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 $E\ll 1$, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the $E^{1/2}$ Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder $\mathscr{C}$, is geostrophic. Stewartson identified a complicated nested layer structure on $\mathscr{C}$, which comprises relatively thick quasigeostrophic $E^{2/7}$- (inside $\mathscr{C}$) and $E^{1/4}$- (outside $\mathscr{C}$) layers. They embed a thinner ageostrophic $E^{1/3}$ shear layer (on $\mathscr{C}$), which merges with the inner sphere Ekman layer to form the $E^{2/5}$-equatorial Ekman layer of axial length $E^{1/5}$. Under appropriate scaling, this $E^{2/5}$-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.