HAL Id: jpa-00219288
https://hal.archives-ouvertes.fr/jpa-00219288
Submitted on 1 Jan 1979
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
DRIFT-DISSIPATIVE WAVES AND ENSUEING STEADY STATE OF A PLASMA COLUMN
M. Evrard, A. Messiaen, P. Vandenplas, G. van Oost
To cite this version:
M. Evrard, A. Messiaen, P. Vandenplas, G. van Oost. DRIFT-DISSIPATIVE WAVES AND ENSUE-
ING STEADY STATE OF A PLASMA COLUMN. Journal de Physique Colloques, 1979, 40 (C7),
pp.C7-617-C7-618. �10.1051/jphyscol:19797299�. �jpa-00219288�
JOURNAL DE PHYSIQUE CoZZoque C7, suppZdment au n07, Tome 40, JuiZZet 1979, page C7- 617
DRIFT-DISSIPATIVE WAVES AND ENSUE3NG STEADY STATE OF A PLASMA COLUMN
M.P. Evrard, A.M. Messiaen, P.E. Vandenplas and G. Van Oost.
Laboratoire de Physique des PZasms
-
k b o r a t o r i w n voor PZasmafysica, Association '2URATOM-Etat beZgerr Associatie Euratom-BeZgische Stuat", EcoZe RoyaZe MiZitaire-
1040 BrusseZs-
KoninkZijke MiZitaire SchooZ.1. INTRODUCTION. Spontaneously excited low-£re- relation for the drift waves- in the quasistatic quency oscillations1 with a frequency of the order approximation. A first calculation which does not of the characteristic drift frequency w* are iden- take collisions and mean particle drift velocities tified as drift-dissipative waves2 and are respon- into account leads to
sible for the anomalous diffusion observed in a ci
w (1)
weakly ionized helium plasma column produced by a helical microwave discharge source (see Fig. 1).
where k,, is the wavenumber parallel to Eo, kL is Plasma parameters are : density, 5 x 10~-8x10~~crn-~;
the radial wavenumber and n/r the azimuthal one ; gas pressure, 1-7~10-' Torr ; electron temperature
wci is the ion~cyc~otron frequency and
vS
is the Te, 3-15 eV ; ion temperature Ti, 0.2-1.4 eV ; ma-ion-acoustic speed. The last term is the extra gnetic field B 850-2200 Gs.
0' nonuniform term with respect tothe dispersion re-
2. THEORY. The plasma consists of hot electrons and hot ions in the presence of background neutral particles and is described by means of the linea- rized two-fluid equations in which T and Ti are introduced by scalar pressure laws and in which only electron-neutral and ion-neutral momentum transfer collisions are included. The radial tem- perature inhomogeneity is negligible with respect to the density inhomogeneity. The bounded plasma problem is solved in the actual cylindrical geome- try.
The static fluid model is characterized by a radial Gaussian equilibrium density profile,
-ar2
<N>(r) = <N> e
,
where <N> is the equilibrium density on the column axis. Assuming quasineutra- lity, strong field approximation, classical ambi- polar diffusion and symmetry of revolution, one obtains from the equilibrium momentum equations the radial component of the static electric field, Eor = (KTi/e<N>) (a<N>/ar). The value of Eor is experimentally verified and found to be negligible for the calculation of the drift wave dispersion relation. We have furthermore checked that the additional possible static effect resulting from the drift wave induced diffusion does not modify the aforesaid value of Eor.-iwt
Using the e perturbed linearized two- fluid continuity and momentum equations3 and the Poisson equation, we calculate the dispersion
lation of the ion-acoustic waves in the uniform plasma ~ o l u m n ~ * ~ . This dispersion relation provides an approximate value of kL and when ki~;/w:~ << 1, one retrieves the standard Kadomtsev dispersion relation6.
When we further take collisions and drift velocities into account, we then obtain a disper- sion relation which can be written as a polynomial of the sixth degree in w with complex coefficents7.
Only one of the six complex roots satisfies the condition for drift waves : Iwl << wci
,
Im w > o.The behaviour of the real and imaginary parts of this root is plotted as a function of kA in Fig. 2 and shows the existence of a region of unstable modes. The maximum of the imaginary part occurs at a kl-value which is in good agreement with the value obtained from equation (1) when one inserts the observed most strongly excited Re w-value of the drift wave spectrum. The importance of taking the diamagnetic drift velocity of the ions into account is very pronounced. Indeed, the neglect of the diamagnetism of the ions with respect to that of the electrons results in the disappearance of the unstable region of Fig. 2. This is
demonstrated in Fig. 3 by gradually decreasing the value of the ion diamagnetic velocity by decreasing Ti, the other plasma parameters remaining constant.
The steady state density distribution in the plasma column is calculated from the diffusion equation and is given by :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797299
where pl i s t h e 1 - t h e r o o t of t h e B e s s e l f u n c t i o n J o , d i s t h e r a d i u s o f t h e m e t a l vacuum chamber, DL and Dl, a r e t h e anomalous d i f f u s i o n c o e f f i c i e n t s
t h e v a l u e s of which a r e c o n s i d e r e d i n t h e n e x t s e c t i o n and C1 i s a c o n s t a n t d e t e r m i n e d by means of t h e boundary c o n d i t i o n a t z = o.
3. EXPERIMENTAL RESULTS. S e v e r a l measurements have been c a r r i e d o u t which i d e n t i f y s p o n t a n e o u s l y e x c i - t e d waves a s d r i f t - d i s s i p a t i v e waves. The o b s e r v e d waves p r o p a g a t e m a i n l y i n t h e a z i m u t h a l d i r e c t i o n , i n t h e same s e n s e a s t h e e l e c t r o n d i a m a g n e t i c d r i f t v e l o c i t y , and have an a z i m u t h a l mode number n = 1. The v a l u e o f kl d e r i v e d from d i s p e r s i o n e q u a t i o n (1) i s c o r r o b o r a t e d by t h e measurement of t h e r a d i a l b e h a v i o u r of t h e wave a m p l i t u d e (Fig.
4 ) . A x i a l l y , t h e waves have a wavelength o f t h e o r d e r o f t h e l e n g t h o f t h e plasma column and t h e y p r o p a g a t e from t h e plasma s o u r c e towards t h e end p l a t e , i n t h e d i r e c t i o n of
B . F i n a l l y , t h e depen- dence of t h e wave f r e q u e n c y on T i s i n agreement w i t h t h a t o f a d r i f t wave. The anomalous d i f f u s i o n c o e f f i c i e n t DL r e s u l t i n g from t h e i n s e r t i o n o f t h e e x p e r i m e n t a l d a t a i n t o t h e t h e o r e t i c a l formulae6 and t h e e x p e r i m e n t a l v a l u e o f Dl, l e a d t o a s t e a d y s t a t e d e n s i t y d i s t r i b u t i o n ( e q u a t i o n 2) of t h e plasma column which i s i n v e r y s a t i s f a c t o r y agreement w i t h t h e measured one. The o b s e r v e d anomalous d i f f u s i o n and t h e e n s u e i n g s t e a d y s t a t e d e n s i t y p r o f i l e i n t h e plasma column a r e t h e r e f o r e e x p l a i n e d by means of t h e o b s e r v e d d r i f t waves.
4. REFERENCES
.
1. See e . g . Chen F.F. (1964) Phys. F l u i d s
1,
949.2. See e.g. S e l f S.A. (1970) J . Plasma P h y s i c s ,
6,
693.
3 . Dubois D.M., Messiaen A.M. and Vandenplas P.E.
(1972) P r o c e e d i n g s o f t h e 5 t h European Conferen- c e on C o n t r o l l e d F u s i o n and Plasma P h y s i c s , Grenoble, F r a n c e , v o l . I , p. 114.
4. Messiaen A.M. and Vandenplas P.E. (1973) Plasma Phys. l5, 505.
5. B h a t n a g a r V.P., Van Oost G., Messiaen A.M. and Vandenplas P.E. (1976) Plasma Phys.
18,
525.6. Kadomtsev B.B. (1965) Plasma T u r b u l e n c e . Acade- mic P r e s s , New York.
7. E v r a r d M.P., Messiaen A.M., Vandenplas P.E. and Van Oost G . , s u b m i t t e d f o r p u b l i c a t i o n t o Plasma Phys.
F i g . 1
-
Schematic diagram of t h e e x p e r i m e n t a l a p p a r a t u s .F i g . 2
-
T h e o r e t i c a l l y computed b e h a v i o u r of t h e r e a l and t h e ima- g i n a r y p a r t o f t h e d r i f t wave f r e q u e n c y a s a f u n c t i o n o f kL.
F i g . 3
-
T h e o r e t i c a l l y computed b e h a v i o u r o f t h e growth r a t e o f t h e d r i f t i n s t a b i l i t y a s a f u n c t i o n of t h eF i g . 4
