Decay instability of lower hybrid wave in a cold homogeneous plasma

Volume 55A, number 7

PHYSICS LETTERS

26 January 1976

DECAY iNSTABILiTY OF LOWER HYBRID WAVE iN A COLD HOMOGENEOUS PLASMA S. BUJARBARUA and Y.S. SATYA Physical Research Laboratory, .4 hmedabed 380009, India Received 29 October 1975 We have pointed out the possibility of parametric excitation of a low frequency wave, propagating almost perpendicular to the magnetic field in a cold homogeneous plasma, by a lower hybrid pump wave. The growth rate and threshold power are calculated.

Heating of plasmas by applying the oscillating electric field near lower hybrid resonance has attracted considerable attention recently [1]. The advantage is that sufficient power is available at low frequencies and also ions are heated directly. At sufficient power parametric excitation of various low frequency modes is possible. It has been pointed out that in a cold homogeneous plasma the lower hybrid pump can decay into another lower hybrid and ion ion hybrid wave [2] or into two lower hybrid waves [3]. To excite the ion ion hybrid mode, it is essential to have a plasma with at least two species of ions. On the other hand the decay into two lower hybrid modes is possible only if the pump wave number is finite and the dipole approximation is not valid. In the present cornmunication, however, we wish to point out the excitation of another low frequency mode propagating almost normal to the magnetic field in a homogaseous cold plasma with single ion species by a lower hybrid pump under the dipole approximation. The dispersion relation thisa mode given 2 =linear ~e~i’~I’~2 (~‘e v 2ofand small isdampby w 5) ing ~ (~e+ vj~,where is the collision frequency, = eB 0/Mjc is the cyclotron frequency andj = i,e. The other symbols have their usual meaning. Suffix II refers to the direction along the background magnetic field B0.mode It is important to note that this is a low frequency such that the frequency is well below — ~

—

It is to be noted that in a cold homogeneous plasma with a single species of ions, this is the only low frequency mode propagating in a small cone normal to the magnetic field. Therefore the excitation of this mode by the high frequency pump should be important. We have calculated the threshold power and the growth rate of this instability for the lower hybrid pump under the dipole approximation. The modified dispersion relation in the presence of high frequency field reduces to [5] 1 1 1 e(w) = —J~(p)x~(w)Xe(W) I + 1(1) Le(~.,1.wo) e(w—w0)J where E(w+nw0) 1+X.(w+nwo)+~(w+nwo);n0, ±1, I and +

~.

Xj~~CA)~ — ~~pi

~= 2

—

[

w+nw0 2— ~ w+nw0 [(~ +nw0) 1 x~(~+nwo) =

r —

1 k2 (~+nw 0)

~‘+n~

pe —~ W+flWnL(’+

k~f

~2_~2 ~O’ e

1

—ik (w+nw —~

2+p2] 1/2 ~ 1, where = w +ip~,~‘by=k,W+IPe X and ,.it3are defined (R and p [X 01 Roe) = A sin t + p costhe w0t, R01 is the excursion jthw species under influence of pump fieldlength E = forcos 0t (dipolefirst approximation), and J1 is the Bessel function of the order. ‘~

—

the2O ion= cyclotron frequency which in turn requires k~f/k2~MeIMi. cos For small collisions wethe cansmall writedamping the frequency 20 and rate as [4] as ~e~ic05 ~‘L ~ +v~).This puts the restriction over the angle of propagation as M~/M~cos2O (Ve_V~2I4~Zefui. ~

~‘

The dispersion relation (1) can be reduced to a simple form 409

Volume 55A, number 7

~2 +2iwvL—w?

PHYSICS LETTERS

AO

=

(2)

~2_(w+ipH)2’

where ~L and VH are low frequency and high frequency (lower hybrid) damping rates respectively, ~=W~J—W~ ~ is the frequency mismatch of pump w 0 from the lower hybrid_frequency M k~ 1+ ~ =

WPi

V~(1+~

k2)/(

~);

K is the measurement2Me of the poer such that andapplied w~= ~Ze&2ikii/k. K =Eq. J~(p) ki~Mi/k (2) w is ~ similar to Nishikawa’s general equation [6]. For p = 0, we get the linear dispersion relation as discussed above. Following Nishikawa the two types of the solution are possible. (i) Re w 0: This purely growing instability grows only for 6 <0. The threshold is minimum at 6 and is given by Eomp ~ 8M~vHc &~~/ekwo when the pump field E 0 is applied in the direction ofkX B0. The maximum growth rate of this instability at suffi3 ciently large applied power is given by ~m (K/2)’/ 1’L’ ~H’ °‘L~ (ii) Re w ~ 0: The minimum threshold in this case is given byE~.~ (2vL/WL)E2 whenE 0 II kX B0, Omp1’H~ at the W~ forWL WL~ ~~ we obtain the On frequency the other hand for threshold as E~m~ (8 ~ ~‘LVHR4’L) E~mpwhen II kX B 0 at the frequency ‘--±(w~+~ PL ~H) 1/2~

26 January 1976

The maximum growth rate well above threshold is ym=(V’~I2)(K/4)1”3at the frequency’~.±(~)(K/4)U3. Thus we see that even in the cold homogeneous plasma with a single ion species, the lower hybrid pump can excite a parametric instability in a narrow cone perpendicular to the magnetic field. For minimum threshold, the instability will be excited in the E0)< B0 direction. This type of instability may be of relevanceplasma. to the heating and confinement of thermonuclear It is a pleasure to thank Dr. A. Sen for taking deep interest in this work.

References

—

~‘

‘~

410

[11 AK. Sundaram

and P.K. Kaw, Nuclear Fusion 13 (1973)

901, and references therein. [2] (1973) P.K. Kaw Y.C. Lee, Fluids Phys. 16(1973)155; E. Ott, J.B.and 270. McBride and Phys. J.H. Orens, Fluids 16 [3] E. Ott, Phys. Fluids 18(1975)566. [4] A.I. Akhiezer et al., Collective oscillations in a plasma (1967) p. 144. 37; Vol. p. A.B. 1, Mikhailovskii, Theory of plasma instabilities (1974) Sen and P. Kaw, Nuclear Fusion 15 (1975) 195.

[51 Y. Satya, A.

[61 K. Nishikawa,

J. Phys. Soc. Japan 24(1968)1152.