Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
379–382
J. Plasma Fusion

A Model for 3He Rich Events in Solar Flares
TOIDA Mieko and OKUMURA Hayato
Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received: 9 December 2003 / Accepted: 20 January 2004)

Abstract
A model for 3He rich events in solar flares, which was recently proposed with attention to nonlinear effects ofcurrent-driven instabilities, is extended to study energies of 3He ions. Selective acceleration of 3He ions is byelectrostatic H cyclotron waves with frequencies near 2X He ( X He is the cyclotron frequency of 3He). Theoreticalexpressions for wavenumbers, k and k 9 , of the H cyclotron waves and for the maximum energy of 3He ions arepresented. As initial electron drift speed v d increases, the value of k 9 decreases; which leads to the increase in themaximum 3He energy. The theory agrees with simulation results obtained by a two-dimensional, electrostatic, particlecode. It is also predicted that when the initial electron drift energy is of the order of 10 keV, 3He ions are accelerated toenergies of the order of MeV/n.
33Keywords:
current-driven instabilities, particle acceleration, particle simulation, 3He rich events

1. Introduction

with a wide range of frequencies and that selectiveacceleration of 3He ions would be difficult [6].
Recently, in order to solve this discrepancy, nonlinear
development of current-driven instabilities and associatedenergy transport have been investigated by means of particle
simulations [8,9]. In ref. [9], the simulations demonstrated
that strong currents can cause selective acceleration of 3Heions though electrostatic H cyclotron waves with ~ - 2X He .
These waves are excited via nonlinear effects of theinstabilities. Furthermore, the condition where ~ - 2X Hewaves become dominant was theoretically studied. It waspredicted that if an electron beam with its initial drift energy
higher than 1 keV exists, the enhancement of energetic 3He
ions would be produced.
In this paper, we present a theory for the maximum
energy of 3He ions. We then show that if the initial electronenergy is of the order of 10 keV, 3He ions are accelerated toenergies of the order of MeV/n. These energies are the sameorder as the observed values in the 3He rich events.
In Sec. 2, we outline the nonlinear development ofstrong current-driven instabilities. In Sec. 3, we present atheory predicting the wavenumbers of H cyclotron waves
with ~ - 2X He and the maximum energy of 3 He ionsaccelerated by these waves. We also show that the theory isin good agreement with simulation results. We then compare
the theory with the observations.

Intensive studies have been made on 3He rich events insolar flares ([1] and references therein). These events arecharacterized by enhancement of energetic 3He ions; theabundance ratio 3He/4He in particles with energies of theorder of MeV/n sometimes exceeds unity, although this ratio
in the solar corona is usually + 10-4.
The 3He rich events are associated with enhancement of
1-100 keV electrons [2] and posses impulsive hard X-ray
emission in the early phases of flares [3,4]. A standard
interpretation of this hard X-ray emission is that electrons are> 10 keV and then stream
initially accelerated to the energies +along the magnetic field toward and into the chromosphere,
producing bremsstrahlung via interaction with ambient
protons. Soft X-ray emission from heated plasma is observed
to follow the hard X-ray emission with a little time delay[3,4]. This indicates that the electrons exist initially as anintense energetic beam with relative cold temperature( v d > v Te , where v d is the stream velocity along the magnetic
field and v Te is the thermal velocity) and are later heated.
As a mechanism for the 3He rich events, instabilities
driven by electron currents are believed to be important, andseveral theoretical models have been proposed based on thelinear stability theories [5-7]. There are, however, someconflicts between the observations and these models. For
example, these models assumed that the instabilities are
driven by weak currents with v d < v Te , although strong
currents with v d > v Te exist in the early phases of the flares.
On the basis of the linear stability theory, it was expected thatthe strong currents would destabilize many kinds of waves

333©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: toida@phys.nagoya-u.ac.jp1379

Toida M. and Okumura H., A Model for 3He Rich Events in Solar Flares

2. Nonlinear development of current-driven
instabilities

Te + m e v /2.

|E k|

2

We consider instabilities by electron currents withv d > v Te , assuming that the electrons have a shifted Maxwell
distribution. According to the linear theory, Buneman waves
are the most unstable [10], while H cyclotron waves arealmost stable. Figure 1 shows typical time variations of a
Buneman wave and a H cyclotron wave; these were obtainedby a two-dimensional electrostatic particle code [9]. The
bottom panel shows time variations of electron temperatures,
Te and T e 9 , where subscripts
and 9 represent quantities
parallel and perpendicular to the magnetic field, respectively.
The temperatures are normalized to their initial values.
As predicted by the linear theory, the Buneman waves
grow at first. However, they quickly saturate owing toelectron trapping [11]. The electron trapping drasticallychanges the shape of electron velocity distribution functionf e (v ) and significantly broadens its width. Because of thisprocess, the parallel electron temperature Te significantlyrises. The saturation level of Te is estimated as0.880.4

H cyclotronwave

T / T0Te
500200400600800 1000

ω pe tFig. 1 Time variations of amplitudes | E k |2 of a Buneman
wave and a H cyclotron wave, and time variations ofelectron parallel and perpendicular temperatures.

k - 2X He /v d .

(5)3The ratio of k 9 to kis written as
1/2(2)

k 9 m H n e 2X He - X H- m e nH cm & 1.
kX He
3

3

These waves grow to large amplitudes and strongly influenceenergy transport to ions.
In ref. [9], it was theoretically and numerically shownthat the H cyclotron waves with ~ - 2X He and ~/k - v dwould eventually grow to the largest amplitudes if> 10TH . These ~ - 2X He waves selectively accelerate
Te +3He ions.

(6)

Hence, the waves propagate nearly perpendicular to theambient magnetic field.
Next, we consider particle orbits in a monochromatic H
cyclotron wave with ~ - 2X He and estimate the maximum
energy of 3He ions under two assumptions; 1) the wave
amplitude does not change in time, 2) the amplitude is sosmall that the stochastic acceleration [12] does not occur.
For the resonant particles with ~ - 2X He - k v - 0 ,we have a constant of motion, J n0 (k 9 t) cos p , ([12] andreferences therein), with J2 the Bessel function of the second
order. Here, the magnetic field is set to be in the z direction.
The wave is assumed to propagate in the (x, z) plane, and thequantity p is defined as333

3. Energy of 3He ions3

3.1 Theory
First, we give theoretical expressions for wavenumbers,
k and k 9 , of the H cyclotron waves with ~ - 2X He . As
described in Sec. 2, the H cyclotron waves are destabilized ina plasma with T e > T H . In this plasma, the waves with small
k 9 , k 9 t H < 1, have great growth rates [9]. The dispersion
relation of these waves is given by3~ - f1 +4

Te

On the other hand, ion temperatures rise little. Thus, aplasma with Te > T H is produced. The change in f e (v ) thendestabilizes the H cyclotron waves which were marginal inthe initial state. The phase velocities of these waves are~/k - v d .

12Bunemanwave

010(1)

2d*0.01.2n H Te
k2 t 2 pX .
2ne TH 9 H H

p / ni - (~ - k v ) t + k 9 x g0 + k z 0 ,(7)where i is the gyration phase, and x g0 and z 0 denote theinitial position. The maximum Larmor radius t Max satisfies(3)| J 2 (k 9 t Max )| = | J 2 (k 9 t 0 ) cos p 0 |,We then obtain k 9 of the ~ - 2X He waves as(8)3RVS ne TH 2X3He - XH W
k9 t H + Seo, WXHS nH mev
WTX
2d

where t 0 the initial Larmor radius.
We suppose that k 9 t 0 % 1. Then, the right-hand side ofeq. (8) is very small, and t Max can be estimated from the firstzero point of J 2 as

(4)

t max - 5/k 9 .
where eq. (1) is used. As v d increases, k 9 decreases. Weestimate k from eq. (2) as(9)The particles have this maximum Larmor radius, almost
independent of the initial gyration phase i 0 or the initial
2380

Toida M. and Okumura H., A Model for 3He Rich Events in Solar Flaresposition, x g and z 0 .
The ratio of the maximum energy to the initial thermalenergy is written as
22vMaxvTHX 2He25,
22 2(k 9 t H ) vT He X 2HvT He
33(10)

3where v TH and v T He are the initial thermal velocities of H
and 3He ions, respectively. Equation (10) shows that in thewave with k 9 t H % 1 , the maximum 3He energy is muchgreater than the initial energy. As k 9 decreases, themaximum 3He energy increases.
Using eq. (4), we approximately write eq. (10) as
32m e vd2vMaxme v 2X He
25++ 50 T d ,2T
HeHe (2X He - X H )
vT He
3

3

3

33Fig. 2 Wavenumber k 9 of the greatest amplitude wave as afunction vd .

(11)

where T He is the initial 3He temperature. The maximum 3Heenergy is proportional to the initial electron drift energy m e v d2 .
3Vmax /VT3He1000

22

3.2 Particle simulation
We perform simulations by means of a two-dimensional(two space and three velocity components) electrostatic
particle code with full ion and electron dynamics. The plasma
is assumed to consist of electrons, H, 4He, and 3He ions withthe abundance of 3He being small. Initially, the electrons
have a shifted Maxwell distribution with v d . We solve theinitial value problem of this system.
According to the theory, the wavenumbers k 9 of the H
cyclotron waves and the maximum 3He energy depend on v d .
We here present the results obtained by the simulations withvarious values of v d (for other parameters, see ref. [9]).
When v d is sufficiently large, the H cyclotron waves
with ~ - 2X He eventually grow to the greatest amplitudes.
Figure 2 shows the wavenumber k 9 of the greatest amplitude
wave as a function of v d . The dots are simulation results, andthe solid line represents the theory, eq. (4). The simulation
results are in good agreement with the theory.
Figure 3 shows the maximum 3He energy as a functionof k 9 of the ~ - 2X He wave. The solid line represents thetheoretical value estimated from eq. (10). The simulation
results are also in good agreement with the theory.

1000.20.4

k0.50.6

HFig. 3 Wavenumber k 9 of the greatest amplitude wave as afunction vd .

than the typical flare size, 109 cm. Hence, we neglect effect of
inhomogeneity of the backgound plasma on the wave.
Electromagnetic component in the wave is negligible.
In the extremely long wavelength region, electrostatic H
cyclotron waves would be connected to magnetosonic waves[13]. Hence, we can write the condition where theelectrostatic approximation is valid as33(13)~/k 9 < v A ,where v A is the Alfvén speed. This condition is well satisfied
because ~/k 9 + 10 6 cm/s and v A + 10 7 cm/s.
Substituting eq. (12) into eq. (10), we estimate the ratioof the maximum 3He energy to the initial thermal energy as

3.3 Comparison with observations
It has been reported that the 3 He rich events are> 10 ke V. We
accompanied by the energetic electrons with +suppose that initial electron drift energy is of the order of 10keV. We then estimate the wavenumber of an electrostatic H
cyclotron wave that would be destabilized by these electrons.
The maximum energy of 3He ions accelerated by this wave isalso given. We here assume that B=100 G, T H = 100 eV,
n H + 10 8 cm-3, and the initial 3He temperature T He = 100 eV.
From eqs. (4) and (5), we obtain ( k , k 9 ) of the wave
with ~ = 2X He and ~/k = v d as

2v max+ 10 4 .
2v T He(14)3This indicates that the maximum 3He energy is of the order ofMeV/n, which is the same as the observed energies of 3Heions in the 3He rich events [1, 2].

34. Summary3(k t H , k 9 t H ) + (0.0011, 0.045).

0.3

A model for 3He rich events in solar flares has beendeveloped with attention to nonlinear effects of strong
current-driven instabilities. We gave a theory for thewavenumbers of the H cyclotron waves with ~ - 2X He and(12)The parallel and perpendicular wavelengths are of the orderof 105 cm and 104 cm, respectively. They are much smaller

33381

Toida M. and Okumura H., A Model for 3He Rich Events in Solar Flares

the maximum energy of 3He ions accelerated by these waves.
As v d increases, the value of k 9 decreases and the maximum
3He energy thus increases. The theory is in good agreementwith the simulation results. We have also theoretically shownthat if the initial electron energy is of the order of 10 keV,
3He ions are accelerated to energies of the order of MeV/n.

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