A modified quasigeostrophic formulation for weakly nonlinear barotropic flow with large-amplitude depth variations

When the β effect is weak, rapidly rotating weakly nonlinear barotropic flow is subject to the "strong" Taylor-Proudman theorem, which severely restricts the horizontal divergence. This constraint is independent of the boundary conditions, and applies even if there are large depth variations. Under these conditions there is an important class of flows in which unbalanced Ekman pumping allows nondivergent flow across depth contours. This paper demonstrates that this type of flow over large-amplitude topography can be modelled surprisingly well by a slightly modified quasigeostrophic vorticity equation, which includes the leading effects of advection, lateral viscosity and β and retains the simplicity and most of the conservation properties of the standard quasigeostrophic equations. It is used in a test case to model flow in a basin in which the depth vanishes at the lateral boundaries, and the calculated flow is in good agreement with that measured in a laboratory realisation of the same system. This formulation could be useful for numerical modelling or analysis of weakly nonlinear Ekman-dominated barotropic flow over steep topography in other models with small β, such flow in some laboratory models