A mathematical representation of random gravity waves in the ocean

We give a mathematical representation of random ocean surface waves in the gravity-wave regime. The so-called random gravity waves are treated as an asymptotic phenomenon when the wind pressure acting on the surface and the dissipation become negligible. We adopt a phenomenological model for the wind pressure such that it excites a surface consisting of wind-driven sea and swell. Starting from the Navier-Stokes equations, we derive a general system of the first-order perturbation equations governing the surface waves, and solve them with this wind pressure as the excitation. The resulting solution is decomposed into a part which is asymptotically dominant and another which is asymptotically negligible. The former consists of two groups: one which is a sum of superpositions of uncorrelated plane waves having approximate dispersion relations and the other a sum of random plane waves with their wavenumbers and frequencies approximately satisfying the dispersion relation. They correspond to the dominant parts of the wind-driven sea and the swell, respectively. Finally, we derive a limiting form of the directional-frequency spectrum in the gravity-wave regime.