Nonlinear modulation of randomly distributed upper-hybrid modes in plasmas

Volume 57A, number 4

PHYSICS LETFERS

28 June 1976

NONLINEAR MODULATION OF RANDOMLY DISTRIBUTED UPPER-HYBRID MODES IN PLASMAS K.H. SPATSCHEK Fachbereich Physik, Universitelt Essen, 4300 Essen, F.R. Germany Received 28 April 1976 A new type of modulations of an ensemble of random waves near the upper-hybrid resonance frequency is investigated. For the linearly unstable cases, the growth rates are obtained and exact nonlinear stationary distributions are presented.

It is well-known [1] that a system of random phased high-frequency waves can be unstable with respect to lowfrequency perturbations. Recently [2], the linear dynamics of an ensemble of upper-hybrid waves has been studied using lower-hybrid as well as ion cyclotron modes as low-frequency perturbations. As expected, the growth rates are larger for smaller modulation frequencies. Obviously, the fastest modulation process will dominate in the system. It is therefore worthwhile to consider perturbations with even lower real frequencies. Therefore, in this letter we consider adiabatic modulations. Besides the corresponding linear growth rates the possible final nonlinear stationary distributions of the upper-hybrid turbulence are presented. For the reason of simplicity, we use a one-dimensional model for the upper-hybrid modes propagating perpendicular (x-direction) to the external magnetic field B0 = B0i The dynamics of upper-hybrid turbulence is described 2>/4~wH,i.e., by equation for the plasmon distribution Nk = (IEkI ONkthe wave-kinetic öNk W~e a~ne aNk —-——=0. (1) at ox 2~~o~’~H ax Ok Here, ~~pe is the electron plasma frequency, ~ = (~‘~e + c)Be)112 is the u~per-hybridresonance frequency, WBe is the electron gyrofrequency, a = veAw2pe/w~,with A = 3W~jI(W~e~‘~e)’ and Lie is the thermal speed of the electrons. The turbulence is modulated through perturbations 6ne in the electron density (ne = n 0 + ~e) which are assumed to vary on a large scale compared to the phase variation of a upper-hybrid mode. We that<2WBe. a >0 2~~Be corresponding to positive group dispersion; negative group dispersion is obtained note for WH for The “~~H > density changes ~ are caused by the ponderomotive force. The latter acts on the electrons and ions and can be written in the form [3]

0 [I’~’p/”~H__~ 2 (1E12) 1

F.—x— ii ~‘ ax ~

-

a r

~i

i—z—---i

_~4J l6irn

0j

Oz

[~,2

(1E12> 1

I,

_,4167rn0

je,i.

(2)

J

2> is the mean square turbulent electric field determined by the strength of the turbulence. It is related Here, to Nk (1E1 through the conservation of norm, i.e.,

~I2

=

2wHLfNkdk,

(3)

where L is the length of the system. Taking into account the adiabatic response of the plasma, we obtain from (2)

ep e

o Te

—

~

i~4

4EI~) ‘~e~

l6ir

‘

—n ep 1

0

7

‘~i

(1E12)

(4ab)

T~w~ 16iT~ 333

Volume 57A, number 4

PHYSICS LETTERS

28 June 1976

Assuming quasineutrality and using the cold dispersion relation for upper-hybrid waves, we can eliminate the

ambipolar potential p to obtain 2

c~fl n 0

(F

l6irn0(7+7;)

Inserting (5) into (1) and using (3) we get aNk

aNk

aNk

-~-=0,

—+ak-+~(~fNkdk)

(6)

with j3 = L w~,eI167Tflo( 7~÷7). Eq. (6) governs the dynamics of upper-hybrid turbulence in a nonlinear dispersive medium. Nonlinear equations of that type have been analysed by many authors, [e.g. 4.] We briefly summarize the main results. A linear analysis, using .

Nk

=

Nf~+ Nk exp(iKx

(7)

i~t),

—

yields the dispersion relation 1

r

(8) 2~ 3W~)/3v2W2K dk. 48irn0(7~+ 7;)v~ k &2wn(w For positive dispersive upper-hybrid modes unstable long-wavelength perturbations exist. The corresponding growth rate y is +

~

=

\/iKV~W2

—

—

[(IEI2)/32irn

(9)

1’2/wn(w~~3w2B)h12.

0(7~ + Ti)]

Negative dispersive upper-hybrid modes can be unstable with respect to short-wavelength perturbations. The corresponding growth rate is 3Vu~w~K~

—

-

96irv~2n

~

(w~~—3w~~)(IEI

0(7~ + 7;)

i

(10)

2)

H(~~

~

where a Gaussian of width ~ has been assumed for the zeroth order plasmon distribution. Possible nonlinear final states of these instabilities can be represented by trapped BGK solutions. One obtains N k = 1611o(lTi)[3tc(FL12) LW2 Ll6~n(T + 7~jw~(w2 3C4e) for positive dispersion and k2X~<(1

~‘~e1°~e)

9v4w4k21 i(Le 3~4e)2J 1/2

(11)

(IEI2)/48~n

0(7~ + 7;). The space dependence of the electric field energy density is still arbitrary and can be chosen to be of the form of an evelope soliton. For negative dispersion one obtains k

N

2 [(IEI~)OO ~_W~e)’/2L L2HWH e°~pe’~

\/~1rW~j2(3W~ =

—

32irn~(7~ + 0)

(12)

with

H = w~e((IEI2) (jEI2>

2W2ek2/2WH(3W~e—

—

0~)I32irn0(7 + ~ 3V Now, (lEt2) can be chosen to have a space dependence like an envelope hole. In conclusion, we have investigated modulations of upper-hybrid turbulence treating the plasma response as adiabatic. It turns out that the growth rates of the corresponding instabilities can be larger than those obtained —

334

Volume 57A, number 4

PHYSICS LETTERS

28 June 1976

previously. Stationary nonlinear solutions in the linear unstable region are trapped BGK solutions. This work has been done within the activities of the Sonderforschungsbereich 162 “Plasmaphysik Bochum/ JUlich”.

References [1] A.A. Vedenov, A.V. Gordeev and L.I. Rudakov, Plasma Phys. 9 (1967) 719. [2] P.K. Shukla and M.Y. Yu, Phys. Lett. A56 (1976) in press. [3] G.J. Morales and Y.C. Lee, Phys. Rev. Lett. 35 (1975) 930. [4] A. Hasegawa, Phys. Fluids 18 (1975) 77.