Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up

Abstract

The unsteady boundary layer of a rotating, stratified, viscous, and diffusive flow along an insulating slope is investigated using theory, numerical simulation, and laboratory experiment. Previous work in this field has focused either on steady flow, or flow over a conducting boundary, both of which yield Ekman-type solutions. After the onset of a circulation directed along constant-depth contours, Ekman-type flux up or down the slope is opposed by buoyancy forces. In the unsteady, insulating case, it is found that the cross-slope transport decreases in time as (t/τ)^−½ where \[ \tau = \frac{1}{S^2f\cos\alpha}\left(\frac{1/\sigma + S}{1+S}\right), \] may be called the ‘shut-down’ time. Here S = (N sin α/f cos α)², f is the Coriolis frequency, α is the slope angle, N is the buoyancy frequency, and σ is the Prandtl number. Subsequently the along-slope flow, $\hat{v}$, approximately obeys a simple diffusion equation \[ \frac{\partial\hat{v}}{\partial t} = \nu\left(\frac{1/\sigma + S}{1+S}\right)\frac{\partial^2\hat{v}}{\partial\hat{z}^2}, \] where t is time, ν is the kinematic viscosity, and $\hat{z}$ is the coordinate normal to the slope. By this process the boundary layer diffuses into the interior, unlike an Ekman layer, but at a rate that may be much slower than would occur with simple non-rotating momentum diffusion. The along-slope flow, $\hat{v}$, is nevertheless close to thermal wind balance, and the much-reduced cross-slope transport is balanced by stress on the boundary. For a slope of infinite extent the steady asymptotic state is the diffusively driven ‘boundary-mixing’ circulation of Thorpe (1987). By inhibiting the cross-slope transport, buoyancy virtually eliminated the boundary stress and hence the ‘ fast’ spin-up of classical theory in laboratory experiments with a bowl-shaped container of stratified, rotating fluid.