THE SIDEBAND INSTABILITY

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Submitted on 1 Jan 1977

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THE SIDEBAND INSTABILITY

R. Franklin

To cite this version:

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ment une instabilite lineaire faisceau-plasma, la densite du faisceau et sa vitesse etant dkfinis par I'accCICration non lineaire initiale donnee par I'onde aux particules du plasma.

Abstract.

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Computational and experimental results are given for the sideband instability in which the time (or space) evolution of the situation is explicitly recognized. It is shown that the results are consistent with a model which regards the instability as essentially a linear beam plasma instability the beam density and drift velocity being determined by the initial nonlinear acceleration of plasma particles by the wave.

This instability occurs when a large amplitude coherent electron plasma wave is excited at some point in a plasma and is characteriscd by there being a band of frequency which grows as the wave propa- gates, the band being close to but lower in frequency than the original wave. It was first reported by Whar- ton, Malmberg and O'Neil [I] and subsequently has been intensively studied experimentally [2-81. There have been in addition a number of theoretical models proposed [9-151 and more recently computational studies [I 6-21].

Many of the theoretical models unfortunately did not adequately describe essential features of the experimental situation and it is only recently that a concensus has emerged as to the underlying mecha- nism. It has been to some extent disappointing to discover that the important features arise neither from trapped particles nor from parametric processes but from the (linear) beam-plasma instability, the beam arising from initially resonant electrons which are accelerated as the plasma wave damps. Thus the setting up of plasma conditions is essentially a transi- ent process as had been anticipated by Brinca [lo] and by Bussac et al. [12]. This is shown in figures 1 and 2 where the observed damping of the originally excited wave is modelled theoretically by an equation of the form

1 W B t

-

- sin o, t x exp

-

2 n

where y , is the linear Landau damping rate and o, = k,(ecpo/m)'r2, which for temporal variation

FIG. 1.

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Motion of particles in phase space near the phase velocity of vn of a wave whose amplitude varies as q(t)/q(O) showing that a group of particles in the initial trapping region (shown dashed) is accelerated and becomes untrapped. The

parameter 2 = ~ L / u B = 0.5.

incorporates the essential features of a large amplitude stable wave as found by Sugihara and Kami- mura [22] and others. Figure 1 shows phase space trajectories for particles initially resonant with the wave indicating that there is a region of phase space from which particles with a speed initially less than that of the wave are accelerated and subsequently move faster than the wave. Figure 2 shows the spa- tially averaged distribution function at different times for the conditions of figure 1 and indicates how for appropriate values of the parameters wave amplitude cp and wave phase velocity v, the distribution function evolves to a form which is recognizably linearly unstable.

Figures 3, 4 and 5 show experimental measurements of the time-averaged distribution function under

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R. N. FRANKLIN

FIG. 2. - The calculated time evolution of the electron distri- bution function corresponding to the data of figure 1 with

va = 3.12 vee at time intervals of r n / 2

--

nlwn

.

/ _No_Wave

34 32 Z ~ ~ Z ~2 5 24 23 L C 22 n TO + ( M H ~ ) 0.5 1.0 1.5 2 0 2 5 3 0 3.5 eV

Energy E

FIG. 3. - Experimental measurements of the electron energy distribution in a plasma column as a function of the parameter probe voltage V at fixed frequency f = 30 MHz and distance from the exciter x = 54 cm. The bar indicates the phase velocity of the large amplitude wave.

I , , - No Wave

0 5 1.0 1.5 2 0 2.5 3 0 3 5ev

FIG. 4.

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Similar measurements to figure 3 with V = 2V, x = 54 cm as a function of wave frequency f.

typical plasma conditions with systematic variation of the parameters, cp, v, and distance from the exciter x. It is seen (a) that there is a relatively well-defined threshold for the distribution to show a region of positive gradient ; (b) there is a range of phase velo- cities over which the effect is significant ; (c) that the

FIG. 5. - Similar measurements to figures 3 and 4 with V = 2 V and f = 30 MHz as a function of distance from the

exciter x.

distribution function evolves to a relatively time- invariant form as anticipated in figure 2. Under these circumstances it is appropriate to model the plasma by an unperturbed distribution function with a beam passing through it and some results have been given elsewhere. In order to more closely model the experimental situation, we have taken the beam and plasma to be Maxwellians with thermal speeds veb and o,, and a relative drift u,, and relative densities n, and n,. The physical situation in a uniform plasma has three natural parameters, n,/n,, v,,/v,,, and u,,/v,,, and has been discussed by O'Neil and Malm- berg [23] and Self, Shoucri and Crawford [24]. If

finite geometry is included then there is the further parameter all, the ratio of plasma radius to Debye length for a column in free space.

We have solved the dispersion relation for different values of these parameters to determine the five least damped modes. Apart from the determination of stability, this is useful for comparison with test wave propagation which is a powerful diagnostic tool.

Figure 6a, b, c shows the variation with beam density of the dispersion and the corresponding distribution functions. For n,/n, = the beam and plasma modes scarcely interact and the plasma fun- damental and first order radial modes are the least damped. For n,/n, = interaction is apparent but any measurement of test wave propagation would show that the least damped wave was very close to the plasma unperturbed mode. However, for n,/n, = 10 - there are three branches which compete for being the least damped and two regions of instability. Figures 7a, b, c show the influence of beam drift velocity with the other parameters constant. For

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R. N. FRANKLIN

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C6-6 R. N. FRANKLIN

mode is almost nowhere the least damped mode and the spectrum of instability is approaching the two- stream case.

Figures 8u, b, c show the variation with thermal spread of the beam. For sufficiently large spread the distribution function is monotonic and the least damped wave is essentially the plasma mode. While for a narrow beam the plasma mode is seen only at low frequencies and the least damped mode is other- wise a beam mode be it slow or fast. With this back- ground one can turn to test wave measurements to see whether they confirm this model, and also consider how growth rates and frequencies should depend upon the basic parameters of the experiment, rp and v,.

Doing the latter first it is intuitively easy to see that

where N is of the order 1, that v,, varies approximately as ( e r p / m ) 1 f 2 and n,/n, = f(u,, rp), and varies approximately as

2 2

(q)

' I 2 exp

-

v,/voe

.

'no,,

Under these circumstances it means that the variation of growth rate with amplitude is going to have a complicated dependence and thus the earlier results reporting different values are not necessary inconsis- tent. On the other and the frequency of maximum growth rate will vary so that

as observed.

Measurements of test wave propagation have been reported which show the propagation of two modes of comparable damping rates and of situations in which one mode damped while another grew [6, 71. An alternative is to map out the dispersion of the least damped mode. For the case of figure 6b and c

for instance, it is expected that there would be a discontinuity close to o/o,, = 1.0, while from

FIG. 9. - Measured dispersion of the least damped test wave in the presence of a large amplitude wave. The unperturbed dispersion is shown as a solid line (a) fD = 48 MHz f = 43.5 MHz

and corresponds to a large drift and large beam density, (b)

f, = 27 MHz, f = 30 MHz with a lower drift and beam density.

0

Frequency (5MHz l div)

FIG. 10. - The spectrum of fluctuations in a plasma under conditions similar to figure 9b but at very high wave amplitude when the spectrum (modified by column resonances) is essen-

tially that of the beam-plasma instability.

Energy E

FIG. 11. - The distribution function and spectrum of fluctuations in a plasma in which two large amplitude waves of different frequency fo, f1 propagate. f1 = 24 MHz and is varied in amplitude 0, 1.0 and 2.0 V while fo = 34 MHz and is constant at 2.0 V. The sideband instability is quenched and the deforma- tion of the distribution function removed when the amplitudes

are comparable.

figure 8c, under the condition of narrow beam width, the least damped mode at high frequencies is the beam fast mode.

Figures 9a and b display dispersion diagrams with these characteristics under different conditions, while figure 10 shows a broad band of unstable frequencies corresponding to the near ' two-stream ' case of figure 7c. A study of the form of the distribution

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WHARTON, C. B., MALMBEKG, J. H. and @NEIL, T. M., Phys. Fluids 11 (1 968) 176 1.

VAN HOVEN, G. and JAHNS, G., Phys. Fluids 18 (1975) 80.

JAI-INS, G. and VAN HOVEN, G., Phys. Fluids 18 (1975) 214.

FRANKLIN, R. N., HAMBERGER, S. M., LAMPIS, G. and SMITH, G. J., PrOc. ROY. SOC. A 347 (1975) 1.

BARBIAN, E. P., Proc. VIth Europ. Conf. Controlled Fusion and Plasma Physics, Moscow (1973), p. 465.

STARKE, T. P. and MALMDERG, J. H., Phys. Rev. Lert. 37

(1976) 505.

SATO, N., POPA, G., MARK, E., SCHRITTWISSEK, R. and MRAVLAG, E., Phys. Rev. Lett. 37 (1976) 1684.

FRANKLIN, R. N., Proc. VIIIth Int. Summer School, Dubrovnik, 1976, Invited Papers, p. 679.

[I31 KRUER, W. L., DAWSON, J. M. and SUDAN, R. N., Phys. Rev. Lett. 23 (1969) 838.

1141 MIMA, K. and NISHIKAWA, K., J. Phys. SOC. Japan, 30

(1971) 1722 and 33 (1972) 1669.

1151 GOLDMAN, M. V. and BERK, H. L., Phys. Fluids, 14 (1971) 801.

[16] KRUER, W. L. and DAWSON, J. M., Phys. Fluids 13 (1970) 2747.

1171 ROSEN, B., SCHMIDT, G. and KRUER, W. L., Phys. Fluids

15 (1972) 2001.

1181 MATSUDA, Y. and CRAWFORD, F. W., Phys. Fluids 18

(1975) 1346.

1191 CANOSA, J. and WRAY, A., Phys. Fluids 19 (1976) 1958. [20] FRANKLIN, R. N. and MACKINLAY, R. R., Phys. Fluids

19 (1976) 173.

_.

,

[9] B ~ M B E R G , M. W. and BERK, H. L., J. Plasma Phys. 9 [21] DENAVIT, J. and KRUER, W. L., Phys. Fluids 14 (1971) 1782. (1973) 235. 1221 SUGIHARA. R. and KAMIMURA, T., J. Phys. Soc. Japan

[lo] BRINCA, A. L., J. Plasma Phys. 7 (1972) 385. 33 (1972) 206.

1111 BUD'KO, N. I., KARPMAN, V. I. and SHKLYAR, D. R., 1231 O'NEIL, T. M. and MALMBERG, J. H., Phys. Fluids 11

Zh. Eksp. Teor Fiz. 61 (1971) 1463. (1969) 1754.

[I21 BUSSAC, M. N., MENDONCA, I., PELLAT, R. and Roux, A., [24] SELF, S. A., SHOUCRI, M. M. and CRAWFORD, F. W., J.