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