Inertial, barotropic and baroclinic instabilities of the Bickley jet in two - layer rotating shallow water model

International audienceWe undertake a detailed study of inertial instability of the barotropic Bickley jet and its nonlinear saturation in the 2 layer rotating shallow water (RSW) model on the f -plane, and compare it with the classical barotropic and baroclinic instabilities. We start with analytical and numerical investigation of the linear stability problem under hypothesis of strict homogeneity in the along-flow direction ("symmetric instability"). The unstable modes are identified and their growth rates and phase velocities are determined. The dependence of the instability on Rossby and Burger numbers of the jet is investigated. The nonlinear development of the instability is then studied with the help of high-resolution well-balanced finite-volume numerical code recently developed for multi-layer RSW, which is initialized with the most unstable mode found from the linear stability analysis. It is shown that symmetric inertial instability is saturated by reorganization of the mean flow, without full homogenization of the anticyclonic region, where the unstable modes reside. We then study along the same lines the fully two-dimensional problem, and compare the results with the one-dimensional analysis. The barotropic instability competes with inertial instability in this case. We show that for sufficiently strong anticyclonic shears the inertial instability still has the dominant growth rate in the long-wave sector. The "symmetric" one-dimensional inertial instability turns to be a liming case of the ageostrophic baroclinic instability. Yet, the "asymmetric" inertial instability at small but non-zero wavenumbers has the higher growth rate. We study again the nonlinear development of the most unstable inertial mode, and show that homogenization of the region of strong anticyclonic shear of the flow takes place on average. The reorganization of the flow reveals a high degree of complexity, with coherent structure formation and inertia-gravity wave emission, both giving rise to the substantial enhancement of ageostrophic motions. We compare these processes for inertial baroclinic and barotropic instabilities