UC San Diego

In this thesis we study nonlinear and breaking deep-water surface waves. First, we consider the vorticity generated by an individual breaking wave, drawing on classical literature on vortex generation by impulsive forcing. We employ this theory to develop a scaling argument for the relationship between the generated circulation and the variables characterizing the breaking wave. This model is then compared to limited laboratory experiments, and good agreement is found.

We next pursue a related problem, namely the partitioning of energy in the breaking induced currents, between the turbulent and mean flow. This is the inverse problem to the vortex generation model, as we work backwards from the structure of the induced flow, using existing results on vortex dynamics to find the energy necessary to generate the (half) vortex ring induced by breaking. This yields a theoretical model for the ratio of the energy in the mean flow currents to the total energy lost from the wave field, in terms of the characteristic variables of the breaking wave. This relationship is then examined numerically, using a direct numerical simulation of the two-phase air-water Navier-Stokes equations, and agreement between the model, the numerical experiments, and limited available laboratory data is found.

One approach to breaking is through the focusing of wave packets. Here, we theoretically and numerically examine weakly nonlinear narrow-banded wave packets. By employing moment evolution equations of the modified nonlinear Schrodinger equation (MNLSE), we derive new predictions for the geometry, kinematics, and dynamics of focusing wave packets. In particular, we predict that as the wave group focuses: the group velocity increases; the packet leans forward; and the energy equipartition (between kinetic and potential) breaks down. These results are then corroborated by numerical integration of both the MNLSE and the fully nonlinear evolution equations for irrotational inviscid deep-water surface gravity waves.

Finally we present several ongoing projects related to nonlinear and breaking surface waves. First, we present a Lagrangian for deep-water surface gravity waves and discuss its numerical implementation. Next, a virial theorem for surface gravity waves is derived. Finally, we derive the criterion for the Benjamin-Feir instability based on a variance identity for the nonlinear Schrodinger equation.