Derivation and optimization of Boussinesq equations for internal waves with current effects in a two-fluid system

A set of Boussinesq equations for simulating internal waves propagating in a two-fluid system in the presence of parallel currents is theoretically derived in this paper. Major mathematical techniques used are referred to Liu [1,2] for the case without current effects. In the derived equations, two undetermined parameters for denoting the positions of velocity potentials which makes the equations more flexible. Most past equations under different assumptions can be recovered by simplifying present equations. In the following linear analysis, the two flexible parameters can be determined by matching the linear dispersion relation with that of Stokes wave theory with the help of the Padé approximation. By choosing the parameters appropriately, Boussinesq equations can be applicable in a much wider range of layer thicknesses. Based on present results, numerical simulation will be performed in the future.