A consistent phase-averaged model of the interactions between surface gravity waves and currents

We formulate a model of the two-way interactions between surface gravity waves and ocean currents. The model couples the transport of wave action in the four-dimensional (horizontal) position–wavevector phase space with the Craik–Leibovich system for the currents. Coupling is via the Doppler shift in the dispersion relation governing action transport, and wave pseudomomentum in the Craik–Leibovich system. The velocity in the Doppler shift is a vertical integral of the Lagrangian mean velocity of the currents, with a weight that is consistent with the vertical structure of the pseudomomentum. This consistency ensures conservation of momentum and energy in the coupled wave–current system. The conservation properties of the wave–current model stem from an underlying variational structure. We derive this structure from that of the rotating Euler equations for an incompressible fluid with free surface by introducing a Lagrangian wave–mean decomposition, making simplifying approximations, and Whitham averaging. The wave–mean decomposition is constructed such that the Lagrangian mean velocity is non-divergent and satisfies a rigid-lid boundary condition. We apply the wave–current model to the problem of generation of inertial oscillations by surface waves originally considered by Hasselmann.