Bidirectional Whitham type equations for internal waves with variable topography

Abstract

In the context of oceanic internal gravity waves, one of the most widely-used theories on examining wave dynamics and interpreting observational data is the Korteweg–de Vries (KdV) equation. Nonetheless, the characters of unidirectional propagation and unbounded phase and group velocities restrict its application to some general cases (Benjamin et al., 1972). Thus, using the Dirichlet–Neumann operator with the rigid-lid approximation, we derive both bidirectional and unidirectional Whitham type equations in the Hamiltonian framework, which retain the full linear dispersion relation of the Euler equations. The effect of topography is also incorporated in modeling due to its practical relevance, although the invoked scaling plausibly excludes the accommodation of a significant bottom variation. There are no analytic solutions of internal solitary waves explicitly given in the newly proposed equations, even though these equations possess a concise form. Therefore, a modified Petviashvili iteration method is implemented to obtain the numerical solutions to circumvent this difficulty. Facilitated by these techniques, several numerical experiments are investigated and compared among different models: the KdV equation, the Whitham type equations, and the primitive equations. The discrepancies and similarities between the various models jointly indicate the advantage of full dispersion and bidirectional propagation and, thus, the effectiveness of the Whitham type equations.