A New Formulation Of Boussinesq-type Wave-current Equations

In coastal waters, water waves and currents often coexist and interact with each other. The currents include vortex currents and the breaking wave-driven currents(such as the longshore currents and rip currents) and so on. The existence of current can change the wave length and amplitude (Doppler shift);cause wave refraction, diffraction and breaking. Especially,when the current is strong,The effects on wave transformation will be substantial. These effects can result in overall redistribution of wave energy. Boussinesq equations are the appropriate model applicable for these problem of wave-current interaction due to its nonlinear feature.Based on Euler equations, Zou(2000) have developed a new Boussinesq model which can consider the strong current. As the presence of currents and the wave-currents interaction will lead the change of the mean depth. Especially,The mean depth change may be large for strong currents and large bathymetry variation. The present study presents a new form of enhancement to remedy this problem. By changing the physical velocity in the former equations to a computational velocity and adding higher order terms (O(εμ4)) to the equations, the drived model can meet the requirement of incorporating the mean depth change due to currents in the resulting wave-current model and have accurate to the Pade[4,4] expansion of the Stokes’linear dispersion relation. The latter method is similar to the approach of MS98but with the total mean depth replacing the still water depth,which leads to the dispersion relation of a Pade[4,4] expansion with the total mean depth hr=h+ηc being the water depth variable. In the present model, the finite difference scheme is used to discretize the spatial derivatives and the Adams-Bashforth-Moulton scheme is employed to perform the time integration. Incident wave boundary and absorbing boundary with sommerfeld boundary are applied at the incident and open boundary respectively. The Shapiro filter is applied to filt the high frequency oscillations in the process of numerical solution.Wave propagation and transformation in water with uniform depth,vortex current,rip current and current over submerged bar are numerically simulated and analyzed. Comparing the simulated results with published numerieal result,experimental results and analytical results, vertified the present numerieal model. The results of considering the mean water depth change caused by current and results without considering the mean water depth change were compared.it show that the wave-length of considering the mean water depth change is shorter than the result of unconsidered the mean water depth change, and the wave amplitude is longer, it is also showed that if the mean water changer caused by current is small,the difference between the results of the considering the mean water change or unconsidering the mean water change is small,and the larger the mean water change caused by current.the larger the differce the result. So,the present model can give a more accurate result when the current is strong.