Waves a heavy, viscous, incompressible, electrically conducting fluid of variable density, in the presence of a magnetic field

Abstract

The equilibrium of a horizontal layer of a heavy incompressible fluid of variable density p 0 in the vertical direction is stable or unstable according as d p o /dz (z being the upward vertical) is everywhere negative or is anywhere positive. In the unstable case, the rate n at which the system departs from equilibrium depends on the total wave number k of the initial disturbance, and there is, in general, one mode characterized by n m and k m which grows more rapidly than any other. In the stable case, after an initial disturbance the equilibrium may be restored either periodically or aperiodically, depending on the value of k . The periodic type of motion gives rise to horizontally propagated ‘gravity waves’. It is the purpose of this paper to examine the influence of viscosity and hydromagnetie forces on the hydrodynamical motion produced by a small disturbance of the aforementioned equilibrium situation. The appropriate perturbation theory is developed initially for any density field p 0 (z) and kinematical viscosity v ( z ) for a fluid of constant electrical conductivity cr e.m.u. and magnetic permeability k in the presence of a uniform magnetic field of strength H 0 in the direction of gravity, acceleration g . The solution is expressed in the form of integrals, and is shown to be characterized by a variational principle. Based on the variational principle an approximate solution is obtained for the special case of a fluid of finite depth d stratified according to the law p 0 = p 1 exp ( βz ), and for which v is constant. It is shown that if n and k are measured in suitable units, they are related by an equation involving three dimensionless parameters: R = ( ηπs /2 dV A ), and B = ( gβd 2 2 dV A ) and B = ( gβd 2 /π 2 s 2 V 2 A ), where η = (4πKo) -1 , V 2 A = (kH 2 0 /4πP 1 ), and s is an integer involved in the description of the velocity field. Explicit solutions may be obtained in three cases, namely, (a) R ->oo (b) B = 0, (c) R = 0. Case (a) has been discussed in a previous paper. Other eases will be discussed in a future paper. Only cases (b) and (c) are considered here. In case (6) we have the problem of hydromagnetic waves damped by viscosity and electrical resistance; the properties of these waves are described. In case (c), we limit ourselves to an ideal conductor. When B > 0 the equilibrium is unstable; the influence of viscosity and hydromagnetic forces on the mode of maximum instability is briefly discussed. When B < 0, the equilibrium is stable. If — B < 2, S 2 > 1, or — B > 2, S 2 > (4(1 — B) 3 /21 B 2 ), the equilibrium is always restored aperiodically. Otherwise, waves can be generated, but only when the wavelength lies within a certain range. These weaves combine the properties of gravity waves and hydromagnetic waves.