HYDROMAGNETIC RAREFACTION WAVES

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Submitted on 1 Jan 1979

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HYDROMAGNETIC RAREFACTION WAVES

J. Allen, L. Wickens

To cite this version:

J. Allen, L. Wickens. HYDROMAGNETIC RAREFACTION WAVES. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-547-C7-548. �10.1051/jphyscol:19797264�. �jpa-00219250�

JOURNAL DE PHYSIQUE CoZZoque C7, suppldment au n07, Tome 40, JuiZZet 1979, page C7- 547

HYDROMAGNETlC RARff ACTION WAVES

J.E. Allen and L.M. Wickens.

Department o f Engineering Science, The University o f Oxford, Parks Road, Oxford, EngZand.

An analytic solution for t h e free expansion of v e l o c i t y i n t h e x d i r e c t i o n and a l l q u a n t i t i e s a collision-free strongly magnetised plasma is

presented. The solution obtained is of i n t e r e s t i n connection with s o l a r f l a r e s and l a s e r produced plasmas. It i s shown t h a t a wavefront develops which t r a v e l s a t twice t h e f a s t magneto - ^acoustic

speed,limiting t h e ion emission velocity t o t h i s value. This contrasts with t h e case of t h e f r e e

expansion of an unmagnetised plasma'') ,where t h e maximum ion emission v e l o c i t y i s determined by

a r e assumed t o vary i n t h e x d i r e c t i o n alone, For a s u f f i c i e n t l y strong magnetic f i e l d t h e magnetic f l u x i s frozen i n t o t h e plasma s o t h a t B/n = const. = Bo /no ( 3 ) a t a f l u i d p a r t i c l e . Since t h e plasma electrons a r e t i e d t o t h e f i e l d l i n e s thermal conduction from t h e source plasma i s strongly i n h i b i t e d and an adiabatic expansion may a l s o be assumed so t h a t

p/nY = const. ( 4)

The expansion has two degrees of freedom s o

other e f f e c t s . Y = 2 and eqn. 4. becomes

kT/n = const. = kT,/n,

The i n i t i a l conaitions f o r t h e problem are-a ( 5 )

Substituting equations 3. and 5. i n t o eqn. 1.

quasineutral cold-ion collision-free plasma s l a b

then gives strongly magnetised i n t h e z d i r e c t i o n by a

magnetic f i e l d IJ,. The source plasma electron

t

density and temperature a r e noand To respectively where "0 = ( k+

<>

'* ^L

(see f i g . 1 ) . speed and y,,; B*/(&W , ^:*C ( ~ z ~ T , / M ) ?

n p ~ w

;?

-.

XnO X ( ~ - 3 ) s ; v C & =

F i g w e 1. a-3 n a) ^{( 7 )}

Schematic of i n i t i a l conditions and

n 9 + ( v -5 = \ o

Neglecting e l e c t r o n i n e r t i a t h e ion momentum 6 9

no = n i = n ) For non zero solutions t o 7. and 8. we have ( 1 ) t h a t

a nd (v - 3 ) = v, ( 9 )

and from 8. and 9. we have

@ + b n v ) ^a 0

whe're n and Ze/M a r e t h e ion number density and (10)

charge t o mass' r a t i o r e s p e c t i v e l y ; ~ i s t h e ion I n t e g r a t i n g 10. from t h e undisturbed plasma out i n t o t h e r a r e f a c t i o n expansion gives

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797264

The important aspect of t h e s o l u t i o n 11. i s t h e existence of a wavefront a t 9 = 2yl.,beyond which no ions exist. F i a r e 2. shows a n o m a l i s e d p l o t of ion d e n s i t y ' i n t h e r a r e f a c t i o n expansion against x / ( v t ) . ^{A t any} s p e c i f i e d time fig.2. thus represents t h e ion d e n s i t y as a f m c t i o n of

normalised distance. The negative values of 5 ^{= x / t}

correspond t o a wavefront t r a v e l l i n g back i n t o t h e source plasma a t t h e f a s t magneto - a c o u s t i c speed.

From equations 9. and 11. we a l s o have v = ( 2 1 3 ) ~ d + 9ivh] (12) corresponding t o a l i n e a r increase i n i o n v e l o c i t y with distance a t a s p e c i f i e d time. A t t h e rare- f a c t i o n wavefrontfs2v,,so eqn. 12. then gives 2v,.as t h e maximum ion emission velocity.This i s i a marked contrast t o t h e case of a f r e e l y expanding plasma with no magnetic field,where no maximum i o n emission v e l o c i t y e x i s t s . I n reference

( 2 ) s i m i l a r f e a t u r e s a r e obtained numerically.

Finally we note t h a t t h e s o l u t i o n obtained i s analagous t o t h a t found i n c o l l i s i o n dominated gas dynamics (3)

.

Figure 2.

Ion density i n r a r e f a c t i o n expansion.

REFERENCES

( 1 ) . GUREVICH, A. V., PARIISKAYA, L. V. and PITAEVSKII, L. P., 1966, J.E.T.P., 22, 449.

ALLEN, J. E. and ANDREWS, J.G. ,1970, J. Plasma Phys. 4, 187.

CROW, 3 . E.,AUER, P.L. and ALLEX,J. E., 1975,J; Plasma Phys. 14,65.

WICKENS, L. M., ALLEN, J. E. and RUMSBY, P. T.

1978 ,Phys. Rev. Lett. 2, ^243.

WICKENS, L. M. and AtLEN,J.E.

3

.

( 2 ) . GUREVICH, A. V. ,PARIISKAYA,L.V. end PITAEVSKY L

.

( 3 ) . ZELDOVICH, I a . B. and .RAIZER - 'Physics of Shock Waves.. . ^'- Vol. 1. ,1966 ,Academic Press.

This work i s supported by t h e Rutherford Laboratory of t h e Science Research Counci1,UK.