Stochastic Transport and Wave Interactions for Multiscale Surface Gravity Waves: Part II: Kinetic Theory and Ocean-Wave Applications

Abstract

Building on the stochastic variational framework established in the companion paper, we investigate here the linearized stochastic water-wave system, consisting of a large-scale stochastic wave dynamics coupled to transport dynamics for the small-scale correlation modes. Within this framework, we develop, in the deep-water regime, a kinetic theory for surface gravity waves interacting with unresolved stochastic velocity fields. An energy analysis yields a wave-action kinetic equation exhibiting two distinct regimes: a diffusive scattering regime and a quartic interaction regime with structural similarities to Hasselmann--Zakharov theory. In the present framework, these effective quartic interactions arise through stochastic transport of unresolved fluctuations by the large-scale flow rather than through classical intrinsic resonant nonlinearity. Scaling laws are derived for the diffusion tensor and the effective growth rate, revealing a Miles-type production--dissipation mechanism. Using JONSWAP spectra, we then compare the strength of stochastic transport and classical Hasselmann interactions. For realistic oceanic values of unresolved velocity variance ($σ_u \approx 0.1\,\mathrm{m\,s^{-1}}$) and decorrelation time ($τ_c \approx 10\,\mathrm{s}$), stochastic transport is found to compete with, and often exceed, classical four-wave interaction rates over broad spectral ranges. The transport intensity $S=σ_u^2τ_c$ emerges as a key parameter controlling the transition between interaction regimes. These results suggest that unresolved stochastic transport may play a substantially larger role in spectral evolution than is commonly represented in operational wave models, and motivate the inclusion of transport-induced source terms alongside standard resonant interaction closures.