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

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.