FLUID THEORY OF PLASMA DOUBLE-LAYERS

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FLUID THEORY OF PLASMA DOUBLE-LAYERS

J. Levine, F.W. Crawford

To cite this version:

J. Levine, F.W. Crawford. FLUID THEORY OF PLASMA DOUBLE-LAYERS. Journal de Physique

Colloques, 1979, 40 (C7), pp.C7-139-C7-140. �10.1051/jphyscol:1979768�. �jpa-00219474�

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JOURNAL DE PHYSIQUE Colloque C7, suppldment a u n07, Tome 40, J u i l l e t 1979, page

C7-

139

FLUID THEORY OF PLASMA D W B L E l A Y E R S

J.S.

Levine and F.W. Crawford.

I n s t i t u t e f o r Plasma Research, Stanford U n i v e r s i t y , S t a n f o r d C a l i f o r n i a 94305 U.S.A.

A double-layer c o n s i s t s of two a d j a c e n t , oppo- s i t e l y - c h a r g e d space-charge l a y e r s , and may occur n e a r a m a t e r i a l boundary, o r w i t h i n t h e plasma vol- ume ( s e e [I]-[3] f o r reviews of double-layer pheno- mena i n l a b o r a t o r y and s p a c e plasmas). Here, we s h a l l analyze a s t e a d y - s t a t e double-layer s e p a r a t - ing two plasmas of d i f f e r e n t d e n s i t i e s and tempera- t u r e s , and r e l a t e i t s l e n g t h and p o t e n t i a l d r o p t o t h e plasma parameters.

S e v e r a l t h e o r e t i c a l models have t r e a t e d t h e double-layer a s a r e g i o n of high e l e c t r i c f i e l d and monotonic p o t e n t i a l v a r i a t i o n . The t o t a l p o t e n t i a l d r o p is g e n e r a l l y t a k e n t o b e l a r g e r t h a n t h e p l a s - ma thermal energy, and t h e e l e c t r i c f i e l d is con- f i n e d t o t h e double-layer, implying t h a t t h e c h a r g e v a r i a t i o n i n t e g r a t e s t o zero. Cold plasma, f l u i d , and k i n e t i c t h e o r y approaches have been d i s c u s s e d

121.

THEORY

We t r e a t t h e double-layer a s a t r a n s i t i o n be- tween two uniform s e m i - i n f i n i t e plasmas; Plasma 1 a t p o t e n t i a l c$ = 0

,

and Plasma 2 a t c$ = c$

>

0

.

Four p o p u l a t i o n s of p a r t i c l e s a r e assumed ? s e e Fig. 1 ) : e l e c t r o n s , t r a n s m i t t e d through t h e double- l a y e r from Plasma

I,

i o n s t r a n s m i t t e d from Plasma 2, and i o n s i n Plasma 1 and e l e c t r o n s i n Plasma 2 t h a t a r e r e f l e c t e d by t h e double-layer. The t r a n s m i t t e d p a r t i c l e s c o n s t i t u t i n g t h e plasma c u r r e n t d r i f t toward t h e double-layer, and a r e a c c e l e r a t e d a d i a - b a t i c a l l y ; t h e r e f l e c t e d p a r t i c l e s a r e r e f l e c t e d i s o t h e r m a l l y . We normalize charged p a r t i c l e d e n s i - t i e s t o t h e t r a n s m i t t e d e l e c t r o n d e n s i t y , and e n e r - g i e s t o t h e t r a n s m i t t e d e l e c t r o n d r i f t energy be- f o r e a c c e l e r a t i o n ,

where e i s t h e magnitude of t h e e l e c t r o n i c charge,

%

t h e mass of e l e c t r o n s o r i o n s

(a

= e , i ) ,

%B

i s t h e d e n s i t y ,

QB

t h e temperature, and vap t h e d r i f t v e l o c i t y i n Plasma 1 o r 2

(B

= 1,2).

The c h a r g e d e n s i t y , p

,

i s g i v e n by (2) below.

When Ti = 0

,

t h e f i r s t e x p o n e n t i a l i s 1 f o r @=0, and 0 f o r @

>

0. When 3 =0, t h e second exponen- t i a l i s 1 f o r @ A 0 , andeO f o r @

<

$O

.

We must s o l v e Poisson's e q u a t i o n w i t h p g i v e n by (2), and s a t i s f y t h e c o n d i t i o n s t h a t p and t h e e l e c t r i c f i e l d , E

,

v a n i s h a t @ = 0 and iP0

,

^and

t h a t @ should vary monotonically from 0 t o $0

.

These determine @

,

^and

^Ti

a s f u n c t i o n s

N, 9, ^T.3 a.8 3,

.

T y p i c a l s p a t i a l v a r i a t i o n s e, 1, e

a r e shown i n Fig. 1.

+PLASMA

1

PLASMA

2 - b

FIG. 1. SPATIAL VARIATIONS OF @ E, _p .(~=l.l, &=0.8,

-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979768

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Bounds on N, Q,

re,

T ~ , 3 and Ti g i v i n g p h y s i - c a l l y a d m i s s i b l e s o l u t i o n s cgn b e e s t a b l i s h e d : ( a ) when t h e r e f l e c t e d p a r t i c l e s a r e cold ( 5 , , ~ ~ = 0 ) , p i s d i s c o n t i n u o u s a t @=O,mO

,

and we o n l y r e q u i r e

$ Q > I > N , o r $ Q < ~ < N . ( 3 ) ( b ) When Te,Zi

#

0, p i s a continuous f u n c t i o n of

; f o r a monotonic p o t e n t i a l we r e q u i r e dp/d@

<

⁰

a t @=O,mO

.

^{For $0}⁺^m

,

t h e s e bounds s i m p l i f y t o Ti +

3re <

1 , Te + 3Ti

<

Q

,

( 4 ) which a r e modified Bohm c o n d i t i o n s f o r c o l l e c t i o n of e l e c t r o n s and ions, r e s p e c t i v e l y , through a s h e a t h . More r e s t r i c t i v e e x p r e s s i o n s o b t a i n f o r f i n i t e Go

.

S o l u t i o n s must s a t i s f y

qi , Te

²⁰

.

^These

c o n d i t i o n s a r e found t o be no more s t r i n g e n t t h a n t h o s e a l r e a d y c o n s i d e r e d . For a l l c o n d i t i o n s t o b e s a t i s f i e d , r e f l e c t e d p a r t i c l e s must b e p r e s e n t on b o t h s i d e s of t h e double-layer, a s we have assumed.

I LOW POTENTIAL I \ \

BOUNDARY

FIG. 2. CONDITIONS FOR DOUBLE-LAYER F i g u r e 2 shows t h e r e g i o n of N,Q space where double-layer s o l u t i o n s can b e found. The boundary f o r @O -. m

,

^$Q ⁼¹

,

^{i s}independent of t h e p a r t i c l e temperatures. The low p o t e n t i a l boundary moves t o N=l f o r c o l d p a r t i c l e s . The p o t e n t i a l v a r i e s along t h e low p o t e n t i a l boundary, b u t i s n e v e r l e s s t h a n t h e r e f l e c t e d p a r t i c l e temperature.

I n g e n e r a l , P o i s s o n ' s e q u a t i o n must b e solved n u m e r i c a l l y t o f i n d

m O ,

b u t f o r a l l p a r t i c l e s c o l d we o b t a i n

I n Fig. 3 ( a ) t h e t e m p e r a t u r e s a r e i n d i v i d u a l l y i n c r e a s e d from zero. The c u r v e s f o r

're

and TI i n c r e a s e t o t h e l i m i t s e t by ( 4 ) . For

ri

^-.^m

, m0 -

⁰

^.

^The^5, c u r v e d e c r e a s e s u n t i l no a d m i s s i b l e s o l u t i o n can b e found. For N

<

1

,

r e v e r s i n g t h e r o l e s of e l e c t r o n s and i o n s g i v e s t h e same q u a l i t a - t i v e v a r i a t i o n s .

-- .

LENGTH OF DOUBm-uYER

When t h e r e f l e c t e d p a r t i c l e s a r e n o t cold, t h e y p e n e t r a t e t h e double-layer, and e l e c t r i c a l n e u t r a l i - t y o b t a i n s only a t Z = f m. A s a c r i t e r i o n of t h e l e n g t h o v e r which most of t h e p o t e n t i a l s t e p o c c u r s ; we d e f i n e L ( s e e Fig. 1 ) a s t h a t d i s t a n c e o v e r which t h e e l e c t r i c f i e l d e v a l u a t e d a t GO/2 would have t o extend t o produce a s t e p of iPo

,

L = ( P ~ / I E ( @ ~ / ~ ) I , (6) F i g u r e 3(b) shows t h e t e m p e r a t u r e v a r i a t i o n of L.

DISCUSSION

Our p r e d i c t i o n s may b e compared w i t h c o n d i t i o n s

c h a r a c t e r i s t i c of l a b o r a t o r y and space plasmas.

Rough e s t i m a t e s f o r a double-layer i n a dquble- .p,lasma d e v i c e a r e [4] : N

=

1.2, Q

=

^0.5,

re =

^0.2,

r i w O . l = T e , T i - 0 . 3 Hence, 1 0 = 4 . 9 , L w 5 . 6 . For a d e n s i t y of 1 0 ~ c m - ~ , and e l e c t r o n s t r e a m i n g energy of 1 eV i n Plasma 1, a s t e p of 2 0 V, 0.8 cm long is p r e d i c t e d , compared with

3-15

V and

3-5

^cm

observed.

To compare our t h e o r y with c o n d i t i o n s d u r i n g an aurora, We assume a c u r r e n t of 1 p ~ / m 2 c a r r i e d by a 1 0 0 eV e l e c t r o n beam above t h e double-layer.

For N =

1.5,

Q = 0.4, 7, = Ti = Te = T i = 0.1, we p r e d i c t iP0 = 10.3, L = 7.4, i . e a 2 kV s t e p i n 0 . 7 km. The average e l e c t r i c f i e l d is f i v e t i m e s g r e a t e r t h a n r e p o r t e d

[5],

b u t t h e measurements may not have b'een made i n t h e c e n t e r of t h e double-

l a y e r . I f t h e c u r r e n t above t h e double-layer i s c a r r i e d by 1 keV e l e c t r o n s , t h e s t e p i n c r e a s e s t o 2 0 kV, w h i l e t h e e l e c t r i c f i e l d o n l y i n c r e a s e s t o 5.0 V/m. These p o t e n t i a l s a r e of t h e o r d e r of magnitude n e c e s s a r y t o account f o r o b s e r v a t i o n s of high-energy e l e c t r o n p r e c i p i t a t i o n

[3].

This work was s u p p o r t e d by t h e NSF and t h e NASA. Thanks a r e due t o Dr. D. B.

1 l i d

f o r many f r u i t f u l d i s c u s s i o n s .

REFERENCES

[I] ~ o r v g n , S. : Astrophysics and Space S c i e n c e L i b r a r y ( i n p r e s s ) .

[ 2 ] C a r l q v i s t , P. : Astrophysics and Space Science L i b r a r y ( i n p r e s s ) .

[

33

Shawhan, S.D., ~'a'lthammar, C.-G., and Block, L.P., J. Geophys. Res.

83,

1049 (1978).

[ 41 Quon, B. H., and Wong, A.Y., Phys. Rev. L e t t e r s

a 1393 (1976)

[

53

Mozer, F.S

.,

Carlson, C.W., Hudson, M. K., Torbert, R.B., Parady, B., Yatteau, J., and Kelley, M.C., Phys. Rev. L e t t e r s

38,

2% (1977).