Axisymmetric selective withdrawal in a rotating stratified fluid

Abstract

In this paper we consider the axisymmetric flow of a rotating stratified fluid into a point sink. Linear analysis of the initial value problem of flow of a linearly stratified fluid into a point sink that is suddenly switched on shows that a spatially variable selective withdrawal layer is established through the outward propagation of inertial shear waves. The amplitude of these waves decays with distance from the sink; the e-folding scale of a given mode is equal to the Rossby radius of that mode. As a consequence, the flow reaches an asymptotic state, dependent on viscosity and species diffusion, in which the withdrawal-layer structure only exists for distances less than the Rossby radius based on the wave speed of the lowest mode, R 1 . If the Prandtl number, Pr , is large, then the withdrawal layer slowly re-forms in a time that is O (δ 2 i κ -1 ), such that it extends out much farther to a distance that is O ( R 1 Pr δ 2 i δ -2 e ) rather than O ( R 1 ). Because there is no azimuthal pressure gradient to balance the Coriolis force associated with the radial, sinkward flow, a strong swirling flow develops. Using scaling arguments, we conclude that this swirl causes the withdrawal-layer thickness to grow like $(ft)^{\frac{1}{3}}$ , such that eventually there is no withdrawal layer anywhere in the flow domain. Scaling arguments also suggest that this thickening takes place in finite-size basins. These analyses of swirl-induced thickening and diffusive thinning can be combined to yield a classification scheme that shows how different types of flows are possible depending on the relative sizes of a parameter J , which we define as fQ ( Nhv ) -1 , E (the Ekman number fh 2 v -1 ), and Pr .