FINITE LARMOR RADIUS EFFECT ON THERMAL INSTABILITY OF A COMPRESSIBLE PLASMA

R. C. SI-IARMA, K. C. SHARMA

Department o f Mathematics, Himachal Pradesh University, Simla - 171005, India

The thermal instability of a compressible plasma in the presence of a uniform vertical magnetic field is studied to include the effects of finiteness of the ion Larmor radius. When the instability sets in as stationary convection, both the compressibility and the finite Larmor radius are found to have stabilizing effect. The sufficient conditions for the nonexistence of overstability are investigated.

I. I N T R O D U C T I O N

A detailed account of the theoretical and experimental study of the onset of B6nard convection in incompressible fluids has been given by CHANDRASEKHAR [1]. The effects of the finiteness of the ion Larmor radius, which exhibits itself in the form of a magnetic viscosity in the fluid equations, have been studied by JUKES [2], VANDA- KUROV [3], ROBERTS a n d TAYLOR [4] and ROSENBLUTH, KRALL a n d ROSTOKER [5].

The stabilizing effect of finite Larmor radius on plasma instabilities has been demon- strated by the above authors. MELCHIOR and PoPowIc~ [6] have studied the finite Larmor radius effect on the Kelvin-Helmholtz instability of a fully ionized plasma while that on the Rayleigh-Taylor instability has been discussed by SINGH and HANS [7]. SHARMA [8] has studied the finite Larmor radius effect on the thermal instability of a plasma in the presence of a vertical magnetic field. The thermal instability in plasma in the presence of a horizontal magnetic field to include finite Larmor radius effects has been studied by SHARMA and PRAKASH [9]. In all the above studies, the use of Boussinesq approximation has been made throughout, which states that the density may be treated as a constant in all the terms in the equations of motion except the term in the external force. For the case of incompressible fluids, the Boussinesq approximation is. well justified.

When the fluids are compressible, the equations governing the system become quite complicated. To simplify the set of equations governing the flow of compres- sible fluids, SPIEGEL and VERONIS [10] have made the following assumptions:

(i) The depth of fluid layer is much smaller than the scale height as defined by them and

(it) the fluctuations in temperature, pressure and density, introduced due to motion, do not exceed their static variations.

Under the above assumptions, the flow equations are the same as those for in- compressible fluids except that the static temperature gradient is replaced by its excess over the adiabatic.

Czech. J. Phys. B28 [1978] 1101

R. C. Sharma et aI.: Finite Larmor radius ef fect . . .

The object of the present paper is to study the effect of finite Larmor radius on the thermal instability of a compressible plasma. This layer is heated from below such that a steady adverse temperature gradient/~'(= IdT/dzl) is maintained. The effects of compressibility and finite Larmor radius are more realistic in stellar atmospheres and astrophysical situations. These aspects form the subject matter of the present paper.

2. F O R M U L A T I O N OF THE P R O B L E M A N D P E R T U R B A T I O N EQUATIONS

Consider an infinite horizontal layer of compressible, viscous, heat-conducting and finitely conducting fluid of thickness d in which a uniform temperature gradient fi'(= - d T / d z ) is maintained. Let the origin be on the lower boundary z = 0 with the axis of z perpendicular to it along the vertical. The fluid is acted on by a vertical magnetic field H(0, 0, H) and gravity force g(0, 0, - 9 ) .

SVImEL and VzRONIS [10] expressed any space variable say Z, in the form

(1) )C = Xm + )~0(z) + Z'(X, y, z, t) ,

where X,. stands for the constant space distribution of )~, Zo is the variation in X in the absence of motion and Z'(x, y, z, t) stands for the fluctuations in X due to the motion of the fluid.

The initial state is one in which the velocity, temperature, pressure and density at any point in the fluid are respectively given by

(2) where according to

(3)

q = 0 , r = r ( z ) , p = p(z) , e = 0(z ) ,

SVIEG~L and VERON~S [10]

T(z) = --fl'z + To,

f, p(z) = Pro-- g (0,. + Oo) d z ,

o

0(z) = 0~[1 - ~ m ( r - r,.) + K,"(p - p , . ) ] ,

p,, and 0m stand for the constant space distribution of p and 0 and To, 0o stand for the temperature and density of the fluid at the lower boundary z = 0.

Let q(u, v, w), 0, 6P, h and 60 denote respectively the perturbations in velocity, temperature, stress tensor P, magnetic f ield/- /and density e; g/cv, #, v, x', ~, cp stand for the adiabatic gradient, the viscosity, the kinematic viscosity, the thermal conduc- tivity, the thermal diffusivity and the specific heat at constant pressure respectively. Then the linearized hydromagnetic perturbation equations, under the above-men- tioned approximations, are

1102 Czech. J. Phys. B 28 [1978]

(4)

(5)

(6)

(7)

(8)

where

and

R. C. Sharma et al.: Finite Larmor radius effect...

~3q#t ( 1 ) VSP - gc~O q- v Veq q- #e " (V X h) X H

V . q = O ,

__~h _- (H . V) q + rl V2h Ct

V . h = 0 ,

~0 ( ~ ' - ) + x V Z O , - - = #-- w &t cp

1 ~ , , = - - = ~ , say,

T=

v = I~/Q=,

X ~--- ~ r / Q m C p .

In writing equation (4), use has been made of the Boussinesq equation of state

(9) : 6Q = - a ~ = 0 ,

where ~ is the coefficient of thermal expansion. For the vertical magnetic field H(0, 0, H), the stress teflsor P, taking into account

the finite ion gyration radius (VANDAr~tn~OV [3]), are

( 1 0 ) e x x = p - eVo + , Px~ = P r~ = eVo ,

P~z =P~x = -2~Vo Ov + , P r y = P + QVo + ,

, �9

Here p is the scalar part of pressure and Qvo = NT/4o~n, where co~ is ion gyration frequency, while N and T denote, respectively, the number density and temperature of the ion.

3. DISPERSION RELATION

Analyzing the disturbances into normal modes, we assume that perturbation quantities are of the form

(11) [w, O, (, h=, ~] = [W(z), O(z),Z(z),K(z),X(z)]exp(ikxX + ikry + n o ,

Gzech. J. Phys. B 28 [197S] 1103

R. C. Sharma et al.: Finite Larmor radius effect...

where /r ky are the wavenumbers along the x and y directions respectively, k = = ( k ~ +..yjkzw 2 is the resultant wave number and n is the frequency of oscillation, which is, in general, a complex constant.

Using expression (11), equations (4 ) - (8) give

(12)

(13)

(14)

( 1 5 )

(~end~ (ff~dv 2) (D 2 - a2)(O 2 - a 2 - a) W + ~ , 4 n o m v / ( D z - a 2) D K - a20 -

= -

(16)

where we have written a =" kd , a = nd2 /v , P t = v / x , P2 = v/~, G = (cp/g) f l ' , x / d =

= x ' , y / d = y ' , z / d = z ' , and D = d / d z ' . ~ and r are, respectively, the vertical components of the vectors curl q and curl h given by

= ik~v - i k y U , ~ = ik~hy - i lcyh~.

Eliminating Z, X, K and O between equations (12) -(16) , we obtaifi

(17) [ (D 2 - a 2 ) ( D 2 - a 2 - 0-) { ( D 2 - a 2 - t r ) ( D 2 - a 2 - p 2 i f ) - - QD 2) - I..

- - Q ( D 2 - a2){(O 2 - a 2 - a ) (O 2 - a 2 - p2 a) - Q D 2 } ( D 2 - a 2 - p l a ) D 2 W +

+ R a 2 { ( V 2 - a z - a) ( D 2 - a 2 - p 2 a ) - Q D 2 } ( 1 ) 2 - a 2 - p2a) W = 0 ,

where R = g e f l ' d 4 / v z stands for the Rayleigh number and Q = # e H 2 d 2 [ 4 ~ e . y r l

denotes the Chandrasekhar number.

1104 Czech. d. Phys. B 28 [1978]

R. C. Sharma et aL: Finite Larmor radius effect...

Consider the case in which both the boundaries are free and the medium adjoining the fluid is nonconducting. Though this case is a little unrealistic but is appropriate for atmospheric physics. The case of two free boundaries is the most appropriate for stellar atmospheres (SPIEGEL [11]).

Regardless of the nature of bounding surfaces, we must require that

(18) O = 0 and W = 0 for z' = 0 and 1;

for, the surfaces z' = 0 and 1 are maintained at constant temperatures and as such they can suffer no change; and it is also clear that the normal component of the velocity must vanish on these surfaces.

Since no tangential stresses act on free surfaces and since the scalar pressure has no transverse component, the components Px~ and Pyz of the viscous stress tensor

(19) p x z = p ~ z + ~ x x and p y z = / t ~ z + ~ y ]

must vanish. Since w vanishes (for all x and y) on the bounding surface, it follows from (19) that

du ~v (20) - - O, on the free surface.

~z ~z

From the equation of continuity (5) differentiated with respect to z and using (I 1), (20); we get

(21) D 2 W = 0 on the free surface.

Since

= ~v ~u

~x ~gy '

it follows from equations (11), (18) and (20)that

(22) DZ = 0 on the free surface.

Dropping the dashes for convenience and using (18), (21) and (22), one can show that all the even derivatives of Wmust vanish for z = 0 and 1 and hence the proper solution of equation (17) characterizing the lowest mode is

(23) W = Wo sin ~z ,

where Wo is a constant.

Czech. J. Phys. B 28 [1978] 1105

R, C. Sharma et al.: Finite Larrnor radius ef fect . . .

Substituting (23) in equation (17) and letting a 2 = rc2x, R 1 = R/(rd), Q1 = Q/~2, we obtain the dispersion relation

(24) Rt = ( ~ - 1)"

. [ (1 + x)(1 + x + iai)(1 + x + iplal)(1 + x + ip2o-,)+ Q,(1 + x)(1 + x + iplal) +

l x(1 + x + ip2al)

N ( 2 - x)2(I + x + imp,) (1 + x + ip2~1)-[

where N = (Vo/V) z and i~r 1 = ~r/~ 2.

4. THE S T A T I O N A R Y CONVECTION

When the instability sets in as stationary convection, the marginal state will be characterized by a = 0. Putting a = 0, the dispersion relation (24) reduces to

( )[ (25) R I = ~ G 1 +Xx {(i + x) z + Q,} + ~(-I- q--x)){+- Q-~- J

For fixed values of Q1 and N, let the nondimensional number G accounting for the compressibility effects be also kept as fixed, then we find that

where Rc and _~c denote respectively the critical Rayleigh numbers in the absence and presence of compressibility. Thus the effect of compressibility is to postpone the onset of thermal instability. Hence we obtain a ~ stabilizing effect of compressi- bility.

From equation (25), it follows that

(27) dRx - ( G ) (2 - x)z (l + x)z dN ~ x{(1 + x)2 + Q,},

which is always positive. G has to be greater than one, as otherwise that would mean negative Rayleigh number which is futile. This shows that the finiteness of the ion Larmor radius has a stabilizing effect on the system, We thus conclude that the com- pressibility and finite Larmor radius effects are stabilizing on thermal instability. The effect of accounting for compressibility and finite Larmor radius on stellar objects is so as to help (or to provide a sort of rigidity to) the stellar objects to retain their shapes.

1106 Czech. J. Phys. B 28 [1978]

R. C. Sharma et al.: Finite Larmor radius e f fec t . . .

5. THE CASE OF OVERSTABILITY

In this section we discuss the possibility of whether instability may occur as an overstability. For overstability, cr 1 in equation (24) is real. Equating the real and imaginary parts of equation (24), we obtain

(28)

.[0 4- x) {(1 4- x) 2 - Plfl} {(1 4- x) 2 + p~fl} 4- Q1(1 4- x) {(1 4- x) 2 4- plP2fl} +

x{0 + x)~+ p ~ }

4- N ( 2 - x)2[-{(1 4- x) 2 _-p ,p2 f l } ( ( l+ x)2+_Ql_ -_ P2fl) +_ fl(1 4- x)2( 1 4- Pl)(Pl 4- P2)].] ~[{(1 + x)2 + ~ _ p~}2 + ~(1 + x): 0 + p2)~] _!'

and

(29) 0 = { ( 1 + x ) 2(1 + Pl)}{(1 + x ) 2 +p2fi} + QI(1 + x ) 2 ( p l . _ P2) +

x{0 + x) 2 + pi~}

+ N ( 2 - x ) ~ 0 + x){(1 + x) 2 + p ~ } [(1 + x)~(p~ - 1) + ~ + p2(p~- 1)~ + pipit] x[{(1 4- x) 2 4- Q1 - P2fl} 2 + fl(1 4- x) 2 (1 4- pc) 2]

where fl = o-~ z. It can be seen from equation (29) that the conditions for the nonexistence of over-

stability are p~ > p 2 and P l > 1 i.e. ~ < v and x < r / . z < v and ~ < t / are, therefore, sufficient conditions for the nonexistence of overstability, the violation of which does not necessarily imply occurrence of overstability.

. In the absence of finite Larmor radius effects, the sufficient condition for the nonexistence of overstability was • < r/ (CHANDRASEKHAR [1], p. 183), whereas the introduction of finite Larmor radius effects adds the further condition, z < v, for nonexistence of overstability.

Received 25. 7. 1977.

References

[1] CHANDRASEKItAR S., Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, Chap. IV, 1961.

[2] JUKES J. D., Phys. Fluids 7 (1964), 52. [3] VANDAKUROV Ju. V., Prik. Mat. Mech. 1, 28 (1964), 69. [4] ROnERTS K. V., TAYLOR J. B., Phys. Rev. Letters 8 (1962), 197. [5] ROSENBLUTIR M. N., KRALL N., ROSTOKER N., Nucl. Fus. Suppl. Pt. 1, (1962), 143. [6] MELCHIOR H., PoPowIcH M., Phys. Fluids 11 (1968), 581. [7] SINGI-I S., HANS H. K., Nucl. Fus. 6 (1966), 6. [8] SnARMA R. C., Nuovo Cimento 20B (1974), 303. [9] SI-tARMA R. C., PRAKASH K., Z. Naturforsch. 30a (1975), 461.

[10] SPIEGEL E. A., VERONIS G., Astrophys. J. 131 (1960), 442. [11] SPIEGEL E. A., Astrophys. J. 141 (1965), 1068.

Czech, J. Phys. B 28 [1978] 1 107