Overturned internal capillary-gravity waves

A vortex sheet formulation of irrotational, incompressible Euler flow is used to compute periodic traveling waves at the interface between two constant-density, two-dimensional fluids, including waves with overturned crests. Branches of traveling waves are computed via numerical continuation, which are jointly continuous in the physical parameters: Bond number, Atwood number and mean shear. Global branches are computed, for various choices of parameters, illustrating the termination criteria of the global bifurcation theorem of Ambrose et al. (2015). The dependence of the branches, and their termini, on the physical parameters are probed via a boundary continuation method. Bifurcation surfaces are computed; these surfaces are both overturned and self-intersecting. The connection between the second harmonic of a Stokes' wave expansion and the shape of these surfaces is discussed.