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.