Large amplitude internal solitary waves in a two-layer system of piecewise linear stratification

We study large amplitude internal solitary waves in a two-layer system where each layer has a constant buoyancy frequency (or Brunt–Väisälä frequency). The strongly nonlinear model originally derived by Voronovich [J. Fluid Mech. 474, 85 (2003)] under the long wave assumption for a density profile discontinuous across the interface is modified for continuous density stratification. For a wide range of depth and buoyancy frequency ratios, the solitary wave solutions of the first two modes are described in detail for both linear-constant and linear-linear density profiles using a dynamical system approach. It is found that both mode-1 and mode-2 solitary waves always point into the layer of smaller buoyancy frequency. The width of mode-1 solitary waves is found to increase with wave amplitude while that of mode-2 solitary waves could decrease. Mode-1 solitary wave of maximum amplitude reaches the upper or lower wall depending on its polarity. On the other hand, mode-2 solitary wave of maximum amplitude can reach the upper or lower wall only when the interface is displaced toward the shallower layer; otherwise, the maximum wave amplitude is smaller than the thickness of the deeper layer. Streamlines and various physical quantities including the horizontal velocity and the Richardson number are computed and discussed in comparison with the recent numerical solutions of the Euler equations by Grue et al. [J. Fluid Mech. 413, 181 (2000)].