Diffusion of surface gravity waves by submesoscale turbulence at the sea surface

Surface gravity waves are an important mechanism by which the ocean exchanges energy, momentum, heat, and gases with the overlying atmosphere. Sea-surface currents modify the wavenumber, direction, and amplitude of the waves, with the potential to significantly affect the spatial variability of the wave field, including the statistics of wave breaking. Although the effect of currents on waves under the WKB approximation is well known, the sparseness of ocean current observations makes it difficult to explicitly account for wave-current interactions in numerical surface wave models. Thus, instead of explicit resolutions of sea-surface currents, a statistical approach is required. Here we use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the sea-surface velocity are stationary and homogeneous, we derive an expression for a diffusivity tensor of wave action. With only the assumption of spatial homogeneity, the action diffusion tensor can be computed from the two-point sea-surface velocity autocorrelation tensor. But for isotropic currents further simplifications are possible so that the action diffusivity is expressed in terms of the kinetic energy spectrum of the flow. A Helmholtz decomposition of the sea-surface currents into rotational (vortical) and irrotational (compressible) components shows that the compressible and irrotational component of the sea-surface velocity field has no effect on the action diffusivity. Our analytic result for the diffusivity is validated by Monte Carlo ray-tracing simulations and averaging over an ensemble of stochastic velocity fields.