Weakly nonlinear wave energy flux and radiation stress

It is known that the wave action propagated in spectral wave models is a small steepness approximation of the observable wave action. For relevant steepness, we need higher order corrections to get a proper representation of the sea states [Janssen, 2009]. For the same reasons, other diagnostic variables should be corrected. Based on the fifth order Stokes solution obtained by Fenton [1985], Jonsson and Arneborg [1995] showed the importance of higher order corrections for determining the energy properties of long crested waves.

Proceeding from Longuet-Higgins and Stewart [1960], assuming a mean stream velocity, we see that how, using Krasitskii [1994] canonical transformations, we can derive general 2D weakly non linear corrections to the rate of transfer of energy across a surface fixed in space. For a monochromatic wave, the resulting equations are compared with truncated expressions given by Jonsson and Arneborg [1995], confirming that second order contributions (in terms of wave energy) can be relevant, depending on steepness and relative water depth.
After applying a proper statistical closure, the derived equations can be used to correct the wave energy properties of wave models spectra, for example to refine the informations transferred to a coupled circulation model.

John D. Fenton. A Fifth-Order Stokes Theory for Steady Waves. Journal of Waterway, Port, Coastal, and Ocean Engineering, 111(2):216–234, 1985. ISSN 0733-950X.
Peter a. E. M. Janssen. On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech., 637(November):1–44, 2009. ISSN 1469-7645.
Ivar G. Jonsson and Lars Arneborg. Energy properties and shoaling of higher-order stokes waves on a current. Ocean Engineering, 22(8):819–857, 1995. ISSN 00298018.
Vladimir P. Krasitskii. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech., 272(-1):1–20, 1994. ISSN 0022-1120.
M. S. Longuet-Higgins and R W Stewart. Changes in the form of short gravity waves on long waves and tidal currents. Journal of Fluid Mechanics, 8(04): 565–583, 1960.