The stability of a barotropic zonal jet aligned with zonal topography on the beta-plane is investigated. The topography is assumed to be spatially periodic, with a period much smaller than the width of the jet. Thus, the potential vorticity (PV) is non-monotonic, varying on the small topographic length scale. The problem is examined both by linear normal-mode analysis and by direct numerical simulations.

The following results are obtained. If the topography is sufficiently weak, the growth-rate of the most unstable normal mode has two maxima. The long-wave maximum occurs at wavelengths comparable to the width of the jet, and is described by the asymptotic theory of Benilov (2000). The short-wave maximum occurs at wavelengths comparable to the scale of the topography. The nonlinear evolution of the flow is in this case similar to that in the case of a flat bottom, i.e. the jet begins to meander and breaks up into separate vortices.

For moderate or strong topography, long-wave disturbances are stable, as predicted by Benilov's (2000) asymptotic theory, while short-wave instabilities are still present. The instabilities are strongest near the lines of maximum shear. In the nonlinear simulations the flow becomes turbulent in strips along these lines, and the PV within the strips homogenises. Thus, a small-scale mean flow component develops that cancels the topographic contribution to the PV. As the two turbulent strips grow wider they begin to interact, resulting in a secondary instability. The initial condition for this instability is a smooth PV field that varies on the large scale of the jet rather than on the small scale of the topography. The subsequent evolution is again similar to that of a jet over a flat bottom: large-scale meandering and break-up of the jet into vortices.