Nonlinear development of inertial instability in a barotropic shear

Inertial instability is investigated numerically in a two-dimensional setting in order to understand its nonlinear stage and saturation. To focus on fundamental mechanisms, a simple barotropic shear U(y) = tanh y on the f-plane is considered. The linear stability problem is first solved analytically, and the analytical solutions are used to benchmark numerical simulations. A simple scenario of the nonlinear development of the most unstable mode was recurrently observed in the case of substantial diffusivity: while reaching finite amplitude the unstable mode spreads laterally, distorting the initially vertical instability zone. This process produces strong vertical gradients which are subsequently annihilated by diffusion, making the flow barotropic again but with the shear spread over a wider region. In the course of such evolution, unexpectedly, strong negative absolute vorticity anomalies are produced. In weakly diffusive simulations, the horizontal spreading of the unstable motions and the enhancement of the anticyclonic vorticity extremum persist, but small-scale motions/instabilities render the flow considerably more complex. It is known that the barotropic component of the final state can be predicted from the conservation of momentum. Our simulations confirm the relevance of this simple prediction in the cases investigated regardless of resolution and diffusion. The baroclinic component of the final state is also analyzed and three types of structures are identified: persistent stationary stratification layers, subinertial waves trapped in the anticyclonic shear, and freely propagating inertia-gravity waves. The subinertial waves and the stratification staircase have clear signatures and can therefore help to identify the regions that have undergone inertial instability.