Nonlinear surface gravity waves on a shear layer

Abstract

Abstract Content of image described in text. We study large-amplitude travelling gravity waves on the free surface of an infinitely deep fluid with a shear layer of constant vorticity extending to a finite depth. A boundary-integral formulation is developed by using conformal mapping to describe the two free boundaries of the problem and is solved numerically using a pseudospectral method. The linear wavespeed c c $c$ satisfies a cubic dispersion relation, which essentially arises from the combination of two gravity modes and a shear mode. For certain ranges of parameters, the collision of these modes leads to linear instability. The nonlinear solutions corresponding to one of the three values of c c $c$ were studied by Milinazzo & Saffman (1990 J. Fluid Mech. 216, 93–101). These solutions are dominated by the surface gravity mode; however, the solutions associated with the remaining two values of c c $c$ were overlooked. We study these new solutions in detail and find distinct properties from those calculated by Milinazzo & Saffman, such as the emergence of large-amplitude waves on the vorticity interface and changes in the relative phase between the two free boundaries. Moreover, bifurcations between shear and gravity modes are shown to emerge between solution branches of disparate wavenumbers under resonance conditions, which results in significant qualitative changes in the solution profiles across very small regions of parameter space.