Numerical simulations of the flow of a continuously stratified fluid, incorporating inertial effects

Abstract

A high-resolution spectral numerical scheme is developed to solve the two-dimensional equations of motion for the flow of a density stratified, incompressible and inviscid fluid. This method incorporates the inertial terms neglected in the Boussinesq approximation. Thus it aims, inter alia, to extend the numerical simulations of Rottman et al. (J. Fluid Mech. 306 (1996) 1) and Aigner et al. (Fluid Dyn. Rev. 25 (1999) 315). The validity of the numerical model is tested with two applications. The first application is the resonant flow over isolated bottom topography in a channel of finite depth, which has been studied extensively in the Boussinesq approximation. The inclusion of inertial effects, that is the influence of the stratification on the acceleration terms discarded in the Boussinesq approximation, allows the comparison of the solution to the unsteady governing equations with the fully nonlinear, but weakly dispersive resonant theory of Grimshaw and Yi (J. Fluid Mech. 229 (1991) 603). This paper focuses on topography of small-to-moderate amplitudes and slopes, and for conditions such that the flow is close to linear resonance for the first internal wave mode. The vertical position of wave breaking is determined. The second application is the propagation of large-amplitude internal solitary waves with vortex cores, again in a channel of finite depth. The existence and permanence of these types of waves derived by Derzho and Grimshaw (Phys. Fluids 9(11) (1997) 3378) is verified. Furthermore, the time-dependent solution provides measurements of the structure of the vortex core and maximum adverse velocity at the top boundary.