Compressible Ekman–Hartmann boundary layers

Abstract

We consider the effect of compressibility on mixed Ekman–Hartmann boundary layers on an infinite plane (z = 0), in the presence of an external magnetic field oblique to the boundary. The aim is to investigate the influence of the magnetic pressure on the fluid density, and hence, via mass conservation, on the mass flow into or out of the boundary layer. We find that if the z-component of vorticity in the main flow, immediately above the boundary layer, is negative, then there is a competition between Ekman suction and the magnetic pressure effect. Indeed, as the magnetic field strength is increased, the magnetic pumping may overcome the Ekman suction produced by anti-cyclonic main flow vortices. Such a mechanism, based on the competition between these effects, may be of importance for understanding the dynamics of the magnetic field in stellar (or planetary) interiors. For the solar tachocline, we find that the analysed magnetic pressure effect is unlikely to play a significant role; however, we give examples of what changes in the assumed scalings would be necessary for the effect to become important.