The stability of zonal jets in a rough-bottomed ocean on the barotropic beta plane

The author considers the stability of a barotropic jet on the beta plane, using the model of a "rough-bottomed ocean" (i.e., assuming that the horizontal scale of bottom irregularities is much smaller than the width of the jet). An equation is derived, which governs disturbances in a sheared flow over one-dimensional bottom topography. such that the isobaths are parallel to the streamlines. Interestingly, this equation looks similar to the equation for internal waves in a vertically stratified current, with the density stratification term being the same as the topography term. It appears that the two effects work in a similar way, that is, to return the particle to the level (isobath) where it "belongs" (determined by its density or potential vorticity). Using the derived equation, the author obtains a criterion of stability based on comparison of the mean-square height of bottom irregularities with the maximum shear of the current. It is argued that the influence of topography is a stabilizing one, and it turns out that "realistic" currents can be stabilized by relatively low bottom irregularities (30-70 m). This conclusion is supported by numerical calculation of the growth rate of instability for jets with a Gaussian profile.