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THEORETICAL

MODELS

OF LOCALIZED

ELECTROSTATIC

STRUCTURES IN THE AURORAL MAGNETOSPHERE A.V. Volosevich’, Yu.1 Galperid, ’ Mogilev

State University,

’ Space Research

Mogilev,

Institute,

Moscow,

F.M. Truhachev’ Belarus

Republic

Russia

ABSTRACT

A 3D fully nonlinear

theory of ion acoustic

solitary

structures

is constructed

in three-component

plasma

con-

sisting of non-thermal electrons, a hot ion background and a cold ion beam. The theory is based on the derivation of a modified Korteweg-deVries-Zakharov-Kuznetzov (KdV-ZK) equation. The ion motion is treated in

the MHD approximation

with self-consistent

distributions

and dynamics of all plasma components.

Some nu-

merical examples are presented for plasma conditions pertinent to the subauroral magnetosphere

and outer

plasmasphere in the stage of refilling after a magnetic storm. 0 2002 Published by Elsevier Science Ltd on behalf of COSPAR.

INTRODUCTION Small-scale solitary structures begin to be one of the most interesting

new features observable by the recent

high resolution rocket and satellite measurements in the aurora1 magnetosphere. These structures were observed on VIKING, FREIA, FAST and POLAR [Mozer et al., 1980, Ergun et al., 1998a, Franz et al., 1998, 20001. They can also be seen on the high-resolution

data from INTERBALG2

[Lefeure et al., 19991.

These structures move mainly, or strictly, along the magnetic field. Some of them have field-aligned comparable to the ion-acoustic

velocity in respect to the accompanying

velocity

ion beam, then they may be identified

as ion acoustic structures [Berthomier et al., 19981. At the same time, both at these and at higher altitudes, solitary structures are seen which are moving with much faster velocities comparable to the accompanying electron beam drift speed. These are identified as electron acoustic solitary structures [Franz et al., 1998, 2000]. New measurements

of electric field components

and El/El, ratios at much higher altitudes by GEOTAIL and

POLAR was shown that solitary structures have 3D geometry. The aim of this study was to consider the formation of three-dimensional multi-component

plasmas

THEORETICAL

MODEL

The particle distribution

3D localized structures for small but finite amplitudes of the electric potential in

functions for the different populations

considered below were assumed as follows: 1) The hot non-thermal electrons with the distribution

of the charged particles in the plasma model

function in the form:

.f,(o)=c(1+~~~/u~-2~)2~xp(-(u~/v:-2a))/2), where

@ = e9 /

T, is the normalized electrostatic potential,

1677

(1) V,

=

kTe /me is the thermal velocity, k - the

1678

A. V. Volosevich et al

Boltzmamr constant, c = n, /(l + 3a)

2~ug ,r, - electron temperature,

me - electron mass. The normalized

electron density is given by n,

= n,,(l-

PO, + [email protected]‘).exp(

Q),

(2)

where p = 4a I 1 + 3a , nor is the undisturbed electron density, and a is an arbitrary parameter which defines the shape of the distribution function and must be chosen from experiment. This form of the distribution function is convenient for description of various observed particle distributions. For a = 0 it reduces to a shifted maxwellian, and for cc-+ 1 it tends to look as two counter-streaming beams with a cold core distribution. 2) The hot (background) positive ion population

with the density n,”and temperature

T” is in thermal equilib-

rium. These ions obey the Boltzmann relation n,h = n,h,exp(-ey,/Th)=lZOhjexp(-~y),y

= TeIT,h

(3)

3) Cold ions may be described by the MHD system of equations with temperature

T’

((T = T” / TJ and bulk

velocity V0along the magnetic field (cold ion beam). Evidently, the Poisson equation guaranties that no violent deviations from quasi-neutrality

will arise, however the electric fields are strong enough and self-consistent

with

the particle distributions and motions. This self-consistency is essential because it selects the viable quasi-stable waveforms and velocities of the moving electrostatic structures. KdV-ZK Equation A reductive perturbation method is used to derive an evolutionary 3D equation of Korteveg - de Vries - Zakharov - Kuznetzov (KdV-ZK) for this problem [Infeld 1985, Zakharov and Kuznetzov 1974, Volosevich and Galperin 20001, Then for the first order contribution to the electrostatic potential we obtain the following equation :

am+a,@ a

a A,@=0

az

dz

where a; = [c;'(3y2 -50,/9)-S2d,]/2y&1; 6=n~,ln,,;u; = 6/2P;c,2;

g =v-v,;

p1 =/&y(l-6);

c,=l-p,,d,

(4)

a? =1-y2(1-&;p;2

hr= q4c,' 16 ,fit, = wc,iw,, w,,- ion Larmor frequency

ai =a; +q3/2Qf;

(5)

=50,/3+6/c,;

and w,, -the ion plasma frequency.

Eq. (4) describes the formation of a 3D electrostatic structure in a three-component

plasma in dependence of the

particular plasma parameters. This equation can be analyzed by traditional methods. Some conclusions

of impor-

tance for plasma diagnostics in situ can, however, already be drawn by analyzing the coefficients of the Eq.(5).

Size and Oblateness Ratio R of Ion Acoustic Holes._Consider now the properties of the 3D solitary waves propagating along the direction of the magnetic field. In the simple 1D case the solution is the classical soliton It is evident from here that @ =Qoch-*(z-MA) with the amplitude @,=3M/a, and width A=2 a2 /M the amplitude of the soliton is positive for a, > 0 and negative for ai < 0. For the 3D stationary waves we obtain from Eq.(4) a reduced equation:

(6) where

q,=z

normalized

Mla,;r,=x to the ion acoustic

Mla,;r,=y

Mla,,

.;1 = a, 12M;G

= (Da, l2M , M is the structure

velocity

speed (the Mach number).

Thus the interplay of the plasma parameters VO,fi x 6, 0 defines the range of existence of a solution as well as the character of the solitary structure. Without loss of generality it may be assumed that M > 0. The Eq. (6) has a

Electrostatic

large number of solutions. 2 when r i=

ones

Structures in the Aurora1 Magnetosphere

Simple forms will be spherically rx2+rv2.

From

(C23!a2j1’2 =(l+bjujfjlu~j)1’2 =(l+bi&l&.)li2; the simplest

For

length(&

symmetric

Eq.(6)

2 ), the longitudinal >> yLi

solutions and cylindrically

we

estimate:

(LII,LI -field-aligned

case bi =:1 , if the ion Larmor radius rL,

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symmetric

R=LJL,,

and perpendicular

is small in comparison

=

scales ).

with the ion Debye

and transverse sizes of the soliton are nearly equal, so the soliton is spheri-

,$ <<,2., then rL >> q , i.e. the solitary structure is flattened, or oblate (R > l), being DL 0 mostly perpendicular to the magnetic field. Another predicted property of the ion-acoustic structures is the possibility to have both positive and negative pocal. If conversely,

tentials. As was described above, the sign of the coefficient al fixes the sign of the potential. As in the 1D case, if 2 > 0, the soliton is ‘compressive’, and if jl< 0, the soliton is ‘rarefactive’. Some dependences

of the oblatness

c ::p_:I, ,

and

scale along magnetic fields of the solitary structures on the plasma parameters were shown on Figure1

cl

Ri

37

24

r

,,’

11

6

1

OO

0 33

0 67

1

On

6

plasma parameters

cr =3, l3 = 0.5, fl, = 3; 1 - y = 0.01, 2- y ~0.2,

3 33

6 61

IO

‘i

Fig. 1. a) The oblateness ratio R, and b) scale along magnetic fields L, 1 for ion-acoustic

Q, = wc, /u,,

7

structures as 6 for the

3 - y = 0.4; c) EIiE=l/R

as function of

for plasma parameters y=O. 1, p = 0.4, 6= 0.5; 1 - 0 = 0.1, 2 - 0 = 1, 3-a = 3.

The soliton dimensions

also can be found from Eq.(5) for particular plasma conditions.

As is clear from above,

they are of order of several Debye lengths and/or several ion Larmor radii. DISCUSSION In this study of 3D ion acoustic solitary structures a three-component plasma model is considered which consists of hot non-Maxwellian electrons described by a rather flexible distribution function, hot ions of Boltzmann-type distributions, and a cold ion beam of temperature T, and velocity Vo.Such a plasma environment corresponds, for example, to the early refilling stage in the subauroral magnetosphere. Indirect inferences from observations of solitary structures have indicated [Ergun et al., 1998a, 1998b] that at high aurora1 altitudes (say, along a typical FAST orbit) these structures have comparable field-aligned and perpendicular scales (L~I- LJ. However measurements of electric field components and EL/El1ratios at much higher altitudes by GEOTAIL and POLAR were consistent with a larger ratio R. Indeed, recently it was shown by statistical analysis of a large data base on high quality electron phase-space hole measurements from POLAR at different altitudes (and thus in a

1680

A. V. Volosevich er al.

wide range of derived values of R) that R scales mostly as the ratio of averaged electric field amplitudes

et al., 2000]. The resulting model curve for R-’ fitted very well the averaged data on

.(mi /m,)“* >>l and the two scalings differ by a large numerical

factor, while

the scaling with the electron density and magnetic field is the same. It needs to be noted that the measured quantity in [Franz et al., 20001. Evidently, we hope that ion-acoustic structures will be found in the subauroral plasma environments, or elsewhere, that fit the above plasma model. The development of this theory for different plasma models, in particular, involving an electron beam, will allow to make more sound predictions on the differences between ion- and electron-acoustic structures. CONCLUSIONS l.A three-dimensional fully nonlinear theory of ion acoustic solitary structures is constructed in a threecomponent plasma which consists of non-thermal electrons, hot ion background, and cold ion beam. The theory is based on the derivation of the KdV-ZK equation from the above distribution function for electrons, and the MHD description of the ion motion with self-consistent distribution and motion of all the plasma components. 2.In the frame of this theory solutions for ion-acoustic phase-space holes are obtained in a wide range of plasma parameters and forms of electron distribution functions. Characteristics of ion-acoustic solitary structures and nonlinear waves in the gyrotropic case include waveforms of density and electric field, temperatures, tieldaligned velocity, field-aligned and perpendicular scales Figure 1b and the oblateness ratio R =LI/L 1, Figure 1a. 3.The plasma model chosen corresponds to the plasma conditions in the subauroral magnetosphere (outer plasmasphere) in the stage of early refilling after a storm. Here polar wind ion outflows produce cold ion beams passing through a hot plasma background, while field-aligned currents carried by electrons can be either present or absent. By now, however, to ‘our knowledge, no such structures were reported in this magnetospheric region during refilling. ACKNOWLEDGMENTS The work was supported in part by NASA ,JURRISS grant NAG58638

and grant N-X013 18 by FBR of Belarus

Republic. The authors thank referees for their assistance in evaluating this paper. REFERENCES Berthomier M., R. Pottelette, M. Malingre, J.Geophys. Res., 103, N13,426 (1998). Ergun RE, C.W.Carlson, J.P. McFadden, F.S. Mozer, G.T. Delory, Geophys.Res.Lett., 25,2025 ( 1998a) Ergun,R.E. and et al., Phys.Rev.Lett., 81,826 (1998b). Franz, J.R., P.M. Kintner, and J.S. Pickett, Geophys. Res. Lett., 25, 1277 (1998). Franz, J.R., P.M. Kintner, C.E. Seyler, J.S. Pickett and J.D. Scudder, 27, 169 (2000). Infeld E.,J.Plasma Physics, 33(2), 171 (1985). Lefeuvre,F., M. Parrot, J.L. Rauch, B. Poirier, A. Massonetet al., Ann.Geophys., 16, 1117 (1998). Mozer,F.S., C.A. Cattell, M.K. Hudson, R.L. Lysak, M.A. Temerin, Space Sci.Rev., 27, 155 (1980). Volosevich, A.V., and Yu.1. Galperin, Cosmic Research, English version, 38, N5, 514 (2000). Zakharov,V.E. and E.A. Kuznetzov, Sov. Phys- JETP, 39,285 (1974).