Withdrawal of a stratified fluid from a rotating channel

Abstract

The flow of a stratified fluid toward a line sink in a rotating channel of finite width and depth is studied. The withdrawal flow is shown to be established by a set of Kelvin shear waves trapped within a distance of Nh/fn from the right-hand side wall (f > 0) looking in the direction of propagation, where n = 1, 2,… is the vertical mode number. In addition there are a set of waves (Poincaré modes) which propagate away from the sink with a cross-channel modal structure. The withdrawal flow has a boundary-layer structure: far from the right-hand wall the flow resembles that of McDonald & Imberger (1991), whereas close to the right-hand wall the development of the vertical structure of the withdrawal flow resembles that of the non-rotating case due to the presence of Kelvin shear waves. In a narrow channel Kelvin shear waves dominate the establishment of the withdrawal flow. The withdrawal flow is investigated for large times compared to the inertial period, where it is shown that the width of the boundary layer is of the same order as the distance downstream from the sink. The flow within the boundary layer is unsteady as the withdrawal layer thickness δ continues to collapse indefinitely, while outside the boundary layer it is steady with δ ∼ fL/N, L being the horizontal lengthscale downstream from the sink. A scaling analysis is developed for the narrow channel case in which the cross-channel velocity can be ignored. The results are applied to actual field data, where it is shown that the effect of rotation may explain why previous non-rotating theories have been inaccurate in predicting withdrawal layer thickness.