Large internal solitary waves on a weak shear

Abstract

Large amplitude solitary internal waves of permanent form propagating in a stratified shallow fluid between the free surface and a horizontal bottom are described by the amplitude equation obtained by a regular asymptotic procedure, which incorporates a complicated nonlinearity and Korteweg–de Vries (KdV) dispersion. It is discussed how the structure of stratification and shear affects wave properties. The particular case of a constant buoyancy frequency and a quadratic polynomial for the ambient shear for the flow under free surface is considered in detail analytically. It is shown that for such profiles, the equation for the wave amplitude reduces to the mixed-modified KdV equation and finite amplitude waves obey it up to the breaking level. Rogue waves could appear in this case, and the condition for their generation is identified. More complicated shear profiles lead to higher-order nonlinearities, which produce the multiscaled pyramidal wave patterns, asymmetric bores, and various instabilities. Such wave structures are studied numerically. An analytical bore-like solution having both exponential and algebraic asymptotes is presented.