The interaction between water currents and the dispersion of surface waves

When wind blows over water, the associated water current can be modeled as an inviscid parallel flow, U(z), where z is the vertical coordinate. At the water surface, waves propagate due to two restoring forces: gravity and surface tension. The phase speed, c, of a wave with wavenumber k in quiescent water was found by George Bidell Airy in the 19th century. The water current can change the relation between c and k in a non-trivial manner, because the shear induced vorticity. We regard the waves as neutral perturbations of the flow, and solve the Rayleigh equation using asymptotic techniques along the lines as those used in a similar problem [1]. We obtain new dispersion relations, c=c(k), as perturbation series whose first few terms show good agreement with the numerical solutions. The competition between the shear and gravity/surface tension is expressed by the Froude/Weber number, which have a profound influence on how strongly the current affects the propagation of waves. We find a modification of the Stokes drift which may have consequences on the Langmuir circulation, and more generally on the spread of pollutants on water surfaces.

[1] Bonfils et al. (2022).