Dispersive effects in Rossby-wave hydraulics

Abstract

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.