PARAMETRIC DECAY OF LOWER HYBRID WAVES

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PARAMETRIC DECAY OF LOWER HYBRID WAVES

S. Pesic

To cite this version:

S. Pesic. PARAMETRIC DECAY OF LOWER HYBRID WAVES. Journal de Physique Colloques,

1979, 40 (C7), pp.C7-553-C7-554. �10.1051/jphyscol:19797267�. �jpa-00219253�

JOURNAL DE PHYSIQUE CoZZoque C7, supptgment au n07, Tome 40, JuiZZet 2979, page C7- 553

PARAMETRIC DECAY Of LOWER HYBRID WAVES

S.S.

Pesic.

Ddpartement de Physique du Plasma e t Za Fusion 85 X

-

38041 GrenobZe Cedex, France.

I

-

INTRODUCTION.

The parametric instabilities induced by pump field with a driving frequency in the range of lower hybrid (LH) and electron plasma (EP) frequen- cies (aCi < u0 < min(wpe,

wee))

have received a great deal of attention (the present theoretical and ex- perimental state of the art has been recently revie- wed in 111). The crucial problem of LH plasma hea- ting is the penetration of the launched waves to the plasma core without significant pump depletion in the outer, low-density plasma layer, which can lead to a reduction of the heating efficiency. In the present paper we analyse the nature of the parame- tric instabilities induced in ohmically heated toroi- dal discharges.

2

-

DISPERSION RELATION.

To approximate the physical conditions of the parametric interaction in ohmically heated to- roidal discharges, we consider a two-component wea- kly inhomogeneous low-B cylindrical plasma confined by a magnetic field with shear. The spatial varia- tion of the pump field and the confining magnetic field is neglected. The oscillations under conside- ration are assumed to be quasi-longitudinal with perpendicular wavelengths much smaller than the scale lengths of plasma inhomogeneities. Within the framework of the local wave dispersion descrip- tion and the weak pump field approximation (ab2<<1), the linear dispersion relation governing the para- metric coupling can be represented in the well- known form 12, 3/,

-t

* ' *

^%2

D(k,w ,r) = ,r) + Xi(~r~,r)~~+Xe(~,w*,r)l

-1'

*-

^-1

' *

(E (k,w u0,r) + E ( k , ~ +w0,r)) = 0 (1)

+

'

where

2

^{= w}

+

iy, ~(k,u,r) E I + C ~(k,w,r) and

-f a

x

(k,w,r) is the susc*eptibility of species a in

01

weakly-inhomogeneous low-collisional magnetized

plasma, ,.

ContrbZe'e, Service Ign

-

Centre d1Etudes NueZIaires

Here @(la,zna> =

5

In(Aa) exp (-Xa)Z(zna),

2 2 2

kllvta and A

a

= k

J.

^{v ta} / 2 uCa. The components of the wave vector of oscillations in the used coordinate system are defined as :

kc = COS@ k -sin4 kZ, k = cos@ k + sin@ k4, where

@ N

'

^-+

4

=*(BoZ7 go). The finite pump wave number is in- cluded to the lowest order in the dispersion rela- tion (1) by defining the sideband permittivity as

~ ( 3

^kll

+ kNo, ^w * ⁺

wo, r) 141. Furthermore, the presence of the longitudinal plasma current is taken into account by replacing w* in the electron suscep-

-t -+

tibilities by

2 -

k u where uD is the average ve- D

locity of electrons with respect to the ions. For a plane-polarized wave the coupling parameter ab is defined by,

.a: = (e k E ~ / ~ ~ w : ) ~ f(e,~,c), _{( 3 )}

-t -t

where 8 = +k, B

,c

⁼

9 s 3

^and 5 is the angle

O " o ' O + +

between the planes (k, Bo) and (Eo, go). In the pre- sent analysis the threshold electric field is mini- mized with respect to the angle 5 by taking the ma- ximum value of the function f ( 0 , 5 , 5 ) for a given

m coupling parameter 12, 31.

3

-

^RESULTS.

In this section we shall present results of detailed numerical analysis of the dispersion re- lation (1). In the low-density plasma region

( W ~ / W ~ ~ > Z ) the stability boundaries are characteri- zed by two pronounced minima of the coupling para- meter ab. The first minimum of ab occurs ?t zero detuning of the driving frequency from exact reso- nance, while the second, less pronobced minimum is associated with the resonant decay into an ion Bernstein or LH wave. Low minimum threshold electric fields of the resonant LH pump wave (E

=

50 Vlcm)

0 s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797267

associated with the first minimum are found. Insta- bilities with relatively small parallel phase veloci- ty (k 2/k2 >> m /m.) are preferentially excited in

/I ^{e 1}

the low-density plasma region. The foregoing discus- sion is illustrated by Fig. 1 on which we have repre- sented the variation of

%

(abt is the coupling pa-

t

rametar corresponding to the marginal stability) and the associated normalized frequency u/uci with khD.

The variation of the normalized growth rate y/u ci and the associated u/uCi with ab for different kXD values is also shown. The plasma parameters choosen are relevant to the low-density plasma layer near the wave launching structure : u0/aLH=5, ne/Bo 2 =

3 . 5 ~ 1 0 ' ~ m - ~ ~ - ~ , T =T. = 30 eV and the scale length e 1

ofinhomogeneitiesL = = 4 c m a n d L T = 1 cm.

ne L ~ i

Note that the level Eos = 5 kV/cm is displayed in the abt versus kX curve and is labelled by circles

D

on the y/uci versus ab curve. For pump electric field intensities of a few kV/cm the highest growth rates obtained (y=Cf(wCi)) are those related to the decay of the pump into a cold LH wave and an IC or nonresonant quasimode; Thus, in addition to the high- density plasma region, the decay processes involving quasimodes also dominate the decay spectrum excited for 2 <w /wLH ? 7. Due to the presence af current in the plasma, low frequency (w < uci) negative ener gy waves, which represent parametrically driven ion cyclotron current instabilities, are excited in the longwavelength part of the decay spectrum (khD <0.1 in Fig. 1). Although abt is somewhat lower in cur- rent carrying plasma, the pump field has a stabili- zing effect on the excitation of these instabilities since it eliminates the most unstable, large k

N instabilities.

It is interesting to note that at large w0/wLH values and for small parallel phase velocities

(a/% 7 0.9 vt ), the derivative ReD can become W

negative in a Finite a,,-range, ab, 7 a < a (the bt b2 values a are labelled by points in Fig. 1). The-

"2

refore the expansion of the dispersion relation i n a power series near the real roots (w, k) cannot be + used in the evaluation of the growth rate of quasi- modes at large wo/wLHvalues. Otherwise the applica- bility of this expansion is limited to a narrow ran- ge of field intensities near the threshold value in

-t

which for a given k the frequency v a r i ~ s slightly with ab and y << w. Note that by increasing the collision frequency ab may become larger than ab

t 2

For large kz/kzo values the pump wave de- cays into an IC quasimode and two LH waves.

This four wave decay process is followed by an abrupt increase of a as well as by a frequency increase.

There are two branches in the dispersion curves be- bt hind this kXD region : the principal forward mode which is an ion Bernstein (IB) quasimode and a low- frequency (w < wci) nonresonant quasimode. By follo- wing the dispersion curve of the principal mode, one concludes that in the short-wavelength region (kX >

D 0.2) and for u0/uLH < 2 the sideband becomes a hot IB wave. Due to their small group velocity and fast growth rates, the spatial amplification of hot IB waves dominates. An important energy transfer from the pump wave to IB waves and nonresonant quasimodes leading to a bulk particle heating occurs at uo/uLB<

2. In connection with this we note that for moderate pump fields the convective losses in the outer plas- ma layer are sufficiently large to allow an efficient wave penetration to the plasma core.

REFERENCES.

I. PORKOLAB M., Nucl. Fusion

18

(1978) 367.

2. PESIC S.S, EUR-CEA-FC 981.

3. PESIC S.S. Joint Varenna-Grenoble Symp. (1978)16L 4. PORKOLAB M., Phys. Fluids

20

(1977) 2058.

4 a/ Wci