Higher–order hydrodynamic modulation: theory and applications for ocean waves

Under the two-scale hydrodynamic model for ocean surface waves, short waves are modulated hydrodynamically by long waves. An exact numerical simulation of the two-scale hydrodynamic process shows that the most commonly used modulation transfer function (MTF), which is a linear approximation, does not capture all of the features caused by the inherent nonlinear nature of the physical processes involved. We rederive the linear MTF and generalize it to include local acceleration and finite depth effects. The phase of the linear MTF is shown to be independent of the direction of long modulating waves. This is an artefact of the linearization of the nonlinear equations. A higher-order theory is also derived based on the truncated Hamiltonian for long modulating waves and dissipation by wave-wave interaction for modulated waves. This new theory includes higher-order derivatives of the source functional and, therefore, short-wave dissipation. Consequently, the phase of the modulation depends on the relative direction of long and short waves. It is shown that while the linear hydrodynamic MTF leads to higher-order statistics equivalent to the bispectrum, the new second-order MTF induces the trispectrum of surface elevation. A succinct derivation for the third-order MTF is given for completeness.