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On the equatorial Ekman layer

Published online by Cambridge University Press:  30 August 2016

Florence Marcotte ,
Emmanuel Dormy  and
Andrew Soward
Florence Marcotte
Affiliation:
MAG (ENS/IPGP), LRA, Département de Physique, Ecole Normale Supérieure, 24, rue Lhomond, F-75231 Paris CEDEX 05, France
Emmanuel Dormy
Affiliation:
MAG (ENS/IPGP), LRA, Département de Physique, Ecole Normale Supérieure, 24, rue Lhomond, F-75231 Paris CEDEX 05, France
Andrew Soward*
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
*
†Email address for correspondence: andrew.soward@ncl.ac.uk
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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.

JFM classification

Boundary Layers: Boundary layer structure Boundary Layers: Free shear layers Geophysical and Geological Flows: Rotating flows

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Papers
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Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 395 - 435
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© 2016 Cambridge University Press 

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Footnotes

‡

Present address: Département de Mathématiques et Applications, CNRS UMR-8553, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France

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