indices with largeamplitude standard deviations.
[33] In contrast, most of the data points for the small
maxCCs are distributed in the STD(AEobs) range between 0
and 100 nT. As indicated in the discussion of Figure 2a, themaxCCs of the AE indices for 89 of the 247 d were less than
0.5. For 57 of the 89 d (64%), the STD(AEobs) were lessthan 100 nT. This means that the simulated AE indices do
not correlate well with the observed AE indices when the
STDs are small. We discuss the reasons for this signature
below.

4. Discussion and Summary[34] In previous studies, the relationships between thevariation in the AE index and the solar wind are analyzed
using the linear prediction filter [Bargatze et al., 1985]. This
study implied that the solar wind-magnetosphere couplingshould be described by nonlinear dynamics. Klimas et al.
[1996] reviewed autonomous analysis and nonlinear inputoutput data-based methods for solar wind-magnetosphere
coupling problems. They showed that unloading process
should be included for realistic replications of the AL index.
Using the Faraday Loop model, Baker et al. [1997] also
showed that a realistic model including the loading andunloading process is necessary to describe the generalsubstorm dynamics, as reflected in the AL index. Thesestudies used mathematical models that describe the solar
wind-magnetosphere coupling and demonstrated specific
events. In the present study, we use the MHD simulation
including the S-M-I coupling and statistically analyze alarge number of events.
[35] We have shown that the AE indices derived from the
MHD simulation using real-time solar wind data as upstream boundary conditions are accurate enough to be usedin predicting the AE indices for the following hour. The
comparison of the simulated and observed AE indices
generally showed a good correlation. However, the correlation of these two indices is not good for 36% of the days
analyzed. What is the reason for such inconsistency?
[36] We first consider this question from the simulation
model side. One possible reason for the inconsistency is thatthe geomagnetic north and geographical north of the coordinate system of the present simulation are defined on thesame axis as the spherical coordinate system [Den et al.,2006]. We think this is the most significant problem in our

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simulation model. The agreement of the geographic andgeomagnetic coordinates causes some deviation in thecurrent flow in the polar ionosphere and leads to errors incalculating the AE index. The simulation model should be
modified to correct this.
[37] Here, we consider the solar wind structure to discuss
the effect on the statistical results shown in Figure 5. The
relationship between the time lag and the solar wind speed
shown in Figure 3 indicates that the time lag increases as thesolar wind speed decreases, although the correlation coefficient of 0.53 might not be valid. We used real-time solar
wind data at a point L1 upstream of the Earth. Although thesolar wind properties monitored at the L1 point generally
represent the solar wind conditions near the magnetopause[e.g., Paularena et al., 1998; Richardson et al., 1998], thespatial structure of the solar wind may affect the time lag.
Solar wind structure was recently investigated using multispacecraft observations in near-Earth interplanetary space[Eastwood et al., 2002; Lepping et al., 2003]. These studies
compared the solar wind data measured by the ACE
spacecraft with those measured by IMP-8 and Cluster
satellites and showed a difference significant enough toaccount for the spatial structure of the heliospheric current
sheet and MHD discontinuities. Such an inhomogeneity ofthe solar wind structure becomes more significant when
there are large-amplitude variations in the solar wind speed
for 24 h. Thus modifying the solar wind property during the
propagation from the L1 point to the upstream boundary of
the simulation may reduce the correlation between the timelag and the solar wind speed. This may also affect thesignature of the maxCC shown in Figure 5. Although thesolar wind properties observed at the L1 point are statistically correlated with those reported near the Earth by thestudies mentioned above, Paularena et al. [1998] also
indicated that the average solar wind flux correlation
between the two spatially separated spacecraft decreases
as the standard deviation of the solar wind flux decreases.
This suggests that the IMF and the plasma parameters maychange significantly during the propagation from ACE to
the upstream boundary of the simulation, reducing the
maxCCs for days when the observed AE indices have small
standard deviations.
[38] Next, we consider the properties of the AE, AU, AL,
and AO indices. In Figure 4, the simulated and observed
indices show good correlations for the AE and AL indices.
The AL index mainly represents substorm activities because
the westward electrojet associated with substorm expansion
is generally observed as a sudden decrease in the AL index.
The good correlation between simulated and observed AL
indices indicates that the present simulation model reproduces the substorm expansion and the resulting westward
electrojet currents well.
[39] The occurrence frequencies of maxCCs of the AU
and AO indices show quite different distributions from thoseof the AE and AL indices. Days with AU and AO indiceswith maxCCs over 0.5 represent 37% and 4% of the totaldays, respectively. What does this variation in AU indicesmean? Unlike the westward electrojet, the eastward ionospheric currents do not flow in the confined region during
the substorm expansion [Kamide et al., 1994, 1996]. Also,
the intensity of the AU index and the ionospheric potential
in the dusk cell are generally no greater during the substorm

A03S10than under nonsubstorm conditions [Weimer, 1999]. It has
also been reported [Weimer, 1999] that the AU index is astatistical linear function of the polar cap potential. However, the mechanism and causes of the AU variation seemnot to have been observationally well proven. The lower
correlation between the simulated and observed AU indices
may indicate that the processes that cause the AU variations
are not correctly included in the simulation. This should be
more carefully investigated in future work.
[40] Although both AE and AO indices are deduced fromthe combination of the AU and AL indices, the correlation of
the simulated and observed AO indices is statistically much
lower than that of the AE index. This can be explained by
the overestimation of the variation in the simulated indices.
When the simulated AU index is larger (smaller) than theobserved AU index, the simulated AL is also larger (smaller)
than the observed AL index. (The AL is usually negative. A
larger AL means a smaller absolute AL value.) In this case,the overestimated variations of the simulated AL and AU
indices cancelled each other out by the subtraction used toderive the AE index and enhanced by the addition used toderive the AO index. As a result, the correlations between
the simulated and observed AE (AO) indices are better(worse).
[41] Finally, we consider the reasons for the larger fluctuations in the simulated AE index. Figure 5 indicates thatthe simulated and observed AE indices show worse correlations when the standard deviations of the observed AE
indices are less than 100 nT. These fluctuations may be theresult of the difference of the magnetospheric response to
the solar wind between the actual magnetosphere and thesimulation. The other factor in the fluctuations may arise
from the time resolution of the present simulation. The time
step depends on the CFL (Courant-Friedrichs-Lewy) conditions, but it is usually much shorter than 1 s. We used the
1 min value for the simulated index, but it may be that the
shorter time scale fluctuations appear in the simulated AE
index rather than in the observed index.
[42] Tanaka [2000] showed that the MHD simulations
that assumed that the magnetotail becomes more diffuse as
it goes further down tail can reproduce substorms. In his
simulation, the southward IMF caused the typical signaturesof the substorm growth phase, which are plasma sheet
thinning, increase in the size of the auroral oval, andincrease in the FAC in the polar ionosphere. This means
the substorm onset may be directly caused by the dipolarization of the magnetic fields, which is hastened by thenorthward turning of the IMF. The onset of the substorm
occurs as an abrupt change in the pressure distribution in thenear-Earth plasma sheet and moving of the convection flow
into the inner magnetosphere. The signatures of the substorm expansion phase (dipolarization, plasma injection into
the inner magnetosphere, and increase in the intensities of
the FACs connected to the nightside ionosphere) were well
reproduced in the simulations.
[43] These simulation results were obtained under ideal
solar wind conditions to enlarge the magnetospheric convection associated with the substorm. In contrast, the AE
indices derived from real-time MHD simulation using realtime solar wind parameters reflected actual solar wind
disturbances, which consist of fluctuations and discontinuities with various ranges of temporal and spatial variations.

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Thus the inconsistency in the statistics between the simulated and observed AE indices shown in Figures 4 and 5
may be caused by the magnetospheric responses to therealistic solar wind disturbances, which are not included in
the simulation. The present result indicates that the simulation responds sensitively to small fluctuations in the IMF
and dynamic pressure of the solar wind. Sometimes, small
substorm-like enhancements of the AE indices appear in thesimulation even if the observed AE shows no substorm
signatures. Uritsky et al. [2001] indicated from a selforganized criticality model that the scaling features in the
AE index are independent of the solar wind input for a timescale shorter than 3.5 h. Our results also show that shortperiod fluctuations in the solar wind do not appear in theobserved index, but that they do appear in the simulation.
Thus the real S-M-I coupling system seems to have afunction that absorbs some of the short-period perturbationsin the electromagnetic energy flowing from the solar wind
into the magnetosphere by some mechanism that is notincluded in our MHD simulation and that dampens theresulting fluctuations in the AE indices compared with thesimulation.
[44] The particle effect and phenomena smaller than themesh size were not included in the simulation model.
Moreover, bursty events whose timescale is shorter than
the time step of the simulation may not be reproduced wellin the present simulation. These phenomena may affect theresults. If we include such effects, the simulation may
reproduce the AE index more accurately. These problems
should be resolved in the future.
[45] We have shown that simulated AE indices are fairly
well correlated with observed AE indices. We have also
examined when and how they did not agree. The disagreement between the simulated and observed AE indices has
given us a significant clue about the processes of substorms
and S-M-I coupling. A more detailed comparison between
the simulation using the real solar wind data and the various
kinds of observations in the magnetosphere should be donein the future.
[46] Acknowledgments. The Quick-look (QL) AE indices used in thisstudy were provided by World Data Center for Geomagnetism, Kyoto. The
authors also acknowledge the observatories that record geomagnetic data
for the AE index. Data from AE stations in Russia have been made available
through the international project, RapidMag, with participants from theWorld Data Center for Geomagnetism, Kyoto University (WDC Kyoto); theNational Institute of Information and Communications Technology (NICT),Japan; Arctic and Antarctic Research Institute (AARI), Russia; the Institutefor Dynamics of Geospheres (IDG), Russian Academy of Science; theGeophysical Institute (GI), the University of Alaska; and the Johns Hopkins
University Applied Physics Laboratory (JHU/APL). The real-time magnetic
field (MAG) and solar wind (SWEPAM) data measured by the ACE
spacecraft were provided by the Space Environment Center (SEC) of the
National Oceanic and Atmospheric Administration (NOAA).
[47] Amitava Bhattacharjee thanks the reviewers for their assistance inevaluating this paper.

References
Allen, J. H., and H. W. Kroehl (1975), Spatial and temporal distributions ofmagnetic effects of auroral electrojets as derived from AE indices,
J. Geophys. Res., 80, 3667.
Baker, D. N., A. J. Klimas, D. Vassiliadis, T. I. Pulkkinen, and R. L.
McPherron (1997), Reexamination of driven and unloading aspects ofmagnetospheric substorms, J. Geophys. Res., 102, 7169.
Bargatze, L. F., D. N. Baker, R. L. McPherron, and E. W. Hones (1985),
Magnetospheric impulse response for many levels of geomagnetic activity, J. Geophys. Res., 90, 6387.

A03S10Davis, T. N., and M. Sugiura (1966), Auroral electrojet activity index AE
and its universal time variations, J. Geophys. Res., 71, 785.
Den, M., et al. (2006), Real-time Earth magnetosphere simulator with threedimensional magnetohydorodynamic code, Space Weather, 4, S06004,
doi:10.1029/2004SW000100.
Eastwood, J. P., A. Balogh, and M. W. Dunlop (2002), Cluster observationsof the heliospheric current sheet and an associated magnetic flux rope andcomparisons with ACE, J. Geophys. Res., 107(A11), 1365, doi:10.1029/
2001JA009158.
Elsen, R. K., and R. M. Winglee (1997), The average shape of the Magnetopause: A comparison of three-dimensional global MHD and empiricalmodels, J. Geophys. Res., 102, 4799.
Fedder, J. A., and J. G. Lyon (1995), The Earth’s magnetosphere is 165 RE
long: Self-consistent currents, convection, magnetospheric structure, andprocesses for northward interplanetary magnetic field, J. Geophys. Res.,
100, 3623.
Janhunen, P. H., and E. J. Koskinen (1997), The closure of Region-1 fieldaligned current in MHD simulation, Geophys. Res. Lett., 24, 1419.
Kamei, T., M. Sugiura, and T. Araki (1981), Auroral electrojet (AE) indicesfor January-June 1978, in Data Book 3, World Data Cent. for Geomagn.,
Kyoto Univ., Japan.
Kamide, Y., et al. (1994), Ground-based studies of ionospheric convection
associated with substorm expansion, J. Geophys. Res., 99, 19,451.
Kamide, Y., W. Sun, and S.-I. Akasofu (1996), The average ionosphericelectrodynamics for the different substorm phases, J. Geophys. Res., 101,
99.
Kitamura, K., H. Kawano, S. Ohtani, A. Yoshikawa, K. Yumoto, and the
Circum-pan Pacific Magnetometer Network Group (2003), Quasi-periodic
substorms during recovery phase of magnetic storm for space weather
study, paper presented at Int. Symp. on Inf. Sci. and Electr. Eng. 2003,
Kyushu Univ., Fukuoka, Japan.
Kitamura, K., H. Kawano, S. Ohtani, A. Yoshikawa, and K. Yumoto(2005), Local-time Distribution of low and middle latitude ground magnetic disturbances at sawtooth injections of April 18 – 19, 2002, J. Geophys. Res., 110, A07208, doi:10.1029/2004JA010734.
Klimas, A. J., D. Vassiliadis, D. N. Baker, and D. A. Roberts (1996), The
organized nonlinear dynamics of the magnetosphere, J. Geophys. Res.,
101, 13,089.
Lepping, R. P., C.-C. Wu, and K. McClernan (2003), Two-dimensional
curvature of large angle interplanetary MHD discontinuity surfaces:
IMP-8 and WIND observations, J. Geophys. Res., 108(A7), 1279,
doi:10.1029/2002JA009640.
Lyons, L. R., G. T. Blanchard, J. C. Samson, R. P. Lepping, T. Yamamoto,and T. Moretto (1997), Coordinated observations demonstrating external
substorm triggering, J. Geophys. Res., 102, 27,039.
Lyons, L. R., D.-Y. Lee, C.-P. Wang, and S. B. Mende (2005), Global
auroral responses to abrupt solar wind changes: Dynamic pressure, substorm, and null events, J. Geophys. Res., 110, A08208, doi:10.1029/
2005JA011089.
Narock, A., L. Rastaetter, M. Kuznetsova, and M. Hesse (2004), Realtime
magnetosphere simulation at CCMC, Eos Trans. AGU, 85(47), Fall Meet.
Suppl., Abstract SA43B-06.
Paularena, K. I., G. N. Zastenker, A. J. Lazarus, and P. A. Dalin (1998),
Solar wind plasma correlations between IMP 8, INTERBALL-1 andWIND, J. Geophys. Res., 103, 14,601.
Raeder, J. R., J. Walker, and M. Ashour-Abdalla (1995), The structure of
the distant geomagnetic tail during long periods of northwsard IMF,
Geophys. Res. Lett., 22, 349.
Richardson, J. D., F. D. Dashevskiy, and K. I. Paularena (1998), Solar windplasma correlations between L1 and Earth, J. Geophys. Res., 103, 14,619.
Shi, Y., E. Zesta, L. R. Lyons, A. Boudouridis, K. Yumoto, and K. Kitamura(2005), Effect of solar wind pressure enhancements on storm time ring
current asymmetry, J. Geophys. Res., 110, A10205, doi:10.1029/
2005JA011019.
Shi, Y., E. Zesta, L. R. Lyons, K. Yumoto, and K. Kitamura (2006), Statistical study of effect of solar wind dynamic pressure enhancements on
dawn-to-dusk ring current asymmetry, J. Geophys. Res., 111, A10216,
doi:10.1029/2005JA011532.
Song, P., D. L. De Zeeuw, T. I. Gombosi, C. P. T. Groth, and K. G. Powell(1999), numerical study of solar wind-magnetosphere interaction for
northward interplanetary magnetic field, J. Geophys. Res., 104, 28,361.
Spicer, D. S., S. T. Zalesak, R. Lohner, and S. Curtis (1996), Simulation of
the magnetosphere with a new three dimensional MHD code and adaptive
mesh refinement: Preliminary results, Adv. Space Res., 18, 253.
Takahashi, K., C. Meng, T. Kamei, T. Kikuchi, and M. Kunitake (2004),
Near-real-time auroral electrojet index: An international collaboration
makes rapid delivery of auroral electrojet index, Space Weather, 2,
S11003, doi:10.1029/2004SW000116.
Tanaka, T. (1994), Finite volume TVD scheme on an unstructured grid
system for three-dimensional MHD simulation of inhomogeneous sys-

9 of 10

A03S10KITAMURA ET AL.: AE INDICES DERIVED FROM MHD SIMULATION

tems including strong background potential fields, J. Comput. Phys., 111,
381.
Tanaka, T. (1995), Generation mechanisms for magnetosphere-ionospherecurrent systems deduced from a three-dimensional MHD simulation of
the solar wind-magnetosphere-ionosphere coupling processes, J. Geophys. Res., 100, 12,057.
Tanaka, T. (1999), Configuration of the magnetosphere-ionosphere convection system under northward IMF conditions with nonzero IMF By,
J. Geophys. Res., 104, 14,683.
Tanaka, T. (2000), The state transition model o the substorm onset,
J. Geophys. Res., 105, 21,081.
Tanaka, T. (2001), Interplanetary magnetic field By and auroral conductance effects on high-latitude ionospheric convection patterns, J. Geophys. Res., 106, 24,505.
Tóth, G., et al. (2005), Space Weather Modeling Framework: A new tool forthe space science community, J. Geophys. Res., 110, A12226,
doi:10.1029/2005JA011126.
Uritsky, V. M., A. J. Klimas, and D. Vassiliadis (2001), Comparative studyof dynamical critical scaling in the auroral electrojet index versus solar
wind fluctuations, Geophys. Res. Lett., 28, 3809.
Walker, R. J., T. Ogino, J. Raeder, and M. Ashour-Abdalla (1993), A global
magnetohydrodynamic simulation of the magnetosphere when the inter-

A03S10planetary magnetic field is southward: The onset of magnetotail reconnection, J. Geophys. Res., 98, 17,235.
Weimer, D. R. (1999), Substorm influence on the ionospheric electric potentials and currents, J. Geophys. Res., 104, 185.
White, W. W., G. L. Siscoe, G. M. Erickson, Z. Kaymaz, N. C. Maynard,
K. D. Siebert, B. U. O. Sonnerup, and D. R. Weimer (1998), The magnetospheric sash and the cross-tail S, Geophys. Res. Lett., 25, 1605.
Zwickl, R. D., et al. (1998), The NOAA Real-Time Solar-Wind (RTSW)
system using ACE data, Space Sci. Rev., 86, 633.


S. Fujita, Meteorological College, Asahicho, Kashiwa, Chiba, 277-0852,
Japan.
K. Kitamura, Tokuyama College of Technology, Gakuendai, Shunan,
Yamaguchi, 745-8585, Japan.
M. Kunitake, H. Shimazu, H. Shinagawa, and S. Watari, National
Institute of Information and Communications Technology, Nukuikitamachi,
Koganei, Tokyo, 184-8795, Japan.
T. Tanaka, Department of Earth and Planetary Sciences, Kyushu
University, Hakozaki, Higashiku, Fukuoka, 812-8581, Japan.

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S07, doi:10.1029/2007JA012707, 2008

for

Full
ArticleNumerical studies on three-dimensional earthward
fast plasma flows in the near-Earth plasma sheet by
the spontaneous fast reconnection model
K. Kondoh1 and M. Ugai1
Received 2 August 2007; revised 4 October 2007; accepted 30 October 2007; published 28 March 2008.

[1] The spontaneous fast reconnection model is applied to the earthward fast flow events
observed in the near-Earth plasma sheet. Here, the earthward fast flow events include bothof bursty bulk flow events and flow burst events. In order to apply it directly to actual
observations, virtual probes are located in the plasma sheet region in the three-dimensional
simulation domain so that we can directly observe the temporal variations of plasma
quantities in accordance with the growth and proceeding of the fast reconnection. In thismodel, magnetic reconnection drastically evolves and Alfvénic fast plasma jet flows in thevery restricted narrow channel, and a large-scale plasmoid is formed ahead of the fastplasma jet. The results of virtual observation of these evolutions are found to be in good
agreement with actual satellite observations. At the same time, in the lobe region,
travelling compression regions (TCRs) are observed in connection with the fast flow
events. The temporal profiles of magnetic fields detected by the virtual probes are also ingood agreement with actual satellite observations. It is concluded that the earthward fast
flow events and earthward TCR events result from the fast reconnection mechanism.
Citation: Kondoh, K., and M. Ugai (2008), Numerical studies on three-dimensional earthward fast plasma flows in the near-Earthplasma sheet by the spontaneous fast reconnection model, J. Geophys. Res., 113, A03S07, doi:10.1029/2007JA012707.

1. Introduction[2] It is widely accepted that magnetic reconnection plays
crucial role on large dissipative events, such as geomagnetic
substorms and solar flares [Shibata, 1999, 2004]. In spaceplasmas of extremely large magnetic Reynolds number,magnetic reconnection can convert magnetic energy toplasma energies most effectively by slow shocks. A possible
fast reconnection configuration, involving standing slow
shocks, was first proposed by Petschek [1964]. In thisrespect we have proposed the spontaneous fast reconnection
model. This model predicts that the fast reconnection
mechanism spontaneously develops from inside the systemby the self-consistent interaction between plasma microscopic processes and macroscopic reconnection flows[Ugai, 1984, 1986]. The previous MHD simulations havedemonstrated that the spontaneous fast reconnection model
works quite effectively even in three-dimensional situations[Ugai and Kondoh, 2001; Ugai et al., 2004, 2005]. Oncethe fast reconnection builds up, an Alfvénic fast flow iscaused, and if the fast flow is obstructed, the reconnected
field lines are piled up, and a magnetic loop is formed. A
magnetic loop in our simulations is allowed to be formed
because of a wall boundary assumed at one edge of acurrent sheet system [Ugai et al., 2003].
1
Electrical and Electronic Engineering and Computer Science, Graduate
School of Science and Engineering, Ehime University, Matsuyama, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012707$09.00[3] Here, we are interested in the flow enhancements (Vx ^300 km/s) in the near-Earth (X ^ 30 RE) plasma sheet
which are called bursty bulk flows (BBFs) or flow bursts(FBs) [Angelopoulos et al., 1992]. The existence of theseearthward fast flow events and sharp compression of themagnetic field in the near-Earth plasma sheet during active
magnetospheric conditions have been pointed out by a lotof individual case studies [Sergeev et al., 2000, 2001;
Angelopoulos et al., 1992]. In particular, Sergeev et al.
[2000] showed a narrow transient flow jet using fivespacecraft measurements. They showed that the 3 Re
wide fast plasma jet propagates from ^40 Re in the midtail,and is able to reach the inner magnetosphere at 6.6 Re in
about 10 min.
[4] Statistical studies (superposed epoch analyses) havealso been carried out in order to understand the generalcharacteristics of BBF [Ohtani et al., 2004; Angelopoulos etal., 1992; Schodel et al., 2001]. To construct the superposed
epoch, it is necessary to define the reference time. The
physical quantity of each event is superposed about thereference time. Previous studies used some reference times
based on the flow velocity [Angelopoulos et al., 1992;
Ohtani et al., 2004] or the magnetic field [Ohtani et al.,2004] or the electric field [Schodel et al., 2001]. By thisaveraging method, any fluctuation in the parameter ispreserved in the average, whereas fluctuations shifting intime are averaged out. Before constructing a superposed
epoch, consideration should be given to the selection of
event. Most of studies define the earthward BBF event as
the interval during which (1) the ion plasma beta b i > 0.5,

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and (2) the ion bulk velocity is continuously greater than
100 km/s and the maximum value is greater than 300 km/s.
In each analysis, the other conditions are added in order tomuch strictly exclude suspicious events. They used the dataset, which are satisfied the above conditions, observed in X >31 Re from the midtail region to the near-Earth region.
[5] These previous studies have shown that earthward
BBFs are generally accompanied by dipolarization of theplasma sheet, and BBFs are directed either earthward or
tailward. BBFs are therefore often considered signatures ofbursty and localized reconnection in the plasma sheet[Ohtani et al., 2004; Schodel et al., 2001]. However, there
is still no agreement about their origin [Angelopoulos et al.,
1992; Lui, 1996].
[6] The main theme of the present paper is to investigate
the relationship between fast magnetic reconnection andearthward fast flow using MHD simulations on the basis of
the spontaneous fast reconnection model. At first, we will
explain the simulation model in section 2 and show thethree-dimensional evolution of magnetic reconnection andmagnetic loop in section 3.1. Then, we will introduce an
example of the results on superposed epoch analysis for theearthward fast flow events in section 3.2 and compare oursimulation results with it using virtual satellite observationin simulation domain in section 3.3. It is well known thatthese earthward fast flow events are accompanied withearthward moving traveling compression regions (TCRs)[Slavin et al., 2003]. Therefore we will introduce an
example of the magnetic field measurements in earthward
moving TCRs and compare our simulation results with itusing virtual satellite measurements as well as the case ofearthward fast flow events and show the connection between the earthward fast flow events and the earthward
moving TCRs using the simultaneous virtual satellite observations in section 3.4. Finally, section 4 is a summary anddiscussion.

are in good agreement with the situation inferred from theobservation results of earthward flow [Shiokawa et al.,1997]. This wall boundary is, of course, not the real one
but is simply assumed in the present paper.
2.1.1. Basic Equations[9] Our interest is directed to the evolution of fast flows,
and the MHD approximations are valid for these macroscopic phenomena. The compressible MHD equations areDr¼ rr uDt
Du
¼ rP þ J  BrDt@B r  ðu  BÞ ¼ r  ðhJÞ@tDer
¼ Pr u þ hJ2
Dt
B
J¼r
m0
r B¼0

P þ B2x ¼ 1 þ b 0

2.1. Simulation Modeling[8] Simulation model in this study is the same as that in
the previous simulations on the three-dimensional magnetic
loop [Ugai et al., 2003] except for the initial density
condition. It is assumed in this study that the initial
temperature is constant everywhere, while the initial density
was constant in the previous study. Our previous magnetic
loop simulations using the wall boundary showed that thefast magnetic reconnection produces fast reconnection jet,and it flows in plasma sheet and is suddenly braked at theboundary between the dipolar and tail-like magnetic fields
because of the counterward pressure force, and the reconnection jet is decelerated [Ugai et al., 2003]. This situation
may be consistent with the earthward flow, and these results

ð1Þ

where, D/Dt @/@t + u r; the gas law, P = (g  1)re isassumed (e is the internal energy per unit mass, and g is the
specific heat ratio with g = 5/3 assumed here (an adiabaticcase)), as is Ohm’s law, E + u  B = hJ (h may be aneffective resistivity). The basic equations (1) are transformed to a conservation-law form, and the modified
Lax-Wendroff scheme is used for the numerical computation [Ugai, 1987].
2.1.2. Initial-Boundary-Value Problem[10] As an initial configuration, the one-dimensional antiparallel magnetic field B = [Bx(y), 0, 0] is assumed as: Bx(y) =sin(py/2) for 0 < y < 1; Bx(y) = 1 for 1 < y <Y1; Bx(y) = cos[(y Y1)p/1.2] for Y1 < y < Ym(= Y1 + 0.6); Bx(y) = 0 for Ym < y;also, Bx(y) = Bx(y) for y < 0. The plasma pressure P(y)initially satisfies the pressure-balance condition,

2. MHD Simulations[7] In the present study, we are interested in the underlying physical mechanism of earthward fast flow evolution,
so that the simulation model will be idealized and simplified
in view of precise numerical computations. Also, theearthward fast flow is closely related to the formation ofmagnetic loop (or magnetic dipolarization), so that thepresent simulation model is similar to the one of the 3-D
magnetic loop dynamics [Ugai et al., 2003].

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ð2Þ

where b 0 is the ratio of plasma pressure to the magnetic
pressure in the ambient magnetic field region 1 < y < Y1, sothat P(y = 0) = 1 + b 0 initially (in the present study, b 0 =0.15 is taken). Initially, fluid velocity u = (0, 0, 0), and,
constant temperature T = P/r = 1 + b0 is assumed, so thatthe plasma density r initially satisfiesrð yÞ ¼ Pð yÞ=ð1 þ b 0 Þ;

ð3Þ

The normalization of quantities, based on the initialquantities, is self-evident; distances are normalized by thehalf-width of the current sheet d0, B by the field strength in
B2x0ﬃ/(2m0), and r by ri =the magnetic field region Bx0, P bypﬃﬃﬃﬃﬃﬃﬃﬃ
r(y = 0); also, u by VAx0(= Bx0/ m0 ri ), time t by d0/VAx0,
current density J by J0 = Bx0/(m0 d0), and so forth. Note that
the Alfvén velocity in the ambient magnetic fieldregion(1 <ﬃ
pﬃﬃﬃﬃ
y < Y1) at initial state is given by VAe = VAx0/ re (re is thedensity there).
[11] Here, the conventional symmetry boundary conditions are assumed on the (x, y), (y, z), and (z, x) planes.
Initially, Bz = 0 because of this symmetry boundary on the(x, y) plane. Hence the computational region can be restrict-

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Figure 1. The relationship between coordinate systems,
usual GSM one and that used in our three-dimensional
simulations.
ed to the first quadrant only and taken to be a rectangularbox, 0 < x < Lx, 0 < y < Ly, and 0 < z < Lz; also, forsimplicity, the conventional symmetry boundary conditionis assumed on the outer boundary plane x = Lx, and on theother boundary planes(y = Ly and z = Lz) the free boundary
conditions are assumed.
[12] Current-driven anomalous resistivities have been studied theoretically and experimentally [Lui, 2001; Treumann,
2001; Ono et al., 2001]. Here, as in the 2-D model, theanomalous resistivity model is assumed in the form,
hðr; t Þ ¼ kR ½Vd ðr; t Þ  VC 
¼0

for Vd > VC ;for Vd < VC

ð4Þ

where Vd(r, t) = jJ(r, t)/r(r, t)j is the relative electron-ion
drift velocity, and VC may be a threshold for microinstabilities, and the fast reconnection evolution for different
functional forms and parameters has been studied [Ugai,
1984, 1992; Ugai and Kondoh, 2001; Ugai and Zheng,2005]. Here, kR = 0.003 and VC = 12 are taken.
[13] Ugai [1986] demonstrated the current-sheet thinningdrastically proceeds when there is no resistivity in t > 4 dueto the similar effect to the pinch one. Here, the ion-electron
drift velocity Vd is in inverse proportional to the half-widthof current sheet d, Vd / 1/d. Therefore anomalous resistivity
builds up when d  d0/VC from equation (4), where d0 is theinitial half-width of the current sheet. If we assume thatthe anomalous resistivity builds up when the half-width ofthe current sheet d becomes the ion inertial length li, VC =12 means that initial half-width of the current sheet d0 
10  li.
[14] In order to disturb the initial static configuration, a
localized resistivity model is assumed around the point (Lx,0, 0) in the 3-D form,hi3hðrÞ ¼ h0 exp ðð x  Lx Þ=kx Þ2  jyj=ky ðjzj=kz Þ3

ð5Þ

A03S07Here, we take kx = ky = 0.8 and h0 = 0.02 in the manner
similar to the previous 2-D simulations. Also, kz provides
the 3-D effects. Previous studies showed that the kz specifies
the effective extent of diffusion region in the z-direction,and magnetic reconnection cannot effectively grow in thecase of small kz, kz < 3 say, [Kondoh et al., 2006], so thatkz = 5 is taken in the present study.
[15] The disturbance (5) is imposed only in the initialtime range 0 < t < 4, and the anomalous resistivity model (4)is assumed for t > 4. Hence the fast reconnection mechanism may be triggered at x = Lx in this model.
[16] It should be noted that sufficiently small mesh sizes
are required for precise computations of the spontaneous
fast reconnection evolution, so that we assume Dx = 0.04,
Dy = 0.015, and Dz = 0.1. Also, we take the magnetic fieldregion size Y1 = 4, and the whole computational region size
is assumed to be Lx = 10, Ly = 9.6 and Lz = 9.8.
[17] Here, we should notice that the positive x, y,z-directions in this paper correspond to the tailward,
southward, and eastward directions in the usual GSM
coordinate system, respectively, as shown in Figure 1. The
coordinate system whose axis names are parenthesized inthis figure is that employed in our simulation. Hereafter, thiscoordinate system is used in this paper.

3. Results
3.1. General Remarks[18] At first, let us show the general results of threedimensional MHD simulations. Initiated by the disturbance(5) given at x = Lx in the finite extent jzj < kz = 5, the current
sheet thinning occurs near the x = Lx, giving rise to distinct
enhancement of the current density. Once the drift velocity
Vd exceeds the given threshold (4) at time t  27, fastreconnection drastically grows as nonlinear instability because of the positive feedback between the anomalous
resistivity and the reconnection flow. Ahead of the Alfvénic
reconnection jet flowing to negative x-direction, a largescale 3-D plasmoid is formed and propagates, and theplasmoid collides with the x = 0 wall boundary at t  39,
giving rise to a 3-D magnetic loop. Here, we define the‘‘plasmoid’’ as high pressure plasma wrapped by the reconnected magnetic filed lines.
[19] Now, we show the resulting three-dimensional configurations of some physical quantities. Figure 2 shows themagnetic field configuration (light lines) and plasma pressure configuration(isosurfaces of P = 1.3, 1.45, 1.6 andcontour lines in the x  z plane) at different times t = 36.75(Figure 2, left), 37.5 (Figure 2, middle), and 38.25 (Figure 2,
right). The grid size shown in Figure 2 is taken to be 2.0. The
magnetic field lines shown in this figure are traced from thepoints on the x  z plane; namely, all of these field lines are
the reconnected magnetic field lines. The spatial interval of
the start point of these field lines in the z-direction is 0.4. The
grid size and the spatial interval are taken to be the same also
in the similar figures that will appear. Most of the start pointsof magnetic field lines in this figure exist in the region z ]1.6; that is, the magnetic reconnection mainly occurs in z ]1.6 near the x = Lx. The 3-D plasmoid propagates to thenegative x-direction as it grows. This plasmoid is pushed by
the slow shock formed behind it, and the high pressure region

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Figure 2. Magnetic field (light lines) and plasma pressure (isosurfaces (P = 1.3, 1.45, 1.6) and contour
lines in the x  z plane) configurations at time t = 36.75, 37.5, 38.25. All the similar figures that willappear will be shown in this way.
is produced in the plasmoid, whereas the plasma pressure
decreases in the upstream region of it.
[20] We find that the 3-D configuration of plasma density
is considerably different from that of plasma pressure; inparticular, those configurations inside the plasmoid are quite
different. Figure 3 shows the magnetic field configurations(light lines) and the plasma density configurations (isosurfaces of r = 1.02, 1.07 and contour lines in the x  z plane)at the times same as those in Figure 2. The magnetic fieldlines are the same as those in Figure 2. We see that theplasma density is rather low in the plasmoid, where
the plasma pressure is very high there. This means thatthe plasma temperature in the plasmoid is high. Dense plasma
in the region ahead of the plasmoid, shown by the multiisosurfaces, consists of the initial current sheet plasma. Thisregion is enclosed by the magnetic field lines that have beenreconnected. Therefore the high-density region does not
significantly expand in the y-direction, whereas the plasmoid(high-pressure region) tends to expand in the y-direction. In theregion behind the plasmoid (x(t = 38.25) ^ 4.0) and jzj ] 2.0,
the plasma density is low near the x-axis in connection
with the fast reconnection jet and strongly depends on thez-position, whereas the plasma pressure is almost uniform
in the z-direction.
[21] The plasma density configuration behind the plasmoid is closely related to the fast plasma flow configuration.
Figure 4 shows the magnetic field configurations (lightlines) and the plasma flow configurations (isosurfaces of
Vx = 0.4, 1.2, 2.0 and arrows in the x  y and x  zplane) in the similar manner as before. It is clear that thefast plasma jet flows in the very narrow channel, and theplasma density is low in the channel where the plasma flow
is fast. Very fast plasma jet jVxj > 1.2 is especially localizedin jzj < 1.0, although the magnetic reconnection occurs in z ]1.6 near x = Lx. The relatively slow plasma jet, jVxj  0.4,
flows in the somewhat wide channel, jzj < 2.0. After theplasma arrives at the magnetic loop top, it flows mainly along
the magnetic loop boundary.
[22] In order to see the temporal evolution of fast plasma
flow, Figure 5a shows the plasma flow profile along thex-axis at different times. The horizontal dotted line
indicates the Alfvén velocity VAe (2.7) in the ambient
lobe magnetic field region (1 < y < Y1). As time passes, themaximum velocity of the reconnection outflow jVxj rapidly

Figure 3. Magnetic field (light lines) and plasma density (isosurfaces (r = 1.02, 1.07) and contour lines
in the x  z plane) configurations at time t = 36.75, 37.5, 38.25.
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Figure 4. Magnetic field (light lines) and plasma flow (isosurfaces of x-component (Vx = 0.4, 1.2,2.0) and arrows in the x  y, x  z plane) configurations at time t = 36.75, 37.5, 38.25.
increases until t  39. After t  39, reconnection outflow
reaches the Alfvén velocity and the profile has the flat
plateau.
[23] Next, let us consider the variation of Vx at different
positions in the z-direction in order to see the z-directional
range of the narrow channel of the fast plasma jet. Figure 5bshows the x-directional variations of jVxj along the different lines at y = 0 in the positive z-direction at time t = 42,
when the magnetic loop has fully developed and the
maximum velocity attains the lobe Alfvén velocity. We
see that the outflow velocity becomes notably smaller forthe larger z-positions; only in jzj < 1.0, it becomescomparable with lobe Alfvén velocity VAe. In fact, in jzj >1.5, the maximum value is less than 30% of VAe, so thatthe spatial range in the z-direction of this narrow channel
may be jzj ] 1.0, and that of the region observable fast flow
may be jzj ] 2.0.
[24] As mentioned above, these four physical quantities(P, r, Vx, B) have different time evolutions in each other,and three-dimensional evolutions of plasma pressure anddensity configurations are different from those of plasma
flow and reconnected magnetic field lines configurations.
The extent, especially in the z-direction, of fast flow andreconnected field lines are restricted in the narrow channel
differently from that of dense plasma region and highpressure region.
3.2. Comparison With Superposed Epoch Analysis[25] Here, let us introduce the actual satellite observations. Since the fast flow events have individually different
characteristics, we may compare the simulation results withthe results of statistical analyses of satellite observations.
Superposed epoch analysis introduced in section 1 has been
conducted in order to understand the generic magnetic fieldand plasma parameter change in the course of the fastplasma flow in the plasma sheet.
[26] Figure 6 shows the stack plots of the various physicalquantities superposed over the 818 fast earthward flow
events reported by Ohtani et al. [2004]. These events wereobserved at 31 < XGSMA < 5 RE during October 1993 toJuly 2001 by Geotail satellite. The reference time in thisanalysis, which is shown by the vertical solid line (dTV = 0)in this figure, is the start of the fast flow. The horizontal axis

covers the 20-min interval centered at the reference time.
We should notice that the coordinate system in this figure is
GSM system differently from that used in our simulations.
In order to prevent the confusion due to this difference
between coordinate systems, we use the coordinate systemin our simulation in text and we put the variables inparenthesis, which is corresponding one in our coordinate
system, in this figure.
[27] At first, let us focus to the time profiles of the flow
velocity component jVxj (Figure 6a), the three magnetic
field components Bx, jByj, jBzj (Figure 6b), the plasma
density r (Figure 6c) and the plasma pressure P(Figure 6e). The peak times, when these intensities reachpeak values, are different from each other, and the order of
their appearance time is r, P, jByj, and jVxj. The peak time of

Figure 5. (a) Profiles of jVxj(x, y = z = 0) along the x-axis
at different times, and (b) x-directional profiles of jVxj(x, y =0, z) at t = 42 at different z locations.

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A03S07y-direction and outside the fast flow narrow channel inthe z-direction.
[30] In their paper, they also examined the temporal
structure of the fast flow using simulations, and comparedthem with the observations. In their simulations, the convergent flows generated by two X lines produces a significant enhancement of density and strong density peak in themagnetic island. Virtual satellites, which are located in thesymmetry plane at the center of the current sheet, observed astrongly bipolar signature in By with a sudden enhancementin the density. They mentioned the discrepancies in r
variation between the observations and the simulations.
The superposed profile of r showed a transient enhancement followed by a sharp decrease. On the other hand, themodeled density decrease was much more gradual than thepreceding density enhancement. They mentioned two possible reasons, (1) the distance from the reconnection site and(2) large density difference between the central plasma sheet
and the lobes. Another discrepancy in r is that the peakcoincides with the minimum of By in the superposed epoch
analysis, which can also be confirmed for some individual
events. Namely, the density starts to decrease before thereference time.

Figure 6. Various magnetic field and plasma quantities
superposed over the 818 fast earthward flow events in GSM
coordinate system. The start of the fast flow is used as areference time (from Ohtani et al. [2004]). The variables inparenthesis indicate the corresponding ones in our coordinate system.
plasma pressure is coincident with the reference time. Bx
decreases in association with the fast flow, then it falls down
to the lowest point around the reference time, and then it
recovers to the same level as jByj in association with theslowing down of the fast flow. jByj stays at levels higher
than the previous levels after overshooting, namely the local
magnetic field dipolarizes.
[28] The ion temperature Ti (Figure 6d) increases aroundthe reference time. The decrease in r overcompensates forthat increase of Ti. The value of bi (Figure 6f) decreased in
the course of the fast flow; however, we note that theminimum value of b i is 4.0. Judging from the above, they
inferred that the near-Earth reconnection is the responsible
mechanism for most flow events.
[29] The magnetic field component Bx is nonzero (2.5 ]jBxj ] 4.5) and the maximum flow velocity is only 300 km/s.
From this, we can infer that the major part of these events
are observed at the region off the neutral sheet in the

3.3. Virtual Observation[31] In order to compare the simulations directly with thesatellite observations, let us consider such a virtual probethat is located at a spatial point (x, y, z) in the plasma sheet.
For simplicity, we assume that the probe does not move inspite of the progress of time. Then, the virtual probe canreadily detect the temporal changes of plasma quantities at
its location on the basis of the simulation results. In what
follows, we examine what the virtual probe observes as thefast reconnection mechanism builds up and proceeds.
[32] As mentioned in section 3.2, it seems that a lot of
events are observed in the region off the neutral sheet in
the y-direction and outside the fast flow narrow channel inthe z-direction. Therefore the virtual probe is located at theposition (x = 4.0, y = 0.4. z = 0.6), and we compare thevirtual probe observations with the actual satellite observations. Figure 7 shows the temporal variations of theplasma flow velocity (Figure 7, top), magnetic field components (Figure 7, middle), plasma pressure and plasma
density (Figure 7, bottom) detected by the virtual probe
located at the above position. Solid vertical line indicatesthe peak time TP of plasma pressure P, and the other dottedvertical lines indicate the peak times Tr, TB, TV of r, jByj, andjVxj, respectively. The order of these peak times are consistent with those in Figure 6. Namely, first, the plasma density
r slightly increases until t = Tr, and then abruptly decreases.
Second, the plasma pressure P largely increases until t = TP,and then abruptly decreases. Third, the y-component ofmagnetic field jByj sharply increases from t  Tr until t =TB, and then decreases to the higher level than the previous
level until t  TV and keeps the level. Finally, thex-component of plasma flow velocity jVxj gradually
increases and sharply increases at t  TP, and then decreases
after t = TV.
[33] The magnetic field component Bx starts to decreaseat the same time with the start of the enhancement of jVxj,
and then slightly recovers from Tr, and then falls down
again at TB, and it starts to recover at the peak time of jVxj.

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Figure 7. Time profiles of jVxj, Bx, jByj, jBzj, P, and robserved at position x = 4.0, y = 0.4, z = 0.6.
The former slight recovery of Bx, whose amount of enhancement is much smaller, synchronizes with the enhancement of jByj. The latter recovery of Bx means that the virtual
probe goes out from the narrow channel of fast plasma jet.
This profile of Bx and the relationship between Bx and Vx
variations are also consistent with those in actual satellite
observations. Therefore, the earthward fast plasma flow canbe regarded to be the direct outcome of the fast reconnection
evolution itself.
3.4. Traveling Compression Regions[34] Earthward fast plasma flow in the plasma sheet isoften accompanied by traveling compression regions(TCRs) in the magnetic lobe region [Slavin et al., 1984,
1993, 2002]. Ugai and Zheng [2006a, 2006b] examined
tailward moving TCRs in the Earth’s magnetotail using thesimilar model with that in this paper. They demonstrated
that the general features of TCR, obtained by the simulations, are in good agreement with the actual satellite
observations, and the virtual satellite which is locatedbehind the plasmoid formation observes the unipolar structure of the southward field component. Then, they concluded
that the TCR event is directly caused by the fast reconnection
mechanism itself. In their model, free boundary is usedinstead of the wall boundary adopted in our simulation, sothat the plasmoid propagates freely without any obstacle,
whereas it collides with (x = 0) wall boundary in our present

A03S07model. Also, it is important to examine simultaneously
earthward moving TCR in this model. In this section, we
examine particularly the connection between earthward fastplasma flow and earthward moving TCR.
[35] At first, we show an example of an earthward
moving TCR in the Cluster FGM measurements on 10
September 2001 reported by Slavin et al. [2005] in Figure 8.
In this figure, we put the variables in parenthesis, which iscorresponding one in our coordinate system and use thecoordinate system in our simulation in text in order toprevent the confusion in the same way of the Figure 6.
The peak compression in Bx (Figure 6, top), which isindicated by the vertical dashed line, occurs near the centerof the DBy variations. The profile of By is unipolar, and it
stays at level higher than the previous level after theovershooting. On the other hand, Bx stays at lower level
after the overshooting. It means that the lobe magnetic fieldlines gently slope down tailward.
[36] In order to compare the simulation results directly tothe satellite observations, let us consider such a virtual
satellite that is located at a spatial point (x, y, z) in the 3-D
simulation domain in the similar manner to the previous
investigations on the fast flow events in the plasma sheet. Topfour panels in Figure 9 show the temporal variations of the
field components, Bx, By,Bz and B, observed at x =4.0, y = 1.8, z = 1.8 in our simulation domain. We thenfind that the characteristic compression signature in Bx anddraping signature in By are consistent with the actual
satellite observations in Figure 8. The unipole variationof By and tailward slope down of lobe magnetic field arealso consistent. Inferring from the results of Ugai andZheng [2006a, 2006b], these unipole variations of By in
actual and virtual satellite observations are attributed to thesatellite location, namely, these satellites were locatedbehind the plasmoid formation. The variation of Bz in thisfigure synchronizes with that of By. On the other hand, noclear variation is seen in Bz in Figure 8. We find, however,that such variation in Bz as seen in Figure 9 is usuallyfound out in the other cases [Slavin et al., 2005]. In order
to examine the relation to earthward fast plasma flow, thetemporal variations of the plasma pressure P (third andsecond panel from the bottom), observed at x = 4.0, y =0.4, z = 1.8 (third panel from the bottom) and z = 0.6(second panel from the bottom) are shown in Figure 9.
The time of peak compression in Bx is coincident with thepeak time of plasma pressure enhancement in both ofbottom two panels. At the location just under the probe
position in the magnetic lobe region (third panel from thebottom), the enhancement of plasma pressure is not solarge, while the profile of plasma pressure varies muchlarger at the location inner the position (second panel fromthe bottom). This three-dimensional effect seems to beeffective to the compression of the lobe magnetic field inextensive region in the z-direction. This compression oflobe magnetic field is due to the passage of plasmoid in
the plasma sheet, so that this TCR observation is closely
relation to the earthward fast flow event.

4. Summary and Discussion[37] In this paper, the spontaneous fast reconnection
model was applied to the earthward fast flow events

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Figure 8. An example of earthward moving travelingcompression region in the Cluster measurements on
10 September 2001 in GSM coordinate system (from Slavin
et al. [2005]). The variables in parenthesis indicate thecorresponding ones in our coordinate system.
observed in the near-Earth plasma sheet. The 3-D plasma
configurations found in the present simulation are summarized as follows: (1) The fast reconnection jet, which attains
the Alfvén velocity VAe, flows in the very narrow channel,
and the decrease in the plasma density becomes much larger
in the region where the flow velocity is faster. (2) The
plasmoid is formed ahead of the fast reconnection jet anddevelops in three dimensions, and the plasma inside theplasmoid becomes high pressure and high temperature.
(3) The plasma pressure in the fast flow channel tends todecrease, where the temperature (P/r) is remarkably enhanced. (4) The dense plasma ahead of the plasmoid, which
consists of the initial plasma sheet plasma, is confined by
the initially reconnected magnetic field lines. (5) The highpressure region and the high density region tend to expandin three dimensions, whereas the fast flow and the intense
magnetic field, accumulated behind the plasmoid, remain to
be restricted in the very narrow channel.
[38] On the basis of the simulation results, virtual probes
were located in the plasma sheet and the lobe region, so thatthey could directly observe the temporal variations of theseplasma quantities. The observations by these virtual probes
were in good agreement with the actual satellite observations on both events in the very fine structure. In particular,
the sequence of peak times of the plasma quantity variations

A03S07

was a complete agreement. The order of this is Tr, TP, TB,
and TV. This order reflects the results of three-dimensional
plasma configurations mentioned above.
[39] Now, let us discussed about the plasma density
profile in the course of fast flow. We have introduced thediscrepancy between the observations and simulations in thework of Ohtani et al. [2004], that is, the superposed profile
of r showed a transient enhancement followed by a sharp
decrease, and the density starts to decrease before thereference time. In their model, they assumed multi X line,
so that the plasma density enhances in the magnetic island.
As the result of it, the density does not decrease before theplasma pressure enhancement. On the other hand, theprofile of plasma density in our simulation was in the good
agreement with superposed profile, since the plasma density
increase before the start of fast flow and decrease in thecourse of fast flow.
[40] Ohtani et al. [2004] compared the earthward flows
with the tailward ones using the satellite observations. They
found that the ion temperature increases for the earthward
flow and decreases for the tailward flow, and inferred that
this contrast may be attributed to different physical conditions surrounding the flow, that is, the associated structure
gets confined in the x-direction for the earthward flow. In
fact, the superposed profile of Ti obtained for earthward
flow in Figure 6 shows a increase in the course of fast flow.
On the other hand, the profile of plasma temperature T,
which is observed at x = 4.0, y = 0.4, z = 0.6 in our
simulation domain, shown in the bottom panel in Figure 9
similarly shows a increase in the course of fast flow. We
note that the peak time of plasma pressure was coincident
with the start of fast flow. Hence it may be reasonable tomodel the earthward flow by the wall boundary, whichsignificantly confine the reconnection flow in the x-direction.
[41] It may be important to explicitly give the normalization units employed in the present simulation. Somelaboratory experiments detected such an anomalous resistivity when the half-width of the current sheet becomes
smaller than the ion inertial length li [Yamada, 2001; Onoet al., 2001]. In the present model, the current-driven
anomalous resistivity builds up when the width of thecurrent sheet becomes thin to about one-tenth of the initialwidth as mentioned in section 2.1.2. Hence the present unit
length d0 may be considered to be about 10li. If wetypically take the field strength in the lobe region Bx0 =20 nT, the number density in the center of plasma sheet
n0 = 1 cm3 and li  1000 km, we have the unit velocity
VAx0 = 450 km/s; therefore, the present unit time becomes
d0/VAx0  20 s. Applying these quantities to Figures 7
and 9, we see that the simulation results are consistent withthe actual observations in Figures 6 and 8, respectively.
[42] In summary, we have demonstrated that the earthward fast flow event and TCRs signatures are directly
caused by the fast reconnection mechanism. The virtual
probe observations in the simulation domain are in the good
agreement with the actual satellite observations, despite thesimulation model is simplified and idealized. For instance,
the present simulations are done in the first-quadrant domain, so that the initial shear field in the plasma sheet is
not considered; also, a realistic geomagnetic initial dipole
field is not considered. These conditions may be important

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A03S07computer program was tested and run at the Computational Centers of
Nagoya and Kyoto University.
[44] Amitava Bhattacharjee thanks the reviewers for their assistance inevaluating this paper.

References

Figure 9. Traveling compression region in the virtual
probe measurements in our simulation domain. Top threepanels show the magnetic field variations in the lobe region,and bottom three panels show the plasma pressure andtemperature variations in the plasma sheet during the same
period.

in the real time predictions, which will be taken into accountin the future work.
[43] Acknowledgments. This work was supported by Grant-in-Aids
for Creative Scientific Research ‘‘The Basic Study of Space Weather
Prediction’’ (17GS0208, Head Investigator: K. Shibata) from the Ministryof Education in Japan, Mitsubishi Foundation, RISH of Kyoto University,and Solar Terrestrial Environment Laboratory of Nagoya University. The

Angelopoulos, V., W. Baumjohnn, C. F. Kennel, F. V. Voroniti, M. G.
Kivelson, R. Pellat, R. J. Walker, H. Luhr, and G. Paschmann (1992),
Bursty bulk flows in the inner central plasma sheet, J. Geophys. Res., 97,4027.
Kondoh, K., M. Ugai, and T. Shimizu (2006), Three dimensional computer
simulation of magnetic loop associated with magnetic reconnection, Adv.
Space Res., 37, 1301.
Lui, A. T. Y. (1996), Current disruption in the Earth’s magnetosphere:
Observations and models, J. Geophys. Res., 101, 13,067.
Lui, A. T. Y. (2001), Multiscale model for substorms, Space Sci. Rev., 95,
325.
Ohtani, S.-I., M. A. Shay, and T. Mukai (2004), Themporal structure of thefast convective flow in the plasma sheet: Comparison between observations and two-fluid simulations, J. Geophys. Res., 109, A03210,
doi:10.1029/2003JA010002.
Ono, Y., M. Inomoto, Y. Ueda, T. Matsuyama, and Y. Murata (2001), Fastcompression of a current sheet during externally driven magnetic reconnection, Earth Planets Space, 53, 521.
Petschek, H. E. (1964), Magnetic field annihilation, in The Physics of Solar
Flares, edited by W. N. Hess, NASA Spec. Publ., SP-50, 425.
Schodel, R., W. Baumjohann, R. Nakamura, and T. Mukai (2001), Rapid
flux transport and plasma sheet reconfiguration, J. Geophys. Res., 106,
8381.
Sergeev, V. A., et al. (2000), Multiple-spacecraft observation of a narrowtransient plasma jet in the Earth’s plasma sheet, Geopys. Res. Lett., 27,851.
Sergeev, V. A., et al. (2001), Substorm and convection bay compared:
Auroral and magnetotail dynamics during convection bay, J. Geophys.
Res., 106, 18,843.
Shibata, K. (1999), Evidence of magnetic reconnection in solar flares and aunified model of flares, Astrophys. Space Sci., 264, 129.
Shibata, K. (2004), Physics of Magnetic Reconnection in High-Temperature
Plasmas, edited by M. Ugai, pp. 193 – 237, Research Signpost, India.
Shiokawa, K., et al. (1997), Braking of high-speed flows in the near-Earthtail, Geophys. Res. Lett., 24, 1179.
Slavin, J. A., R. P. Lepping, and D. N. Baker (1984), Substorm associatedtraveling compression regions in the distant tail-ISEE-3 geotail observations, Geopys. Res. Lett., 11, 657.
Slavin, J. A., et al. (1993), ISEE 3 observations of traveling compression
regions in the Earth’s magnetotail, J. Geophys. Res., 98, 15,425.
Slavin, J. A., et al. (2002), Simultaneous observations of earthward flow
bursts and plasmoid ejection during magnetospheric substorms, J. Geophys. Res., 107(A7), 1106, doi:10.1029/2000JA003501.
Slavin, J. A., et al. (2003), Cluster four spacecraft measurements of small
traveling compression regions in the near-tail, Geophys. Res. Lett., 30(7),1362, doi:10.1029/2002GL016411.
Slavin, J. A., et al. (2005), Cluster observations of traveling compression
regions in the near-tail, J. Geophys. Res., 110, A06207, doi:10.1029/
2004JA010878.
Treumann, R. A. (2001), Origin of resistivity in reconnection, Earth Planets
Space, 53, 453.
Ugai, M. (1984), Self-consistent development of fast magnetic reconnection
with anomalous plasma resistivity, Plasma Phys. Controlled Fusion, 26,
1549.
Ugai, M. (1986), Global dynamics and rapid collapse of an isolated currentsheet system enclosed by free boundaries, Phys. Fluids, 29, 3659.
Ugai, M. (1987), MHD simulations of fast reconnection spontaneously
developing in a current sheet, Comput. Phys. Commun., 49, 185.
Ugai, M. (1992), Computer studies on development of the fast reconnection
mechanism for different resistivity models, Phys. Fluids, 4, 2953.
Ugai, M., and K. Kondoh (2001), Computer studies on the spontaneous fastreconnection evolution in various physical situations, Phys. Plasmas, 8,1545.
Ugai, M., and L. Zheng (2005), Conditions for the fast reconnection
mechanism in three dimensions, Phys. Plasmas, 12, 092312.
Ugai, M., and L. Zheng (2006a), Parametric studies on traveling compression regions observed in the Earth’s magnetotail, Phys. Plasmas, 13, 1.
Ugai, M., and L. Zheng (2006b), Modeling of traveling compression regions in the Earth’s magnetotail by the spontaneous fast reconnection
model, Phys. Plasmas, 13, 032901.
Ugai, M., K. Kondoh, and T. Shimizu (2003), Computer simulations on
three-dimensional magnetic loop dynamics by the spontaneous fast reconnection model, Phys. Plasmas, 10, 357.

9 of 10

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Ugai, M., K. Kondoh, and T. Shimizu (2004), Computer studies on thethree-dimensional spontaneous fast reconnection model as a nonlinear
instability, Phys. Plasmas, 11, 1416.
Ugai, M., K. Kondoh, and T. Shimizu (2005), Spontaneous fast reconnection model in three dimensions, Phys. Plasmas, 12, 042903.

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Yamada, M. (2001), Review of the recent controlled experiments for studyof local reconnection physics, Earth Planets Space, 53, 509 – 519.


K. Kondoh and M. Ugai, Electrical and Electronic Engineering andComputer Science, Graduate School of Science and Engineering, Ehime
University, 790-8577, Matsuyama, Japan. (kondo@cs.ehime-u.ac.jp)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S09, doi:10.1029/2007JA012506, 2008

for

Full
ArticleFlux enhancement of the outer radiation belt electrons
after the arrival of stream interaction regions
Yoshizumi Miyoshi1 and Ryuho Kataoka1,2Received 30 April 2007; revised 20 June 2007; accepted 19 July 2007; published 18 January 2008.

[1] The Earth’s outer radiation belt electrons increase when the magnetosphere issurrounded by the high-speed solar wind stream, while the southward interplanetary
magnetic field (IMF) is also known as an important factor for the flux enhancement.
In order to distinguish the two different kinds of solar wind parameter dependence
statistically, we investigate the response of the outer belt to stream interaction regions(SIRs). A total of 179 SIR events are identified for the time period from 1994 to 2005.
We classify the SIR events into two groups according to the so-called ‘‘spring-toward
fall-away’’ rule: IMF sector polarity after the stream interface is toward in spring or away
in fall (group A) and vice versa (group B). According to the Russell-McPherron
effect, groups A and B have a significant negative and positive offset of the IMF Bz after
the stream interface, respectively. Comparing groups A and B by superposing about thestream interface, only IMF Bz dependence can be obtained because the other solar windparameters change in the same manner. As a result, the greatest flux enhancement is foundin the highest-speed streams with a southward offset of the IMF Bz, indicating that only
the solar wind speed by itself is not a sufficient condition for the large flux enhancement.
It is also found that the large flux enhancement tends to be associated with weakgeomagnetic activities with minimum Dst of about 50 nT on average, implying that theexistence of intense magnetic storms is not essential for the flux enhancement.
Citation: Miyoshi, Y., and R. Kataoka (2008), Flux enhancement of the outer radiation belt electrons after the arrival of stream
interaction regions, J. Geophys. Res., 113, A03S09, doi:10.1029/2007JA012506.

1. Introduction[2] Understanding and modeling the flux enhancement of
the outer belt electrons is a central topic of the space
weather research because the energetic electrons can cause
anomalies of the satellites that provide weather observations,communications, and other important services to modern life
[Baker et al., 1987, 1998; Kataoka and Miyoshi, 2006]. Theenergetic electrons of the outer belt are accelerated up to theenergy range of a few MeV in the inner magnetosphere.
Classically, the inward radial diffusion and the betatronacceleration are thought to be a fundamental process toaccelerate the energetic electrons [Schulz and Lanzerotti,
1974; Elkington et al., 1999; Li, 2006]. Wave-particle
interaction is also known as another essential mechanism
to accelerate the energetic electrons [Summers et al., 1998;
Meredith et al., 2001; Miyoshi et al., 2003; Horne et al.,2006]. Both processes are expected to be effective to work
simultaneously for the acceleration [O’Brien et al., 2003],
although there is still no consensus about the relative
importance.
1
Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya,Japan.
2
Now at Computational Astrophysics Laboratory, Institute of Physics
and Chemical Research, Saitama, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012506$09.00[3] The outer radiation belt fluxes are not stable, showing
a somewhat systematic change especially during magnetic
storms; rapid decrease in the main phase and gradualincrease in the recovery phase. Reeves et al. [2003] showedthat about a half of magnetic storms produce the flux
enhancement of the outer belt. Miyoshi and Kataoka[2005] found that the flux enhancement is controlled by
the interplanetary driver of storms, such as coronal massejections (CMEs) and stream interaction regions (SIRs),
suggesting that the large amplitude fluctuation of theinterplanetary magnetic field (IMF) directions which has asignificant southward component within the high-speed
coronal hole stream following a SIR is important for theacceleration of energetic electrons. In fact, Kataoka andMiyoshi [2006] showed that a strong flux enhancement at
geosynchronous orbit followed the 5 of 6 SIR-associatedstorms, while only 20 of 49 CME-associated storms havesuch strong enhancements. The solar wind structure dependence possibly places observational restrictions on theeffective acceleration mechanism.
[4] Another important observational restriction is thesolar wind parameter dependence. It has been well known
that the outer belt flux enhancements are well correlatedwith the high-speed solar wind [Paulikas and Blake, 1979;
Baker et al., 1986; O’Brien et al., 2001; Weigel et al., 2003].
Several case studies have shown that the southward IMF isanother important parameter for the acceleration [Blake etal., 1997; Iles et al., 2002; Miyoshi et al., 2007].

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MIYOSHI AND KATAOKA: RADIATION BELT RESPONSE TO SIR[5] It is important that the statistical research of the solar
wind parameter dependence should take into account thesolar wind structure dependence at the same time, althoughmost of the previous studies did not handle this point
comprehensively. One straightforward way to do that iscollecting a number of SIR events to find a pure solar windparameter dependence on the flux enhancement without
structure dependence. McPherron and Weygand [2006]
examined a number of SIR events to clarify some statisticalproperties of geomagnetic activities and fluxes of the outer
belt, while they have not shown parameter dependences on
the flux enhancement. In this paper, we show the pure solar
wind parameter dependence for the flux enhancement in theouter belt associated with the SIR statistically.

2. Data Analysis[6] SIRs are the regions of compressed plasma formed at
the leading edge of high-speed streams as they interact withthe preceding slow solar wind. In this study, the stream
interfaces of SIRs are detected from the OMNI-2 data (http://omniweb.gsfc.nasa.gov) using the simple criteria of (1) alarge change in the azimuth angle of the solar wind velocity(>5°/hour), (2) positive slopes in the speed and temperatureand negative slope in density, and (3) the magnetic fieldstrength higher than the weekly average. We use theseparameters because stream interfaces are usually characterized by these parameters [Gosling et al., 1978] and using
these parameters work successfully to detect a number of
stream interfaces [McPherron and Weygand, 2006]. We
further restrict our selection of the events with relatively
stable magnetic fields for 72 hours after the stream interface,where 75% of the magnetic fields directing in the away
quadrant (the IMF azimuthal angle from the x axis ranges
from 90° to 180°) or 75% of the magnetic fields directed in
the toward quadrant (the IMF azimuthal angle from the xaxis ranges from 270° to 360°). As a result, a total of 179 SIR
events are identified for the time interval from 1994 to 2005.
It is worthwhile to note that in this paper we identify the SIR
events regardless of the storm amplitude.
[7] In order to categorize the SIR events into different
IMF Bz conditions, we consider the Russell-McPherron
effect [Russell and McPherron, 1973]. The RussellMcPherron effect controls the IMF Bz of GSM coordinate
through the projection of the IMF By of GSE coordinate.
According to the Russell-McPherron effect, geomagnetic
activities are enhanced during the time intervals of so-called‘‘spring-toward fall-away’’ (STFA) because the IMF polarities of toward and away have a large projection componentof the southward IMF around the spring and fall equinox,
respectively. The southward component of IMF in a highspeed stream is always significantly suppressed or enhanced
because of the STFA rule, because of comparable amplitudes of both the Russell-McPherron effect and the Alfvénic
IMF fluctuation of a high-speed stream.
[8] According to the STFA rule, we classify the SIR
events into two groups: IMF sector after the stream interface
is toward in spring (February, March, April, May) or away
in fall (August, September, October, November) (group A)
and vice versa (group B). As a result of this categorization,
two sets of SIR events with the same solar wind structure ofdifferent IMF offset after the stream interface can be

A03S09obtained. Hereafter we call the group A as SBZ events,
and the group B as NBZ events. The total numbers of the
SBZ and NBZ events are 75 and 43, respectively.
[9] Figure 1 shows the result of superposed epoch analysis about the solar wind parameters and geomagnetic
activities. The reference time (t = 0.0; time unit is shownin days) of the superposed epoch analysis is chosen at thetime of a stream interface.
The fluctuationﬃ of the magnetic
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
field dB is calculated as s2Bx þ s2By þ s2Bz , where sBx, sBy ,and sBz are the standard deviations of the x, y, and z
components, respectively. The typical solar wind parameterchanges associated with a SIR can be found, i.e., speed andtemperature increase accompanied with density drop, following high-speed stream [e.g., Gosling et al., 1978;
Richardson et al., 2006; McPherron and Weygand, 2006].
Following the STFA rule, the average IMF Bz is systematically shifted to southward after t = 0.0 day in SBZ events,
while the IMF Bz is shifted to northward in NBZ events.
Modest Kp activity continues for a few days during SBZ
events, while the Kp index is significantly less enhanced in
NBZ events. Although the AE index has not been available
during the whole period, continuous Kp activities indicate
the continuous substorms and enhanced convections, i.e.,
High Intensity Long Duration Continuous AE Activity(HILDCAA)-like activities [Tsurutani and Gonzalez, 1987;
Tsurutani et al., 2006]. The solar wind speed is slightly
higher in NBZ events, causing slightly hot and tenuous
plasma parameters, although the slightly different plasma
parameters do not affect the essential results of this paper.
[10] As a direct measurement of the outer belt variation,
we use the flux of >2 MeV electrons observed in situ by the
GOES satellites at geosynchronous orbit. In addition, theradiation belt electrons over a wide range of L value arecontinuously monitored by the low-altitude NOAA/POES
satellites. Note that L value in this study is McIlwain’s L
derived from IGRF. We use NOAA 12 and 15 satellites,
orbiting 0730 – 1930 local time meridians, before and after 1
July 1998, respectively. We use the 90° detectors mostly
measuring the local mirroring >300 keV electrons. Though>300 keV electron detectors are also sensitive to >440 keV
protons [see Evans and Greer, 2000], we confirmed that thecontamination is not a severe problem by examination of the240– 800 keV ion measurement.

3. Results[11] First, we show that the flux enhancement in SBZ
events is faster and stronger than that in NBZ events.
Figures 2a and 2b show the superposed flux of >2.0 MeV
electrons for SBZ and NBZ events as observed by the GOES
satellites with the superposed Dst index. About 0.5 day
before the stream interface, the electron flux begins todecrease in both groups. The flux decrease in SBZ events is
relatively faster than that in NBZ events. Figure 2a shows thatthe average flux begins to increase just after the stream
interface during SBZ events. The flux recovers to the preSIR level at t = 1.0 day and then increases above the pre-SIR
level. Figure 2b shows that the flux continues to decrease
till one day after the stream interface, and then gradually
recovers to the pre-SIR level at t = 2.0 day during NBZ
events. After t = 3.0 day, although the flux is over the pre-2 of 9

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MIYOSHI AND KATAOKA: RADIATION BELT RESPONSE TO SIR

Figure 1. Superposed time series of solar wind parametersand geomagnetic activities. SBZ and NBZ events are shownin red and blue, respectively (see text for the categorizationin detail). Shown are (top to bottom) magnetic field strength
B, southward magnetic field Bz, fluctuation of the magnetic
field dB, solar wind speed V, the logarithm of density N, thelogarithm of proton temperature T, and the Kp and Dst
indices.
SIR level, the amount of the flux enhancement is relatively
weak.
[12] Secondly, we examine the location of the outer belt
using the data from NOAA/POES satellites. Figures 3a

A03S09and 3b are the superposed L  t diagrams of the radiation
belt electrons for SBZ and NBZ events, respectively.
Similar to the GOES observations, the flux of the outerbelt electron begins to decrease just before the stream
interface. Note that the flux decrease occurs only at theouter belt at L > 4.0 and the flux does not change at the
inner portion. Figure 3a shows that flux recovers quickly tothe pre-SIR level and then increases over the pre-SIR level
during SBZ events. The flux enhancement after the stream
interface at the lower L shell is faster than that at the higher
L shell. Figure 3b shows that the flux gradually recovers to
the pre-SIR level after the stream interface during NBZ
events. As same as Figure 3a, the flux recovers to the pre-SIR
level till t = 1.0 day at L < 4.5, while the recovery takes muchtime at the outer portion. Similar to the GOES observations
in NBZ events (Figure 2b), the flux recovers to the pre-SIR
level and the amount of the flux enhancement is relatively
weak.
[13] The above results showed that the flux enhancementin the outer radiation belt is significantly controlled by the
STFA rule. As a next examination, in order to clarify theeffect of the solar wind speed for the flux enhancement, we
further categorize SBZ events into two different sets interms of the solar wind speed. We calculate the average
solar wind speed Vsw for 72 hours after the stream interface.
In SBZ events, 37 events are identified as the solar windspeed more than 500 km/s (hereafter, SBZ-fast), while
38 events are identified as the solar wind speed less than
500 km/s (SBZ-slow). Comparing SBZ-fast and SBZ-slow
events, we can investigate the effect of the solar wind speed
for the evolution of the outer belt with the same southward
IMF offset.
[14] The solar wind parameters and geomagnetic activities
for SBZ-fast and SBZ-slow events are shown in Figure 4.
The offsets of the southward IMF in both groups are almost
the same, while the solar wind speed after the stream
interface is clearly different from each other. In SBZ-fast
events, the solar wind speed increases up to 600 km/s on
average and remains above 500 km/s till t = 2.0 day. The
temperature of the solar wind in SBZ-fast events is higher
than that in SBZ-slow events, which is consistent withstatistical results that the solar wind temperature is wellcorrelated with the solar wind speed [e.g., Borovsky et al.,
1998; Borovsky and Steinberg, 2006a]. Although the Kp
index of both groups shows continuous activities after the
stream interface, the amplitude of the Kp index in SBZ-fast
events is larger than that in SBZ-slow events. As same as
the variation of the Kp index, the Dst index in SBZ-fast
events is smaller than that in SBZ-slow events and keeps
small values for several days after the stream interface.
[15] The solar wind speed difference can be clearly seenin Figures 5 and 6 where the format is the same as Figures 2
and 3, respectively. It is interesting to note that the flux
recovery and enhancement are very different between
SBZ-fast and SBZ-slow events. The flux recovery starts at
t = 0.0 day in SBZ-fast events, while the recovery start at t =0.5 day in SBZ-slow events as shown in Figure 5. The flux
enhancement is faster and stronger in SBZ-fast events,
which is similar variation to the SIR-driven storms of
Dst < 100 nT [Miyoshi and Kataoka, 2005]. During
SBZ-fast events, the flux recovers to the pre-SIR levelwithin one day after the stream interface, and the enhanced

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A03S09Figure 2. Superposed time series of the logarithm of the flux of >2.0 MeV electrons as observed by
the GOES satellites for (a) SBZ and (b) NBZ events. The bottom plots show the superposed Dst index.
The gray region indicates the standard deviation.
flux is about ten times of the pre-SIR level at t = 1.0 day.
In contrast, the flux recovery takes a longer time in SBZslow events, where the flux recovers to the pre-SIR levelat t = 2.0 day, and the increase after t = 2.0 day isrelatively weak. In SBZ-fast event, the flux increase takes
place so quickly at L = 4.5 and then the region of the flux
increase spreads over the outer belt as shown in Figure 6a.
The flux increases more than the pre-SIR level within one
day after the stream interface. Again, there exists the Lshell dependence for the period of the flux recovery andenhancement: the flux enhancement at the lower L shell isfaster than that at the higher L shell. In SBZ-fast events,
after the recovery to the pre-SIR level, the flux continues
to increase and the amount of enhancements is large. On
the other hand, in SBZ-slow events, again the flux
recovers to the pre-SIR level at t = 1.0 day but the flux
enhancement is significantly weak.
[16] The above results are summarized in Figure 7,
showing a scatter diagram of the daily maximum flux ofgeosynchronous orbit at t = 3.0 day for all events. Figure 7ashows the SIR events that Vsw is faster than 500 km/s, andFigure 7b shows the SIR events that Vsw is slower thanFigure 3. Superposed L-t diagram of >300 keV electrons as observed by the NOAA/POES satellites for(a) SBZ and (b) NBZ events. Average values after taking the logarithm of the flux are color coded. The
bottom plots show the superposed Dst index. The gray region indicates the standard deviation.
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A03S09tion between fluxes of pre-SIR and post-SIR considering the
STFA rule and solar wind speed because it has beenreported that prestorm and poststorm fluxes are highly
uncorrelated [Reeves et al., 2003]. Figure 8 shows the
maximum >2 MeV electron flux at geosynchronous orbit
for 72 hours after the stream interface against the maximum
flux for 72 hours before the stream interface. Figure 8a
corresponds to events that Vsw is faster than 500 km/s, andFigure 8b is for events with Vsw below 500 km/s. Colors are
the same as Figure 7. It is found that the correlation between
the fluxes of pre-SIR and post-SIR depends on the solar
wind speed and the STFA rule. Correlation coefficients for
SBZ, NBZ, and other SIR events are 0.36, 0.77, 0.42 forfast stream events as shown in Figure 8a, and 0.44, 0.82,
and 0.86 for slow stream events as shown in Figure 8b,
respectively. The correlations of fast stream events arebasically smaller than that of slow stream events. The
smallest correlation (0.36) is seen in the SBZ-fast events;
the increased fluxes are almost independent of the pre-SIR
fluxes. The highest correlation (0.77) in the fast stream
events is seen in NBZ events, indicating that the flux after
the SIR does not largely increase or decrease relative to thepre-SIR level. In the slow stream events, the high-correlation
coefficients (more than 0.8) are found in NBZ events andother SIR events; flux variations are small in the events
between before and after the SIR. This may suggest that theslow stream without the southward IMF offset do not workfor the net flux variation at geosynchronous orbit. The
correlations of other SIR events for fast and slow stream
events are natural, regarding the events as somewhere
between SBZ and NBZ events.

4. Discussion

Figure 4. Superposed time series of solar wind parametersand geomagnetic activities. SBZ-fast and SBZ-slow events
are shown in red and orange, respectively (see text for the
categorization in detail). The format is the same as Figure 1.
500 km/s. It is clearly demonstrated that the flux enhancement in the outer belt depends on the STFA rule. Comparing
Figures 7a and 7b, the dependence of solar wind speed isalso apparent. The greatest flux enhancement is achieved ina high-speed coronal hole stream with a southward IMF
offset of a few nT.
[17] Before finishing to show the results, from a different
point of view, it may also be interesting to see the correla-[18] The main result of this paper is that the southward
IMF as well as the solar wind speed controls the flux
enhancement of the outer belt in the SIR events. The mosteffective solar wind stream to produce the large flux
enhancement is the highest-speed stream with the southward offset of IMF because of the STFA rule. The dependence on the STFA rule itself is not a new finding. Similar
results about the outer belt flux variation caused by the
STFA rule were reported by McPherron [2006], but they did
not investigate the solar wind speed dependence. Although
the excellent correlation for the solar wind speed is consistent
with past studies [Paulikas and Blake, 1979; Baker et al.,
1986; O’Brien et al., 2001; Weigel et al., 2003], the result of
this study indicates for the first time that only the solar windspeed by itself is not a sufficient condition for the large
acceleration.
[19] The significant role of the STFA rule for the flux
enhancement would be reasonable because the small amplitude southward IMF within a coronal hole stream, whichcannot be a driver for the storm main phase, can be a driverfor continuous substorms and enhanced convections, socalled HILDCAAs [Tsurutani and Gonzalez, 1987; Tsurutani
et al., 2006]. The southward IMF offset is a preferable
condition for substorms and enhanced convections, while
the northward IMF offset suppresses such activities in themagnetosphere. There have been several reports on the effectof substorms and enhanced convections for the flux enhancement in the outer belt through the nonadiabatic acceleration

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A03S09Figure 5. Superposed time series of the logarithm of the flux of >2.0 MeV electrons as observed by
the GOES satellites for (a) SBZ-fast and (b) SBZ-slow events. The format is the same as Figure 2.
[Meredith et al., 2002; Miyoshi et al., 2003, 2007; Lyons etal., 2005]. Meredith et al. [2002] suggested that the gradualacceleration of electrons during storms can be effective when
there are periods of prolonged substorm activity during thestorm recovery phase. Lyons et al. [2005] claimed that theenhanced convection leads to the enhanced dawnside chorus
waves which accelerate the relativistic electrons. Further,
they suggested that the large-amplitude IMF Bz fluctuation
which has a large southward component and large IMF jByj
with a high solar wind speed are expected to lead to enhanced
convection. Miyoshi et al. [2007] examined two magnetic
storms driven by typical SIRs occurred in November 1993, in
which the IMF sector polarities were different from eachother: the outer belt electrons strongly increased in one storm,

but did not increase in another storm. Dynamics of seedelectron population and chorus waves were quite different
between two storms because of different activities of substorms and enhanced convection, and these differences
produced the different evolution of the outer belt.
[20] The results of superposed epoch analysis of the
NOAA data (Figures 3 and 6) showed that flux enhancements during the events take place at L > 4. That is, the SIR
generally causes the flux enhancement at the outer portionand is less effective in the flux variation at the inner portion
such as the slot region. Since the Dst index during the SIR
events is greater than 50 nT on average, this is consistent
with the statistical result between the peak L shell and the
Dst index [O’Brien et al., 2003]. It is worthwhile to note

Figure 6. Superposed L-t diagram of >300 keV electrons as observed by the NOAA/POES satellites for(a) SBZ-fast and (b) SBZ-slow events. The format is the same as Figure 3.
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A03S09Figure 7. Scatter diagram of the daily maximum flux of geosynchronous orbit at t = 3.0 days for SIR
events from 1994 to 2005. Horizontal axis is the 72 hour average of GSM Bz after the stream interface.
Shown are (a) fast and (b) slow SIR events (see text for more detail). Red and blue circles correspond to
SBZ and NBZ events, and green circles show other SIR events.
that the solar wind driver dependences of peak L shell
control the solar cycle variation of the outer belt. Since thenumber of SIRs with fast coronal hole streams increases
during the solar declining phase, it is expected that the flux
at the outer portion of the outer belt increases largely during
the declining phase. Moreover, CME-driven great storms
with average minimum Dst of 200 nT cause the flux
enhancement at the inner portion [Miyoshi and Kataoka,
2005] and the number of CME-driven great storms
increases during the solar active period [e.g., Kataoka andMiyoshi, 2006], so it is also expected that the flux at the
inner portion increases during the solar maximum. This isconsistent with the observational results: the outer portionof the outer belt develops during the solar declining phase,
while the inner portion develops during the solar active
period [Miyoshi et al., 2004].
[21] Here we discuss the relationship between relativistic
electron flux enhancements and magnetic storm amplitudes.
The statistical examination showed that the flux enhancementof the outer belt does not depend on the storm amplitude[Reeves et al., 2003]. Recently, Kim et al. [2006] showedthat the average of minimum Dst indices for which therelativistic electron flux increased is above 50 nT, andthey are normally classified as weak magnetic storms or

Figure 8. Flux of geosynchronous orbit for 72 hours before and after the stream interface. Horizontal
axis is the pre-SIR flux, and vertical axis is the post-SIR flux. Shown are (a) fast and (b) slow SIR events(see text for more detail). Red and blue circles correspond to SBZ and NBZ events, and green circles
show other SIR events.
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absence of magnetic storms. This is consistent with our
results as shown in Figure 5a: the superposed Dst index for
SBZ-fast events, in which the greatest flux enhancement in
the outer belt was observed, is greater than 50 nT. Since theperiod of a high-speed stream that produces the flux enhancement typically corresponds to the storm recovery
phase, the flux enhancement is independent of the storm
main phase which is driven by different structure such as
CMEs or SIRs. McPherron and Weygand [2006] also pointed
out that different solar wind structures relative to a stream
interface of SIRs are important for the flux enhancements.
Thus intense magnetic storms are not essential for the flux
enhancement in the outer belt. This is consistent withdiscussions of Meredith et al. [2002]. Moreover, time delay
between the storm main phase and the flux enhancement of
the outer belt is readily understood from this consideration.
[22] The Dst effect should be important for the loss ofelectrons during the main phase [Kim and Chan, 1997]. The
adiabatic Dst effect, however, can be negligible because in
the superposed epoch analysis in this study (e.g., Figure 2),
the loss of the outer belt were observed around the SIR
when the Dst index did not show significant evolution. As
shown by Green et al. [2004] and Kim et al. [2006], whenrelativistic electrons decrease at geosynchronous orbit, theambient magnetic field is less dipolar, implying that a
distortion of the ambient magnetic field causes a net
reduction in the outer belt. The result in this study suggeststhat the SIR is one of the interplanetary loss drivers,
probably leading to nonadiabatic loss due to the distortionof the magnetic field. This is consistent with the report of
Onsager et al. [2002] showing that several nonadiabatic
flux decreases at geosynchronous orbit are correlated withdiscontinuities in the solar wind. Note that the loss associated with the SIR during weak magnetic storms or absenceof magnetic storms takes place only around the outerportion at the outer belt, and the inner portion is lesssensitive. This suggests that the field distortion associatedwith the SIR would be typically limited at the outer portion.
[23] The small correlation between pre-SIR and post-SIR
fluxes in SBZ events in Figure 8 results from resetting theouter belt by rapid loss during the storm main phase andsubsequent large enhancements during the recovery phase.
One of the mechanisms for the rapid loss of the outer belt in
SBZ events (Figure 5a) may be caused by enhanced pitch
angle scattering within plasmapause drainage [Borovsky andSteinberg, 2006b]. According to the paper by Borovsky andSteinberg [2006b], SBZ events are likely to produce socalled ‘‘calm’’ just prior to the SIR and ‘‘calm’’ stimulates
the formation of plasmapause drainage by the preconditioning effect. The high correlation found in NBZ events is alsointeresting. During NBZ events, the outer belt electrons
show slow decrease and recovery to the pre-SIR level,
suggesting that the loss balances with the acceleration.
Although this high correlation in NBZ events may not be
consistent with Reeves et al. [2003], we have to note that
their statistical correlation analysis does not involve most of
our NBZ events because the Dst minima in NBZ events are
somewhat large, and hard to detect the events as storms.
[24] The obtained fact that the flux enhancement of theouter belt is controlled by both the solar wind speed and the
STFA rule would be important for application of the space

A03S09weather forecast. One can predict which coronal hole stream
produces the large flux enhancement by looking at the solar
wind speed and the IMF sector polarity. In fact, thediagrams for the probabilistic forecast are presented by
Miyoshi and Kataoka [2008], which predict probability of
the flux enhancement associated with the SIR events on thebasis of the STFA rule and the solar wind speed. Furthermore, the results shown in this paper can give a possible
observational restriction to determine the acceleration mechanism of the energetic electrons in the outer belt, and cancontribute as a basic research to future inner magetosphericsatellite missions such as RBSP (US), ORBITALS (Canada),and ERG (Japan).

5. Conclusion[25] We evaluated the solar wind parameter dependenceof the outer belt to find that both the IMF Bz and solar windspeed are important for the large flux enhancements in SIR
events. The STFA rule plays a significant role to control theflux enhancement via the IMF Bz offset of large-amplitude
IMF fluctuations in a high-speed stream, that is, only thesolar wind speed by itself is not a sufficient condition for thelarge acceleration. The largest flux enhancement is found in
the highest-speed streams with a southward IMF offset. The
strong flux enhancements tend to be associated with weakgeomagnetic storms with minimum Dst of about 50 nT on
average.
[26] Acknowledgments. The OMNI2 data are provided by NSSDC.
The NOAA and GOES data are provided by NOAA. The work by R.K. was
supported by a research fellowship of the Japan Society for the Promotionof Science for Young Scientists and a research fellowship of Special
Postdoctoral Research Program at RIKEN. This work was supported by
the Grant-in-Aid for Creative Scientific Research ‘‘The Basic Study ofSpace Weather Prediction’’ (17GS0208, Head Investigator K. Shibata) from
the Ministry of Education, Science, Sports, Technology, and Culture ofJapan, and the Housou Bunka Foundation, Japan.
[27] Amitava Bhattacharjee thanks Larry Lyons and another reviewer
for their assistance in evaluating this paper.

References
Baker, D. N., R. W. Klebesadel, P. R. Higbie, and J. B. Blake (1986),
Highly relativistic electrons in the Earth’s outer magnetosphere: 1. Lifetimes and temporal history 1979 – 1984, J. Geophys. Res., 91, 4265 –4276.
Baker, D. N., R. D. Belian, P. R. Higbie, R. W. Kebesadel, and J. B. Blake(1987), Deep dielectric charging effects due to high energy electrons in
the Earth’s outer magnetosphere, J. Electrost., 20, 3 – 19.
Baker, D. N., J. H. Allen, S. G. Kanekal, and G. D. Reeves (1998),
Disturbed space environment may have been related to pager satellite
failure, Eos Trans. AGU, 79, 477.
Blake, J. B., D. N. Baker, N. Turner, K. W. Ogilvie, and R. P. Lepping,(1997), Correlation of changes in the outer-zone relativistic-electron population with upstream solar wind and magnetic field measurements,
Geophys. Res. Lett., 24, 927 – 929.
Borovsky, J. E., and J. T. Steinberg (2006a), The freestream turbulence
effect in solar-wind/magnetosphere coupling: Analysis through the solar
cycle and for various types of solar wind, in Recurrent Magnetic Storms:
Corotating Solar Wind Streams, Geophys. Monogr. Ser., vol. 167, edited
by B. Tsurutani et al., pp. 59 – 76, AGU, Washington, D. C.
Borovsky, J. E., and J. T. Steinberg (2006b), The ‘‘calm before the storm’’in CIR/magnetosphere interactions: Occurrence statistics, solar windstatistics, and magnetospheric preconditioning, J. Geophys. Res., 111,
A07S10, doi:10.1029/2005JA011397.
Borovsky, J. E., M. F. Thomsen, and R. C. Elphic (1998), The driving of theplasma sheet by the solar wind, J. Geophys. Res., 103, 17,617 – 17,640.
Elkington, S. R., M. K. Hudson, and A. A. Chan (1999), Acceleration ofrelativistic electrons via drift resonant interactions with toroidal-mode
Pc5 ULF oscillation, Geophys. Res. Lett., 26, 3273 – 3276.

8 of 9

A03S09

MIYOSHI AND KATAOKA: RADIATION BELT RESPONSE TO SIR

Evans, D. S., and M. S. Greer (2000), Polar orbiting environmental satellite
space environment monitor: 2. Instrument description and archive data
documentation, NOAA Tech. Memo. OAR SEC-93, NOAA, Boulder,
Colo.
Gosling, J. T., J. R. Asbridge, S. J. Bame, and W. C. Feldman (1978), Solarwind stream interface, J. Geophys. Res., 83, 1401 – 1412.
Green, J. C., T. G. Onsager, T. P. O’Brien, and D. N. Baker (2004), Testing
loss mechanisms capable of rapidly depleting relativistic electron flux inthe Earth’s outer radiation belt, J. Geophys. Res., 109, A12211,
doi:10.1029/2004JA010579.
Horne, R. B., N. P. Meredith, S. A. Glauert, A. Varotsou, D. Boscher, R. M.
Thorne, Y. Y. Shprits, and R. R. Anderson (2006), Mechanisms for theacceleration of radiation belt electrons, in Recurrent Magnetic Storms:
Corotating Solar Wind Streams, Geophys. Monogr. Ser., vol. 167, edited
by B. Tsurutani et al., pp. 151 – 174, AGU, Washington, D. C.
Iles, R. H. A., A. N. Fazakerley, A. D. Johnstone, N. P. Meredith, and
P. Buhler (2002), The relativistic electron response in the outer radiation
belt during magnetic storms, Ann. Geophys., 20, 957 – 965.
Kataoka, R., and Y. Miyoshi (2006), Flux enhancement of radiation belt
electrons during geomagnetic storms driven by coronal mass ejections
and corotating interaction regions, Space Weather, 4, S09004, doi:10.1029/
2005SW000211.
Kim, H.-J., and A. A. Chan (1997), Fully adiabatic changes in storm time
relativistic electron fluxes, J. Geophys. Res., 102, 22,107 – 22,116.
Kim, H.-J., K. C. Kim, D.-Y. Lee, and G. Rostoker (2006), Origin ofgeosynchronous relativistic electron events, J. Geophys. Res., 111,
A03208, doi:10.1029/2005JA011469.
Li, X. (2006), The role of radial transport in accelerating radiation belt
electrons, in Recurrent Magnetic Storms: Corotating Solar Wind Streams,
Geophys. Monogr. Ser., vol. 167, edited by B. Tsurutani et al., pp. 139 –
174, AGU, Washington, D. C.
Lyons, L. R., D.-Y. Lee, R. M. Thorne, R. B. Horne, and A. J. Smith(2005), Solar wind – magnetosphere coupling leading to relativistic electron energization during high-speed streams, J. Geophys. Res., 110,
A11202, doi:10.1029/2005JA011254.
McPherron, R. L. (2006), Statistical properties of the solar wind and IMF at
1 AU, Eos Trans., AGU, 87(52), Fall Meet. Suppl., Abstract SH43C-01.
McPherron, R. L., and J. Weygand (2006), The solar wind and geomagnetic
activity as a function of time relative to corotating interaction regions, in
Recurrent Magnetic Storms: Corotating Solar Wind Streams, Geophys.
Monogr. Ser., vol. 167, edited by B. Tsurutani et al., pp. 125 – 138, AGU,
Washington, D. C.
Meredith, N. P., R. B. Horne, and R. R. Anderson (2001), Substorm
dependence of chorus amplitudes: Implications for the acceleration ofelectrons to relativistic energies, J. Geophys. Res., 106, 13,165 – 13,178.
Meredith, N. P., R. B. Horne, R. H. A. Iles, R. M. Thorne, D. Heynderickx,
and R. R. Anderson (2002), Outer zone relativistic electron acceleration
associated with substorm-enhanced whistler mode chorus, J. Geophys.
Res., 107(A7), 1144, doi:10.1029/2001JA900146.
Miyoshi, Y., and R. Kataoka (2005), Ring current ions and radiation belt
electrons during geomagnetic storms driven by coronal mass ejections
and corotating interaction regions, Geophys. Res. Lett., 32, L21105,
doi:10.1029/2005GL024590.
Miyoshi, Y., and R. Kataoka (2008), Probabilistic space weather forecast of
the relativistic electron flux enhancement at geosynchronous orbit,
J. Atmos. Sol. Terr. Phys., in press.

A03S09Miyoshi, Y., A. Morioka, H. Misawa, T. Obara, T. Nagai, and Y. Kasahara(2003), Rebuilding process of the outer radiation belt during the 3 November
1993 magnetic storm: NOAA and Exos-D observations, J. Geophys. Res.,
108(A1), 1004, doi:10.1029/2001JA007542.
Miyoshi, Y. S., V. K. Jordanova, A. Morioka, and D. S. Evans (2004),
Solar cycle variations of the electron radiation belts: Observations andradial diffusion simulation, Space Weather, 2, S10S02, doi:10.1029/
2004SW000070.
Miyoshi, Y., A. Morioka, R. Kataoka, Y. Kasahara, and T. Mukai (2007),
Evolution of the outer radiation belt during the November 1993 storms
driven by corotating interaction regions, J. Geophys. Res., 112, A05210,
doi:10.1029/2006JA012148.
O’Brien, T. P., R. L. McPherron, D. Sornette, G. D. Reeves, R. Friedel, and
H. J. Singer (2001), Which magnetic storms produce relativistic electronsat geosynchronous orbit?, J. Geophys. Res., 106, 15,533 – 15,544.
O’Brien, T. P., K. R. Lorentzen, I. R. Mann, N. P. Meredith, J. B. Blake, J. F.
Fennell, M. D. Looper, D. K. Milling, and R. R. Anderson (2003),
Energization of relativistic electrons in the presence of ULF power and
MeV microbursts: Evidence for dual ULF and VLF acceleration, J. Geophys. Res., 108(A8), 1329, doi:10.1029/2002JA009784.
Onsager, T. G., G. Rostoker, H.-J. Kim, G. D. Reeves, T. Obara, H. J.
Singer, and C. Smithtro (2002), Radiation belt electron flux dropouts:
Local time, radial, and particle-energy dependence, J. Geophys. Res.,
107(A11), 1382, doi:10.1029/2001JA000187.
Paulikas, G. A., and J. B. Blake (1979), Effects of the solar wind on
magnetospheric dynamics: Energetic electrons at the synchronous orbit,
in Quantitative Modeling of Magnetospheric Processes, Geophys.
Monogr. Ser., vol. 21, edited by W. P. Olson, pp. 180 – 202, AGU,
Washington D. C.
Reeves, G. D., K. L. McAdams, R. H. W. Friedel, and T. P. O’Brien (2003),
Acceleration and loss of relativistic electrons during geomagnetic storms,
Geophys. Res. Lett., 30(10), 1529, doi:10.1029/2002GL016513.
Richardson, I. G., et al. (2006), Major geomagnetic storms (Dst  100 nT)
generated by corotating interaction regions, J. Geophys. Res., 111,
A07S09, doi:10.1029/2005JA011476.
Russell, C. T., and R. L. McPherron (1973), Semiannual variation of geomagnetic activity, J. Geophys. Res., 78, 92 – 108.
Schulz, M., and L. Lanzerotti (1974), Particle Diffusion in the Radiation
Belts, Springer, New York.
Summers, D., R. M. Thorne, and F. Xiao (1998), Relativistic theory ofwave-particle resonant diffusion with application to electron accelerationin the magnetosphere, J. Geophys. Res., 103, 20,487 – 20,500.
Tsurutani, B. T., and W. D. Gonzalez (1987), The cause of high-intensity
long-duration continuous AE activity (HILDCAAs): Interplanetary
Alfvén wave trains, Planet. Space Sci., 35(4), 305 – 412.
Tsurutani, B. T., et al. (2006), Corotating solar wind streams and recurrent
geomagnetic activity: A review, J. Geophys. Res., 111, A07S01,
doi:10.1029/2005JA011273.
Weigel, R. S., A. J. Klimas, and D. Vassiliadis (2003), Precursor analysis
and prediction of large-amplitude relativistic electron fluxes, Space
Weather, 1(3), 1014, doi:10.1029/2003SW000023.


R. Kataoka, Computational Astrophysics Laboratory, Institute of Physics
and Chemical Research (RIKEN), Saitama 351-0198, Japan. (ryuho@
riken.jp)
Y. Miyoshi, Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya 464-8601, Japan. (miyoshi@stelab.nagoya-u.ac.jp)

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ClickHere

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S05, doi:10.1029/2007JA012516, 2008

for

Full
ArticleA quantitative MHD study of the relation amongarcade shearing, flux rope formation, and eruption
due to the tearing instability
D. Shiota,1 K. Kusano,2 T. Miyoshi,3 N. Nishikawa,4 and K. Shibata5
Received 1 May 2007; revised 12 November 2007; accepted 11 January 2008; published 15 March 2008.

[1] The quantitative relationship between the magnetohydrodynamic (MHD) activity ofsolar coronal arcade and the magnetic helicity injection, which is caused by shearing
motion, has been investigated, using azimuthally symmetric model of MHD simulation.
We have calculated several cases in which the width of the shearing region is varied andexamined the relationship between the magnetic arcade dynamics and magnetic helicity
evolution. As a result, it is found that as the shearing motion is imposed on narrower
regions along each side of the magnetic inversion line, the magnetic arcade can be easily
destabilized by the resistive tearing mode. However, in this case, even thoughreconnection driven by the tearing mode produces plasmoids, the plasmoid elevation isalmost in proportion to the total amount of magnetic helicity contained in the arcade, and it
is too slow to explain the trigger process of coronal mass ejections (CMEs). On the other
hand, in the case where the shearing motion is imposed on the entire region, muchlarger magnetic helicity injection is required to injected arcade in order to destabilize thesystem, compared to practical helicity injection measured in the solar corona. The results
suggest that it may be difficult to trigger a CME just by the axisymmetric shearing
motion and that some other mechanisms should be involved in the triggering process of a
CME. The results also imply that the relation between the magnetic helicity and theoverlying magnetic flux can be a key parameter for the CME occurrence.
Citation: Shiota, D., K. Kusano, T. Miyoshi, N. Nishikawa, and K. Shibata (2008), A quantitative MHD study of the relation amongarcade shearing, flux rope formation, and eruption due to the tearing instability, J. Geophys. Res., 113, A03S05,
doi:10.1029/2007JA012516.

1. Introduction[2] Coronal mass ejections (CMEs) are the most explosive phenomena in the solar corona, whose triggering
mechanism is one of the major problems in the solar
physics. In addition, it is also important for the understanding of CMEs and space weather forecasting to make clear akind of critical condition to determine whether CMEs canbe launched as the result of eruptions of coronal fields.
[3] CMEs are defined as ejections of a large amount ofmass within magnetic flux, which are produced by eruptionsof the coronal magnetic field. In many observations withcoronagraphs, CMEs have been observed to have finecomplicated density structures which are thought to be
traces of magnetic field lines due to freezing of the plasma
1Center for Computational Astrophysics, National Astronomical
Observatory of Japan, Tokyo, Japan.
2
Earth Simulator Center, Japan Agency for Marine-Earth Science andTechnology, Kanagawa, Japan.
3
Facility of Science, Hiroshima University, Hiroshima, Japan.
4Super Computer System Planning and Operations Department, Japan
Agency for Marine-Earth Science and Technology, Kanagawa, Japan.
5
Kwasan and Hida Observatories, Kyoto University, Kyoto, Japan.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012516$09.00

to the magnetic field. Usual structures in CMEs are helicalstructures, i.e., twisted magnetic flux ropes, and in some ofthem there are dense cores corresponding to erupted
chromospheric prominences (see Hundhausen [1999] forreview). This results suggested that the magnetic structuresin prominences are closely related to the CME initialprocesses. Furthermore, because twisted magnetic flux
ropes represent current carrying magnetic flux, they arethought to be the most essential features of CMEs.
[4] Since the discovery of CMEs in the early 1970s,
many attempts to understand the mechanism of CMEs
theoretically or numerically have been made by many solar
physicists. There are some reviews for such historical works[Forbes, 2000; Priest and Forbes, 2002; Lin et al., 2003;
Forbes et al., 2006], and we do not duplicate them here.
Most of theoretical works about CMEs since the 1990s canbe roughly classified into two major categories: flux ropemodels and sheared arcade models.
[5] Studies with the former models focused only on theenergy release process assuming initial conditions including
a twisted magnetic flux rope [Forbes and Priest, 1995; Linand Forbes, 2000; Chen and Shibata, 2000; Kliem and
Torok, 2005]. These works succeeded in explaining the
rapid energy release and acceleration in the initial phaseof CMEs due to loss of equilibrium or stability of the initial

A03S05

1 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

conditions containing high magnetic energy but do notdiscuss how such the initial conditions can be formed.
[6] In contrast, the other group studies the shearing of
the coronal potential magnetic arcades whose magnetic
energy is minimum [Mikic et al., 1988; Steinolfson,
1991; Inhester et al., 1992; Mikic and Linker, 1994; Linker
and Mikic, 1995; Kusano et al., 1995; Choe and Lee, 1996;
Antiochos et al., 1999; Amari et al., 2003; MacNeice et al.,2004]. These works succeed in explaining the energy
buildup and release processes with slow velocity on thelower boundary, starting from initial minimum energy
conditions. In these models, twisted flux ropes are produced as a result of magnetic reconnection in the sheared
arcades, and the flux ropes are ejected without remaining inequilibrium states as with the initial conditions of the flux
rope models.
[7] Because the magnetic configuration of each model is
too simple, these results show no continuity between thesescenarios. To avoid the incompleteness due to the simplification, recent studies tend to simulate as realistically as
possible. Manchester et al. [2004] performed MHD simulation of magnetic buoyant processes and the initiation of
eruptions starting from the initial condition where thetwisted magnetic flux rope embedded below the photosphere. Their results show that the emergence of thetwisted magnetic flux ropes achieves shearing motion of
the initial coronal arcades on the photosphere. Because of
the stratification in the solar interior, the buoyant processproceeds with nonuniform speed due to mass accumulated
in the concave part inside the flux rope. As a result, mostof the poloidal flux has footpoints anchored to the surface
and appears to bind the strongly sheared arcade
corresponding to the axis of the original large-scale flux
rope (also investigated by Magara [2006]). These result
suggests that a strong sheared arcade embedded in a
weakly or unsheared arcade, i.e., a magnetic configuration
where its shear component is localized, could be acommon magnetic configuration in an emerging flux
region.
[8] On the other hands, the formation process of a coronal
twisted flux rope has been studied with the model ofmagnetic field evolutions [van Ballegooijen and Martens,
1989; MacKay and van Ballegooijen, 2005, 2006a, 2006b].
Those studies suggests that the important process for theformation is ‘‘magnetic cancellation’’ of a sheared arcade
which is achieved by the motion of opposite magnetic
elements on the photosphere, toward the polarity inversion
line. The cancellation process leads to formation of a current
sheet and then reconnection, which results in the formation
the flux rope. However, these authors did not include thedynamical effect of current sheet formation (inhibition ofreconnection), and the effect on CME occurrence of theimpulsive energy release by reconnection On the basis ofthese results, in this paper, we developed a classical arcade
shearing model in a simple axisymmetric configuration toinvestigate the questions: how coronal twisted flux rope canbe formed via magnetic reconnection and what is anappropriate condition for CME occurrence. Specifically,
we use a single potential arcade across the equator andimpose shearing motion on its bottom boundary.
[9] In this study, we focus on the relation between thecondition for a CME and the following quantities: magnetic

A03S05helicity, magnetic flux, and electric resistivity. Magnetic
helicity [Berger and Field, 1984] defined as
Z
H¼

A BdV :

ð1ÞThe magnetic helicity is an important quantity related to the
topology of magnetic field and the free magnetic energy
where B and A are magnetic field and the corresponding
vector potential. In order to exclude their gauge arbitrary,
we use relative helicity uniquely defined asZ
Hrel ¼ 
B  Bp dV ;A þ Apð2Þ

where Bp and Ap are the potential magnetic field and thecorresponding vector potential, whose details are describedin section 2.2. It is widely accepted that a CME is theprocess in which magnetic helicity in the solar corona is
expelled into interplanetary space, and therefore theexistence of magnetic helicity is the necessary condition
for a CME occurrence. However, how the minimum amountof helicity leads to a CME is not clear. Magnetic flux is
another key quantity for determining whether an eruption ina sheared arcade can evolve into a CME or not. That is, if
enough magnetic flux exists outside of the reconnecting
arcade, it can confine and stabilize the ejected plasmoid(flux rope) by magnetic tension force. Therefore the amountof overlying flux is thought to be an essential quantity for a
CME occurrence. Finally, resistivity is known as theessential quantity to determine the magnetic reconnection
process and the growth rate of a tearing-mode instability. In
this study we try to make clear the quantitative relations
among these quantities and the condition for a CME
occurrence.
[10] Here we briefly summarize the contexts of this paper.
In the next section, we describe the detailed methodology ofthe numerical simulation. In section 3, the numerical resultsof different shear and different resistivity cases are shown.
We discuss the quantitative relations suggested by thenumerical results in section 4. Finally, we summarize thispaper in section 5.

2. Numerical Model
2.1. Numerical Scheme[11] In a numerical simulation performed in this work, wesolve the following MHD equations using a finite volume
method with the HLLD nonlinear Riemann solver [Miyoshi
and Kusano, 2005], the third-order TVD MUSCL, and theRunge-Kutta time integration:

2 of 14@r
þ r ðrvÞ ¼ 0;
@t

ð3Þ

@rv
þ r ðrvv  BB þ pT ^eÞ ¼ rg;
@t

ð4Þ

@B
þ r ðBv  vB þ y^eÞ þ r  ðhJÞ ¼ 0;@t

ð5ÞSHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05

A03S05@eþ r ½ðe þ pT Þv  ðv BÞB ¼ 0;@t

ð6Þnumerical r B small so it does not change the solution
significantly. The essential point of the method of Dedner et
al. [2002] is that introducing a new variable y and equation(7) with the divergence of equation (5)@yþ c2p r B þ cd y ¼ 0;@t

ð7Þ@ ðr BÞ
þ r2 y ¼ 0;
@t

where
J ¼ r  B;

e¼

ð8Þ

rv2
p
B2
þ
 rY;
þ
2
g1 2

pT ¼ p þ
0

B2
;
2

ey;r
ey;q
ey;f

ex;r
^e ¼ @ ex;q
ex;f

1
ez;r
ez;q A
ez;f

ð9Þ

ð10Þ

ð11Þ

GM r0
;
R p0

ð12Þ

g ¼ G0 r=r3 ;

ð13Þ

GM
:
r

ð14Þ

G0 ¼

Y¼

Note that we introduce a new variable y and thecorresponding equation (7) in order to remove numericalr B from the domain of interest. Detail of this method
is described in the next paragraph. All other variables aredefined as standard usages. The vector potential, which isused for the calculation of magnetic helicity, is also time
integrated with the following equation:
@A
¼ E;
@t

ð15Þ

E ¼ v  B þ hJ:

ð16Þ

where

[12] Here we briefly introduce the divergence-free correction method [Dedner et al., 2002]. Equations (4), (5), and(6) are derived assuming the Solenoidal condition of themagnetic field; i.e., these equations have the following
neglected right-hand terms: (r B)B, (r B)v, and(r B)(v B), respectively. Generally, in the result of thetime integration of equation (5) in a three-dimensional
simulation, r B cannot be kept to zero, and the resulting
finite r B generates numerical errors through the neglected
terms above. Because an accumulation of the errors causes
wrong solutions, it is very important to keep the magnitude of

ð17Þ

we can establish a hyperbolic and parabolic system aboutr B and y. The coefficient cp in equation (7) is the
propagation speed of numerical r B, and the coefficient
cd is the diffusion coefficient of y which has finite value
only near the outer boundary (r > 3). When nonzero valuesof r
B are generated somewhere in the numerical
domain, this method distributes that value to the all
direction as a wave, thus reducing its local magnitude.
Furthermore, when the wave approaches the outerboundary, the amplitudes of both r B and y reduces
rapidly. On the other hand, the curl procedure of equation(5) vanishes the second term of the right-hand side, i.e., theterm due to the correction method. Hence this method does
not directly affect the evolution of the current density,
which is significant in resistive MHD processes. Thismeans that the correction method developed for ideal MHD
process can be adopted to resistive MHD simulations.
[13] Parameters are chosen to realistic value: the solar
radius R = 6.998  1010 cm for length, sound speedpcﬃﬃﬃﬃﬃﬃﬃﬃﬃ
s forﬃ
velocity, free-fall time t 0 = R /cs for time, and B0 = 8pp0
for magnetic field. When these values are chosen to be
characteristic values of the solar corona, such as n0 = 1 
109 cm3 and T0 = 2 MK with a specific heat ratio g = 1.02,
the normalization parameters become cs = 1.83  102 kms1, t 0 = 3.81  103 s, and B0 = 3.72 Gauss.
[14] Thermodynamic effects, such as coronal heating andradiative cooling, are neglected in this study, and we assume
the specific heat ratio g is set to be 1.02 for simplicity. The
electric resistivity h is set to be uniform: zero from cases A
to E, 104 for case F, 105 for case G, and 106 for case H.
However, when the width of the current sheet becomes as
small as the grid size, TVD limiter, installed in order toavoid numerical oscillations at discontinuities, behaves asanomalous resistivity which could lead to fast reconnection.
[15] The numerical domain is set to a sector in thespherical coordinate, i.e., (Rmin < r < Rmax, 0 < q < p, 0 <f < 2p/Nd) where Rmin = 1, Rmax = 5, and Nd = 256. The
domain is discretized by (Nr, Nq, Nf) = (510, 255, 1) grid
points which are distributed nonuniformly to enhance theresolution in the equatorial and lower regions. The radial
and latitudinal location of grid points are determined so thatits first and second derivatives of the grid coordinate
become continuous between uniform and nonuniform
distributions, as follows:rðiÞ ¼

Rmin þ dr1 i
Rmin þ ar i3 þ br i2 þ ðcr þ dr1 Þi þ dri  ici > icð18Þ

where dr1 = (Rmax  Rmin)er/Nr, 0 < er  1, and

3 of 14

0

0
1
1
1
ar
B br C Rmax  Rmin  dr1 Nr B 3ic C
B C¼
B 2 C:@ cr A@ 3i AðNr  ic Þ3
c
i3c
dr

ð19Þ

SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S058p=2 þ dq1 j0  aq fcos½bq ðj0 þ jc Þ  1g>>>>
cq ðj0 þ jc Þ2ðjh  j0 < jc Þ<ðjc  j0  jc Þqð jÞ ¼ p=2 þ dq1 j0>00>> p=2 þ dq1 j þ aq fcos½bq ðj  jc Þ  1g>:
þcq ðj0  jc Þ2ðjc < j0  jh Þ;[17] Once Bp is obtained, using the gauge as Ap,r = 0, we
can get the vector potential Ap by integration of Bp = r 
Ap. Assuming limr!1Ap = 0, we get a simple solution for
the vector potential:
Ap;q ¼

ð20Þ

where j0 = j  jh, jh = (Nq + 1)/2, dq1 = peq/(Nq + 1), 0 <eq  1, and
0

1 0
1
aq
ðp=2  dq1 jh Þ=2p2@ bq A ¼ @A:
2p=ðNq  jc  jh Þ2cq
ðp=2  dq1 jh Þ=ðNq  jc  jh Þ

A03S05

(  
)
N X
n
X
 Pnm ðcos qÞ
m 1 nþ1  m
gn sin f  hm
cos
f
n
n r
sin q
n¼1 m¼0

ð27Þ
(  
)
N X
n
X
 dPnm ðcos qÞ
1 1 nþ1  m
m
gn cos f þ hn sin f:
Ap;f ¼ n r
dq
n¼1 m¼0

ð21Þ

ð28Þ

Parameters (ic, jc, er, eq) are set to be (0.4 Nr, 0.0625 Nq,
0.125, 0.0625), so that minimum and maximum grid sizeare (drmin, drmax,dqmin, dqmax) = (9.8  104, 3.5  102,
p/4096, 2.5  102).

[18] In this study, the mode limit N is set to 64 and only
the axisymmetric component (m = 0) is used. Thus Bf andAq vanish. The coefficients g and h are determined from theboundary condition "#2.2. Initial Condition[16] Pressure, velocity, and density throughout the numerical domain are determined by assuming hydrostatic
equilibrium at uniform temperature. Potential magnetic field
is obtained from a boundary value problem of Laplace
equation for F [Altschuler and Newkirk, 1969], defined asfollows:r Bp ¼ r ðrFÞ ¼ r2 F ¼ 0:

ð22Þ

With the outer boundary at infinity, the solution isdetermined by the boundary condition at the solar surface(r = 1). When the normal component on the surface Bp,r0 is
given as Neumann condition of F, we get the solution
( 
)
N X
n
X
 m
1 nþ1  mm
F¼
gn cos f þ hn sin f Pn ðcos qÞ ; ð23Þr
n¼1 m¼0q  qc 2q  qctanh;
Bp;r0 ðqÞ ¼ B0 c exp 
qr
qd

where the normalized coefficient c is determined as max(Bp,r0(q)) = B0 = 3.0, and (qc,qr,qd) = (0.5p, 0.1p, 0.05p).
2.3. Boundary Conditions[19] The bottom boundary (r = 1) is a line-tied boundaryon which the normal component of the magnetic field (29)is conserved. If we assume that the photosphere can betreated as a no-slip wall, then the normal and meridional
components of velocity vr and vq are set to be zero. In orderto energize the magnetic field, we impose only azimuthal(toroidal) shearing motion vy,sh(q, t) to the footpoints of thearcade. We carried out five different simulations, cases A
through E, with the five different patterns of shear motion,
as shown in Figure 1. Only in case A, is the shearing motion
imposed on the whole area between the magnetic poles, thatis,where Pmn (cos q) are associated Legendre polynomials, andmgmn and hn are coefficients obtained by spherical harmonics
analysis. Then we get the potential magnetic field as,
Br ¼ 

N X
nX
n¼1 m¼0()
 nþ2
 m
 m
1
ðn þ 1Þ
gn cos f þ hm
sin
f
P
ð
cos
q
Þ
; ð24Þ
n
n
r

( 
)
N X
n
X
 dPnm ðcos qÞ1 nþ2  mm
gn cos f þ hn sin f
Bq ¼
;
r
dq
n¼1 m¼0vy;sh ðq; t Þ ¼ v0 Rðt Þ(  ) Pnm ðcos qÞ1 nþ2  mmm
gn sin f  hn cos f:
Bf ¼ 
r
sin q
n¼1 m¼0

Bp;r0 ðqÞ
;
B0

ð30Þ

where v0 is maximum speed and R(t) = min(t, 1) is a rampfunction. The function is useful for reducing the fluctuation
due to a sudden velocity injection. The maximum velocity
v0 is set to be 0.03; that is, about 1% of Alfvén wave in thearcade region. In cases B through E, only a limited region
near the magnetic neutral line between the latitudes qp,max
and qp,min, where max(Br) and min(Br), respectively, aresheared by means of the following forms:
8<0
for q < qp;min or qp;max < qvy;sh ðq; t Þ ¼ v0 RðtÞ @f ðqÞ; ð31Þ
:for qp;max  q  qp;min
@q

ð25Þ
N X
n
X

ð29Þ

in case B,

ð26Þ

(
f ðqÞ ¼ )2
jBp;r0 ðqÞj 2
2
1 1
Bp;r0 ðqÞ sgn Bp;r0 ðqÞ :
B0


ð32Þ
4 of 14SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05Figure 1. Velocity profiles of the boundary shearing
motion.
In case C,(
f ðqÞ ¼

 )2
jBp;r0 ðqÞj 2
1 1
Bp;r0 ðqÞ:
B0

ð33Þ

In case D,(
f ðqÞ ¼ )2

jBp;r0 ðqÞj 21 1
sgn Bp;r0 ðqÞ :
B0

ð34Þ

Finally, in case E,(
f ðqÞ ¼2 )2
jBp;r0 ðqÞj1  1  min 2;1
sgn Bp;r0 ðqÞ :
B0


ð35Þ

We have also carried out other simulations with different
finite resistivities in cases F through H, with the same shear
pattern used for case D.

3. Results
3.1. No Overlying Flux Case[20] In case A the imposed shearing motion has a distribution that is proportional to the radial component of themagnetic field shown in equation (30). This case is similar
to that of Choe and Lee [1996]. All the magnetic flux in thearcade is sheared, and there is no overlying arcade which
confines the nonpotential flux injected by the shearing motion.
[21] The time evolution of the magnetic field and thetoroidal current density is shown in Figure 2 in which themagnetic field lines are illustrated as the contours of
the toroidal flux function defined as rAf. The evolution of
the arcade is almost the same as that of rectangular casestudied by Choe and Lee [1996]. As in their simulation, theA03S05

arcade quasi-statically evolves by three steps; in the firststep the toroidal component of the magnetic field increases
without expanding, in the second step the arcade begins to
expand vertically, and in the third step the vertically
expanded arcade begins to thin and form a vertical current
sheet. Note that the ‘‘vertical’’ direction means to be parallelto the opposite direction of the gravity, and hence itcorresponds to the radial direction. As a result, the initialpotential arcade evolves into a vertically elongated sheared
arcade with an inverse Y-shape current structure (seeFigure 2). Further shearing motion leads to continual
thinning of the current sheet. When the width of the current
sheet approaches the grid scale, numerical diffusion
becomes significant and a tearing mode instability ensues.
The reconnection which results from the instability forms
a magnetic island (a helical flux rope in three dimensions(3-D)), as shown in Figure 2d; this flux rope subsequently ejected into interplanetary space.
[22] Figure 3 shows the time evolution of the toroidal flux
function and the toroidal current density. In the axisymmetricspherical coordinates system, the flux function evolving intime is rAf(r, q, t). Figure 3 plots the contour of the flux
function at the equator determined rAf(r, p/2, t) = const
with the solid lines and indicates the toroidal current densitywith grey. The contours of the flux function correspond to
the trajectories of plasma frozen in the same magnetic fieldlines across the equator, and the vertical distance from acontour to the next contour are inversely proportional to thestrength of Bq.
3.2. Partially Sheared Cases[23] In cases B, C, D, and E, shearing motions are
imposed only on the footpoints of the inner region of theinitial potential arcade. From case B to case E, the area
where shear motion is imposed becomes narrower (seeFigure 1). The behavior in these cases is similar to thatfound by Choe and Cheng [2000]. In contrast to case A
described in the previous subsection, there are two distinct
flux systems; i.e., one corresponds to the overlying
unsheared arcade and the other to the inner sheared arcade.
Figure 4 shows the current density distribution and contoursof the flux function displaying magnetic field lines. In
Figure 4, the overlying flux region can be recognized as
the region where Bq is dominant and field lines areconcentrated, while the sheared region is where Bf isdominant and the distances among the field lines are large.
Between these two regions there is a separatrix surface
where the current is enhanced slightly as shown in Figure 4.
[24] A vertical current sheet similar to case A is also
formed above the inversion line inside the sheared arcade(see Figure 4b). An inverse Y-shape current structure also
forms, and the upper sheet of this structure eventually
becomes tearing unstable. When this happens an X point
forms at the center of the current sheet. The reconnection at
the X point produces a magnetic island corresponding to ahelical flux rope in 3-D. However, the speed of the helicalflux rope soon decreases because the system approaches a
new equilibrium state. Indeed, the newly formed helical flux
rope overruns the new equilibrium state and bounces aroundit (see Figure 3). In the downward phase of the bounce, ahorizontal current sheet between the flux rope surface and5 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05Figure 2. Meridional distribution of the toroidal current density (Jf) in case A. Arrows show thevelocity field meridional component. Solid lines projected the magnetic field lines.
the sheared arcade is formed in which reconnection repetitively occurs.
[25] In the other cases (B, C, and E), the evolution issimilar to those described above, except for the start times
and intervals of the repeated disruption of the sheared arcade.
Figure 3 shows the time evolution of the current density
distribution above the equator in cases B, C, D, and E.
3.3. Partially Sheared Case With Finite Resistivity[26] In cases F, G, and H, a shearing motion whose
distribution is the same as that in case D is imposed. Fromcase F to case H, the resistivity becomes smaller raging
from 104 to 106.
[27] In the case H (h = 106), the evolution is almost thesame as that in case D. This suggests that effective numerical diffusion is of the order of 106.
[28] In case F, a large resisitivity case, the evolution of themagnetic field is much different from the cases describedabove (Figure 5). The evolution of the field in this case issimilar to that in case D during the initial phase whencurrent distribution is smooth (first and second stages
discussed in section 3.1). However, as the sheared arcade
starts to develop a vertical current sheet (in the third stage),the evolution differs significantly from that in case D. As
shown in Figure 5, an extended vertical current sheet isnever formed. Instead an elongated sheared arcade (like an
inverse V-shape) is formed during this simulation. This
result suggests that the timescale of the current dissipationis shorter than the Alfven timescale in the sheared arcade inthis case.
[29] The evolution in case G is the intermediate casebetween cases D and F. As shown in Figure 3, the start timeof the tearing instability is much later than in case D. The
reason seems to be the same as for case F and that theinverse V-shape arcade forms before the tearing start time.
In contrast to case F, however, reconnection due to thetearing instability is sufficient to form a flux rope. Theinterval of the repetitive reconnection events are shorter
than those in case D.

4. Discussion
4.1. Evolution of Energy, Helicity, and Flux
[30] The change of helicity can be written as follows[Priest and Forbes, 2000, chap. 8] with the use of the vector
potentials Ap of equations (27) and (28):
dH
¼ 2
dt

Z

Z
E BdV þV

Ap  E ndS

ð36Þ

S

then
dH
¼ 2
dt

6 of 14ZZ
hJ BdV þV
B Ap ðv nÞ  v Ap ðB nÞdS:

S

ð37Þ

SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05

A03S05Figure 3. Time evolution of the toroidal current density (Jf) above the equator (q = p/2) in each case.
Solid lines are trajectories of the magnetic field.
In this study, Ap = Ap,f ef and v = vsh,f ef on the bottom
boundary. Therefore equation (37) reduces to
dH
¼
dt

Z
vsh;f Ap;f Bp;r0 dS:

ð38Þ

S

which is constant as long as the shearing motion is constant.
This relation is confirmed in Figure 6a which shows thetime evolution of the total magnetic helicity in the entire
numerical domain in each case.

[31] On the other hand, the time evolution of the magnetic
energy (Figure 6b) shows three stages in the evolution. In
the first stage, the magnetic energy increases until it is about
twice that of the potential magnetic energy; in the second
stage, the rate of energy increase gradually slows; and in thethird stage, the rate of energy increase becomes almost
constant. In the third stage, we find many fluctuations ofboth the magnetic and kinetic energies. These correspond to
the disruption of the arcade due to the tearing-mode
instability in the central current sheet. These three evolution
stages correspond to those of the topological evolution

7 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05Figure 4. Current density distribution and its time evolution in case D. The meaning of each panel is thesame as that of Figure 2.

described in the previous section. This behavior is alsosimilar to that of Choe and Cheng [2000].
[32] There are a few observations which succeed in
detecting features that result from tearing-mode instability
such as plasmoid in the current sheet [e.g., Ko et al., 2003;
Lin et al., 2005; Lin et al., 2007] and numerically illustrated
by Riley et al. [2007]. However, these features were foundin the current sheet behind some CMEs where reconnection
might continue and accelerate the CMEs. Therefore thesefeatures were thought to be not the results of instability in asheared arcade which discussed in this study, but the resultsof some instability of a reconnecting current sheet, such as
secondary tearing [Shibata and Tanuma, 2001].
[33] The relation of kinetic energy to helicity (Figure 6e)shows that the maximum kinetic energy produced by thetearing instability appears to have an upper limit (pink line)
which is a function of helicity. This result suggests that theupper limit of released magnetic energy is closely related tothe total helicity or energy.

4.2. Relation of Evolution to Energy, Helicity, and Flux[34] As discussed in section 1, it is thought that themagnetic fluxes in various domains are an important factor
in determining whether a CME occurs or not. We firstexamine how much initial potential magnetic flux is involved in the sheared arcade. The initial potential magnetic
flux is calculated by the following surface integral as
Yp ¼

1
2

Z
jBp;r0 jdS;

ð39Þ

S

where S is the bottom boundary. Considering the intersection between the surface and the separatrix, the bottom
boundary is divided into two areas Sov and Ssh. The total
poloidal magnetic flux involved in sheared arcade andoverlying arcade as follows:

8 of 14

Ysh ¼

1
2

Z
jBp;r0 jdS;
Ssh

ð40Þ

SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05

A03S05Figure 5. Snapshots of the current density toroidal (Jf) component distribution in the meridional plane(f = 0) in case F. Solid lines are projected magnetic field lines.

1
Yov ¼
2

Z
jBp;r0 jdS:

ð41Þ

Sov

The results are tabulated in Table 1. It is found that a largest
part of the magnetic flux is involved in the overlying arcade.
This is the reason why the newly formed flux rope did noteject like a CME in cases B-E. If the condition for CME
ejection depends on both the magnetic helicity and theoverlying magnetic flux, then there may be some relation
between them. The dimensionless parameteraCME ¼

Y2ov
H

ð42Þ

could be a key parameter for CME triggering. In case A theflux rope appears to ejected to the outer boundary as a CME(before it reached the boundary the simulation was halted),
while in other cases it clearly increases and bounces aroundan equilibrium point. From Table 1 we get0ðcaseAÞ < aCME < 1:7ðcaseBÞ:ð43Þ

Because this limitation is too broad, more investigation isnecessary.
[35] In order to examine how the total helicity is related tothe evolution of the field, we compare the evolution in cases

A to E based on the total helicity amount. Figure 7 plots theheights of the separatrix and the center of the verticalcurrent sheet versus the total helicity. It is found that theheights of the separatrix in each case, are almost the same,
which suggests that the vertical extension of the sheared
arcade is closely correlated to the total amount of helicity;
i.e., the rise velocity of the separatrix is correlated to thehelicity injection rate at the bottom boundary, as long as themagnetic helicity is conserved. As described in section 3.2,
a plasmoid (a twisted flux rope) are generated as the resultof the nonlinear evolution of tearing-mode instability, andthe repeating eruption due to the tearing instability feeds themagnetic flux to the plasmoid in each of cases B through E.
Because the flux ropes are confined within the separatrix,
which is closely related to the total magnetic helicity, theelevation of the flux ropes is also related to the totalmagnetic helicity. Figure 8[36] The growth rate of the tearing-mode instability
depends on the aspect ratio of the current sheet, i.e., theheight and width of the sheet. As discussed above, theheight of the sheared arcade is closely related to the total
helicity. Comparing two fields with the same helicity butwith different sheared region widths, the growth rate in thefield whose sheared region is the most narrow should be thelargest. In fact, Figure 3 shows that the interval of theeruption becomes shorter from case B to case E. This result
is consistent with that of Kusano et al. [1995].

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A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

Figure 6. (a) Time profile of total magnetic helicity, (b) total magnetic energy, and (c) total kinetic
energy in each case. Also shown is (d,e) the total magnetic energy and the kinetic energy relative to thetotal helicity. The unit of energy is 1.88  1032 erg. Also shown is (f) the maximum current density in thevertical current sheet relative to the total helicity.

10 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05

A03S05

Table 1. Fraction of the Magnetic Flux of the Sheared Arcade, the Overlying Arcade, its Square, and the Total Helicity When the
Simulation is Stopped
Case

Sheared Arcade

Overlying Arcade

Overlying Arcade2

Final Helicity

A
B
C
D
E

5.59
1.06
0.76
0.60
0.34

0
4.53
4.83
4.99
5.25

0
20.52
23.33
24.90
27.56

14.72
12.11
10.88
10.59
5.03

[37] However, when the area of the sheared region issmall, the amount of the overlying flux is large. As
discussed above, the amount of overlying flux contributes
to confine the eruption and stabilize the newly formed flux
rope as the result of reconnection. Therefore the flux rope
does not erupt but finds a new equilibrium state in the lower
corona. The necessary condition for this to occur can be
determined by the relation between the magnetic helicity
and the square of the overlying flux. Hence there is a kindof trade-off between the condition for the tearing instability
and that for the eruption of a CME. Consequently, a vast
amount of helicity is needed for eruption in an axisymmetricsystem. In the real solar corona the amount of magnetic
helicity injected into the corona is much smaller than thatused in this simulation [Kusano et al., 2002; Maeshiro etal., 2005; Yamamoto et al., 2005]. This suggests that it may
be difficult to trigger a CME just by the axisymmetricshearing motion and that some other mechanisms should be
involved in the trigger process of CME, such as helicity
annihilation [Kusano et al., 2004], flux cancellation, or
emergence [Chen and Shibata, 2000].
[38] From the point of view of magnetic flux, we compare
our results with those of other works. Note that theirconcepts are different from this study because these studies
concentrated on only the way to cause CMEs. However, thecomparisons give us important suggestions.

[39] Mikic and Linker [1994] performed an early study ofarcade shearing in the spherical geometry. In the study, theinitial magnetic field is a dipole field and shearing motion is
imposed on a localized region beside the equator. It is
common to our studies that the shearing motion is imposed
only in the inner loops of the initial magnetic field, but asignificant difference is that the initial magnetic field is the
dipole field. Our results suggest that the confinement ofsheared flux by an overlying magnetic flux is an important
process for determining whether a CME can occur. However, in the case of the dipole field, the spatial scale gap
between the overlying unsheared magnetic flux and thesheared flux is so large (comparable to the solar radius) thatmagnetic tension force of the unsheared flux is muchsmaller than our cases. Thus the spatial scale of the region
involved in the CME may be one of the key factors fordetermining the CME occurrence condition.
[40] The breakout model [Antiochos et al., 1999;
MacNeice et al., 2004] is also an important model of
CME initiation. In these studies, the initial magnetic
field is a quadrupole field and a shearing motion is
imposed on the inner region beside the equatorial inversion
line where the magnetic field radial component becomes 0.
As shear is injected to the inner arcade, it swells upward,
similar to our results and others. The essential point of themodel is that the direction of the magnetic field further above
the inner arcade is opposite so that reconnection can strip themagnetic flux which confines the innermost sheared flux.
This result and our results suggest that the ambient magnetic
field structure may be also one of key factors for determining
CME occurrence condition. By the same reason, the effect ofopen field due to the solar wind should be taken into account.
4.3. Effect of Resistivity[41] Electric resistivity is another factor governing thetearing instability. The fastest growth rate is
w¼

Figure 7. Variations of height of the separatrix between
the sheared and unsheared portions of the arcade (shown by
diamond symbol) and the center of the central current sheet(shown by cross symbol) versus the total magnetic helicity.

1
td tA

1=2
;

ð44Þ

where t d = l2/h and t A = l/VA are the diffusion and Alfvéntimescales, and l is the width of the current sheet. Therefore
in a case with a larger resistivity, the tearing instability willgrow faster. This finding is confirmed by the fact theinterval for the reconnection events in case G is shorter thanin case D.
[42] On the other hand, the larger resistivity can preventthe formation of a current sheet. A current sheet is formedby the thinning of a current containing a sheared arcade.
However, if the current diffusion due to resistivity is faster11 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

Figure 8. (a) Time profile of total magnetic helicity, (b) total magnetic energy, and (c) total kinetic
energy in each case. Also shown are (d,e) the total magnetic energy and the kinetic energy relative to thetotal helicity. The unit of energy is 1.88  1032 erg. Also shown is (f) the maximum current density in thevertical current sheet relative to the total helicity.

12 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

A03S05than the current enhancement due to thinning, the current
sheet is never formed, as in case F.
[43] In cases A to E we set explicit resistivity to be 0, andtherefore the evolutions are caused due only to the effective
resistivity, numerical diffusion. In cases F, G, and H theresistivity values are set to be 104, 105, and 106,respectively. The evolution in case D is almost the sameas that in case H, while it is a little different from that in case
G. This result suggests that the effective resistivity due tonumerical diffusion is larger than 106 and less than 105.
[44] In our simulation (case G for example) the magnetic
Reynolds number is Rm = VA L/h  1/h  105 for the spatial
scale of the active region 0.3 R . In the solar corona,considering just the Spitzer type resistivity, the magnetic
Reynolds number is 1014 for an active region. Furthermore,
Lin et al. [2007] derived the electric resistivity from therecent observations of reconnecting current sheets [Ko etal., 2003; Lin et al., 2005]. The obtained magnetic
diffusivities of quiet corona and anomalous resistivity inthe cgs units hdif = 4 phe/c are
hdif ;c ¼ ð1:7  4:6Þ  106 cm2 s1ð45Þ

and hdif,a = (1.1  4.2)  102 cm2s1. The corresponding
magnetic Reynolds numbers Rm = VAL/hdif  csR /h are
Rm,c = (7.6  2.8)  1023 cm2s1 and Rm,a = (12.  3.1) 
1015 cm2s1 for the spatial scale of the active region. Hencethe resistivity in this study is much larger than that in thesolar corona because of the following limitation of thenumerical simulation. In the numerical simulation the widthof the current sheet is limited to the minimum grid scale.
This limitation makes the numerical diffusion large, so thetearing instability grows easily. Future investigations withdifferent resolutions will be necessary to extrapolate ourresults to real coronal conditions. We need more understanding about the anomalous resistivity which might becaused by a plasma microinstability.

5. Conclusion[45] We have investigated the quantitative relations between arcade shearing, flux rope formation, and eruption
with a 2.5-dimensional axisymmetric MHD simulation. We
studied several cases with different velocity profiles of theshearing motion and with different resistivity value cases. In
the case in which all magnetic flux is sheared, a verticalcurrent sheet is formed, and then reconnection due totearing-mode instability in the sheet forms helical flux ropeand leads to an eruption.
[46] On the other hand, in the cases in which only the
inner magnetic flux is sheared, a different structure isformed. This structure consists of both sheared andunsheared flux, and it has a vertical current sheet. As theresults of reconnection in the current sheet, a helical flux
rope is formed, but this flux rope remains in an equilibrium
state and is not ejected. Further injection of shear leads torepeating eruptions of the sheared arcade, which feedtoroidal flux to the flux rope. As the results, although theflux rope was elevated gradually, the flux rope has not beenejected as a CME until the end of the simulation. The results
suggest that it may be difficult to trigger a CME just by the

A03S05axisymmetric shearing motion and that some other mechanisms could be involved in the trigger process of a CME.
[47] Resistive cases have also been studied. The results
suggest that a sufficiently large resistivity can prevent thecurrent sheet from forming. However, somewhat smaller but
still large values enhance the growth rate of the tearing
instability.
[48] We examined the relations among evolution, magnetic
helicity magnetic energy, and fraction of magnetic flux. It isfound that the amount of magnetic helicity is closely related
to the topology of the shear configuration and the magnitudeof kinetic energy released in a confined eruption.
[49] The parameter ranges in this study are insufficient tomake clear the quantitative relations. In order to extrapolate
our results to real coronal conditions, we will investigate theissue with wider parameter range and different resolutions
in our future studies.
[50] Acknowledgments. The numerical calculation was carried out,
in part, using the Earth Simulator (ES) in Japan Agency for Marine-Earth
Science and Technology (JAMSTEC), under the ES project ‘‘Developmentof Macro-Micro Interlocked Simulation Algorithm’’ (PI: K. Kusano). The
other part of the calculation was carried out on a VPP5000 at the Center for
Computational Astrophysics (CfCA) of the National Astronomical Observatory of Japan. 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, and by the Grant-in-Aid
for Scientific Research (B) (19340180, Head Investigator: K. Kusano) fromthe Japan Society for the Promotion of Science (JSPS).
[51] Amitava Bhattacharjee thanks Terry Forbes and another reviewer
for their assistance in evaluating this paper.

References
Altschuler, M. D., and G. Newkirk (1969), Magnetic fields and the structureof the solar corona. I: Methods of calculating coronal fields, Sol. Phys., 9,
131 – 149, doi:10.1007/BF00145734.
Amari, T., J. F. Luciani, J. J. Aly, Z. Mikic, and J. Linker (2003), Coronalmass ejection: Initiation, magnetic helicity, and flux ropes. I. Boundary
motion-driven evolution, Astrophys. J., 585, 1073 – 1086, doi:10.1086/
345501.
Antiochos, S. K., C. R. DeVore, and J. A. Klimchuk (1999), A model forsolar coronal mass ejections, Astrophys. J., 510, 485 – 493, doi:10.1086/
306563.
Berger, M. A., and G. B. Field (1984), The topological properties of magnetic helicity, J. Fluid Mech., 147, 133 – 148.
Chen, P. F., and K. Shibata (2000), An emerging flux trigger mechanism forcoronal mass ejections, Astrophys. J., 545, 524 – 531, doi:10.1086/
317803.
Choe, G. S., and C. Z. Cheng (2000), A model of solar flares and theirhomologous behavior, Astrophys. J., 541, 449 – 467, doi:10.1086/309415.
Choe, G. S., and L. C. Lee (1996), Evolution of solar magnetic arcades. II.
Effect of resistivity and solar eruptive processes, Astrophys. J., 472,
372,doi:10.1086/178070.
Dedner, A., F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg(2002), Hyperbolic divergence cleaning for the MHD equations, J. Comput.
Phys., 175, 645– 673.
Forbes, T. G. (2000), A review on the genesis of coronal mass ejections,J. Geophys. Res., 105, 23,153 – 23,166.
Forbes, T. G., and E. R. Priest (1995), Photospheric magnetic field evolution and eruptive flares, Astrophys. J., 446, 377,doi:10.1086/175797.
Forbes, T. G., et al. (2006), CME theory and models, Space Sci. Rev., 123,
251 – 302, doi:10.1007/s11214-006-9019-8.
Hundhausen, A. (1999), Coronal mass ejections, in The Many Faces of theSun: A Summary of the Results From NASA’s Solar Maximum Mission,edited by K. T. Strong et al., p. 143, Springer, New York.
Inhester, B., J. Birn, and M. Hesse (1992), The evolution of line-tied
coronal arcades including a converging footpoint motion, Sol. Phys.,138, 257 – 281.
Kliem, B., and T. Török (2006), Torus instability, Phys. Rev. Lett., 96(25),255,002, doi:10.1103/PhysRevLett.96.255002.
Ko, Y.-K., J. C. Raymond, J. Lin, G. Lawrence, J. Li, and A. Fludra (2003),
Dynamical and physical properties of a post-coronal mass ejection current sheet, Astrophys. J., 594, 1068 – 1084, doi:10.1086/376982.

13 of 14

A03S05SHIOTA ET AL.: MHD STUDY OF ARCADE SHEARING

Kusano, K., Y. Suzuki, and K. Nishikawa (1995), A solar flare triggering
mechanism based on the Woltjer-Taylor minimum energy principle,
Astrophys. J., 441, 942 – 951, doi:10.1086/175413.
Kusano, K., T. Maeshiro, T. Yokoyama, and T. Sakurai (2002), Measurement of magnetic helicity injection and free energy loading into the solar
corona, Astrophys. J., 577, 501 – 512, doi:10.1086/342171.
Kusano, K., T. Maeshiro, T. Yokoyama, and T. Sakurai (2004), The trigger
mechanism of solar flares in a coronal arcade with reversed magnetic
shear, Astrophys. J., 610, 537 – 549, doi:10.1086/421547.
Lin, J., and T. G. Forbes (2000), Effects of reconnection on the coronalmass ejection process, J. Geophys. Res., 105, 2375 – 2392.
Lin, J., W. Soon, and S. L. Baliunas (2003), Theories of solar eruptions: A
review, New Astron. Rev., 47, 53 – 84.
Lin, J., Y.-K. Ko, L. Sui, J. C. Raymond, G. A. Stenborg, Y. Jiang, S. Zhao,
and S. Mancuso (2005), Direct observations of the magnetic reconnection
site of an eruption on 2003 November 18, Astrophys. J., 622, 1251 – 1264,
doi:10.1086/428110.
Lin, J., J. Li, T. G. Forbes, Y.-K. Ko, J. C. Raymond, and A. Vourlidas(2007), Features and properties of coronal mass ejection/flare current
sheets, Astrophys. J., 658, L123 – L126, doi:10.1086/515568.
Linker, J. A., and Z. Mikic (1995), Disruption of a helmet streamer byphotospheric shear, Astrophys. J., 438, L45 – L48, doi:10.1086/187711.
Mackay, D. H., and A. A. van Ballegooijen (2005), New results in modelingthe hemispheric pattern of solar filaments, Astrophys. J., 621, L77 – L80,
doi:10.1086/428904.
Mackay, D. H., and A. A. van Ballegooijen (2006a), Models of the largescale corona. I. Formation, evolution, and liftoff of magnetic flux ropes,
Astrophys. J., 641, 577 – 589, doi:10.1086/500425.
Mackay, D. H., and A. A. van Ballegooijen (2006b), Models of the largescale corona. II. Magnetic connectivity and open flux variation, Astrophys.
J., 642, 1193 – 1204, doi:10.1086/501043.
MacNeice, P., S. K. Antiochos, A. Phillips, D. S. Spicer, C. R. DeVore, andK. Olson (2004), A numerical study of the breakout model for coronalmass ejection initiation, Astrophys. J., 614, 1028 – 1041, doi:10.1086/
423887.
Maeshiro, T., K. Kusano, T. Yokoyama, and T. Sakurai (2005), A statistical
study of the correlation between magnetic helicity injection and soft X-ray
activity in solar active regions, Astrophys. J., 620, 1069 – 1084,
doi:10.1086/426806.
Magara, T. (2006), Dynamic and topological features of photospheric andcoronal activities produced by flux emergence in the Sun, Astrophys. J.,
653, 1499 – 1509, doi:10.1086/508926.

A03S05Manchester, W., IV, T. Gombosi, D. DeZeeuw, and Y. Fan (2004), Eruption of
a buoyantly emerging magnetic flux rope, Astrophys. J., 610, 588 – 596,
doi:10.1086/421516.
Mikic, Z., and J. A. Linker (1994), Disruption of coronal magnetic fieldarcades, Astrophys. J., 430, 898 – 912, doi:10.1086/174460.
Mikic, Z., D. C. Barnes, and D. D. Schnack (1988), Dynamical evolution ofa solar coronal magnetic field arcade, Astrophys. J., 328, 830 – 847,
doi:10.1086/166341.
Miyoshi, T., and K. Kusano (2005), A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208,
315 – 344, doi:10.1016/j.jcp.2005.02.017.
Priest, E., and T. Forbes (2000), Magnetic Reconnection, 612 pp., Cambridge Univ. Press, Cambridge, UK.
Priest, E. R., and T. G. Forbes (2002), The magnetic nature of solar flares,
Astron. Astrophys. Rev., 10, 313 – 377, doi:10.1007/s001590100013.
Riley, P., R. Lionello, Z. Mikiæ, J. Linker, E. Clark, J. Lin, and Y.-K. Ko(2007), ‘‘Bursty’’ reconnection following solar eruptions: MHD simulations and comparison with observations, Astrophys. J., 655, 591 – 597,
doi:10.1086/509913.
Shibata, K., and S. Tanuma (2001), Plasmoid-induced-reconnection andfractal reconnection, Earth Planets Space, 53, 473 – 482.
Steinolfson, R. S. (1991), Coronal evolution due to shear motion, Astrophys.
J., 382, 677 – 687, doi:10.1086/170755.
van Ballegooijen, A. A., and P. C. H. Martens (1989), Formation and eruptionof solar prominences, Astrophys. J., 343, 971 – 984, doi:10.1086/167766.
Yamamoto, T. T., K. Kusano, T. Maeshiro, T. Yokoyama, and T. Sakurai(2005), Magnetic helicity injection and sigmoidal coronal loops, Astrophys. J., 624, 1072 – 1079, doi:10.1086/429363.


K. Kusano, Earth Simulator Center, Japan Agency for Marine-Earth
Science and Technology, Yokohama, Kanagawa, Japan.
T. Miyoshi, Facility of Science, Hiroshima University, Higashi-Hiroshima,
Hiroshima, Japan.
N. Nishikawa, Super Computer System Planning and Operations
Department, Japan Agency for Marine-Earth Science and Technology,
Yokohama, Kanagawa, Japan.
K. Shibata, Kwasan and Hida Observatories, Kyoto University,
Yamashina, Kyoto, Japan.
D. Shiota, Center for Computational Astrophysics (CfCA), National
Astronomical Observatory of Japan, 2-21-1 Ohsawa, Mitaka, Tokyo, Japan
181-8588. (shiota@cfca.jp)

14 of 14

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A10110, doi:10.1029/2007JA012513, 2007

for

Full
ArticleParametric decay of circularly polarized Alfvénwaves in the radially expanding solar wind
Shin Tanaka,1 Tatsuki Ogino,1 and Takayuki Umeda1
Received 1 May 2007; revised 24 June 2007; accepted 7 August 2007; published 30 October 2007.

[1] We present a magnetohydrodynamic (MHD) simulation study of the parametric decayof the circularly polarized Alfvén waves propagating in the radial outflow of the solar
wind. Assuming the transonic wind solution as an initial condition, we continuously
injected monochromatic circularly polarized Alfvén waves from the inner boundary at
the lower corona, and simulated the wave propagation. The injected Alfvén waves are
subject to the parametric decay, and density fluctuations in the solar wind plasma growrapidly at a specific region. The location of the most unstable region depends on theamplitude and frequency of injected Alfvén waves. We found that the unstable region ofsimulation results can be well estimated by considering a localized dispersion relation in
the frame of reference moving with the background solar wind.
Citation: Tanaka, S., T. Ogino, and T. Umeda (2007), Parametric decay of circularly polarized Alfvén waves in the radiallyexpanding solar wind, J. Geophys. Res., 112, A10110, doi:10.1029/2007JA012513.

1. Introduction[2] Alfvén waves are ubiquitous phenomena in spaceplasmas, and are commonly observed in various situations
such as the solar wind [Belcher and Davis, 1971] and theinterplanetary shocks [Agim et al., 1995]. Since Alfvénwaves are non-compressional transverse magnetohydrodynamic (MHD) modes, they can propagate a long distancecompared to compressional MHD modes which can easily
damp. Because of this nature, Alfvén waves are thought
to propagate along the open magnetic flux tube in apolar coronal hole and to play an important role in theacceleration of the high-speed solar wind.
[3] It is well known that circularly polarized Alfvénwaves with a finite amplitude are an exact solution to thenonlinear MHD equations, and are unstable to a decay
process in the presence of the density fluctuation [Sagdeev
and Galeev, 1969]. The pump Alfvén waves decay into
the forward-propagating sound waves and the backwardpropagating Alfvén waves due to a nonlinear three-wave
coupling. Goldstein [1978] and Derby [1978] derived thegeneral dispersion relation for fundamental modes from
ideal MHD equations with finite b. Recent observation of
the solar wind suggests the presence of outward- andinward-propagating Alfvén waves [Bavassano et al.,2000]. The parametric decay was thought to be one oftriggering mechanisms of inward-traveling Alfvénic modes
and an MHD turbulence in the solar wind.
[4] Recently, several one- and multidimensional MHD
simulations have been performed to examine the parametricdecay of Alfvén waves in a homogeneous medium with1
Solar-Terrestrial Environment Laboratory, Nagoya University, Aichi,
Japan.

both the periodic and open boundaries [Del Zanna et al.,
2001; Malara et al., 2000; Ghosh and Goldstein, 1994;
Ghosh et al., 1994]. Pruneti and Velli [1997] and Turkmani
and Torkelsson [2004] consider the propagation of Alfvénwaves in a gravitationally stratified medium in a slab
geometry. In their simulations, Alfvén waves are injectedfrom the one side of boundaries in open systems. However,they assumed a static background medium as an initial state.
[5] The propagation of Alfvén waves in the solar wind
has been extensively studied by many authors throughnumerical simulations in association with the coronal heating
and wind acceleration [Ofman, 2004; Ofman and Davila,
1998; Lau and Siregar, 1996; Buti et al., 1998; Buti et al.,1999]. Suzuki and Inutsuka [2006] performed an MHD
simulation from the photosphere toward the solar wind,and concluded that low-frequency Alfvén waves can establish both the coronal high temperature and the high-speed
solar wind. Most of these studies focused on a quasi-steadystate after injected Alfvén waves and compressive waves
were saturated through linear and nonlinear processes suchas wave steepening, reflection, and possibly parametricdecay.
[6] In the present study, we focus on a growing stage ofthe parametric decay of Alfvén waves in the radiallyexpanding medium before the quasi-steady state. Circularly
polarized monochromatic Alfvén waves are injected into aspherically symmetric transonic wind [Parker, 1963]. The
injected Alfvén waves propagate outward along the radiallyexpanding flow tube of the solar wind. When the amplitude
is sufficiently large, Alfvén waves are subject to theparametric decay, generating outward-propagating soundwaves and inward-propagating Alfvén waves. We analyze
the dispersion relation to determine the heliocentric distanceat which the compressive fluctuation grows most rapidly by
the parametric decay.

Copyright 2007 by the American Geophysical Union.
0148-0227/07/2007JA012513$09.00

A101101 of 7TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

A10110[7] The paper is organized as follows. In Section 2, we
introduce the model equation of Alfvén waves in the solar
wind, the initial and boundary condition. In Section 3, weshow the results of MHD numerical simulations for a few
parameters. In Section 4, we discuss the parametric decay
for numerical results and compare with the linear theory.

2. Simulation Model[8] In order to model the radial expansion of the solar
wind, we solve the following ideal MHD equations in a
one dimensional spherical coordinate in a heliocentric
distance r:@r0 @ 0þ ðr vr Þ ¼ 0
@t @r
@ 0
@  0 2@jB0 j2
ðr vr Þ þr vr ¼ 
p0 þ ?
2m0
@t
@r
@r

ð1Þ

!
þ r0

jv0? j2 2p0
þþ r0 gr3r
ð2Þ

@ 0 0  @  0 0 @  0 r v? þ
vr r v? ¼
CB?
@t
@r
@r

ð3Þ

"
!#

@e0 @
jB0? j2@   000¼vr e þ p þ
Br v? B0? þ r0 vr g
þ
@r
@t @r
2m0
ð4Þ

@B0?
@  0
¼
vr B?  Br v0?
@t
@r

ð5Þ

where r0 r2r, p0 r2p, e0 r2e, v0? rv? and B0? rB?
with r, p, vr, v?, B? and m0 being the density, pressure,radial component of velocity, tangential component ofvelocity, tangential component of magnetic field, andmagnetic permeability in vacuum, respectively. Here g =GMS/r2 is a gravitational acceleration where G and MS are agravitational constant and a solar mass, respectively, and
2

e¼

r 2
p
jB? j
;
v þ jv? j2 þ
þ
2 r
g1
2m0

ð6Þ

is the total energy of the fluid, where g is the polytropic
index. The radial component of the magnetic field Br isgiven by Br = Cr2 where C is a constant.
[9] We solve Equations (1) – (5) with the PIC-MOCCT
scheme [S. Tanaka et al., submitted to Comput. Phys.
Commun.], which consists of the PIC scheme [T. Umeda,
submitted to Comput. Phys. Commun.] and the MOCCT[Hawley and Stone, 1995]. The PIC scheme is a nonoscillatory, positivity preserving and conservative numericalinterpolation scheme for solving one-dimensional advection
equations, which is used to solve advection phases of the
MHD equations in the left hand side of Equations (1)– (4).
The MOCCT solves the characteristics of Alfvén waves,maintaining r B = 0, which is applied to the induction

A10110Equation (5) and terms for the magnetic stress, i.e., the righthand side of Equation (3) and the first term in the right handside of Equation (4). In the present study, we adopted the
PIC-MOCCT scheme is because this scheme does not need to
introduce an additional diffusion term to suppress numericaloscillations. This scheme can treat Alfvén waves moreaccurately than previous non-oscillatory numerical schemesfor MHD equations such as TVD schemes [e.g., Fukuda
and Hanawa, 1999; Tanaka, 1994].
[10] The simulation domain is taken from r = 1 RS to r =40 RS on an uniform mesh with 16384 grid points. As an
initial condition, we used a steady transonic wind solution[Parker, 1963] for r, p and vr. We used the following solar
wind parameters: T0 = 1.8  106 K, n0 = r0/m = 1.0  108cm3 and Br0 = 5.0 G, where T0, n0, Br0, r0 and m are
the temperature, number density, radial component of themagnetic field, density at the lower boundary (r = 1 RS) anda proton mass, respectively. We imposed a very small
random initial noise (105 to the initial backgroundprofiles) to the initial density, pressure and radial velocity
to give compressive fluctuations.
[11] In the present study we set g = 1.1. Note that theproperties of the wind solution for the initial condition
depend on g. With g = 1.1 we have the radial velocity vr 
300 km/s at the upper boundary r = 40 RS, while we havevr  400 km/s with g = 1.05. The previous coronal models
used, for example, g = 1.05 [e.g., Linker et al., 1990; Linker
et al., 1999]. We have performed several test runs withdifferent g, and found that physical processes were essentially the same.
bulk[12] Figure 1 shows the initial profiles of thepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
velocity of the solar wind vr, sound speed vS = gp=r,pﬃﬃﬃﬃﬃﬃﬃﬃ
and Alfvén speed vA = Br/ m0 r, respectively.
[13] We impose the following boundary conditions for thetangential component of the velocity and magnetic field toinject the monochromatic circularly polarized Alfvén waves
at the lower boundary (r = 1 RS):
v? ¼ dv0 sin w0 t eq þ cos w0 t ef

ð7Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
B? ¼ v? m0 r0

ð8Þ

where dv0, w0, and r0 are the velocity amplitude, frequency
of injected Alfvén waves, and the density at the lower
boundary, respectively. Here, eq and ef are the unit vectorsof tangential components. Equations (7) and (8) give therelationship between the velocity amplitude and magnetic
field amplitude of the outward-propagating Alfvén waves.
Note that since we use ideal MHD equations the circularpolarization does not affect the present simulation results.
[14] The generation mechanisms of Alfvén waves are
expected to be due to solar activities near the solar surface.
Cranmer and Ballegooijen [2005] consider the turbulent
motion of the foot point of the solar magnetic field as asource of waves. Parhi et al. [1997] simulated a propagationof MHD waves which are constantly driven by photosphericfoot point motions in the context of the coronal heating.
Sturrock [1999] suggests that magnetic reconnections in thechromosphere can generate MHD waves.

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A10110TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

Figure 1. Initial profiles of the radial velocity of the solar
wind vr (solid line), sound speed vS (dashed line), Alfvénspeed vA (dashed-dotted line), and plasma b (dotted line).
The velocities vr, vS and vA are normalized by the soundspeed at the lower boundary, r = 1 RS.
[15] The lower boundary condition for r, e, and vr are
fixed at the initial values. Note that r, e and vr should bedetermined from the characteristic equations to avoid anunphysical wave reflection [Steinolfson and Nakagawa,
1976]. However, we confirmed that the wave reflection at
the lower boundary is less than 3% for the present parameters. At the upper boundary (r = 40 RS), we impose theopen boundary condition where the spatial derivatives of all
the variables are set to be zero. Since the bulk velocity ofthe solar wind exceeds both the sound speed and the Alfvénspeed near the upper boundary as shown in Figure 1,
unphysical reflections, which might occur at the upperboundary, is considered to be negligibly small.

3. Simulation Results[16] In the present study, the amplitude and frequency of
injected Alfvén waves are varied as different simulation
runs. Detailed parameters are shown in Table 1.
[17] Figure 2 shows the profiles of the number density n(a), the radial velocity components vr (b), and the ratio of
the tangential magnetic field component to the ambient
magnetic field jB?j/Br (c) at time t = 430 min. for Run A,
respectively. As injected Alfvén waves propagate outward, vr and r become larger than their initial values.
At r  30 RS, we found discontinuities in vr and n. These
are generated by the outward-directed magnetic pressure
due to the discontinuous variation of the magnetic field
amplitude at the wavefront shown in Figure 2c. Aroundr  18 RS, there appear fluctuations in the velocity, density,and the tangential magnetic field. These fluctuations propagate both outward and inward as they grow. This indicatesthe parametric decay, in which outward-propagating soundwaves and inward-propagating Alfvén waves are excited by
the large-amplitude Alfvén waves.
[18] To confirm the inward-propagating Alfvén waves,
we plot the q component of the Elsässer vectors, Z±q =pﬃﬃﬃﬃﬃﬃﬃﬃ
vq Bq/ m0 r in Figures 2d and 2e. The Elsässer vectors Z+q
and Zq correspond to amplitude of the outward- and inward-

A10110propagating Alfvén waves, respectively. The profile of Z
qshows the apparent generation of inward-propagating
Alfvén waves in the unstable region.
[19] There are two possible mechanisms to generate theinward-propagating Alfvén waves. One is the parametricdecay and the other is the wave reflection due to theinhomogeneity of background medium. The wave reflection
becomes important when the frequency of Alfvén waves w0
is smaller than the spatial gradient of the Alfvén speed
dvA/dr [An et al., 1990]. Figure 2e shows that there also
exist inward-propagating components outside the unstableregion. However, the frequency of injected Alfvén waves
are w0  10 mHz in Runs A-C, while the maximum valueof dvA/dr is 1.4 mHz. Thus the amplitude of inwardpropagating waves generated by the reflection is considerably small compared to that by the parametric decay. It isalso noted that when Alfvén waves are linearly polarized,
the variation of the magnetic pressure can generate secondary compressive modes [e.g., Hollweg, 1971] which leads toa reflection of Alfvén waves. However, since we inject
circularly polarized Alfvén waves, the compressive motions
are negligibly small compared to the transverse motion until
the parametric decay evolves.
[20] Note that the beat instability is also a possible
mechanism to generate inward-propagating Alfvén waves
in an ideal MHD system for b > 1 [Jayanti and Hollweg,
1993]. The beat instability involves the interaction between
the forward-propagating pump Alfvén waves and backward-propagating lower sideband Alfvén waves. The beatinstability does not occur in the present simulations becauseb < 1 as shown in Figure 1.
[21] To confirm the parametric decay, in Figure 3, we plot
the w  k spectrum with a gray scale obtained by Fouriertransformation of simulation data in the range of 15.56 RS <r < 20.44 RS and 319 min < t < 474 min for (Bq, Bf) and vr.
This range of r covers the most unstable region as seen in
Figure 2, and the range of t covers from an early stage witha steady flow to the saturation stage in which the growing
compressive modes have sufficiently large amplitude. We
found the injected Alfvén waves at K 10 R1
S , W  11 mHz
in Figure 3a, outward-propagating sound waves at kS 
15 R1
S , wS  8 mHz in Figure 3b, and inward-propagating
Alfvén waves at kA  5 R1
S , wA = 2 mHz in Figure 3a.
Note that the frequency in the w  k spectra is Dopplershifted, i.e., w = w0 + kvr, where w and w0 represent thefrequency of waves observed from the rest frame and fromthe moving reference frame with the solar wind velocity vr,
respectively. We found that observed three waves satisfied
the resonant condition K = kS + kA, W = wS + wA for theparametric decay [Sagdeev and Galeev, 1969] by taking
into account the Doppler shift with the radial velocity vr 
240 km/s in the range of the spectrum data (15.56 RS < r <20.44 RS). We also found another wave mode at kA  25 R1S
and wA  19 mHz in Figure 3a. This wave mode satisfies
Table 1. Parameters of Injected Alfvén Waves for Different
Simulation Runs
Run

d v0

w0

A
B
C

30 km/s
20 km/s
30 km/s

10 mHz
10 mHz
20 mHz3 of 7

A10110TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

another resonant condition kA = kS + K and wA = wS + W,
which is associated with the excitation of the outwardpropagating magnetic fluctuation. Since this wave is not a
normal mode, the power of this wave is much smaller thanthat of inward-propagating waves [Goldstein, 1978].

A10110[22] Figure 4 shows the profile of vr for Runs B and C. In
both cases, the injected Alfvén waves generate the fluctuations in a quite similar manner to Run A, and we confirmed
the parametric decay (not shown). However, the positions of
the unstable region are different from each other. In Run
B the amplitude of injected Alfvén waves is smaller than in
Run A. Then the position of unstable region becomes
farther from the lower boundary. In Run C, on the other
hand, the frequency of injected Alfvén waves is higher thanin Run A, Then the position of unstable region becomescloser to the lower boundary. From these results, it isexpected that the position of the unstable region becomescloser to the lower boundary as the amplitude or frequency
of injected Alfvén waves becomes larger.

4. Discussion[23] The present simulation results suggest that compressive fluctuation grows most rapidly at a localized region by
the parametric decay. It is known that a larger amplitude of
Alfvén waves or a lower b lead to higher growth rate of theparametric decay [Goldstein, 1978; Derby, 1978]. In thepresent case the profile of the normalized amplitude of Alfvénwaves b B?0/Br increases as the injected Alfvén waves
propagate outward while the local beta value b
v2S/v2A
decreases (see Figures 1 and 2c). These profiles imply that thegrowth rate of the parametric decay has a maximum value at a
certain position.
[24] We discuss the location and wave number of themost rapidly growing compressive mode due to the parametric decay instability. The dispersion relation for theparametric decay instability in a homogeneous plasma was
first derived by Goldstein [1978] and Derby [1978]. To
consider the dispersion equation in a inhomogeneous background as in a present simulation, we assume so-called
Wentzel-Kramers-Brillouin (WKB) approximation in which
the wavelength of Alfvén waves is small compared to thescale height of the background. We computed the pressure
scale height H = p(@p/@r)1 and the wavelength of injected
Alfvén waves l = 2p (vA + vr)/w0. We found that the
wavelength is shorter than the pressure scale height forr > 4 RS, and is comparable for r < 4 RS, indicating that thecondition of the WKB approximation is well satisfied forr > 4 RS.
[25] In the WKB approximation, we can consider thedispersion equation for a homogeneous medium as a local
dispersion equation at each positions. Let W0 and K be thefrequency and wave number of injected Alfvén waves,which is regarded as the pump wave. Then, we have thelocal dispersion relation for fundamental modes induced by
the injected circularly polarized Alfvén waves with the

Figure 2. Results of Run A at t = 430 min. The radial
dependence of (a) radial velocity vr (km/s), (b) number
density n (cm3), (c) magnitude of the tangential magnetic
field jB?j normalized by local radial magnetic field Br,
and (d)(e) q component of Elsässer vectors Zq± = vq Bq/
pﬃﬃﬃﬃﬃﬃﬃﬃ
m0 r (km/s) are shown. Note the compressive fluctuations
and inward-propagating Alfvén waves growing around r 
18 RS.
4 of 7

A10110TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

A10110

Doppler shift as, w0 = w  kvr and W0 = W  Kvr where w
and W are the frequency of compressive fluctuations andinjected Alfvén waves observed in the rest frame,
respectively. Note that vS, vA, vr, b, K, and W0 are functionsof r.
[26] We solved Equation (8) numerically for a complex
solution w = wr + ig which exhibits an unstable compressivemode, and obtained the growth rate g at each wave number
k and each position r. The values of vS, vA, vr, and b, inEquation (8) are given as a function of r. Note that W isassumed to be equal to w0 which is a frequency of theinjection of Alfvén waves at the lower boundary and isconstant anywhere in a simulation domain while K varies
with r. Then the wave number K for injected Alfvén waves
in the rest frame is given as K = W/(vA + vr). By solving
Equation (8) as described above, we have the locallydefined growth rate g(r, k) for the parametric decay instability.
[27] In the present simulation the pump Alfvén waves are
injected into the background medium which is initially asteady solar wind outflow with the bulk velocity vr(r). It is
expected that the fluid element of the solar wind at a certain
distance undergoes injected waves at a certain time, andthen the density fluctuation in the fluid element will begin togrow due to the instability. We assume that this density

Figure 3. The w-k diagram of simulation data in the rangeof 15.56 RS < r < 20.44 RS and 319 min < t < 474 min. The
horizontal and vertical axes indicate k(RS1) and w(mHz),
respectively. Positive (negative) k correspond to outward(inward-) propagating waves. (a) Fourier amplitude
jB?(k, w)j plotted with a gray scale. Solid line and dottedline indicate the dispersion relation of outward- andinward-propagating Alfvén waves observed in the rest
frame w = (vr + vA)k and w = (vr  vA)k, respectively, where
the solar wind velocity and Alfvén speed are vr = 240 km/s
and vA = 500 km/s, respectively. (b) Fourier amplitude
jvr(k, w)j plotted with a gray scale. Solid line indicates thedispersion relation of outward-propagating sound waves in
the rest frame w = (vr + vS)k where the sound speed is vS =100 km/s. These results show that excited modes satisfy theresonant condition of the parametric decay(see text).
finite amplitude in the solar wind frame [Goldstein, 1978;
Derby, 1978], 02
w  v2S k 2 ðw0  vA k Þðw0 þ vA k þ 2W0 Þðw0 þ vA k  2W0 Þ
¼ b2 v2A k 2 w03 þ w02 vA k  3W02 w0 þ W02 vA k

ð9Þ

where w0 and k are the frequency and wave number ofcompressive fluctuations, respectively. Here, vS and vA are
the sound speed and Alfvén speed of the solar wind plasma,
respectively, and b is a normalized magnetic field amplitudedefined as b
B?/Br where B? and Br are the magnetic
field amplitude of injected Alfvén waves and the background radial magnetic field, respectively. The frequency w0
and W0 are observed in the solar wind frame moving at thevelocity vr, and are expressed by using the relation of the

Figure 4. The radial dependence of the radial velocity vr.
(a) Result of Run B (a case of a smaller dv0 than Run A) at
t = 758 min. The discontinuity due to the magnetic pressure
as shown in Figure 2 does not exist because it propagatedout of the simulation domain. (b) Result of Run C (a caseof a larger w0 than Run A) at t = 257 min. These results
indicate that a larger amplitude and frequency of injected
Alfvén waves make the most unstable region closer to thelower boundary.

5 of 7TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

A10110

A10110Note that A(r, k) is a non-dimensional value corresponding
to an effective amplification factor of the compressive
fluctuation for a specific wave number k.
[28] We plot A(r, k) with the gray scale for Runs A-C in
Figures 5a – 5c. These figures correspond to the spatial
evolution of the amplitude of the compressive fluctuationfor various wave number in the solar wind frame. We
compare the theoretical calculations with the simulation
results. The compressive fluctuation has the largest amplitude at r  18 RS as shown in Figure 2a, and this fluctuation
has the wave number k  15 R1
S as shown in Figure 3b.
The solid line in Figure 5 corresponds to the contour lineof A(r, k) = 21, which pass through the point r  18 RS,
k  15 R1
S . We expect that a wave mode with andamplification factor A > 21 will be dominant due to thespatial growth in the solar wind frame.
[29] Next we compared Figure 4a and Figure 5b for
Run B. The solid line in Figure 5b corresponds to thecontour line of A = 21. This contour line indicates that theunstable mode with the wave number k  17 R1
S grows
most rapidly at r  28 RS. Figure 4a shows that thecompressive fluctuations appear at r = 28  34 RS, whichis in agreement with our theoretical analysis. We also
compared Figure 4b and Figure 5c for Run C. The solidline in Figure 5c corresponds the contour line of A = 21.
This contour line indicates that the unstable mode with thewave number k  25 RS grows most rapidly at r  11 RS.
Figure 4b shows that the compressive fluctuations appear at
r = 10  14 RS, which is again in agreement with ourtheoretical analysis.
[30] From these comparison we found that the profile ofA(k, r) can well estimate the wave number of the most
rapidly growing compressive mode, where the parametricinstability occurs. Note that it is difficult to evaluate theexact amplification factor of the compressive mode becauseof the spatial variation of the growth rate and the background bulk velocity. However, the simulation results are
qualitatively explained by the effective amplification factor
obtained by the local linear dispersion analysis with the
WKB approximation.

5. Conclusion

Figure 5. The plot of the amplification factor A(r, k)
defined by Equation (10) for (a) Run A, (b) Run B, and(c) Run C. The solid lines indicate the contour line for A =21 which corresponds to the amplitude of the compressive
fluctuations seen in Figure 2a.
fluctuation propagates outward at the bulk velocity vr(r).
Here we evaluate the spatial growth of the compressive
fluctuation in the solar wind frame by integrating the local
growth rate as the fluid element of the solar wind moves
from a position r0 to r with the bulk velocity vr,
Aðr; k Þ

Z r

g ðr0 ; k Þ 0
dr :0
r0 vr ðr Þ

ð10Þ[31] We have performed MHD simulations on the propagation of Alfvén waves in the solar wind. Circularly
polarized monochromatic Alfvén waves have been injectedfrom the lower boundary into the transonic wind solution.
Simulation results have shows that the injected Alfvén waves
are subject to the parametric decay, exciting outwardpropagating sound waves and inward-propagating Alfvénwaves. The compressive fluctuations grow most rapidly at a
certain position in the simulation domain. To estimate themost unstable position, we analyzed the local dispersion
relation for the parametric decay instability with the WKB
approximation. We computed the effective amplification
factor with the variation of the local growth rate due to thebackground bulk velocity, which is in good agreement withthe simulation results.
[32] In the present study we have adopted a simplified
model in which circularly polarized monochromatic Alfvénwaves propagate in the radially expanding transonic flow ina one-dimensional system. However, the present result gives

6 of 7

A10110TANAKA ET AL.: PARAMETRIC DECAY OF ALFVÉN WAVES

a quantitative description of nonlinear evolution of Alfvénwaves in the solar wind. Better understanding on the
propagation of Alfvén waves in the solar wind would
contribute the space weather modeling research in associated with the acceleration and heating processes of the solar
wind. For a more realistic description of Alfvén waves
propagation in the solar wind, we need to introduce a
realistic energy equation and multidimensional effects.
These are left as future works.
[33] Kinetic effects are also important. They can change
the growth rate of unstable modes [Gomberoff, 2000;
Gomberoff et al., 2001], which would modify the location
and wave number of the most rapidly growing compressive
mode. Even with the damping effects of ion acoustic waves,however, we think that the decay instability can occur andthat the compressive mode can grow at a certain position as
discussed above. Full kinetic effects for parametric decay
instability are important issues.
[34] Acknowledgments. This work was supported by the Grant-inAid for Creative Scientific Research ‘‘The Basic Study of Space Weather
Prediction’’ (17GS0208, Head Investigator: K. Shibata) from the Ministryof Education, Science, Sports, Technology, and Culture of Japan and partly
by the 21st Century COE Program ‘‘Dynamics of the Sun-Earth-Life
Interactive System(SELIS)’’ at Nagoya University.
[35] Amitava Bhattacharjee thanks Luis Gomberoff and G. Lakhina for
their assistance in evaluating this paper.

References
Agim, Y. Z., A. F. Viñas, and M. L. Goldstein (1995), Magnetohydrodynamic
and hybrid simulations of broadband fluctuations near interplanetaryshocks, J. Geophys. Res., 100, 17,081 – 17,106, doi:10.1029/95JA00527.
An, C.-H., S. T. Suess, R. L. Moore, and Z. E. Musielak (1990), Reflection
and trapping of Alfven waves in a spherically symmetric stellar atmosphere, Astrophys. J., 350, 309 – 323, doi:10.1086/168384.
Bavassano, B., E. Pietropaolo, and R. Bruno (2000), On the evolution ofoutward and inward Alfvénic fluctuations in the polar wind, J. Geophys.
Res., 105, 15,959 – 15,964, doi:10.1029/1999JA000276.
Belcher, J. W., and L. Davis (1971), Large amplitude Alfvén waves in theinterplanetary medium, J. Geophys. Res., 76, 3534 – 3563.
Buti, B., V. Jayanti, A. F. Viñas, S. Ghosh, M. L. Goldstein, D. A. Roberts,
G. S. Lakhina, and B. T. Tsurutani (1998), Nonlinear evolution of
Alfvénic wave packets, Geophys. Res. Lett., 25, 2377 – 2380,
doi:10.1029/98GL01688.
Buti, B., V. L. Galinski, V. I. Shevchenko, G. S. Lakhina, B. T. Tsurutani,
B. E. Goldstein, P. Diamond, and M. V. Medvedev (1999), Evolution of
nonlinear Alfvén waves in streaming inhomogeneous plasmas, Astrophys.
J., 523, 849 – 854, doi:10.1086/307743.
Cranmer, S. R., and A. A. Ballegooijen (2005), On the generation, propagation, and reflection of Alfvén waves from the solar photosphere to the
distant heliosphere, Astrophys. J., 156, 265 – 293, doi:10.1086/426507.
Del Zanna, L., M. Velli, and P. Londrillo (2001), Parametric decay ofcircularly polarized Alfvén waves: Multidimensional simulations inperiodic and open domains, Astron. Astrophys., 367, 705–718, doi:10.1051/
0004-6361:20000455.
Derby, N. F., Jr. (1978), Modulational instability of finite-amplitude, circularly polarized Alfven waves, Astrophys. J., 224, 1013 – 1016,
doi:10.1086/156451.
Fukuda, N., and T. Hanawa (1999), Gravitational and parametric instabilitiesof the interstellar medium in which the Alfvén wave travels, Astrophys. J.,
517, 226 – 241, doi:10.1086/307169.
Ghosh, S., and M. L. Goldstein (1994), Nonlinear evolution of a largeamplitude circularly polarized Alfven wave: Low beta, J. Geophys. Res.,99, 13,351 – 13,362.

A10110Ghosh, S., A. F. Vinas, and M. L. Goldstein (1994), Nonlinear evolution ofa large-amplitude circularly polarized Alfven wave: High beta, J. Geophys. Res., 99, 19,289 – 19,300.
Goldstein, M. L. (1978), An instability of finite amplitude circularly polarized
Alfven waves, Astrophys. J., 219, 700 – 704, doi:10.1086/155829.
Gomberoff, L. (2000), Ion acoustic damping effects on parametric decaysof Alfvén waves, J. Geophys. Res., 105, 10,509 – 10,518, doi:10.1029/
1999JA000281.
Gomberoff, L., K. Gomberoff, and A. L. Brinca (2001), Ion acoustic damping
effects on parametric decays of Alfvén waves: Right-hand polarization,
J. Geophys. Res., 106, 18,713 – 18,720, doi:10.1029/2000JA000384.
Hawley, J. F., and J. M. Stone (1995), MOCCT: A numerical technique forastrophysical MHD, Comput. Phys. Commun., 89, 127 – 148, doi:10.1016/
0010-4655 (95)00190-Q.
Hollweg, J. V. (1971), Density fluctuations driven by Alfvén waves,
J. Geophys. Res., 76, 5155 – 5161.
Jayanti, V., and J. V. Hollweg (1993), On the dispersion relations for parametric instabilities of parallel-propagating Alfvén waves, J. Geophys.
Res., 98, 13,247 – 13,252, doi:10.1029/93JA00920.
Lau, Y.-T., and E. Siregar (1996), Nonlinear Alfven wave propagation in
the solar wind, Astrophys. J., 465, 451 – 461, doi:10.1086/177432.
Linker, J. A., G. van Hoven, and D. D. Schnack (1990), A three-dimensional simulation of a coronal streamer, Geophys Res. Lett., 17, 2281 –
2284.
Linker, J. A., et al. (1999), Magnetohydrodynamic modeling of the solar
corona during Whole Sun Month, J. Geophys. Res., 104, 9809 – 9830,
doi:10.1029/1998JA900159.
Malara, F., L. Primavera, and P. Veltri (2000), Nonlinear evolution ofparametric instability of a large-amplitude nonmonochromatic Alfvénwave, Phys. Plasmas, 7, 2866 – 2877, doi:10.1063/1.874136.
Ofman, L. (2004), Three-fluid model of the heating and acceleration of thefast solar wind, J. Geophys. Res., 109, A07102, doi:10.1029/2003JA010221.
Ofman, L., and J. M. Davila (1998), Solar wind acceleration by largeamplitude nonlinear waves: Parametric study, J. Geophys. Res., 103,
23,677 – 23,690, doi:10.1029/98JA01996.
Parhi, S., B. P. Pandey, M. Goossens, G. S. Lakhina, and P. de Bruyne(1997), MHD study of coronal waves: A numerical approach, Astrophys.
Space Sci., 250, 147 – 162, doi:10.1023/A:1000444410913.
Parker, E. N. (1963), Interplanetary Dynamical Processes, Wiley Intersci.,
New York.
Pruneti, F., and M. Velli (1997), Parametric Decay of Large Amplitude
Alfven Waves in the Solar Atmosphere, in Eur. Space Agency, Spec.
Publ.-404: Fifth SOHO Workshop: The Corona and Solar Wind Near
Minimum Activity, edited by A. Wilson, pp. 623 – 627.
Sagdeev, R. Z., and A. A. Galeev (1969), Nonlinear Plasma Theory, Nonlinear Plasma Theory, Benjamin, New York.
Steinolfson, R. S., and Y. Nakagawa (1976), Dynamical response of thesolar corona. II. Numerical simulations near the sun, Astrophys. J., 207,
300 – 307, doi:10.1086/154494.
Sturrock, P. A. (1999), Chromospheric magnetic reconnection and its possible
relationship to coronal heating, Astrophys. J., 521, 451– 459, doi:10.1086/
307544.
Suzuki, T. K., and S.-I. Inutsuka (2006), Solar winds driven by nonlinear
low-frequency Alfvén waves from the photosphere: Parametric study forfast/slow winds and disappearance of solar winds, J. Geophys. Res., 111,
A06101, doi:10.1029/2005JA011502.
Tanaka, T. (1994), Finite volume TVD scheme on an unstructured grid
system for three-dimensional MHD simulation of inhomogeneous systems
including strong background potential fields, J. Comput. Phys., 111, 381 –390.
Turkmani, R., and U. Torkelsson (2004), Dynamics of circularly polarised
Alfvén waves in a stratified medium, Astron. Astrophys., 428, 227 – 233,
doi:10.1051/0004-6361:20041071.


T. Ogino, S. Tanaka, and T. Umeda, Solar-Terrestrial Environment
Laboratory, Nagoya University, Nagoya, Aichi 464-8601, Japan. (ogino@stelab.nagoya-u.ac.jp; stanaka@stelab.nagoya-u.ac.jp; umeda@stelab.nagoyau.ac.jp)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S02, doi:10.1029/2007JA012432, 2008

for

Full
ArticleNonlinear force-free modeling of the solar coronalmagnetic field
T. Wiegelmann1
Received 30 March 2007; revised 23 May 2007; accepted 18 July 2007; published 8 January 2008.

[1] The coronal magnetic field is an important quantity because the magnetic fielddominates the structure of the solar corona. Unfortunately, direct measurements of coronalmagnetic fields are usually not available. The photospheric magnetic field ismeasured routinely with vector magnetographs. These photospheric measurements are
extrapolated into the solar corona. The extrapolated coronal magnetic field depends on
assumptions regarding the coronal plasma, for example, force-freeness. Force-freemeans that all nonmagnetic forces like pressure gradients and gravity are neglected. Thisapproach is well justified in the solar corona owing to the low plasma beta. One has
to take care, however, about ambiguities, noise and nonmagnetic forces in thephotosphere, where the magnetic field vector is measured. Here we review
different numerical methods for a nonlinear force-free coronal magnetic field
extrapolation: Grad-Rubin codes, upward integration method, MHD relaxation,
optimization, and the boundary element approach. We briefly discuss the main features ofthe different methods and concentrate mainly on recently developed new codes.
Citation: Wiegelmann, T. (2008), Nonlinear force-free modeling of the solar coronal magnetic field, J. Geophys. Res., 113, A03S02,
doi:10.1029/2007JA012432.

1. Introduction
1.1. How to Obtain the Coronal Magnetic Field?
[2] Information regarding the coronal magnetic field isimportant for space weather application like the onset offlares and coronal mass ejections (CMEs). Unfortunately,
we usually cannot measure the coronal magnetic field
directly, although recently some progress has been made[see, e.g., Judge, 1998; Solanki et al., 2003; Lin et al.,2004]. Because of the optically thin coronal plasma, direct
measurements of the coronal magnetic field have a line-ofsight integrated character and to derive the accurate 3D
structure of the coronal magnetic field a vector tomographic
inversion is required. Corresponding feasibility studies
based on coronal Zeeman and Hanle effect measurements
have recently been done by Kramar et al. [2006] andKramar and Inhester [2006]. These direct measurements
are only available for a few individual cases and usually one
has to extrapolate the coronal magnetic field from photospheric magnetic measurements. To do so, one has to makeassumptions regarding the coronal plasma. It is helpful thatthe low solar corona is strongly dominated by the coronalmagnetic field and the magnetic pressure is orders ofmagnitude higher than the plasma pressure. The quotientof plasma pressure p and magnetic pressure, B2/(2 m0) is
small compared to unity (b = 2 m0 p/B2 1). In lowest-ordernonmagnetic forces like pressure gradient and gravity canbe neglected which leads to the force-free assumption.
Force-free fields are characterized by the equations
ð1Þ

r  B ¼ m0 j;

ð2Þ

r B ¼ 0;

ð3Þ

where B is the magnetic field, j the electric current density
and m0 the permeability of vacuum. Equation (1) implies
that for force-free fields the current density and themagnetic field are parallel, i.e.,
m0 j ¼ aB;

ð4Þ

or by replacing j with equation (2),r  B ¼ aB;

ð5Þ

where a is called the force-free function. To get someinsights in the structure of the space-dependent function a,
we take the divergence of equation (4) and make use ofequations (2) and (3),1Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau,
Germany.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012432$09.00

j  B ¼ 0;

B ra ¼ 0;

ð6Þ

which tells us that the force-free function a is constant on
every field line, but will usually change from one field line

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A03S02to another. This generic case is called nonlinear force-free
approach.
[3] Popular simplifications are a = 0 (current free potential
fields [see, e.g., Schmidt, 1964; Semel, 1967; Schatten et al.,
1969; Sakurai, 1982]) and a = constant (linear force-free
approach [see, e.g., Nakagawa and Raadu, 1972; Chiu andHilton, 1977; Seehafer, 1978; Alissandrakis, 1981; Seehafer,
1982; Semel, 1988]). These simplified models have been inparticular popular owing to their relative mathematical simplicity and because only line-of-sight photospheric magnetic
field measurements are required. Linear force-free fields stillcontain one free global parameter a, which can be derived bycomparing coronal images with projections of magnetic
field lines [e.g., Carcedo et al., 2003]. It is also possible
to derive an averaged value of a from transverse photospheric magnetic field measurements [e.g., Pevtsov et al.,
1994; Wheatland, 1999; Leka and Skumanich, 1999]. Despite
the popularity and frequent use of these simplified models in
the past, there are several limitations in these models (seebelow) which ask for considering the more sophisticatednonlinear force-free approach.
[4] Our aim is to review recent developments of theextrapolation of nonlinear force-free fields (NLFFF). For
earlier reviews on force-free fields we refer to Sakurai[1989], Aly [1989], Amari et al. [1997], McClymont et al.
[1997], and chapter 5 of Aschwanden [2005]. Here we willconcentrate mainly on new developments which took place
after these earlier reviews. Our main emphasis is to study
methods which extrapolate the coronal magnetic field fromphotospheric vector magnetograms. Several vector magnetographs are currently operating or planed for the nearest
future, for example, ground based: the solar flare telescope/
NAOJ [Sakurai et al., 1995], the imaging vector magnetograph/MEES Observatory [Mickey et al., 1996], Big Bear
Solar Observatory, Infrared Polarimeter VTT, SOLIS/NSO[Henney et al., 2006] and space born: Hinode/SOT[Shimizu, 2004], SDO/HMI [Borrero et al., 2006]. Measurements from these vector magnetograms will provide us
eventually with the magnetic field vector on the photosphere, say Bz0 for the normal and Bx0 and By0 for thetransverse field. Deriving these quantities from the measurements is an involved physical process, which includes
measurements based on the Zeeman and Hanle effect, theinversion of Stokes profiles [e.g., LaBonte et al., 1999] andremoving the 180 ambiguity [e.g., Metcalf, 1994; Metcalf etal., 2006] of the horizontal magnetic field component.
Special care has to be taken for vector magnetographmeasurements which are not close to the solar disk, when
the line-of-sight and normal magnetic field component are
far apart [e.g., Gary and Hagyard, 1990]. For the purpose ofthis paper we do not address the observational methods andrecent developments and problems related to deriving thephotospheric magnetic field vector. We rather will concentrate on how to use the photospheric Bx0, By0 and Bz0 toderive the coronal magnetic field.
[5] The transverse photospheric magnetic field (Bx0, By0)
can be used to approximate the normal electric current
distribution by
m0 jz0 ¼@By0 @Bx0

;
@x
@y

ð7Þ

A03S02and from this one gets the distribution of a on thephotosphere by
aðx; yÞ ¼ m0

jz0
:
Bz0

ð8Þ

[6] By using equation (8) one has to keep in mind thatrather large uncertainties in the transverse field component
and numerical derivations used in (7) can cumulate insignificant errors for the current density. The problem
becomes even more severe by using (8) to compute a inregions with a low normal magnetic field strength Bz0.
Special care has to be taken at photospheric polarity
inversion lines, i.e., lines along which Bz0 = 0 [see, e.g.,
Cuperman et al., 1991]. The nonlinear force-free coronalmagnetic field extrapolation is a boundary value problem.
As we will see later some of the NLFFF codes make use of(8) to specify the boundary conditions while other methods
use the photospheric magnetic field vector more directly to
extrapolate the field into the corona.
[7] Pure mathematical investigations of the nonlinear
force-free equations [see, e.g., Aly, 1984; Boulmezaoudand Amari, 2000; Aly, 2005] and modeling approaches
not based on vector magnetograms are important andoccasionally mentioned in this paper. A detailed review ofthese topics is well outside the scope of this paper, however.
Some of the model approaches not based on vector magnetograms are occasionally used to test the nonlinear forcefree extrapolation codes described here.
1.2. Why Do We Need Nonlinear Force-Free Fields?
[8] 1. A comparison of global potential magnetic fieldmodels with TRACE images by Schrijver et al. [2005]
revealed that significant nonpotentially occurs regularly inactive regions, in particular when new flux has emerged inor close to the regions.
[9] 2. Usually a changes in space, even inside one active
region. This can be seen, if we try to fit for the optimal linear
force-free parameter a by comparing field lines with coronalplasma structures. An example is given by Wiegelmann andNeukirch [2002] where stereoscopic reconstructed loops byAschwanden et al. [1999] have been compared with a linear
force-free field model. The optimal value of a changes evensign within the investigated active regions, which is acontradiction to the a = constant linear force-free approach(see Figure 1).
[10] 3. Photospheric a distributions derived from vector
magnetic field measurements by equation (8) show as wellthat a usually changes within an active region [see, e.g.,Régnier et al., 2002].
[11] 4. Potential and linear force-free fields are too simple
to estimate the free magnetic energy and magnetic topology
accurately. The magnetic energy of linear force-free fields is
unbounded in a halfspace [Seehafer, 1978] which makes
this approach unsuitable for energy approximations of thecoronal magnetic field. Potential fields have a minimum
energy for an observed line-of-sight photospheric magnetic
field. An estimate of the excess of energy a configuration
has above that of a potential field is an important quantity
which might help to understand the onset of flares andcoronal mass ejections.

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Figure 1. Linear force – free field model for NOAA 7986 with the best fit for a. (top) A group of loops
with a = 2.5 and (bottom) another group of loops with a = 2.0. The different optimal values of thelinear force-free parameter within one active region are a contradiction to the linear assumption (aconstant) and tell us that a consistent modeling of this active regions requires nonlinear force-free
approach. (This figure was originally published as Wiegelmann and Neukirch’s [2002] Figure 7). Used
with permission of Springer.).
[12] 5. A direct comparison of measured fields in a newly
developed active region by Solanki et al. [2003] withextrapolations from the photosphere with a potential, linear
and nonlinear force-free model by Wiegelmann et al.
[2005b] showed that nonlinear fields are more accurate thansimpler models. Figure 2 shows some selected magnetic
field lines for the original measured field and extrapolations
from the photosphere with the help of a potential, linear andnonlinear force-free model.

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Figure 2. Magnetic field structure of the newly developed active region NOAA 9451. Direct
measurements of the field have been compared with potential, linear, and nonlinear force-free models.
Best agreement has been found for the nonlinear model. (This figure was originally published as part ofWiegelmann et al.’s [2005b] Figure 1. Used with permission of Astronomy and Astrophysics).
[13] These points tell us that nonlinear force-free modeling is required for an accurate reconstruction of the coronalmagnetic field. Simpler models have been used frequently
in the past. Global potential fields provide some informationof the coronal magnetic field structure already, for example,
the location of coronal holes. The generic case of force-freecoronal magnetic field models are nonlinear force-freefields, however. Under generic we understand that a can(and usually will) change in space, but this approach also
includes the special cases a = constant and a = 0. Some
active regions just happen to be more potential (or linear
force-free) and if this is the case they can be described withsimpler models. Linear force-free models might provide arough estimate of the true 3D magnetic field structure if thenonlinearity is weak. The use of simpler models was often
justified owing to limited observational data, in particular if
only the line-of-sight photospheric magnetic field has beenmeasured.
[14] While the assumption of nonlinear force-free fields iswell accepted for the coronal magnetic fields in active
regions, this is not true for the photosphere. The photospheric plasma is a finite b plasma and nonmagnetic forces
like pressure gradient and gravity cannot be neglected here.
As a result electric currents have a component perpendicular
to the magnetic field, which contradicts the force-freeassumption. We will discuss later how these difficulties
can be overcome.

2. Nonlinear Force-Free Codes[15] Different methods have been proposed to extrapolate
nonlinear force-free fields from photospheric vector magnetic field measurements. (1) The Grad-Rubin method was
proposed for fusion plasmas by Grad and Rubin [1958] andfirst applied to coronal magnetic fields by Sakurai [1981].
(2) The upward integration method was proposed by
Nakagawa [1974] and encoded by Wu et al. [1985].
(3) The MHD relaxation method was proposed for general
MHD equilibria by Chodura and Schlueter [1981] andapplied to force-free coronal magnetic fields by Mikic and
McClymont [1994]. (4) The optimization approach wasdeveloped by Wheatland et al. [2000]. (5) The boundary
element (or Greens function like) method was developedby Yan and Sakurai [2000].
2.1. Grad-Rubin Method[16] The Grad-Rubin method reformulates the nonlinear
force-free equations in such a way, that one has to solve awell posed boundary value problem. This makes thisapproach also interesting for a mathematical investigationof the structure of the nonlinear force-free equations. Bineau

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A03S02[1972] demonstrated that the used boundary conditions(vertical magnetic field on the photosphere and a distribution at one polarity) ensure, at least for small values of aand weak nonlinearities the existence of a unique nonlinear
force-free solution. A detailed analysis of the mathematicalproblem of existence and uniqueness of nonlinear force-freefields is outside the scope of this review and is given, forexample, by Amari et al. [1997, 2006].
[17] The method first computes a potential field, which
can be obtained from the observed line-of-sight photospheric magnetic field (say Bz in Cartesian geometry) by different
methods, for example, a Greens function method as described by Aly [1989]. It is also popular to use linear forcefree solvers, for example, as implemented by Seehafer[1978] and Alissandrakis [1981] with the linear force-freeparameter a = 0 to compute the initial potential field. The
transverse component of the measured magnetic field isthen used to compute the distribution of a on the photosphere by equation (8). While a is described this way on theentire photosphere, for both polarities, a well posed boundary value problem requires that the a distribution becomes
only described for one polarity. The basic idea is toiteratively calculate a for a given B field from (6), thencalculate the current via (4) and finally update B from the
Biot-Savart problem (5). These processes are repeated until
the full current as prescribed by the a distribution has been
injected into the magnetic field and the updated magnetic
field configuration becomes stationary in the sense thateventually the recalculation of the magnetic field with
Amperes law does not change the configuration anymore.
To our knowledge the Grad-Rubin approach has been first
implemented by Sakurai [1981]. Here a has been prescribedon several nodal points along a number of magnetic fieldlines of the initial potential field. The method used a finiteelement-like discretization of current tubes associated withmagnetic field lines. Each current tube was divided intoelementary current tubes of cylindrical shape. The magnetic
field is updated with Ampere’s law using a superposition of
the elementary current tubes. The method was limited by
the number of current-carrying field lines, nodal points andthe corresponding number of nonlinear equations (N9) tosolve with the available computer resources more than aquarter century ago.
[18] Computer resources have increased rapidly since thefirst NLFFF implementation by Sakurai [1981] and about adecade ago the Grad-Rubin method has been implemented
on a finite difference grid by Amari et al. [1997, 1999]. Thisapproach decomposes equations (1) – (3) into a hyperbolic
part for evolving a along the magnetic field lines and an
elliptic one to iterate the updated magnetic field fromAmperes law. For every iteration step k one has to solve
iteratively forBðkÞ raðkÞ ¼ 0

ð9ÞaðkÞ jS  ¼ a0 ;ð10Þwhich evolves a in the volume andr  Bðkþ1Þ ¼ aðkÞ BðkÞ ;ð11Þ

A03S02r Bðkþ1Þ ¼ 0;

ð12ÞjS ¼ Bz0 ;Bðkþ1Þ
z

ð13Þ

lim j Bðkþ1Þ j¼ 0;

ð14Þ

jrj!1

where a0± corresponds to the photospheric distribution of afor either on the positive or the negative polarity. The GradRubin method as described by Amari et al. [1997, 1999] hasbeen applied to investigate particular active regions by
Bleybel et al. [2002] and a comparison of the extrapolated
field with 2D projections of plasma structures as seen in
Ha, EUV and X ray has been done by Régnier et al. [2002]
and Régnier and Amari [2004]. The code has also been used
to investigate mutual and self helicity in active regions byRégnier et al. [2005] and to flaring active regions byRégnier and Canfield [2006].
[19] A similar approach as done by Sakurai [1981] hasbeen implemented by Wheatland [2004]. The implemented
method computes the magnetic field directly on the numerical grid from Ampere’s law. This is somewhat simpler andfaster as Sakurai’s approach which required solving a large
system of nonlinear equations for this aim. The implementation by Wheatland [2004] has, in particular, been developed with the aim of parallelization. The parallelization
approach seems to be effective owing to a limited number ofinterprocess communications. This is possible because as
the result of the linearity of Ampere’s law the contributionsof the different current carrying field lines are basicallyindependent from each other. In the original paper Wheatland reported problems for large currents on the field lines.
These problems have been related to an error in current
representation of the code and the corrected code worked
significantly better [see also Schrijver et al., 2006]. The
method has been further developed by Wheatland [2006].
This newest Wheatland-implementation scales with thenumber of grid points N 4 for a N 3 volume, rather than N 6
for the earlier [Wheatland, 2004] implementation. The main
new development is a faster implementation of the currentfield iteration. To do so the magnetic field has been
separated into a current-free and a current carrying part ateach iteration step. Both parts are solved using a discrete
Fast Fourier Transformation, which imposes the required
boundary conditions implicitly. The code has been parallelized on shared memory distributions with OpenMP.
[20] Amari et al. [2006] developed two new versions of
their Grad-Rubin code. The first version is a finite difference method and the code was called ’XTRAPOL.’ This
code prescribe the coronal magnetic field with the help of avector potential A. The code has obviously it’s heritage
from the earlier implementation of Amari et al. [1999], butwith several remarkable differences:[21] 1. The code includes a divergence cleaning routine,
which takes care about r A = 0. The condition r A = 0 is
fulfilled with high accuracy in the new code 109 compared
to 102 in the earlier implementation.
[22] 2. The lateral and top boundaries are more flexible
compared to the earlier implementation and allow a finite Bn

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and non zero a values for one polarity on all boundaries.
This treats the whole boundary (all six faces) as a whole.
[23] 3. The slow current input as reported for the earlier
implementation, which lead to a two level iteration, hasbeen replaced. Now the whole current is injected at onceand only the inner iteration loop of the earlier code
remained in the new version.
[24] 4. The computation of the a characteristics has been
improved with an adaptive Adams-Bashforth integration
scheme (see Press [2002] for details).
[25] 5. The fixed number of iteration loops have been
replaced by a quantitative convergence criterium.
[26] In the same paper [Amari et al., 2006] introduced
another Grad-Rubin approach based on finite elements,
which they called ’FEMQ’. Different from alternative
implementation this code does not use a vector potential
but iterates the coupled divergence and curl system, which
is solved with the help of a finite element discretization. The
method transforms the nonlinear force-free equations into a
global linear algebraic system.
[27] Inhester and Wiegelmann [2006] implemented a
Grad-Rubin code on a finite element grid with staggered
field components [see Yee, 1966] which uses discrete
Whitney forms [Bossavit, 1988]. Whitney forms allow totransform standard vector analysis (as the differential operators gradient, curl and divergence) consistently into thediscrete space used for numerical computations. Whitney
forms contain four types of finite elements (form 0– 3).
They can be considered as a discrete approximation ofdifferential forms. The finite element base may consist ofpolynomials of any order. In its simplest form, the 0-forms
have as parameters the function values at the vertices of thecells and are linearly interpolated within each cell. The 1forms are a discrete representation of a vector field definedon the cell edges. The 2-forms are defined as the fieldcomponent normal to the surfaces of the cells. The 3-forms
are finite volume elements for a scalar function approximation, which represents the average of a scalar over the entire
cell. The 4-forms are related to each other by GRAD (0 to 1form), CURL (1 to 2 form) and DIV (2 to 3 form). As forcontinuous differential forms, double differentiation (CURLof GRAD, DIV of CURL) give exactly zero, independent of
the numerical precision. A dual grid, shifted by half a gridsize in each axis, was introduced in order to allow for
Laplacians. Whitney forms on the dual grid are related toforms on the primary grid in a consistent way.
[28] The Grad-Rubin implementation uses a vector potential representation of the magnetic field, where the vector
potential is updated with a Poisson equation in each
iteration step. The Poisson equation is effectively solved
with the help of a multigrid solver. The main computing
time is spend to distribute a along the field lines with (6).
This seems to be a general property of Grad-Rubin implementations. One can estimate the scaling of (6) by / N4,
where the number of field lines to compute is / N3 and thelength of a field line / N. The Biot-Savart step (5) solved
with FFT or multigrid methods scales only with / N log N.
Empirical tests show that the number of iteration steps until
a stationary state is reached does not depend on the numberof grid points N for Grad-Rubin solvers. We have explained
before, that the Grad-Rubin implementation requires theprescription of a only for one polarity to have a well posed

A03S02mathematical problem. The Inhester and Wiegelmann[2006] implementation allows these choice of boundary
conditions as a special case. In general one does not needto make the distinction between (@V)+ and (@V)- in the new
implementation. A well posed mathematical problem is still
ensured, however, in the following way. Each boundary
value of a is attached with a weight. The final version of aon each field line is then determined by a weighted averageof the a values on both endpoints of a field lines. By thisway the influence of uncertain boundary values, for example, on the side walls and imprecise photospheric measurements can be suppressed.
2.2. Upward Integration Method[29] The basic equations for the upward integration method (or progressive extension method) have been publishedalready by Nakagawa [1974] and a corresponding code has
been developed by Wu et al. [1985, 1990a]. The upward
integration method is a straight forward approach to use thenonlinear force-free equations directly to extrapolate thephotospheric magnetic field into the corona. To do so one
reformulates the force-free equations (1) – (3) in order toextrapolate the measured photospheric magnetic field vector
into the solar corona.
[30] As a first step the magnetic field vector on the lower
boundary B0(x,y,0) is used to compute the z component of
the electric current m0 jz0 with equation (7) and the photospheric a distribution (say a0) by equation (8). With thehelp of equation (4) we calculate the x and y component of
the current density
m0 jx0 ¼ a0 Bx0ð15Þ

m0 jy0 ¼ a0 By0 :ð16Þ[31] We now use equation (3) and the x and y componentof equation (2) to obtain expressions for the z derivatives of
all three magnetic field components in the form
@Bx0
@Bz0
¼ m0 jy0 þ
;
@z
@x

ð17Þ

@By0 @Bz0¼ m0 jx0 ;
@z
@y

ð18Þ

@Bz0
@Bx0 @By0
¼

:
@z
@x
@y

ð19Þ

The idea is to integrate this set of equations numericallyupward in z by repeating the previous steps at each height.
As a result we get in principle the 3D magnetic field vector
in the corona. While this approach is straight forward, easy
to implement and computational fast (no iteration isrequired), a serious drawback is that it is unstable. Several
authors [e.g., Cuperman et al., 1990; Amari et al., 1997]
pointed out that the formulation of the force-free equationsin this way is unstable because it is based on an ill-posed
mathematical problem. In particular one finds that expo-

6 of 19

A03S02

WIEGELMANN: CORONAL MAGNETIC FIELDS

nential growth of the magnetic field with increasing heightis a typical behavior. What makes this boundary value
problem ill-posed is that the solution does not depend
continuously on the boundary data. Small changes or
inaccuracies in the measured boundary data lead to adivergent extrapolated field (see Low and Lou [1990] for amore detailed discussion). As pointed out by Low and Lou
meaningful boundary conditions are required also on theouter boundaries of the computational domain. It is also
possible to prescribe open boundaries in the sense that themagnetic field vanishes at infinity. This causes an additional
problem for the upward integration method, because themethod transports information only from the photosphereupward and does not incorporate boundary information onother boundaries or at infinity. Attempts have been made toregularize the method [e.g., Cuperman et al., 1991;
Demoulin and Priest, 1992], but cannot be considered as
fully successful.
[32] Wu et al. [1990b] compared the Grad-Rubin methodin the implementation of Sakurai [1981] with the upward
integration method in the implementation of Wu et al.
[1990a]. (The authors used a somewhat different nomenclature: The upward integration method was called ’progressive extension method’ and the Grad-Rubin method’iterative method’. That time the term ’iterative method’
was reasonable because Grad-Rubin was the only iterative
approach available, but now, 17 years later, several other
iterative methods are available to compute nonlinear forcefree fields.) The comparison showed qualitatively similar
results for extrapolations from an observed magnetogram,but quantitatively differences. The NLFFF computations
have been very similar to potential field extrapolations,however, too. One reason for this behavior was, that themethod of Sakurai [1981] is limited to small values of aand a ’by eye’ comparison shows that the corresponding
NLFFF field is very close to a potential field configuration.
The field computed with the upward integration method
deteriorated if the height of the extrapolation exceeded a
typical horizontal scale length.
[33] The upward integration method has been recently
reexamined by Song et al. [2006] who developed a new
formulation of this approach. The new implementation uses
smooth continuous functions and the equations are solved inasymptotic manner iteratively. The original upward integration equations are reformulated into a set of ordinary
differential equations and uniqueness of the solution seems
to be guaranteed at least locally. While Demoulin and Priest[1992] stated that ’no further improvement has beenobtained with other types of smoothing functions’ Song etal. [2006] point out that the transformation of the original
partial differential equations into ordinary ones eliminates
the growing modes in the upward integration method, whichhave been reported before by Wu et al. [1990a] andsubsequent papers. The problem that all three componentsof the photospheric magnetic field and the photospheric a
distribution has to be prescribed in a consistent way remains
in principle, but some compatibility conditions to compute a
slowly varying a have been provided by Song et al. [2006].
These compatibility conditions are slightly different for real
photospheric observations and tests with smooth boundaries
extracted from semianalytic equilibria. For the latter kind ofproblems the new formulation provided reasonable results

A03S02

with the standard test equilibrium found by Low and Lou[1990]. The method seems to be also reasonably fast. Of
course, further tests with more sophisticated equilibria andreal data are necessary to evaluate this approach in moredetail.
2.3. MHD Relaxation[34] MHD relaxation codes [e.g., Chodura and Schlueter,
1981] can be applied to solve nonlinear force-free fields aswell. The idea is to start with a suitable magnetic field
which is not in equilibrium and to relax it into a force-freestate. This is done by using the MHD equations in thefollowing form:
uv ¼ ðr  BÞ  B;

ð20ÞE þ v  B ¼ 0;

ð21Þ

@B
¼ r  E;@t

ð22Þ

r B ¼ 0;

ð23Þ

where v is a viscosity and E the electric field. As the MHD
relaxation aims for a quasiphysical temporal evolution of
the magnetic field from a nonequilibrium toward a (nonlinear force-free) equilibrium this method is also called’evolutionary method’ or ’magneto-frictional method’. The
basic idea is that the velocity field in the equation of motion(21) is reduced during the relaxation process. Ideal Ohm’slaw (22) ensures that the magnetic connectivity remains
unchanged during the relaxation. The artificial viscosity v
plays the role of a relaxation coefficient which can bechosen in such way that it accelerates the approach to theequilibrium state. A typical choice is
u¼

1j B j2 ;m

ð24Þ

with m = constant. Combining equations (20), (21), (22) and(24) we get an equation for the evolution of the magnetic
field during the relaxation process,@B¼ mFMHD ;
@t

ð25Þ

with
FMHD ¼ r ½ðr  BÞ  B  B
:
B2

ð26Þ

This equation is then solved numerically starting with agiven initial condition for B, usually a potential field.
Equation (25) ensures that equation (23) is satisfied during
the relaxation if the initial magnetic field satisfies it. (As wewill see below the ’optimization’ approach leads to a similar
iteration equations for the magnetic field, but a different
artificial driving force F.) The difficulty with this method isthat it cannot be guaranteed that for given boundary

7 of 19

WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02conditions and initial magnetic field (i.e., given connectivity), a smooth force-free equilibrium exists to which thesystem can relax. If such a smooth equilibrium does notexist the formation of current sheets is to be expected which
will lead to numerical difficulties. Therefore care has to betaken when choosing an initial magnetic field.
[35] Yang et al. [1986] developed a magneto frictional
method which represent the magnetic field with the help ofEuler (or Clebsch) potentials.
B ¼ rg  rh;

ð27Þ

where the potentials g and h are scalar functions. The
general method has been developed for three dimensional
fields and iterative equations for g(x, y, z) and h(x, y, z) havebeen derived. The Clebsch representation automaticallyensures r B = 0. The method has been explicitly tested in
the paper by [Yang et al., 1986] with the help of an
equilibrium with one invariant coordinate. In principle itshould be possible to use this representation for the
extrapolation of nonlinear force-free fields, but we are notaware of a corresponding implementation. On the basis of
the discussion by Yang et al. [1986] a difficulty seems to bethat one needs to specify boundary conditions for the
potentials, rather than for the magnetic fields. It seems inparticular to be difficult to find boundaries conditions forpotentials which correspond to the transverse component of
the photospheric magnetic field vector. One problem is thatboundary conditions for g and h prescribe the connectivity.
Every field line can be labeled by its (g,h) values. Henceboundary values for g and h establish foot point relations
although the field is not known yet.
[36] The MHD relaxation (or evolutionary) method has
been implemented by Mikic and McClymont [1994] andMcClymont et al. [1997] on the basis of the time-dependent
MHD code by Mikic et al. [1988]. The code uses anonuniform mesh and the region of interested is embedded
in a large computational domain to reduce the influence ofthe lateral boundaries. The method has been applied to
extrapolate the magnetic field above an active region by
Jiao et al. [1997]. The computations have been carried out
with a resolution of the order of 1003 points. A supercomputer was required for these computations that time (10 yearsago), but because of the rapid increase of computer speed andmemory within the last decade this restriction is very probably not valid anymore.
[37] Roumeliotis [1996] developed the so-called stressand relax method. In this approach the initial potential field
becomes disturbed by the observed transverse field component on the photosphere. The boundary conditions are
replaced in subsequently in several small steps and always
relaxed with a similar MHD relaxation scheme as described above toward a force-free equilibrium. The code by
Roumeliotis [1996] has implemented a function w(x,y)
which allows to give a lower weight to regions where thetransverse photospheric field has been measured with lower
accuracy. Additional to the iterative equations as discussed
above, the method includes a resistivity h (or diffusivity) byadding a term h j on the right hand site of Ohms law (21).
This relaxes somewhat the topological constrains of ideal
MHD relaxation, because a finite resistivity allows a kind of

A03S02

artificial reconnection and corresponding changes of theinitial potential field topology. The method has been tested
with a force-free equilibrium found by Klimchuk andSturrock [1992] and applied to an active region measured
with the MSFC vector-magnetograph.
[38] The stress and relax method has been revisited by
Valori et al. [2005]. Different from the earlier implementation by Roumeliotis [1996] the new implementation uses
directly the magnetic field, rather than the vector potential
in order to keep errors from taking numerical deviations
from noisy magnetograms minimal. The solenoidal condition is controlled by a diffusive approach by Dedner et al.
[2002] which removes effectively a numerically createdfinite divergence of the relaxed magnetic field. The new
implementation uses a single stress step, rather than themultiple small stress used by Roumeliotis [1996] to speed
up the computation. The single step stress and relax method
is connected with a suitable control of artificial plasma
flows by the Courant criterium. The authors reported that a
multistep and single-step implementation do not reveal
significant differences. The numerical implementation isbased on the time-dependent full MHD code ’AMRVAC’
by Keppens et al. [2003]. Valori et al. [2005] tested theirnonlinear force-free implementation with a numericallyconstructed nonlinear force-free twisted loop computed byTörök and Kliem [2003].
2.4. Optimization Approach[39] The optimization approach has been developed by
Wheatland et al. [2000]. The solution is found by minimizing the functionalZ
L¼½B2 j ðr  BÞ  B j2 þ j r B j2 d 3 V :

ð28Þ

V

Obviously, L is bound from below by 0. This bound is
attained if the magnetic field satisfies the force-freeequations (1) –(3).
[40] By taking the functional derivatives with respect tosome iteration parameter t we get
)

1 dL
¼
2 dt

Z

@B ~ 3
Fd x V @t

ZS@B ~ 2
Gd x;
@t

ð29Þ

with

F¼r




½ðr  BÞ  B  B
ððr BÞBÞ  B
þ
r

B2
B2

 W  ðr  BÞ  rðW BÞ þ Wðr BÞ þ W2 Bg
W ¼ B2 ½ðr  BÞ  B  ðr BÞB:

ð30Þ

ð31Þ

The surface term vanishes if the magnetic field vector is
kept constant on the surface, for example, prescribed fromphotospheric measurements. In this case L decreases
monotonically if the magnetic field is iterated by

8 of 19

@B
¼ mF:
@t

ð32Þ

WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02[41] Let us remark that FMHS as defined in equation (26)and used for MHD relaxation is identical with the first termon the right-hand-side of equation (30), but equation (30)contains additional terms.
[42] For this method the vector field B is not necessarilysolenoidal during the computation, but will be divergencefree if the optimal state with L = 0 is reached. A disadvantage of the method is that it cannot be guaranteed that thisoptimal state is indeed reached for a given initial field andboundary conditions. If this is not the case then the resulting
B will either be not force-free or not solenoidal or both.
[43] McTiernan has implemented the optimization approach basically as described by Wheatland et al. [2000]
in IDL (see Schrijver et al. [2006] for a brief description of
the McTiernan implementation). This code allows the use ofa nonuniform computational grid. In a code intercomparison
by Schrijver et al. [2006] the IDL optimization code by
McTiernan was about a factor of 50 slower compared to an
implementation in parallelized C by Wiegelmann [2004]. To
our knowledge McTiernan has translated his IDL code into
FORTRAN in the meantime for faster computation (J. M.
McTiernan, personal communication on the NLFFF workshop Palo Alto, June 2006 [see also Metcalf et al., 2007]).
[44] Several tests have been performed with the optimization approach of Wiegelmann and Neukirch [2003]. It has
been investigated how the unknown lateral and top boundary influence the solution. The original optimization approach by Wheatland et al. [2000] has been extended
toward more flexible boundary conditions, which allow@B@t 6¼ 0 on the lateral and top boundaries. This has been
made with the help of the surface integral term in (29) andled to an additional term @B@t ¼ mG on the boundaries. Thisapproached improved the performance of the code for cases
where only the bottom boundary was prescribed. No improvement was found for a slow multistep replacement of
the boundary and this possibility has been abandoned infavor of a single step method. It has been also investigated
how noise influences the optimization code and this study
revealed that noise in the vector magnetograms leads to lessaccurate nonlinear force-free fields.
[45] Wiegelmann [2004] has reformulated the optimization principle by introducing weighting functions One
defines the functionalZ½wB2 j ðr  BÞ  B j2 þw j r B j2 d 3 x;

L¼

ð33Þ

V

where w(x, y, z) is a weighting function. It is obvious that (forw > 0) the force-free equations (1 – 3) are fulfilled when L
is equal zero. Minimization of the functional (34) lead to
@B
~
¼ mF;
@t

ð34Þ

~ ¼ wF þ ðWa  BÞ  rw þ ðWb BÞrw;F

ð35Þ

Wa ¼ B2 ½ðr  BÞ  B;

ð36Þ

Wb ¼ B2 ½ðr BÞB;

ð37Þ

A03S02

with F as defined in (30). With w(x, y, z) = 1 this approach
reduces to the Wheatland et al. [2000] method as describedabove. The weighting function is useful if only the bottom
boundary data are known. In this case we a buffer boundaryof several grid points toward the lateral and top boundary of
the computational box is introduced. The weighting
function is chosen constant in the inner, physical domain
and drop to 0 with a cosine profile in the buffer boundary
toward the lateral and top boundary of the computationalbox. In the work of Schrijver et al. [2006] some tests havebeen made with different weighting functions for the forcefree and solenoidal part of the functional (34), but the best
results have been obtained if both terms got the same
weight. The computational implementation involves thefollowing steps.
[46] 1. Compute start equilibrium (e.g., a potential field)in the computational box.
[47] 2. Replace the bottom boundary with the vector
magnetogram.
[48] 3. Minimize the functional (33) with the help ofequation (34). The continuous form of (34) guaranties amonotonically decreasing L. This is as well ensured in thediscretized form if the iteration step dt is sufficiently small.
The code checks if L(t+dt) < L(t) after each time step. If thecondition is not fulfilled, the iteration step is repeated withdt reduced by a factor of 2. After each successful iteration
step we increase dt slowly by a factor of 1.01 to allow thetime step to become as large as possible with respect to thestability condition.
[49] 4. The iteration stops if L becomes stationary. Sta4
for 100 consecutive
tionarity is assumed if @L@t =L < 1.0 10
iteration steps.
[50] The program has been tested with the semianalytic
nonlinear force-free configuration of Low and Lou [1990]
and Titov and Démoulin [1999] by Wiegelmann et al.
[2006a]. The code has been applied to extrapolate thecoronal magnetic field in active regions by Wiegelmann etal. [2005b, 2005a].
[51] A finite element optimization approach has been
implemented by Inhester and Wiegelmann [2006] using
the Whitney elements as for the Grad-Rubin code (which
has been described above). The optimization method uses
exactly the same staggered finite element grid as describedabove, which is different from the finite difference grids
used in the earlier implementations by Wheatland et al.
[2000], Wiegelmann and Neukirch [2003], and Wiegelmann[2004]. Another difference is that earlier implementations
discretized the analytical derivative of the functional L (28),while the new code takes the numerical more consistentderivative of the discretized function L. All other implementations used a simple Landweber scheme for updating
the magnetic field, which is replaced here by an unpreconditioned conjugate gradient iteration, which at every time
step performs an exact line search to the minimum of L in
the current search direction and additional selects an improve search direction instead of the gradient of the functional L. To do so the Hessian matrix of the functional L iscomputed during every iteration step. An effective computation of the Hessian matrix is possible, because the reformulated function L(s) is a fourth-order polynomial in B andall five polynomial coefficients can be computed in one go.
The code has been tested by Low and Lou [1990] and the

9 of 19

WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02

result of twisted loop computations of the Grad-Rubin
implementation on the same grid.
[52] The optimization code in the implementation ofWiegelmann [2004] has recently been extended toward
using a multiscale implementation. The main difference
from the original code are [see also Metcalf et al., 2007]
as follows. (1) The method is not full multigrid, but
computes the solution on different grids only once, forexample, something like 503, 1003, 2003. (2) The main idea
is to get a better (than potential field) start equilibrium on
the full resolution box. (3) Solution of smaller grids areinterpolated onto larger grids as initial state for the magnetic
field in the computational domain of the next larger box.
The multiscale implementation has been tested as part of acode-intercomparison test by Metcalf et al. [2007] withthe help of solar-like reference model computed by vanBallegooijen [2004] and van Ballegooijen et al. [2007].
[53] The optimization approach has recently been implemented in spherical geometry by Wiegelmann [2007] andtested by Low and Lou [1990]. The original longitudinal
symmetric Low and Lou solution has been shifted by 1/4 ofa solar radius to test the code without any symmetry withrespect to the Suns surface. The numerical implementationis very similar as the Cartesian implementation described byWiegelmann [2004]. The spherical implementation converged fast for low-latitude regions, but the computing timeincreased significantly if polar regions have been included.
It has been suggested to implement the code on a so called’Yin and Yang’ grid as developed by Kageyama and Sato
[2004] to reduce the computing time. The ’Yin and Yang’
grid is suitable for massive parallelization, which is necessary for full-sphere high-resolution NLFFF computations.
2.5. Boundary Element or Greens-Function-Like
Method[54] The boundary integral method has been developedby Yan and Sakurai [2000]. The method relates the measured boundary values with the nonlinear force-free field in
the entire volume by
ci Bi ¼I  @B  @ Y B0 dS;
Y
@n @n
S

ð38Þ

where ci = 1 for points in the volume and ci = 1/2 forboundary points and B0 is the measured vector magnetic
field on the photosphere. The auxiliary vector function isdefined as ¼ diag cosðlx rÞ ; cosðly rÞ ; cosðlz rÞ ;
Y
4pr
4pr
4pr

ð39Þ

and the li, (i = x, y, z) are computed in the original approachby Yan and Sakurai [2000] with integrals over the wholevolume, which define the li implicitly,Z
V

Yi ½l2i Bi  a2 Bi  ðra  Bi ÞdV ¼ 0:

ð40ÞThis volume integration, which has to be carried out forevery point in the volume is certainly very time consuming(a sixth-order process). The li have the same dimension as

A03S02the magnetic field. The existence of the li has been
confirmed for the semianalytic field of Low and Lou [1990]
by Li et al. [2004]. While the work of Li et al. [2004]
 for ashowed that one can find the auxiliary function Y is agiven force-free field in 3D, the difficulty is that Y
priori unknown if only the photospheric magnetic field
vector is given. Yan and Sakurai [2000] proposed an
iterative scheme to compute the auxiliary functions and thenonlinear force-free magnetic field self-consistently. They
use the approximate solution k on the right-hand side ofequation (38) to compute a better solution k + 1 by
ðkþ1Þ

ci Bi

I 
¼
S


ðkÞ
 ðkÞ
 ðkÞ @B  @ Y B0 dS;
Y
@n
@n

ð41Þ

where the initial guess for the magnetic field in the volume
is B = 0 and also the initial @@nY ¼ 0. In principle it would bealso possible to compute a potential field first and derive theauxiliary functions for this field as done by Li et al. [2004]
and iterate subsequently for the nonlinear force-free fields
and the associated auxiliary functions with equation (41).
This possibility has not been tried out to our knowledge
until now, however. The method iterates the magnetic field
until B and @B@n converge. In an inter code comparison by
Schrijver et al. [2006] one iteration step of (41) took about
80 hours for this method and only this one step was carriedout without further iteration. This seems, however, not to be
sufficient to derive an accurate nonlinear force-free solution.
The method has been applied for the comparison with soft
X-ray loops observed with YOHKOH by Wang et al. [2000]
and Liu et al. [2002] and to model a magnetic flux robe by
Yan et al. [2001a, 2001b].
[55] In a new implementation of the boundary element
method by Yan and Li [2006] the auxiliary functions arecomputed iteratively with the help of a simplex method.
This avoids the numerical expensive computation of thevolume integral (40). The boundary element method is stillrather slow if a magnetic field has to be computed in an
entire 3D domain. Different from other method, it allows,
however, to evaluate the NLFFF field at every arbitrary
point within the domain from the boundary data, without therequirement to compute the field in an entire domain. This
is in particular useful if one is interested to compute the
NLFFF field only along a given loop.
[56] He and Wang [2006] investigated the validity of theboundary integral representation for a spherical implementation. The method has been tested with the longitudinalinvariant [Low and Lou, 1990] solution. The spherical
implementation method of this method revealed reasonable
results for smooth modestly nonlinear fields, but a poor
convergence for complex magnetic field structures and large
values of a.

3. How to Deal With Non-Force-Free Boundaries
and Noise?
[57] Given arbitrary boundary conditions of the magnetic
field vector on the photosphere, the solution to the forcefree equations in 3D may not exist. Nonlinear force-freecoronal magnetic field models assume, however, that thesolution exists. It is certainly possible and necessary to

10 of 19

WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02check after or during the computation if a solution has beenfound. In the following we will discuss what we can do if
the measured photospheric data are incompatible with theassumption of a force-free coronal magnetic field.
3.1. Consistency Check of Vector Magnetograms[58] We reexamine some necessary conditions with thephotospheric field (or bottom boundary of a computationalbox). These conditions have to be fulfilled in order to besuitable boundary conditions for a nonlinear force-freecoronal magnetic field extrapolation. An a priori assumption
about the photospheric data is that the magnetic flux fromthe photosphere is sufficiently distant from the lateralboundaries of the observational domain and the net flux isin balance, i.e.,Z
Bz ðx; y; 0Þdxdy ¼ 0:

ð42Þ

S

Molodensky [1969, 1974], Aly [1989], and Sakurai [1989]
used the viral theorem to define which conditions a vector
magnetogram has to fulfill to be consistent with theassumption of a force-free field in the corona above theboundary. These conditions are as follows.
[59] 1. The total force on the boundary vanishesZ

Z
Bx Bz dxdy ¼
S

By Bz dxdy ¼ 0

ð43Þ

S

Z

Z

S

ðB2x þ B2y Þdxdy ¼

S

B2z dxdy:

ð44Þ[60] 2. The total torque on the boundary vanishes
Z

Z
S

xðB2x þ B2y Þdxdy ¼Z

xB2z dxdy;

ð45Þ

yB2z dxdy;

ð46Þ

xBy Bz dxdy:

ð47Þ

S

Z
S

yðB2x þ B2y Þdxdy ¼

SZZ
yBx Bz dxdy ¼
S

A03S02corona) with a plasma b  1, where the force-free assumption might break down. An additional problem is thatmeasurements of the photospheric magnetic vector fieldcontain inconsistencies and noise. In particular the transverse components (say Bx and By) of current vector magnetographs include uncertainties.
[62] The force-free field in a domain requires the Maxwell stress (43) – (47) to sum to zero over the boundary. If
these conditions are not fulfilled a force-free field cannot befound in the volume. A faithful algorithm should therefore
have the capability of rejecting a prescription of the vector
field at the boundary that fails to produce zero net Maxwell
stress. A simple way to incorporate these conditions would
be to evaluate the integrals (43) – (47) within or prior to the
NLFFF computation and to refuse the vector field if theconditions are not fulfilled with sufficient accuracy. Current
codes do run, however, although if feeded with inconsistentboundary data, but they certainly cannot find a force-freesolution in this case (because it does not exist). This
property of current codes does, however, not challenge thetrustworthiness of the algorithms, because the force-free andsolenoidal conditions are checked in 3D, for example, withthe help of the functional L as defined in (28). A non zero
value of L (within numerical accuracy) tells the user that aforce-free state has not been reached. In principle it would
be possible that the codes do refuse to output the magnetic
field in this case. For current codes this is not automaticallycontrolled but responsibility of the user.
[63] Unfortunately current measurements of the magnetic
field vector are only available routinely in the photosphere,
where we have a finite b plasma and nonmagnetic forces
might become important. The force-free compatibility conditions (43) – (47) are not fulfilled in the photosphere, butthey should be fulfilled in the low b chromospheric andcoronal plasma above. The question is if we still can use thephotospheric measurements to find suitable consistentboundary conditions for a nonlinear force-free modeling.
Such an approach has been called preprocessing of vector
magnetograms.
3.2. Preprocessing[64] The preprocessing routine has been developed byWiegelmann et al. [2006b]. The integral relations (43) – (47)have been used to define a 2D functional of quadratic forms:

S

Lprep ¼ m1 L1 þ m2 L2 þ m3 L3 þ m4 L4 ;[61] In an earlier review, Aly [1989] has mentioned
already that the magnetic field is probably not force-freein the photosphere, where B is measured because the plasmab in the photosphere is of the order of 1 and pressure andgravity forces are not negligible. The integral relations(43) – (47) are not satisfied in this case in the photosphereand the measured photospheric field is not a suitableboundary condition for a force-free extrapolation. Metcalf
et al. [1995] concluded that the solar magnetic field is notforce-free in the photosphere, but becomes force-free only
at about 400 km above the photosphere. Gary [2001]
pointed out that care has to be taken when extrapolating
the coronal magnetic field as a force-free field from photospheric measurements, because the force-free low corona issandwiched between two regions (photosphere and higher

ð48Þ

where
2
L1 ¼ 4Xp!2
B x BzþX!2
B y BzþpX

B2z  B2x  B2y

!2 3
5;

p

ð49Þ
"
L2 ¼

11 of 19X

!2
xðB2z  B2x  B2y Þ

p

þ

X
p

þ
!2 #

yBx Bz  xBy Bz

;

X

!2
yðB2z  B2x  B2y Þ

p

ð50Þ

WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02

L3 ¼

"
X

2

ðBx  Bxobs Þ þ

p

X

2

ðBy  Byobs Þ þ

p

X

#
2

ðBz  Bzobs Þ ;

p

ð51Þ
"
L4 ¼

X

#
2

2

2

ðDBx Þ þ ðDBy Þ þ ðDBz Þ :

ð52Þ

p

P
The surface integrals are here replaced by a summation pover all grid nodes p of the bottom surface grid and thedifferentiation in the smoothing term is achieved by theusual five-point stencil for the 2D-Laplace operator. Each
constraint Ln is weighted by a yet undetermined factor mn.
The first term (n = 1) corresponds to the force-balance
conditions (43) –(44), the next (n = 2) to the torque-free
condition (45) – (47). The following term (n = 3) ensuresthat the optimized boundary condition agrees with themeasured photospheric data and the last terms (n = 4)
controls the smoothing. The 2D-Laplace operator isdesignated by D. The aim of the preprocessing procedure
is to minimize Lprep so that all terms Ln if possible are madesmall simultaneously. A strategy on how to find the optimal
yet undefined parameters mn is described by Wiegelmann etal. [2006b]. As result of the preprocessing we get a data set
which is consistent with the assumption of a force-freemagnetic field in the corona but also as close as possible tothe measured data within the noise level.

4. Code Testing and Code Comparisons4.1. Code Testing[65] Newly developed codes for the extrapolation of
nonlinear force-free fields from boundary data have to betested before they are applied to measurements. In principle
any analytical or numerically created solution of the forcefree equations (1) – (3) can be used as a reference case.
One cuts a plane (artificial photosphere, bottom boundary)out of the 3D reference solution (for the pure task of code
testing it is also acceptable to use all six boundaries of thereference solution; these kind of data are not available forreal solar cases of course) and uses the above described
extrapolation codes to reconstruct the magnetic field. The
result of this extrapolation is then compared with thereference to rate the quality of the reconstruction. Unfortunately, it is very hard to find a truly nonlinear 3D solution of(1) – (3) analytically and very few solutions are known. Low
and Lou [1990] (LL) found a class of solutions which havebecome a standard reference for testing NLFFF extrapolation codes. LL found axisymmetric equilibria which areseparable in spherical coordinates. They are self-similar in
the radial coordinate, and the polar angle dependence isdetermined from a nonlinear eigenvalue equation. The
symmetry is broken by cutting out a rectangular chunk of
the solution by using a Cartesian coordinate system which isshifted and rotated with respect to the original coordinate
system in which the LL equilibria are calculated. The
parameters of the LL solutions and the parameters of the
new Cartesian coordinate system allow for a large numberof different situations which can be used for tests. The
original axisymmetric spherical LL solution has also been

A03S02

used (with and without symmetry breaking by shifting theorigin of the coordinate system) to test spherical NLFFF
programs. To our knowledge all recent implementations of
the described NLFFF approaches have been tested with LL,
either immediately in the original code-describing papers orin subsequent works, for example, in a blind-algorithm test
within the NLFFF consortium, as described below.
[66] The MHD relaxation method and the optimization
approach have been compared by Wiegelmann and Neukirch[2003]. Both methods have been applied to the Low and Lou[1990] equilibrium with exactly the same finite difference
grid. The iterative equations for MHD relaxation andoptimization have both the form @B@t ¼ mF but the structureof F is more complicated for optimization than for MHD
relaxation. The MHD relaxation term is indeed identicalwith the first term of the optimization approach. WhileMHD relaxation minimizes only the Lorentz force, theoptimization does additional minimize r B, while adecreasing magnetic field divergence during MHD relaxation [as shown by Wiegelmann and Neukirch, 2003] is theresult of numerical diffusion. Despite the numerical overhead in computing F for the optimization code, optimization
provided more accurate results and faster convergence.
[67] A practical advantage of the MHD approach is thatseveral time-dependent MHD codes are well known andestablished and can be used for the force-free relaxation
discussed here. The inclusion of nonmagnetic forces likepressure gradients and gravity looks straight forward for the
MHD approach. Other methods are usually developed withthe only task of computing nonlinear force-free coronalmagnetic fields, also a generalization toward magnetohydrostatic and stationary MHD equilibria is possible and has
been done for the optimization approach [see Wiegelmann
and Inhester, 2003; Wiegelmann and Neukirch, 2006].
Another advantage of using time-dependent MHD codes
for relaxation is that the computed force-free equilibrium
can be used on the same grid and with the same code as
initial state for time-dependent MHD simulations. One can,in principle, use the force-free equilibria computed with anyof the described method as initial state for time-dependent
MHD simulation, but having the initial equilibrium statealready directly on the MHD grid might be very handy,
because no further adjustments are needed.
4.2. NLFFF Consortium[68] Since the year 2004 activities are ongoing to bring
NLFFF modelers together and to compare the different
existing codes. A workshop series has been organized forthis aim by Karel Schrijver and three workshops took placeso far from 2004 – 2006. The next workshop is planed forJune 2007. As we have been asked to summarize theworkshop results on the CSWM meeting, we give also avery brief overview in the corresponding special issue paper
here. The main results of the first two workshops have been
published by Schrijver et al. [2006]. In this paper six
different NLFFF implementations (Grad-Rubin codes ofAmari et al. [1999] and Wheatland [2004], MHD relaxation
code of Valori et al. [2005], optimization codes by
McTiernan and by Wiegelmann [2004], boundary element
method by Yan and Sakurai [2000]) have been compared.
The codes have been tested in a blind algorithm test with thehelp of the semianalytic equilibrium by Low and Lou [1990]

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in two cases. In case I all six boundaries of a computational
box have been described, and in case II only the bottom
boundary has been described. The comparison of theextrapolation results with the reference solution has been
done qualitatively by magnetic field line plots (shown here
in Figure 3 for the central region of case II) and quantitatively by a number of sophisticated comparison matrices.
All NLFFF fields agreed best with the reference field for the
low-lying central magnetic field region, where the magnetic
field and electric currents are strongest and the influence ofthe boundaries lowest. The code converged with speeds thatdiffered by a factor of one million per iteration steps. (Thecodes run on different machines, have been written indifferent programming languages and used different compilers. A real test of the exact computing time would
comprise a proper operation count, for example, the numberof fixed point additions and multiplications per iteration
step.) The fastest-converging and best-performing code was
the Wheatland et al. [2000] optimization code as implemented by Wiegelmann [2004]. Recent implementations of
the Grad-Rubin code by Amari et al. [2006] and Inhesterand Wiegelmann [2006] and a new implementation of the
upward integration method by Song et al. [2006] did notparticipate in the blind-algorithm intercomparison by
Schrijver et al. [2006], but these three new codes have beentested by the authors with similar measures and revealed
similar accuracy as the best performing codes in the blind
algorithm test. It seems that the somewhat more flexible
boundary conditions used in the Grad-Rubin approaches ofAmari et al. [2006] and Inhester and Wiegelmann [2006] areresponsible for the better performance compared to theearlier implementation by Amari et al. [1999], which has
been used in the blind algorithm test.
[69] The widely used LL equilibrium contains a verysmooth photospheric magnetic field and an extended current
distribution. It is therefore also desirable to test NLFFF
codes also with other, more challenging boundary fields,
which are less smooth, have localized current distribution
and to investigate also the effects of noise and effects fromnon-force-free boundaries. A somewhat more challenging
reference case is the equilibrium found by Titov andDémoulin [1999] (TD). Similar to LL, the TD equilibrium
is an axisymmetric equilibrium. The TD model contains a
potential field which is disturbed by a toroidal nonlinear
force-free current. This equilibrium has been used fortesting the MHD relaxation code (G. Valori and B. Kliem,
personal communication, 2006) and the optimization codefrom Wiegelmann et al. [2006a].
[70] Any numerically created NLFFF model might besuitable for code testing, too. It is in particular interesting to
use models, which are partly related on observational data.
Very recently, van Ballegooijen et al. [2007] used line-ofsight photospheric measurements from SOHO/MDI to compute a potential field, which was then disturbed by insertinga twisted flux robe and relaxed toward a nonlinear force-freestate with a magnetofrictional method as described by vanBallegooijen [2004]. The van Ballegooijen et al. [2007]
model is not force-free in the entire computational domain,
but only above a certain height above the bottom boundary(artificial chromosphere). On the lowest boundary (photosphere) the model contains significant nonmagnetic forces.
Both the chromospheric as well as the photospheric mag-

A03S02

netic field vector from the van Ballegooijen et al. [2007]
model have been used to test four of the recently developed
extrapolation codes (one Grad-Rubin method, one MHD
relaxation code and two optimization approaches) in asecond blind algorithm test by Metcalf et al. [2007]. While
the NLFFF consortium paper, part I [Schrijver et al., 2006],
used a domain of just 643 pixel, the part II paper used a
computational domain of 320  320  258 pixel andmodern NLFFF codes where able to compute the nonlinear
force-free field in such relatively large boxes within a few
hours for a moderate parallelization on only 1– 4 processors
and a memory requirement of 2.5– 4 GB of RAM. This very
recent code comparison shows a major improvement regarding computing time and suitable grid sizes within lessthan 3 years. On the first NLFFF consortium meeting in
2004, box sizes of some 643 have been a kind of standard orcomputing times of some 2 weeks have been reported for1503 boxes. We briefly summarize the results of Metcalf et al.
[2007] as: (1) NLFFF extrapolations from chromospheric
data recover the original reference field with high accuracy;(2) when the extrapolations are applied to the photosphericdata, the reference field is not well recovered; and (3)preprocessing of the photospheric data improve the result,but the accuracy is still lower as for extrapolations from thechromosphere.

5. Conclusions and Outlook[71] Within the last few years the scientific community
showed a growing interest into coronal magnetic fields.
(Publications containing the phrase ’coronal magnetic
fields’ in title or abstract have been cited less than about
50 times per year until the early 1990s and this number
increased to about 150 citations per year in 2004. A peakyear was 2006 (last year) with more than 300 citations(source: ISI Web of Knowledge, March 2007)). The development of new ground based and space born vector magnetographs provide us measurements of the magnetic field
vector on the suns photosphere. Accompanied from these
hardware development, software has been developed to
extrapolate the photospheric measurements into the corona.
Special attention has recently been given to nonlinear forcefree codes. Five different numerical approaches (GradRubin, upward integration, MHD relaxation, optimization,boundary elements) have been developed for this aim. It isremarkable that new codes or major updates of existingcodes have been published for all five methods within thelast two years, mainly in the last year (2006). A workshop
series (NLFFF consortium) since 2004 on nonlinear forcefree fields has recently released synergy effects, by bringing
modelers of the different numerical implementations together to compare, evaluate and improve the programs. Severalof the most recent new codes and utility programs (e.g.,
preprocessing) have at least been partly inspired by theseworkshops. The new implementations have been tested withthe smooth semianalytic Low-Lou equilibrium and showedreasonable agreement with this reference field. While all
methods aim for a reconstruction of the coronal magnetic
field from the photospheric magnetic field vector, the way
how these measurements are used to prescribe the boundaries of the codes is different.

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Figure 3. Evaluation of six nonlinear force – free codes. The (a) reference solution has been
compared with extrapolations with (b, c) optimation, (d) MHD relaxation, (e, f) Grand – Rubin, and(g) boundary element. For comparison, (h) linear-force-free and (i) potential fields are shown, too.
The images show the central domain of the model. Only the bottom boundary has been provided forthe extrapolation. (This figure was originally published as Schrijver et al.’s [2006] Figure 4. Used
with permission of Springer.)

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WIEGELMANN: CORONAL MAGNETIC FIELDS

A03S02[72] 1. MHD relaxation and optimization use Bx0, By0, Bz0
on the bottom boundary. This overdetermines the boundary
value problem. Both methods are closely related and compute the magnetic field in a computational box with
@B
¼ mF;
@t

ð53Þ

where the structure of F is somewhat different (theoptimization approach has more terms) for both methods.
Usually a potential field is used as initial state for both
approaches, also the use of a linear force-free initial state ispossible. Recently a multiscale version of optimization has
been installed, which uses a low-resolution NLFFF field as
input for higher-resolution computations. Specifying theentire magnetic field vector on the bottom boundary is an
overimposed problem and a unique NLFFF field (or asolution at all) requires that the boundary data fulfill certain
consistency criteria. A recently developed preprocessing
routine helps to find suitable consistent boundary data frominconsistent photospheric measurements. Earlier and current
comparisons showed a somewhat higher accuracy for the
optimization approach. A practical advantage of the MHD
approach is that in principle any available time-dependent
MHD code can be adjusted to compute the NLFFF field.
[73] 2. The Grad-Rubin approach uses Bz0 and the distribution of a computed with equation (8) for one polarity,
which corresponds to well posed mathematical problem. A
practical problem is that the computation of a requires
numerical differences of the noisy and forced transverse
photospheric field Bx0, By0 with (7) leading to inaccuracies
in the normal electric current distribution and in a. Forsmooth semianalytic test cases this is certainly not a
problem, but real data require special attention (smoothing,
preprocessing, limiting a 6¼ 0 to regions where Bz0 is above
a certain limit) to derive a meaningful distribution of a.
While the method requires only a for one polarity, thecomputation from photospheric data provide a for bothpolarities. We are not aware of any tests on how well
NLFFF solutions computed from a prescribed on thepositive and negative polarity coincide. It is also unclear
how well the computed transverse field components on thebottom boundary agree with the measured values of Bx0,
By0. (In principle Bx0, By0 may have an additional field (Bx0,
By0) + (@ x@ y) 8 without making a difference for a and hence
for the Grad-Rubin result.) More tests on this topics arenecessary, including the recently installed possibility toprescribe a for both polarities and adjust the boundary bya weighed average of a on both polarities to fulfill
equation (6). As initial state the Grad-Rubin method uses
a potential field, which is also true for MHD relaxation andoptimization.
[74] 3. The upward integration and the boundary element
method prescribe both all components of the bottom boundary magnetic field vector and the a distribution computedwith equation (8). This approach over imposes the boundary
and Bx0, By0, Bz0 and a have to be consistent which eachother and the force-free assumption. This is certainly not a
problem at all for smooth semianalytic test equilibria andstrategies to derive consistent boundary data from measured
data have been developed recently. Different from the three

A03S02

approaches discussed above, upward integration and boundary element methods do not require to compute first an
initial potential field in the computational domain. It is wellknown that the upward integration method is based on an
ill-posed problem and the method has not been considered
for several years, but a recent implementation with smooth
analytic functions might help to regularize this method. First
tests showed a reasonable results for computations with thesmooth semianalytic Low-Lou solution.
[75] The boundary element method has the problem to bevery slow and an earlier implementation of this method
could not reach a converged state for a 643 boxed used in
the NLFFF consortium paper, part I, owing to this problem.
A new ’direct boundary method’ has been developed, which
seems to be faster than the original ’boundary element
method’, but still slower compared with the four other
NLFFF approaches if the task is to compute a 3D magnetic
field in an entire 3D domain. Different from all otherdescribed methods the boundary element approach allows
to compute the nonlinear force-free field vector at any
arbitrary point above the boundary and it is not necessary
to compute the entire 3D field above the photosphere. Thismight be a very useful feature if one is interested incomputing the magnetic field only along a single loop and
not interested in an entire active region.
[76] The new implementations of upward integration andboundary element method show both reasonable results forfirst tests with the smooth semianalytic Low and Lou
equilibrium. Further tests with more sophisticated equilibria, for example, a solar-like test case as used in the
NLFFF consortium paper, part II, would be useful tocome to more sound conclusions regarding the feasibilityof these methods.
[77] Most of the efforts done in nonlinear force-freemodeling until now concentrated mainly on developing
these models and testing their accuracy and speed withthe help of well known test configuration. Not too many
applications of nonlinear force-free models to real data are
currently available, from which we learned new physics.
One reason was the insufficient access to high-accuracy
photospheric vector magnetograms and a second one werelimitations of the models. Force-free field extrapolation is amere tool, if properly employed on vector magnetograms, it
can help to understand physical, magnetic field dominatedprocesses in the corona. Both the computational methods aswell as the accuracy of required measurements (e.g., with
Hinode, SDO) are rapidly improving. Within the NLFFF
consortium we just started (since April 2007) to apply thedifferent codes to compute nonlinear force-free coronalmagnetic fields from Hinode vectormagnetograms. This
project might provide us already some new insights aboutcoronal physics.
[78] To conclude, we can say that the capability of
Cartesian nonlinear force-free extrapolation codes has rapidly increased in recent years. Only 3 years ago most codes
run usually on grids of about 643 pixel. Recently developed
or updated codes (Grad-Rubin by Wheatland, MHD relaxation by Valori, optimization by Wiegelmann, optimization
by McTiernan) have been applied to grids of about 3003
pixel. Although this increase of traceable grid sizes iscertainly encouraging, the resolution of current and near-

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future vector magnetographs (which of course measure only
data in 2D!) is significantly higher. We should keep in mind,
however, that the currently implemented NLFFF codes havebeen only moderately parallelized using only a few processors. The CSWM conference, where this paper has beenpresented, took place at the ’Earth simulator’ in Yokohama,
which contains several thousands of processors used for
Earth-science computer simulations. An installation of
NLFFF codes on such massive parallel computers (which
has been briefly addressed on NLFFF consortium meetings)
combined with adaptive mesh refinements might enable
drastically improved grid sizes. One should not underestimate the time and effort necessary to program and installsuch massive parallelized versions of existing codes. As fulldisk vectormagnetograms will become available soon
(SOLIS, SDO/HMI) it is also an important task to take aspherical geometry into account. First steps in this direction
have been carried out with the optimization and boundary
element methods. Spherical NLFFF geometries are currently still in it’s infancy and have been tested until now onlywith smooth semianalytic Low and Lou equilibria andrequire further developments.
[79] Attention has also recently been drawn to the problem that the coronal magnetic field is force-free, but thephotospheric one is not. Tests with extrapolations fromsolar-like artificial photospheric and chromospheric measurements within the NLFFF consortium paper, part II,
revealed that extrapolations from the (force-free) chromospheric field provide significantly better results as extrapolations using directly the (forced) photospheric field.
Applying a preprocessing program on the photosphericdata, which effectively removes the nonmagnetic forces,
leads to significantly better results, but they are not as good
as by using the chromospheric magnetic field vector as
boundary condition. An area of current research is thepossibility to use chromospheric images to improve thepreprocessing of photospheric magnetic field measurements. Improvements in measuring the chromospheric magnetic field directly [e.g., Lagg et al., 2004] might further
improve to find suitable boundary conditions for NLFFF
extrapolations. Force-free extrapolations are not suitable,
however, to understand the details of physical processes on
how the magnetic field evolves from the forced photosphere
into the chromosphere, because nonmagnetic forces areimportant in the photosphere. For a better understandingof these phenomena more sophisticated models which takepressure gradients and gravity (and maybe also plasmaflow) into account are required. Some first steps have been
done with a generalization of the optimization method byWiegelmann and Neukirch [2006], but such approaches arestill in their infancy and have been tested so far only withsmooth MHD equilibria. It is also not entirely clear how
well necessary information regarding the plasma (density,pressure, temperature, flow) can be derived from measurements. Nonmagnetic forces become important also in quiet
sun regions [Schrijver and van Ballegooijen, 2005] and in
the higher layers of the corona, where the plasma b is of theorder of unity. Coronagraph measurements, preferably fromtwo viewpoints as provided by the STEREO mission,
combined with a tomographic inversion might help here
to get insights in the required 3D structure of the plasma
density. One should also pay attention to the combination of

A03S02extrapolation methods, as described here, with measurements of the Hanle and Zeeman effects in coronal lines
which allows the reconstruction of the coronal magnetic
field as proposed in feasibility studies of vector tomography
by Kramar et al. [2006] and Kramar and Inhester [2006].
Other measurements of coronal features, for example,
coronal plasma images from two STEREO viewpoints,
can be used for observational tests of coronal magnetic
field models. Using two viewpoints provide a much more
restrictive test of models as images from only one view
direction. While a nonlinear force-free coronal magnetic
field model helps us to derive the topology, magnetic fieldand electric current strength in coronal loops, they do notprovide plasma parameters. One way to get insights regarding the coronal plasma is the use of scaling laws to model
the plasma along the reconstructed 3D field lines andcompare correspondent artificial plasma images with realcoronal images. Schrijver et al. [2004] applied such an
approach to global potential coronal magnetic fields andcompared simulated and real coronal images from one
viewpoint. A generalization of such methods toward theuse of more sophisticated magnetic field models and coronal images from two STEREO viewpoints will probably
provide many insights regarding the structure and physics ofthe coronal plasma. An important challenge is for example
the coronal heating problem. The dominating coronal magnetic field is assumed to play an important role here,
because magnetic field configuration containing free energy
can under certain circumstances reconnect Priest [1996,
1999] and supply energy for coronal heating. Priest et al.
[2005] pointed out that magnetic reconnection at separatorsand separatrices plays an important role for coronal heating.
Nonlinear force-free models can help here to identify themagnetic field topology, magnetic null points, separatrices
and localized strong current concentration. While magnetic
reconnection [see, e.g., Priest and Schrijver, 1999] is a
dynamical phenomenon, the static magnetic field models
discussed here can help to identify the locations favorable
for reconnection. Time sequences of nonlinear force-freemodels computed from corresponding vector magnetograms
will also tell wether the topology of the coronal magnetic
field has changed due to reconnection, even if the physics ofreconnection is not described by force-free models. Sophisticated 3D coronal magnetic field models and plasma
images from two viewpoints might help to constrain thecoronal heating function further, which has been done so far
with plasma images from one viewpoint [Aschwanden,
2001a, 2001b] (by using data from Yokoh, Soho and Trace).
[80] Acknowledgments. I would like to thank the organizers of
CSWM for inviting me to give a review talk on nonlinear force-freemagnetic field modeling. The author thanks Bernd Inhester for usefulcomments and acknowledge inspiriting discussions with many other
NLFFF modelers on three workshops organized by Karel Schrijver (NLFFF
consortium) in Palo Alto between 2004 and 2006 and at various otheroccasions. This work was supported by DLR grant 50 OC 0501.
[81] Amitava Bhattacharjee thanks Boon-Chye Low and another reviewer for their assistance in evaluating the manuscript.

References
Alissandrakis, C. E. (1981), On the computation of constant alpha forcefree magnetic field, Astron. Astrophys., 100, 197 – 200.
Aly, J. J. (1984), On some properties of force-free magnetic fields in infinite
regions of space, Astrophys. J., 283, 349 – 362, doi:10.1086/162313.

16 of 19

A03S02

WIEGELMANN: CORONAL MAGNETIC FIELDS

Aly, J. J. (1989), On the reconstruction of the nonlinear force-free coronalmagnetic field from boundary data, Sol. Phys., 120, 19 – 48.
Aly, J. J. (2005), A uniqueness result for a simple force-free magnetic fieldsubmitted to a topological constraint, Astron. Astrophys., 429, 15 – 18,
doi:10.1051/0004-6361:20041547.
Amari, T., J. J. Aly, J. F. Luciani, T. Z. Boulmezaoud, and Z. Mikic (1997),
Reconstructing the solar coronal magnetic field as a force-free magnetic
field, Sol. Phys., 174, 129 – 149.
Amari, T., T. Z. Boulmezaoud, and Z. Mikic (1999), An iterative method
for the reconstructionbreak of the solar coronal magnetic field. I. Method
for regular solutions, Astron. Astrophys., 350, 1051 – 1059.
Amari, T., T. Z. Boulmezaoud, and J. J. Aly (2006), Well posed reconstruction of the solar coronal magnetic field, Astron, Astrophys., 446, 691 –
705, doi:10.1051/0004-6361:20054076.
Aschwanden, M. J. (2001a), Revisiting the determination of the coronalheating function from Yohkoh data, Astrophys. J., 559, L171 – L174,
doi:10.1086/323788.
Aschwanden, M. J. (2001b), An evaluation of coronal heating models for
active regions based on Yohkoh, SOHO, and TRACE observations, Astrophys. J., 560, 1035 – 1044, doi10.1086/323064.
Aschwanden, M. J. (2005), Physics of the Solar Corona: An Introduction
with Problems and Solutions, 2nd ed., Springer, New York.
Aschwanden, M. J., J. S. Newmark, J.-P. Delaboudinière, W. M. Neupert,
J. A. Klimchuk, G. A. Gary, F. Portier-Fozzani, and A. Zucker (1999),
Three-dimensional stereoscopic analysis of solar active region loops. I.
SOHO/EIT observations at temperatures of (1.0 – 1.5)  106 K, Astrophys. J., 515, 842 – 867, doi:10.1086/307036.
Bineau, M. (1972), Existence of force-free magnetic-fields, Commun. Pure
Appl. Math., 25, 77 – 84.
Bleybel, A., T. Amari, L. van Driel-Gesztelyi, and K. D. Leka (2002),
Global budget for an eruptive active region. I. Equilibrium reconstruction
approach, Astron. Astrophys., 395, 685 – 695, doi:10.1051/00046361:20021332.
Borrero, J. M., S. Tomczyk, A. Norton, T. Darnell, J. Schou, P. Scherrer,
R. Bush, and Y. Lui (2006), Magnetic field vector retrieval with HMI, in
Solar Polarization 4, ASP Conf. Ser., vol. 358, edited by R. Casini andB. W. Lites, p. 144, Astron. Soc. of the Pac., San Francisco, Calif.
(Available at http://arxiv.org/abs/astro-ph/0602130)
Bossavit, A. (1988), Mixed finite elements and the complex of Whitney
forms, in The Mathemetics of Finite Elements and Applications VI, edited
by J. Whiteman, pp. 137 – 144, Academic Press, London.
Boulmezaoud, T. Z., and T. Amari (2000), On the existence of non-linear
force-free fields in three-dimensional domains, ZAMP, 51, 942 – 967.
Carcedo, L., D. S. Brown, A. W. Hood, T. Neukirch, and T. Wiegelmann(2003), A quantitative method to optimise magnetic field line fitting ofobserved coronal loops, Sol. Phys., 218, 29 – 40.
Chiu, Y. T., and H. H. Hilton (1977), Exact Green’s function method ofsolar force-free magnetic-field computations with constant alpha. I. Theory and basic test cases, Astrophys. J., 212, 873 – 885.
Chodura, R., and A. Schlueter (1981), A 3D code for MHD equilibrium andstability, J. Comput. Phys., 41, 68 – 88.
Cuperman, S., L. Ofman, and M. Semel (1990), Extrapolation of photospheric potential magnetic fields using oblique boundary values—A simplified approach, Astron. Astrophys., 227, 583 – 590.
Cuperman, S., P. Demoulin, and M. Semel (1991), Removal of singularities
in the Cauchy problem for the extrapolation of solar force-free magnetic
fields, Astron. Astrophys., 245, 285 – 288.
Dedner, A., F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and
M. Wesenberg (2002), Hyperbolic divergence cleaning for the MHD
equations, J. Comput. Phys., 175, 645 – 673.
Demoulin, P., and E. R. Priest (1992), The properties of sources and sinksof a linear force-free field, Astron. Astrophys., 258, 535 – 541.
Gary, G. A. (2001), Plasma Beta above a solar active region: Rethinking theparadigm, Sol. Phys., 203, 71 – 86.
Gary, G. A., and M. J. Hagyard (1990), Transformation of vector magnetograms and the problems associated with the effects of perspective and theazimuthal ambiguity, Sol. Phys., 126, 21 – 36.
Grad, H., and H. Rubin (1958), Hydromagnetic equilibria and force-freefields, paper presented at 2nd International Conference on Peaceful Usesof Atomic Energy, Int. At. Energy Agency, Geneva.
He, H., and H. Wang (2006), The validity of the boundary integral equation
for magnetic field extrapolation in open space above a spherical surface,
Mon. Not. R. Astron. Soc., 369, 207 – 215, doi:10.1111/j.13652966.2006.10288.x.
Henney, C. J., C. U. Keller, and J. W. Harvey (2006), SOLIS-VSM
solar vector magnetograms, in Solar Polarization 4, ASP Conf. Ser.,vol. 358, edited by R. Casini and B. W. Lites, p. 92, Astron. Soc. of
the Pac., San Francisco, Calif. (Available at http://arxiv.org/abs/astro-ph/
0701122)

A03S02Inhester, B., and T. Wiegelmann (2006), Nonlinear force-free magnetic field
extrapolations: Comparison of the Grad Rubin and Wheatland Sturrock
Roumeliotis algorithm, Sol. Phys., 235, 201 – 221, doi:10.1007/s11207006-0065-x.
Jiao, L., A. N. McClymont, and Z. Mikic (1997), Reconstruction of thethree-dimensional coronal magnetic field, Sol. Phys., 174, 311 – 327.
Judge, P. G. (1998), Spectral lines for polarization measurements of thecoronal magnetic field. I. Theoretical intensities, Astrophys. J., 500,
1009 – 1022.
Kageyama, A., and T. Sato (2004), ‘‘Yin-Yang grid’’: An overset grid in
spherical geometry, Geochem. Geophys. Geosyst., 5, Q09005,
doi:10.1029/2004GC000734.
Keppens, R., M. Nool, G. Tóth, and J. P. Goedbloed (2003), Adaptive meshrefinement for conservative systems: Multi-dimensional efficiency evaluation, Comput. Phys. Commun., 153, 317 – 339.
Klimchuk, J. A., and P. A. Sturrock (1992), Three-dimensional force-freemagnetic fields and flare energy buildup, Astrophys. J., 385, 344 – 353,
doi:10.1086/170943.
Kramar, M., and B. Inhester (2006), Inversion of coronal Zeeman andHanle observations to reconstruct the coronal magnetic field, Mem.
Soc. Astron. Ital., 78, 120. (Available at http://arxiv.org/abs/astro-ph/0612639)
Kramar, M., B. Inhester, and S. K. Solanki (2006), Vector tomography forthe coronal magnetic field. I. Longitudinal Zeeman effect measurements,
Astron. Astrophys., 456, 665 – 673, doi:10.1051/0004-6361:20064865.
LaBonte, B. J., D. L. Mickey, and K. D. Leka (1999), The imaging vector
magnetograph at Haleakala: II. Reconstruction of Stokes spectra, Sol.
Phys., 189, 1 – 24.
Lagg, A., J. Woch, N. Krupp, and S. K. Solanki (2004), Retrieval of the fullmagnetic vector with the He I multiplet at 1083 nm: Maps of an emerging
flux region, Astron. Astrophys., 414, 1109 – 1120.
Leka, K. D., and A. Skumanich (1999), On the value of ‘aAR’ from vector
magnetograph data: I. Methods and caveats, Sol. Phys., 188, 3 – 19.
Li, Z., Y. Yan, and G. Song (2004), Properties of the boundary integral
equation for solar non-constant-a force-free magnetic fields, Mon. Not. R.
Astron. Soc., 347, 1255 – 1265, doi:10.1111/j.1365-2966.2004.07309.x.
Lin, H., J. R. Kuhn, and R. Coulter (2004), Coronal magnetic field measurements, Astrophys. J. Lett., 613, L177 – L180, doi:10.1086/425217.
Liu, Y., X. P. Zhao, J. T. Hoeksema, P. H. Scherrer, J. Wang, and Y. Yan(2002), On formation of the sigmoidal structure in solar active region
NOAA 8100, Sol. Phys., 206, 333 – 346.
Low, B. C., and Y. Q. Lou (1990), Modeling solar force-free magnetic
fields, Astrophys. J., 352, 343 – 352.
McClymont, A. N., L. Jiao, and Z. Mikic (1997), Problems and progress incomputing three-dimensional coronal active region magnetic fields fromboundary data, Sol. Phys., 174, 191 – 218.
Metcalf, T. R. (1994), Resolving the 180-degree ambiguity in vector magnetic field measurements: The ‘minimum’ energy solution, Sol. Phys.,155, 235 – 242.
Metcalf, T. R., L. Jiao, A. N. McClymont, R. C. Canfield, andH. Uitenbroek (1995), Is the solar chromospheric magnetic field forcefree?, Astrophys. J., 439, 474 – 481, doi:10.1086/175188.
Metcalf, T. R., et al. (2006), An overview of existing algorithms for resolving the 180-degree ambiguity in vector magnetic fields: Quantitative
tests with synthetic data, Sol. Phys., 237, 267 – 296, doi:10.1007/
s11207-006-0170-x.
Me t c a l f , T. R . , M. L. DeR o s a , C. J . S c h r i j v e r, G . B a r n e s ,A. VanBallegooijen, T. Wiegelmann, M. S. Wheatland, G. Valori, andJ. M. McTiernan (2007), Non-linear force-free modeling of coronal magnetic fields. II. Modeling a filament arcade from simulated chromosphericand photospheric vector fields, Sol. Phys., in press.
Mickey, D. L., R. C. Canfield, B. J. Labonte, K. D. Leka, M. F. Waterson,and H. M. Weber (1996), The imaging vector magnetograph at Haleakala,Sol. Phys., 168, 229 – 250.
Mikic, Z., and A. N. McClymont (1994), Deducing coronal magnetic fields
from vector magnetograms, in Solar Active Region Evolution: Comparing Models with Observations, ASP Conf. Ser., vol. 68, edited by K. S.
Balasubramaniam and G. W. Simon, pp. 225-232, Astron. Soc. of the
Pac., San Francisco, Calif.
Mikic, Z., D. C. Barnes, and D. D. Schnack (1988), Dynamical evolution ofa solar coronal magnetic field arcade, Astrophys. J., 328, 830 – 847,
doi:10.1086/166341.
Molodensky, M. M. (1969), Integral properties of force-free fields, Soviet
Astron., Engl. Transl., 12, 585 – 588.
Molodensky, M. M. (1974), Equilibrium and stability of force-free magnetic field, Sol. Phys., 39, 393 – 404.
Nakagawa, Y. (1974), Dynamics of the solar magnetic field. I. Methodof examination of force-free magnetic fields, Astrophys. J., 190, 437 –440.

17 of 19

A03S02

WIEGELMANN: CORONAL MAGNETIC FIELDS

Nakagawa, Y., and M. A. Raadu (1972), On practical representation ofmagnetic field, Sol. Phys., 25, 127 – 135.
Pevtsov, A. A., R. C. Canfield, and T. R. Metcalf (1994), Patterns ofhelicity in solar active regions, Astrophys. J. Lett., 425, L117 – L119,
doi:10.1086/187324.
Press, W. H. (2002), Numerical Recipes in C++: The Art of Scientific
Computing, 1002 pp., Cambridge Univ. Press, New York.
Priest, E. (1996), Coronal heating by magnetic reconnection, Astrophys.
Space Sci., 237, 49 – 73.
Priest, E. R. (1999), Heating the solar corona by magnetic reconnection,
Astrophys. Space Sci., 264, 77 – 100.
Priest, E. R., and C. J. Schrijver (1999), Aspects of three-dimensional
magnetic reconnection (invited review), Sol. Phys., 190, 1 – 24,
doi:10.1023/A:1005248007615.
Priest, E. R., D. W. Longcope, and J. Heyvaerts (2005), Coronal heating at
separators and separatrices, Astrophys. J., 624, 1057 – 1071, doi:10.1086/
429312.
Régnier, S., and T. Amari (2004), 3D magnetic configuration of the Ha
filament and X-ray sigmoid in NOAA AR 8151, Astron. Astrophys., 425,
345 – 352.
Régnier, S., and R. C. Canfield (2006), Evolution of magnetic fields andenergetics of flares in active region 8210, Astron. Astrophys., 451, 319 –
330, doi:10.1051/0004-6361:20054171.
Régnier, S., T. Amari, and E. Kersalé (2002), 3D coronal magnetic fieldfrom vector magnetograms: Non-constant-alpha force-free configurationof the active region NOAA 8151, Astron. Astrophys., 392, 1119 – 1127.
Régnier, S., T. Amari, and R. C. Canfield (2005), Self and mutual magnetic
helicities in coronal magnetic configurations, Astron. Astrophys., 442,
345 – 349, doi:10.1051/0004-6361:20053509.
Roumeliotis, G. (1996), The ‘‘stress-and-relax’’ method for reconstructing
the coronal magnetic field from vector magnetograph data, Astrophys. J.,
473, 1095 – 1103.
Sakurai, T. (1981), Calculation of force-free magnetic field with non constant alpha, Sol. Phys., 69, 343 – 359.
Sakurai, T. (1982), Green’s function methods for potential magnetic fields,
Sol. Phys., 76, 301 – 321.
Sakurai, T. (1989), Computational modeling of magnetic fields in solar
active regions, Space Sci. Rev., 51, 11 – 48.
Sakurai, T., et al. (1995), Solar flare telescope at Mitaka, Publ. Astron. Soc.
Jpn., 47, 81 – 92.
Schatten, K. H., J. M. Wilcox, and N. F. Ness (1969), A model of interplanetary and coronal magnetic fields, Sol. Phys., 6, 442 – 455.
Schmidt, H. U. (1964), On the observable effects of magnetic energy storage and release connected with solar flares, in The Physics of Solar
Flares, edited by W. N. Hess, pp. 107 – 114, NASA, Washington, D. C.
Schrijver, C. J., and A. A. van Ballegooijen (2005), Is the quiet-sun corona
a quasi-steady, force-free environment?, Astrophys. J., 630, 552 – 560,
doi:10.1086/431754.
Schrijver, C. J., A. W. Sandman, M. J. Aschwanden, and M. L. DeRosa(2004), The coronal heating mechanism as identified by full-sun visualizations, Astrophys. J., 615, 512 – 525, doi:10.1086/424028.
Schrijver, C. J., M. L. DeRosa, A. M. Title, and T. R. Metcalf (2005), The
nonpotentiality of active-region coronae and the dynamics of the photospheric magnetic field, Astrophys. J., 628, 501 – 513, doi:10.1086/
430733.
Schrijver, C. J., M. L. Derosa, T. R. Metcalf, Y. Liu, J. McTiernan,
S. Régnier, G. Valori, M. S. Wheatland, and T. Wiegelmann (2006),
Nonlinear force-free modeling of coronal magnetic fields part i: A quantitative comparison of methods, Sol. Phys., 235, 161 – 190, doi:10.1007/
s11207-006-0068-7.
Seehafer, N. (1978), Determination of constant alpha force-free solar magnetic fields from magnetograph data, Sol. Phys., 58, 215 – 223.
Seehafer, N. (1982), A comparison of different solar magnetic field extrapolation procedures, Sol. Phys., 81, 69 – 80.
Semel, M. (1967), Contribution à létude des champs magn\’ etiques dansles régions actives solaires, Ann. Astrophys., 30, 513.
Semel, M. (1988), Extrapolation functions for constant-alpha force-freefields—Green’s method for the oblique boundary value, Astron. Astrophys., 198, 293 – 299.
Shimizu, T. (2004), SolarB Solar Optical Telescope (SOT), in The Solar-B
Mission and the Forefront of Solar Physics, ASP Conf. Ser., vol. 325,
edited by T. Sakurai and T. Sekii, p. 3, Astron. Soc. of the Pac., San
Francisco, Calif.
Solanki, S. K., A. Lagg, J. Woch, N. Krupp, and M. Collados (2003),
Three-dimensional magnetic field topology in a region of solar coronalheating, Nature, 425, 692 – 695.
Song, M. T., C. Fang, Y. H. Tang, S. T. Wu, and Y. A. Zhang (2006), A newand fast way to reconstruct a nonlinear force-free field in the solar corona,
Astrophys. J., 649, 1084 – 1092, doi:10.1086/506249.

A03S02

Titov, V. S., and P. Démoulin (1999), Basic topology of twisted magnetic
configurations in solar flares, Astron. Astrophys., 351, 707 – 720.
Török, T., and B. Kliem (2003), The evolution of twisting coronal magnetic
flux tubes, Astron. Astrophys., 406, 1043 – 1059, doi:10.1051/00046361:20030692.
Valori, G., B. Kliem, and R. Keppens (2005), Extrapolation of a nonlinear
force-free field containing a highly twisted magnetic loop, Astron. Astrophys., 433, 335 – 347.
van Ballegooijen, A. A. (2004), Observations and modeling of a filamenton the Sun, Astrophys. J., 612, 519 – 529, doi:10.1086/422512.
van Ballegooijen, A. A., E. E. Deluca, K. Squires, and D. H. Mackay(2007), Modeling magnetic flux ropes in the solar atmosphere, J. Atmos.
Terr. Phys., 69, 24 – 31, doi:10.1016/j.jastp.2006.06.007.
Wang, H., Y. Yan, T. Sakurai, and M. Zhang (2000), Topology of magnetic
field and coronal heating in solar active regions. II. The role of quasiseparatrix layers, Sol. Phys., 197, 263 – 273.
Wheatland, M. S. (1999), A better linear force-free field, Astrophys. J., 518,
948 – 953, doi:10.1086/307301.
Wheatland, M. S. (2004), Parallel construction of nonlinear force-freefields, Sol. Phys., 222, 247 – 264.
Wheatland, M. S. (2006), A fast current-field iteration method for calculating nonlinear force-free fields, Sol. Phys., 238, 29 – 39, doi:10.1007/
s11207-006-0232-0.
Wheatland, M. S., P. A. Sturrock, and G. Roumeliotis (2000), An optimization approach to reconstructing force-free fields, Astrophys. J., 540,
1150 – 1155.
Wiegelmann, T. (2004), Optimization code with weighting function for thereconstruction of coronal magnetic fields, Sol. Phys., 219, 87 – 108.
Wiegelmann, T. (2007), Computing nonlinear force-free coronal magnetic
fields in spherical geometry, Sol. Phys., 240, 227 – 239, doi:10.1007/
s11207-006-0266-3.
Wiegelmann, T., and B. Inhester (2003), Magnetic modeling and tomography: First steps towards a consistent reconstruction of the solar corona,
Sol. Phys., 214, 287 – 312.
Wiegelmann, T., and T. Neukirch (2002), Including stereoscopic information in the reconstruction of coronal magnetic fields, Sol. Phys., 208,
233 – 251.
Wiegelmann, T., and T. Neukirch (2003), Computing nonlinear force
free coronal magnetic fields, Nonlinear Processes Geophys., 10, 313 –322.
Wiegelmann, T., and T. Neukirch (2006), An optimization principle for thecomputation of MHD equilibria in the solar corona, Astron. Astrophys.,457, 1053 – 1058, doi:10.1051/0004-6361:20065281.
Wiegelmann, T., B. Inhester, A. Lagg, and S. K. Solanki (2005a), How to
use magnetic field information for coronal loop identification, Sol. Phys.,
228, 67 – 78, doi:10.1007/s11207-005-2511-6.
Wiegelmann, T., A. Lagg, S. K. Solanki, B. Inhester, and J. Woch (2005b),
Comparing magnetic field extrapolations with measurements of magnetic
loops, Astron. Astrophys., 433, 701 – 705.
Wiegelmann, T., B. Inhester, B. Kliem, G. Valori, and T. Neukirch (2006a),
Testing non-linear force-free coronal magnetic field extrapolations with
the Titov-Démoulin equilibrium, Astron. Astrophys., 453, 737 – 741,
doi:10.1051/0004-6361:20054751.
Wiegelmann, T., B. Inhester, and T. Sakurai (2006b), Preprocessing ofvector magnetograph data for a nonlinear force-free magnetic field reconstruction, Sol. Phys., 233, 215 – 232.
Wu, S. T., H. M. Chang, and M. J. Hagyard (1985), On the numericalcomputation of nonlinear force-free magnetic fields, in Measurementsof Solar Vector Magnetic Fields, NASA Conf. Publ., 2374, 17 – 40.
Wu, S. T., M. T. Sun, H. M. Chang, M. J. Hagyard, and G. A. Gary (1990a),
On the numerical computation of nonlinear force-free magnetic fields,
Astrophys. J., 362, 698 – 708.
Wu, S. T., M. T. Sun, and T. Sakurai (1990b), A comparison between
progressive extension method (PEM) and iterative method (IM) for magnetic field extrapolations in the solar atmosphere, Mem. Soc. Astron. Ital.,
61, 477 – 484.
Yan, Y., and Z. Li (2006), Direct boundary integral formulation for solar
non-constant-a force-free magnetic fields, Astrophys. J., 638, 1162 –
1168, doi:10.1086/499064.
Yan, Y., and T. Sakurai (2000), New boundary integral equation representation for finite energy force-free magnetic fields in open space above theSun, Sol. Phys., 195, 89 – 109.
Yan, Y., M. J. Aschwanden, S. Wang, and Y. Deng (2001a), Evolution ofmagnetic flux rope in the active region NOAA 9077 on 14 July 2000, Sol.
Phys., 204, 27 – 40, doi:10.1023/A:1014265123601.
Yan, Y., Y. Deng, M. Karlický, Q. Fu, S. Wang, and Y. Liu (2001b), The
magnetic rope structure and associated energetic processes in the 2000
July 14 solar flare, Astrophys. J. Lett., 551, L115 – L119, doi:10.1086/
319829.

18 of 19

A03S02

WIEGELMANN: CORONAL MAGNETIC FIELDS

Yang, W. H., P. A. Sturrock, and S. K. Antiochos (1986), Force-free magnetic fields: The magneto-frictional method, Astrophys. J., 309, 383 –
391, doi:10.1086/164610.

A03S02

Yee, K. S. (1966), Numerical solition of initial boundary value problems
involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas
Propag., AP-14(4), 302 – 307.


T. Wiegelmann, Max-Planck-Institut für Sonnensystemforschung, MaxPlanck-Strasse 2, D-37191 Katlenburg-Lindau, Germany. (wiegelmann@
mps.mpg.de)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A03S11, doi:10.1029/2007JA012588, 2008

for

Full
ArticleKey points model for polar region currents
Wen-Yao Xu,1 Geng-Xiong Chen,1 Ai-Min Du,1 Ying-Yan Wu,1 Bo Chen,1
and Xiao-Can Liu1
Received 26 June 2007; revised 10 November 2007; accepted 19 December 2007; published 29 February 2008.

[1] The equivalent ionospheric electric currents in the polar region mainly consist of the
DP1 and DP2 systems. The former involves a westward electrojet around midnight, while
the later involves a two-cell system with foci on the morningside and eveningside,
respectively. In space weather prediction and nowcasting, sophisticated models of the polar
currents are needed, but concise and convenient models are also useful to predict or nowcast
the principal characteristics of the current systems, such as intensity and position. In thispaper, we propose a ‘‘key points model’’ for outlining the basic features of the polar region
current system for different disturbance levels. The ‘‘key points model’’ (or KP model)
includes six key points of the current system: the centers of two DP2 cells, the maximum
densities of the eastward and westward electrojets, and the maximum densities of thenorthward and southward currents. Each of six key points is described by three parameters:
intensity, local time, and latitude. The AE-dependences of the 18 parameters are deduced
from the equivalent current systems for every 5 min during a 2-d period (18–19 March
1978). The KP model reveals systematic variations of the current systems. When AE
increases, the currents and the current densities are simultaneously enhanced linearly,
and most of the key points concentrate towards midnight. In addition, when AE
increases, the key points K2 and K4 for the evening current cell move equatorward,
while the key points K1 and K3 for morning cell move poleward.
Citation: Xu, W.-Y., G.-X. Chen, A.-M. Du, Y.-Y. Wu, B. Chen, and X.-C. Liu (2008), Key points model for polar region currents,
J. Geophys. Res., 113, A03S11, doi:10.1029/2007JA012588.

1. Introduction[2] The ionospheric electric currents in the polar region
and associated magnetic disturbances are essential manifestation of solar wind-magnetosphere-ionosphere coupling andare what space weather prediction concerns with [Akasofu,
1979; Chen et al., 2000]. The equivalent currents in the polar
region mainly consist of the DP1 and DP2 systems [Clauer
and Kamide, 1985; Kamide and Kokubun, 1996]. The former
involves a westward electrojet around midnight, associatedwith the substorm current wedge that reflects the unloading
process in the magnetic tail. The latter is a two-cell systemwith the foci of the cells on the morningside and eveningside
and which is driven by large-scale magnetospheric convection [Akasofu, 1964; Rostoker, 1969, 1996; Rostoker et al.,1987]. During quiet periods, the DP2 currents dominate thewhole polar current pattern. During disturbed conditions,especially in the expansion phase of substorms, the DP1currents are drastically intensified, while at the same time, asteady enhancement of the DP2 currents is observed [Sun etal., 1998, 2000].

1
Institute of Geology and Geophysics, Chinese Academy of Sciences,
Beijing, China.

[3] Magnetic data from observatories distributed in thepolar region have been widely used to deduce the ionospheric equivalent current system, from which the three
dimensional current system with field-aligned currents canbe inferred [Kamide et al., 1981]. Using the ground-basedmagnetometer data from 71 stations distributed along six
meridian chains at high latitudes during the period of theInternational Magnetosphere Study (IMS), Kamide et al.
[1982] calculated equivalent ionospheric current systems in
the polar region for every 5 min of 17– 19 March 1978. The
current systems demonstrate the dominant DP1 and DP2components [Kamide and Kokubun, 1996], although they
undergo very complicated variations in both pattern andintensity (Figure 1).
[4] The concept of two-component process of magnetospheric substorm was proposed by Akasofu [1979], andhas been extensively studied over the past few decades[Rostoker et al., 1987; Kamide and Kokubun, 1996; Sun etal., 1998, 2000]. The first component involves the directly
driven process, resulting in a two-cell convection andcurrent system in the polar regions. The second component
is unloading process, producing a current wedge and an
ionospheric westward electrojet around midnight. Kamide
and Kokubun [1996] illustrated two pairs of the electric
potential cells in a schematic diagram, representing theeffects of enhanced plasma convection and substorm
expansion.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012588$09.00

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A03S11Figure 1. Magnetic indices AE, AL, and AU during 18– 19 March 1978. The insets illustrate typicalpolar ionospheric equivalent current systems for three phases of a substorm, deduced from ground-basedmagnetic records by Kamide et al. [1982].

[5] A quantitative separation of the components DP1 andDP2 in the current systems have been attempted by Sun etal. [1998, 2000]. They applied the NOC method (method of
Natural Orthogonal Components) to the data supplied by
Kamide et al. [1982] and obtained a series of polar current
eigenmodes. Comparison of the eigenmode patterns withthe current systems of the driven and unloading processes
suggested that the first and second modes correspond toDP2 and DP1, respectively.
[6] During the past couple of decades, several sophisticated models of the polar currents have been established,which are certainly needed for understanding magnetosphericionospheric coupling and space weather predictions. In
addition, concise and convenient models are also usefulfor predicting or nowcasting most essential characteristicsof the current system, such as current intensity, pattern, andposition of the current center. These models require relatively few inputs and much easier available data and areable to promptly give the principal features of the current
system with detailed structures omitted. The purpose of thispaper is to propose a convenient and simplified model for
nowcasting basic characteristics of the polar currents.

2. Data and Analysis[7] The data used in this paper are 576 ionosphericequivalent current systems (IECS) computed every 5 minover the interval 18– 19 March 1978 [Kamide et al., 1981].
The IECS cover a latitudinal range of 50° – 90° at 1°intervals and a longitudinal range of 0°– 360° (or 0000 –2400 LT) at 15° intervals (or 1 h). The IECS is assumed toflow in the ionosphere at a distance of a + h from the centerof the Earth, where a is the Earth’s radius and h is the heightof the ionosphere above the Earth’s surface. It is represented
by current function (in amperes) as follows:J ðq; lÞ ¼/
X

Jn ðq; lÞð1Þ
1 2n þ 1 a þ h n e
Un ðq; lÞ
m0 n þ 1
r

ð2Þ

n¼0

Jn ðq; lÞ ¼ 

P e
where
Un(q, l) is external magnetic potential deduced
from 71 stations along six meridian chains in the Northern
Hemisphere. The southward and eastward components ofthe current density (in amperes/m) can be calculated from J
byIq ðq; lÞ ¼

1 @J ðq; lÞr sin q @lð3Þ

1 @J ðq; lÞ
r
@q

ð4Þ

Il ðq; lÞ ¼ [8] The IECS pattern is represented by contours of thecurrent function, as shown in Figure 1. In general, the IECS
structure is rather complicated. Omitting small-scale details
and random distortions, one can recognize the two-cellcurrents of DP2, which converge at the Harang discontinuity just before midnight, then flowing sunward across thepolar cap, eventually diverging near the forenoon cusp
region, returning to their respective electrojets. At theconvergence (or divergence) point there should be major

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XU ET AL.: KP MODEL FOR POLAR REGION CURRENTS

A03S11Figure 2. Average ionospheric equivalent (top) current systems and (bottom) current densities for AE =600– 800 nT (left) and AE > 800 nT (right). The ‘‘key points’’ in the model are indicated by K1, K2, K3,
K4, K5, and K6 (see text).
northward (or southward) currents. It is noted that the realcurrent system in the polar region is three dimensional,
including field-aligned currents [Baumjohann, 1983].
[9] Figure 2 shows two average equivalent current patterns for AE = 600 –800 nT and AE > 800 nT, which are
obtained by superposing 88 and 27 individual patterns,
respectively, in order to eliminate complicated irregularfluctuations and small-scale structures. It is noted in
Figure 2 that the principal features of the polar current
system are described by six ‘‘key points’’: (1) K1, the centerof the clockwise cell of the DP2 currents in morning sector,(2) K2, the center of the counterclockwise cell of the DP2currents in evening sector, (3) K3, the point of the maximum
westward current density in morning sector, (4) K4, the pointof the maximum eastward current density in evening sector,(5) K5, the point of the maximum northward current density
near Harang discontinuity, (6) K6, the point of the maximum
southward current density near the cusp region.
[10] Each key point is described by three parameters:
current intensity (total current J for K1 and K2, and current
density I for K3, K4, K5, and K6), magnetic local time T,
and geomagnetic latitude F. All of the 18 parameters
determine the geometry and strength of the polar region
currents.
[11] In order to eliminate irregular fluctuations andrandom distortions in the current patterns, the 576 samples
are divided into 10 groups according to AE ranges: thefirst group for AE = 0  100 nT, the second for AE = 100 
200 nT. . . . . ., the 10th for AE = 900  1000 nT. Then the
average current pattern for each AE group is calculated bysuperposing all current functions included in this group. We
confine our analysis to the range AE = 0  1000 nT because
only three samples are beyond this range. The average
current systems so obtained are illustrated in Figure 3,
where the AE range, sample number, and contour interval
for each AE group are indicated.
[12] It is noted in Figure 3 that the principal feature of thecurrent system is a two-cell structure, including a major
clockwise cell (assigned as positive) in the early morningand a minor counterclockwise cell around evening (assigned
as negative). The total currents of the two cells increase withincreasing AE.
[13] Figure 4 illustrates the distributions of E-W current
density for the 10 AE groups, where westward current is
assigned as positive (red color). The most outstanding
features are westward electrojet in midnight-morning sector
and eastward electrojet (blue color) in afternoon-evening
sector.
[14] It is interesting to note a ‘‘double peak’’ structure of
the westward electrojet in Figure 4 for AE  200 –300 and
900– 1000, one near midnight and another near dawn. This
conforms to the view that westward electrojet has twocontributors: the directly driven westward electrojet in
DP2 with its peak near dawn and the substorm expansion
phase westward electrojet in DP1 with its peak near
midnight. The other ranges show that when AE increases,
the midnight peak increases much faster than the dawn peak
and two peaks combine together, forming an enhanced
electrojet belt. In other words, during disturbance periods(typically, substorms), enhancement of DP1 due to the

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XU ET AL.: KP MODEL FOR POLAR REGION CURRENTS

A03S11Figure 3. Average equivalent current systems for 10 AE groups. The numbers at the top of each figure
are AE range, sample number, and contour interval for the group.
unloading process is much more significant than theincrease of DP2. Similar double peak structure can be seen
in eastward electrojet, especially for intensely disturbed
periods.
[15] As a first step in the analysis, the 18 parameters of
the six key points for each of the 576 samples are foundfrom its current pattern. Then the ‘‘average parameters’’
are calculated for each of the 10 AE groups. Finally, theAE dependence of each parameter is deduced by linear
fitting.
[16] Figure 5 shows the obtained AE dependences of the
18 parameters for six key points. The regression coefficientsare listed in Table 1.
[17] It is noted in Figure 5 and Table 1 that when AE
increases, the total current JCCW of the evening cell K2 isenhanced at a rate of 0.100 kA/nT, while a much greater

enhancement rate of 0.806 kA/nT is seen in the current
JCW of the morning cell K1, implying the important role ofthe unloading process in the formation of the equivalentpolar current system during disturbances. A similar contrast is noted in the current densities IE (K4, enhancement
rate 0.431 A/km/nT) and IW (K3, enhancement rate
1.382 A/km/nT). In addition, the N-S current densities IN
and IS for K5 and K6 are simultaneously enhanced
linearly.
[18] When AE increases, all key points systematicallymove in both local time (T), and geomagnetic latitude (F),as shown by arrows in Figure 6. It is noted in Figure 6 thatall key points, except for K3 (the maximum westward
current density IW in morning sector), concentrate towards
midnight with increasing AE. At the same time, the key

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A03S11Figure 4. Same as Figure 3, but for the average E-W current density. Different color scales are fordifferent figures.
points K2, K4, and K6 shift equatorward, while the key
points K1, K3, and K5 shift poleward.
[19] The most remarkable movements occur for K2 (theevening current cell) and K4 (the eastward electrojet),which are associated with equatorward expansion of theauroral oval during substorms [Akasofu, 1977]. When AE
increases from 0 nT to 1000 nT, K2 moves toward midnightby 2.9 h and equatorward by 10°, while K4 moves towardmidnight by about 1 h and equatorward by 7°.
[20] The poleward shift of the key points K1 and K3 is apeculiar feature in present model. A possible explanation is
as follows. As mentioned above, the westward electrojet
comes from two contributors: the directly driven westward
electrojet and the substorm expansion phase westward
electrojet. They have different latitudinal motions when
AE increases. Since substorm expansion phase electrojet

contributes significantly to AE during strong activity, thehigh-latitude edge of the expanded oval in midnight sector
shifts poleward, although directly driven electrojet in themorning sector shifts equatorward. New research by Ahn etal. [2005] has found that the region of brightest auroras does
not coincide with the area of strongest magnetic perturbations. The auroral electrojet is found to remain locatedadjacent to and poleward of the region of bright auroral
luminosity.

3. Discussion
3.1. Advantage of the KP Model[21] Compared with some sophisticated models of magnetospheric activity with many inputs and outputs, the KP
model developed in this paper is a concise, convenient, and

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Figure 5. Linear fitting of AE dependences of 18 parameters for six key points. The left column shows
current intensities, the middle column shows local times of the key points, and the right column showstheir geomagnetic latitudes. For clarity, the horizontal dotted lines indicate the times of noon, midnight,
dawn, and dusk in the middle and the latitude 700° on the right.

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A03S11

Table 1. Regression Coefficients of the Key Point (Ki, i = 1, 2. . . 6)
Parameters
Fitting y = a0 + a1x
x = AE index, nT
Parametersa0a1

JCW, kA
TCW, h
FCW, deg

K1 (CW) Clockwise Cell
7.9
1.7
70.9

0.806
0.00172
0.00396

JCCW, kA
TCCW, h
FCCW, deg

K2 (CCW) Counterclockwise Cell
88.6
10.6
74.2

0.1000.002850.0100IW, A/km
TW, h
FW, deg

K3 (W) Maximum Westward Density
10.0
0.54
67.1

1.382
0.00080.0011

IE, A/km
TE, h
FE, deg

K4 (E) Maximum Eastward Density
148.1
8.7
70.4

0.4310.00136
0.00626

IN, A/km
TN, h
FN, deg

K5 (N) Maximum Northward Density
4.7
3.9
71.3

0.54390.001130.00147

IS, A/km
TS, h
FS, deg

K6 (S) Maximum Southward Density
83.7
8.9
76.7

0.2539
0.000309
0.00312

simplified model. After the equivalent current systems are
obtained from the magnetic data at 71 stations, the input tothe model is AE index, which is easily and timely available
for users. The output of the model is an equivalent current

A03S11system in the polar region, which is essential to descriptionof the space environment. This model can be used to
nowcast the most important characteristics of the polar
current system, such as the overall current pattern, current
intensity, and the key point positions, although the detailed
structure, irregular fluctuations, and random distortions in
the current system are omitted.
3.2. Limitation of the KP Model[22] The KP model is a concise, convenient, and simplified model, involving computation of the equivalent ionospheric current, instead of the real three-dimensional current
system. In fact, field-aligned currents play a key role inpolar region currents and associated magnetic disturbances.
Using the KRM method or MIT technique, one can deduce
the three dimensional current system, which requires anassumed or experimentally determined ionospheric conductivity model and involves complicated calculations.
[23] It should be pointed out that the magnetic measurements from which the current systems are inferred comefrom a widely scattered distribution of magnetometer stations that are irregularly distributed globally, and in someregions there is very poor coverage. Therefore one should
not expect to believe in small-scale details of the equivalentcurrent structure where there are no data to constrain the
results. Fortunately, the KP model only concerns with largescale structures, averaging out the small-scale details whencalculating average current patterns for different AE ranges.
3.3. Input and Output[24] In this paper, the AE index is chosen as the input tothe KP model after comparison with AL and AU. By thedefinition of the auroral electrojet indices, AU is dependent
on the maximum eastward electrojet, represented by IE of
key point K4. However, the present analysis shows that the

Figure 6. AE dependence of the key point positions. The arrows indicate movement tendencies of the
key points with increasing AE.
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correlation coefficient of IE with AU is less than those withAL and AE. One of possible reasons is the large movementof the key point K4 (see Figure 6). During strong substorms,
K4 moves equatorward beyond the observation of AE
stations, which are confined in a relatively narrow geomagnetic latitudinal belt from 60.7° to 70.8°. In such cases, themagnitude of AU reaches ‘‘saturation,’’ although auroral
currents may continue increasing [Akasofu, 1977; Kamide
and Akasofu, 1983]. The saturation phenomenon in AE, AU,
and AL indices lowers reliability of large values of AE, and also
limits extension of the KP model to very intense substorms.
Since the data used in this paper are limited, the present KP
model is valid for AE < 1000 nT. A further extension of the KP
model to much more intense substorms requires more data
covering AE > 1000 nT. However, as indicated above, thisextension would be limited by low reliability of large AE andits saturation during strong substorms.
[25] Another possible candidate input parameter is the PC
index, that represents the equivalent current JPC flowing
across the polar cap and which can be estimated with datafrom only one station [Vennerstrom and Friis-Christensen,
1987].

4. Summary[26] 1. The equivalent ionospheric current system in thepolar region always shows dominant DP1 and DP2 components. This is the basis of the key points model for the polar
region current system.
[27] 2. Six key points in the KP model are the centers of
the two current cells in DP2, the maximum eastward andwestward electrojets, and the maximum northward andsouthward current densities. These points determine thebasic features of polar region current system.
[28] 3. As AE increases, the total currents for K1 and K2
and the current intensities for K3, K4, K5, and K6 aresimultaneously enhanced linearly.
[29] 4. As AE increases, most of the key points concentrate
toward midnight and, at the same time, shift equatorward.
The most remarkable movement occurs at the evening
current cell (K2) and the eastward electrojet (K4). When
AE increases from 0 nT to 1000 nT, the key point K2 moves
toward midnight by 2.9 h and equatorward by 10°, while K4
moves toward midnight by about 1 h and equatorward by 7°.
[30] 5. A peculiar feature in present model is poleward
shift of the key points K1 and K3. A possible explanation is
the poleward expansion of substorm expansion phase electrojet during strong activity.

A03S11[31] Acknowledgments. The authors would like to thank W. Sun for
supplying the data used in the present study. This study is supported by the
grant NO.40436016 from the National Natural Science Foundation ofChina, the grant NO.2006CB806305 from the National Key Basic Research
Science Foundation of China, and the KIP pilot project kzcx3-sw-144 of theChinese Academy of Sciences.
[32] Zuyin Pu thanks Wallace Campbell and another reviewer for theirassistance in evaluating this paper.

References
Ahn, B.-H., G. X. Chen, W. Sun, J. W. Gjerloev, Y. Kamide, J. B. Sigwarth,and L. A. Frank (2005), Equatorward expansion of the westward electrojet during magnetically disturbed periods, J. Geophys. Res., 110, A01305,
doi:10.1029/2004JA010553.
Akasofu, S.-I. (1964), The development of the auroral substorm, Planet.
Space Sci., 12, 273.
Akasofu, S.-I. (1977), Physics of Magnetospheric Substorms, pp. 381 – 472,
D. Reidel, Boston.
Akasofu, S.-I. (1979), Dynamics of the Magnetosphere, 447 pp., D. Reidel,
Boston.
Baumjohann, W. (1983), Ionospheric and field-aligned current systems in
the auroral zone: A concise review, Adv. Space Res., 2, 55.
Chen, H.-F., G.-X. Chen, F.-L. Peng, and W.-Y. Xu (2000), Analysis of Sqp
current system by using corrected geomagnetic coordinates, Chin. J.
Polar Sci., 11, 59.
Clauer, C. R., and Y. Kamide (1985), DP1 and DP2 current systems for the
March 22, 1979 substorm, J. Geophys. Res., 90, 1343.
Kamide, Y., and S.-I. Akasofu (1983), Notes on the auroral electrojet indices, Rev. Geophys., 21, 1647.
Kamide, Y., and S. Kokubun (1996), Two-component auroral electrojet:
Importance for substorm studies, J. Geophys. Res., 101, 13,027.
Kamide, Y., A. D. Richmond, and S. Matsushita (1981), Estimation ofionospheric electric field, ionospheric currents and field-aligned currents
from ground magnetic records, J. Geophys.Res., 86, 801.
Kamide, Y., et al. (1982), Global distribution of ionospheric and fieldaligned currents during substorms as determined from six IMS meridian
chains of magnetometers: Initial results, J. Geophys.Res., 87, 8228.
Rostoker, G. (1969), Classification of polar magnetic disturbances, J. Geophys. Res., 74, 5161.
Rostoker, G. (1996), Phenomenology and physics of magnetospheric substorms, J. Geophys. Res., 101, 12,955.
Rostoker, G., S.-I. Akasofu, W. Baumjohann, Y. Kamide, and R. L.
McPherron (1987), The roles of directly input of energy from the solar
wind and unloading of stored magnetotail energy in driving magnetospheric substorms, Space Sci. Rev., 46, 93.
Sun, W., W.-Y. Xu, and S.-I. Akasofu (1998), Mathematical separation of
directly driven and unloading components in the ionospheric equivalentcurrents during substorms, J. Geophys. Res., 103, 11,695.
Sun, W., W.-Y. Xu, and S.-I. Akasofu (2000), An improved method todeduce the unloading component for magnetospheric substorms, J. Geophys. Res., 105, 13,131.
Vennerstrom, S., and E. Friis-Christensen (1987), On the role of IMF B-y ingenerating the electric field responsible for the flow across the polar cap,
J. Geophys. Res., 92, 195.


B. Chen, G.-X. Chen, A.-M. Du, X.-C. Liu, Y.-Y. Wu, and W.-Y. Xu,
Institute of Geology and Geophysics, Chinese Academy of Sciences,
Beijing 100029, China. (wyxu@mail.igcas.ac.cn)

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A04S91, doi:10.1029/2007JA012632, 2008

for

Full
ArticleSome ubiquitous features of the mesospheric Fe and Na layer borders
from simultaneous and common-volume Fe and Na lidar observations
Fan Yi,1,2,3 Shaodong Zhang,1,2,3 Xianchang Yue,1,2,3 Yujin He,1,2,3 Changming Yu,1,2,3
Chunming Huang,1,2,3 and Wei Li1,2,3
Received 7 July 2007; revised 28 February 2008; accepted 5 March 2008; published 12 April 2008.

[1] High-accuracy atom density profiles, obtained by the simultaneous and commonvolume Fe and Na lidar measurements at Wuhan, China (30.5°N, 114.4°E), reveal someubiquitous features of the Fe and Na layers on their borders. The Fe and Na lower boundaries
show consistently a delicate stratification in which the lower boundary of the Fe layer isin general slightly higher than or coincident with that of the Na layer, with an overallmean altitude difference being about 0.2 km. Despite the existence of considerable
vertical displacements, the two lower boundaries vary always following almost the same
track. The overall correlation coefficient between them is as high as 0.96. Thisubiquitous delicate stratification of the measured lower boundaries (nearly coincident
density cutoff ) suggests strongly that the undersides of Fe and Na layers are controlled by
the same or very similar processes. The upper boundary of the Na layer is always several
kilometers higher than that of the Fe layer. A relatively weak positive correlation is
also persistently observed between the two upper boundaries. Weak sporadic layering events
frequently occur on the upper extent of the metal layers. They may impair the correct
determination of the upper boundaries of the normal metal layers and consequently
weaken the correlation. Both the Fe and Na layers often show an evidently steeper
density gradient on the underside than on the upper extent, and the borders of the Felayer are clearly steeper than those of the Na layer. The explanation to these ubiquitousfeatures needs further experimental and modeling efforts.
Citation: Yi, F., S. Zhang, X. Yue, Y. He, C. Yu, C. Huang, and W. Li (2008), Some ubiquitous features of the mesospheric Fe andNa layer borders from simultaneous and common-volume Fe and Na lidar observations, J. Geophys. Res., 113, A04S91, doi:10.1029/
2007JA012632.

1. Introduction[2] Many species of free metal atoms gather in themesopause region from 80 to 110 km to form layers (Fe,Na, K, Ca, and Li) detectable by ground-based lidars. These
metal atoms are believed to result from meteoric ablation.
On the basis of independent lidar measurements from many
locations, the average properties and seasonal variations ofthese layers have been respectively investigated. Fe is themost abundant of the mesospheric metallic atoms, with anannual mean column abundance roughly twice that of Na[Kane and Gardner, 1993]. The mean column abundancesof K and Ca are about two orders smaller [Eska et al., 1999;
Gerding et al., 2000]. The mean peak heights (or meancentroid heights) of these metal atom layers exhibit somedifference. The Fe layer has the lowest peak height(88 km) among these metal atom layers. In comparison
with the peak of the Fe layer, the Na peak is about 3 km
1School of Electronic Information, Wuhan University, Wuhan, China.
Key Laboratory of Geospace Environment and Geodesy, Ministry of
Education, Wuhan, China.
3
State Observatory for Atmospheric Remote Sensing, Wuhan, China.
2

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JA012632$09.00

higher, while both the Ca and K peaks are roughly 2 –3 km
higher [Kane and Gardner, 1993; Eska et al., 1998; Plane etal., 1999; Gerding et al., 2000; Friedman et al., 2002]. At
midlatitudes the Na layer is the widest of these layers, with afull width at half-maximum (FWHM) over 10 km. The Fe
and K layer FWHMs are 1 – 2 km narrower than that of the
Na layer. The Ca layer is the narrowest among these layers.
The seasonal variations of these layers show some similarities as well as differences. At midlatitudes, the Fe and Na
layers have early winter maxima and midsummer minima incolumn abundance, while the K and Ca layers display asemiannual variation with a prominent second maximum in
midsummer [Eska et al., 1998; States and Gardner, 1999;
Gerding et al., 2000; Plane, 2003]. The centroid heights ofthese layers are characterized by a strong semiannual
variation. The K layer has the lowest altitude in winter[Kane and Gardner, 1993; Eska et al., 1998], whereas theNa, Fe, and Ca layers have the lowest altitude in summer[Kane and Gardner, 1993; Gerding et al., 2000]. The layer
widths of all layers show maxima in winter. At low
latitudes, the seasonal behaviors of the K and Fe layers
are clearly different from the corresponding results observedat midlatitude [Friedman et al., 2002; Raizada and Tepley,2003]. In particular, the centroid height and RMS width of

A04S91

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the Fe layer are nearly out of phase with respect to its
midlatitude counterparts [Raizada and Tepley, 2003], while
the Fe column abundance lags that at midlatitude by about
3 months. The latitudinal differences in Fe layer properties
have been explained by the latitudinal differences in theannual variations of [O], [O3], and [O2], which result in the
different season dependence of the Fe chemistry at twolatitudes.
[3] Several one-dimensional gas-phase chemical models
corresponding to different metal species such as Na, Fe, K,
and Ca have been respectively developed to explain theseobservational results [Helmer et al., 1998; Plane et al.,
1999; Eska et al., 1999; Gerding et al., 2000; Delgado etal., 2006]. The Fe and Na models among others successfully reproduced some primary features of the normal Fe
and Na layers observed at midlatitudes [Helmer et al., 1998;
Plane et al., 1999], for instance, the layer shapes and theseasonal variations in the layer parameters. However, the Kmodel could satisfactorily reproduce the seasonal behaviorof the K layer observed at a midlatitude site only after the
wintertime K influx from meteoric ablation had been
reduced by 30% compared to the summer [Eska et al.,1999]. Furthermore, the Fe and Na layers observed during
the austral winter at the South Pole could be well modeled if
the Fe and Na influxes from meteoric ablation were increased by a factor of 1.7 for these months [Gardner et al.,2005]. These results appear to reflect two facts: the first isthat chemistry alone cannot explain all the seasonal differences [Raizada et al., 2004], and second, we currently lacka quantitative description on the deposition flux of themetals (the influx of extraterrestrial material and ablation
efficiency of various metal species).
[4] In fact, the idea that meteoric deposition is the major
source of mesospheric metal atoms is currently appealing inqualitative way because hitherto there is no direct andindirect measurement for the total vertical flux of any
mesospheric metal species produced by meteoric ablation.
In the existing chemical models, the total vertical flux foreach metal species was specified to a value which resulted
in good agreement between the observed and simulated
metal atom profiles [Helmer et al., 1998; Gardner et al.,2005]. In order to understand the formation, structure, andvariability of the metal atom layers, one needs to havequantitative knowledge about the source, such as the micrometeor flux as a function of time and geographic grid, thedistributions of micrometeor particle masses, mass densities
and velocities, the chemical composition of micrometeor
particles, the evaporation processes including their dependences on particle mass, density, velocity, direction, andaltitude [Gerding et al., 1999]. Janches et al. [2006] havemade a modeling effort in constructing a global micrometeor input function based on the radar observations ofionized meteor input rate. The model assumes that thedetected particles come from three radiant distributions.
The apex source flux makes up 70% of the total numberof micrometeoroid particles, while the other 30% is provided by the combined distribution of two other sources which
are concentrated 80° in ecliptic longitude to each side of theapex. The modeling results reproduce successfully some
diurnal variation features of the ionized meteor input rates
observed by high power and large-aperture radars at different locations in different seasons. For instance, the ionized

A04S91meteor input rate observed at each location almost always
shows a consistent increase throughout each night with amaximum near dawn. The magnitude of the rate variation
during each night is beyond one order. Different from thenocturnal variation feature of the observed ionized meteor
input rate, the observed Fe and Na abundances do not show
a consistent increase during each night [Kane and Gardner,
1993]. Moreover, in contrast to more than one order of thevariations in the ionized meteor rate during each night, theFe and Na abundances show often a relatively small
nocturnal variation. In order to clarify whether there was aconnection between the ionized meteor input and the Fe andNa layer abundances, we compared the temporal variationsof the ionized meteor input rate and the total Fe and Na
abundances observed simultaneously by closely colocatedmeteor radar and two resonance fluorescence lidars at
Wuhan [Yi et al., 2007]. It was found that the ionized
meteor input rate recorded by radar had no clear-cut relation
with the abundance enhancements of the mesospheric metal
atom layer. This is accordant with the observations by
Gerding et al. [1999], who discerned the inconsistency
between the occurrences of the lidar-observed atom meteor
trail enhancements and the increased ionized meteor activity
recorded by forward meteor scatter radio stations.
[5] As pointed out by Janches et al. [2006], in the modelof the mesospheric Na layer, Plane [2004] used a micrometeor input function including seasonal and diurnal variations based on the radio meteor survey (ionized meteor
input rate) by Yrjölä and Jenniskens [1998]. However, theobservations mentioned above indicate that the ionized
meteor input rate has no clear-cut relation with the metal
atom layer abundance. Therefore it is necessary to give an
explanation for this phenomenon. A meteor radar only
samples a subset of the mass/velocity distribution of entering meteoroids. If the idea that meteoric ablation is the
major source of the mesospheric metal atoms is still maintained, this phenomenon appears to imply that the features(mass/velocity distribution or flux, etc.) of the subset
associated with the ionized meteoric trails differ in generalfrom those of that makes primary contribution to the metal
atom layers. Hence we conjecture that the micrometeoroids,that are too small to yield the ionized meteoric trails
observed by meteor radars but have a large population,
may be the source of the mesospheric metal atoms.
[6] Since the formation of the metal atom layers is related
to the input of gas-phase metal materials from meteoroid
ablation, dynamic transport and chemical reactions, exploring the relationship between various layers in the mesopause region may help clarify which process is playing a
predominant role in the layer formation, particularly given
the lack of quantitative information on the metal deposition
flux. The simultaneous and common-volume observationsof the normal Fe and Na layers at a midlatitude site showedthat even though the individual density profiles appearedquite different, the temporal variations of Fe and Na layer
parameters (abundances, centroid heights, and RMS widths)
were evidently similar in most cases [Kane and Gardner,
1993]. This phenomenon was ascribed to dynamic effects[Kane and Gardner, 1993]. From concurrent measurementsof the charged dust and atomic metals (Fe and Na) in thepolar winter mesosphere by sounding rocket and lidars,some interesting relations between the charged dust and the2 of 9

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YI ET AL.: FEATURES OF THE Fe AND Na LAYERS

Fe layer have been found [Lynch et al., 2005; Gelinas etal., 2005]. There exists a strong correlation between the
dust density and the Fe density for an altitude range from82 to 95 km over all four rocket flights [Lynch et al.,2005]. Although the dust layer lay a few kilometers lower
than the Fe layer, its shape was similar to the shape of the
Fe layer [Gelinas et al., 2005]. The two species appeared
to move in unison during the three rocket launches on one
night, with the layers compressing and tracking downward
together. These relations are surprising, as the density
distribution and variation of the dust layer depend on
local mesospheric dynamics (eddy diffusion, neutral wind,etc.) and characteristics of the incident meteoroids (velocity, composition, etc.) [Hunten et al., 1980; Kalashnikova
et al., 2000] rather than the gas-phase chemistry. Obviously, clarifying the relations needs further experimental
and modeling efforts.
[7] Upper extent and underside are critical regions of anormal metal atom layer, where metal atoms manifest atransition from absence to presence (or vice versa). Suchtransition regions may facilitate identifying the practicalphysics and chemistry behind the observed layer structure
because therein some production mechanisms of the metal
atoms likely begin or cease to operate or some removal
and/or conversion actions are likely strong enough tocause a cutoff of metal atom density. Therefore exploring
the behaviors at the metal atom layer boundaries may help
to recognize the processes leading to the formation of thenormal metal layer. Recently, it has been revealed bycomparing simultaneous and common-volume observations of different resonance fluorescence lidars that thetopside layers above 110 km for different metal atoms
have remarkably similar seasonal characteristics [Höffner
and Friedman, 2004, 2005]. In addition, a qualitative
agreement is also found for K at two distinct latitudes(54°N and 18°N) [Höffner and Friedman, 2004]. The
seasonal characteristics differ clearly from those of themain layer below. The calculated metal abundance ratios
are fairly constant over an altitude range within the topsidelayers throughout all seasons, which are in reasonably
good agreement with metal abundance ratios determinedfrom simultaneous measurements inside single meteor trails
within the main layer. A comparison of metal densities at
113 km altitude with known meteor showers indicates astrong influence of shower meteoroids on the topside layers.
Moreover, the metal density enhancement in summer correlates well with the seasonal variation of sporadic micrometeoroid input independent of meteor showers. Taking theseevidences into consideration, Höffner and Friedman [2004]
argued a direct link between ablating meteoroids and thetopside metal layers. The findings about the metal layer
topside [Höffner and Friedman, 2004, 2005] are calling for
more attention to the metal layer upper extent and underside.
[8] Lidar measurements also reveal another layer form
called ‘‘sporadic metal layers.’’ The first finding about them
was from São José Dos Campos (23°S, 46°W) by Clemesha
et al. [1978]. Sporadic metal layers are characterized bylarge density enhancements in a narrow altitude range[Hansen and von Zahn, 1990; Clemsha, 1995; Clemsha etal., 1999; Yi et al., 2002]. They occur usually in the upper
part of the normal metal layers and have no clear-cut
relation with the normal layers. The high correlation in time

A04S91and space between Nas and Es layers led many scientists to
explore a possibility of Nas layer formation via the neutralization of a reservoir of Na+. Taking the idea proposed by
Cox and Plane [1998], Collins et al. [2002] successfully
simulated the sporadic layering events observed simultaneously by the Na lidar and incoherent scatter radar at
Arecibo. Their modeling work showed an excellent agreement with observations in five out of a total of six casestudies. However, some additional observational facts areworth noting. It has been observed that the amount of
Na+ ions in Es layers is not sufficient to form large Nas layers[Kane et al., 1993]. An observation at Arecibo Observatory
showed that the electron density peak of a Es layer lagged
behind that of the correlated Nas layer [Friedman et al.,2000]. This phenomenon makes the Es an unlikely source forthe formation of sporadic atomic layers because it appeared
to contradict the conventional cause/effect relation in the ion
reservoir neutralization mechanism. The simultaneous andcommon-volume Fe and Na lidar observations indicate that
the Fes and Nas layers occur in overlapping altitude ranges
and moved following almost the same track [Yi et al., 2007].
On occasion, the Fes and Nas layers exactly simultaneously
reach their maximum peak densities at nearly the same
altitude. These observational results suggest that Fes andNas layers come from the same source substance and may
be formed via the same or similar atomization process. For
the majority of the sporadic Na layers observed by a Na
wind/temperature lidar at low latitudes, their temperatures
were significantly warmer than the mean [Qian et al., 1998].
The narrow-band Na lidar measures the Na atom temperature. There exists a possibility that the Na atoms in the
sporadic Na layers are freshly generated by more violent
meteoric ablation and then have evidently higher temperature than the mean. A systematic study on the variations ofoccurrence of sporadic Ca and Ca+ layers indicates that theneutral and ionized Ca layers are not closely correlated[Gerding et al., 2001]. This result is inconsistent with therecombination mechanism that sporadic layers are producedby the neutralization of metal ions. On the other hand, it
seems to support that the neutral and ionized metal layers
may come from different meteoric source if the metal atoms
and ions are believed to come from the direct meteoric
ablation. Obviously, clarifying the formation mechanism ofsporadic layering events needs further observational andmodeling efforts.
[9] In order to study the mesospheric Fe and Na layers at
30°N, we established two independent resonance fluorescence lidars [Yi et al., 2007]. The two lidar systems have thesame altitude and time resolutions (96 m and 5 min) as wellas the identical starting and ending times for each sampling.
This allows us to make a detailed comparison between Fe
and Na density profiles. From one year of simultaneous andnearly common-volume Fe and Na lidar measurements at
Wuhan (30.5°N, 114.4°E), we have found that the lower
boundaries of the normal Fe and Na layers consistently
show a quasi-overlap [Yi et al., 2007]. In addition, thenormal Na layer always showed a few kilometers more
extension on its top side than the normal Fe layer. Thesefeatures can be clearly seen also from a comparison ofsimultaneously observed Fe and Na layers at high, mid andlow latitudes respectively [Kane and Gardner, 1993; Lynch
et al., 2005; Shibata et al., 2006]. Thus these features may

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A04S91

A04S91Figure 1. Sequences of density profiles measured during the night of 7 – 8 December 2004 for (top) Feand (bottom) Na. Note that the undersides of the normal Fe and Na layers followed nearly the exact
movements and occurred at very similar altitudes.
be universal, in which case they may provide a connection
between the two different metallic species and reflect thephysics and/or chemistry in their formation. Therefore, here
we examine further the relationship between the normal Fe
and Na layers on their underside and upper extent.

2. Observational Results[10] The Fe and Na density data employed in this study
come from the 195 h simultaneous and nearly commonvolume Fe and Na lidar measurements at Wuhan during
26 nights from 8 March 2004 to 14 March 2005, whichhave also been used for studying the relationship between
sporadic Fe (Fes) and Na (Nas) layers [Yi et al., 2007].
Figure 1 gives a comparison between the time sequencesof Fe and Na density profiles measured during the night of
7 – 8 December 2004. On this winter night, despite theexistence of a weak single Fes layer near 93.0 km around0140 LT the absence of any high-altitude sporadic layering
event (100 km or above) makes it possible to determine
the upper boundaries of the normal layers. As seen in
Figure 1, the main bodies of the simultaneously observed

Fe and Na layers appeared quite different at any given
time, but their undersides followed one another nearly
exactly. The normal Fe layer is several kilometers narrower than the Na layer nearly at all time, and thisdifference was primarily reflected in the extent of theupper edge of the layer.
[11] For quantitatively characterizing the border characteristics of the normal Fe and Na layers, we define their
upper and lower boundaries in terms of a given small
density criterion. In more detail, the upper (or lower)boundary for each density profile is defined as such analtitude where the density value of the normal layer
becomes equal to or less than the given density criterion
as one moves from the layer inside to its high (or low)border. An idealized choice of the density criterion is thecutoff point of the normal layer (null density point).
However, near the cutoff altitude the lidar detected photon
counts are close to a background noise level so that theupper (or lower) boundary obtained in terms of the nulldensity criterion may have a large uncertainty. For choosing
a suitable density criterion we calculate the detection thresholds of Fe and Na atoms between about 75 and 110 km for

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Figure 2. (top) Upper and (bottom) lower boundary
altitudes of the normal Fe and Na layers as a function oftime on the night of 7– 8 December 2004. Note that the twolower boundaries varied following almost the same track
and the lower boundary of the Fe layer was in generalslightly higher than or coincident with that of the Na layer,with a mean altitude difference being about 0.3 km. The
upper boundary of the Fe layer was always lower than thatof the Na layer, with a mean altitude difference being about
4 km.
all density profiles shown in Figure 1. It is noticed that theirmaxima are 23 cm3 (Fe) and 7 cm3 (Na), respectively.
Thus, the density criterion for both the Fe and Na layers is
set to 30 cm3. According to the above definition, the upperand lower boundaries of the normal Fe and Na layers are
determined respectively and plotted versus time in Figure 2.
As seen from the lower panel, the lower boundaries of the
Fe and Na layers show consistently a delicate stratificationthat the lower boundary of the Fe layer is in general slightly
higher than or coincident with that of the Na layer, with amean altitude difference being about 0.3 km (three exceptional time points, i.e., 2130, 2135, and 2235 LT, may result
from the low signal-to-noise of our Fe lidar). The two lower
boundaries exhibit also considerable vertical displacements
on a timescale of a few tens of minutes, particularly with adisplacement magnitude of 4.8 km between 2115 and2230 LT, which modulate the delicate stratification. Despite
the existence of the considerable vertical displacements, thetwo lower boundaries vary following almost the same trackwith a correlation coefficient being as high as 0.93. Here,it is interesting to examine whether the observed DC offset
in low boundary altitudes is a geophysical phenomenon or

A04S91

an effect of zenith pointing errors between the two lidar
beams. By using a new method developed by our group [Liuet al., 2005], the optical axes of our Fe and Na lidar
telescopes have been adjusted to point to the zenith accurately (with a maximum error of 0.2 mrad in off-zenith angle).
For each lidar system in our group there is an autocollimating
device above the computer-controlled gimbal-mounted
mirror that directs the laser beam toward to the zenith.
The autocollimating device guarantees the zenith pointing
error of a laser beam being within 0.15 mrad. Hence, themaximum altitude difference induced by the zenith pointing
error between the two lidar beams would be roughly
1.8  103 m that is too small to yield the observed DC offset
of in altitudes (hundreds of meters or more). Therefore, theobserved DC offset in low boundary altitudes results fromgeophysical cause. The above observed results suggest
strongly that the undersides of the Fe and Na layers arecontrolled by the same or very similar mechanisms.
[12] As shown in the upper panel of Figure 2, the upperboundary of the Fe layer is obviously lower than that of the
Na layer, with a mean altitude difference of about 4 km.
The correlation coefficient between them (the two upperboundaries) has a value of 0.50, which is obviously lower
compared to that of the lower boundaries. However, if thedensity criterion is set to 200 cm3 the correlation coefficient grows to 0.72, while the corresponding value for the
lower boundaries increases to 0.98. This result appears toindicate that at those altitudes more close to the null density
points, the stronger noise fluctuation would reduce thecorrelation between the upper (or lower) boundaries of
the Fe and Na layers. In order to understand why thecorrelation between the two upper boundaries is obviously
weaker than that between the two lower boundaries, wehave surveyed all the Fe and Na density profiles shown in
Figure 1. It is found that the upper extent of the metal layers
is characterized by the frequent occurrence of weak density
enhancements in contrast to the underside. They actually
represent sporadic layering events with very small absolutepeak densities [Yi et al., 2007]. Weak density enhancements
occurring frequently on the upper extent may impair thecorrect determination of the upper boundaries of the normal
metal layers. This is confirmed by a fact that the calculated
upper boundaries usually show obviously stronger irregularfluctuations on small temporal scales than the lower boundaries (see Figure 2). Therefore, the low correlation between
the two upper boundaries is at least partly due to these weakdensity enhancements prevailing on the upper extent of themetal layers.
[13] It is noted from Figure 2 that the upper boundary
appears to be uncorrelated with the lower boundary for eachof the metal layers. Combining them and using our boundary condition, we obtain the mean widths of 22.7 km for the
Fe layer and 27.0 km for the Na layer. Hence, the Na layer
is on average about 4.3 km wider than the Fe layer, and thisdifference is mostly reflected in the extent of the Na layer
top. For further discerning the relationship between the Fe
and Na layer structures on the underside and upper extent, atypical example of the individual Fe and Na density profiles
is presented in Figure 3. It is observed from Figure 3 thatboth of the layers show an evidently steeper density gradienton the underside than on the upper extent, and both the Fe
underside and upper extent are clearly steeper than those of

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A04S91or sink for Fe and Na atoms as conjectured earlier byTilgner and von Zahn [1988]. At the lower boundaries of
the Fe and Na layers, the consistently negative and smallvalues of the measured Fe and Na scale heights reflect
the significant gradients in the source and/or sink for Fe
and Na atoms near the lower boundary.
[14] As an example from summertime, Figure 4, which isbased on the sequences of the Fe and Na density profiles
shown by Yi et al. [2007] (Figure 1), displays the temporalvariations of the upper and lower boundaries for the normal
Fe and Na layers observed on the night of 28– 29 July 2004.
Since the maximum detection thresholds for all the Fe andNa density samplings (5-min) are 13 cm3 and 5 cm3,
respectively, the density criterion here is still taken to be
30 cm3. Note that the upper boundaries of the normal
layers are uncertain between 0015 to 0500 LT because of
the presence of the high-altitude Fes and Nas layers [Yi et al.,2007]. It is clear from the upper panel of Figure 4 that theupper boundary of the Fe layer is over 8 km lower than thatof the Na layer before 0015 LT. The two upper boundaries
exhibit some features similar to those observed in thewintertime example, i.e., strong irregular fluctuations anda low correlation compared with the lower boundaries. Just

Figure 3. Fe and Na density profiles measured at 2325 LT
on 7 December 2004. Note that the upper boundary of the
Fe layer is 5.2 km lower than that of the Na layer, while
the lower boundary of the Fe layer is 0.6 km higher thanthat of the Na layer. Both of the layers show an evidently
steeper density gradient on the underside than on thetopside, and both the Fe layer underside and topside areclearly steeper than those of the Na layer.
the Na layer. In order to ascertain whether the density
gradient features are prevalent over this night, we havecalculated the density scale heights around the upper andlower boundaries for all the Fe and Na density profiles
shown in Figure 1. As a result, out of the total 81 pairsof the Fe and Na density profiles, 72 (88%) have thedensity gradient features. The mean density scale heights
around the upper boundary are 0.36 ± 0.15 km for Fe
and 1.2 ± 0.75 km for Na, while the corresponding
values around the lower boundary are 0.18 ± 0.12 km
and 0.27 ± 0.16 km. At the upper boundary of themetal atom layers near 106 km, the diffusion for eachmetal constituent is believed to be dominated by both
molecular and eddy effects. Molecular diffusion tends todrive the density falloff of any given constituent to its
natural scale height, while eddy diffusion tends to drive
the density falloff to the background atmospheric scaleheight [Heinselman, 1999]. The natural scale heights for
Fe and Na atoms have the values of 3.2 and 7.9 km,
respectively, at 106 km altitude, which are comparable tothe background atmospheric scale height of 6.7 km at thisaltitude. Noticing the fact that the measured Fe and Na
scale heights around their upper boundaries are consistently far less than those values of the natural andbackground atmospheric scale heights, we believe thatthe altitude variation in Fe and Na densities on the layer
upper extent must represent some features of source and/Figure 4. (top) Upper and (bottom) lower boundary
altitudes of the normal Fe and Na layers as a function oftime on the night of 28– 29 July 2004. Note that the twolower boundaries varied following almost the same track
and the lower boundary of the Fe layer was in generalslightly higher than or coincident with that of the Na layer,with a mean altitude difference being about 0.2 km. The
upper boundaries of the normal layers are uncertain between
0015 to 0500 LT because of the presence of the highaltitude Fes and Nas layers.

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YI ET AL.: FEATURES OF THE Fe AND Na LAYERS

A04S91Figure 5. Scatterplot showing the relationship between the lower boundary altitudes of the normal Fe
and Na layers based on the 1075 pairs of the high-accuracy Fe and Na density profiles from the 12 nights.
The color of each open circle denotes the repetition number on this data point. The straight line is a linear
regression fit to the data. The correlation coefficient is 0.96, indicating that it is a ubiquitous phenomenon
that the lower boundaries of normal Fe and Na layers follow nearly the same movement. Note that thelower boundary of the Fe layer mostly lies at a slightly higher altitude than or coincides with that of the
Na layer.
like the wintertime example above, the lower boundaries ofthe summertime layers displayed consistently the delicate
stratification that the low boundary of the Fe layer was
mostly slightly higher than or accordant with that of the Na
layer, with a mean altitude difference being about 0.2 km(six exceptional time points may result from the noise
fluctuation, see Figure 4, bottom). The two lower boundaries also follow nearly the same movement with a correlation coefficient as high as 0.96. The delicate stratification
appears to be modulated by the movement. As shown in the
low panels of Figure 2 and 4, the consistently subtle
stratification that the lower boundary of the Fe layer is in
general slightly higher than or coincident with that of the Na
layer makes it unlikely that the correlated variations of themeasured lower boundaries are simply caused by gravity
waves alone because at the lower boundaries near 80 km
altitude the vertical mixing of constituents by turbulence
generated by gravity wave breaking tends to spoil thestratification. The gradient features shown in most individual Fe and Na profiles (89%) near the upper and lower
boundaries are also consistent with those observed from thewintertime example, i.e., there exists an evidently steeper
density gradient on the underside than on the upper extent,
and the Fe layer borders are clearly steeper than those of the
Na layer. In the summertime example, the mean density
scale heights around the upper boundary are 0.40 ± 0.23 km
for Fe and 1.7 ± 0.73 km for Na, while the corresponding
values around the lower boundary are 0.16 ± 0.03 km and0.22 ± 0.10 km. The scale height values are comparablewith those in the wintertime example.
[15] In order to identify the statistical significance of theabove observed results, we have further surveyed all thesimultaneous Fe and Na data. Note that out of a total of
26 observation nights, 12 nights (including two examples

shown above) have the maximum detection thresholds lessthan 30 cm3 due to a variational atmospheric transmission
from night to night. These 12 nights with high detection
accuracy (high signal-to-noise ratio) allow us to explore thebehaviors of the layer boundaries close to the density cutoff
altitudes. Figure 5 contains a scatter plot showing therelationship between the lower boundaries of the normal
Fe and Na layers based on the 1075 pairs of the highaccuracy Fe and Na density profiles from the 12 nights. The
straight line represents a linear regression fit to the data. The
overall correlation coefficient is 0.96, indicating that it is aubiquitous phenomenon that the lower boundaries of thenormal Fe and Na layers follow nearly the same movement.
Figure 5 reveals also that the lower boundary of the Fe layer
mostly lies at a slightly higher altitude than or coincides
with that of the Na layer, with a mean altitude difference
being about 0.2 km. This is clearly seen in 847 (79%) out of
the total 1075 pairs of the Fe and Na density profiles. The
remaining 228 pairs (21%) may be ascribed to the noisecontamination or tiny structure on underside. The correlation coefficient between the upper boundaries of the Fe andNa layers is low (0.37) because of the impairment of weakdensity enhancements occurring frequently on the upper
extent of the metal layers. However, it is clear that theupper boundary of the Fe layer is always several kilometers
lower than that of the Na layer in the absence of highaltitude sporadic layers. Our statistics reveals again that
there is no clear-cut correlation between the upper andlower boundaries for each of the metal layers.
[16] Both the normal Fe and Na layers over Wuhan
display an evidently steeper density gradient on underside
than on upper extent as seen from the two examples above.
The feature is consistent with the earlier observations athigh and middle latitudes [Tilgner and von Zahn, 1988;

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YI ET AL.: FEATURES OF THE Fe AND Na LAYERS

Kane and Gardner, 1993]. In order to quantitatively determine the consistency of the feature at our site, we havecalculated the density gradients near the upper and lower
boundaries for all the high-accuracy Fe and Na density
profiles in the absence of sporadic layers around the upper
extent of the normal layers. As a result, the feature (steeper
density gradient on the underside than on the upper extent)
is seen for 90% of all the Fe profiles and for 86% of all the
Na profiles. In addition, it is also noticed from the Fe andNa density profiles that the Fe layer generally has obviously
steeper density gradients on its borders than the Na layer(e.g., Figure 3). This agrees with the available observational
fact at other midlatitude sites that the Fe layer has anextremely small bottom scale height compared with the
Na layer [Plane, 2003].
[17] For the convenience of depicting the observed layerboundaries, we have introduced a digital definition. Itshould be emphasized that the chosen density criterion(30 cm3) is an extremely small quantity (comparable tothe minimum detectable density) rather than an arbitrarily
chosen value. The validity of the definition has been tested
further by comparing the calculated boundary altitudes withall the individual pairs of the Fe and Na profiles. It is
indicated that the altitude relations between the calculated
Fe and Na boundaries are consistent with those seen innearly all the individual pairs of the Fe and Na profiles.
Furthermore, the lower boundary altitude relation, that thelower boundary of Fe layer is in general slightly higher thanor coincident with that of Na layer, is also consistent withthe observed fact that the Fe underside is generally obviously steeper than that of the Na layer.
[18] In addition, only from a visual observation of all thenights of the sequences of the Fe and Na density profiles
observed simultaneously, we can clearly judge that theundersides of the normal Fe and Na layers follow nearly
exact movement and occur at nearly the same altitude(e.g., Figure 1). In this case, the correlation coefficient is
not sufficient to depict the relation between the Fe and Na
lower boundaries. Thus, the undersides of Fe and Na
layers are likely to be formed by the same or very similar
processes.

3. Summary[19] The high-accuracy atom density profiles, obtained by
the simultaneous and common-volume Fe and Na lidar
measurements at Wuhan, China over 1 year, reveal someprevalent relations between the Fe and Na layer borders.
The lower boundaries of the Fe and Na layers show
consistently a delicate stratification in which the lower
boundary of the Fe layer is in general slightly higher thanor coincident with that of the Na layer, with an overall mean
altitude difference being about 0.2 km. The delicate stratification is often modulated by the highly correlative verticalmovements of the two lower boundaries. The overallcorrelation coefficient between the two lower boundary
altitudes is as high as 0.96, indicating that the delicate
stratification is a ubiquitous phenomenon. These results
suggest strongly that the undersides of Fe and Na layers
are controlled by the same or very similar processes. The
upper boundary of the Na layer is always several kilometers
higher than that of the Fe layer. A relatively weak positive

A04S91

correlation is also persistently observed between the two
upper boundaries. Weak sporadic layering events frequently
occur on the upper extent of the metal layers. They may
impair the correct determination of the upper boundaries of
the normal metal layers and consequently weaken thecorrelation. Both the Fe and Na layers often show anevidently steeper density gradient on the underside thanon the upper extent, and the borders of the Fe layer are
clearly steeper than those of the Na layer. The explanation
to these observed relations needs further work in bothobservation and modeling.
[20] Acknowledgments. The research is supported jointly by the
National Natural Science Foundation of China through grants 40474059
and 40731055 and Program for Changjiang Scholars and Innovative
Research Team in University (PCSIRT0643). The authors appreciate theanonymous referees for their enlightening comments and suggestions.
[21] Zuyin Pu thanks the reviewers for their assistance in evaluating
this paper.

References
Clemesha, B. R. (1995), Sporadic neutral metal layers in the mesosphereand lower thermosphere, J. Atmos. Terr. Phys., 57, 725 – 736.
Clemesha, B. R., V. W. J. H. Kirchhoff, D. M. Simonich, and H. Takahashi(1978), Evidence of an extraterrestrial source for the mesospheric sodium
layer, Geophys. Res. Lett., 5, 873 – 876.
Clemesha, B. R., P. P. Batista, and D. M. Simonich (1999), An evaluation of
the evidence for ion recombination as a source of sporadic neutral layers
in the lower thermosphere, Adv. Space Res., 24, 547 – 556.
Collins, S. C., et al. (2002), A study of the role of ion – molecule chemistry
in the formation of sporadic sodium layers, J. Atmos. Sol. Terr. Phys., 64,
845 – 860.
Cox, R. M., and J. M. C. Plane (1998), An ion-molecule mechanism forthe formation of neutral sporadic Na layers, J. Geophys. Res., 103,6349 – 6359.
Delgado, R., B. R. Weiner, and J. S. Friedman (2006), Chemical model
for mid-summer lidar observations of mesospheric potassium over the
Arecibo Observatory, Geophys. Res. Lett., 33, L02801, doi:10.1029/
2005GL024326.
Eska, V., J. Höffner, and U. Von Zahn (1998), The upper atmospherepotassium layer and its seasonal variability at 54°N, J. Geophys. Res.,
103, 29,207 – 29,214.
Eska, V., U. von Zahn, and J. M. C. Plane (1999), The terrestrial potassium
layer (75 – 110 km) between 71°S and 54°N: Observations and modeling,
J. Geophys. Res., 104, 17,173 – 17,186.
Friedman, J. S., S. A. Gonzalez, C. A. Tepley, Q. Zhou, M. P. Sulzer,
S. C. Collins, and B. W. Grime (2000), Simultaneous atomic and ion
layer enhancements observed in the mesopause region over Arecibo
during the Coqui II sounding rocket campaign, Geophys. Res. Lett.,
27, 449 – 452.
Friedman, J. S., S. C. Collins, R. Delgado, and P. A. Castleberg (2002),
Mesospheric potassium layer over the Arecibo Observatory, 18.3°N
66.75°W, Geophys. Res. Lett., 29(5), 1071, doi:10.1029/2001GL013542.
Gardner, C. S., J. M. C. Plane, W. Pan, T. Vondra, B. J. Murray, and
X. Chu (2005), Seasonal variations of the Na and Fe layers at theSouth Pole and their implications for the chemistry and general circulation of the polar mesosphere, J. Geophys. Res., 110, D10302,
doi:10.1029/2004JD005670.
Gelinas, L. J., K. A. Lynch, M. C. Kelley, R. L. Collins, M. Widholm,
E. MacDonald, J. Ulwick, and P. Mace (2005), Mesospheric charged
dust layer: Implications for neutral chemistry, J. Geophys. Res., 110,
A01310, doi:10.1029/2004JA010503.
Gerding, M., M. Alpers, J. Höffner, and U. von Zahn (1999), Simultaneous K
and Ca lidar observations during a meteor shower on March 6 – 7, 1997, at
Kühlungsborn, Germany, J. Geophys. Res., 104, 24,689 – 24,698.
Gerding, M., M. Alpers, U. von Zahn, R. J. Rollason, and J. M. C. Plane(2000), Atmospheric Ca and Ca+ layers: Midlatitude observations andmodeling, J. Geophys. Res., 105, 27,131 – 27,146.
Gerding, M., M. Alpers, J. Höffner, and U. von Zahn (2001), Sporadic Ca
and Ca+ layers at mid-latitudes: Simultaneous observations and implications for their formation, Ann. Geophys., 19, 47 – 58.
Hansen, G., and U. von Zahn (1990), Sudden sodium layers in polar latitudes, J. Atmos. Terr. Phys., 52, 585 – 608.
Heinselman, C. J. (1999), Auroral effects on meteoric metals in the upper
atmosphere, Ph.D. dissertation, Stanford Univ., Stanford, Calif.

8 of 9

A04S91

YI ET AL.: FEATURES OF THE Fe AND Na LAYERS

Helmer, M., J. M. C. Plane, J. Qian, and C. S. Gardner (1998), A modelof meteoric iron in the upper atmosphere, J. Geophys. Res., 103,
10,913 – 10,925.
Höffner, J., and J. S. Friedman (2004), The mesospheric metal layer
topside: a possible connection to meteoroids, Atmos. Chem. Phys., 4,
801 – 808.
Höffner, J., and J. S. Friedman (2005), The mesospheric metal layer topside: Examples of simultaneous metal observations, J. Atmos. Sol. Terr.
Phys., 67, 1226 – 1237.
Hunten, D. M., R. P. Turco, and O. B. Toon (1980), Smoke and dust
particles of meteoric origin in the mesosphere and stratosphere, J. Atmos.
Sci., 37, 1342 – 1357.
Janches, D., C. J. Heinselman, J. L. Chau, A. Chandran, and R. Woodman(2006), Modeling the global micrometeor input function in the upper
atmosphere observed by high power and large aperture radars, J. Geophys.
Res., 111, A07317, doi:10.1029/2006JA011628.
Kalashnikova, O., M. Horanyi, G. E. Thomas, and O. B. Toon (2000),
Meteoric smoke production in the atmosphere, Geophys. Res. Lett., 27,
3293 – 3296.
Kane, T. J., and C. S. Gardner (1993), Structure and seasonal variability of
the nighttime mesospheric Fe layer at midlatitudes, J. Geophys. Res., 98,
16,875 – 16,886.
Kane, T. J., C. S. Gardner, Q. Zhou, J. D. Mathews, and C. A. Tepley(1993), Lidar, radar and airglow observations of a prominent sporadic Na/
sporadic E layer event at Arecibo during AIDA-89, J. Atmos. Terr. Phys.,55, 499 – 511.
Liu, B., F. Yi, and C. M. Yu (2005), Methods for optical adjustment in lidar
systems, Appl. Opt., 44, 1480 – 1484.
Lynch, K. A., L. J. Gelinas, M. C. Kelley, R. L. Collins, M. Widholm,
D. Rau, E. MacDonald, Y. Liu, J. Ulwick, and P. Mace (2005), Multiple
sounding rocket observations of charged dust in the polar winter mesosphere, J. Geophys. Res., 110, A03302, doi:10.1029/2004JA010502.
Plane, J. M. C., C. S. Gardner, J. Yu, C. Y. She, R. R. Garcia, and
H. C. Pumphrey (1999), Mesospheric Na layer at 40°N: Modeling andobservations, J. Geophys. Res., 104, 3773 – 3788.
Plane, J. M. C. (2003), Atmospheric chemistry of meteoric metals, Chem.
Rev., 103, 4963 – 4984.
Plane, J. M. C. (2004), A new time-resolved model for the mesospheric Na
layer: Constraints on the meteor input function, Atmos. Chem. Phys., 4,627 – 638.

A04S91

Qian, J., Y. Gu, and C. S. Gradner (1998), Characteristics of the sporadic
Na layers observed during the airborne lidar and observations of
Hawaiian airglow/airborne noctilucent cloud (ALOHA/ANLC-93)
campaigns, J. Geophys. Res., 103, 6333 – 6347.
Raizada, S., and C. A. Tepley (2003), Seasonal variation of mesosphericiron layers at Arecibo: First results from low-latitudes, Geophys. Res.
Lett., 30(2), 1082, doi:10.1029/2002GL016537.
Raizada, S., C. A. Tepley, D. Janches, J. S. Friedman, Q. Zhou, andJ. D. Mathews (2004), Lidar observations of Ca and K metallic layers from
Arecibo and comparison with micrometeor sporadic activity, J. Atmos. Sol.
Terr. Phys., 66, 595 – 606.
Shibata, Y., C. Nagasawa, M. Abo, T. Maruyama, S. Saito, and T. Nakamura(2006), Lidar observations of sporadic Fe and Na layers in the mesopause
region over equator, J. Meteorol. Soc. Jpn., 84A, 317 – 325.
States, R. J., and C. S. Gardner (1999), Structure of the mesospheric Na
layer at 40°N latitude: Seasonal and diurnal variations, J. Geophys. Res.,
104, 11,783 – 11,798.
Tilgner, C., and U. von Zahn (1988), Average properties of the sodium
density distribution as observed at 69°N latitude in winter, J. Geophys.
Res., 93, 8439 – 8454.
Yi, F., S. D. Zhang, H. J. Zeng, Y. J. He, X. C. Yue, J. B. Liu, H. F. Lv, andD. H. Xiong (2002), Lidar observations of sporadic Na layers over Wuhan(30.5°N, 114.4°N), Geophys. Res. Lett., 29(9), 1345, doi:10.1029/
2001GL014353.
Yi, F., S. Zhang, C. Yu, Y. He, X. Yue, C. Huang, and J. Zhou (2007),
Simultaneous observations of sporadic Fe and Na layers by two closely
colocated resonance fluorescence lidars at Wuhan (30.5°N, 114.4°E),
China, J. Geophys. Res., 112, D04303, doi:10.1029/2006JD007413.
Yrjölä, I., and P. Jenniskens (1998), Meteor stream activity. VI: A survey ofannual meteor activity by means of forward meteor scattering, Astron.
Astrophys., 330, 739 – 752.


Y. He, C. Huang, W. Li, F. Yi, C. Yu, X. Yue, and S. Zhang, Department
of Electronic Engineering, Wuhan University, Wuhan, 430079, China.
(yf@whu.edu.cn)

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