Wave-current interaction based on an enhanced Boussinesq approach

This paper focuses on the derivation of Boussinesq-type equations for the combined motion of waves and currents in shallow water areas. The ambient current is assumed to be uniform over depth and to have a magnitude as large as the shallow water wave celerity, allowing for the consideration of wave blocking of fairly long waves. The temporal variation of the current is ignored, while the spatial variation is assumed to take place on a larger scale than the wave-length scale. Boussinesq-type equations for the combined wave-current motion are derived by explicit use of four scales representing the particle velocity, the surface elevation of the total wave-current motion, the wave-nonlinearity and the wave-dispersion. Firstly, equations are derived in terms of the depth-averaged velocity to obtain a generalization of the equations of Yoon and Liu (1989) [Yoon, S.B., Liu, P.L.F., 1989. Interaction of currents and weakly nonlinear water waves in shallow water. J. Fluid Mech. 294, 71-92.] to allow for stronger currents. Secondly, these equations are formulated in terms of the velocity variable at an arbitrary z-location. This results in Pade [2, 2] dispersion characteristics retaining the correct form of the Doppler shift, i.e. corresponding to a Pade [2, 2] expansion in the wave number of the squared intrinsic celerity for the fully dispersive linear theory. For vanishing currents, these equations reduce to the those of Nwogu (1993) [Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coastal Ocean Eng. ASCE 119 (6), 618-638.]. Finally, this formulation is enhanced to achieve Pade [4, 4] dispersion characteristics. Model results for the case of monochromatic and bichromatic waves being fully or partly blocked by opposing currents are given and the results are shown to be in reasonable agreement with theoretical calculations based on the wave-action principle. (C) 1998 Elsevier Science B.V.