Nonlinear and dispersive free surface waves propagating over fluids with weak vertical and horizontal density variation

We consider the change in fluid density in a depth-integrated long-wave model. By allowing horizontal and vertical variation of fluid density, a depth-integrated model for long gravity waves over a variable-density fluid has been developed, where density change effects are included as correction terms. In particular, a two-layer fluid system is chosen to represent vertical density variations, where interfacial wave effects on the free surface are accounted for through direct inclusion of the velocity component of the interfacial wave. For the numerical implementation of the model, a finite-volume scheme coupled with an approximate Riemann solver is adopted for leading-order terms while cell-centred finite-volume methods are utilized for others. Numerical tests in which the density field is configured to vary either horizontally or vertically have been performed to verify the model. For horizontal variation of fluid density, a pneumatic breakwater system is simulated and fair agreement is observed between computed and measured data, indicating that the current induced by the upward bubble flux is responsible for wave attenuation to some degree. To investigate the effects of internal motion on the free surface, a two-layer fluid system with monochromatic internal wave motion is tested numerically. Simulated results agree well with the measured and analytical data. Lastly, nonlinear interactions between external- and internal-mode surface waves are studied numerically and analytically, and the model is shown to have nonlinear accuracy limitations similar to existing Boussinesq-type models.