Lattice Boltzmann method for variable density shallow water equations

A lattice Boltzmann method is developed for solution of a form of the shallow water equations that is suitable for flows which are fully mixed in the vertical direction but have variable density in the horizontal plane. In the present approach, double distribution functions are applied: one for the shallow water flows and the other for the mass transport. Direct coupling between the water flow and mass transport is achieved by updating the flow density from the concentration during simulation. Accuracy and applicability of the model are demonstrated by two numerical tests: the stationary hydrostatic equilibrium of liquid of variable density in a tank with non-uniform bed terrain, and the horizontal diffusion of species with an initial Gaussian distribution of concentration in a uniform flow field.