The Equatorial Ekman Layer

TL;DR

This paper investigates the structure and behavior of the Equatorial Ekman layer in rotating spherical flows at low Ekman numbers, providing asymptotic analysis and numerical solutions that extend Stewartson's classical theory.

Contribution

The study extends Stewartson's asymptotic analysis by developing higher-order solutions for the Equatorial Ekman layer and implementing a numerical model with non-local boundary conditions.

Findings

The far-field solution matches well with numerical results.

The E^{2/5} layer problem can be formulated independently of E.

The extended similarity solution improves understanding of the layer structure.

Abstract

The steady incompressible viscous flow in the wide gap between spheres rotating about a common axis at slightly different rates (small Ekman number E) 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 limit E<<1, was undertaken by Stewartson (J. Fluid Mech. 1966, vol. 26, 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 C, is geostrophic. Stewartson identified a complicated nested layer structure on C, which comprises relatively thick quasi-geostrophic E^{2/7} (inside C) and E^{1/4} (outside C) layers. They embed a thinner E^{1/3} ageostrophic shear layer (on 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. Accordingly, the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial co-ordinate z) solution. We present a numerical solution, which uses 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.