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Submitted on 1 Jan 1979
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NONLINEAR WAVES IN A MAGNETIZED PLASMA WAVEGUIDE
J. Lynov, P. Michelsen, H. Pécseli, J. Juul Rasmussen, H. Sugai
To cite this version:
J. Lynov, P. Michelsen, H. Pécseli, J. Juul Rasmussen, H. Sugai. NONLINEAR WAVES IN A MAG- NETIZED PLASMA WAVEGUIDE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-571-C7-572.
�10.1051/jphyscol:19797276�. �jpa-00219263�
CoZloque C?, suppZQment au n07, Tome 40, J u i Z l e t 2979, pageC7- 571
NONLINEAR WAVES IN A MAGNETIZED PLASMA WAVEGUIDE
3.6'. Lynov, P. Michelsen, H.L. PBcseli, J. Juul Rasmussen and H. ~ucjai*.
XAssociation EUKATOM-RIS@ National Laboratory Physics Department DK-4000 Roskilde-Dewnark.
Department of EZectricaZ Engineering, Nagoya University, Nagoya-Japan.
1;JTRODUCTION: Theoretical investigations of nonlinear modulation of stronalv diswersive electrostatic plasma oscillations /I] have demonstrated the importance of including the effect of nonlinear Landau damping, i-e., the resonant particles at the group velocity of the wave. The contribution of these particles modify a nonlinear SchrG-.
dinger (NLS) equation for the wave envelope by a nonlocal term. For electron waves in unbounded plasmas only the resonant ions are generally of importance in this connec- tion / 2 / . In the present paper we report investigations of electron plasma waves in bounded plasmas where the dispersion re- lation is modified so that resonant elec- trons play the dominant role. We consider waves in a strongly magnetized (w < < w )
plasma waveguide with radius r pe~onsi8&
ring the ions as a stationary 8;iform neu- tralizing background the linear dispersion relation for electron plasma waves becomes:
(o/k) 2 . = u;/(kj+kZ)+ 3.7: where ve = J g while k is the axial wave number and k
2.4/r0. We consider only the lowest orbe:
radial mode. For many laboratory exper- iments of interest w/k >> ve so linear Landau damping is negligible. Moreover, particle trapping may also be ignored for weakly nonlinear waves. However for certain k, the group velocity v will be close to ve
.
We derive a modif ie% NLS equation taking into account these resonant par- ticles, and present preliminary exper-- imental results supporting our results.THEORY: Our basic equations are the one- dimensional. electron Vlasov equation and Poisson's equation modified to take into account the finite geometry
a
2 $p -
$ = n-lwith lfdv = n, lfodv = 1 where fo(v) is the unperturbed electron velocity distribution function. The perturbed density n is nos- malized with no, v with u ? w /kL, t with w - I I x with kil and
+
wivh u;E/e. The cgefficient a = 0.72 originates from the expansion in radial eigenmodes /3/ where only the lowest one is considered here. To study the long time amplitude variation of electron plasma waves we use a multiple time-scale analysis and proceed in a manner very much similar to the one employed in /2/. Substituting an expansion of (I?,+) in powers of E into (1)-(2) we obtain a set of equations for each power of E. By success- ively removing secularity-producing terms, we finally obtainas the nonsecularity condition to third order, where T = c2t and 5 = E (x-v t)
.
Equation (3) has the form of a modyfied NLS (eq.) where the nonlocal term accounts for the effect of resonant particles at v /1,2/. The coefficient p E pdv /dk, wzife q and s are complicated functi8ns of k. A NLS eq. in the cold plasma limit was derived previously /3/. The present equa- tion differs not only in the nonlocal term but also in the significantly modified coefficient q,in particular we have a sign rev,ersal of q even for moderately small k ,
see fig. 1 where q and s are s h ~ w n as functions of k for the case,where fo(v) is chosen to be a Maxwellian. The dotted line shows q in the cold plasma approximation / 3 /
-
We now introduce the real functions
p ( t ; , ~ ) and O(~,T)--tki-ough a = pexp(i6) and note that (3) has a plane wave solution:
P = po = const, 6 = Oo = -qp;t = 6wt, where 6w accounts for the frequency shift. Note that 6w is independent of s. To examine the stability of (3) we modulate this solution by introducing a small perturbation:
1
\
Fig. 1
The linearized dispersion relation obtained from (3) determines Q = Rr (K)
+
ir ( K ),
where
Eqs. ( 4 ) show that the resonant particles at v make the electron plasma wave un- stabye for any amplitude modulation, con- trary to the case when this effect is ab- sent (i.e. s E O), where a necessary con- dition for instability is p.q>O. In the case s = 0 the instability is purely grow- ing in the group velocity frame of refer-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797276
ence (GR), i.e., the frequency has no real part. Considering an initial-value problem where the modulation is applied at t = 0 with wave number K, the instability will manifest itself in the lab. frame as two growing sidebands shifted from the carrier- wave frequency by +Kv (Fig. 2, where only the effect of amplitu8e modulation is shown). However, in the case s # 0 the fre- quency has a real part, and for p-s<O we find that I'>O for disturbances with Rr/K>O and r<0 for Rr/K<O. Considering again an initial-value problem, we see that the two modes (one growing and one damping) in GR give rise to two lower- and two upper-side- bands in the laboratory frame as shown schematically in Fig. 2.
EXPERIMENT. Preliminary experimental re- sults on the electron wave modulation were obtained in the Riso Q-machine in single- ended operation. The plasma is confined ra- dially by a strong magne ic field 0.4 T, has a density =.1-107 cm-' and Te 2 0.2 eV, i.e., u /ve = 15. By applying a modulated oscilla?ion to the floating cold end-plate, we excited an amplitude-modulated electron plasma wave. The wave propagation was in- vestigated by an axially movable Langmuir probe connected to a capacitive amplifier.
Fig. 3a shows the evolution of the fre- quency spectrum for a modulated wave at large amplitude (only nearest sidebands are shown). The sidebands grow with distance.
Fig. 2
The growth rate for both sidebands were found to scale roughly with the square of the applied amplitude Fig. 3b, where the growth rate of the lower sideband is plotted.
To interpret these results in terms of the presented theory, we note that (3) is more suited to an initial-value problem than to a boundary-value one applicable to the experimental situation. This is espe- cially true because the nonlocal term must take another form when applied to the boundary-value problem. However, at Least qualitatively, one should expect a k-spec- trum with multiple sidebands like the w-
spectrum in Fig. 2 (s # 0 ) . The frequency spectrum, on the other hand, consists of the two "applied" sidebands both growing in agreement with measurements (Fig. 3a).
Furthermore the spatial growth rate K = T/vg, i.e., approximately Ri a (: f ~ q . 4b), which also agrees with the m@kure- ments. By varying the carrier frequency we
found that the growth rate increased for v approaching ve as expected from (4b) and
9
wo12n = 36 MHz Q12n = 2 MHz
Fig. 3a Fig. 3b
Fig. 1.
We have ensured that alternative inter- pretations /4,5/ can be ruled out in the present experiment.
References
/1/ Y.H. Ichikawa and T. Taniuti, J. Phys.
Soc. Jap.
2 ,
513 (1973)/2/ K.B. Dysthe and H.L. Pscseli, Plasma Phys. 19, 931 (1977)
/3/ J. ~ u u ~ ~ a s r n u s s e n , Plasma Phys. 20., 997 (1978)
/4/ G. van Hoven and G. Jahns, Phys. Fluids 18, 80 (1975)
/5/
XN.
Franklin, R.R. MacKinlay, P.D.Edqley, and D.N. Wall, Proc. R. Soc.
Lond.
