Blow-up instability in shallow water flows with horizontally-nonuniform density

The mechanisms of instability, whose development leads to the occurrence of the collapse (blow up), have been studied in the scope of the rotating shallow water flows with horizontal density gradient. Analysis shows that collapses in such models are initiated by the Rayleigh–Taylor instability and two scenarios are possible. Both the scenarios evolve according to a power law $(t_{0}-t)^{\gamma}$, where $t_{0}$ is the collapse time, with $\gamma=-1,-2$, and $\gamma=-2/3,-1$ for the isotropic and anisotropic collapses, respectively. The rigorous criterion of collapse is found on the base of integrals of motion.