NONLINEAR INTERACTION OF WAVES IN A HOT INHOMO . GENEOUS MAGNETIZED PLASMA

This study is concerned with the theory of parametric coupling of waves in a hot inhomogeneous magnetized plasma in which the temperature gradient has been taken into account. The general dispersion relation and the polarization of the ordinary and the extra-ordinary wave modes are discussed. The eigen-mode solutions of the coupled differential equations for the wave amplitudes are obtained in the terms of the so called three wave interaction matrix elements. The theory of nonlinear wave-wave interactions, which has been extended to the case of an inhomogeneous magnetized plasma, is used to determine the threshold value of the electric field and the frequency shift. The results of this paper are also compared 326 NONLINEAR INTERACTION OF WAVES with the other known results. It is shown that the findings of this study are in excellent agreement with the results of earlier investigators.

with the other known results.It is shown that the findings of this study are in excellent agreement with the results of earlier investigators.KEY WORDS AND PHRASES.Parametric instaby, Waves in inhomogeneo plasma, wave-wave interactions, Threshold electric field and frequency shift.
In recent years there has been considerable interest in the theory of parametric instability in an inhomogeneous plasma [1-8] because of its fundamental role in the study of weak plasma turbulence, and of many important physical applications.In their recent research-expository and survey article on parametric henomena in a plasma, Galeev and Sagdeev [9] reviewed the theory of parametric instabilities in an inhomogeneous plasma using the generalized Mathleu equation as a model equation.They have also presented the latest advances of the nonlinear theories of parametric instabilities based on the ideas of weakly turbulent plasma.
Perkins and Flick [I0] have made an interesting study of parametric instabilities in an inhomogeneous plasmas, and then calculated the threshold electric field.It is shown that the value of the threshold electric field increases in an inhomogeneous plasma because energy propagates away from the unstable region by electron plasma waves.Thus an additional energy loss occurs and is solely responsible for the increase of the threshold electric field not observed in a homogeneous plasma.
Eubank [ii] experimentally confirmed the theoretical prediction of Perkins and Flick.In a recent paper, Kroll, Ron and Rostoker [12] have suggested that the nonlinear resonance of two transverse lectromagnetic waves whose frequencies differ slightly by the electron plasma frequency can be applied to excite longitudinal electron plasma oscillations.Montgomery [13] observed certain mathematical inaccuracy and physical limitations of the work of Kroll, Ron and Rostoker, and then analyzed the problem of nonlinear wave interactions in plasma with laser beams.
Using the perturbation method of Krylov-Bogliubov-Mitropolskii, Montgomery obtained the amplitude-dependent frequency shift and wave number shift with physical significance.It was shown that the resonant excitation of longitudinal plasma oscillations is possible by the transverse electromagnetic waves.Etievant, Ossakow, Ozizmir and Su [14] have investigated the nonlinear wave-wave interactions of electromagnetic waves in an infinite homogeneous plasma.It is interesting and important to take into account the effects of density and temperature gradients on the above problem.
The present study deals with the theory of parametric coupling of waves in a hot inhomogeneous magnetized plasma in which the temperature gradient has been taken into consideration.The general dispersion relation and the polarization of two different wave modes are investigated.The eigen-mode solutions of the coupled equations for the wave amplitudes are obtained in terms of the three wave inter- action matrix elements.The theory of nonlinear wave-wave interactions is used to determine the threshold electric field and the frequency shift.The results of this analysis are found to be in excellent agreement with those of earlier workers.

BASIC EQUATIONS FOR TWO PLASMA MODEL.
In two plasma model, the equations of motion and the continuity equation for each kind of component are in the usual notations given by Dv_n Ze V (-Tap) eB where is the cyclotron frequency, v is the collision frequency of each cm a component and B B z is the external magnetic field The subscript u stands for O e or i corresponding to electrons and ions respectively, Z e is the charge of an electron or ion and is the Boltzmann Constant.(2.7) 3. THE FIRST ORDER EQUATIONS.
In the first order approximation, equations (2.1) (2.2) can be written as, .VTg) .VTg) represents the first order terms in (--where ('---i Assuming the density gradient is small, we take the space and time dependent of the first order quantities as exp[i(k r-tot).The equation (3.1) can then readily be solved to obtain and is the ratio of the two specific heats.
In the second order approximation, we obtain from equations (2.1) Using the space and time dependence of all second order quantities in the form exp[i(_k r mt)] we find Z_e Using the Maxwell equations and eliminating the second order magnetic field B2, it follows that Substituting the values of v 2 in equation (4.4), we obtain We assume a plane wave solution with an amplitude varying in the direction of propagation and write _E 2 a A (x,t) exp[i(k r mt)] (5.1) where A is a slowly varying function of x and t because the nonlinearity is assumed to be weak.
We next drop the second derivatives of A in comparison to k2A and m2A, and then use the linear dispersion relation D E 2 0 to obtain the final result (u2 x +)A2 AI A3 V213 (-m21-mllm3) exp{i(-x A k + t A m)}, (u 3 -+ )A 3 A I A 2 V321 (-m31m21-ml) exp{i(-x A k + t A m)} (5.3) (5.4) where Ak k 3 (k_2 + kl) and Am m3-(ml + m2) (5.5ab) We write the matrix elements for a I a 3 a and a are defined in (3.15ab).We consider , k, Ak real and A 3 is the pump of fixed amplitude.The small amplitudes are described by equations (5.2) (5.4).We take A's to be  Neglecting i A m -it follows that the solution of (59) by W.K B method t assumes the form I[A I exp{i I g(t)dt} + B 2 exp{-i g(t)dt}] o o (5.11)   with i g IV123 V213 A32 (A 24 i (5.12) When the wave packet drifts along x the time-increment dt can be written as dt (dx) ) to obtain xt AI [exp g(x) dx] ) (5.13) o where the integration is limited to the instability zone.It is interesting to note that the threshold value of the wave can be calculated when g < 0 Am (5.14) A3=

2/V123V213
Thus the value of the frequency shift can be obtained from (5.Similarly, considering the space dependence only, one can calculate the wave number shift from equations (5.2) (5.4).
The general features of density gradient and magnetic field in the nonlinear interactions of plasma oscillations have been investigated.This is important in connection with its use as an optical density probe suggested by Kroll, Ron and

(
dependence of Pao UO due to any external electric field.The zeroeth order solution of (2.1) gives T < Vo (t)e e e when I is the unit dyadic and _c is c

9 )( 3
In the case of wave propagation along with x-direction (k k), we obtain the dispersion relation Det D(k, m) o, There are two independent solutions of equation (3.10) in the form the extra-ordinary wave, where Dxx, Dyy, are the elements of the matrix D.The first order fields for the different modes (ordinary and extra-ordinary) FOR THE INTERACTING MODES.
is in good agreement with that ofMontgomery 13]