Tidal wave propagation in a rotating conducting fluid with a magnetic field

Abstract

I discussing the propagation of tidal waves in a nonrotating system, Lamb has shown that viscosity damps waves that are capable of propagating and inhibits the propagation of such waves if the viscosity is large enough. This is because of the dissipation of energy due to viscosity. However, in a rotating system, viscosity may play a much different role. It is shown by Duty that, in a rotating system, viscosity (if it is large enough) can interact with rotation and encourage tidal wave propagation instead of damping it. The physical reason for this lies in the fact that under certain circumstances viscosity plays the dual role of damping disturbances and at the same time extracting energy from the mean flow to feed into the disturbance, thus producing instability. This mechanism is discussed at length by Lin in connection with boundary-layer instability. On the other hand, Chandrasekhar, while studying the thermal instability of a layer of a rotating electrically conducting fluid in the presence of a magnetic field, has shown that rotation can interact with the magnetic field to produce instability. Although in some respects the effects of rotation and a magnetic field acting separately are alike, viz., they both inhibit the onset of instability, they do not necessarily reinforce each other when acting together. On the contrary, they may sometimes oppose each other. Thus viscosity enhances the onset of instability if rotation is present, and it is also known that a magnetic field imparts to the fluid some characteristics of viscosity. Hence it follows that, even though the two acting separately inhibit the onset of instability, they might conspire to produce instability when acting jointly. Again rotation brings into play a component of vorticity along its direction, and, for large rotation, streamlines are closely wound spirals with motions mainly confined to planes perpendicular to the direction of rotation. But a magnetic field does not induce such a component of vorticity, and the tension in the lines of force tends to prevent motions transverse to them. This shows that the two fields do not produce comparable effects. From these considerations it is reasonable to think that, like thermal instability, a rotation interacting with a magnetic field might give rise to an enhanced tidal wave propagation. The present note reveals that for large Hartman numbers (as in the case of liquid mercury) rotation and magnetic field do give rise to such an instability.