A simple model of Rossby-wave hydraulic behaviour

Abstract

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.