A geostrophic-like model for large-Hartmann-number flows

Abstract

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 ($\hbox{\it Ha} \,{\gg}\, 1$). 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. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii (Isv. Akad. Nauk SSSR Mekh. Zhidk Gaza, vol. 3, 1968, p. 3) play a role similar to that 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. 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 $\hbox{\it Ha}^{-1/4}$. Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd & Walker (J. Fluid Mech. vol. 84, 1978, p. 471), 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 $\hbox{\it Ha}^{-1/4}$ 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 Walker & Ludford (J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.