Shear instability in shallow water

This study considers the linear stability of shear flows in shallow water. It explores instabilities related to the classical incompressible (Rayleigh) instability, and those caused by the over-reflection of surface gravity waves. Numerical solutions of the linear stability problem are presented, together with analytical arguments elucidating the role of finite potential vorticity gradients. The slow development of marginally unstable modes is considered for almost inviscid flows. This is described by an evolution equation for the amplitude of the unstable mode, coupled to a critical-layer potential vorticity equation. This reduced system presents a compact description of the linear stability problem and allows exploration of viscous effects.