Generation of electrostatic nonlinear wave structures

Phys. Chem. Earrh (C), Vol. 25, No. l-2, pp. 31-34,200O

0 1999 Elsevier Science Ltd

Pergamon

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Generation of Electrostatic Nonlinear Wave Structures T. Burinskaya, E. Indenbom and V. Pivovarov Space Research Institute, 117810 Profsoyuznaya 84132, Moscow, Russia Received 7 September 1998; revised 4 February 1999; accepted 13 April 1999

them (cold component), T,, is much smaller than the temperature of the other hot component, Th. Because of the typical values of the PSBL parameters, plasma may be treated as umnagnetized for studies of the electre static waves driven by ion beams in the PSBL ($&river and Ashour-Abdalla, 1987). The electron sound mode is a normal mode of any plasma with two electron camp+ nents with different temperatures. Within the approximation VT~ << w/k << VTh, where VT~ is a thermal velocity of cold (CX= c) and hot (a = h) electrons, the dispersion relation is given by:

The development of the electron sound instability driven by warm ion beams propogating along the magnetic field lines in the Earths magnetotail is examined in the weak nonlinear approximation. It is shown that the instability evolution can be described with the Korteveg-de Vris equation involving a weak wave growth and dumping due to resonance wave particle interactions. Analytical estimates and numerical calculations show the existence of quasi-stationary se lutions in the form of solitary and nonlinear periodic waves with amplitudes of the order of experimentally observed z 10v3 + 10w4V/m. 0 1999 Eisevier Science Ltd. All rights reserved.

Abstract.

1

w = cok - pk3, where:

Introduction.

It has been known for a long time that a broadband electrostatic noise (BEN) is often met in the plasma sheet boundary layer (PSBL) and the auroral region. Recent measurements made with fast waveform capture detectors show that this wave activity is not a noise in a common sense. In reality BEN emissions represent complex nonlinear structures of three basic types: soliton-like waveforms, double-soliton and complex nonlinear waves (Matsumoto et al., 1994). The goal of our work is to investigate the generation and the dynamics of electrostatic nonlinear structures induced by electron sound instability. Although our study is primarily related to PSBL, the main results are applicable also to the other regions of the Earth’s magnetosphere.

2

Analytical

(1)

(2)

CC?, =

bhd%/nh

Here c,, is the electron sound phase velocity without ion’s correction, TDa is a Debye radius of corresponding plasma component and rg is the compound ion-electron Debye length. For investigation of the electron sound stability we have used the Penrose criteria (Penrose, 1960) and direct numerical solutions of the full dispersion equation. Panels (a) of Fig. 1 and 2 show instability regions for different ion thermal velocities, VT~, versus the ion bulk velocity, Vi, and the wave frequency, w. It is clearly seen that there are two general types of growth rate behavior exhibited in Fig. l(b) and 2(b). Panels (b) show numerical solutions of the full dispersion equation for the parameters corresponding to dashed line sections

results.

We consider a one-dimensional, homogeneous, collisionless plasma composed of warm ions propagating along the magnetic field lines and two nondrifting electron components assuming that the temperature of one of Correspondence to: T.Burinskaya 31

32

T. Burinskaya et al.: Generation of Electrostatic Nonlinear Wave Structures

4

/

Kc,,

00

1.6-

-0.01/

0.0

0.2

0.4

k.1

Fig. 1. Panel (a) shows instabilty regions for d&rent V=i/cea on the plane of normalized ion bulk velocity Vi/&, and W/C+, for nc/m = 0.03, ~&r~h = 0.25. Panel (b) shows the growth rate dependence corresponding to a dashed line section for V~i/c~,, = 0.2. The dispersion coetticient /3 = 0.2.

drawn in panels (a). It is worth to note that a dispersion coefficient /3 is substantially different from a hydrodynamics coefficient given in (2) due to the kinetic effects. In Fig. l(b) the unstable region starts from zero wave numbers. Hereafter this type of growth rate dependence will be referred as a long wave unstable case while the growth rate dependence depicted in Fig. 2(b) we will call a long wave stable case. Two nonzero roots of growth rate (y = 0) are lettered Jcr and k2 and wavenumbers are normalized to a characteristic length X = (P/L+,~)“~. For numerical simulations we have fitted the growth rate dependence in the form:

Y =

a Ikl(kl - lkl) (k2- lkl>

0.1

0.0

‘&

(3)

where the coefficient a is determined by the growth rate maximum value. For the long wave stable case we assume k2 = -ICI = ko. Within the 1-D approach the nonlinear dynamics of small amplitude waves governed by dispersion relation similar to (1) may be described by the following equation in a coordinate system moving with an electron sound phase velocity cc (Karpman , 1973):

This equation is a Korteveg-de Vris (KdV) equation involving a weak wave growth and damping through the

0.2

0.3

0.4 k.3L 0.5

Fig. 2. Panel (a) shows instabilty regions for different V~i/c,, on the plane of normalized ion bulk velocity Vi/c,, and w/w,,, for n,/no = 0.1, T&TDh = 0.14. Panel (b) shows the growth rate dependence corresponding to a dashed line section for VT~/C,, = 0.1. The dispersion coefficient p = 0.069.

linear operator f:

(5) Here the following dimensionless variables are used:

77 =

E = 7

=

3r~ Sn 2Xn C 2 cot

(6)

7

twpc

In such a manner t and 7 are dimensionless time and space coordinates respectively and 77is a normalized cold electron density variation. It is worth to note that in the equation (4) the dispersive term is determined by the compound Debye length and the nonlinear term results from the hydrodynamics nonlinearity of cold electrons. It is well known that the original KdV equation with a zero right hand part has a class of solitary solutions:

‘I’ = cosh2 ((;“-

- 12/A2, c~) /A) ’ ” -

c = 4/A2,

(7)

where A is a soliton width, The modification of this solution by a weak wave growth and damping can be estimated from the energy balance condition that must be fulfilled in the steady state: r(k)q.2(k)dk=0 --oo

(8)

T. Burinskaya et al.: Generation of Electrostatic Nonlinear Wave Structures

33

!I!+-7 0

500

5

1500

1000

z=700 1

I

Fig. 4. The density profile snapshot for the pulse evolution in the long wave stable case when the steady soliton condition is not fuMilled (plasma parameters are the same as in Fig. 2(b)).

0.5

1? 0

3

Fig. 3. Time dynamics of pulse shaped initial perturbation for the long wave unstable case (plasma parameters are the same as in Fig. l(b)).

Substitution of the model growth rate from (3) and the original soliton form from (7) gives the relationship between a soliton width and characteristic wavenumbers. For the long wave unstable case we have a single steady soliton solution complying to: k2A2 w 2.5 0

(9)

For the long wave stable case the energy balance condition reduces to a quadratic equation: IPA -C (3) kiks - 4

+24- (4) (ICI+ kz) $

where C is the Riemann has solutions for:

k,2+k$2C.klk2,

C=

-51(5)

zeta function.

55 (5) 5 (3)

c2 (4

= 0, (10)

This equation

- 2 w 3.32

(11)

We will call this expression as a steady soliton condition. When it is fulfilled there are two solutions. The first solution with wave numbers falling around kl is unstable while the other with wave numbers falling around k2 is stable as follows from the energy balance analysis.

Numerical

results.

It is well known that solutions of the original KdV equation are strongly dependent on initial conditions. So we have solved numerically the investigated KdV equation (4) involving a weak wave growth and damping for two types of different initial conditions taken as a pulse or noise. The pulse dynamics for the long wave unstable case is shown in Fig. (3). Early in the evolution the initial pulse grows up to a stable amplitude while the formation of a dispersion tail occurs. The soliton moves right in the phase velocity direction while the dispersion tail expands left. As time proceeds a front edge of the expanding dispersion tail achieves the stable soliton amplitude and new soliton separation starts. The secondary solitons move right like the initial one. The pulse dynamics for the long wave stable case, when the steady soliton condition (11) is fulfilled, is quite similar to that shown in Fig. (3), while it is crucially different when the steady soliton condition violated. In this case the initial spike dumps leaving the dispersion tail. The dispersion tail grows up in amplitude and expands left. Finally the wave amplitude achieves a steady value while the expansion proceeds. The snapshot of the dispersion tail at some point following the amplitude saturation is shown in Fig. 4. The noise evolution for the long wave unstable and stable cases leads to distinctly different final states. For the unstable case the wave dynamics exhibits features of complex wave behavior shown in Fig. 5(the upper panel) by contours of constant cold electron density.. Although the system energy reaches a stable value, when a characteristic scale length has grown up to the saturation scale, the wave dynamics persists to be quasi chaotic. The bottom panel shows snapshot of the nonlinear wave structure in the energy saturation state. Fig. 6 shows the noise evolution snapshots for the long wave stable case. It is clearly seen that as time proceeds the dump ing of long scale perturbations leads to the regularization and yields a periodical nonlinear structure. The

34

T. Burinskaya et al.: Generation of Electrostatic Nonlinear Wave Structures

0.005

v

0

-0.005

II 0

50

100

0

50

100

’

r

I

150

200

150

200

250

0.005

77

0

-0.005 I

I

I

I

I,

250

6 0.44

r=lOOO

0.3 0.2 0.1

Fig. 5. The upper panel shows contours plots of constant cold electron density variations for noise evolution for the long wave unstable case (plasma parameters are the same as in Fig. 1). The bottom panel presents the density prolile snapshot at some point following the saturation.

110

-0.1 -0.2

, ., 0

., 50

.

.

, . . . 100

9

I..

150

I.

I..

200

8.

I

250

5

length of the unstable region governs the regularization time and the deviation of the steady solution from harmonic oscillations. The characteristic scale of the final structure lies in the unstable region.

3.1

Discussion,

The nonlinear dynamics described by the KdV equation involving a weak wave growth and damping is investigated. It is shown that the existence of moving nonlinear dissipative wave structures is possible. Parameters of these structures are strongly determined by the wave number dependence of the growth rate. The instability evolution is highly dependent on the initial conditions and may lead to solitary spikes as well as complex nonlinear periodic or chaotic waves. Waveforms obtained during numerical simulations are quite similar to those captured by GEOTAIL. Characterical length of the numerical solutions is of the order of several hot electron Debye radii and the electric field amplitude can be evaluated for the typical PSBL parameters as: E w (T&z)

/ (enhA) EJ 10m3 f 10v4V/m,

(12)

that is of the same order in magnitude as observed by GEOTAIL.

Fig. 6. Time dynamics of initial noise perturbation for the long waves stable case (plasma parameters are the same as in Fig. 2)

The KdV equation under consideration describes not only the electrostatic nonlinear wave structures driven by warm ion beams in the PSBL. It can be applied to the amoral region where an electron sound instability may be driven by electron beams. Moreover as this equation involves all basic features of waves with a sound-like dispersion relation it can be used to describe much wider class of wave phenomena. Acknowledgments. This research was supported by the grants of Russian Foundation for Basic Research (RFBR) 96-15-96723, 97-02-16489 and INTAS 96-2346. References Karpman V.I., Nonlinear waves in dispersed media, Nauka, Moscow (in Russian), 1973. Matsumoto, H. et al, Electrostatic Solitary Waves (ESW) in the Magnetotail: BEN Wave Forms Observed by GEOTAIL, Gwphys. Res. Lett., 21. p.2915, 1994. Penrose O., Electrostatic Instabilities in a Uniform Non-Maxwellian Plasma, Phys. Fluids, 3, p.258, 1960. Schriver D. and M. Ashour-Abdalla, Generation of High-F++ quency Broadband Electrostatic Noise: the Role of Cold Electrons, J. Gwphys. Res., 92, p.5807, 1987.