Approximations of the Wave Action Equation in Strongly Sheared Mean Flows

Abstract:

Spectral wave models based on the wave action equation typically use a depth uniform current to account for the current effect on waves. The choice of this depth uniform current is formally arbitrary; recently, several studies have made use of the depth-weighted current due to Skop (1987) or Kirby & Chen (1989). These applications do not typically take into account the change in the expression for the wave action flux, which results from the dependence of the depth-weighted current on wavenumber k.  Recently, Dong and Kirby (2014) have derived a multiple-scale asymptotic theory for the case of wave-current interaction with strongly sheared mean flows. At leading order, the Rayleigh equation governs the wave motion, and a wave action equation (identical to that of Voronovich, 1976) follows from consideration of a solvability condition at second order. Incorporation of this result in existing wave models would require extensive code revisions and is left for future work. Instead, we investigate the suitability of the perturbation solutions by examining predictions of wave action density and flux in comparison to results from the full theory. To facilitate this, a revision to the perturbation solution of Kirby & Chen (1989) has been derived based on a strong current; weak shear approach. Wave action and action flux are evaluated using both depth averaged and depth-weighted approximations, and are compared with numerical solutions based on the Rayleigh equation. In particular, deriving an expression for group velocity as the derivative of frequency with respect to k suggests that an additional term k∂u/∂k should appear. This term has not been accounted for in existing extensions to the depth-uniform current equations. To illustrate the importance of this effect in the absolute group velocity, a comparison of group velocity with and without the additional term is made. The results show that significant errors are incurred in neglecting the additional term, pointing to the need for a more careful application of the approach in existing models.