A high-order nonlinear Boussinesq-type model for internal waves over a mildly-sloping topography in a two-fluid system

Abstract

With a rigid lid assumption on the upper boundary in a two-fluid system, a new high-order Boussinesq-type model for internal waves over a mildly-sloping topography is derived. The model is formulated in terms of computational velocity vectors defined at mid-water depth in each fluid. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the new equations. High accuracy in linear, nonlinear, shoaling and kinematic aspects is achieved and the model is applicable from deep to shallow water. For an equivalent water depth in upper and lower fluid, the model can be applied up to kh2 ≈ 10 (where k is the wave number and h2 is the water depth of the lower fluid) in linear dispersion, up to kh2 ≈ 7.0 in the second nonlinear property within 2% error, and up to 0 < kh2 < 10 in the linear shoaling property. Compared with most existing Boussinesq-type models, the accuracy of the horizontal and vertical velocity profiles along the water column is improved significantly and is applicable to up to kh2 ≈ 4.48.