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. 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 Benilov's (J. Phys. Oceanogr. vol. 30, 2000, p. 733) asymptotic theory. The short-wave maximum occurs at wavelengths comparable to the scale of the topography or at a shorter one. 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 a stronger topography, long-wave disturbances are stable, as predicted by Benilov's (2000) asymptotic theory, whereas short-wave instabilities are still present. The instabilities are strongest near the lines of maximum shear. In nonlinear simulations, the flow becomes turbulent within narrow strips along these lines, and potential vorticity there homogenizes. As the strips grow wider, they begin to interact, and 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. In the presence of topography, however, the vortices are ‘filled’ with short-wave turbulence, and break-up occurs later.