T H E R M O S O L U T A L - C O N V E C T I V E I N S T A B I L I T Y OF A

C O M P O S I T E S T E L L A R A T M O S P H E R E 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

R. C. S H A R M A a n d V E E N A K U M A R I

Department of Mathematics, Himachal Pradesh University, Shimla, India

(Received 20 November, 1990)

Abstract. The thermosolutal-convective instability of a composite stellar atmosphere is considered in the presence of variable horizontal magnetic field and coUisional effects. The criteria for monotonic instability are obtained which generalize the criterion derived for thermal-convective instability in the absence of above effects.

1. Introduction

The convective instability (in which motions are driven by buoyancy forces) of a thermally unstable atmosphere has been termed as 'thermal-convective instability' by Defouw (1970). He has generalized the Schwarzschild criterion for convection to include departures from adiabatic motion and has shown that a thermally unstable atmosphere is also convectively unstable, irrespective of the atmospheric temperature gradient.

Defouw (1970) has established that the convective instability will set in if

1 D ' = - - ( L ~ - - p ~ L p ) + ~/r < 0 , (1)

c.

where L is the heat-loss function which depends only on its density and temperature; L T and Lp, partial derivatives of L with respect to temperature T and density p both evaluated in the equilibrium state. Cp is the specific heat at constant pressure; ~, the thermal diffusivity; c~, the coefficient of thermal expansion; and k, the wave number of the perturbation.

The effects of a uniform rotation and a uniform magnetic field on thermal-convective instability of a stellar atmosphere have been studied separately by Defouw (1970) and simultaneously by Bhatia (1971). In astrophysical situations concerning stellar atmos- pheres, it has been found that the above inequality is a sufficient condition for monotonic instability in both cases. Quite frequently it happens that the plasma is not fully ionized but, instead, may be permeated with neutral atoms. Str0mgren (1939) has reported that ionized hydrogen is limited to certain rather sharply bounded regions in space sur- rounding- for example, O-type stars and clusters of such stars - and that the gas outside these regions is essentially non-ionized. The medium has, therefore, been idealized as a composite mixture of a hydromagnetic (ionized) component and neutral component, the two interacting through mutual collisional (frictional) effects, by Hans (1968). The

Astrophysics and Space Science 184: 43-48, 1991. �9 1991 Kluwer Academic Publishers. Printed in Belgium.

44 R. C. SHARMA AND V. KUMARI

thermal hydromagnetic instability of a partially-ionized plasma, for incompressible and compressible cases, has been studied by Sharma (1976) and Sharma and Misra (1986).

Usually the magnetic field has a stabilizing effect on the instability. However, 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. In stellar atmospheres and interiors, the magnetic field may be variable and may altogether alter the nature of the instability. The Coriolis force also plays an important role on the stability of stellar atmospheres.

The conditions under which the convective motions are important in stellar atmos- pheres are generally far removed from the consideration of a single-component fluid and rigid boundaries. It is, therefore, desirable to consider a two-component fluid or one gas component acted on by a solute concentration gradient and free boundaries. In the case of a two-component fluid, buoyancy forces can arise not only from density differences due to variations in temperature but also due to variations in solute concentrations. The problem of the onset of thermal-convective instability is of great importance on account of its applications to atmospheric physics and astrophysics, especially in the study of ionosphere and outer layers &the solar atmosphere. Sharma and Sharma (1984) have studied the thermosolutal-convective instability in a stellar atmosphere.

In the present paper, we consider the thermosolutal-convective instability (the ther- mal-convective instability in the presence of stable solute gradient) of a composite stellar atmosphere in the presence of a variable horizontal magnetic field and collisional effects. The criteria for monotonic instability are obtained which generalize the criterion derived for thermal-convective instability in the absence of above effects.

2. Formulation of the Problem and Perturbation Equations

Consider an infinite horizontal composite layer consisting of finitely conducting hydro- magnetic incompressible fluid of density p and a neutral gas of density Pal, subjected to a stable solute concentration gradient and acted on by gravity force g (0, 0, - g) and a variable horizontal magnetic field H(Ho(z), 0, 0). This layer is heated from above such that a steady temperature gradient fl ( --- dT/dz) is maintained. The layer is soluted from below such that a steady solute concentration gradient fl' ( = - dC/dz) is maintained. If we regard the model under consideration we assume that both the ionized fluid and the neutral gas behave like continuum fluids and that the effects on the neutral component resulting from the presence of gravity and pressure are neglected. The magnetic field interacts with the ionized component only.

The first law of thermodynamics may be written as

dT K p dp C ~ - - = - L § 7 2 T + - - , (2)

dt p p2 dt

INSTABILITY OF A COMPOSITE STELLAR ATMOSPHERE 45

where C v, K, T, t, V 2, and p denote, respectively, the specific heat at constant volume, the thermal conductivity, the temperature, the time, the Laplacian operator, and the pressure. The second-order (viscous dissipation) term has been neglected from Equation (2). Then, following the Boussinesq approximation, the equations governing the motion of the mixture of the hydromagnetic fluid and a neutral gas and Maxwell's equations are:

- - + ( q . V ) q = - - - V p + g 1 + + vV2q + & Po

1 + - - [ (v x H) x HI + p~v~ (q,~ _ q) , (3)

4rCpo Po

aqa - v~(qd - q), (4) •t

V.q = 0, ( 5 )

8H - V x (q x H) + t/V2H, (6)

Ot

and

V. H = 0, (7)

0C - - + ( q . v ) c = ~ ' v 2 c (g ) Ot

p = p o [ 1 - c~(T- T o ) + ~ ' ( C - C o ) ] . (9)

The suffix zero refers to values at the reference level z = 0, p, p, C, q, ~', ~c', and v( = t~/Po) stand for pressure, density, solute concentration, velocity, coefficient of solvent expansion, the solute diffusivity, and the kinematic viscosity, respectively. % qu, and Vc denote the resistivity, the velocity of the neutral gas, and the collisional frequency between the two components of the composite medium.

Let bp, bp, h(hx, hy, hz), q(u, v, w), 0, and 7 denote the perturbations in pressure p, density p, magnetic field H, velocity, temperature, and solute concentration, respectively, so that the change in density bp, caused by the perturbations 0 and y in temperature and solute concentration is given by

tip = - Po( c~O - ~' 7)- (10)

Equations (3)-(8) on linearization then give

1

Oq~t - Pol 7~p + vV2q + 4~p--o - - [ ( 7 x h ) x H + ( V x H ) x h ] +

bp pdv~ + g -- + (q~/- q), (l i)

Po Po

4 6 R. C. SHARMA AND V. KUMARI

a q d _

~t - - - - ~(qd-- q), (12)

ah - V X ( q X H) + r/VZh, (13)

at

a7 -- fl'W + tCtV2~, (14) at

V. h = O, (15)

V.q = 0. (16)

Following Defouw (1970), the linearized perturbation form of Equation (2) is

ao 1 - - + - - (L r - p ~ L p ) O - x 7 2 0 = - fl + g W, (17) at c~

where Cp is the specific heat at constant pressure and the thermal diffusivity x ( = K/poCv) is assumed to be constant.

Here we consider the case in which both the boundaries are free as well as perfect conductors of both heat and solute concentration. The density changes arise principally from thermal effects. The case of two free boundaries is the most appropriate for stellar atmospheres (Spiegel, 1965).

The boundary conditions appropriate for the problem are

a 2 w w - - 0= e= o, (18) &2

= 0 and hx, hy, h z are continuous with an external vacuum field. ~ denotes the z-component of current density.

3. Dispersion Relation and-Discussion

If we analyze in terms of normal modes, we seek solutions whose dependence on space- and time-c0ordinates is of the form

sinkzz e x p ( i k x x + ikyy + n t ) , (19)

wheren is the growth rate, k = (k~ + k2y + k~ ) 1/2 is the wave number of the perturbation and k z = src/d, s being any integer and d is the thickness of the layer.

If we eliminate qd between Equations (t 1)and (12) and by use of expressions (10) and (19), we obtain

n ' q = - - - 7 b p + v V 2 q + - - ~h z + j H o + po 4~po -&z \ ax Oy 7)

t_ t \ & az/ ~ z 3 J - g ( ~ ~ ~'~), (20)

INSTABILITY OF A COMPOSITE STELLAR ATMOSPHERE 47

where

~~ I Pd (21) n' =n 1+ and % - n + vc/ Po

If we eliminate u, v, bp, hx, hy, h~, O, and 7 between Equations (13)-(17) and (20) and using expression (19), we obtain the dispersion relation

n 5+A4n 4 + A 3 n3+A2n 2 + A I n + A o = 0 , (22)

where

A 4 = D' + (v + x' + ~)k 2 + Vc(1 + 0~o) ,

A 3 = vc(1 + %) {(t/+ /r 2 + D'} + (t/+ /r ' + vk 2) + vvck 2 +

+ r l#k 4+ vkZD ' + F' fl+ g : + F"fl' +k~V~ 1 lc2. ]

A2 = vc(l + O~o) {(rl + re')+ tc'rlk2}k2 + {vk2D' + F ' ( f l + ~ ) } •

• {vc + (tl + K')k 2} + x'qk4(D ' + vk 2) + VVck4(t '] + 1r

kVJ kV

+ D' r"fl ' + kx V~ k2J) ,

A1 = x' qk4{D' v~( 1 + ~0) + vv~k 2} + k2(v~ + D') x

x F " / ; ' r / + , ~ , o ~ . A 1 - + kV)

+ ~c'tlk4{vk2D ' + F' ( f i + ~ ) } + vc(,+ ~ ' )+

+D" r" f l +k~V~ 1 - v~,

Ao= v ~ q k 2 [ ( v # k 4 + F " f l ' ) D ' + F' ( f l+ ~ ) tc'k2] +

+ vci 'k 2 1 - ,

48 R. C. SHARMA AND V. KUMARI

and

r ' = go~(k~ + k2y) F" - g~ + k2) k 2 ' k 2

and VA 2 = Hg/4rcpo denotes the square of the Alfv6n velocity. When

D ' < 0 and I ( v ~ c ' k 4 + F ' ~ ' ) D ' l > F ' ( f l + ~ p ) ~ C ' k 2, (24)

the constant term in (22) is negative. Equation (22), therefore, involves one change of sign and, hence, contains one positive real root. The occurrence of positive root implies monotonic instability. We thus obtain the criteria for the thermosolutal-convective instability of a composite stellar atmosphere in the presence of variable horizontal magnetic field and collisional effect to be unstable if

D ' < 0 and ( w ' k 4 + F " f l ' ) D ' [ > F ' ( f i + ~ p ) t c ' k 2.

The Schwarzschild criterion is applicable in the interior of a star, where the photon mean free path is small, the assumption that the motion is adiabatic is justified. The departure from adiabatic motion may be significant in the outer layers of a stellar atmosphere, where the effective heat transfer is no longer prevented by opacity. The Schwarzschild criterion for convection has been generalized to include departures from adiabatic motion by Defouw (1970). The stellar chromospheres, coronae, and the interstellar medium may exhibit thermal-convective instability.

For astrophysical situations described above, the solute gradient (e.g., helium), the variable magnetic field, and the collisional effects play an important role. The thermal- convective instability of a composite stellar atmosphere including these effects has been considered. The criteria for monotonic instability have been obtained which generalize the criterion derived for thermal-convective instability in the absence of above effects.

References

Bhatia, P. K.: 1971, Publ. Astron. Soc. Japan 23, 181. Chandrasekhar, S.: 1981, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York. Defouw, R. J.: 1970, Astrophys. J. 160, 659. Hans, H. K.: 1968, Nucl. Fusion 8, 89. Kent, A.: 1966, Phys. Fluids 9, 1286. Sharma, R. C.: 1976, Physica 81C, 199. Sharma, R. C. and Misra, J. N.: 1986, Z. NatuoCorsch. 41a, 729. Sharma, R. C. and Sharma, K. N.: 1984, Acta Phys. Hungarica 56, 47. Spiegel, E. A.: 1965, Astrophys. J. 141, 1068. Str6mgren, B.: 1939, Astrophys. J. 89, 526.