Contrib. Plasma Phys. 29 (1989) 6, 617-626

Gravitational Instability of a Rotating Viscous Thermally Conducting Plasma

VIJAY MEHTA and P . K. BHATIA

Department of Mathematics and Statistics, University of Jodhpur, Jodhpur (India)

Abstract

This paper deals with the gravitational instability of an infinite homogeneous viscous rotating plasma of finite electrical conductivity in the combined presence of effects of Hall currents, finite Larmor radius (FLR) and thermal conductivity. The ambient magnetic field is assumed to be uniform and acting along the vertical direction. Both longitudinal and transverse modes of wave propagation have been studied. It is shown that JEAN’S criterion determines the gravitational instability even in the presence of the effects of thermal conductivity, viscosity, finite electrical conductivity, FLR, rotation and Hall currents. Further it is found that while FLR, viscosity and rotation have a stabilizing influence, both the thermal and the electrical conductivities have a destabilizing influence on the gravitational instability of a plasma.

1. Introduction

The problem of gravitational instability of an infinite homogeneous medium was first considered by JEANS El]. He has given a criterion that an infinite homogeneous self- gravitating atmosphere is unstable for all wave numbers k less than JEANS’ wave num- ber

where e is density, S is velocity of sound in the gas and G is gravitational constant. A comprehensive account of the investigations of this problem, as carried out by

various authors under different conditions, has been given by CHANDRASEIZRAR [2]. He has shown that JEANS’ criterion remains unchanged by the separate or simultaneous inclusion of a uniform rotation and a uniform magnetic field. PACHOLCZYK and STODOL- K I E ~ C Z [3] have studied the effects of viscosity and finite electrical conductivity on the gravitational instability of an interstellar medium such as H I region. These investiga- tions have been carried out be using single fluid hydromagnetic equations which assume that the Larmor radius of charged particles are effectively zero and the corresponding Larmor frequencies are regarded as infinitely large. In many astrophysical situations such as interstellar and interplanetary plasmas, the approximations of zero Larmor radius and infinite Larmor frequency is not valid. Several authors namely ROSENBLUTH, KRALL and ROSTOKER [4], ROBERTS and TAYLOR [5] and JUKES [6] have pointed out the importance of the effects of finite Larmor radius (FLR) on plasma instabilities. BHATIA [7] has studied the gravitational instability to include the effects of uniform rotation, Hall currents, finite electrical conductivity and FLR.

5*

618 VIJAY MEHTA, BHATIA, P. K., Gravitational Instability

The gravitational instability of an anisotropic plasma has been considered by GLID- noN [8]. He has shown that the instability of such type of plasma is depended upon the magnetic field. The effect of rotation has been included by BHATIA [9] on this problem of gravitational instability of an anisotropic plasma and it has been shown there that the instability criterion is depended on rotation also.

The gravitational instability of anisotropic two component plasma has been analysed by HERRNEQQER [lo]. The problem has been extended by CHHONKAR and BEATIA [ll] to include the effects of FLR, magnetic resistivity and Hall currents.

In the absence of the effects of FLR and Hall currents, TASSOTJL [12] has studied the effects of thermal conductivity, viscosity, rotation and electrical conductivity on the gravitational instability of a homogeneous isotropic plasma. BHATIA and MLU~ESHWARI [ 131 have more recently investigated the effects of viscosity, FLR, finite electrical con- ductivity and Hall currents on the gravitational instability of an infinite homogeneous rotating plasma in the absence of effects of thermal conductivity.

It may, therefore, be of interest and is the object of the present paper to study the combined influence of the effects of viscosity, FLR, finite electrical conductivity, Hall currents and thermal conductivity on the gravitational instability of an infinitely extend- ing homogeneous rotating plasma.

2. Perturbation Equations

We consider the motion of an infinite homogeneous, viscous, uniformly rotating ther- mally conducting plasma of finite electrical conductivity in the presence of the effects of Hall currents and FLR. The prevailing magnetic field is taken to be uniform and acting along the vertical direction.

The relevant linearized perturbation equations are

a - Sp = -e(V * u) at (3)

a - (6p - YS26Q) = yxV2(Sp - S26,O) at (5)

whereu(u, v, w), h(h,, hy, hz), SP and 6~ denote the perturbations respectively in velocity, magnetic field €I, stress tensor P and density Q , Q(S,, Sg, 3) is rotation, d 0 is the per- turbation in the gravitational potential 0, ,LA is the coefficient of viscosity and 7 is the coefficient of magnetic resistivity, N is the electron number density and e is the charge of an electron. Also y denotes an adiabatic exponent and x is the coefficient of thermal conductivity. In equation (5), Sp is the perturbation in the scalar pressure p . For the uniform vertical magnetic field H(0, 0, H ) , the components of stress tensor Pii in the presence of FLR effects have been given by ROBERTS and TAYLOR [5]. Using these components and by analysing the disturbance in terms of normal modes in which the

Contrib. Plasma Phys. 29 (1989) 6 619

dependence on space coordinates x , z and time t is of the form

exp (ik,x + ik , + in t ) (6)

where k2 = kZ2 + kZ2 is the square of wave number and n (may be complex) is the fre- quency of the disturbance.

Eliminating 6e, 60, 6p and writing

S2k2(iny - yXk2) - Ge( in - yXk2) ( i n - yxk2)

$2 = (7)

wo obtain six equations which can be written in matrix form as

[A1 [BI = 0 (8)

where [ A ] is a sixth order square matrix and [B] is a single column matrix whose ele- ments are u, v, w, h,, h, and h,. The elements of [ A ] are

1 ik,2$2 i k a A,, = - A,, = i n + yo (k2 + 7 kZ2) - x,

Aq4 = i n + v k 2 , Ap6 = 0

k,k,H A55 = i n + q k 2 , A - -- Ne 56 -

kZ2H A54 = - Ne ’

where vo = ,u/e is coefficient of kinematic viscosity and v incorporates the FLR effect in these equations. The condition that (8) has non-trivial solution gives us the required dispersion relation. We obtain the dispersion relation for both the longitudinal and transverse modes of propagation.

620 VIJAY MEHTA, BHATIA, P. K., Gravitational Instability

3. Dispersion Relation

Mode I : Longitudinal propagation (k, = k). Case I: 9, = Qu = 0, QZ = 9. For this case we see that the dispersion relation splits into the three factors

in + vk2 = 0 (9)

W4 - 2W3k2(vo + q ) + W2[k4{~02 + q2 + 4~07 + L2) + 4(9 - yk2)2

+ 2k2Va2] - W[kP(k%ov + V,') (YO + v ) + 47k'(Q - ~k')' + ksv0L2] + k4[(v2 + L2) {4(9 - ~ k ~ ) ~ + k4yO2}

+ V,2(Va2 + (Q - vk2) L + 2k2?7Yo}] = 0 where

QJ2 = S2k2 - GQ, n = iW, L = H / N e , V: = H 2 / e , (12)

V , being Alfv6n velocity. Equation (9) corresponds to a viscous type of damped mode modified by finite electri-

cal conductivity. From equation (10) we note that when QJ2 < 0, there is always a negative real root of W . Consequently the plasma is unstable when this inequality holds. Equation (11) has either positive real roots or complex roots. The real roots correspond to stable modes. If the roots are complex, Re ( W) is always positive since the equation satisfied by Re ( W ) turns out to be one which has its coefficients alternately positive and negative implying thereby stability. Hence JEANS' criterion remains unchanged in this case.

In order to see the influence of effects of viscosity and thermal conductivity on the growth rate of the unstable mode, numerical calculations were performed to locate the roots of equation (10). For these calculations we take the numerical values of the physi- cal parameters, which correspond to the conditions in the galexies :

e = 1.7 x G = 6.658 x lo-" (kg)-1 m3 s - ~

S2 = 2.5 x lo8 m2 S-2

V,2 = 5.0 x lo8 m2 s - ~ .

kg m-3

The values of the critical wavelength for the gravitational instability of a homogeneous infinitely extending plasma are of the order of 1020 m. We have, therefore calculated the roots of the dispersion relation (10) for different values of the parameters vo and x char- acterizing respectively viscosity and thermal conductivity, taking multiples of as the values for the wave number. These calculations are presented in Fig. 1 where we have plotted the growth rate (negative real part of W after multiplying by 104) against wave number k (after multiplying by 1020) for vo = 1.0, 6.0, 11.0 (after multiplying by 104) and x = 6.0, 11.0, 16.0 (after multiplying by 104). In these calculations y = 5/3 is fixed. It is seen from Fig. 1 that the growth rate decreases for same k as the value of para- meters v,, increases. The growth rate also increases for same k as the value of parameter

Contrib. Plasma Phys. 29 (1989) 6 621

0.9

0. a

0.7

0.2

0.1

0 Wave number

Big. 1. Plot of growth rate (negative real part of W multiplied by lo4) against wave number k (multiplied by lozo) for different values of v,, (multiplied by 104) and x (multiplied by lo4) for the fixed values y = 5/3.

x increases. Thus the viscosity has stabilizing influence while thermal concluct,ivity has destabilizing influence on the growth rate of the gravitational instability of a homo- geneous rotating isotropic plasma. Case 11: Q, + 0, Q, + 0, Qz = 0. In this case the dispersion relation splits into two factors, given by (9) and

W7 - B,W6 + B5W5 - B4W4 + B3W3 - B2W2 + BXW - Bo = 0 (13) where

7 B, = 2k2 ( r j + 3 vo) - yxk2

20 11 3 3

r j 2 + L2 + 4~' + - v O ~ + - y o 2 ) + 2k2Va2 + 4QT2 + Qj'

- 2k2 (q + vo) yxk2

622 VIJAY MEHTA, BHATIA, P. K., Gravitational Instability

5 4 - {$v0k4 (7. + yo2 + 4rv0 + - 2 L2 + 4v2) + - 3 vo(2k2Va2 + 3QT2)

" 8 + vokP(Va2 + k2vyo) (V + vo) + 5 vo2k'L2 yxk2

B, = k4Qj2(V,4 + 2k27~oVO2 - k%Vo2L + k4(72 + L2) ( ~ 0 2 + 4~2)) 4 - - k4VO2qQT2 + 5 k6vo Voz( Voz + 27v0 - k2vL) + 2k4Va2(7 + vo) QJ

) 2 8

3

{i + 2 k ' ~ o ( 7 ~ f L2) (5 k"Vo2 + - k e y 2 f 2 8 ~ ' + Qj2

+ 2k'7(vo2 + 4v2) QJz y%k2 1 Bo = -k4Qj2{vo4 + 2 k 2 ~ ~ o V o 2 - k2VVa2L + k4(72 + L2) (YO' + 4 ~ ~ ) ) yxk2

and QT2 = Qz2 Qu2.

When QJz < 0, equation (13) has always one negative real root. Hence there is always an unstable mode of wave propagation when QJz < 0. However, when QJ2 > 0, equa- tion (13) has either all positive real roots or a positive real root and pairs of comples conjugate roots. The real roots will give stable modes. If the roots are complex the equa- tion satisfied by Re ( W) comes out to the one in which the coefficients are alternately positive and negative. The values of Re ( W ) are, therefore, all positive and this implies that the plasma is always stable when QJ2 > 0.

We may thus conclude that the gravitational instability of a uniformaly rotating conducting plasma is determined by Jeans' criterion even in the presence of the effects of FLR, viscosity, Hall currents and thermal conductivity.

Contrib. Plasma Phys. 29 (1989) 6 622

Mode 11: Transverse propagation (kz = 0, k, = k ) Case I: Q, = 0, Qv = 0, Qz = Q. For this case the dispersion relation can be expressed as the product

4 (in + qk2) (in + qvok2) [ W4 - W3k2 (F Y + 7 - yx)

+ W 2 k J 2 + k2Va2 + ( 2 0 - vk2)' -

- W{QJa + ( 2 9 - Vk')'} (k27 + y~k') - {Qj' + (2Q - Vk2)'} PYX (14)

The first factor of equation (14) correspond to a damped mode modified by magnetic resistivity. While the second factor corresponds to a damped mode modified by kine- matic viscosity. The third factor has always a negative real root if

QJ2 + (2Q - Yk')2 < 0. (15)

Consequently plasma is unstable when inequality (15) holds. If

Q j 2 + (2Q - yk')' > 0 , (16)

the third factor of (14) has either all positive real roots or all complex roots or real roots and pair of complex conjugates. The real roots correspond to stability. If the roots are complex the equation satisfied by Re ( W ) turns out to one having alternately positive and negative coefficients. The values of Re ( W ) are, therefore, all positive. Hence the plasma is stable in this case.

Inequality (15) depends only on effects of rotation and FLR and is independent of effects of thermal conductivity as well as of viscosity and electrical conductivity. In order to see the influence of the effects of rotation, finite electrical conductivity, FLR, viscosity and thermal conductivity on the growth rate of unstable mode, computations of negative real roots of third factor of equation (14) were performed for different values of the parameters. These calculations were also performed for the same numerical values of the physical parameters, corresponding to the galexies, as given above for eq. (10). The results are given in Figs. 2 and 3, where again the growth rate is plotted (after multi- plying by 1Oa) against wave number (after multiplying by 1020). Figs. 2 and 3 show that FLR viscosity and rotation have stabilizing influence while thermal conductivity and electrical conductivity have destabilizing character. Case 11: Qz =+ 0, Q, =+ 0, Q, = 0. In this case the dispersion relation splite into two factors given by (9) and

624 VIJAY MEHTA, BHATIA, P. K., Gravitational Instability

Wow@ number

Fig. 2. Plot of growth rate (negative real part of W multiplied by 104) against wave number k (multiplied by 1020) for different values of Y,,, Y and R (multiplied by 10') for the fixed values I ] = 1.0, x = 1.0 (multiplied by 10") and y = 5/3. The curves 1-9 correspond to values

(1) Yo = 1.0, Y = 1, n = 1.0

(3) Yo = 11.0, Y = 1.0, a = 1.0

(6) Y = 11.0, Y,, = 0.1, a = 1.0

(8) (2) yo = 6.0, Y = 1.0, f2 = 1.0

(4) Y = 1.0, Yo = 0.1, n = 1.0

(7) n = 2.0, Yo = 0.1, Y = 1.0

(9) a = 12.0, Yo = 0.1, Y = 1.0 = 7.0, yo = 0.1, Y = 1.0

(5) Y = 6.0, yo = 0.1, 0 = 1.0

Contrib. Plasma

Q i.. e s 3 e Q

625

Wave number 'Fig. 3. Plot of growth rate (negative real part of W multiplied by 104) against wave number k (multiplied by lozo) for different values of x and 71 (multiplied by lo4) for the fixed value of y = 5/3.

When QJ2 < 0, equation (17) has always a real negative root consequently the consid- ered plasma is unstable when QJ2 < 0. However when QJz > 0, the roots of equation (17) are either all positive real or all complex or real and pairs of complex conjugates. The real roots correspond to stable modes. If the roots are complex the equation satisfied by Re ( W ) again turns out to be one in which coefficients are alternately positive and negative. The values of Re ( W ) are therefore, all positive and the plasma is consequently stable for this case.

We thus conclude that the gravitational instability of an infinite homogeneous ther- mally conductingmedium is determined by Jeans' criterion. We also conclude that FLR viscosity and rotation have stabilizing influence while thermal and electrical conduc- tivities have destabilizing influence on the gravitational instability of the homogeneous infinitely extending isotropic plasma.

References

[l] JEANS, J. H., Phil. Trans. Roy. SOC. London A199 (1902) 1. [ 21 CHANDRASEKHAR, S., Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Ox-

[3] PACHOLCZYK, A. G., and STODOLKIEWICZ, J. S., Acta Astronomica 10 (1960) 1. [4] ROSENBLUTH, M. H., KRALL, N. A., and ROSTOKER, N., Nucl. fusion Supplement Pt. 1

[5] ROBERTS, K. V., and TAYLOR, J. B., Phys. Rev. Lett. 8 (1962) 197. [6] JURES, J. D., Phys. Fluids 7 (1964) 52. [7] BHATIA, P. K., Nuovo Cimento 59B (1969) 228.

ford (1960).

(1962) 143.

626 VIJAY MEHTA, BHATIA, P. K., Gravitational Instability

[8] GLIDDON, J. E. C., Astrophys. J. 146 (1966) 583. [9] BHATIA, P. K., Phys. Fluids 10 (1967) 1652.

[lo] HERRNEOQER, F., J. Plasma Phys. 8 (1972) 393. [ll] CFIHONRAR, R. P. S., and BHATIA, P. K., J. Plasma Phys. 18 (1977) 273. [12] TASSOUL, J., Bull. Acad. r. Belgique (Sciences) 49 (1963) 935. [13] BHATI.4, P. K., and MAHESHWARI, s. L., Contrib. Plasma Phys. 26 (1986) 111.

Received June 2, 1988; f ind manuscript received Deecmber 7,1988