Magnetohydrodynamic shock production and current sheet diffusion

MHD SHOCK PRODUCTION AND CURRENT SHEET DIFFUSION

T h e s i s by Alan L. Hoffman

Xn P a r t i a l F u l f i l l m e n t o f t h e Requirements For t h e Degree o f

Doctor o f Philosophy

C a l i f o r n i a I n s t i t u t e of Technology Pasadena, C a l i f o r n i a

1967

ACKNOWLEDGMENTS

The a u t h o r w i s h e s t o e x p r e s s h i s a p p r e c i a t i o n t o P r o f e s s o r H. W. Liepmann, h i s a d v i s o r , who f u r n i s h e d eon- t i n u i n g a d v i c e and encouragement t h r o u g h o u t t h e p e r i o d t h e r e s e a r c h was c a r r i e d o u t . I n a d d i t i o n , t h e h e l p o f P r o f e s - s o r G, B. Whitham i n o b t a i n i n g a p h y s i c a l i n s i g h t i n t o t h e problem i s g r a t e f u l l y acknowledged. The a s s i s t a n c e of h i s

w i f e Bonnie i n c o r r e c t i n g t h e m a n u s c r i p t and M r s . G e r a l d i n e K r e n t l e r , who t y p e d t h e m a n u s c r i p t i s d e e p l y a p p r e c i a t e d .

S i n c e r e t h a n k s a r e a l s o due t o J. A. S m i t h f o r h i s eon- s t r u c t i v e c r i t i c i s m o f t h e o r i g i n a l m a n u s c r i p t and h i s a i d i n b r i n g i n g t h e m a n u s c r i p t t o i t s f i n a l , more csmpre- h e n s i b l e form. The a u t h o r i s i n d e b t e d t o t h e N a t i o n a l

S c i e n c e F o u n d a t i o n f o r i t s f i n a n c i a l a s s i s t a n c e , and a l s o t o t h e O f f i c e o f Naval Research which s u p p o r t e d t h e

ABSTRACT

Current s h e e t s i n i n v e r s e pinch MHD shock t u b e s

e x h i b i t t h e s t r a n g e property of forming shocks i n t h e very r e a r of t h e s h e e t when a c c e l e r a t i n g heavy gases. When a c c e l e r a t i n g l i g h t g a s e s , shocks a r e formed f u r t h e r t o t h e f r o n t i n t h e s h e e t , b u t i n no c a s e do t h e shocks s e p a r a t e from t h e d r i v i n g c u r r e n t s h e e t . This " p i s t o n dragging shock" e f f e c t is explained on t h e b a s i s of a s i n g l e - f l u i d model w i t h v a r i a b l e c o n d u c t i v i t y , Shocks a r e shown t o always form w i t h i n c u r r e n t s h e e t s which move a t supersonic speeds w i t h r e s p e c t t o t h e d r i v e n g a s , The r e l e v a n t para- meters f o r determining t h e shock p o s i t i o n a r e t h e Mach number and t h e magnetic Reynolds number. Large magnetic Reynolds numbers and small Mach numbers enhance forward shock formation. !These c o n d i t i o n s a r e obtained i n l i g h t gases with high speeds of sound. S i m i l a r i t y methods a r e developed t o e s t i m a t e gas c o n d u c t i v i t i e s , e l e c t r o n tempera- t u r e s . and degrees of i o n i z a t i o n f o r t h e experiments which a r e conducted. In hydrogen t y p i c a l e l e c t r o n temperatures of 4 ev a r e produced by t h e ohmic h e a t i n g , b u t twice t h i s value i s shown necessary t o achieve s e p a r a t i o n a t t h e c u r r e n t s h e e t speeds of 2-3 c m / ~ s e c used. Higher c u r r e n t s h e e t speeds produce shocks i n t h e r e a r s f t h e c u r r e n t

s h e e t where s e p a r a t i o n can never occur. The c o r r e c t method

separation are given. The success of single-fluid methods in explaining plasma phenomena is especially notable, and these methods can be extended to other similar problems. Based on these methods, multiple-fluid and microscopic

PART TITLE Acknowledgments

A b s t r a c t

T a b l e o f C o n t e n t s L i s t o f F i g u r e s L i s t o f T a b l e s I . I n t r o d u c t i o n

PAGE

ii

iii

v

v i i

I I . Q u a l i t a t i v e Behavior o f D i f f u s i v e F o r c e F i e l d s Used t o A c c e l e r a t e Gases

2 . 1 E q u a t i o n s o f Motion With a Source Term 8

2.2 y = 3 Approximation 12

2 . 3 S o l u t i o n f o r a C o n c e n t r a t e d Force 1 7

111. E f f e c t s o f C u r r e n t S h e e t D i f f u s i o n

3 , l I n d u c t i o n E q u a t i o n s With a Magnetic ,

F i e l d 2 1

3 . 2 S i m i l a r i t y S o l u t i o n f o r F i n i t e

C o n d u c t i v i t y i n C y l i n d r i c a l Geometry 28 3.3 Numerical R e s u l t s

PV. Exper Pments

4 . 1 D e s c r i p t i o n o f Apparatus 43

4.2 P r e i o n i z a t i o n Survey 4 6

4 . 3 A x i a l Magnetic F i e l d s 51

4 . 4 NumerieaP Techniques

-

E s t i m a t i o n o f C o n d u c t i v i t y and Degree o f I o n i z a t i o n

4 . 5 Argon 64

TABLE OF CONTENTS ( c o n t . )

PART TITLE

4 . 7 Hydrogen

V. I n t e r p r e t a t i o n and Atomic Q u a n t i t i e s 5.1 E l e c t r o n C o l l i s i o n s and E l e c t r o n

Temperature

5 . 2 I o n and N e u t r a l C o l l i s i o n C r o s s S e c t i o n s

VP. Summary and ConcPusions Append i c e s

A. Full D e l t a F u n c t i o n Force S o l u t i o n B. C o n c e n t r a t e d Force Program

@, C y l i n d r i c a l S i m i l a r i t y S o l u t i o n Program R e f e r e n c e s

T a b l e s F i g u r e s

PAGE

LIST OF FIGURES

1. C h a r a c t e r i s t i c d i a g r a m s f o r a f l u i d d r i v e n by a d i s - t r i b u t e d f o r c e l o c a t e d i n t h e shaded r e g i o n . The r e s u l t a n t s h o c k s a r e shown by a heavy l i n e .

2 . D i s t u r b a n c e p r o f i l e s c o r r e s p o n d i n g t o t h e c h a r a c t e r i s - t i c diagrams f o r l a r g e t i m e s . The s t e a d y f l o w r e g i o n s a r e i n d i c a t e d .

3 . F o r c e diagram f o r t h e n o r m a l i z e d shock v e l o c i t y a s a f u n c t i o n o f t h e n o r m a l i z e d p i s t o n f o r c e f o r a p i s t o n which i s moving a t Mach 1 0 w i t h r e s p e c t t o t h e g a s u p s t r e a m o f t h e shock.

4, One d i m e n s i o n a l s i m p l i f i e d model o f a c u r r e n t s h e e t w i t h f i e l d and f l o w q u a n t i t i e s i n d i c a t e d i n b o t h

l a b o r a t o r y and c u r r e n t s h e e t c o o r d i n a t e s .

5. C u r r e n t and d e n s i t y d i s t r i b u t i o n s i n an i n v e r s e p i n c h a s o b t a i n e d from a c o n i c a l s o l u t i o n , a s a f u n c t i o n o f t i m e t * = ~ t / k = l / q , Small Mach number.

6. C u r r e n t and d e n s i t y d i s t r i b u t i o n s i n an i n v e r s e p i n c h a s o b t a i n e d from a c o n i c a l s o l u t i o n , as a f u n c t i o n of t i m e t * =

~ t - / f r =

l / q , Large Mach nurnber ,

7 , I n v e r s e p i n c h geometry i n c l u d i n g c i r c u i t p a r a m e t e r s . 8. S i m j l a r i t y s o l u t i o n showing t h e r e l a t i o n s h i p s between

J, B e t and B.

9 , Shock ahd c u r r e n t s h e e t t r a j e c t o r i e s f o r p r e i o n i z e d hydrogen.

10. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 500 y Hg BO KV.

l P . Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r p r e i o n i z e d argon. 500 p Hg 90 MV.

12. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 1 2 5 p H g l 0 K V .

13. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r p r e i o n i z e d a r g o n , 125 p Hg 1 0 KV.

L I S T OF FIGURES ( c o n t . )

Actual o s c i l l o s c o p e t r a c e s f o r argon a t 175 p and 84 KV. Time s c a l e 1 wsec/div. Upper t r a c e

A8,

1 v/div. Lower t r a c e p r e s s u r e , . 5 v/div.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 100 p Hg 7 KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 100 p Hg 8% KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 100 p Qg 10 KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 170 p Hg 10 KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 1 7 0 p H g 1 2 K V .

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 170 p Hg 14 KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r argon. 400 y Hg 14 KV.

Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r helium, Current s h e e t t r a j e c t o r i e s f o r Pow d e n s i t y helium.

5. Actual osciPPoscope t r a c e s f o r hydrogen a t 700 p, and 6 KV. Time s c a l e , 1 p,sec/div. Upper t r a c e Be.

1 v/div, Lower t r a c e p r e s s u r e , - 2 v/divo

26. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r hydrogen. 700 p Hg 6 KVe

2 7 . Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r hydrogen. 700 p Iig 7% KV.

28* Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r hydrogen. 700 p Hg 9 KV,

29. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r hydrogen. 700 p Hg 10% KV.

L I S T O F F I G U R E S ( c o n t . )

31. Shock and c u r r e n t s h e e t t r a j e c t o r i e s f o r hydrogen. 290 CL Hg 9 KV.

32. C u r r e n t s h e e t t r a j e c t o r i e s f o r low d e n s i t y hydrogen. 33. Shock and c u r r e n t s h e e t t r a j e c t o r i e s i n hydrogen w i t h

t h e main c a p a c i t o r bank c h a r g e d t o 7% KV and f i r e d 1 . 4 p s e c a f t e r t h e p r e i o n i z a t i o n bank which i s charged t o 6 W .

LIST OF TABLES

Table 1 . Summary of e l e v e n computer runs f o r t h e c o n i c a l s o l u t i o n t o an i n v e r s e p i n c h under v a r i o u s

o p e r a t i n g c o n d i t i o n s . N o t a t i o n i s shown i n

f i g u r e 5 .

Table 2 . Argon Experimental Data

Table 3 . Hydrogen Experimental Data

Table 4 . Equilibrium Conditions Behind a Shock i n Hydrogen

I . INTRODUCTION

I n r e c e n t y e a r s t h e u s e o f body f o r c e s t o produce and c o n t r o l t h e motion of f l u i d s h a s r e c e i v e d g r e a t a t t e n t i o n . The most u s e f u l body f o r c e i s of c o u r s e t h e

Jxll

f o r c e

produced by d i s c h a r g i n g a c u r r e n t through a conducting f l u i d i n t h e presence of a p e r p e n d i c u l a r magnetic f i e l d . For f l u i d s such a s l i q u i d m e t a l s t h e problems a r e w e l l

f o r m u l a t e d , though by no means e a s i l y s o l v e d , t o t h e e x t e n t t h a t c l a s s i c a l f l u i d mechanics t e c h n i q u e s may be employed, The a d d i t i o n a l complications caused by t h e body f o r c e and t h e e q u a t i o n s governing i t s behavior form t h e f i e l d of Magnetohydrodynamics, which, a s %he name i m p l i e s , i s

s t a n d a r d hydrodynamics i n t h e presence of magnetic f i e l d s ' which i n f l u e n c e t h e f l u i d flow,

The h i g h temperatures necessary t o produce conduc- t i v i t y i n compressible f l u i d s , such a s g a s e s , u n f o r t u n a t e l y produce a d d i t i o n a l e f f e c t s , and t h e f o r m u l a t i o n of a

k i n e t i c t h e o r y approach t o r a r e f i e d , low temperature g a s flows. The importance of c o r r e c t l y f o r m u l a t i n g t h e problem becomes e v i d e n t when one c o n s i d e r s t h e s i g n i f i c a n t s i m p l i -

f i c a t i o n s t h a t can b e achieved i f t h e plasma can i n any way b e considered a f l u i d , even a multi-component one, and t h e

powerful t e c h n i q u e s of f l u i d mechanics b e brought t o b e a r . I t i s t h e purpose of t h i s paper t o i n v e s t i g a t e a f i e l d of r e s e a r c h t h a t h a s s u f f e r e d from b o t h a Pack of f o r m u l a t i o n a s a problem g r e a t a b l e by macroscopic f l u i d e q u a t i o n s , and a Back of understanding of t h e c o r r e c t macroscopic b e h a v i o r . The demonstration of t h e a b i l i t y t o extend f l u i d mechanics t o new r e g i o n s s f i n t e r e s t , and t o e x p l a i n new phenomena, i s perhaps more important t h a n t h e a c t u a l e l u c i d a t i o n of t h i s p a r t i c u l a r problem.

I

The probgern t o b e i n v e s t i g a t e d i s t h a t o f a c c e l e r a t - i n g plasmas through t h e u s e o f l a r g e a r e d i s c h a r g e s which a r e a c t e d upon by t h e i r own magnetic f i e l d s , The geometries a r e many, and vary according t o t h e purposes of t h e a c c e l - e r a t i o n , which a r e t h r e e f o l d ; p r o p u l s i o n , f u s i o n , and shock p r o d u c t i o n . The f i r s t two do n o t depend e x p l i c i t l y on t h e plasma b e i n g a f l u i d P The t h i r d , however, i s dependent on t h e a p p r o p r i a t e n e s s of a f l u i d d e s c r i p t i o n s i n c e a conven- t i o n a l shock wave i s of n e c e s s i t y a f l u i d phenomenon in- v o l v i n g t h e h y p e r b o l i c n a t u r e of t h e macrsscopie f l u i d e q u a t i o n s of motion. MiD shock t u b e s , a s t h e s e shock

on t h e e f f e c t i v e n e s s of t h e a r c d i s c h a r g e , o r c u r r e n t s h e e t , on a c t i n g a s a s o l i d p i s t o n and sweeping up a l l t h e gas i n

i t s p a t h , a s does a conventional shock tube p i s t o n . I f t h e c u r r e n t s h e e t does a c t a s an impermeable p i s t o n , then i t would be expected, by analogy t o t h e standard shock t u b e , t h a t a shock would be d r i v e n out i n f r o n t of i t , and t h e normal shock tube r e l a t i o n s would apply. However, it has been found t h a t i n a l l c a s e s where a shock does form, it does s o w i t h i n t h e c u r r e n t s h e e t and never s e p a r a t e s from

i t . This i s s o even though, based on t h e macroscopic equations of motion, t h e c u r r e n t s h e e t should be t h i n compared t o t h e d i s t a n c e a shock would be expected t o s e p a r a t e from a conventional s o l i d p i s t o n . Because of t h i s , and i n s p i t e of t h e f a c t t h a t a shock d i d form,

experimenters were quick t o claim t h a t t h e f l u i d equations d i d not apply, and t h e microscopic behavior m u s t be held accountable. This microscopic behavior, involving p a r t i c l e o r b i t s and endothermic c o l l i s i o n s , i s important and w i l l be considered i n a l a t e r c h a p t e r , b u t it c a n only be i n v e s t i - gated s e n s i b l y once t h e g r o s s behavior s f t h e v a r i o u s

d e v i c e s has been e s t a b l i s h e d . This i s a l s o t r u e f o r the propulsion a p p l i c a t i o n s , and t o some e x t e n t may even apply t o t h e f u s i o n devices. For t h e MHD shock t u b e s , t h e micro- s c o p i c behavior w i l l be considered a s a p e r t u r b a t i o n ,

MHD a c c e l e r a t o r s must b e d i s t i n g u i s h e d from t h e o l d e r a r c h e a t i n g d e v i c e s , s u c h a s t h e "TI' t u b e , which employ an ohmic h e a t i n g e n e r g y s o u r c e and a r e sometimes a i d e d by a

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d r i v i n g f o r c e p r o v i d e d by a b a c k s t r a p . The newer

-

p a r a l l e l p l a t e , c o a x i a l (MAST), and i n v e r s e p i n c h shock t u b e s make u s e of a g x g s o u r c e t e r m a s t h e p r i m a r y d r i v i n g mechanism. I n f a c t , i n t h e s e newer d e v i c e s t h e ohmic

h e a t i n g , which a c t s d i r e c t l y on t h e e l e c t r o n s , h a s s o l i t t l e e f f e c t on t h e a c c e l e r a t i o n o r h e a t i n g o f t h e heavy p a r t i c l e s , t h a t it c a n b e n e g l e c t e d . T h i s i s due t o t h e l a r g e mass d i f f e r e n c e between t h e e l e c t r o n s and t h e i o n s o r n e u t r a l s which p r e c l u d e s e n e r g y t r a n s f e r t o t h e h e a v i e r p a r t i c l e s i n t h e t y p i c a l l y s h o r t o p e r a t i n g t i m e s i n t h e s e d e v i c e s . S i n c e t h e e l e c t r o n s c a r r y t h e c u r r e n t , t h e f o r c e

a c t s d i r e c t l y on them, b u t t h e s h o r t Debye l e n g t h e f f e c t i v e - l y c o u p l e s them t o t h e i o n s i n t r a n s f e r r i n g momentum. The i o n s , i n t u r n , a r e coupled t o any n e u t r a l s p r e s e n t t h r o u g h e i t h e r a l a r g e e l a s t i c o r c h a r g e exchange c r o s s s e c t i o n . These a s s u m p t i o n s w i l l . b e used a s a f i r s t approximation t o j u s t i f y t h e u s e o f a macroscopic t r e a t m e n t o f t h e plasma f l o w and shock f o r m a t i o n phenomenon, b u t l a t e r , d e v i a t i o n s from t h e s e c o n d i t i o n s w i l l b e c o n s i d e r e d .

o n l y t h e shock i t s e l f , assuming e i t h e r a s i n g l e - f l u i d model w i t h g i v e n f i n i t e c o n d u c t i v i t i e s s u c h a s M a r s h a l l ' s ( R e f . 1)

o r more c o m p l i c a t e d m u l t i - f l u i d b e h a v i o r w i t h c h e m i c a l r e a c t i o n s t y p i f i e d by Gross

'

work (Ref. 2 ) , The second t y p e i s t h e snowplow, o r i n f i n i t e c o n d u c t i v i t y s i m i l a r i t y s o l u t i o n o f G r e i f i n g e r and Cole ( R e f . 3 ) which t r e a t s t h e problem a s a whole. However, t h i s method s t i l l d o e s n o t h i n g t o e l u c i d a t e t h e i n t e r a c t i o n between the driving f o r c e and t h e r e s u l t a n t shock, s i n c e t h e d r i v i n g f o r c e f i e l d i s

assumed t o b e i n f i n i t e l y t h i n . S i n c e t h i s f o r c e f i e l d shock i n t e r a c t i o n i s t h e primary c h a r a c t e r i s t i c of a l l MHD

shock t u b e s , it must b e accounted f o r i n any macroscopic f i r s t a p p r o x i m a t i o n o f t h e f l o w problem. The p a t t e r n t o b e foPlowed i n t h i s p a p e r w i l l b e a s e r i e s o f r e f i n e m e n t s on t h e s i m p l e s t model one c a n c o n s t r u c t which s t i l l e x h i b i t s t h e g r o s s b e h a v i o r o b s e r v e d i n e x p e r i m e n t s . The more

advanced r e f i n e m e n t s w i l l depend on t h e e x p e r i m e n t a l r e s u l t s t h e m s e l v e s f o r e v a l u a t i n g t h e m i c r o s c o p i c i n f l u e n c e s which a r e t o o c o m p l i c a t e d t o e s t i m a t e by t h e o r e t i c a l methods a l o n e , The v a l i d i t y o f t h e o r i g i n a l assumptions can t h e n b e

e v a l u a t e d t o d e t e r m i n e t h e c o n s i s t e n c y s f t h e method. In o r d e r t o i n v e s t i g a t e t h e d e t a i l s o f t h e shock f o r m a t i o n problem, a s i m p l e , one d i m e n s i o n a l , s t a b l e , and r e p e a t a b l e experiment i s d e s i r e d . The i n v e r s e p i n c h

over t h e pinch, p a r a l l e l p l a t e , and c o a x i a l geometries f o r t h i s p a r t i c u l a r problem. I t has t h e f u r t h e r advantage of y i e l d i n g a s i m i l a r i t y s o l u t i o n w i t h a c o n s t a n t speed shock

and p i s t o n f o r a l i n e a r l y r i s i n g d i s c h a r g e c u r r e n t , which can e a s i l y be produced experimentally. The s o - c a l l e d snow- plow speed derived from a simple momentum balance, assuming a l l t h e gas t o be swept i n t o a t h i n s h e e t , w i l l be t h e

c h a r a c t e r i s t i c v e l o c i t y of t h e problem. I n i t i a l experiments i n t h e i n v e r s e pinch y i e l d e d t h e somewhat s t a r t l i n g r e s u l t t h a t a shock was always l o c a t e d w i t h i n t h e c u r r e n t s h e e t , and f o r heavy q a s e s was i n f a c t l o c a t e d i n t h e very r e a r - most p o r t i o n of t h e s h e e t . This would correspond t o a

p i s t o n dragging a shock behind i t , and f o r t h i s reason s i m - p l e f l u i d mechanical arguments were considered inadequate t o e x p l a i n t h e a c t i o n of t h e device. Exotic models such a s a c u r r e n t c a r r y i n g e l e c t r o n s h e e t dragging an ion s h e e t were proposed, b u t were q u i c k l y r u l e d o u t from energy con- s i d e r a t i o n s . More p l a u s i b l e arguments c o n s i d e r i n g ion s l i p (Ref. 6 ) were proposed t o d e a l w i t h t h e growth of t h e c u r r e n t s h e e t , b u t they could i n no way account f o r t h e observed shock formation phenomena.

shock w i t h i n t h e c u r r e n t s h e e t w i l l be t r e a t e d p u r e l y from f l u i d mechanical arguments, which w i l l be shown adequate t o e x p l a i n t h e observed phenomena. These arguments w i l l a l s o b o t h q u a l i t a t i v e l y and q u a n t i t a t i v e l y account f o r t h e

11. QUALITATIVE BEHAVIOR OF DIFFUSIVE FORCE FIELDS USED TO ACCELERATE GASES

2 . 1 ~ q u a t i o n s o f Motion With a Source Term

The s i m p l e s t model f o r g a s a c c e l e r a t i o n by

2 x 2

body f o r c e s must a t l e a s t a c c o u n t f o r two t h i n g s ; t h e d i s t r i b - u t e d n a t u r e o f t h e f o r c e f i e l d and i t s p e r m e a b i l i t y . To o b t a i n an i n t k i t i v e f e e l i n g f o r t h e p o s s i b l e phenomena, an i d e a l , i n v i s c i d , and non-heat c o n d u c t i n g f l u i d w i l l b e con- s i d e r e d s u b j e c t t o a momentum s o u r c e t e r m i n i t s e q u a t i o n s o f motion. A l l shock p r o d u c i n g d e v i c e s r e q u i r e some such s o u r c e t e r m t o produce g r a d i e n t s i n t h e f l o w v a r i a b l e s , which w i l l s t e e p e n i n t o a narrow r e g i o n o f h i g h g r a d i e n t s due t o t h e h y p e r b o l i c n a t u r e o f t h e e q u a t i o n s . The c l a s s i - c a l shock t u b e p i s t o n c a n b e t h o u g h t o f a s e i t h e r a s t e p f u n c t i o n s o u r c e o f mass o r a d e l t a f u n c t i o n of a p p l i e d f o r c e . Exploding w i r e s , l a s e r s , and MID a c c e l e r a t o r s make u s e s f mass, e n e r g y , and momentum r e s p e c t i v e l y a s t h e

p r i m a r y s o u r c e t e r m s t o produce shock waves. The momentum s o u r c e t e r m c o n s i d e r e d i n t h i s s i m p l e c a s e w i l l b e one d i m e n s i o n a l , and o f c o n s t a n t b u t a r b i t r a r y s h a p e .

R e f e r e n c e s w i l l . b e made t o t h e

- -

JxB body f o r c e i n a con- d u c t i n g plasma, b u t t h e f o l l o w i n g d i s c u s s i o n a p p l i e s

e q u a l l y w e l l t o any p o s s i b l e body f o r c e i n an i d e a l f l u i d . S e c t i o n 111 w i l l t a k e i n t o a c c o u n t t h e b e h a v i o r s f t h e

n a t u r e o f t h e problem, and which w i l l a l l o w a n a l y t i c s o l u t i o n s t o b e d e r i v e d t o h e l p v i s u a l i z e t h e shock t u b e b e h a v i o r .

For tfie c a s e o f a f o r c e f i e l d t r a v e l i n g t h r o u g h t h e f l u i d a t some c o n s t a n t speed U , t h e mass and qomentum

e q u a t i o n s a r e s i m p l e t o w r i t e . However, c a r e must b e t a k e n i n w r i t i n g t h e e n e r g y e q u a t i o n t o make it c o n s i s t e n t w i t h t h e assumption o f ohmic h e a t i n g h a v i n g no e f r e c t on t h e heavy p a r t i c l e s ( o r g r o s s f l u i d ) . I g n o r i n g v i s c o s i t y and t h e r m a l c o n d u c t i v i t y , t h e e q u a t i o n s o f motion c a n b e w r i t t e n a s f o l l o w s :

w i t h s t a n d a r d n o t a t i o n . The t o t a l r a t e o f work done by t h e e x t e r n a l f o r c e f i e l d i s Uf ( x

-

U t )

,

o f which u f ( x

-

U t ) g o e s d i r e c t l y i n t o a c c e l e r a t i n g t h e f l u i d . The r e m a i n d e r ,

(U

-

u ) f ( x

-

U t ) i s an e f f e c t i v e f r i c t i o n due t o t h e f o r c e f i e l d moving f a s t e r t h a n t h e f l u i d e T h i s mayP o r may n o t b e accounted f o r i n t h e h e a t i n p u t t e r m q, depending on whether t h e h e a t o f f r i c t i o n i s o r is n o t t r a n s f e r r e d t o t h e f l u i d . For t h e c a s e o f a

Jxg

f o r c e c a u s e d b y a moving c u r r e n t

which allows t h e c u r r e n t t o d i f f u s e through t h e gas. I t i s shown i n s e c t i o n I11 t h a t t h i s f r i c t i o n i s j u s t t h e ohmic d i s s i p a t i o n and should be ignored i n t h e flow equations f o r t h e heavy p a r t i c l e s . I n t h a t case t h e e l e c t r o n s a c t a s a s e p a r a t e h e a t absorbing medium. I n t h i s g e n e r a l d i s c u s s i o n t h e h e a t of f r i c t i o n is a l s o assumed not t o go i n t o t h e f l u i d under c o n s i d e r a t i o n and q is s e t equal t o zero, Thus, t h e energy equation reduces t o t h e simple form:

where s i s t h e entropy, A s long a s no shock i s formed t h e entropy remains c o n s t a n t .

Using t h e f a c t t h a t t h e entropy i s constant t o r e l a t e t h e d e n s i t y p t o t h e sound speed a, and t r a n s f e r r i n g t o a moving frame X = x

-

Ut, u" IJ

-

_{u ,} t h e mass and momentsum equations become:

The i n i t i a l v a l u e problem i s t h e important case t o consider where rp = qo, $ =

4,

a t t = 0 f o r a l l X. However, it i s u s e f u l t o look f i r s t f o r steady s o l u t i o n s s i n c e t h e i n i t i a l value problem w i l l c o n t a i n r e g i o n s of steady flow. Steady

s o l u t i o n s can only be w r i t t e n i m p l i c i t l y , and a r e most

e a s i l y expressed i n t h e p h y s i c a l v a r i a b l e s u' and a. Call- ing n = 2 / ( ~

-

I),

t h e steady s o l u t i o n s are:

These two e q u a t i o n s , p l u s t h e standard shock jump c o n d i t i o n s g i v e a f u l l d e s c r i p t i o n of t h e flow i n r e g i o n s where t r a n - s i e n t s from t h e i n i t i a l s t a r t i n g p r o c e s s a r e not important. To t r e a t t h e t r a n s i e n t s , which include t h e phenomenon of shock formation, t h e f u l l time dependent equations (1) and

( 2 ) must be used with a numerical scheme employed t o i n t e g r a t e along b o t h s e t s of c h a r a c t e r i s t i c s q and $ ,

2.2

Y

= 3 Approximation

For a g a s w i t h Y = 3 t h e c h a r a c t e r i s t i c e q u a t i o n s u n c o u p l e and o n l y (1) is n e c e s s a r y t o d e s c r i b e t h e f l o w i n t e r m s of

v ,

a l t h o u g h ( 2 ) i s of c o u r s e n e c e s s a r y t o o b t a i n t h e p h y s i c a l q u a n t i t i e s u @ and a . E q u a t i o n (1) c a n now b e

w r i t t e n :

and t h e c o r r e s p o n d i n g s t e a d y s o l u t i o n i s s e e n t o have two b r a n c h e s :

>8

f o r cpm

<o

( 6 ) .

X

where qm i s t h e v a l u e o f cp a t X = =. It i s immediately o b v i o u s t h a t t h e r e i s a c r i t i c a l t o t a l f o r c e H =

L

h(X)dX and t h a t t h e e q u a t i o n w i l l have d i f f e r e n t s o l u t i o n s depend-

2

i n g on whether H i s g r e a t e r o r less t h a n

%CL

I n t h e X, t c o o r d i n a t e s t h e f l u i d v e l o c i t y i s u ' , (cp = a

-

u ' ) , s o t h a t t h e f l o w is s u p e r s o n i c f o r cp < 0 and s u b s o n i c f o r cp > 8 .

2

For s u p e r s o n i c f l o w and H l e s s t h a n e q u a t i o n ( 6 ) must a p p l y , s i n c e t h e e f f e c t s s f t h e t r a n s i e n t s a r e swept down- s t r e a m . For s u b s o n i c f l o w o r H l a r g e t h i s i s no l o n g e r t h e c a s e and t h e e f f e c t s o f t r a n s i e n t s must b e c o n s i d e r e d ,

mere a r e t h u s t h r e e c a s e s o f i n t e r e s t f o r t h e i n i t i a l v a l u e problem with cp, = a

-

W t

0 1) Po > 0 : 2) cp, < 0 ,

2

( 5 ) can be w r i t t e n as:

dX along = cp

and t h e c h a r a c t e r i s t i c diagrams f o r t h e s e c a s e s a r e drawn i n f i g u r e 1 f o r a f i n i t e f o r c e region (shown shaded). The shapes of t h e r e s u l t a n t d i s t u r b a n c e s a r e shown i n f i g u r e 2

f o r some reasonably l a r g e time. The subsonic case (cpo > 0 ) i s t h e one which would be i n t u i t i v e l y expected with t h e f o r c e f i e l d causing a disturbance i n f r o n t of it which

immediately s t e e p e n s i n t o a shock. However, when t h e f o r c e f i e l d i s moving a t a supersonic speed (cpo < 0 ) . t h e shock, i f one occurs a t a l l , i s formed w i t h i n t h e f o r c e f i e l d and i t s p o s i t i o n depends on b o t h t h e s i z e s f t h e f o r c e f i e l d and t h e i n i t i a l s l o p e of t h e c h a r a c t e r i s t i c s ( v a l u e of p o ) .

I n f a c t , f o r t h e weak f i e l d of case 3 , it i s seen t h a t t h e s o l u t i o n c o n s i s t s only of an expansion and compression

r e g i o n , without a shock ever b e i n g formed, I t i s a notable r e s u l t t h a t t h e compression r e g i o n d o @ s n 8 t steepen, a s it would be expected t o when d i s s i p a t i v e mechanisms a r e not

included, The steady s o l u t i o n s a r e a p p l i c a b l e i n these r e g i o n s , which always include t h e region of applied f o r c e - Shock speeds can b e simply d e r i v e d from t h e p h y s i c a l

Although t h e c h a r a c t e r i s t i c diagrams were drawn f o r

Y

= 3 , t h i s s p e c i a l c h o i c e i n no way a l t e r s t h e b e h a v i o r o f t h e s o l u t i o n s . The d e c o u p l i n g o f t h e c h a r a c t e r i s t i c

e q u a t i o n s d o e s n o t i n f l u e n c e t h e f u n c t i o n o f t h e c h a r a c t e r - i s t i c s i n c a r r y i n g i n f o r m a t i o n , b u t it d o e s a l l o w one s e t t o b e drawn i n d e p e n d e n t o f t h e o t h e r . T h i s s h o u l d n o t b e c o n f u s e d w i t h a s i m p l e wave s o l u t i o n where o n l y one s e t o f c h a r a c t e r i s t i c s c a r r i e s i n f o r m a t i o n . The a p p r o x i m a t i o n

Y

= 3 c a n even b e j u s t i f i e d i n c e r t a i n c a s e s f o r r e a l g a s e s where one s e t o f c h a r a c t e r i s t i c s i s l e s s i n f l u e n c e d by t h e

a p p l i e d f o r c e s t h a n t h e o t h e r ( R e f , 5 )

.

One i m p o r t a n t r e s u l t t o n o t i c e which i s i n d e p e n d e n t of

Y

i s t h e p o s i t i o n o f t h e e x p a n s i o n wave f o r c a s e 2. The r e a r m o s t c h a r a c k e r i s t i c r e m a i n i n g i n t h e r e g i o n o f a p p l i e d f o r c e must b e v e r t i c a l , a s l o n g a s it i s o u t o f t h e i n f l u - e n c e o f t h e e x p a n s i o n i n t h e $ c h a r a c t e r i s t i c s . T h i s con- d i t i o n w i l l be d i s c u s s e d f u l l y i n t h e f o l l o w i n g s u b s e c t i o n . When it i s s a t i s f i e d , t h e c h a r a c t e r i s t i c d i r e c t i o n cp, which d o e s n ' t c o n t a i n Y , must b e z e r o a t t h e r e a r o f t h e f o r c e f i e l d . T h i s c o r r e s p o n d s t o t h e Chapman-Jouget c o n d i t i o n o f t h e f l o w b e i n g j u s t s o n i c b e h i n d an e x p l o s i v e l y produced shock wave, and i s n e c e s s a r y t o s o l v e f o r t h e complete f l o w f i e l d i n t h e l i m i t i n g c o n d i t i o n o f a c o n c e n t r a t e d f o r c e .

it i s d o u b t f u l i f a t r u e shock i s e v e r formed. I n t h e

i n v e r s e pinch experiments d e s c r i b e d i n s e c t i o n 4 , t h e shock p o s i t i o n was a c c u r a t e l y d e f i n e d by t h e u s e of a piezo-

e l e c t r i c p r e s s u r e probe. ( A shock always formed when t h e n e u t r a l - n e u t r a l mean f r e e p a t h was s m a l l compared t o t h e

c u r r e n t s h e e t dimensions.) From t h e simple model. p r e s e n t e d h e r e , t h e important parameters governing t h e shock formation p o s i t i o n can be a s c e r t a i n e d . Normalizing t h e force, and t h e v e l o c i t i e s ,

The c o n d i t i o n t h a t = 0 a t Xs. t h e shock formation p o s i t i o n , can be w r i t t e n :

where MF = u/ao i s t h e f o r c e f i e l d Mach number. A s MF

down t o s o n i c v e l o c i t i e s . This can a l s o b e seen from c a s e

2 of f i g u r e 1 where t h e i n i t i a l l y more h o r i z o n t a l c h a r a c t e r -

-

i s t i c s (cpo ^ I , MF + - ) t a k e l o n g e r t o b e t u r n e d around. A second important parameter i s obviously t h e v a l u e of H and a l s o t h e d i s t r i b u t i o n and width o f t h e f o r c e f i e l d

h ( X ) . I n t h e more d e t a i l e d numerical s o l u t i o n s , t h e p i s t o n w i l l be assumed impermeable and H w i l l b e w e l l d e f i n e d , b u t t h e width w i l l be a r b i t r a r y and w i l l be c h a r a c t e r i z e d by t h e magnetic Reynolds number which governs t h e d i f f u s i o n of t h e c u r r e n t s h e e t , The a c t u a l dependence on t h e Mach number w i l l a l s o become more pronounced. Before going i n t o t h i s

2 . 3 S o l u t i o n f o r a C o n c e n t r a t e d F o r c e

I n t h e l i m i t o f d e l t a f u n c t i o n a p p l i e d f o r c e s , t h e p r e v i o u s problem can b e e x t e n d e d t o g a s e s w i t h a r b i t r a r y

Y and t h e r e s t r i c t i o n o f a mass dependent body f o r c e can b e l i f t e d . Of c o u r s e , no i n f o r m a t i o n c a n b e o b t a i n e d a s t o shock f o r m a t i o n p o s i t i o n w i t h i n t h e f o r c e f i e l d . The u p p e r l i m i t on t h e a p p l i e d f o r c e i s j u s t t h e p i s t o n p r e s s u r e i n an o r d i n a r y shock t u b e i n which a l l t h e d r i v e n g a s i s

a c c e l e r a t e d t o t h e p i s t o n o r f o r c e f i e l d v e l o c i t y . F o r c e s P e s s t h a n t h i s v a l u e must c o r r e s p o n d t o l e a k a g e t h r o u g h t h e f o r c e f i e l d , A good a n a l o g y f o r t h i s c a s e i s t h a t o f

moving a s c r e e n t h r o u g h a g a s . Again, t h e d i f f e r e n c e i n s c r e e n and g a s v e l o c i t i e s c o n s t i t u t e s an e f f e c t i v e

f r i c t i o n , b u t t o b e c o n s i s t e n t , t h i s h e a t a d d i t i o n must b e i g n o r e d s i n c e o n l y t h e e l e c t r o n s a r e a f f e c t e d . I n f a c t , s o l v i n g t h e problem w i t h and w i t h o u t h e a t a d d i t i o n l e a d s t o a p p r o x i m a t e l y t h e same r e s u l t s , e x c e p t t h a t t h e g a s i n t h e e x p a n s i o n wave b e h i n d t h e p i s t o n i s h o t t e r when t h e f r i c t i o n t e r m i s i n c l u d e d . P h y s i c a l l y , t h e s e l e a k y p i s t o n s o l u t i o n s c o r r e s p o n d t o c a s e s where a l l t h e g a s h a s n o t been f u l l y swept u p , and t h e r e i s e v i d e n c e o f t h i s e x p a n s i o n wave i n t h e i n v e r s e p i n c h e x p e r i m e n t s .

t o t h e mass a c c u m u l a t i o n b e h i n d t h e s h o c k , b u t i n f r o n t o f t h e p i s t o n . Two c r i t e r i a can i n f l u e n c e t h e t e n d e n c y o f a shock t o s e p a r a t e from a d i f f u s e , n o n - s o l i d p i s t o n . They a r e mass a c c u m u l a t i o n ahead o f t h e s h o c k , and l e a k a g e t h r o u g h t h e p i s t o n , b o t h e f f e c t s b e i n g i m p o s s i b l e f o r a s o l i d p i s t o n . The f i r s t o f t h e s e w i l l b e i n v e s t i g a t e d i n c o n j u n c t i o n w i t h t h e d i s c u s s i o n on c u r r e n t s h e e t s p r e a d i n g i n s e c t i o n 111 b u t t h e second e f f e c t can b e d e a l t w i t h i n t h e p r e s e n t d i s c u s s i o n . The r a t e o f s e p a r a t i o n depends on t h e amount of l e a k a g e , and hence on t h e s i z e o f t h e t o t a l f o r c e , Only t h e p h y s i c a l l y i m p o r t a n t s u p e r s o n i c e a s e

( c a s e 2 ) w i t h shock f o r m a t i o n w i l l b e c o n s i d e r e d and it i s t h e r e f o r e o f i n t e r e s t t o d e t e r m i n e t h e minimum t o t a l f o r c e n e c e s s a r y t o form a shock ( r e l a t e d t o t h e o l d c o n d i t i o n

2 H > J i r p O L

When t h e minimum f o r c e F (where f ( x

-

U t ) = F6 ( x

-

U t ) i s exceeded, t h e s o l u t i o n w i l l c o n s i s t o f an o r d i n a r y shock moving f a s t e r t h a n t h e f o r c e f i e l d , a u n i f o r m r e g i o n between t h e two, and a n e x p a n s i o n r e g i o n b e h i n d t h e f o r c e f i e l d . The jump c o n d i t i o n s a c r o s s t h e f o r c e a r e d e r i v e d from t h e mass, momentum, and e n e r g y e q u a t i o n s w r i t t e n i n c o n a e r v a t i o n

-

2 2

-

-

I n f o r c e f i x e d c o o r d i n a t e s , u' = 1

-

u , t h e e q u a t i o n s become :

I n f i n d i n g t h e s o l u t i o n , a shock v e l o c i t y V i s chosen s u c h t h a t U 5 V I {-

* *

.U]+-i and t h e v a l u e o f

-

4 _M~

F needed t o produce it i s d e t e r m i n e d . The maximum v a l u e o f V c o r r e s p o n d s t o a s o l i d p i s t o n w i t h no l e a k a g e and t h e

c o r r e s p o n d i n g maximum f o r c e F w i l l b e j u s t t h e nsn-dimen- s i o n a l p i s t o n p r e s s u r e f o r an i d e a l shock t u b e . For a g i v e n f o r c e f i e l d Mach number and a g i v e n shock v e l o c i t y t h e c o n d i t i o n s b e h i n d t h e shock a r e known. However, t h e r e

-

-

a r e o n l y t h r e e jump c o n d i t i o n s t o s o l v e f o r p , u \ and b e h i n d t h e f o r c e f i e l d , p l u s t h e f o r c e

F,

One a d d i t i o n a l c o n d i t i o n i s r e q u i r e d , which i n a n o r d i n a r y shock t u b e i s p r o v i d e d by matching t o t h e downstream e x p a n s i o n wave i n t h e d r i v e r g a s . Here, t h e c o r r e c t c o n d i t i o n , f o r t h e c a s e when t h e f o r c e f i e l d i s moving f a s t e r t h a n t h e e s c a p e speed

2

of

-

Y - l "0' i s t h a t cp = u

+

a

-

U = 0 b e h i n d t h e f o r c e f i e l d -

t h e f u l l s o l u t i o n behind t h e f o r c e f i e l d i s found. For t h e p r e v i o u s y = 3 approximation, t h e escape speed i s j u s t ao,

and a l l s u p e r s o n i c f o r c e f i e l d s move f a s t e r t h a n t h i s speed, s o i t s e f f e c t i s ne;er n o t i c e d a s a lower l i m i t on t h e

simple s o l u t i o n .

The c ~ n d i t i o n behind t h e f o r c e can b e w r i t t e n :

Since

a

= a / ~ can b e r e l a t e d t o t h e o t h e r v a r i a b l e s , t h e problem i s completely determined. A simple program, shown

i n appendix B, g i v e s a p l o t of

F

as a f u n c t i o n of t h e shock speed = V/U f o r a given

Y

and MF = IJ/aoe

%

i s denoted by RMO i n t h e program and KK i s a c o n t r o l , The r e s u l t s a r e

111. CURRENT SHEET DIFFUSION 3 . 1 I n d u c t i o n E q u a t i o n w i t h a Magnetic F i e l d

The p r e v i o u s s e c t i o n h a s proceeded a s f a r a s p o s s i b l e b a s e d on a model w i t h an a r b i t r a r y and non-spreading f o r c e

f i e l d . Even w i t h t h i s model, a more d e t a i l e d knowledge o f t h e f o r c e f i e l d i s r e q u i r e d t o d e t e r m i n e t h e shock f o r m a t i o n p o s i t i o n a s d e f i n e d by e q u a t i o n ( 8 ) . A l s o , t o a c c o u n t f o r t h e l a c k o f s e p a r a t i o n , some knowledge o f t h e f o r c e f i e l d growth is r e q u i r e d s i n c e it h a s been shown t h a t l e a k a g e , a l o n e , c a n n o t a c c o u n t f o r t h i s f a i l u r e t o s e p a r a t e , Using t h e s i n g l e - f l u i d model, a form s f t h e i n d u c t i o n e q u a t i o n must b e i n c l u d e d i f one w i s h e s t o i n v e s t i g a t e t h e s p r e a d i n g o f t h e c u r r e n t s h e e t i n MHE) shock t u b e s . Howevpr, due t o t h e

l a r g e m a g n e t i c f i e l d s which a r e p r e s e n t , t h e e l e c t r o n c y c l o t r o n f r e q u e n c y w i l l b e l a r g e r t h a n t h e e l e c t r o n

c o l l i s i o n f r e q u e n c y , and it i s n o t immediately obvious what e f f e c t t h e c o n d u c t i v i t y o f t h e g a s w i l l have on t h e c u r r e n t d i f f u s i o n . For t h i s r e a s o n a b r i e f d e r i v a t i o n o f t h e

i n d u c t i o n e q u a t i o n w i l l b e p r e s e n t e d a t t h i s t i m e , a l o n g w i t h a d e r i v a t i o n o f t h e r e l a t i o n s h i p between ohmic h e a t i n g

and f o r c e f i e l d f r i c t i o n .

t h e f i e l d f o r c e s must j u s t b a l a n c e t h e c o l l i s i o n l o s s e s . T h e r e f o r e ,

where ne i s t h e e l e c t r o n number d e n s i t y . pe i s t h e e l e c t r o n p r e s s u r e , and

Gi

and

gen

a r e t h e momentum l o s s t e r m s f o r c o l l i s i o n s w i t h i o n s and n e u t r a l s . When t h e i o n and

n e u t r a l v e l o c i t i e s a r e s m a l l compared t o t h e e l e c t r o n v e l o c i t y ,

ITV,

i n t h e c u r r e n t c a r r y i n g d i r e c t i o n , t h e c o l l i s i o n t e r m s can b e w r i t t e n :

P =-n v m v -en e en e-e

v _{e i} and ven a r e t h e r e s p e c t i v e c o l l i s i o n f r e q u e n c i e s f o r t h e e l e c t r o n c o l l i s i a n s w i t h i o n s and n e u t r a l s . C a l l i n g

vT

=

v

+

v

and d e f i n i n g an e l e c t r i c a l r e s i s t i v i t y , e i e n

and c a l l i n g t h e c u r r e n t ;B = -n e v

,

e q u a t i o n ( 9 ) c a n b e e -e

w r i t t e n :

A A

C a l l i n g

2

= Bk, where k i s a u n i t v e c t o r i n t h e z d i r e c t i o n , and d e f i n i n g a c y c l o t r o n f r e q u e n c y w c = eB e q u a t i o n (11)

t a k e s t h e form:

W

C A I

-

-

u l x B

+

-

1 v p e l . ( 1 2 ) J + -

-

_v

_-

Jxk =

-

V T

-

-

T nee

T h i s somewhat a r b i t r a r y n o t a t i o n c a n b e used t o d e f i n e a t e n s o r c o n d u c t i v i t y s o t h a t :

W

, a t a k e s t h e form: C a l l i n g a? =

-

V T =

T h i s d e s c r i p t i o n i s u s e f u l when computing f i e l d q u a n t i t i e s b u t i s e x t r e m e l y c o n f u s i n g when d i s c u s s i n g d i s s i p a t i v e mechanisms s u c h a s ohmic h e a t i n g and d i f f u s i o n s i n c e t h e

i n c r e a s e d r e s i s t i v i t y q fiZ due t o t h e c y c l o t r o n f r e q u e n c y T

i s n o t d i s s i p a t i v e . I n d e r i v i n g t h e i n d u c t i o n e q u a t i o n , e q u a t i o n (11) w i l l b e u s e d r a t h e r t h a n e q u a t i o n s (13) and

( 1 4 )

-

I n l a b o r a t o r y c o o r d i n a t e s _e3 r e m a i n s unchanged and E =

E'

-

Wxg

.

Assuming t h a t t h e r e i s no c h a r g e s e p a r a t i o n

-

-

Using Maxwell's e q u a t i o n s ,

vxg =

-

and

ox2

= Cl& t

i g n o r i n g s m a l l e r t e r m s , and t a k i n g t h e c u r l of e q u a t i o n

( 1 5 ) # I

rl

_P

-

vx,,, = v x [ ~ v x E ]

+

ox(;;,

a t

(2x2

-

ope)

1

.

( 1 6 )

vo e

For t h e one d i m e n s i o n a l problem w i t h t h e d i r e c t i o n s d e f i n e d i n f i g u r e 4 , and v a r i a t i o n o n l y i n t h e x d i r e c t i o n

e q u a t i o n ( 1 6 ) t a k e s t h e s c a l a r form:

I t i s i m p o r t a n t t o n o t i c e t h a t f o r t h e c a s e when Jx = 0 t h e l a s t t e r m i n e q u a t i o n ( 1 6 ) , which g r e a t l y a f f e c t s t h e c o n d u c t i v i t y a s d e f i n e d by e q u a t i o n (13), h a s no e f f e c t o n t h e d i f f u s i o n o f t h e c u r r e n t s h e e t . I n f a c t S c h l u t e r

shows i n g e n e r a l ( R e f . 2 1 ) t h a t t h i s i s t r u e f o r a f u l l y i o n i z e d g a s where

-

Jxg

can b e w r i t t e n a s t h e g r a d i e n t o f t h e t o t a l p r e s s u r e , T h i s c a n a l s o b e e x t e n d e d t o p a r t i a l l y i o n i z e d g a s e s where t h e heavy p a r t i c l e s a r e a c c e l e r a t e d by g r a d i e n t s i n p r e s s u r e and e l e c t r i c p o t e n t i a l which a r e b a l a n c e d by t h e 2 x 3 f o r c e s Q i r r o t a t i o n a l f l o w )

.

I t was mentioned p r e v i o u s l y t h a t t h e h e a t i n p u t t e r m q in t h e e q u a t i o n s sf motion f o r a c u r r e n t s h e e t f o r c e f i e l d

t h e c u r r e n t s h e e t i n f i g u r e 4 . The t o t a l e n e r g y added t o t h e s t e a d y f l o w i s j u s t J O E ' . Using e q u a t i o n (11):

s i n c e

-

JxB

-

and Ope a r e p e r p e n d i c u l a r t o J. Thus a g a i n , t h e n o n - d i s s i p a t i v e p a r t of t h e r e s i s t i v i t y has no e % f e e t . For

2 a c o n d u c t i n g f l u i d

E'

w O and t h e ohmic h e a t i n g q T J i s a p p r o x i m a t e l y e q u a l t o t h e f r i c t i o n t e r m u'JB. For an

i n f i n i t e l y c o n d u c t i n g f l u i d , "qT = '0. t h e r e can b e no

r e l a t i v e v e l o c i t y u' between t h e f l u i d and f i e l d . For t h e u n s t e a d y c a s e :

2

J'E = qTS

+

U J B w U J B

.

- -

"qTJ2 i s e x a c t l y t h e f r i c t i o n h e a t i n p u t t e r m q which was i g n o r e d i n s e c t i o n P I .

Thus :

I t i s s e e n t h a t t h e H a l l p a r a m e t e r 0 c r e a t e s a s t r o n g e l e c t r i c f i e l d Ex and t h a t t h e e l e c t r o n c u r r e n t i s due t o ExB d r i f t . E q u a t i o n ( 2 0 ) c a n b e u s e d t o c a P c u l a t e t h e

- -

e l e c t r o n number d e n s i t y ne. S o r r e l l ( R e f . 6 ) e s s e n t i a l l y makes u s e of t h i s e q u a t i o n t o c a l c u l a t e t h e d e g r e e o f

i o n i z a t i o n , a l t h o u g h h i s r e a s o n i n g i s more p h y s i c a l . I f t h e e l e c t r o n s c a r r y t h e c u r r e n t J t h e n a f o r c e J B w i l l

Y ' Y

b e e x e r t e d on them, T h i s f o r c e w i l l b e t r a n s f e r r e d t o t h e i o n s t h r o u g h t h e e l e c t r i c f i e l d Ex. T h e r e f o r e

J

J Bdx =

y l n i e E x d x a c r o s s t h e c u r r e n t s h e e t where ni i s t h e i o n c o n c e n t r a t i o n and e q u a l t o ne. S i n c e t h e i o n s must

e v e n t u a l l y a c c e l e r a t e a l l t h e g a s by c o l l i s i o n s , t h e above e q u a t i o n c a n b e w r i t t e n i n t e r m s o f t h e v o l t a g e Vx a c r o s s t h e s h e e t ,

o r c a l l i n g a = ni/n f o r c o n s t a n t a ,

from gross momentum balance for any experiment. Sorrel1

uses equation (21) to great advantage to calculate average

degrees of ionization and confirms these measurements with

Stark broadening data where possible. His results will be

used in section IV to confirm some of the methods developed

there to estimate electron temperatures and degrees of

3 . 2 S i m i l a r i t y S o l u t i o n f o r F i n i t e C o n d u c t i v i t y i n C y l i n d r i c a l Geometry

The i n d u c t i o n e q u a t i o n ( 1 7 ) a l o n g w i t h t h e f l u i d e q u a t i o n s o f motion p r e v i o u s l y u s e d i n s e c t i o n I1 form t h e second P e v e l o f a p p r o x i m a t i o n t o b e i n v e s t i g a t e d . The most c r i t i c a l a d d i t i o n a l p a r a m e t e r i n t r o d u c e d is t h e e l e c t r i c a l c o n d u c t i v i t y u =

-

which g o v e r n s t h e t h i c k n e s s o f t h e

TT

c u r r e n t s h e e t , , The two o t h e r plasma t r a n s p o y t p a r a m e t e r s , t h e v i s c o s i t y and t h e t h e r m a l c o n d u c t i v i t y , a r e i g n o r e d , s o t h a t t h e m a t h e m a t i c a l model d e s c r i b e s a f o r c e f i e l d o f

f i n i t e d i m e n s i o n , b u t an i n f i n i t e l y t h i n shock d i s c o n t i - n u i t y . I n e f f e c t , t h i s a l l o w s t h e f l o w v a r i a b l e s p , u, and p t o have d i s c o n t i n u i t i e s , b u t r e q u i r e s t h e m a g n e t i c f i e l d B t o b e c o n t i n u o u s . The shock w i l l b e o r d i n a r y i n t h e g a s dynamic s e n s e , and n o t a s o - c a l l e d MHD shock. The f l o w e q u a t i o n s t h u s e x h i b i t l o c a l h y p e r b o l i c p r o p e r t i e s , b u t become b a s i c a l l y p a r a b o l i c due t o t h e d i f f u s i o n e q u a t i o n

f o r B. For t h i s r e a s o n tinusual p r o p e r t i e s must b e e x p e c t e d , s u c h a s t h e compression u p s t r e a m of t h e shock mentioned i n s e c t i o n H I .

heavy p a r t i c l e g a s temperature given by t h e flow e q u a t i o n s . There i s n o t enough time f o r t h e e l e c t r o n s t o e q u i l i b r a t e w i t h t h e heavy s p e c i e s . The e l e c t r o n temperature i s t h e n

o n l y a f u n c t i o n of t h e ohmic h e a t i n g , t h e i o n i z a t i o n l o s s e s , r a d i a t i o n , and many o t h e r e f f e c t s t h a t a r e t o o complicated t o t r e a t t h e o r e t i c a l l y . The method t o b e followed h e r e i s t o assume c o n d u c t i v i t y d i s t r i b u t i o n s and t h e n u s e t h e

experimental r e s u l t s t o e s t i m a t e t h e accuracy s f the

assumptions. I t i s found t h a t t h e g e n e r a l b e h a v i o r o f MED shock producing d e v i c e s i s n o t t o o dependent o n t h e e x a c t c o n d u c t i v i t y d i s t r i b u t i o n chosen, b u t r a t h e r on t h e average of t h e d i s t r i b u t i o n . I n some c a s e s , where t h e shock i t s e l f i n f l u e n c e s t h e c o n d u c t i v i t y , t h e d e v i a t i o n from t h e theo- r e t i c a l model i s obvious and t h e e f f e c t of t h e shock on t h e c o n d u c t i v i t y can b e determined. Thus, t h i s simple model even s e r v e s a s a powerful t o o l i n i n v e s t i g a t i n g e f f e c t s t h a t would seem t o l i m i t i t s v a l i d i t y . I t s main s i g n i f i c a n c e i s

t h a t it g i v e s one a s t a n 6 a r d w i t h which t o compare e x p e r i - mental r e s u l t s , t h a t h a s s o f a r been l a c k i n g i n problems such

I n o r d e r t o compare t h e t h e o r e t i c a l r e s u l t s w i t h t h e geometry o f t h e i n v e r s e p i n c h ( F i g . 7 ) , t h e e q u a t i o n s o f motion a r e w r i t t e n below i n c y l i n d r i c a l c o o r d i n a t e s . The mass, momentum, i n d u c t i o n , and e n t r o p y e q u a t i o n s are:

where B is t h e v a l u e of t h e m a g n e t i c f i e l d i n t h e 8 d i r e c t i o n . Except f o r t h e c y l i n d r i c a l geometry and t h e c o u p l e d d i f f u s i o n e q u a t i o n f o r t h e momentum s o u r c e t e r m , t h e e q u a t i o n s a r e i d e n t i c a l t o t h e o n e s used i n t h e g e n e r a l a n a l y s i s o f s e c t i o n I. Even f o r a s p e c i f i e d a , e q u a t i o n s

( 2 2 ) a r e e x t r e m e l y d i f f i c u l t t o s o l v e s i n c e t h e y c o n t a i n h y p e r b o l i c c h a r a c t e r i s t i c s , b u t are e s s e n t i a l l y p a r a b o l i c , The problem can b e s i m p l i f i e d by l o o k i n g f o r s i m i l a r i t y s o l u t i o n s . G r e i f i n g e r and Cole ( R e f . 3 ) have done t h i s f o r e s s e n t i a l l y t h e same e q u a t i o n s w i t h a = Assuming

i n c l u d e d an a x i a l m a g n e t i c f i e l d which r e m a i n s p r o p o r t i o n a l t o t h e d e n s i t y and d o e s n ' t c o m p l i c a t e t h e problem. With a = m , e q u a t i o n s ( 2 2 ) y i e l d s i m i l a r i t y s o l u t i o n s i n any

v a r i a b l e

5

a r / t n ' , n ' b e i n g d e t e r m i n e d by t h e boundary

c o n d i t i o n s . For a f i n i t e c o n d u c t i v i t y , s i m i l a r i t y c a n a l s o

be a c h i e v e d f o r any n ' , b u t c h o o s i n g n ' w i l l d e t e r m i n e t h e r a d i a l o r t i m e dependence o f 0 - The most n a t u r a l c h o i c e

f o r t h e c o n d u c t i v i t y would be a = c o n s t .

,

which c a n be

f i t t e d i n t o a s i m i l a r i t y s o l u t i o n o n l y f o r n g =

%,

( T h i s is due t o t h e p a r a b o l i c d i f f u s i o n sf t h e c u r r e n t s h e e t

A r a&.) However, t h i s t y p e o f s i m i l a r i t y c o r r e s p o n d s t o a

. -

c o n s t a n t d r i v i n g c u r r e n t , f o r , c h o o s i n g B i n t h e form =

(

5

and l o o k i n g a t t h e s o l u t i o n f o r a f i x e d l a r g e t i m e b e h i n d t h e c u r r e n t s h e e t , B must approach a P / r '

r B

dependence, Hence B 4

- -

a

-

.

A l l i n v e r s e p i n c h e s u s e

t ,2

r

5

a s i n u s o i d a l d r i v i n g c u r r e n t p r o v i d e d b y a s i m p l e c a p a c i t o r d i s c h a r g e i n t o an i n d u c t i v e c i r c u i t . T h i s can be approxi- mated by a l i n e a r c u r r e n t f o r t h e i n i t i a l p h a s e and r e q u i r e s

r 1

a

5

r / t dependence, y i e l d i n g B 4

- -

a:

-

'

b e h i n d t h e

t 5 2 r

c u r r e n t s h e e t , The c o n s t a n t c u r r e n t s o l u t i o n i s l e s s

i n t e r e s t i n g from a p r a c t i c a l v i e w p o i n t s i n c e t h e r e s u l t a n t shock waves would p r o p a g a t e a t non-constant s p e e d s . How- e v e r , t o f i t a s i m i l a r i t y . s o l u t i o n t o t h e r e a l i s t i c ,

B i n e a r l y i n c r e a s i n g d r i v i n g c u r r e n t , a o a B / r dependence

is r e q u i r e d .

a l i n e a r l y expanding c u r r e n t s h e e t due t o t h e c o n s t a n t l y expanding d i f f u s i v e l e n g t h s c a l e p r o p o r t i o n a l t o l / a . T h i s l i n e a r s p r e a d i n g i s o b s e r v e d e x p e r i m e n t a l l y , and can b e accounted f o r by two f a c t o r s . (1) I f t h e c u r r e n t s h e e t o r i g i n a l l y forms w i t h a f i n i t e w i d t h , t h e p a r a b o l i c s p r e a d -

i n g may n o t b e d i s t i n g u i s h a b l e from a l i n e a r growth o v e r t h e l e n g t h s c a l e s i n v o l v e d . Even i f t h e breakdown r e g i o n i s narrow, t h e s p r e a d i n g a t Barge radii w i l l s t i l l a p p e a r l i n e a r o v e r s m a l l d i s t a n c e s . ( 2 ) The c o n d u c t i v i t y depends on t h e e l e c t r o n t e m p e r a t u r e , which, i n t u r n , depends on t h e ohmic h e a t i n g . A s t h e c u r r e n t s h e e t d i f f u s e s , t h e c u r r e n t d e n s i t y d e c r e a s e s and t h e ohmic h e a t i n g i s r e d u c e d , y i e l d i n g an approximate l / r dependence i n t h e c o n d u c t i v i t y . T h i s can a l s o b e t h o u g p t o f a s a d e c r e a s e o f t h e d i r e c t e d e l e c t r o n v e l o c i t y n e c e s s a r y t o c a r r y t h e d e c r e a s i n g c u r r e n t d e n s i t y J, l i k e w i s e y i e l d i n g a P / r dependence i n t h e c o n d u c t i v i t y , Obviously t h e f i r s t f a c t o r i s i m p o r t a n t , b u t an o b s e r v a t i o n o f t h e e x p e r i m e n t a l c u r r e n t s h e e t growth, a s s e e n i n

f i g u r e s 1 4 , 24, and 31, shows t h a t t h e l i n e a r i t y c a n n o t b e f u l l y accounted f o r by t h i s i n a b i l i t y to d i s t i n g u i s h a p a r a b o l i c growth r a t e from a l i n e a r o n e o T h e r e f o r e , t h e

B a %/r r e l a t i o n h a s m e r i t o t h e r t h a n j u s t making t h e

s e c t i o n IV where t h e s i m i l a r i t y s o l u t i o n s t o be developed here w i l l be used t o i n t e r p r e t t h e experimental r e s u l t s and t o e s t i m a t e t h e conductivity. The extreme u s e f u l n e s s of t h e s i m i l a r i t y d e s c r i p t i o n w i l l then become e v i d e n t . S u f f i c e it t o say h e r e , t h a t it i s a good approximation f o r a t l e a s t p o r t i o n s of t h e flow f i e l d , and w i l l give quanti- t a t i v e r e s u l t s a s t o shock formation p o s i t i o n , b u t c a r e must

be exercised i n i n t e r p r e t i n g these r e s u l t s .

u r _r

O O

and a s i m i l a r i t y v a r i a b l e q =

-

Choosing u =

-

_r

U t where U i s some c h a r a c t e r i s t i c speed r e l a t e d to t h e f o r c e f i e l d speed used e a r l i e r , equations ( 2 2 ) can be w r i t t e n i n t h e following s i m p l i f i e d form:

The normalized v a r i a b l e s used are:

*

Also, Rm = p 0 r U i s assumed constant. b u t t h e equations

0 o 8

which have a special significance. Greifinger and Cole

expressed their simplified version of equations (22) in

alu

terms of a stream function Y defined by

-

= (p/po)rp br

a Y

- = 2

at

-

(p/po)urp and a variable @ = 2Y/r which they showed

to be equal to the expression in equation (24). Then,

is the ratio of the mass between q(or r) and the center,

divided by the mass originally there. Thus @ ( m ) = 1 and

@(q,) = 0 , where qc is the contact surface between gas and

vacuum at which

u

= q,. S is the converted pressure term

whose magnitude depends on the parameter K, which is given

by the initial conditions. Calling "a" the sound speed

JyP/p. S can be shown by simple substitution to be the

inverse sf the local Mach number squared:

Thus the characteristic Mach number MF is again found to be important, and the additional parameter governing the

might have been expected. The e x a c t Mach number dependence w i l l b e s e e n from numerical c a l c u l a t i o n s , b u t i t s

q u a l i t a t i v e e f f e c t can e a s i l y b e determined. Equations ( 2 3 )

a r e singular at S = 1, as ehould be expected when t h e

incoming flow i s d e c e l e r a t e d t o s o n i c v e l o c i t y i n c u r r e n t s h e e t f i x e d c o o r d i n a t e s . Allowance must be made f o r a d i s c o n t i n u i t y , which d i c t a t e s t h a t a shock must b e p r e s e n t . For a l a r g e v a l u e of X ( s m a l l &IF), S w i l l approach u n i t y q u i c k l y and t h e shock w i l l be l o c a t e d a t t h e f r o n t of t h e

c u r r e n t s h e e t . Again, t h i s i s t h e same s u p e r s o n i c choking n o t i c e d i n t h e f o r c e f i e l d model.

The only d i f f i c u l t y i n numerically i n t e g r a t i n g e q u a t i o n s ( 2 3 ) i s knowing how t o s t a r t . G r e i f i n g e r and Cole expressed t h e i r e q u a t i o n s i n terms of t h e independent v a r i a b l e @, p u t i n a shock a t cP = 1 , and s t a r t e d i n t e g r a t -

i n g a t t h e known c o n d i t i o n s behind t h e shock. I n t h e

p r e s e n t c a s e t h e shock p o s i t i o n i s unknown and must be found by an i t e r a t i v e procedure. I n terms of i , t h e e q u a t i o n s e x h i b i t a s a d d l e p o i n t behavior a t @ = 1 , and it i s impossi- b l e t o s t a r t t h e i n t e g r a t i o n procedure t h e r e . However, i n

A

-

Rmq terms of q , a l i n e a r i z e d s o l u t i o n (-fl ) = ~ e i s

T

a p p l i c a b l e when q + r n , and t h e i n t e g r a t i o n can proceed from

t h i s i n i t i a l p a t h . A governs t h e magnitude of t h e d r i v e c u r r e n t and w i l l b e r e l a t e d t o it through t h e a c t u a l