Influence of an Axial Magnetic Field on the Steady Linear Ekman Boundary Layer

Abstract

A hydromagnetic version of the Ekman boundary layer is developed in a simple form in order to study how the geophysically important Ekman suction velocity is affected by magnetic fields. The problem treated consists of a viscous, incompressible, conducting fluid in the presence of an infinite, flat, insulating boundary which rotates at speed Ω0. Outside the boundary layer, the fluid rotates uniformly with speed Ω1 = Ω0 (1 + ε), and there is a uniform magnetic field aligned with the rotation axis. An expansion in powers of ε, the Rossby number, together with von Kármán similarity, leads to an exact solution which to first order in ε, describes a continuous transition between pure Ekman flow and a rotating analog of Hartmann flow. The magnetic field is found to inhibit Ekman suction; yet, such a boundary layer may still exert a strong influence on the outer flow because of a new feature that replaces the suction, namely an induced axial current outside the boundary layer. This “Hartmann current,” not present in the conventional Hartmann problem, owes its existence to the fluid rotation.