Based on a variable-coefficient Kadomtsev–Petviashvili (KP) equation, the topographic effect on the wave interactions between two oblique internal solitary waves was investigated (JFM, 2018). In the absence of rotation and background shear, the model set-up featuring idealised shoaling topography and continuous stratification was motivated by the large expanse of continental shelf in the South China Sea. When the bottom is flat, the evolution of an initial wave consisting of two branches of internal solitary waves can be categorised into six patterns depending on the respective amplitudes and the oblique angles measured counterclockwise from the transverse axis. Using theoretical multi-soliton solutions of the constant-coefficient KP equation, we select three observed patterns and examine each of them in detail both analytically and numerically. The effect of shoaling topography leads to a complicated structure of the leading waves and the emergence of two types of trailing wave trains.

Nevertheless, the KP equation (at least theoretically) requires the variations in the transverse direction is one-order smaller than those in the propagating direction, and more importantly, it is an uni-directional equation,  which indicates that the KP equation is plausibly inadequate for some general cases. Recently, we have derived an bi-directional propagating isotropic model, modified Benney-Luke (mBL) equation, for the description of 3D internal solitary waves and we used it to investigate the internal solitary wave-wave interactions further.

References:

1. C. Yuan, Z. Wang, “On diffraction and oblique interactions of horizontally two-dimensional internal solitary waves”, Journal of Fluid Mechanics 936, A20 (2022).

2. C. Yuan, Z. Wang, X. Chen, “The derivation of an isotropic model for internal waves and its application to wave generation”, Ocean Modelling 153, 101663 (2020).

3. C. Yuan*, R. Grimshaw, E. Johnson, Z. Wang, “Topographic effect on oblique internal wave–wave interactions”, Journal of Fluid Mechanics 856, 36–60 (2018).