Boussinesq Equations with Improved Doppler Shift and Dispersion for Wave/Current Interaction

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Abstract
Boussinesq-type equations with improved dispersion characteristics for the combined motion of waves and currents are introduced. 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 vary on a larger scale than the wave-length scale. Boussinesq-type equations are derived by explicit use of four scales ν, δ, ϵ and μ representing the particle velocity and the surface elevation of the total wave-current motion, as well as the wave-nonlinearity and the wave-dispersion, respectively. Firstly, equations are derived in terms of the depth-averaged velocity to obtain a generalization of the equations of Yoon & Liu (1989) to allow for stronger currents. Secondly, these equations are formulated in terms of the velocity variable at an arbitrary z-location resulting in an improved dispersion relation which corresponds to a Padé [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 equations of Nwogu (1993). Finally, this formulation is enhanced to achieve Padé [4,4] dispersion characteristics. Model results for 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.