Convergence of Multilayer Nonhydrostatic Models in Relation to Boussinesq-Type Equations

Abstract

Multilayer nonhydrostatic models have gained popularity in the computation of ocean wave processes, but a recent attempt to make a connection to the Boussinesq-type approach raises some fundamental issues that require a rigorous assessment beyond the conventional framework of approximation. Such an endeavor is readily feasible for depth-integrated nonhydrostatic models because of the existence of an equivalent Boussinesq form. An examination of the governing equations shows that defining the nonhydrostatic pressure at layer interfaces for depth integration gives rise to a leading-order approximation of the dispersion relation distinct from the Taylor-series expansion. Introducing a parameterized nonhydrostatic pressure profile or increasing the number of layers is akin to retaining high-order terms in the Boussinesq-type equations with comparable rational function approximations of linear and nonlinear wave properties. Although both approaches converge to the exact solution at high order, the spatial derivatives in nonhydrostatic models always remain at first order. The simple numerical framework along with grid refinement techniques enable application of a single model over a wide range of spatial scales for practical application.