A non-equilibrium system in a steady state: wind waves in the open ocean

We derive scaling laws for the steady spectrum of wind excited waves, assuming two inviscid fluids (air and water) and no surface tension, an approximation valid at large speeds. In this limit there exists an unique (small) dimensionless parameter $\epsilon$, the ratio of the mass densities of the two fluids, air and water, independently of the wind speed. The smallness of $\epsilon$ allows to derive some important average properties of the wave system. The average square slope of the waves is of order $|ln (\epsilon^2)|^{-1}$, a small but not very small quantity. This supports the often used assumption of small nonlinearity in the wave-wave interaction. We introduce an equation to be satisfied by the two-point correlation of the height fluctuations.