Steady Hydromagnetic Boundary Layer near a Rotating, Electrically Conducting Plate

Abstract

Gilman and Benton's study of the steady fluid motions and magnetic fields induced by a differentially rotating, electrically insulating infinite flat plate in the presence of a uniform magnetic field is generalized by allowing the electrical conductivity of the plate to be an arbitrary function of distance from the fluid‐plate interface. It is found that the important measure of the conductivity of the plate is the ratio of plate conductance to conductance of a layer of fluid whose thickness is one Ekman depth. The form of the steady Ekman‐Hartmann layer is found to be completely independent of the conductivity of the plate. The steady perturbation magnetic field within the plate is found to be entirely azimuthal and its magnitude is directly proportional to the conductance ratio. For values of parameters approximating conditions within the earth, the perturbation field may not be small. This azimuthal field causes an axial electric current to be drawn into the plate from the fluid, effectively strenghtening the coupling between fluid and plate. If the conductance ratio is large, as it is within the earth, spin‐up is accomplished primarily by this current and the Ekman‐Hartmann layer no longer plays a dominant role in coupling the fluid with its boundary.