On the equatorial Ekman layer

Abstract

The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit E≪1, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the E1/2 Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder C, is geostrophic. Stewartson identified a complicated nested layer structure on C, which comprises relatively thick quasigeostrophic E2/7- (inside C) and E1/4E1/4- (outside C) layers. They embed a thinner ageostrophic E1/3 shear layer (on C), which merges with the inner sphere Ekman layer to form the E2/5-equatorial Ekman layer of axial length E1/5. Under appropriate scaling, this E2/5-layer problem may be formulated, correct to leading order, independent of E. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate z) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite z to account for the far-field behaviour. Adopting z−1 as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.