It is well known that wind blowing over the ocean creates waves, thereby transferring momentum into the ocean. But if there is a pre-existing wave, does that affect the rate of air-sea momentum transfer? Does the initial perturbation to the flat sea state that eventually becomes a bona fide wave affect its own response to the wind? A nonlinear stability analysis in the Lagrangian frame, extending Miles' (1957) classical theory of wind-wave generation, answers these questions. By explicitly accounting for the actual velocity experienced by a particle (the Lagrangian drift), the analysis predicts a significant suppression of wave growth with increasing wave steepness, providing a new explanation for prior experimental results showing this trend. See the preprint here (currently in revision), or this webpage I made to present this work at the Division of Fluid Dynamics.

The LANS-α equations have been used successfully as a turbulence closure model in many contexts, including in atmosphere and ocean models. The mathematical roots of these equations can be traced back to Jean Leray's 1934 work, regularizing the nonlinear term in Navier-Stokes to prove the existence of weak solutions. The LANS-α equations are thus fundamentally different from other turbulence closure models, as they modify the nonlinear term rather than the dissipation. However, there have been numerical stability issues in some implementations of the system, with the kinetic energy growing without bound. This work analyzes the energy in the framework of a fast singular limit, which also provides leading-order inertia-gravity wave interactions. See the recent preprint here (currently under review).