The growth of waves by wind as a problem in nonequilibrium statistical mechanics

Abstract: In 1948, Casimir predicted a net attractive force between two perfectly conducting parallel plates due to electromagnetic vacuum fluctuations. By analogy, the interaction of two ships on a wavy sea has been named Maritime Casimir effect. This is an example of force generation in non-equilibrium systems. Lee, Vella and Wettlaufer showed it to be oscillatory as it is induced by the sharply peaked energy spectrum measured in the sixties by Pierson and Moskowitz for a fully developed sea; a sea whose state is independent of the distance over which the wind blows and the time for which it has been blowing. The aim of this project is to construct a theory for that spectrum and understand how the Maritime Casimir effect emerges from wind-wave interaction. Waves in the absence of wind, so-called water waves, are mainly characterized by dispersion and weak non-linearity. The coupling of both results in the instability of a wave packet to side-band perturbations in deep water. The growth rate can be calculated thanks to a non-linear Schrödinger equation, which is a universal model for weakly non-linear waves in a dispersive medium. Furthermore, this instability can be understood in the even more general framework of resonant wave-wave interaction. The evolution of deep water gravity waves is actually a sum of four-wave interaction processes and triadic interactions should be added for capillary waves. That evolution is strongly affected by the presence of turbulent wind because it transfers energy to the waves. The growth rate of wind waves was calculated by Miles in 1957 on the basis of the weak air-water coupling. His formula involves the solution of the hydrodynamic Rayleigh equation at the critical level, which is the height at which the phase speed of the wave is equal to the wind speed. We develop an efficient numerical scheme to compute it and then compare the theory with the observational data compiled by Plant. We eventually propose asymptotic solutions of the Rayleigh equation for a generic wind profile, which will be useful to get a better understanding of the experimental results.