Fully nonlinear solitary waves in a layered stratified fluid

Abstract

Fully nonlinear solitary waves in a layered stratified fluid, each layer with a constant Brunt–Väisälä frequency, are investigated. The stream function satisfies the Helmholtz equation in each layer and is expressed in terms of singularity distributions. As the Green function, a combination of Bessel functions of order zero, of the second and first kind is advocated. Computations performed for two- and three-layer cases show that the wave speed increases with increasing stratification of the top layer. The thickness of the pycnocline increases with wave amplitude when the top layer is homogeneous but decreases when the top layer is stratified. The wave width depends little on the pycnocline thickness. The fluid velocity may exceed the wave speed in the upper part of the water column when the top layer is stratified, but is always smaller than the wave velocity if the top layer is homogeneous. A large vertical excursion of the individual isopycnals contributes to a small Richardson number $Ri$. The smallest value of $Ri$ is observed in the main body of the fluid. Solitary waves of increasing strength are investigated until the wave-induced fluid velocity equals the wave speed, or the minimal $Ri$ becomes smaller than one quarter. The results may support experimental studies of breaking internal solitary waves.