A mathematical explanation for how wind generates waves was given almost 70 years ago by John W. Miles, and has remained a leading theory since then. The idea is that an initial, sinusoidal perturbation on the ocean surface interacts resonantly with background wind shear at a critical level in the air, causing an instability that translates into wave growth. There have been numerous extensions to this theory, but the role of the Lagrangian mean flow -- the velocity a fluid parcel actually experiences -- in this instability mechanism was not understood. Importantly, the Lagrangian mean flow also contains wave-induced currents. See the recent preprint here.

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 (preprint coming soon).