Short-wave, localized disturbances in jets, with applications to flows on a beta plane with topography

Abstract

We examine the stability of jets over topography on the so-called barotropic beta plane (which models oceanic currents in mid-latitudes). Attention is focused on disturbances with a large wave number, for which an asymptotic solution of the normal-mode eigenvalue problem is presented. It is demonstrated that short-wave modes, if they exist, are localized in narrow strips “centered” at local extrema of the velocity profile U(y) (y is the transverse variable). It is further shown that an extremum, say y=yl, can support a short-wave mode only if the vorticity gradient and potential vorticity (PV) gradient at yl are of opposite signs. If they are, the leading-order solution of the eigenvalue problem describes a mode with a phase speed slightly larger than U(yl), if yl is a maximum; or slightly smaller than U(yl), if yl is a minimum. In other words, the mode does not have critical levels in the vicinity of yl, although it may have them elsewhere, at a distant point. If that indeed happens, an additional condition is required to guarantee the existence of the mode: namely, the PV gradient at yl and that at the critical level must have opposite signs. If they do, the mode exists and is weakly unstable (the phase speed has a small imaginary part). Thus, a change of sign of the PV gradient does not necessarily destabilize the flow; and in order to guarantee instability, the PV gradient should have opposite signs at the “important” points, i.e., the localization point and critical level. The asymptotic results are tested against the numerical solution of the exact normal-mode eigenvalue problem. The former and the latter are in good agreement, and not only for large wave numbers, but also for moderate ones. It is also demonstrated that our approach can be applied to other cases of jets in fluids and plasmas.