Rapid computations of steep surface waves in three dimensions, and comparisons with experiments

A novel fully nonlinear, rapid method for computations of ocean surface waves in three dimensions is outlined in this chapter. The essential step is to use Fourier transform to invert the integral equation over the ocean surface that solves the Laplace equation. This leads to a relation for the normal velocity of the free surface that is useful for iterations. This relation has a global contribution that is obtained by FFT and local contribution that is evaluated by rapidly converging integrals in the horizontal plane. The global part, evaluated by FFT, captures the most essential parts of the wave field. Together with an efficient time integration of the prognostic equations, where the linear part is integrated analytically and a time variable step size control is used for the nonlinear part, this results in a highly rapid computational strategy. Methods for efficient nonlinear wave generation and absorption are outlined. Conservation of various quantities of the wave field and convergence are discussed. Fully nonlinear computations of very large (rogue) wave events at sea, like the Camille and Drauper waves, are compared to laboratory measurements of the waves using Particle Image Velocimetry. The method is used to exemplify the generation, propagation and shoaling of very long wave phenomena like tsunamis, and class I and class II instabilities in water of infinite and finite depth. Both steady and oscillatory crescent wave patterns are predicted up to breaking. Competition between class I and class II instabilities is discussed.