Hall effects on thermal instability in a rotating layer of a conducting fluid

Abstract

This paper discusses the effects of Hall currents on the convective instability of a rotating layer of a conducting fluid in the presence of a vertical magnetic field. It is found that the Hall currents are always destabilizing in the absence of rotation but tend to stabilize the flow as soon as the rotation exceeds certain limits. An analytical condition for nonexistence of overstability is also derived.In a collision dominated plasma (wr << 1, where w is the gyration frequency and z is the time of collision of a charged particle), electrical conductivity becomes isotropic and as such a scalar quantity. However, conductivity becomes anisotropic if the medium is rarefied and/or a strong magnetic field is present. The charged particles are in a way tied to the lines of force and their mobility transverse to the magnetic field is checked to some extent. If an electric field is applied transverse to the magnetic field, the whole current will not flow along the electric field, but a part will flow in a direction perpendicular to the electric and magnetic field. This latter part is known as the Hall current. Flow of Hall currents plays a vital role in problems of MHD power generation and in several astrophysical situations.The problem of thermal instability in a rotating layer of a conducting fluid in the presence of a uniform vertical magnetic field has been discussed exhaustively by Chandrasekhar (1961, chap. V).Recently Gupta (1967) investigated the Hall effects on the thermal instability of a horizontal layer of a conducting fluid in the presence of a uniform vertical magnetic field and found that Hall currents are destabilizing. In this paper we investigate the Hall effects on the thermal instability when the layer is subjected to a uniform rotation and permeated by a strong magnetic field.where J, E, H, V are respectively the current density, the electric field, the magnetic field, and the velocity vector. Further, p is the magnetic permeability and o0 is the electiical conductivity given by e2nr/m,, where e, 11, and n1, are the charge, number density, and mass of an electron.Eliminating the electric field from [ I ] and Maxwell's equations, we obtain the modified magnetic diffusion equation where q = 1/4npoo is the magnetic diffusivity. We choose a Cartesian system of coordinates with origin at the lower surface of the layer and z axis taken in the vertical direction.From the equations of conservation of mass, momentum, and the magnetic diffusion equation [2], we have Consider a horizontal layer of a conducting fluid which is in a state of uniform rotation and is [5] (g -qV2)h3 = H 0 -av3 Ho 853 heated from below. A uniform magnetic field H , az 4nen az ' acts in the vertical direction. The linearized hydromagnetic equations and the usual Maxwell [6] (k -V V~) V ' V , = ga($ + $) equations remain unchanged (cf. Chandrasekhar 1961, chap. V). However, Ohm's law gets modified because of the incorporation of the term am3 PHO a -2Q -+ ---v 2 h 3 , arising out of Hall currents and is as follows: az 4np az