修士論文：
多成分プラズマにおける重イオンの波動による新しい加熱過程
水田孝信
指導教官 星野真弘 教授
東京大学大学院理学系研究科地球惑星科学専攻(２００２年２月２８日)

概要

テスト粒子シミュレーションを用いて、有限振幅の電磁流体 (MHD) 波動中
での、陽子より重い荷電粒子 (重イオン ) の選択的強い加熱を研究した。電子、陽子、重イオンで構成される３成分プラズマ中では、波動モード の一つは、長波長において、Alfvén 速度より非常に速い位相速度を持つ。この波動と、他のAlfvén 波動と共鳴することにより、重イオンは背景磁場に対して、垂直方向に強く非等方に加熱される。我々は、３成分プラズマ中での、重イオンの新たな加熱モデルを提案し 、その新しいモデルがテスト粒子シミュレーションと良くあう事を示す。新しいモデルを検証するため、２つの波の振幅依存性について調べた。この新しい加熱過程は、電子と陽子のみで構成される２成分プラズマの単純な延長では説明出来ないと考えられる。

i謝辞

本研究は星野真弘教授の指導の下で行いました。教授の多大なご指導にこころより感謝いたします。また、この論文を丁寧に査読され、有用で的確な助言を頂き、感謝いたします。
寺澤敏夫教授には有用な議論、助言を頂き感謝いたします。また、東京大学ＳＴＰ (太陽地球間物理) グループのスタッフ、学生の方々に感謝いたします。
また、岩垂奨学会には多大なご支援を頂き、感謝いたします。奨学会の発展に大きく貢献された故岩垂孝一氏に深く感謝するとともに 、ご冥福をお祈りいたします。

ii

Master Thesis:
New wave heating process of heavy ions in multi-component
plasmasTakanobu Mizuta
Supervisor: Prof. Masahiro Hoshino
Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo(February 25, 2002)

Abstract
We discuss preferential, strong heating of heavier charged particles than protons in large amplitude magnetohydrodynamic (MHD) waves by using test
particle simulations. In a 3 component plasma that consists of electrons, protons and heavy ions, one of the wave branches has phase velocity larger thanthe Alfvén velocity in a long wavelength regime. We ﬁnd that the resonant interaction between this wave and the other Alfvén wave leads the heavy ions tostrong anisotropic heating in perpendicular direction to an ambient magnetic
ﬁeld. We propose a new heating model for the heavy ions in the 3 component
plasma, and the new model takes a good agreement with the test particle
simulations. To conﬁrm the new model, we also study the dependence of theamplitudes of these two waves. We consider that this new heating process is
not explained by a simple extension of the plasma heating process understoodin a 2 component plasma which consists of electrons and protons.

iii

ACKNOWLEDGMENTS

The present thesis has been achieved being supervised by Professor M. Hoshino, theauthor would like to express sincere and hearty appreciation to him for his guidance andgreat help to this work and for his careful reading of the manuscript and useful suggestion
for revision.
The author would like to thank Professor T. Terasawa for valuable discussion and comments. The author is also very much grateful to staﬀs and graduate students of STP (solar
terrestrial physics) group at the University of Tokyo.

iv

Contents

概要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

謝辞 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

I

INTORODUCTION

1II

PARTICLE MOTION IN ONE EMIC WAVE

4

A

Dispersion relation for the multi-component plasma . . . . . . . . . . . .

4

B

Pitch-angle scattering in a EMIC wave . . . . . . . . . . . . . . . . . . . .

5

III

IV

V

VI

NUMERICAL STUDY

8

A

Simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

BResult for protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9C

Result for heavy ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

THEORY

12

A

Explanation by ε and χ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12B

Purely electric and magnetic waves theory . . . . . . . . . . . . . . . . . .

13

DEPENDENCE OF WAVE AMPLITUDES

17

A

Dependence of wave amplitudes for heavy ions . . . . . . . . . . . . . . .

17

BCase of weak SPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

CCase of strong SPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

D

Dependence of wave amplitudes for protons . . . . . . . . . . . . . . . . .

20

E

Dependence of wave amplitudes in the 2 component plasma . . . . . . . .

20

SUMMARY AND DISSCUSIONv

22

References

24

Tables26

Figures27

vi

I. INTORODUCTION

Preferential heating of heaver charged particles than protons is observed in several plasma
circumstances. The observations of the Ultraviolet Coronagraph Spectrometer (UVCS) operating onboard the Solar and Heliospheric Observatory (SOHO) satellite indicate that in
the polar coronal holes the temperature ratio between O5+ and protons is much lager thantheir mass ratio1 , TO5+ /Tp > mO5+ /mp , where T is the temperature, m is the mass, respectively. (The subscripts “O5+ ” and “p ” stand for the O5+ and the protons, respectively.) The
observations also indicate that O5+ has a highly anisotropic temperature: the temperaturein the perpendicular direction to the magnetic ﬁeld is 10 - 100 times hotter than that in theparallel direction to the magnetic ﬁeld. This observation imply the existence of preferential
heating processes of heavy ions to the perpendicular direction for the magnetic ﬁeld. Fast
Auroral SnapshoT (FAST) and Freja satellites observed preferential heating of He+ ionsto the perpendicular direction for the terrestrial magnetic ﬁeld in the terrestrial auroral
region2,3 . This heating occurs in association with electromagnetic ion cyclotron (EMIC)
waves, which implies that the He+ are heated through a cyclotron resonance with the waves.
For these studies, it is very important to understand the wave-particle interaction for theheavy ions.
Many authors have examined preferential heating of heavy ions. Tanaka4 has performed
numerical simulations and indicated that the heavy ions are preferentially heated in EMIC
waves driven by thermal anisotropic instability of protons. He discovered that the velocities
of most heavy ions direct to the electric ﬁeld of the excited waves, and the heavy ions arestrongly heated to the perpendicular direction for an ambient magnetic ﬁeld by experiencing
the electric ﬁeld of the waves. The quasi linear theory5 , assuming that the phase between
particles and waves is random, cannot explain this heating because this heating occurs bypersisting a speciﬁc phase relation between the heavy ions velocity and waves. In fact, inthe framework of the quasi linear theory under a weak turbulence, it has been analyticallyshowed that the heating process of the heavy ions does not have a qualitative diﬀerence from1the heating process of the protons6 . These facts indicate that the phase between particles
and waves is quite important information for the strong heating.
Isenberg7 has pointed out that heavy ions can be eﬀectively heated by the second order
Fermi acceleration, because they can interact with EMIC mode in two resonant points while
protons can interact with the EMIC mode in only one resonant point in the solar wind.
Mizuta and Hoshino8 have indicated that there is a very fast EMIC mode in a 3 component
plasma that consists of electrons, protons, and heavy ions, which mode does not exist in
the 2 component plasma that consists of electrons and protons, and this mode plays animportant role for the preferential heating of the heavy ions. Theoretical studies such as
beam instability, parametric instability and so on in the multi component plasma indicatethat several wave modes can be excited, which modes are never excited in the 2 component
plasma9–11 . These facts indicate that, what the heavy ions resonate with the waves whichcannot resonate with the protons causes the preferentiality of the heavy ions.
Theses studies implied that properties of wave-particle interaction in the multi component plasma are qualitatively diﬀerent from that in the 2 component plasma, and simply
extension of the theory on the 2 component plasma will not explain the phenomena in themulti component plasma. For understanding the preferential heating, it is important to examine the wave-particle interaction of the heavy ions in the multi-component plasma taking
account of the phase relation between the heavy ions and the waves.
In this thesis, we discuss by using test particle simulations that heavy ions are preferentially and strongly heated by the resonant interaction between a high-frequency EMIC
wave and a low-frequency MHD wave if the two waves have almost same Poynting ﬂuxes inthe 3 component plasma. We propose a new wave heating model explaining the preferential
and strong heating of heavy ions, and show that the model is good agreement with the test
particle simulations.
The plan of this thesis is as follows. In Section II, we brieﬂy review the dispersion
relation in the multi component plasma and the pitch-angle scattering in one coherent EMIC
wave. In Section III, we develop test particle simulations and show details of result in the2case of protons and heavy ions, respectively. In Section IV, we propose a new model for
preferential and strong heating of heavy ions in the multi component plasma. In Section V,
we examine the dependence of the wave amplitudes, and we show that the new model takegood agreement with the test particle simulations.

3II. PARTICLE MOTION IN ONE EMIC WAVE

A. Dispersion relation for the multi-component plasma

We only treat circularly polarized waves propagating along the ambient magnetic ﬁeld
(B0 ) in a cold plasma. We assume that the wave frequency (ω) is much smaller than theelectron cyclotron frequency (Ωe ). The dispersion relation in the 2 component plasma thatconsists of electrons and protons is




ω
ω2
= VA2 1 −
,
2
k
Ωp

(1)

where k is the wave number, VA is the Alfvén velocity and Ωp is the proton cyclotron
frequency. We deﬁne the sign of ω and the phase velocity Vph ≡ ω/k as follows: whena wave is left-hand circularly polarized (right-hand circularly polarized), we take ω > 0(ω < 0), when the wave having phase velocity parallel (antiparallel) to the ambient ﬁeld,
we take Vph > 0 ( Vph < 0 ). The top of Figure 1 shows this dispersion relation. The
brunch having ω < 0 is a whistler mode, and that having ω > 0 is an EMIC mode. Whenk  Ωp /VA (the reciprocal of the proton inertial length), these two modes degenerate to
the non-dispersive shear Alfvén wave which dispersion relation is given as ω/k = VA . As
k → ∞, the frequency of the EMIC mode asymptotically approaches Ωp , while the whistler
mode extends to the wave brunch having the lager velocity than the Alfvén velocity.
The dispersion relation in the 3 component plasma that consists of electrons, protons
and heavy ions isΩh f
k 2 VA2(1 − f )Ωp= 1,+−Ωp − ω
Ωh − ω
ωΩp(2)

where f ≡ qh nh /(|qe |ne ) (q is the charge of the particle, m is the mass, n is the number
density) and the subscript denotes “e”, “p” and “h” denote electrons, protons and heavyions, respectively. The bottom of Figure 1 shows this dispersion relation. The EMIC mode
separates two brunches, Lp mode (the brunch approaching ω → Ωp ) and Lh mode (thebrunch approaching ω → Ωh ). When k  Ωp /VA , Lh mode is nondispersive shear Alfvén4

wave like 2 component plasma. On the other hand, Lp mode has ﬁnite frequency (ωco ) whenk → 0. This cut-oﬀ frequency ωco is,ωco = f Ωp + (1 − f )Ωh .

(3)

Lp mode has very faster phase velocity than Alfvén velocity and has very long wavelength
near the cut-oﬀ frequency. We call the Lp mode near the cut-oﬀ frequency “Super Alfvénic
EMIC wave” (SPA) and call the Lp mode having not so faster phase velocity than Alfvénvelocity and Lh mode “Sub Alfvénic EMIC wave” (SBA). Lh mode is categorized into the> Ωp /VA is categorized
SBA regardless of wave number k. Roughly speaking, Lp mode at k ∼into the SBA and Lp mode at k → 0 is categorized into the SPA. It is very important that
the SPA never exists in the 2 component plasma. Following sections, we show that there
is a qualitative big diﬀerence between the properties of wave-particle interaction in the 2component plasma and that in the 3 component plasma.

B. Pitch-angle scattering in a EMIC wave

The behavior of a test particle in a monochromatic wave is well studied so far12,13 . The
test particles obey the following equations of motion,dv = ηΩv ⊥ sin ψ,
dt
dv ⊥
ω
= −ηΩ(v  − ) sin ψ,
dt
k
ηΩ ω
dψ
( − v  ) cos ψ,= −k(v  − VR ) +
dt
v⊥ k

(4)
(5)
(6)

where v  and v ⊥ are the test particle velocity components parallel and perpendicular to theambient magnetic ﬁeld, respectively. Ω is the gyro frequency for the test particle, ψ is thediﬀerence between the phase angle of v⊥ and that of the wave magnetic ﬁeld (B⊥ ), andη ≡ B⊥ /B0 . VR is the resonant velocity deﬁned as
VR =

ω−Ω
.
k
5

(7)

From these equations, we obtain the two constants of motion,
ω 2) + v 2⊥ ≡ ε,k1Ω(v  − VR )2 − ηv ⊥ cos ψ ≡ χ.
2k(v  −(8)(9)

Eq. (8) means the energy conversation in the wave frame and shows that the particle motions
are constrained to a constant ε circle in (v  , v ⊥ ) space (Figure 2 a). Eq. (9) shows that thetest particles satisfying the cyclotron resonant condition (v   VR ) are orbiting around the “trapping circle ” which is trapped orbit of χ constants near v   VR and ψ  0 in the (ψ, v  )space while nonresonant particles (v   VR ) move straightly parallel to the ψ axe (Figure 2b). The resonant particles are pitch-angle scatted because of v  oscillation by orbiting thetrapping circle, while the nonresonant particles are not pitch-angle scattered since v  of theparticles are not changed. v  of resonant particles are changed from v   VR − Vt to VR + Vt
within orbiting around the trapping circle, where resonant width Vt is,


Vt =

ηΩv ⊥
.
k

(10)

The resonant particles are orbiting around the trapping circle with trapping frequency Ωt ,Ωt =

ηΩkv ⊥ .

(11)

Due to these two constants, the pitch-angle scattering of resonant particles is constrained toa ε circle within v  = VR − Vt and VR + Vt (Figure 2 c). Resonant particles sense the wave
phases from ψ/π  −0.5 to 0.5 (Figure 2 b), which is corresponding to a half of all phases(2π). Therefore, the particles sense the wave phases corresponding to a half of wavelength
in the real space (Figure 2 d).
Figure 3 shows the dispersion relation and the resonant lines (ω = kVR + Ω) for protons
and heavy ions in the 3 component plasma. The particles having v  = VR resonate with thewaves on the crossing points where the resonant lines intersect the dispersion relation in thebottom of Figure 1. In the case of protons, particles resonate with two SBAs, which are Lp
and Lh modes. In the case of heavy ions, particles resonate with the SPA (Lp mode) and6

SBA (Lh mode). This fact leads heavy ions to the preferential heating. It is known that if
two waves are counter-streaming with same speed exist, ε is not constant and the particles
approximately move from one ε curve to another curve at random, which leads to the energy
diﬀusion in the velocity space, the so-called the second order Fermi acceleration14,15 . In thecase of protons, two SBAs are counter-streaming and have the about same speed, therefore
resonant protons are understood as the second order Fermi acceleration. In the case ofheavy ions, the SPA and SBA have very diﬀerent wave speed and wavelength, thus resonant
heavy ions experience a qualitative diﬀerent acceleration mechanism from the case of the
protons. In this thesis, we discuss that the resonance of the SPA plays an important role on
the preferential heating of the heavy ions.

7III. NUMERICAL STUDY

A. Simulation model

We consider a one-dimensional system with the main axis x. In the system, we supposethat there exist two waves with which protons (heavy ions) having v  = VA resonate. We
choose α particles (Ωh = Ωα = Ωp /2) as heavy ions. Given the two resonant waves, we
integrate in time the Lorentz equation for ions,dvv=q E+ ×B
m
dt
c



;

dr
= v,
dt

(12)

where the electric and the magnetic ﬁelds are respectively given by E = (0, Ey , Ez ), andB = (B0 , By , Bz ), with

 By 

=



Bz


2
j=1



 sin(kj x − ωj + φj ) B ⊥ ,j cos(kj x − ωj + φj )(13)

and the transverse electric ﬁeld is determined from k × E = ωB/c, Faraday’s low, wherec is the speed of light. The following test particle discussion neglects the reaction of theparticles on the waves. At ﬁrst, we show the standard wave-particle interaction in the twocoherent SBAs for protons. We study next the wave-particle interaction in the coherent
SPA and SBA for heavy ions. The simulation parameters are as follows: we set VR = VA ,then k and ω are determined so as to satisfy the resonant condition, (ω + Ω)/k = VR . To
specify the wave amplitudes, we assume that the two waves have the same Poynting ﬂux.
Faraday’s law, E⊥ = {ω/(ck)}B⊥ = (Vph /c)B⊥ where E⊥ is the wave amplitude of theelectric ﬁeld, requires the wave amplitudes η Lh and η Lp to satisfy η 2Lh Vph,Lh = η 2Lp Vph,Lp . The
simulation parameters are given in Table I. The number of test particles n is 100,000, theinitial velocities of test particles are v   VR = VA and v ⊥ ∼ 0, the thermal velosity is
0.02VA to the any direction. The initial positions are randomly distributed in space in orderto have random phases with the wave.
Figure 2 shows the positions of the test particles in velocity space at Ωp t = 100 andthe trajectory of a typical test particle from Ωp t = 0 to 100. In the case of run 1 which8

is the simulation for the protons, the test particles diﬀuse across the constant ε curves of
the two waves. In the case of run 2 which is the simulation for the heavy ions, however,the particles are not constrained to constant ε curves. They are eﬃciently heated in theperpendicular direction (note that the scale is not the same). This kind of particle motion
cannot be explained in a conventional pitch angle diﬀusion model in a 2 component plasma.
In the next subsections, we discuss more detail.

B. Result for protons

In this subsection, we discuss the result of run 1. Figure 4 (a) shows time evolution ofv ⊥ distribution for all test particles. The horizontal axis is time normalized by Ω−1
p , thevertical axis is v ⊥ normalized by VA and color counter is the particle number density witha logarithm scale. As time goes on, particles are slowly diﬀused, and are heated up toV ⊥ ∼ 3VA in 100Ωp . Figure 4 (b) shows time evolution of pitch-angle (the angle between
the ambient ﬁeld and particle velocity). We deﬁne the sign of the pitch-angle is positive
when v  > 0 and is negative when v  < 0. The particles are quickly pitch-angle scattered.
No particle have negative pitch angle because an initial v  is +VR and the particles are notstrongly scattered to v  < 0. One can ﬁned a cyclic dense curve in Figure 4 (a) and (b).
It means that some particles constrained to energy constant ε curves in one wave frame.
There are also a dense lines extended to high v ⊥ in Figure 4 (a), they are regarded as lucky
particles in the sense that they are successively accelerated by resonating with two waves.
Figure 4 (c) shows time evolution of the phase angle (ψ) between v ⊥ and B ⊥ of the wave
1. The particles experience various phases. We turn the detail explanation to the next
section, here we give a brief explanation with Figure 4 (d). A particle resonating with wave
1 (dense curve) senses the wave phases corresponding to a half of the wavelength of the wave
1 ( λ1 ), and the particle also experience various phases of the wave 2 (light curve) because
waves 1 and 2 have similar wavelength, λ1  λ2 . Therefore, particles experience various
phases for both waves, and this fact leads that one can regard the particle motions to be9

stochastic.

C. Result for heavy ions

In this subsection, we discuss a result of run 2. Figure 5 (a) shows time evolution ofv ⊥ distribution same as Figure 4 and theoretical result8 (dashed line), v ⊥ = η SPA Vph , SPA Ωt.
The theory agrees very well with the simulation. The next section, we discuss the detailabout the theory. Particles are “strongly” heated, which means that v ⊥ (the average ofv ⊥ for all particles) linearly increases very faster than that in the case of run 1 (note thatthe scale of v ⊥ of Figure 5 (a) is bigger than that of Figure 4 (a)) and the variance ofv ⊥ , (1/n)

j (v ⊥,j − v ⊥ )2increase very slower than v ⊥ , where a subscript “j ” denotes

each particle. Figure 5 (b) shows time evolution of the pitch angle. The particles arequickly accelerated towards the perpendicular direction, and the particles are not pitchangle scattered.
Figure 5 (c) shows time evolution for ψ SPA , an angle between v ⊥ and B ⊥ of the SPA.
Particles are very quickly “bunched” near ψ SPA /π  0.5. The phase behavior is describedby diﬀerent terms, “phase trapping” and “phase bunching”. “Phase trapping” means abounded phase oscillation of an individual resonant particle. On the other hand, “phasebunching” describes a state in which phases of many resonant particles are bunched arounda certain wave phase12 . The time scale of the bunching is about Ω−1
p . ψ SPA /π  0.5 isthe parallel to E ⊥ of the SPA. Therefore, during the interaction most particles are always
resonating with the SPA electric ﬁeld and the particles are strongly heated to the direction to
the electric ﬁeld of the SPA, which is the perpendicular direction to the ambient magnetic
ﬁeld. On the other hand, other few particles are deaccelerated from Ωp t  60 (also see
Figure 5 c), which are trapped only by the SPA and do not resonate with SBA. We turnthe detail explanation to the next section, here we give a brief explanation with Figure 5(d). A particle trapped by the SBA senses the wave phases corresponding to a half of the
wavelength of the SBA, and the particle experiences a speciﬁc phase for the SPA (light curve)

10

because the SPA have very long wavelength than that of the SBA, λSPA

λSBA . During thetrapping by short wave SBA, particles are bunched to the direction to the electric ﬁeld of
long wave SPA. Therefore the particles are strongly heated by experiencing the SPA electricﬁeld. This heating cannot be explained as the second ordered Fermi acceleration because of
the speciﬁc phase bunching. The next section, we propose new heating model and obtain
good agreement with the simulation result.

11

IV. THEORY
A. Explanation by ε and χ

In the case of run 1, two SBAs have about the same resonant width Vt . The top of
Figure 6 is an illustration of typical particle motion in the velocity space, and the bottomof Figure 6 is that in (ψ, v  ) space. We draw constant ε and Vt curves for two waves in the
top of Figure 6, and the constant χ curves in Figure 6 (b). The particle are stochasticallytrapped by one of two waves, therefore the particles are sometimes trapped by one of two
waves and sometimes trapped by another wave. Some particles are successively accelerated
by resonating with two waves and some particles are decelerated by the two waves. Since
fate of particles, accelerated or decelerated, is a matter of probability, the behavior of theparticles becomes diﬀusive.
In the case of run 2, two waves have very diﬀerent Vt , the SPA have larger Vt than thatof the SBA. The top of Figure 7 is an illustration of typical particle motion in the velocity
space, and the bottom of Figure 7 is that in (ψ, v  ) space. The particles trapped by the SBA
cannot orbit around the SPA trapping circle because v  of the particles are bounded withinsmall variation by the χ constant curves of the SBA. Since the particles cannot orbit aroundthe SPA trapping circle, thus ψSPA is not changed (Figure 7 bottom). The resonant particles
are accelerated to the parallel direction only by B ⊥ because the force of wave electric ﬁeld E ⊥has the perpendicular direction. Because particles are trapped by a wave B ⊥ varying v  toorbit around the trapping circle, the particles are trapped by the wave having stronger B ⊥ .
In the case of run 1, since B ⊥ of two waves are about same, particles sometimes orbit aroundthe trapping circle of one wave and sometimes orbit around that of another wave. On theother hand, in the case of run 2, B ⊥,SBA is very greater than B ⊥,SPA . (Note that the SBA and22= Vph ,SBA ηSBA
, and Vph ,SPA
SPA have same Poynting ﬂux, Vph ,SPA ηSPA

Vph ,SBA .) Therefore,

most particles are orbiting around the trapping circle of the SBA. An important point isthat SPA has larger trapping circle (Vt ,SPA > Vt ,SBA ) though particles are trapped by the SBA

12because the SBA has stronger magnetic ﬁeld than that of the SPA (B ⊥,SBA > B ⊥,SPA ). From
the above discussions, we can get the conditions of the occurrence to the “strong” heatingsuch as the case of run 2,
B ⊥ ,SPA < B ⊥ ,SBA

(14)λSPA B ⊥ ,SPA > λSBA B ⊥,SBA(15)

We examin Eqs. (14) and (15) by simulations in various parameters in Section V.

B. Purely electric and magnetic waves theory

In this subsection, we discuss about strong heating more quantitatively. In the case ofrun 2, the simulation for heavy ions, the SPA has a larger electric ﬁeld and weaker magnetic
ﬁeld than the SBA since,η SBA Vph,SBA
E⊥,SBA= 2,
E⊥,SPAη SPA Vph,SPA
B⊥,SBA
η SPA=
 0.2.
B⊥,SPA
η SBA

(16)
(17)

To simplify the discussion, we treat a model in which the SPA is taken as purely electric,
while the SBA is assumed purely magnetic. We therefore neglect the magnetic ﬁeld in the
SPA and the electric ﬁeld in the SBA. In other words, we treat “the purely electric wave”
which brings only the electric ﬁeld and “the purely magnetic wave” which has only themagnetic ﬁeld. (These situations may be realized in the cut-oﬀ frequency with k = 0 andthe resonance frequency with k → ∞, respectively.) For propagation along the ambient
magnetic ﬁeld, the test particles obey the following equations of motion:

13

dv 
= Ωη B v ⊥ sin ψ B ,dtdv ⊥= Ω(η E c cos ψ E − η B v  sin ψ B ),dtdψ B
= −kB (v  − VR )dtΩ− (η E c sin ψ E + η B v  cos ψ B ),v⊥dψ E
= −kE (v  − VR )dtΩ− (η E c sin ψ E + η B v  cos ψ B ),
v⊥

(18)
(19)

(20)

(21)

where η E = E⊥ /B0 , η B = B⊥ /B0 (subscript “E ” stands for the purely electric ﬁeld and “B ”
describes the purely magnetic wave.)
Let us consider the case that only purely electric wave exists. Substituting η B = 0 in
Eqs.(18)-(21), these equations become
dv 
= 0,
dt
dv ⊥
= AE cos ψ E ,
dt
dψ E
AE
= −k(v  − VR,E ) −sin ψ E ,
dt
v⊥

(22)
(23)
(24)

where we deﬁne a theoretical heating ratio, AE ≡ η E Ωc. Eq.(22) indicates v  is a constantof time. We are interested in the behavior of resonant particles (v  = VR ). Eq.(24) becomes
dψ E
AE
=−
sin ψ E ,
dt
v⊥

(25)

showing that ψ E decreases for ψ E > 0 (−π < ψ E < π), while ψ E increases for ψ E < 0. When
ψ E  0, v ⊥ increases from Eq.(23) and ψ̇ E → 0 from Eq.(24). Therefore, ψ E for resonantparticles is approaching zero, reaching the stable equilibrium at ψ E = 0. Resonant particles
are quickly bunched to ψ E = 0. The time scale of the particle to be bunched to ψ E = 0 is
v⊥
.
AE

(26)

v⊥
 Ω−1
p ,
η SPA ΩVph,SPA

(27)

TE =
In the case of run 2, TE is given by
TE 

14

where we use that because of Faraday’s law η E c corresponds to η SPA Vph,SPA , and v ⊥ = 0.1VA
as a typical initial value. This indicates that the particles are immediately bunched at
ψ E = 0. Substituting ψ E = 0 in Eq.(23) we have v ⊥ = AE t, with v ⊥ = 0 initially. The
resonant particle velocity increases linearly with the constant heating rate, AE .
Next, let us consider the case that only purely magnetic wave exists. By substituting
η E = 0 in Eqs.(18)-(21), we obtain the constants of motion,v 2 + v 2⊥ = ε(28)

η B Ωv ⊥ cos ψ B
1(v  − VR )2 −
≡ χ.
2
kB

(29)

The purely magnetic wave has the same properties as the monochromatic wave. The resonantparticles are trapped by the purely magnetic wave and are pitch-angle scattered. Because
ψ B = 0 of the center of the trapping circle, ψ B oscillates near ψ B  0.
Now, we discuss the case that both the purely electric wave and the purely magnetic
wave coexist. We assume that v  is close to resonant velocity VR . From the previous
discussion, ψ E becomes zero quickly and ψ B oscillates near ψ B  0. We discuss the particle
motions having ψ E = 0 and ψ B  1 after the initial time evolution. Furthermore, we
assume η E c

η B v  ψ B because η SPA Vph,SPA > η SBA VR in the case of run 2 (η E c corresponds

to η SPA Vph,SPA , η SBA VR ∼ η B v  and ψ B  1). We therefore neglect the second term of theright hand side in Eq.(19). We assume the particles have a large v ⊥ /v  . Thus we can drop
the last term of the right hand side in Eq.(21). In this approximation we ﬁnally getv  = Vt,B ψ B0 sin(ωt,B t) + VR ,(30)v ⊥ = AE t,(31)

ψ B = ψ B0 cos(ωt,B t),(32)


where Vt,B is the trapping velocity for the purely magnetic wave, Vt,B ≡ η B Ωv ⊥ /k B , ωt,B is√the trapping frequency for that, ωt,B ≡ η B Ωv ⊥ k B and ψ B0 is the initial value of ψ B . Theseequations show that v  and ψ B are determined by the purely magnetic wave and a simple
harmonic oscillator while v ⊥ is described by the purely electric wave alone and increases
15

linearly AE . We have showed that (31) agree with the numerical result in the case of run 2
in Section III.

16

V. DEPENDENCE OF WAVE AMPLITUDES

A. Dependence of wave amplitudes for heavy ions

In this section, we perform simulations in various parameters of two waves in the case ofrun 2 (for heavy ions). In the previous sections, we assume that two waves have the same
Poynting ﬂuxe (P f ). Here, we study the dependence of the ratio of Poynting ﬂux of two
waves, P f SPA/P f SBA , and investigate the eﬀect of the resonant velocity VR . When VR ischanged, we must take diﬀerent k and ω of two resonant waves (Figure 1 bottom). Thus
the ratio of phase velocity, Vph ,SPA /Vph ,SBA is also modiﬁed.
To decide whether strong heating occurs or does not for each simulation, we ﬁt a numerical result on an equation,
v ⊥ = ctα ,(33)

where the bracket “ ” means an average for all particles. When the particles are strongly
heated, one obtains α = 1 from Eq.(31). On the other hand, when particles are only
pitch-angle scattered, one obtains α = 0 because pitch-angle scattering does not diﬀuse theparticle energy (Eq.8). The top of Figure 8 shows an index α as the function of P f SPA /P f SBA
and Vph ,SPA /Vph ,SBA . The horizontal and vertical axes are P f SPA /P f SBA and Vph ,SPA /Vph ,SBA ,respectively. We execute simulations from tΩp = 0 to 1000. We have conﬁrmed that if theratio of Poynting ﬂux of two waves is same, the absolute value of Pounting ﬂux does not
aﬀect the index α. The top of Figure 8 indicates that when the Poynting ﬂux of two waves
are same, the particles are strongly heated. When one of waves is weak, on the other hand,the particles are not heated, and then the particles are pitch-angle scattered by the wave
having lager Poynting ﬂux. The occurrence of the strong heating needs both of the SPA
and SBA and the particles are not strongly heated even if which wave is missing. In the topof Figure 8, white hatched area is satisﬁed Eqs. (14) and (15), which agree very well with
the numerical result. Therefore, we consider that Eqs. (14) and (15) are the necessary andsuﬃcient conditions for the strong heating.
17

The bottom of Figure 8 shows a index c/AE , where c means numerical heating ratio
given by Eq. (33) only when α  1 and AE is theoretical hating ratio given by Eq. (31),respectively. In order to avoid misleading, we only plot c/AE satisfying α > 0.9 (the strong
heating area) because c does not mean heating ratio when α  1, for example c meansa constant velocity, v ⊥ = c when α = 0. In the left part (P f SPA /P f SBA < 1) of thestrong heating area, on ﬁnds c/AE  1 which means that theoretical result AE agrees withnumerical result c. On the other hand, one ﬁnds c/AE < 1 in the right part of the strong
heating area because some particles are strongly heated and others are only pitch-angle
scattered. We discuss about this phenomenon in the following subsection.

B. Case of weak SPA

In this subsection, we discuss the case of λSPA B ⊥,SPA  λSBA B ⊥,SBA , which is one of
the critical parameter whether Eq.(15) is satisﬁed or not. In this case, the SPA has smaller
Poynting ﬂux than that of the SBA, that is to say, the SPA is weaker than the SBA. In the topof Figure 8, a white dashed curve on the left side is satisﬁed with λSPA B ⊥ ,SPA = λSBA B ⊥ ,SBA .
Here, we take Vph ,SPA /Vph ,SBA = 21.7 and P f SPA/P f SBA = 0.098 (black circle in the top of
Figure 8). The top of Figure 9 shows time evolution of v ⊥ from tΩ = 0 to 1000, and thewhite dashed line is the theoretical heating ratio (Eq.31). Before tΩ  200, particles areheated same as theory, but the after, the particles are not strongly heated. The bottomof Figure 9 is an illustration of a typical particle motion in (ψ, v  ) space. In this case, theresonant width of the SPA, Vt ,SPA is same as Vt ,SBA . At ﬁrst, the particles are trapped by the
SBA and phases for the SPA are bunched at E ⊥,SPA , thus the particles are strongly heated.
However, the phase bunching for the SPA is gradually broken because by orbiting aroundthe trapping circle of the SBA the particles can easily obtain v  enough to orbit around thetrapping circle of the SPA (note Vt ,SBA  Vt ,SPA ). The SPA is not strong enough to bunch theparticles at the direction of the electric ﬁeld of the SPA. Therefore, the particles gradually
drop out of the strong heating process.

18

C. Case of strong SPA

We discuss the case of B ⊥,SPA  B ⊥,SBA , which is another critical parameter whether
Eq.(14) is satisﬁed or not. In this case, the SPA has bigger Poynting ﬂux than that of the
SBA, that is to say, the SPA is stronger than the SBA. In the top of Figure 8, a white dashedline on the right side is satisﬁed with B ⊥ ,SPA  B ⊥,SBA . Here, we take Vph ,SPA /Vph ,SBA = 21.7
(same as previous subsection) and P f SPA /P f SBA = 20.5 (black square in the top of Figure8). The top of Figure 10 shows time evolution of v ⊥ from tΩ = 0 to 1000. Most particles
are pitch-angle scattered and energy diﬀused slowly. The frequency of these pitch-angle
scattering indicates that the particles resonate with the SPA. However, some particles arestrongly heated and their heating ratio agrees with the theoretical heating ratio (not shown).
Some particles begin to be strongly heated, and once the heating begins, the particles neverdrop out of the strong heating process. The bottom of Figure 10 is an illustration of a typicalparticle motion in (ψ, v  ) space. In this case, some particles orbit around the trapping circleof the SPA because the magnetic ﬁeld of the SPA is strong as same as that of the SBA,
and particles moves neglecting χ counters for the SBA. Therefore, they are pitch-angle
scatted by the SPA. However, some particles are sometimes trapped by the SBA, and begin
to be phase-bunched near E ⊥,SPA . Such particles never orbit around the trapping circle of
the SPA again because the particles trapped by the SBA cannot obtain v  enough (note
Vt ,SPA

Vt ,SBA ). Therefore, the selected particles are strongly heated all the time after theselection. Let us see the top of Figure 10 again, most particles begin to be strongly heated
at v ⊥  0. The time scale of the particle to be bunched at ψ SPA = 0 is given by Eq.(26).
When v ⊥ is small, particles are rapidly bunched at ψ SPA = 0 and easily begin to be strongly
heating. Therefore, particles are selected every trapping frequency of the SPA and makelines at equal intervals in the top of Figure 10.

19

D. Dependence of wave amplitudes for protons

We also perform simulations in various parameters of two waves in the case of run 1(for protons). The top of Figure 11 shows the index α in the same format as Figure 8.
The strong heating area is very narrower than that in the case of heavy ions. The bottomof Figure 11 shows the index c/AE in the case of protons. Protons are strongly heated
only when the wave parameters are very speciﬁc. Note that the scale of vertical axe is not
same as Figure 8 because the protons cannot resonate with two wave having very diﬀerent
wavelength (Figure 1 bottom). Therefore, very speciﬁc wave parameters are satisﬁed with
Eqs. (14) and (15). We consider that the strong heating occurs not only for the heavy ionsbut also for the protons. The conditions Eqs.(14),(15) are satisﬁed for protons only whenwaves have the speciﬁc amplitudes because the protons cannot resonate with two waves
having very diﬀerent wavelength while these conditions can be easily satisﬁed for the heavy
ions in the 3 component plasma because the SPA having very long wavelength exists.

E. Dependence of wave amplitudes in the 2 component plasma

We also perform simulations in various paramerters for protons in the 2 component
plasma. In the 2 component plasma, there are whistler and EMIC modes (Figure 1 top).
We account the index α in various parameters in the coherent whistler and EMIC waves
resonating with the protons. Figure 12 shows the index α in the same format as Figure 8
except the vertical axe is VR . This result shows that the protons are strongly heated evenin the 2 component plasma when two waves have speciﬁc parameters. The top of Figure 13
shows v ⊥ at P f EMIC /P f Whistler = 0.26 and VR = 3 (black circle in Figure 12). Some particles
are strongly heated and others are pitch-angle scattered, it is similar to the case of the strong
SPA in the previous subsection. In this case, it is diﬀerent from the case of the SPA and SBA
that the EMIC has longer wavelength and weaker electric ﬁeld (slower phase velocity) than
the whistler wave while the SPA has longer wavelength and stronger electric ﬁeld than that20

of the SBA. The heating ratio obtained by numerical result for selected particles agrees withtheoretical heating ratio obtained by whistler electric ﬁeld, and the strength of EMIC electricﬁeld is too weak to explain the numerical heating ratio. However, the selected particles not
may be bunched to the direction of the whistler electric ﬁeld (Figure 13 bottom, the directionof wave electric ﬁeld is ψ ,Whistler /π = 0.5), and the particles are buched at ψ ,Whistler /π  0.2.
It may not be easy to explain the reason.

21

VI. SUMMARY AND DISSCUSION

We have discussed a new heating process of heavy ions in two coherent EMIC waves
having diﬀerent wavelength each other by using the test particle simulations, and we haveproposed the new models which agree with simulations. We have found that particles arestrongly heated in the two coherent circularly polarized waves which are satisﬁed
B ⊥,1 < B ⊥,2 ,(34)

λ1 B ⊥,1 > λ2 B ⊥ ,2 ,(35)

and the heating ratio for the perpendicular direction to the ambient magnetic ﬁeld, AE isAE =B ⊥,1 ω1
Ω,
B0 k1

(36)

where subscript “1 ” and “2 ” describe one of two coherent waves and wave 1 is longer thanwave 2 (k1 < k2 ). The above conditions can be easily satisﬁed for the heavy ions in the 3component plasma, because the cut-oﬀ EMIC wave which has very long wavelength exists.
On the other hand, it is diﬃcult that the above conditions are be satisﬁed for protons, because the protons cannot resonate with the two waves which have very diﬀerent wavelength.
Therefore, heavy ions are strongly heated within various wave parameters while protons areheated within speciﬁc parameters. The dispersion relation in the 3 component plasma leads
to such preferential heating of heavy ions.
We assumed for simplicity two coherent waves model, but we expect that heavy ions arestrongly heated in one coherent long wave and weak turbulence, and if that is right, themodel proposed by this thesis will take more applications to heliospheric and astronomical
phenomena. Now, we are analyzing about this issue. We did not discuss how a coherent long
EMIC wave is excited. One of the possible processes is an anisotropic thermal α particle
beam instability10 . Other possible process is a parametric instability in the multi component
plasma9 . The excitation mechanisms of this mode should be studied further. We are alsoplanning to study the self-consistent interaction between waves and particles using a hybrid
simulation.
22

We have only begun to investigate the potentially rich particle heating process in themulti-component plasma. It is important to study our theoretical model from other aspects
such as the stochastic/chaotic heating process etc.16,17 . This issue will be reported elsewhere.
We expect that this theory can be applied to many heliospheric and astronomical phenomena. In future publications, we hope to show that observed preferential heating agrees
with new heating model proposed in this thesis.

23

REFERENCES
1

S. R. Cranmer, J. L. Kohl, G. Noci, E. Antonucci, G. Tondello, M. C. E. Huber, L.
Strachan, A. V. Panasyuk, L. D. Gardner, M. Romoli, S. Fineschi, D. Dobrzycka, J.
C. Raymond, P. Nicolosi, O. H. W. Siegmund, D. Spadaro, C. Benna, A. Ciaravella, S.
Giordano, S. R. Habbal, M. Karovska, X. Li, R. Martin, J. G. Michels, A. Modigliani,
G. Naletto, R. H. O’Neal, C. Pernechele, G. Poletto, P. L. Smith, and R. M. Suleiman,
Astrophys. J. 511, 481, (1999).

2

E. J. Lund, E. Möbius, L. Tang, L. M. Kistler, M. A. Popecki, D. M. Klumpar, W.
K. Peterson, E. G. Shelley, B. Klecker, D. Hovestadt, M. Temerin, R. E. Ergun, J. P.
McFadden, C. W. Carlson, F. S. Mozer, R. C. Elphic, R. J. Strangeway, C. A. Cattell,and R. F. Pfaﬀ, Geophys. Res. Let. 25, 2049 (1998).

3

R. E. Erlandson, L. J. Zanetti, M. H. Acuña, A. I. Eriksson, L. Eliasson, M. H. Boehm,
and L. G. Blomberg, Geophys. Res. Let. 21, 1855 (1994).

4

M. Tanaka, J. Geophys. Res. 90, 6459 (1985).

5

C. F. Kennel and F. Engelmann, Phys. Fluids 9 2377 (1966).

6

V. K. Jordanova, J. U. Kozyra, and A. F. Nagy, J. Geophys. Res. 101, 19771 (1996).

7

P. A. Isenberg, Space Sci. Rev. 95, 119 (2001).

8

T. Mizuta and M. Hoshino, Geophys. Res. Let. 28, 3099 (2001).

9

T. Kubo, master thesis, University of Tokyo (1986).

10

K. Killen, N. Omidi, D. Krauss-Varban, and H. Karimabadi, J. Geophys. Res. 100, 5835
(1995).

11

S. P. Gary, L. Yin, D. Winske, and D. B. Rsisenfeld, J. Geophys. Res. 105, 20989 (2000).

12

H. Matsumoto, Wave Instabilities in Space Plasmas, D. Reidel Pub. Co. 74, 163 (1979).

24

13

M. Hoshino and T. Terasawa, J. Geophys. Res. 90, 57 (1985).

14

T. Terasawa, Geophysical Monograph, American Geophysical Union, 53, 41 (1989).

15

T. Sugiyama, M. Fujimoto, and T. Mukai, J. Geophys. Res. 106, 21657 (2001).

16

W. Horton and Y. H. Ichikawa, Chaos and Structures in Nonlinear Plasmas, World Scientiﬁc Pub. Co. (1996).

17

R. White, L. Chen and Z. Lin, Phys. Plasmas (in press).

25

TABLESTABLE I. Parameters for the test particle simulations
run particles

wave 1wave 2

1

protons

SBA(B ⊥ /B0 = 0.075, Vph /VA  −1.2) SBA (B ⊥ /B0 = 0.1, Vph /VA  −0.68)

2heavy ions (α) SPA (B ⊥ /B0 = 0.022, Vph /VA  18)

26

SBA (B ⊥ /B0 = 0.1, Vph /VA  −0.83)

FIGURES

FIG. 1. The dispersion relation in 2 component plasma (top) and in 3 component plasma(bottom), the solid lines are the dispersion relations, and the dashed line is the resonance line forheavy ions (α particles) and the dot-dashed line for protons.

27

D

C

E

F

FIG. 2. (a): Constant ε circles in (v , v ⊥ ) space. ε means the energy conversation in the wave
frame. The particle motions are constrained to a constant ε circle. The arrow shows a typicalparticle trajectory in this space. (b): Constant χ curves in (ψ, v  ) space. The arrow shows the
trajectory of a typical resonant particle in this space. (c): An arrow is the trajectory of a typicalresonant particle motion in (v , v ⊥ ) space. We also draw constants ε (thin solid circles) and resonant
width (thick solid curves). The pitch-angle scattering of resonant particles is constrained to a εcircle from v  = VR − Vt to VR + Vt . (d): An illustration of the resonant particle motion in one
coherent wave in the real space. A resonant particle senses the wave phases corresponding to ahalf of wavelength in one coherent wave.

28

FIG. 3. (Left): Test particles at Ωp t = 100 in velocity space. (Right): The trajectory of typicaltest particle (solid lines) and ε constant curves by two waves (thin curves) in the velocity space.
Top is for run 1 (protons) and bottom is for run 2 (heavy ions).

29

C

D

E

PWODGTFGPUKV[PWODGTFGPUKV[F

PWODGTFGPUKV[

FIG. 4. The time evolution of v ⊥ (a), the pitch angle (the angle between the ambient ﬁeld
and particle velocity) (b) and the phase angle (ψ) between v ⊥ and B ⊥ of the wave 1 (c) for all
test particles in the case of run 1 (for protons). The horizontal axis is time normalized by Ω−1p ,the vertical axis is v ⊥ normalized by VA and color counter is the particle number density with alogarithm scale. We deﬁne the sign of the pitch-angle is positive when v  > 0 and is negative whenv  < 0. (d): An illustration of the resonant particle motion in the two coherent SBAs in the real
space in the case of run 1 (for protons). The resonant particle trapped by the wave 1 experiences
various phases of the wave 2 and one can regard the particle motions as stochastic.

30

C

D

PWODGTFGPUKV[PWODGTFGPUKV[

E
F
'ӆ

SBA
SPA

PWODGTFGPUKV[

FIG. 5. The time evolution of v ⊥ distribution and theoretical result (dashed line) (a), the pitch
angle (b), and ψ SPA , an angle between v ⊥ and B ⊥ of the SPA (c) in the case of run 2 (for heavyions). (d): An illustration of the resonant particle motion in the coherent SPA (thin curve) andSBA (dense curve) in the real space in the case of run 2 (heavy ions). Within the resonant particle
being trapped by short wave SBA, the particle always experiences the speciﬁc phase of the SPA.

31

FIG. 6. An illustration of typical particle motion (dashed curve) in the velocity space (top)and in the (ψ, v  ) space (bottom) in the case of run 1 (for protons). We also draw constants εcurves (thin solid circles) and resonant width (thick solid curves) on the top panel. A particle issometimes trapped by one of two waves and is sometimes trapped by another wave.

32

FIG. 7. An illustration of typical particle motion (dashed curve) in the velocity space (top) andin the (ψ, v  ) space (bottom) in the case of run 2 (for heavy ions). We also draw constants ε curves(thin solid circles) and resonant width (thick solid curves) on the top panel. The resonant particle
trapped by the SBA cannot orbit around the trapping circle of the SPA because the particles
cannot obtain the varying of v  enough to orbite around the trapping circle of the SPA, therefore
ψSPA is not changed.

33

+PFGZ#NRJC

E#'

34

FIG. 8. (a): The index α (color) obtained from a ﬁtting equation, v⊥ = ctα (tΩp = 0 → 1000)
in various parameters of two waves in the case of heavy ions. The horizontal and vertical axes are
P f SPA /P f SBA and Vph ,SPA /Vph ,SBA , respectively. White hatched area enclosed white dashed curves
is the strong heating area obtained by the theory discussion in Section IV. We also perform casestudies, the case of weak SPA, λSPA B ⊥,SPA  λ,SBA B ⊥,SBA (black circle) and the case of strong SPA,
B ⊥,SPA  B ⊥,SBA (a black square). (b): The index c/AE in the case of heavy ions, where c means
numerical heating ratio given by v⊥ = ctα and AE is theoretical hating ratio. In order to avoid
misleading, we only plot c/AE satisfying α > 0.9 (the strong heating region) because c does notmean heating ratio when α  1, for example c means a constant velocity, v⊥ = c when α = 0.
White dashed curves are same as (a)35

PWODGTFGPUKV[

FIG. 9. (a): The time evolution of v ⊥ in the case of weak SPA, λSPA B ⊥,SPA  λSBA B ⊥,SBA (black
circle in Figure 8). White dashed line is the theoretical heating ratio. (b): An illustration of atypical particle motion in (ψ, v  ) space.

36

PWODGTFGPUKV[

FIG. 10. (a): The time evolution of v ⊥ in the case of strong SPA, B ⊥,SPA  B ⊥,SBA (blacksquare in Figure 8). (b): An illustration of a typical particle motion in (ψ, v  ) space.

37

+PFGZ#NRJC

E#'FIG. 11. The index α (a) and the index c/AE (b) in the case of protons in the same format as
Figure 8.

38

+PFGZ#NRJC

FIG. 12. The index α in the case of the 2 component plasma in the same format as Figure 8
except the vertical axe is the resonant velocity.

39

PWODGTFGPUKV[PWODGTFGPUKV[
FIG. 13. The time evolution of v ⊥ (a) and ψ ,Whistler (b) in the case of the 2 component plasma(black square in Figure 12).

40