Water wave–current interaction with logarithmic wind-induced current

The effect of wind-induced logarithmic current shear on the wave dispersion relationship, the linear wave velocity, and the Stokes-drift velocity as a function of depth has been theoretically studied in regular deep-water wave conditions by accounting for the vorticity caused by the current. A formula showing how wind-induced current shear affects the wave dispersion relationship is derived. Case studies show that the current shear influences the wavelengths, which affect, e.g., dynamic pressure and wave velocity. In the linear analysis, the logarithmic current shear has a singularity at the mean free surface, leading to unrealistic predictions of the horizontal wave velocity and the Stokes drift velocity in a thin layer near the mean free surface. Since the wind-induced current must satisfy boundary conditions on the instantaneous free surface, strong nonlinearities occur in the predictions of the horizontal wave velocity near the free surface. The effect of current shear on the linear wave excitation loads on a pontoon of a planned floating bridge in Bjørnafjorden is assessed. An error analysis indicates that neglecting current shear introduces acceptable errors, which are comparable in magnitude to those caused by disregarding viscous flow separation around the structure.