Weakly nonlinear focused wave group on arbitrary shear based on second-order theory

Abstract

Using a recently developed second-order theory of  irregular waves on a current varying arbitrarily with depth [1], we study dispersive focussing of wave groups and their dependence on the current profile.  Long-crested wave groups are presumed to propagate obliquely on a flow with non-linear dependence on depth. We investigate the wave surface elevation and wave kinematics of a focused wave group. Nonlinear wave surface elevations vary with the angle between the wave propagation and flow, and it is found that they increase to a maximum where the current increases adversely for larger depth. For wave kinematics, the horizontal wave-induced velocity shows significantly different behaviours due to the presence of shear current.The development of the highest crest as a function of propagation time is studied for a wave group which linearly focusses at a particular position and time in the absence of shear. The adverse shear causes an increase in maximum height. Exponential and linear depth dependence is compared, and a real, measured shear current [2] is used showing the practical importance of the results.The results complement our recent study of weakly nonlinear wave statistics in the presence of arbitrary vertical shear, which showed among other observations, a strongly increased probability of rogue waves in the presence of an adverse vertical shear, in accordance with field observations by Zippel and Thomson [2]. [1] Zheng, Z, Li, Y and Ellingsen, S Å 2023 “Statistics of weakly nonlinear waves on currents with strong vertical shear” Phys. Rev. Fluids (accepted, in press)[2] Zippel, S and Thomson, J 2017 “Surface wave breaking over sheared currents: Observations from the Mouth of the Columbia River J. Geophys. Res.: Oceans 24 127102.