On Rayleigh expansion for nonlinear long water waves

We consider strongly nonlinear long waves on the surface of a homogeneous fluid layer. By modifying the formulation for the high-order spectral (HOS) method for waves in water of finite depth, we present a higher-order nonlinear system for the surface elevation and the velocity potential on the free surface to describe the two-dimensional evolution of large amplitude long waves. It is shown that the resulting system preserves the Hamiltonian structure of the Euler equations and can be transformed to the strongly nonlinear long-wave model for the depth-averaged velocity. Due to truncation of the linear dispersion relation for water waves, both the system for the surface velocity potential and that for the depth-averaged velocity are ill-posed when the order of approximation is odd and even, respectively. To avoid this ill-posedness, fully dispersive models are also proposed. Under the same order approximation, the long-wave model is found more effective for numeral studies of large amplitude long waves than the finite-depth model.