On regularizing the strongly nonlinear model for two-dimensional internal waves

Abstract

To study the evolution of two-dimensional large amplitude internal waves in a two-layer system with variable bottom topography, a new asymptotic model is derived. The model can be obtained from the original Euler equations for weakly rotational flows under the long-wave approximation, without making any smallness assumption on the wave amplitude, and it is asymptotically equivalent to the strongly nonlinear model proposed by Choi and Camassa (1999) [3]. This new set of equations extends the regularized model for one-dimensional waves proposed by Choi et al. (2009) [30], known to be free from shear instability for a wide range of physical parameters. The two-dimensional generalization exhibits new terms in the equations, related to rotational effects of the flow, and possesses a conservation law for the vertical vorticity. Furthermore, it is proved that if this vorticity is initially zero everywhere in space, then it will remain so for all time. This property–in clear contrast with the original strongly nonlinear model formulated in terms of depth-averaged velocity fields–allows us to simplify the model by focusing on the case when the velocity fields involved by large amplitude waves are irrotational. Weakly two-dimensional and weakly nonlinear limits are then discussed. Finally, after investigating the shear stability of the regularized model for flat bottom, the effect of slowly-varying bottom topography is included in the model.