Year: 2016
Type: article
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.
Source: Journal of Fluid Mechanics
DOI: 10.1017/jfm.2016.493
Authors: Florence Marcotte, Emmanuel Dormy, Andrew Soward
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W2290503283