A geostrophic - like model for large Hartmann number flows

A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number (Ha >> 1). This is analogous to the case of low Ekman number rotating flows, with Ekman layers replacing Hartmann layers. This has been at the origin of the homogeneous model for flows in a rotating frame of reference, with its rich structure: geostrophic contours and shear layers of Stewartson [Ste57, Ste66], Munk [Mun50] and Stommel [Sto48]. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii [Kul68] play a role similar to the role of the geostrophic contours. However, a general theory for quasi two-dimensional magnetohydrodynamics is lacking. In this paper, a model is proposed which provides a general framework for quasi two-dimensional magnetohydrodynamic flows. Not only can this model account for otherwise disconnected past results, but it is also used to predict a new type of shear layer, of typical thickness Ha. Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd and Walker [HW78], and the problem of a rectangular cross-section duct flow in a slowly varying transverse magnetic field. For the first problem, the existence of thick shear layers of dimensionless thickness of order of magnitude Ha explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past [WL72] for the so-called “M-shape” velocity profile, which is here understood as an aspect of general quasi two-dimensional magnetohydrodynamics.