VISCOUS DAMPING OF THE MAGNETO-ACOUSTIC OSCILLATIONS, MAO, IN BOUNDED PLASMAS

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VISCOUS DAMPING OF THE

MAGNETO-ACOUSTIC OSCILLATIONS, MAO, IN BOUNDED PLASMAS

Yu. Sayasov

To cite this version:

Yu. Sayasov. VISCOUS DAMPING OF THE MAGNETO-ACOUSTIC OSCILLATIONS, MAO, IN BOUNDED PLASMAS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-585-C7-586.

�10.1051/jphyscol:19797283�. �jpa-00219271�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au no?, Tome

40,

JuiZZet

1979,

page C7- 585

VISCOUS DAMPING OF THE MAGNETO-ACOUSTIC OSCRLA77ONS, MAO, W

BOUNDED

PLASMAS

Yu.

S.

Sayasov.

I n s t i t u t e o f Physics, University o f Fribourg, Fribourg, SwitzerZand.

A sound wave r e f l e c t e d from a s o l i d boundary i s known t o experience s t r o n g a b s o r p t i o n connected w i t h t h e f a c t t h a t i n a narrow boundary l a y e r , where mass v e l o c i t y $ drops q u i c k l y t o zero, marked v e l o c i t y o r a d i e n t s a r i s e . As a r e s u l t t h e viscous f o r c e (an example o f a monoatomic gas)

-+

1

³

? = q

( A V + ~ grad divv),n i s v i s c o s i t y c o e f f i c i e n t , can t a k e t h e r e b i g values thus i n f l u e n c i n g essen- t i a l l y t h e damping mechanism ( i l l , 977). These c o n s i d e r a t i o n s a r e c e r t a i n l y a p p l i c a b l e a l s o f o r F4AO i n bounded plasmas. However, i t seems t h a t such viscous damping o f MA0 i n bounded plasmas due t o t h e h i c h v e l o c i t y g r a d i e n t s a t t h e boun- dary was never i n v e s t i g a t e d . I n what f o l l o w s an account o f t h e o r e t i c a l r e s u l t s obtained i n t h i s d i r e c t i o n i n E21 i s presented.

R e s t r i c t i n g ourselves w i t h cold, monoatomic, q u a s i n e u t r a l , isothermal, homooeneous plasmas i n a magnetic f i e l d so = const, one can w r i t e t h e MHD-equations o f t h e o n e - f l u i d approximation, accounting f o r t h e viscous f o r c e ?, i n t h e form:

u2

where

o

=& i s plasma c o n d u c t i v i t y ,

OI

i s plasma P

frequency, v i s e l e c t r o n c o l l i s i o n frequency,

uce

i s e l e c t r o n c y c l o t r o n frequency,

p

i s mass d e n s i t y . System ( 1 ) must s a t i s f y a t a s o l i d boun- d a r y some c o n s t r a i n t s imposed on t h e mass velo-

c i t y ?. Here are some r e s u l t s f o l l o w i n g from (1) f o r a p a r t i c u l a r case of MA0 i n the'bounded cy- 1 in d r i c a l plasmas.

1 . MA0 i n a l o n g c y l i n d r i c a l plasma column sur- rounded by a d i e l e c t r i c a l boundary o f t h e r a - d i u s a and s i t u a t e d i n an a x i a l magnetic f i e l d Bo.

E x c i t a t i o n o f MA0 o f frequency

OJ

i s performed by a c o i l having t h e same r a d i u s and t h e same l e n g t h as t h e plasma column. (These assumptions c o r r e s - pond t c t h e e x p e r i c e n t s dsscribed e.g. Sn i3,4)).

The o n l y non-zero components o f t h e mass v e l o c i t y

3

v and o f t h e high-frequency magnetic f i e l d 8 ^{a r e}

resp. t h e r a d i a l component vr and a x i a l component Hz. S o l u t i o n s o f ( 1 ) a r e d e f i n e d u n i q u e l y by t h e boundary c o n d i t i o n s vr(a) = 0 and Hz(a) = Hex ( f i e l d generated by t h e c o i l ) . I n t r o d u c i n g dimensionless q u a n t i t i e s

€ 1

=l wvm ,

^E~

=zh

c A

CA =

-- Bo i s A l f v e n v e l o c i t y ) and makina assump-

J4ap

t i o n s < < I , s2

<<

(weak d i s s i p a t i o n ) , o f t e n f u l l f i l l e d i n experiments, one can f o r m u l a t e main r e s u l t as follows. The r a t i o o f t h e magnetic f i e l d amplitudes a t t h e c y l i n d e r a x i s HZ(D) and a t t h e boundary Hex i s g i v e n by

H(O) = 1 wa 1

'+ex J - i J k A = ( I ~

+ -

2 €1) (2) where J o y JI a r e Bessel functions,

A

= J %= J .

€1

vm

For t h e frequencies c l o s e t o t h e frequency

(,, =- qocA , qo =2,4 (Jo(q ) =O) o f t h e f i r s t mag-

0

a

neto-acoustic resonance, MAR, t h e modulus o f t h i s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797283

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r a t i o can be represented i n t h e form N - / H ~ ( 0 ) i - 1

^a

- a o ) 2

1 -

^J

( ) +

Y ~ I - ' ~ ~

1 Hex

0 1

9, ( 3 )

boundary l a y e r thickness p e r t a i n i n g t o these MAO).

I n f l u e n c e o f t h e v e l o c i t y becomes appreciable ac- c o r d i n g t o t h i s formula i f e

^<

² ^, i.e. f o r

6

9 o C ~ "

9 ; 2,9

<<

1 whereas f o r t h e i n f i n i t e plasmas vm

t h e corresponding c o n d i t i o n i s much more s t r i n g e n t :

>

1. For a f u l l y i o n i z e d monoatomic plasma we

Vm

o b t a i n , using expressions f o r t h e e l e c t r o n - i o n c o l l i s i o n frequency

v

and t h e i o n - i o n v i s c o s i t y c o e f f i c i e n t

0

given i n [51, t h e formula

106n

⁺

2,4.106T2

Y =

----

C A T , 2

A114Anr2

^-

where A i s t h e Coulomb l o g a r i t h m , A i s atomic weight, T i s temperature i n eV and n cm-3 i s e l e c t r o n d e n s i t y . (For a = 5 cm,

~ , = 1 0 ~ Gauss, r ! = 1 c 1 5 * = I , viscos!ty be- g i n s t o p l a y an e s s e n t i a l r o l e according t o t h i s expression i f T ; 2 e ~ ) . S i m i l a r conclusion can be drawn a l s o f o r plasmas w i t h comparable concentra- t i o n s o f ions and n e u t r a l s

As f o l l o w s from (3) maximal v a l u e N(u,) = 0 , 8 / ~ corresponding t o t h e f i r s t MAR i s reduced i n a r e - s u l t o f t h e viscous e f f e c t by a f a c t o r (I+-)-'. 2h

90E1 T h i s conclusion seems t o o f f e r a p l a u s i b l e expla- n a t i o n sf a considerable r e d u c t i o n o f t h e expe- r i m e n t a l values o f N(u,) compared w i t h those c a l - c u l ated without accounting f o r the viscous e f f e c t .

(See e.g. Fig. 12 i n [31 , Fig. 10 i n 141).

2. MA0 a r e generated by a s h o r t c o i l i n a l o n g cy- l i n d r i c a l plasma column o f r a d i u s a. (Otherwise t h e assumptions, corresponding t o t h e experiments described e.g. i n [6, 71, a r e t h e same as f o r

n u a t i o n o f t h e a x i a l magnetic f i e l d HZ a t s u f f i - c i e n t l y l o n g distances z from t h e c o i l f o l l o w s t h e formula ~,-e-"',

^K

= ImkZ, where k Z i s a r o o t (having s m a l l e s t imaginary p a r t ) o f t h e equation Jo(k,a) - i x Jl(k, a) = 0 and k,(kz) i s t h e r a d i a l wave-number d e f i n e d e.g. i n [41. (Fe r e t a i n here o n l y t h e ;embers o f t h e o r d e r

A =

ch ^, n e g l e c t i n g

vm

c o r r e c t i o n s t o k, which a r e o f t h e o r d e r o f A).

W e vm

I f -

<<

f (usual MHD-approximation) one can represent k Z i n t h e form: k, = [ q 2 - q:+i(slq2+

2 ~ q ~ ) l ' / ~ , q

=

- @a . For

w

n o t t o o c l o s e t o t h e

C~

frequency

U,

o f the f i r s t MAR

K

= ImkZ - -

^q2e1⁺

^2Aq0 . According t o t h i s 2Jq2 - q;

expression t h e viscous e f f e c t leads t o an 8ddi- t i o n a l damping w i t h d i s t a n c e z described by t h e f a c t o r exp(- ) , which can be v e r y im-

Gz7T

p o r t a n t f o r n o t too small values o f t h e para- meter

A .