On the equatorial Ekman layer

This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.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.F.M. and E.D. have been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01. F.M. acknowledges a PhD mobility grant from Institut de Physique du Globe de Paris. A.M.S. visited ENS, Paris (19–25 October 2014), while F.M. and E.D. visited the School of Mathematics and Statistics, Newcastle University (respectively, 7–25 September 2015 and 25–30 November 2015); the authors wish to thank their respective host institutions for their hospitality and support