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CHARACTERISTICS OF SPATIAL WAVE ECHOES AT THE LOWER HYBRID BRANCH
Ichiro. Mori, H. Leppert, K. Wiesemann
To cite this version:
Ichiro. Mori, H. Leppert, K. Wiesemann. CHARACTERISTICS OF SPATIAL WAVE ECHOES AT THE LOWER HYBRID BRANCH. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-555-C7-556.
�10.1051/jphyscol:19797268�. �jpa-00219254�
JOURNAL DE PHYSIQUE Cottoque C7, suppte'ment au n07, Tome 40, JuiZZet 1979, page CT- 555
CHARACTERISTICS OF SPATIAL WAVE ECHOES AT THE LOWER HYBRID BRANCH
Ichiro. Mori and H.D. ~ e p p e r t " , K. ~ i e s e m e n n * .
XDepartment o f EZectric Engineering, TeehnicaL eoZZege of Tokwhima University, Tokushima Japan.
I n s t i t u t ftir experimentalphysik, Ruhr-Universitiit Bochwn, Boehwn West-&many,
1 . I n t r o d u c t i o n : The phenomenon of plasma wave The f i r s t - o r d e r ( o r l i n e a r ) s o l u t i o n can b e w r i t t e n e
3kwb
/&$,uI).. . .
w . .. . .
(5)echoes [ I ] i n a c o l l i s i o n l e s s plasma i s a r e v e r s i b l e a s Ih Uir t -m%
process i n which macroscopic f i e l d energy c o n v e r t s
M ~ I -
I + + -1 $%~).1ni(&Yll
?-
...
(6)t o p a r t i c l e ' s k i n e t i c energy and v i c e v e r s a , even i n t h e presence of phase-mixing and Landau-damping.
Thereforeechoes possess g r e a t p o t e n t i a l a s diag- n o s t i c t o o l f o r i n v e s t i g a t i o n i n t h e v e l o c i t y s p a c e . E s p e c i a l l y , t h e echoes of lower hybrid branch a r e i n t e r e s t i n g because of h e a t i n g a s p e c t s simultaneous- l y i n v o l v i n g both electron-and ion-dynamics. I n t h i s paper, comparison between t h e t h e o r y and t h e e x p e r i - ments on t h e s p a t i a l lower hybrid echoes have been made.
2 . ~ 0 d e l of s p a t i a l wave echo : The Lower hybridwave belongs t o a ion-cyclotron wave, t h e n t h e g y r a t i n g motion of p a r t i c l e s Ln f i n i t e magnetic f i e l d extends
t h e dimensions of t h e model a u t o m a t i c a l l y . The experimental plasma i s , of course, of f i n i t e r a d i a l e x t e n t , b u t t h e l e n g t h i s l o n g enough t o c o n s i d e r t o be i n f i n i t e , s o t h a t we u s e h e r e f o l l o w i n g model.
W e s t a r t from t h e model of i n f i n i t e r a d i a l e x t e n t
, t h e n t h e e f f e c t of f i n i t e r a d i u s i s i n t r o d u c e d by means of t h e r e s t r i c t i o n of p e r p e n d i c u l a r wave number. I n t h e experiment, t h e background p r e s s u r e i s Torr hydrogen, and t h e plasma having a e l e c t r o n d e n s i t y ne- 3.6 x 1014m-3 i s immersed i n a magnetic f i e l d of 30 mT. Thus t h e plasma can be considered t o b e c o l l i s i o n l e s s and Vlasov e q u a t i o n can b e used, whereas t h e s e l f - c o n s i s t e n t f i e l d s a r e taken i n t o account through P o i s s o n ' s e q u a t i o n . F o u r i e r transforms of t h e s e eq$ations a r e expressed
S u b s t i t u t i n g E q . ( 5 ) , ( 6 ) , i n t o a non-linear p a r t of E q . ( l ) , and i n t e g r a t e i t w i t h r e s p e c t t o 8 , we g e t a second o r d e r d i s t r i b u t i o n f u n c t i o n f(:)in t h e suc- c e s s i v e approximation. I f we s u b s t i t u t e t h i s d i s - t r i b u t i o n i n t o P o i s s o n ' s e q u a t i o n , we can g e t a echo - p o t e n t i a l i n (g,w) space.
3 . C a l c u l a t i n g r e s u l t s : To o b t a i n t h e s o l u t i o n , we u s e f o l l o w i n g assumptions. ( i )We c o n s i d e r only a a x i a l symmetric mode. ( i i ) ~ a d i a l d i s t r i b u t i o n of a p p l i e d p o t e n t i a l ( g r i d p o t e n t i a l ) i s uniform a s g ( r ) = l . (ti( )Landau-damping is n o t s o strong,b/k)21.
(iV)For
fz
, maxwellian d i s t r i b u t i o n c a n be used.( v
) I n e q u a l i t i e s U,.&%-W*, G>>$$G
a r e h e l d . Under t h e s e c o n d i t i o n s , we g e t f i n a l l y t h e second o r d e r e c h o - p o t e n t i a l a sI
- +$$ ,
(GI,= (~nOa=u, etc.u - cu*-lr)ca.
wi+= -
W-@cZ UJ p t 7 (* = us- a x )where wm=wp-wx i s t h e echo-frequency and t h e d i s - t a n c e z* i s normalized by
&
(rzn- zx), t h e d i s t a n c e between two e x c i t i n g g r i d s . A summation on shows a mean w i t h r e s p e c t t o an a n g l e between t h e v e c t o r s+
-Dk and q , and t h e y r e l a t e d a s
& A = #if,
$,,, (
r,
X, t) =+< 9(ns
&$,@-~~].CS C O , ~ ( b ' - ~ ~ ) & / ~ b ~ - aqba+
(bz-@)/2bza = COS+~a-3~.g(r).~[:&(~-Zn)l.~~~ writ
. . .
(3) The e f f e c t of f i n i t e plasma-radius Is t a k e n i n t oEt - i8.e . . . . . .,. . . . . . . . . . . . .
( 4 ) account by t h e r e s t r i c t i o n of q L v a l u e i n t h e i n t e - g r a t i o n w i t h r e s p e c t t o qL. The minimum v a l u eArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797268
of qL is
,,,
( n /D 1, where Dp is a plasma diameter,P : ECHO-POTENTIAL
whereas the maximum value will be determined by 1.0
-
FI- 70 W MF. -120 MHz
Landau-damping. If q , , ~ w /vte (vte: electron &-SO MHz thermal velocity), the wave will be strongly damped Fig. 1. ,
out. From the dispersion relation, it becomes
A m
3.5~ L m a x w J ~ ~ q ~ m a x = J m - . ( w / v t e ) .
v v -
4.Comparison with experiments : The experiments
vf
DISTANCE were performed with an apparatus previously reported
[2]. The active source plasma is generated by non-
-1.0
-
resonant wave excitation by means of a waveguide
-
-plasma-resonator arrangement [ 3 ] at 2.45 GHz.
The plasma streams into a test tube of 150 mm i.d.
and 2.5 m length consisting of glass as well as stainless steel sections. The plasma is confined radially by the magnetic field and it's diameter is determined by the qualtz tube of the plasma gener- ator. Figure 1 shows a typical line shape of the second-order echo at the time of t = .rr /%obtain-
A' ed by computer under the condition Te-5.5 eV,
B = 30 mT, ne-v3.6 x 10l~m-~. Figure 2 gives the
experimental curve together with the theoreticals' I
in the same condit2on'and same scale of distance.
In Fig.2(a), upper trace gives a phase difference
between test and reference signal in a linear scale V showing a sawtoothed shape. 1tk lower end corre-
Fig. 3 (a) Fig. 3 (b)
sponds to -180° phase difference and the upper +180°
-
-
,difference. The lower trace of Fig.2(a) gives the
3
5 - 5 0 M k amplitude A in a logarithmic scale. In Fig. 3(a), -$
.'the dependence of frequency wZ on the maximum echo, w . -amplitude is given, where the echo-frequency
%is fixed, and Fig.3(b) shows the corresponding experi- mental result. Figure 4 gives the spatial position of amplitude-maximum, as the frequency w I isvaried.
In this figure, the theoretical result is shown by 0 50
m w E N C Y fi C U M ] 0
thick line and open circles correspond to experi- ment s
.
In summary this theory of the lower- hybrid-echoes shows good agreement with the experi- ments.This Investigation is partly sponsored by the DFG within the Sonderforshungsbereich 162 "Plasmaphysik
~ o ~ h u m / ~ U l i ~ h " . The authors thank Dr. B.Lammers for his contribution,Prof .H. ~chlGter for encouragement.
Thanks are also due to Mr.Kindermann for his assis- tance in the experimental appatatus.
5.References : [lIR.W.Gould,T.M.O'Neil and J.H.
Malmberg, Phys.Rev.Lett.19 (1967)216.
[Z]H.D.Leppert et al.,~hys.~ett.62~(19j7)501.
[3]H. Beerwald,G. ~ z h m , ~ . Kampmann and B. Schweer,
~hys. Lett. 538(1975) 267.
Theoretical Experimental
F i g . 4
