Nonlinear dynamics over rough topography: barotropic and stratified quasi-geostrophic theory

The weakly nonlinear dynamics of quasi-geostrophic flows over a one-dimensional, periodic or random, small-scale topography is investigated using an asymptotic approach. Averaged (or homogenised) evolution equations which account for the flow-topography interaction are derived for both barotropic and continuously stratified geostrophic fluids. The scaling assumptions are detailed in each case; for stratified fluids, they imply that the direct influence of the topography is confined within a thin bottom boundary layer, so that it is through a new bottom boundary condition that the topography affects the large-scale flow. For both barotropic and stratified fluids, a single scalar function entirely encapsulates the properties of the topography that are relevant to the large-scale flow: it is the correlation function of the topographic height in the barotropic case, and a linear transform thereof in the continuously stratified case. Some properties of the averaged equations, including their Hamiltonian structure, are discussed, and explicit nonlinear solutions in the form of one-dimensional travelling waves are sought. In the barotropic case, previously investigated by Volosov, these obey an integrable second-order differential equation; in the stratified case, and in the absence of β-effect, they instead obey a nonlinear pseudo-differential equation, which reduces to the Peierls–Nabarro equation for sinusoidal topography. The known solutions to this equation provide examples of nonlinear periodic and solitary waves in continuously stratified fluid over topography. The influence of bottom topography on baroclinic instability is also examined using the averaged equations: they allow a straightforward extension of Eady’s model which demonstrates the stabilising effect of topography on baroclinic instability.