J. Plasma Fusion Res. SERIES, Vol.4 (2001) 326-330

Nonlinear Evolution of Current-Driven Instabilities
and Energy Transport to Heavy lons
TOIDA Mieko*, SUGISHIMA Akihiro and OHSAWA Yukiharu
Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received: 5 December 2000 / Accepted: 18 August 2001)

Abstract
Nonlinear evolution of current-driven instabilities is studied through theory and three-dimensional
particle simulation, with special attention to associated potential formation and energy transport among
different particle species. The plasma is assumed to consist of hydrogen (H) and helium (He) ions andelectrons with the electron temperature higher than the ion temperatures. Electrons drift along a uniform
magnetic field with an initial speed equal to the thermal speed, ad, = Dr.. It is found that the second
harmonic H cyclotron wave with parallel phase velocity equal to od eventually becomes dominant, eventhough it is only marginal in the initial state. This wave produces a coherent potential profile and heats
He ions more significantly than H ions.

Keywords:
current-driven instability, ion cyclotron wave, ion acoustic wave, energy transport, multi-ion-species
plasmatemperatures. The electrons drift along a uniform
magnetic field with the initial speed equal to the thermalspeed, o6 = or". The linear theory for these initialconditions predicts that ion acoustic waves and H
cyclotron waves are unstable, with the former having
much greater growth rates. The simulations show thatafter the development of ion acoustic waves and H
cyclotron waves, second harmonic H cyclotron waves
are destabilized owing to the change in the electron

1. Introduction
Current-driven instabilities are believed to playessential roles in heavy-ion heating in space plasmas.
For example, for 3He rich events in solar flares or for O*
heating in the Earth's ionosphere, cyclotron resonanceswith unstable waves are thought to be important, andseveral theoretical models have been proposed [1-5].
However, those studies are mainly based on linear
theories and have not succeeded in quantitatively
explaining observations. To do this, we need to make atheory which self-consistently treats both nonlinear
evolution of instabilities and energy transport among
different particle species.
In the previous paper [6], we studied current-driven
instabilities using a two-dimensional, electrostatic,
particle simulation code with full ion and electrondynamics. The plasma was assumed to consist ofhydrogen (H) and helium (He) ions and electrons withparallel velocity distribution function, f"(o11). Even
though they are only marginally unstable in the initialstate, they eventually grow to the largest amplitudes andstrongly influence energy transport.
In the previous two-dimensional simulation (2D),
the number of unstable H cyclotron waves was small. Inorder to make a theory predicting the dominant wave,
we must study the nonlinear evolution of the instabilities
bv the simulation where a large number of unstablethe electron temperature higher than the ion*

Corresponding author's e-mail : toida@ phys.nagoya-u.ac jp326@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchToida M. el al., Nonlinear Evolution of Current-Driven Instabilities and Energy Transport to Heavy Ions

waves are contained. In this paper, we develop a theory

study the initial value problem of this system.

on nonlinear evolution of the instabilities. Furthermore,
we study associated potential formation and ion heating

We assume that the electron velocity distributionfunction at time / can be written as

by the three-dimensional simulation (3D) where thel,-.- *p(-+)str,,,,1. e)f(u,,.u,.r)= (2nu'r")'^\ .u-t" lnumber of unstable H cyclotron waves is much larger

than in 2D. It is shown that a second harmonic H
cyclotron wave with phase velocity equal to the initial
electron drift speed eventually becomes dominant andproduces a coherent potential profile. The heating ofHe

Then, growth rate of an ion acoustic wave isa3,t7tu/\t,,-ions by the dominant wave is more noticeable than that.-kii = H.Heof H ions.

2. Theory

3i1,,=,,-lCurrent-driven instabilities in a plasma consistingof H, He and electrons are studied with special attention
to the effect of the change in the electron velocity
distribution function. We assume that, initially, the ionsand frequency at is, = L*, ?u""1a2, f ogt,lrn k,,.

distribution function at time / = 0 is taken to benumber; Le. and Le; are the electron and ion Debye
lengths, respectively; and f,(p;) is f,(p,) = I,(1t;) exp(-p,), where 1, is the modified Bessel function of the n-h)..,(-"#) ,',th order. The quantitiy p; is defined ds Fi = klpl with p,
being the gyro-radius, pi = hy'{li.

The subscript ll and I denote quantities parallel andperpendicular to the magnetic field, respectively. We

aslh,1,,=,,0,,(4)Here, the subscript i refers to ion species; ft is the wave

f(utt,uto)

f(t)=a"^fuW*, zr<iui,ll''l(3)

have isotropic Maxwellian velocity distributionfunctions and the electrons drift along a uniform
magnetic field with the speed 'u6. The electron_*IeXD= (2ru'.r")''',r*^ ,lllo)'-The growth rate of n-th harmonic hydrogen
cyclotron wave (n = 1,2,...) is-,7,"*#+#"^,(*)l(5)Here, o) and a,, ate given byo)=ndt,/, *.,,u,[,-.*r", . m. m f;o -r.(r,J)]o, =+l-^r. 2m{lrf^ 11u,1) * ^-l- p t o".t.
n6,ln'""^ ((D_m{ti'The parallel phase velocities of H cyclotron waves arelarger than those of ion acoustic waves.

A'or,(l*11-'.

^ (a_mo""),

(6)),
(7)

lof the instabilities may be summarized as follows. Whenthe electron temperature is higher than the iontemperatures, ion acoustic waves are initially mostAccording to the simulations [6], the developmenttzt

Toida M. et al., Nonlinear Evolution of Current-Driven Instabilities and Energy Transport to Heavy Ions

The energy of the dominant wave is gradually

unstable [7]. They grow firstly and flatten g(t;11) aroundtheir phase velocities, which makes the slope of g(o11)steep in the region of phase velocities of H cyclotrontransferred to some ions that satisfy resonance condition,a - nd2i - kxux = 0. If ions have isotropic Maxwell
velocity distribution function, the increase rates ofwaves. This increases the growth rates of H cyclotron
waves. These waves further change the shape of g(o11),
which destabilizes waves that were marginal in theinitial state. The instabilities of waves with different

perpendicular kinetic energy densities can be given byo{, t n, q?f ,W) exp [_ @a:- r)''l lu lr, ttl='\dtphase velocities give rise to flattening of g(o11) over awide velocity region.
We now discuss what determines the wave thateventually becomes dominant. As we will see later inFig. 2, simulations show that the second harmonic H
cyclotron wave with olkx = ?6 grows to the largest

8 m,'u.r,k,,ki,rr, J'I

where n; is the ion density and E is the electric field.

3. Simulation
We have performed simulations using a three-amplitude, although it is marginal in the initial state.
That the waves with (Dlkx=po are important after thedevelopment of the instabilities may be related to thefact that the difference g(orr, 0) - g(an, t) is the largest at
ul = 'ud, as shown in Fig. 1. If we assume this, then wewould be able to predict which harmonic wave becomes

dimensional, electrostatic particle code with full ion and2x 10"

important, using the values of an given by Eq. (7). Thatis, the n-th harmonic wave with the largest a, and witholkn= po would eventually become dominant.

4Ne{33UJE,,,;
10006e0"1 2000Fig. 2 Power spectrum of fundamental. second and third
harmonic H cyclotron waves as a function of thefrequency, ar, and phase velocity, ay'ku.

o)60(a)(b)

n01,vil// vre
Vtt\fr"302Fig. 1 Evolution of three typical modes and electron
parallel velocity distribution function. Lines (a), (b)and (c) in the upper panel show an ion acoustic

wave, fundamental and second harmonic H
cyclotron waves, respectively. Their wave
Fig. 3 Contour map of electric potential at t4.t = 4800. e

numbers are lk,pu, kltsl = 10.97,0.211, (0.19,0.42l,,and (0.11, 0.73), respectively. Three vertical lines
in the lower panel represent their phase velocities.

and 4 are lengths parallel and perpendicular to

the magnetic field, respectively.
328

Toida M. et cl., Nonlinear Evolution of Current-Driven Instabilities and Energy Transport to Heavy Ions

electron dynamics. A 128 x 256 x 16 grid is used in thex, y, z direction with periodic boundary conditions. The
grid size is the Debye length ,1,p". We chose a uniform
magnetic field such tbat (8,, Bn, Br) = Bo(0.089, 0.49,0.87). This configuration enables us to have ion acoustic
waves and a large number of H cyclotron waves; thenumber of H cyclotron waves in this 3D system is about
ten times as large as that in the previous 2D system [6].
The particle numbers are Ne - 33,554,432, Nr= 27,787, 264, and Ns" = 2, 883,584. The mass ratios are
fttslm. = 100 and ms"lm11 = 4; the charge ratios are qsl
lq"l =This is mode (c). The second harmonic wave with a/k,,= oa €ventually becomes dominant in 3D system as wellas in 2D system, although the numbers of unstablemodes are quite different in these two systems. This

simulation result supports the theory developed in thispaper.

The dominant wave produces a coherent potentialprofile. Figure 3 shows a contour map of the electricfield potential at @pet = 4800. Here, ( and 4 are thelengths parallel and perpendicular to the magnetic field,respectively. They are normalized to,l,p". We can clearly
see the potential profile due to mode (c).

| and qs"lqs = 2: the temperature ratios are Z"/2,= 5.0 and Ts"lTs= l; the electron cyclotron frequency isldl"ll(Dp" = 2.0. The initial electron drift speed is equal tothe electron thermal speed, ?rd = ?re.

The electron drift energy is transferred to the ion
energies through unstable modes; the total energy in thesimulation is conserved. The energy transfer from mode

For these initial conditions, the linear theory
predicts that ion acoustic waves, fundamental and(c) to He ions is noticeable. Figure 4 shows time
variations of kinetic energies of H and He ions. The
dashed and solid lines represent parallel andsecond harmonic H cyclotron waves are unstable, while

He cyclotron waves are stable [6]. The ion acoustic
waves have the greatest growth rates. The second
harmonic waves have only small initial growth rates,
compared with the ion acoustic waves and thefundamental waves (see Fig. 1-3 in ref. [6]). However,the values of a" given by Eq. (7) for the second

perpendicular energies, respectively. After mode (c) isdestabilized, i.e., after aro"/ = 1500, K1 of He ionscontinues to increase more rapidly than that of H ions.

The heating of He ions is caused by the cyclotron
resonance with mode (c). (The acceleration of He ionsis not observed in the simulation.)
Figure 5 shows contour maps of the ion distributionfunctions in the (2r1, uf plane. The velocities of H and
He ions are normalized by 211 artd ?1s", respectively.

harmonic waves are larger than those for thefundamental waves; qc - Ql2)a, for the waves with rd
ktt='ua.

We show in Fig. 1 time variations of theamplitudes of three typical modes and electron parallel
velocity distribution function g(o11). Here the electricfield energies lE1,l2 arc normalized to m"al". Dotted line(a) in the upper panel represents an ion acoustic wavewith (/c11pg, ktpt) = (0.97, O.2l). Dashed line (b) andoY 1.1

solid line (c) show fundamental and second harmonic H
cyclotron waves with (k1ps, kgi = (0.19, O.42) andv(0. I I, 0.73), respectively. The vertical three lines in the1.0

lower panel show phase velocities, alkn, of these threemodes. Note that mode (c) has the phase velocity, a//c11,

equal to o6 and is only marginal in the initial state. It is
destabilized after the development of modes (a) and (b),and grows to a much larger amplitude than the other twomodes. Because of the interaction with these waves,g(o11) eventually has a large plateau region, 0Suxla1"SoY 1.1
Y1.0

1.5.

0Figure 2 shows spectrum of H cyclotron waves as afunction of the frequency and phase velocity.
Fundamental, second, and third harmonic waves areplotted. The spectrum is obtained from all componentsof the electric field for the period from a4"/ = 0 to 6000.
The highest peak is at ol{tH = 2.4 and (olkn)laa = l.g.

2000 4000 6000o)pet

Fig. 4 Time variations ion kinetic energies. The dashedand solid lines show parallel and perpendicularenergies, respectively. Energies are normalized totheir initial values.
329

Toida M. et a/., Nonlinear Evolution of Cunent-Driven Instabilities and Energy Transport to Heavy Ions

4. Summary
We have studied current-driven instabilities in aplasma consisting of H, He and electrons through theory
and three-dimensional particle simulation, with attention

to nonlinear evolution of instabilities and energy

0v1-404vtl-4transport among different particle species. It is foundthat the second harmonic H cyclotron wave with parallel
phase speed equal to the electron drift speed ud G ar")0

eventually becomes dominant and produces a coherent
potential profile. The heating of He ions by theV tl

Fig.5 Contour maps of /H and /"" in the (Yr, v') plane.

dominant wave is more noticeable than that of H ions.

The left panel shows the initial distribution functions; Hand He ions have the same profile. The middle and rightpanels show /g and fg" at apet = 6000, respectively. Thetll L.A. Fisk, Astrophys. J.224,1048 (1978).
l2l M.Ashour-Abdalla and H. Okuda, J. Geophys.

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H ions with o11lo1g = 4.0 are heated perpendicularly.
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Because the resonance takes place in the region of small
parallel velocity, the time rate of change in K1 of He ismuch greater than that of H for ob"t > 1500, as expectedfrom Eq. (8).

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