Development of self-consistent statistical description of ocean waves in the coastal area is an important problem in physical oceanography. On deep water the main nonlinear effect is the four-wave resonant interaction described by Hasselmann kinetic equation for spectrum of wave action. Three-wave interaction becomes also important at finite depth, and comes to dominate in shallow water. Three-wave interactions of gravity waves are non-resonant; they become almost resonant on very shallow water only. This fact makes the development of consistent, well-justified analytical statistical theory of gravity waves at finite depth a difficult problem. It is unlikely to be solved by any heuristic modification of the Hasselmann equation that is written for time evolution of the pair correlation function. For the proper description, one has to derive a coupled system of equations for time evolution of pair and triple correlation functions. This project will derive, justify, and study these equations through the following steps: (1) derive the coupled system of equations for pair and triple correlation functions and make sure that this system preserves energy and on deep water goes to the classical Hasselmann equation; (2) generalize the obtained equation for the case of varying bottom topography and presence of current; (3) develop the numerical code for solution of equation for correlations, including into equations the input from wind and the dissipation of this input due to white-capping; (4) perform a massive numerical simulation of primordial dynamic equations in full 3-dimension geometry and use the obtained data for justification of statistical equations; and (5) on the base of deterministic numerical experiments find the function of dissipation due to white-capping on shallow water.
Intellectual Merit The development of an analytical model for statistical description of waves in shallow water will be a breakthrough in the theory of nonlinear waves, which would have practical consequences such as for coastal wave forecasting. The numerical codes for solution of the coupled equation for pair and triple correlations will be an advancement in computational geophysics. The verification of approximate analytical theory by a more exact and detailed numerical simulation could be a model applied to other topics in the future.
Broader Impacts The method of using the coupled system of equations for correlation functions of different orders for description of wave turbulence is not limited to gravity waves on shallow water. Similar methods can be applied to the theory of long internal gravity waves in ocean and atmosphere, to the theory of Rossby waves in atmosphere of rotating planets. Numerical experiments can improve our understanding of breaking internal waves as well. Some of the results will likely be applicable to different branches of nonlinear wave dynamics such as magnetohydrodynamics, plasma physics, and nonlinear optics.