We investigate weakly two-dimensional weakly nonlinear weakly dispersive surface waves propagating in a turbulent flow over a gradually sloping bottom. The waves are shown to be governed by a turbulently damped variable-coefficient Kadomtsev–Petviashvili equation with periodic boundary conditions. Equations governing the lowest-order mean currents in both directions as well as the equation describing the lowest-order mean surface elevation are also derived. Solutions for the wave equation are found numerically using a Fourier pseudospectral technique in space and finite differencing in the time-like variable.