Instability of a shallow-water potential-vorticity front

A straight front separating two semi-infinite regions of uniform potential vorticity (PV) in a rotating shallow-water fluid gives rise to a localized fluid jet and a geostrophically balanced shelf in the free surface. The linear stability of this configuration, consisting of the simplest non-trivial PV distribution, has been studied previously, with ambiguous results. We revisit the problem and show that the flow is weakly unstable when the maximum Rossby number ${\textsfi R}\,{>}\,1$ . The instability is surprisingly weak, indeed exponentially so, scaling like $\exp[-4.3/({\textsfi R} - 1)]$ as ${\textsfi R}\,{\to}\,1$ . Even when ${\textsfi R}\,{=}\,\sqrt{2}$ (when the maximum Froude number ${\textsfi F}\,{=}\,1$ ), the maximum growth rate is only $7.76\,{\times}\,10^{-6}$ times the Coriolis frequency. Its existence nonetheless sheds light on the concept of ‘balance’ in geophysical flows, i.e. the degree to which the PV controls the dynamical evolution of these flows.