On the equatorial Ekman layer

• Published in: Journal of fluid mechanics Vol. 803; pp. 395 - 435
• Main Authors: Marcotte, Florence, Dormy, Emmanuel, Soward, Andrew
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.09.2016
• Subjects: Boundary conditions; Boundary layer; Boundary layers; Computational fluid dynamics; Conducting fluids; Core flow; Cylinders; Dynamics; Ekman layer; Ekman layers; Fluid flow; Fluid mechanics; Fluids; Geometry; Geophysics; Identification; Incompressible flow; Mathematical models; Numerical models; Planetary cores; Rossby number; Rotating spheres; Scaling; Shear; Shear layers; Similarity solutions; Spheres; Valuation methods; Viscous flow; France; boundary layer structure; rotating flows; free shear layers
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2016.493

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.