parametric excitation of hybrid modes in multispecies plasmas

Contrib. Plasma Phys. 32 (1992) 1, 3745

Parametric Excitation of Hybrid Modes in Multispecies Plasmas

MEENU ASTHANA' and S. GUHA

Department of Physia. Ravirhankm University, Raipur - 492010 (M.P.) India

Abstract

Nonlinear decay of an electromagnetic wave into lower-hybrid and upper-hybrid wave in a plasma containing two types of ions and two temperature electrons has been analytically investigated. Hydrody- namical model of the plasma is used. Nonlinear dispersion relation and growth rates are calculated for parametric decay, modulational and filamentation instabilities. As an application of the investigation growth rates are calculated for typical parameters of both laboratory and space plasmas. Effect of addition of second species of electrons and ions is discussed.

1. Introduction

Multispecies plasmas consisting two species of ions and two electron components exist in the earth's magnetosphere (WILLIAMS [l], GEIS et al. [2]), ionosphere (MENDILLO et al. [3]), solar wind (FIEDMAN et al. [4]) and in laboratory plasmas (GUEST et al. [5]). Heating of plasma using an ordinary electromagnetic wave in a system containing single species of ion and one electron component has been studied by several workers in the past. HASEGAWA and CHEN [6] gave a theory of plasma heating by nonlinear excitation of lower-hybrid resonance. They reported the evidence of effcient ion heating. The linear theory of parametric instabilities due to lower-hybrid heating of tokamak plasmas has been studied by PORKOLAB [A. SATYA et al. [8] investigated anomalous plasma heating at lower-hybrid frequency which again resulted in effcient heating of ions in the plasma. Lower-hybrid drift dissipative instabilities in a multi- ion species plasma has been studied by BOSE et al. [9] under the restrictive condition IT;. B T,.

The availability of high power microwave sources has attracted a great deal of attention of electron heating in tokamak devices (PORKOLAB et al. [lo]). Plasma heating from upper-hybrid mode conversion in an inhomogeneous magnetic field was studied by LIN et al. [ll]. The mode converted at the upper-hybrid layer can attain a very large amplitude limited by dissipation and thermal convection. The electrostatic oscillation near the upper-hybrid mode conversion layer is capable of giving rise to a variety of nonlinear effects. Further, it can explain a number of phenomena in laser produced plasmas near the critical layer where a self generated magnetic field exists, in ionospheric plasmas stimulated by intense radio waves, as well as in laboratory plasmas in which an electron beam generates electrostatic waves. Stimulated scattering of an electromagnetic wave by an upper-hybrid wave in a two-electron temperature plasma has been studied by GUHA et al. [12].

High power electron cyclotron heating (ECH) experiments in mirror machines are expected to start in near future. It is proposed to employ high power densities. Hence in the light of

') Present address: School of Physics Devi Ahilya University, '

Khandwa Road, Indore - 452001, India.

38 ASTHANA, M., GuHA,~.

above review we have investigated theoretically the possibility of excitation of parametric instabilities namely parametric excitation of hybrid modes in a plasma consisting of two types of ions and two components of electrons which has not been done so far. An ordinary wave is made to decay into two electrostatic waves lower-hybrid and upper-hybrid side band modes. The coupling is found to be very strong both the decay modes being electrostatic and also one of them having short wave length and the other long wave length. The short wave length electrostatic fields generated at the upper-hybrid resonance layer due to mode conversion process reduces the convection losses and lead to parametric instabilities which are difficult to realize because of their high thresholds. The parametric process is observed to produce a sub- stantial number of energetic ions localized at the resonance layer which may give rise to ion heating. The heating is not expected to produce any deteriorating effect on plasma continent. The decay process converts the incident wave to lower frequency, allowing it to penetrate deeper into the plasma.

The paper is organized as follows: Using basic equations of the hydrodynamical model, nonlinear wave equations for the hybrid modes consisting of two ion species and two electron components are formulated in Section 2. The second species of ions is more favourable for the occurrence of the lower-hybrid wave whereas the second electron component is found to be more favourable for the upper-hybrid wave. Nonlinear dispersion relation and stability analysis are made in Section 3. The nonlinearity is introduced through a ponderomotive force on electrons which is the cause of coupling of hybrid modes. Section 4 deals with the discussions of the investigation with numerical calculations for typical parameters of laser plasma interaction in mirror machines and space plasmas. Growth rates have been calculated. The main conclusions are stated in Section 5.

2. Mathematical Formalism

We assume that the external magnetic field B, is in the z-direction and the high-frequency ordinary wave (w,, k,) propagates in the x-direction. Thus w i = (kit' + wf) where c is the velocity of light and w p is the electron plasma frequency. The lower hybrid wave (0, k) is taken to be propagating nearly along the x-direction but with a small wave vector k, 4 k, . (rne/rnJ1". Thus o2 x 0:". (1 + k:/k' . m,/m,) where w i H = wfJ(1 + wf,/wf,). The upper-hybrid wave (o,, k,) is assumed to be propagating nearly along y-direction but with a small wave vector k , , 4 k,,, and coto= 0:" + k:,C,Z where 0:" = wf + wf,, w,, is the electron cyclotron frequency and C, electron thermal velocity.

The set of self consistent basic equations for the hybrid modes are

e 3 :c _ - (va x B,) + ( v S . V ) v s - - (vS x at m

an, at - + V . (nave) = 0

Contrib. Plasma Phys. 32 (1992) 1 39

an, at - + V . (npvB) = 0 (4)

V - E = - 4 x e ( n c + nh - nL - nH) (5 ) The subscripts 'a' = L, H represents two types f ions light (4 and heavy (H) and 'B = c, h

represents cold (c) and hot (h) electrons. C; and C; ire the thermal velocity of the corresponding species, respectively. vi and v, represents the average collision frequency of ions and electrons, respectively.

In terms of pondermotive force, the equations of motion (1-2) can be written as ava 1 e C,2Vna

[eE + FPO] + - (va x B,) - viva - -

1 e CiVn,

m mc 9 0

- at M, Ma "a0

at av, - = - - [eE+Fp,] -- ( V , x B,) - v,v, - - (7)

where I; and Fp, represent the ponderomotive force on ions and electrons, respectively. In terms orpotential Fpe,i = -e V@p,,i here GpCsi = eEo. E,/4me,,o,w, is the ponderomotive potential.

2.1. Lower-Hybrid Wave

A LHW is essentially an electrostatic wave near w z wpi propagating in the direction almost perpendicular to the magnetic field. The frequency region lies between the electron and ion gyrofrequencies (wee % w 9 qi). Because the parallel phase velocity is much larger than the electron thermal speed and the perpendicular phase velocity is much larger than the ion thermal speed, one may use the cold plasma approximations and w > k,C, % kC,.

We consider the propagation of LHW (0, k) in the x-z plane with the external magnetic field along the z-direction. Applying perturbation technique and considering the dependence of all the variable quantities as exp i(wt - kx$ - kz i ) , we get the values of perturbed densities from eqs. (1-4) as

where a:,, = 4A nu,# e2/ma., is the plasma frequency of the corresponding species,

and

From eqs. (5 ) and (8) one gets the nonlinear wave eq. for lower-hybrid wave in multi-species plasmas under the cold plasma approximation (i.e., = 0) as


We have considered two types of ions and one type of electrons but neglected the ponderomotive force on ions being small.

EL,@ = PLH Eo. @? (9)

where EL, = (1 - X , - X, - X , ) is the plasma dielectric function for the lower-hybrid wave. [ kf (o + iv,) X , = y +

k o { ( w + iv,) - o;,} w(w + ivi)

k3 1 k i (o + iv,) + o{(o + ivi)2 - ofH} o (o + iv,)

X , = -

are the dielectric susceptibilities of light ions, heavy ions and average single electron species e respectively.

pLH = - i e k / 4 m o o o X,, is the coupling constant of lower-hybrid wave. If there is no pump wave, eq. (9) reduces to the linear dispersion relation of LHW as

obtained by HASEGAWA [6] for a collisional plasma in the frequency regime oCi G o 6 ace. For a collisionless plasma it reduces to the linear dispersion relation as obtained by O n (131 given by

2.2. Upper-Hybrid Wave

It is a high-frequency mode propagating at a frequency o2 z w;, + kyCz where o;, = oi + ofe, o,, is the electron cyclotron frequency. We assume an upper-hybrid wave (o,, k,) to be propagating nearly along the y-direction but with a small component .k,, 4 k,,. The role of ions and collisions is negl5cted instead the role of second electron component is investigated being a high frequency mode.

Applying perturbation technique and considering the dependence of all the variable quantities as exp i (o , t - k , , j - klLi), we get the values of perturbed electron densities as

where

Using (10) with Poisson’s equation (9, one gets the nonlinear wave equation for an upper- hybrid wave as


or & U I l @ l = /-%IHEO@* (1 1)

where .sun = (1 - X, - X,,) is the plasma dielectric function of an upper-hybrid wave. X, = o&/A,k: (k:,/A, + ktZ/o:), X, = o;JA,k: (k:,/A, + k:,/o:) are the dielectric susceptibilities of cold and hot electrons and pvn = - iek/4mooo X , is the coupling constant for the upper-hybrid wave. If in equation (11) the nonlinear terms are put to zero and there is only one type of electrons, it reduces to the linear dispersion relation of the upper-hybrid wave as obtained by GUHA et al. [12].

3. Stability Analysis

We now carry out the stability analysis of the hybrid mode described by coupled nonlinear wave equations (9) and (1 1). We decompose the total electric field into three components, viz., the pump and two side band modes given as

E = Eoexp(ioot - k, . r) + E,exp(iw,t - k , * r) + E,exp(io,t - k, . r ) .

This demands that the frequency and phase matching conditions o,,’ = o k oo and k1,2 = k k k, are satisfied.

3.1. Three-Wave Interact ion

Here one of the side band modes is assumed to be off-resonant. It is a decay instability wherein an electromagnetic pump wave decays into a LHW and high frequency UHW. Combining (9) with the complex conjugate of (1 I), we get the nonlinear dispersion relation as

ELHsH = PLHPUHIEOI’. (12)

We take o = o, + i yD and E , x 0, .$ z 0 for decay modes at resonant interaction. Following SHARMA and TRIPATHI (141 the growth rate for the three-wave decay process is given by

The growth rate in the frequency regime oci c o < w,, under the conditions C’, c 02/ k2 c Ci, k, + k, and plasma is collisionless, comes out as

3.2 . Four-Wave Interact ion

In the case of three-wave parametric instability one of the upper-sideband has been taken to be off resonant. However, there are phenomena, viz., filamentation and modulational instabilities of the pump wave, where both the side bands become important and must be taken into consideration.

42 ASTHANA, M., G w , S .

When both the side bands are resonant say at frequencies ( o ~ , ~ , kIs2), using eqs. (9) and (11) we get the coupled non-linear dispersion relation for a four-wave interaction process as

where E, , E~ are the plasma dielectric functions of upper-hybrid side band modes given by gLIH at frequencies (a,, k,) and (o,, k,) respectively.

is the coupling constant of the upper-hybrid side band with the lower-hybrid wave. The growth rate for the four-wave interaction process can be obtained from

In the following we study the two special cases, viz., modulation and filamentation instabilities of the incident ordinary electromagnetic pump wave in the presence of the megagauss magnetic fields, in laser produced plasma for typical parameters of mirror machines where a multispecies plasma exist and also in the magnetosphere ionosphere coupling of space plasma.

(i) Filamentation instability

When the low-frequency density perturbations of the lower hybrid wave propagate trans- verse to the direction of the pump wave, the nonlinear growth of the density perturbations can break up the initially uniform pump wave into filaments, thereby leading to what is called filamentation instability. This demands k , = k,, = k Z y = 0.

Using (16), after a little simplification, the growth rate of the filamentation instability of the incident electromagnetic wave is

(ii) Modulational instability

To study this we consider that the low-frequency electrostatic perturbation due to lower- hybrid wave to propagate along the direction of the propagation of the pump wave which requires k,, = kzY = k , = 0. Under this condition using (16), we get


4. Discussions

In the present investigation we have considered the parametric interaction of a high frequency electromagnetic wave and hybrid modes in a multispecies magnetoplasma. The parametric instabilities are relevant to laser plasma interaction process in mirror machines and in understanding the wave phenomena in magnetosphere, ionosphere and solar wind plasmas. For the sake of appreciation of the results, numerical calculations have been made.

For the typical set of the parameters of the laser plasma interaction in a mirror machine (SHARMA et al. [15]).

n, -1014~m-3, B0-20 KG, k-104cm-'

- mL - 3600, 5 - 5400 me me

n,- 10" n h - 10"

q-T,-10-20KeV, T,-350-400 KeV

oo !z 1.778 x 1014 rad sec-'(CO, laser), o, = 2.46 x lo4 rad sec-'

and ol, = 1.66 x lo7 rad sec-'.

The growth rates for the decay instabilities are found to be less for the multispecies plasmas. The values are yo - los s- l, vMr - lo4 s-', and vFr - 10' s-l , which are less in comparison to parametric instabilities in a two-electron temperature plasma. Thus the addition of any species is found to decrease the growth rates by one order. The growth rates are found to have sensitive dependence on concentration ratio of hot to cold electrons, masses of the ions, electron collision frequency, pump wave amplitude and magnetic field. Growth rates for different concentration and masses ratios respectively are found to reduce the instabilities. The coupling constant is found to have a sharp variation with electron species which is relevant with the conditions in space plasma discussed below.

Our results can be quite useful in understanding the wave phenomena in the naturally occurring space plasmas namely in the ionosphere, earth's magnetosphere and beyond in the solar wind wherein multispecies plasmas exist widely. These parametric instabilities may lead to strong coupling of magnetosphere-ionosphere, magnetosphere-solar wind and ionosphere-solar wind plasmas. We have calculated the growth rates for the typical parameters of the ionosphere and magnetosphere coupling. MENDILLO and FORBES [3] injected experimentally the highly electromagnetic SF6 in the

F-region of the ionosphere and observed the presence of 0' and SF; ions. The number densities of electrons and ions are considered as

n,,(O+) = 106 c m - 3 n,,(SF;) = lo4 noc N 9.9 x lo5


The other parameters are

v, = 8.21 x 10' sec-' , vi = 4.34 x lo-' sec-', cogL = 1.04 x lo5 sec-', a:,, = 2.4 x lo4 sec-', B = 3 Gauss.

Our system consists of multi-ions and multi-electrons, thereby in view of KARPMAN et al. [16, 171, we take multi-ions from the ionosphere and hot electrons from the magnetosphere with a 'loss-cone' velocity space distribution provoding the free energy for the instability.

For multi electrons we take typical parameters of the magnetosphere (SHARMA et al. ~ 1 ) .

ncK= 1 0 ~ c m - ~ , no,,= 10 cmd3, ~ = 5 . 4 x 10-3 G .

The growth rates for the decay instabilities are

yD = 0.12 x lo6 s - ' , y M r N 2.96 x 10' s - ' , YF,N-0.12 x 103 s - 1 .

5. Conclusions

In conclusion we have made a comprehensive study of the parametric instabilities in multispecies plasmas and it is found that it leads to efficient plasma heating in mirror machines. The multi-species system (containing two types of ions and two-electron components) in itself looks very practical having its relevance both in space and laboratory plasmas. The two hybrid modes namely lower-hybrid and upper-hybrid excited together gets the advantage of two ions and two electrons respectively. The ion motion is found to be important for LHW whereas electron motion is important for UHW. Thus a system consisting of two ions and two electrons strongly couple the two modes. Thus it may lead to the description of strong coupling of magnetosphere-ionosphere, ionosphere-solar wind, solar wind-magnetosphere. It could describe the wave phenomena'in space plasma and explain magnetospheric electrostatic emissions. The addition of second species of ion and electron suppresses the instabilities. The parametric instabilities discussed here change the electromagnetic energy into plasma energy through the generation of plasma waves which leads to hereby heating of the plasmas. In order to achieve non-negligible fusion reaction rates in a thermonuclear plasma, the temperature should be at least 10 KeV. In a tokamak, the inherent ohmic heating is well capable of creating a 1 KeV plasma but the decreasing resistivity makes it difficult to reach much higher temperatures. For the final temperature sboost to 10 KeV, supplementary heating is required to be employed. The wave heating is one of the different supplementary heating methods. The advent of high power laser has opened a new and promising method of wave plasma heating. In this case it is accomplished by the mode conversion process wherein the electromagnetic pump wave interacts with the plasma and couples with the plasma modes. Thus the pump energy is converted into plasma mode energy and consequently, it is imparted to the plasma particles in the form of kinetic energy of the particles. The case considered by us heats the ions directly through lower-hybrid wave. The high frequency upper hybrid wave indirectly heats the ions. Most of the energy transferred from the pump is initially confined to the electrons in the tail. These electrons transfer their energy to the bulk electrons by electron- electron collisions. The electrons ultimately transfer their energy to ions by electron-ion collisions. In this way both electrons and ions can be heated.

Contrib. Plasma Phys. 32 (1992) I 45

Acknowledgements

One of the authors (MEENU ASTHANA) acknowledge thankfully the financial support from C.S.I.R. India for award of Senior Research Fellowship.

References

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Received k m k r 11. 1990; revised manuscript received k m k r 5,199 I

parametric excitation of hybrid modes in multispecies plasmas

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