Interaction of finite-amplitude waves with vertically sheared current fields

A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.