Statistics of weakly nonlinear waves on currents with strong vertical shear

We investigate how the presence of a vertically sheared current affects wave statistics, including probability rogue waves, and apply it to real-world case using measured spectral shear data from Mouth Columbia River. A theory for weakly nonlinear waves valid second order in steepness is derived, used analyze statistical properties surface waves; extends classic by Longuet-Higgins [J. Fluid Mech. 12, 3 (1962)] allow an arbitrary depth-dependent background flow, $U(z)$, with $U$ horizontal velocity along main direction propagation $z$ vertical axis. Numerical statistics are collected large number realisations random, irregular sea-states following JONSWAP spectrum, on linear exponential model currents varying strengths. quantities presented compared range theoretical expressions literature; particular distribution elevation, maxima, crest height; exceedance maximum height among $N_s$ skewness elevation distribution. find that no-shear conditions, opposing ($U'(z)>0$) leads increased nonlinear-wave distribution, while ($U'(z)<0$) has opposite effects. With spectrum profile River estuary Zippel & Thomson Geophys. Res: Oceans 122, 3311 (2017)] our second--order predicts significantly reduced enhanced during ebb flood, respectively, adding support notion need be accounted modelling prediction.