Horizontal 2D fully dispersive nonlinear mild slope equations

A set of horizontal 2D fully dispersive wave equations with fourth order nonlinearity for mild slope bottom is developed. The equations are of integral-differential type, formulated by using a Fourier integral to express the velocity potential and by applying the relation between horizontal and vertical flow velocities on still water level. The bottom variation is treated by introducing the bottom slope terms plus bottom rapidly undulating terms; the former account for the effect of mean slope and the latter for the effect of bottom rapid undulation around the mean slope. Two approaches are proposed for the kernel functions of bottom rapidly undulating terms. The linear version of the model can be seen as the extension of conventional mild slope equation to the broad-banded and nonlinear model. The numerical computation for the integral term of the equations is studied to increase the computational efficiency. The numerical examples are presented to validate the abilities of the model to simulate the nonlinear wave evolution and to illustrate the effects of varying topography on wave motions.