Inhibiting shear instability induced by large amplitude internal solitary waves in two-layer flows with a free surface

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin-Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.