Non-Boussinesq effect on the internal solitary wave propagation under a Lagrangian-like description for various pycnocline thickness conditions

Abstract

The Dubreil–Jacotin–Long (DJL) equation is an exact model for the steady propagation of internal solitary waves (ISWs) under Boussinesq approximation. The aim of this study is to discuss the non-Boussinesq effect on ISWs initialized by DJL equation for various pycnocline thickness conditions. The simulated results show that the non-Boussinesq effect can indeed lead to instability of DJL equation, and the pycnocline thickness will affect the amplitude of re-stable ISWs and the separation process of tail waves. Moreover, in order to extract features of the non-Boussinesq instability and the location of the tail wave separation, we propose an effective analytical approach for the quasi-traveling wave problem inspired by the definition of classical Lagrangian coherent structure (LCS). The improved finite-time Lyapunov exponent with a moving window (FTLEmw) is a Lagrangian-like description with considering the prior information of physical phenomenon. For a steady ISW propagation in the Boussinesq system, the repelling and attracting FTLEmv fields are symmetrical and the field ridge behaves as a continuous spiral topology, which suggests that particles on different streamlines in the flow structure have different rotation periods, while the streamlines are not actually completely closed due to the forward propagation of the wave. During the main unstable stage of non-Boussinesq effect, the topology of separated tail wave which is disconnected from the ISW main part can be clearly distinguished. The tail wave separation locations identified by the improved FTLEmv field and the classical FTLE field are not the same, which illustrates the difference between the two perspectives of flow structure change and fluid particle motion. In addition, the leading edge, trailing edge and main part of ISWs will gradually be divided into three-part FTLEmv topologies in the thicker pycnocline conditions.