Singularity formation during relaxation of jets and fronts toward the state of geostrophic equilibrium

Relaxation of rectilinear jets and fronts towards the state of geostrophic equilibrium (geostrophic adjustment) is studied in three models of increasing complexity. In the one-dimensional rotating shallow water model we perform high-resolution numerical simulations of adjustment and show that wave-breaking and shock formation in the jet core are ubiquitous. We show that adjusted final states are attained in the cases where their existence is not guaranteed by the theory. In the two-layer rotating shallow water model, we show how the baroclinic effects change the scenario of adjustment. We establish criteria of existence and uniqueness of the adjusted states and demonstrate that the adjustment process can lead to a state where the jet traps the fast oscillations. Symmetric instability is shown to appear for strong enough jets. In the full continuously stratified primitive equations model we analyze the final states of the relaxation process by using the Lagrangian variables. We demonstrate that existence of the adjusted state is further restricted as compared to the previous models and show that singularity formation on route to the geostrophic equilibrium is generic.