THE INFLUENCE OF CORIOLIS FORCE ON MAD WAVES

By S. NARASIMHA MURTHY(Department of Mathematics, University of Agricultural Sciences, Bangalore-24)

Received January 24, 1973

(Communicated by Dr. K. Rainakrishnan r.A.sc.)

ABSTRACT

The influence of coriolis force on the propagation of MHD wavesin compressible medium is investigated by using linear stability analysis.In a combined effect we find that, while compressibility gives rise to anacoustic wave, coriolis force results in a modified Alfven wave.

INTRODUCTION

IT is an established fact that a conducting, incompressible fluid gives rise toa new wave known as Alfven wave in the presence of a magnetic field, forany external disturbance (Ferraro and Plumpton, 1966). Lundquist (1949)experimentally demonstrated the existence of such MHD waves. However,the influence of coriolis or geostrophic force on the propagation of waveshas not attracted enough attention.

The coriolis force which plays hardly any role as regards physicalphenomenon on a laboratory scale, may often have a predominant influencein cosmic studies such as in sunspots, magnetic storms. This has been pointedout by Chandrasekhar (1953), on the basis of his investigations on the effectof the coriolis force on problems on thermal instability and on the stabilityof a viscous flow in the presence of magnetic field. Later, Lenhert (1954,1955) discussed the influence of geostropic force on the wave propagationtaking incompressible medium. His results reveal that the phase velocityof the waves is modified due to such a force. The gravitational instabilityof an infinite homogeneous medium in the presence of coriolis force andmagnetic field has been discussed by Chandrasekhar (1954). It is shownthat Jean's Criterion remains unaffected in the presence of magnetic field;even if the system is subjected to rotation, and coriolis force operative.

The aim of the present note is to discuss the influence of geostrophicforce and the role of compressibility on the propagation of waves in a

202

The Influence of Coriolis Force on MHD Waves 203

uniform gas, in the presence of an applied magnetic field. It is found,that acoustic waves are propagated with a velocity of sound ± C and amodified Alfven wave due to coriolis force with phase velocity

± (VA 2 + 2Q2)

where VA is the Alfven wave velocity, 0 is the angular velocity, and K is thewave number.

2. LINEARIZED EQUATION

We consider a progressive plane wave propagated in the direction OZin a gas of infinite electrical conductivity a which is pervaded by a uniform

magnetic field B 0 = (0, 0, B 0), and is rotating around the magnetic lines offorce with a constant angular velocity Q = (0, 0, 0). The orientation ofthe co-ordinate axes chosen by Chandrasekhar (1954) is

-* 4H = (0, Hy , Hi) ; Q = (Qs, Dy, Q)

which is different in the present case.

Denoting the variations in pressure and density by 8p and 8p respectivelyfrom the equilibrium state, the linearized equations of motion are:

4

P ^V = — V (6p) + (curl b) x B o + 2p (V x 0), (2.1)

where b denotes the variation in the magnetic field.

Also, in an adiabatic regime the variations in density and pressure areconnected by the relation

8p=ypPP =e 2 Sp,

where

C = /VP1/

A. A2

(2.2)

204 S. NARASIMHA MURTHY

denotes the velocity of sound. Substituting for 8p in equation (2.1) leads to

P -- c2 O (SP) — (curl b) x B 0 — 2p (V x Q) = 0. (2.3)

The equation of conservation of mass is:

(Sp) + p div. V=0, (2.4)

and the equation of conservation of magnetic flux is:

^b — curl (V x B o) = 0. (2.5)

We suppose that all variables depend on Z and t only. Then the equation-^

div b = 0 implies bz = 0. We neglect centrifugal force not only for simp-licity but for the study of the effects of coriolis force in isolation. Centri-fugal force in general is neglected (Chandraseldur, 1951) in the studies ofastrophysical plasmas— such neglect being partly justified by the small magni-tude of rotational frequency.

The resolutes of equation (2.5) are

bx Bo (2.6)

Eby_.wy (2.7)^t bz

where the suffixes denote the corresponding components in respective direc-tions.

The resolutes of equation (2.3) are

^t µp zz

'y = —0 bby — 2pvx9, (2.9)

P, aatz -l- C$ jz (sp) 0 (2.10)

The Influence. of. Coriolis Force on MHD Waves 205

and the equation of continuity (2.4) is

($p) :-}- p wx - p (2.11)

3. DISPERSION RELATION AND DISCUSSION

To study the wave propagation we take time and space dependence ofall dependent variables as exp. [i (wt - kz)], where w is the frequency, andk the wave number. Thus we obtain the following linearized equations,from (2.6) to (2.11) :

iavx = — (ILP) iKbx + 2vu SQ, (3.1)

iuruy = — (PP) iKby -. 2vxQ, (3,2)

Pwv z = C 2 Kbp, (3.3)

cwap — pKvz =0, (3.4)

wbx = — B oKvz , (3 :5)

ruby = — B OKvy . (3.6)

Eliminating the ratios

Vx .Vy :Vz :bx :by :op

from these equations we obtain the frequency equation:

\K/(`0l 4 (C2 + VA2 +. l(l% 2P2 2 + ( Cz VA2 + 2S2 2 C$ O'

\ K2I `K/ K2 _(3.7)

where

VA=V ffo

µP

is the -AlfVen velocity.

Since w/K denotes the phase velocity of the waves, U, we may write(3.7) as

U' _(ca +v -{ K2) u2 (C2 VAZ -i- 2 KZ C Q) 0.

206 S. NARASIMHA MURTHY

This has four real roots.

Equation (3.8) is the same as the equation (169) of Chandrasekhar(1953) when Sl = 0 in (3.8) and G = 0 in (169). Equation (3.8) can bewritten further as

(U2 — C2) [ u2 — (VA2 +2Q2

KF )J = 0. (3.9)

Thus we have acoustic wave propagated with a velocity of sound t C anda modiLed Alfven wave due to rotation with velocity

(V,.2+ ").

The two waves can be designated as a fast wave and a slow wave whosespeeds bound C, the acoustic wave

af^et^ a0>a.,.

Thus

JVA2

2 f K2 ,

amtoW = — - IVA Z -}- K2aV C-

Hence the role of compressibility and coriolis force is to introduce an acousticwave with longitudinal oscillations and a modified Alfven wave with trans-verse oscillations respectively.

Presently the wave propagation at an arbitrary angle to the directionof magnetic field and the axis of rotation is under invV stigation.

ACKOOWLEDGEMENT

The author is grateful to Prof. P. L. Bhatnagar for his keen interestand kind encouragement. He is thankful for the referee for kind comments.

REFERENCES

1. Ferraro, V. C. A. and An introduction to Magneso-fluid Mechanics, CJeiindoaPlumpton, C. Press, 1966.

The Thence of Coriolis Force on MUD Waves 207

2. Lundquist, S. ... Phys. Rev., 1949, 76 (12), 1805.

3. Chandrasekhar, S. ... Mon. Not. R. astr. Soc., 1953, 113. 667,

4. -- - --- .. Astrophys, 1., 1954, 119, 7.

5. Lenhert, B. .. Ibid., 1954, 119, 647.

6. — -- .. Ibid., 1955, 121, 481.

7. Chandrasekhar, S. .. Ibid., 1953, 118, 116.

8. _.__ -. .. Hydrodynamic and Hydromagnetic Stability, Chap. 13, pp.589-98. Calrendon Press, Oxford.