Rotational flow underlying coupled surface and internal waves. I: Eulerian perspective

In this paper we examine the flow generated by coupled surface and internal small-amplitude water waves in a two-fluid layer model, where we take the upper layer to be rotational (constant vorticity) and the lower layer to be irrotational. The presence of vorticity greatly complicates the underlying analysis, yet it generates a rich array of otherwise unobservable phenomena such as the presence of critical layers, and stagnation points, in the fluid interior. We employ a phase-plane analysis to elucidate the qualitative behaviour of streamlines for a range of different coupled-wave, and vorticity, regimes. Although the water waves considered are linear in the fluid dynamics sense, the dynamical systems which govern their motion are nonlinear.