Approximation of wave action conservation in vertically sheared mean flows

We develop asymptotic expressions for wave action density and action flux, using an extension of Kirby & Chen (1989)'s perturbation solution for weakly-sheared currents allowing for a basic flow with Froude number F = U/root gh = O(1) but with weak vertical shear. The accuracy of the expressions for action density and flux is established by comparison to analytic results for a current with constant shear, and to numerical results for a field case involving a buoyant ebb-tidal plume with strong vertical shear and for a case involving a numerically determined profile for a wind-driven current. We compare our results to those from recent work of Quinn et al. (2017), and find unresolved discrepancies in that prior work. We provide additional suggestions for efficiently implementing the required extensions in coupled wave/circulation models using a Taylor series expansion based on conditions at peak frequency and direction. These results generalize the previous work of Banihashemi et al. (2017) to motions in two horizontal dimensions, and cover the determination of the wave action.