The structure of the Ekman layer for geostrophic flows with lateral shear

Abstract

A generalized Ekman flow is obtained for the boundary layer of a semi-infinite, rotating homogeneous fluid with lateral variation in the horizontal velocity components far from the boundary surface. The fields of motion considered include uni-directional motion with lateral shear, symmetric and elongated eddies, and cols. A solution for the first of these cases is obtained to fifth order: for the others, t o second order. In all cases the magnitude of the vertical velocities induced by convergence in the boundary layer is greater for systems with negative relative vorticity than for systems with positive relative vorticity. The result differs substantially from first-order theory, for which the magnitude of the vertical velocity is proportional to the relative vorticity. A number of meteorological applications are considered.

Horizontal and residual circulations driven by wind stress curl in Tokyo Bay

“…Following Benton et al . [], we assume a solution of the form: …”

“…To understand the effects of centrifugal forces on pumping velocity conceptually, we consider a nonlinear Ekman layer in a simple setting, following previous studies on the von K arm an swirling flow problem [e.g., Rogers and Lance, 1960;Benton et al, 1964;Zandbergen and Dijkstra, 1987]. We consider a horizontally infinite homogeneous fluid in cylindrical coordinates (r, h, z), and assume idealized wind stress s5ðs r ; s h Þ5ð0; q 0 TrÞ and eddy viscosity m 5 m(z), where q 0 is the reference density and T is half the wind stress curl (note q 0 21 r3s52T).…”

“…Due to the form of the forcing, the problem does not depend on h. Writing velocity components in (r, h, z) as (u, v, w) and using dynamic pressure p, the Reynolds-averaged Navier-Stokes equations for a thin Ekman layer become @u @t 1u @u @r 1w @u @z 2 v 2 r 52 1 q 0 @p @r 1fv1 @ @z m @u @z ; where f is the Coriolis parameter. Following Benton et al [1964], we assume a solution of the form:…”

“…Stern [1965], Thomas andRhines [2002], andPedlosky [2008] obtained weakly nonlinear corrections to the linear Ekman layer solution. It appears that a strongly nonlinear Ekman layer has not been considered in an oceanic context, though strongly nonlinear rotating boundary layers have been investigated in a closely related von K arm an swirling flow problem [e.g., Rogers and Lance, 1960;Benton et al, 1964;Zandbergen and Dijkstra, 1987]. In addition to the spin-up of horizontal circulation, to close the circuit of Ekman transport, and to satisfy the conservation of mass, Ekman and nonlinear transport also induces three-dimensional residual flow [Greenspan, 1968].…”