A fully nonlinear Boussinesq model for surface waves. Part 2. Extension toO(kh)(4)

A Boussinesq-type model is derived which is accurate to O(kh)(4) and which retains the full representation of the fluid kinematics in nonlinear surfac e boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity poten tial. In order to achieve a (4,4) Pade representation of the dispersion rel ationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of int ernal kinematics is greatly improved over existing O(kh)(2) approximations, especially in the intermediate to deep water range. The model equations ar e first examined for their ability to represent weakly nonlinear wave evolu tion in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a ran dom sea as well as nonlinear dispersion and stability effects in the nonlin ear Schrodinger equation for a narrow-banded sea state. We then examine num erical properties of solitary wave solutions in shallow water, and compare model performance to the full solution of Tanaka (1986) as well as the leve l 1, 2 and 3 solutions of Shields & Webster (1988).