Generation of waves on a free surface by quadratic shear currents

Surface waves are ubiquitous in nature. They range from large amplitude waves on the ocean, small-scale ripples in a pond, to thin films flowing down an incline in engineering applications. In an oceanic context, they grow by extracting energy from the shear flow, subsequently breaking due to various nonlinear processes, thereby generating sea spray and communicating between the ocean and the atmosphere, dictating the mass, momentum and energy exchanges.

In this study, we look at the growth of waves in a shear flow bounded between a rigid wall and a free surface, using linear stability analysis and characterizing the instabilities in the system. The base state shear flow in water is chosen to be a single-parameter family of quadratic velocity profiles, allowing us to access various kinds of flows ranging from a forward bulging to a backward bulging profile along with intermediate monotonic profiles, which can be concave or convex. We perform both viscous and inviscid calculations for this family of shear flows, using exact analytical, asymptotic and numerical methods.