G R A V I T A T I O N A L I N S T A B I L I T Y OF A C O M P O S I T E AND

R O T A T I N G P L A S M A IN T H E P R E S E N C E OF A V A R I A B L E

M A G N E T I C F I E L D T H R O U G H A P O R O U S M E D I U M

R. C. S H A R M A and A. R A J P U T

Department of Mathematics, Himachal Pradesh University, Shim&, India

(Received 25 February, 1991)

Abstract. The gravitational instability of an infinite homogeneous self-gravitating plasma through porous medium is considered to include, separately, the effects due to rotation and collisions between ionized and neutral components. The dispersion relations are obtained in both cases. It is found that the gravitational instability of a composite and rotating plasma in the presence of a variable horizontal magnetic field through porous medium is determined by the Jeans's criterion.

I. Introduction

The gravitational instability of an infinite homogeneous self-gravitating medium has been discussed by Chandrasekhar (1981). The Jeans's criterion for instability has been found to remain unaffected by the presence, separately or simultaneously, of uniform rotation and uniform magnetic field. A partially-ionized plasma represents a state which often exists in the Universe and there are several situations in which the interaction between the ionized and neutral gas components becomes important in cosmic physics. Str0mgren (1939) has reported that ionized hydrogen is limited to certain rather sharply bounded regions in space surrounding - for example, O-type stars and clusters of such stars - and that the gas outside these regions is essentially non-ionized. Other examples of the existence of such situations are given by Alfv6n's (1954) theory of the origin of the planetary system, where a high ionization rate is suggested to appear from collisions between a plasma and a neutral gas cloud and by the absorption of plasma waves due to ion-neutral collisions such as in the solar photosphere and chromosphere and in cool interstellar clouds (Piddington, 1954; Lehnert, 1959). Lehnert (1972) has found that both ion viscosity and neutral gas friction have a stabilizing influence on cosmical plasma interacting with a neutral gas.

According to Hans (1968), the medium may be idealized as a composite mixture of a hydromagnetic (ionized) component and a neutral component, the two interacting through mutual collisional (frictional) effects. Hans (1968) and Bhatia (1970) have shown that the collisions have a stabilizing effect on the Rayleigh-Taylor instability while Rao and KaIra (1967) and Hans (1968) have found thaL the collisional effects are, in fact, destabilizing for a sufficiently large collision frequency for the Kelvin-Helmholtz configuration. Sharma (1976) and Sharma and Misra (1986) have studied the thermal hydromagnetic instability of a partially-ionized plasma for incompressible and corn pressible cases, respectively. The magnetic field, in general, has a stabilizing effect on

Astrophysics and Space Science 187: 105- i 11, 1992. �9 1992 Kluwer Academic Pubh'shers. Printed in Belgium.

106 R. C. SHARMA AND A. RAJPUT

instability. However, there are a few exceptions are there. For example, Kent (1966) has studied the effect of a horizontal magnetic field which varies in the vertical direction on the stability of parallel flows and has shown that the system is unstable under certain conditions, while in the absence of magnetic field the system is known to be stable. The gravitational instability of a rotating and partially-ionized plasma in the presence of variable magnetic field has been considered by Sharma and Singh (1988) through non-porous medium.

Generally, it is accepted that comets consist of a dusty 'snowball' of a mixture of frozen gases which in the process of their journey changes from solid to gas and vice

versa . The physical properties of comets, meteorites, and interplanetary dust strongly suggest importance of porosity in astrophysical context (McDonnel, 1978). In stellar interiors and atmospheres, the magnetic field may be (and quite often is), variable (and nonuniform) and may altogether alter the nature of the instability. More often than not, the plasma is not fully ionized and may be permeated with neutral atoms and so the collisional effects are important. The Coriolis force also plays an important role in astrophysical situations, the object of the present paper is to study, separately, the effects of collisions and rotation on the gravitational instability through porous medium in the presence of a variable horizontal magnetic field.

2. Composite Plasma and Variable Magnetic Field Through Porous Medium

Consider an infinite homogeneous composite plasma consisting of infinitely conducting ionized component of density p and neutral component of density Pa acted on by a variable horizontal magnetic field H(Ho(z), 0, 0) and gravity force g(0, 0, -g). We assume that both the ionized and neutral components behave like continuum fluids and that the effects on neutral component resulting from the presence of magnetic field, porosity, and the fields of gravity and pressure are neglected. This composite plasma layer is assumed to be flowing through an isotropic and homogeneous porous medium of porosity e and permeability k 1. Then the linearized hydromagnetic perturbation equations governing the motion of the composite plasma through porous medium are

e Ot

1 v 1 V ~ p + T b U - - - q+ [ (Vxh) x H + ( V x H ) x h ] +

P k l

+ PdV~ (qa-- q), (1) pe

~qd _

& - - - - v c ( q d - - q), (2)

e - - 6p= - p V ' q , (3) Ot

bp = c 2 b p , (4)

GRAVITATIONAL INSTABILITY OF A COMPOSITE ROTATING PLASMA 107

72~3U = -47zGbp, (5)

7" h = 0, (6)

#h 1 - V x (q x H ) , (7)

c~t e

where ~p, bp, bU, h(hx, by, hz), q(u, v, w), and qa denote, respectively, the perturbations in density p, pressure p, gravitational potential U, magnetic field H, velocity of ionized (hydromagnetic) component and velocity of neutral component. Pa, vc, Y (= Cp/C,), c ( = x/TP/P), and G stand for density of the neutral component, mutual collisional (frictional) frequency between the two components of the composite plasma, ratio of specific heats, velocity of sound in the medium, and gravitational constant, respectively.

If we follow the normal mode analysis, we assume that the perturbation quantities have the space- and time-dependence of the form

expi(kxx + kzz + at), (8)

where r is the growth rate of the perturbation and kx, k z are the wave numbers of the perturbation along the x- and z-axes.

Equations (1) and (7) with the help of (2)-(6) become

- - + ia 1+ + q - t h z - 7 - + ) H o ia + vc/

+ l c { ( ~ - 0 h x ) H 0 H @ ] = _ a V @ 2 4~GP~s #z J o - hx ?z ) J + T U - j ' (9)

ah ~ { Ow OHo~ av A 9w e - ~ H o - - + w . + ) H o - - + k H o - - , (10)

~t \ ~z ~z J ~x Ox

where ~t,J, ~ Ic denote the unit vectors in x-, y-, and z-directions, s ( = 6p/i) denotes the condensation in the medium and % = Pa/P. Letting k 2 = kx 2 + k ~ , ~,~j2 = c Z k 2 _ 4~zGp, V 2 = Hg/4rcp and by use of (10) and expression (8), Equation (9) becomes

[ ( vd) u, o] - a 3 + i a 2 v c 1 + % + + - - q +

kl

+ Vg(a - ivy) {~k~kzw + jk~v~ 2 + k(k~̂ + 6k~)w} =

a(ia + Vc)e V•ys. (11) k 2

The magnetic field and density are assumed to vary such that V 2 = Hg/4~zp is uniform

] 0 8 R. C. SHARMA AND A, RAJPUT

and the three scalar equations of (11) are

_ 0-3 + i r v c l + % + + 7

vv~e a l u + kxk~V2(o " - ivc) w = kl A

ik~f~2e0-(ia + v c)

k 2

- 0 - 3 + i a 2 v ~ l + c % + ~ / ~ - I v + k ~ g z ( a - i v ~ ) v = O ,

s , (12)

(13)

_ i f 3 + i0-2 V c 1 + O~ o + + W kl

+ (k~ + 6k~)V2(cr - iv~)w =

ikzn~e~r(i~r + vc)

k 2

The divergence of Equation (1 I) and use of (3) yields

[ ( v~) ivv~a 0-2 ~f~2a(i0-+v~)]s+ i 0 "4 + 0 -3 V c 1 + ~o + - kl

+ 2k~(k2x + 3kZ)V2(ia + v~)w = O.

The set of Equations (12)-(15) yields the dispersion relation

{ ( ( Vg~ VVcg" _ 0-3 + iff2 V c 1 + % + + 0-2 _0-2+i0- v c 1 +~o + + k~ )

v, }[ ( v;) + 0 - + k ~ V 2 ( 0 - - i v c ) - i a 6 - 2 vc l + 0%+ a s +

kl

(14)

(15)

i~ 2vvce ( ve)2~ + + f~2 + (k~ + 4k~)V 2 + v c 1 + c~ o + a 4 +

( k l k l ] )

{ ( + Vc(~ 2 + V2k2x + 6k]) + v c 1 + eo + + f2] + k kl

2 2 -- i v c Pc + V k ~ + 6 k a 3 l+c~ o+ (f~2j+ k 2 +

vv~g vv~ V 2 e 6k] + ' ~ - ~ 5k~) a e +- + n~ + k~ + - - (k~ + - kl \ kl k 2

2k~ V f~j v c - t k~ (nf + v 2 k~ + 6kz ~) + k 2 (k~ + 5k~) 0- +

�9 2 2 2 2 7 lkZx V ~'~j V c

+ k ~ (k~ + Sk~)J = o . (16)

GRAVITATIONAL INSTABILITY OF A COMPOSITE ROTATING PLASMA 109

The first factor of Equation (16) gives

i f = 0 ,

which is a neutrally stable mode. Setting icr = W, the second factor of Equation (16) yields

(17)

V~l) Y v c g W2+ v c l + % + W + = 0 . (18) kl

Equation (18) does not admit of a positive real root or a complex root whose real part is positive and so the system is stable. The third factor of Equation (16), on putting i~ = W, gives

( W 3 + v~l + c%+ + +kx zV W + k ~ V 2 v c = 0 , \ k~

(19)

which does not allow any root whose real part is positive and so the system is stable. The last factor of Equation (16), again putting ia = W, yields

v~) w ' ~2vvc~ W 6 + 2 vcl + % + + I . ~ - - + nJ2 +(k~+4kff )V 2+

+

+

( vcl + ~o+ + ~ j 2 + V k ~ + 6 + \ k 1 -

w v2 2 6k~) + x v~ v ~ l + % + (~q2+ k~+ , kl

( Vl~c/~ V2 6k~ 2) - - k l +nJ~+ k~+ +

k x tz2 i"~ 2 } _ oo, ( ~ + 5~z ~) w~ + k 2

v vie 2kZx V 2 f~} vc } U (f2~+ V 2k~+6k~ 2)+ k 2 - (k~+5k~) W+

k 2 rr2t'~2 2 X V t Z J V c

+ k 2 - (k~ + 5 k b = 0 , (20)

where ~)~ < 0, the constant term in Equation (20) is negative. The product of the roots must, therefore, be negative. This implies that at least one root of W ( = ia) is positive. The system is, therefore, unstable.

Thus the gravitational instability of an infinite homogeneous, self-gravitating, and

110 R.C. SHARMA AND A. RAJPUT

partially-ionized plasma through porous medium in the presence of a variable horizontal magnetic field is determined by Jeans's criterion.

3. R o t a t i o n and Var iab l e M a g n e t i c Fie ld T hr ough P o r o u s M e d i u m

Here we consider an infinite homogeneous, infinitely conducting and self-gravitating fluid layer through porous medium acted on by a uniform rotation f~(0, 0, fl) and a variable horizontal magnetic field H(Ho(z), 0, 0). The linearized hydromagnetic per- turbation equations are

1 Oq 1 7 b p + V 6 U v - _ _ q + Ot p k,

1 2 + - - [(7 x h) x H + (V x H) x h] + - (q x f~), (21)

4rip e

together with Equations (3)-(7). By use of expression (8), Equations (3)-(7) and (21) give

ive ) - a 2 + a U - 2if~aV + k~k~V2w ekxO2a - - s = 0 (22)

k 1 k 2 '

iv8 a) - a 2 + - - v + 2iflau + k2V2v = 0 , (23)

k,

and

ive ) - ,r 2 + ,r w + + 6k )V2w - - s = 0 (24)

k 1 k 2 '

i re a2 ~ 2 e ) s _ e - a 3 + - - + + 2ikxf~av 2kz(k ~ + 3k~)V2w = 0 (25) kl

Equations (22)-(25) yield the dispersion relation

W 7 + B 6 W 6 + B s W S + B a W 4 + B 3 W 3 + B 2 W 2 + B 1 W + B o = O , (26)

where we have put ia = W and

4 v g B 6 -

kl

6v282 B s - _ _ _ + flj 2 + 2(kx z + 3k~)V 2 + 4• 2 ,

I 4v2e2~ B4 = ~ve 6(k~ + 3kZ)V 2 + 3fl 2 + ~ j

GRAVITATIONAL INSTABILITY OF A COMPOSITE ROTATING PLASMA 111

( ~'282N} ( V292 ) B , = ~ v 2+ k 2 / \ ~ + n , ~ + v 2k~+6a:~ +

2v282 kx 2 V2f~j 2 (kx 2 + 5k 2) + - - k 2 k~

(n~ + 2V~k~ + 3~ 2) +

k;~'~j V 2 2 + 4 f l 2 \ k 2 + k 2 + k ~ + 6 k J '

B~ ~L ~ (k~+Sk~)+ v ~+ ~ /

2 2 6kZ)+kz~V2 f~f+ k~ + V2k~ +6k~ + • (f~f + v k~+

+4f~ 2 V 2 k ~ + 6 k ~ + k 2 ,]J'

(27)

u ) B, k~V2n~(k~+Sk~) ~ 2 _ _ + S O s +

= ~ k ~ k~ V + k~

2 2. 2~2 k~V v ~ (f]j2 + V 2 k~ + 6k~2),

4 4 2 Bo = (k~ + Ski) .

k 1 k:

When ~ < 0, Equation (26) has at least one positive root. This means that at least one value of ia is positive, meaning thereby instability. Jeans's criterion, therefore, deter- mines the gravitational instability.

References

Alfv6n, H.: 1954, The Origin of the Solar System, Clarendon Press, Oxford. Bhatia, P. K.: 1970, NucL Fusion 10, 383. Chandrasekhar, S.: 1981, Hydrodynamic and Hydromagnetic Stability, Dover PUN., New York. Hans, H. K.: 1968, Nucl. Fusion 8, 89. Kent, A.: 1966, Phys. Fluids 9, 1286. Lehnert, B.: 1959, Nuovo Cimento Suppl. AI3 (Ser. X), 59. Lehnert, B.: 1972, Roy. Inst. Technol. Stockholm Rep. Nos. TRITA-EEP-72-05, 06; and 'Proc. 5th

European Conf. on Controlled Fusion and Plasma Physics', Grenoble, France, August 21-25. McDonnel, J. W.: 1978, Cosmic Dust, John Wiley and Sons, Toronto, p. 330. Piddington, J. H.: 1954, Monthly Notices Roy. Astron. Soc. 14, 638. Rao, S. S. and Kalra, G. L.: 1967, Can. J. Phys. 45, 2779. Sharma, R. C.: 1976, Physica 81C, 199. Sharma, R. C. and Misra, J. N.: 1986, Z. Naturforsch. 41a, 729. Sharma, R. C. and Singh, B.: 1988, Astrophys. Space Sci. 143, 233. Str6mgren, B.: 1939, Astrophys. J. 89, 526.