## Abstract

The application of wave-current interaction theory in ocean circulation models has been extensively developed over the past decade, with formulations extended to three dimensions and based either on radiation stress formulations or on the Craik-Leibovich formulation. However, few of these studies consider the interaction of waves with relatively strongly sheared current, in which current shear can affect linear wave dynamics at leading order. The problem arises from the study of the evolution of highly concentrated sediment plumes developing at the mouth of small mountainous rivers. Although the annually averaged discharge of these small mountainous rivers is trivial compared to large rivers, during the extreme flooding events triggered by typhoon or tropic cyclones, these rivers, most of which located at tectonically active mountain belts, can carry highly concentrated sediment ( up to several g/l in the river plume) into the ocean. The magnitude of river discharge velocity at the river mouth may reach several m/s, comparable to the wave phase speed in coastal water. In addition, these flooding events usually coincide with very energetic wave conditions induced by the storms. Therefore, the interaction of waves with strongly sheared current becomes a very important dynamic process at this kind of river plumes. In our study, we establish a new framework to describe the interaction of small amplitude surface gravity waves and strongly sheared currents, where shear can exist in both vertical and horizontal directions. To begin with, we limit the derivation to the case of a narrow-banded slowly varying wave train propagating shoreward in the coastal ocean outside of the surf zone. Accordingly, our problem is assumed to be finite depth without wave breaking. Later we can extend the formulation to describe a spectrum of surface waves and include wave energy dissipation. In contrast to existing formulations, where waves at most feel a weighted depth-average current which follows from a weak-current, weak-shear approximation, the present formulation allows for an arbitrary degree of vertical shear, leading to a description of the vertical structure of waves in terms of solutions to the Rayleigh stability equation. The resulting formulation leads to a conservation law for wave action, and forcing terms for the description of mean flow using the Craik-Leibovich vortex force formulation. This new framework of wave-current interaction can be applied to numerical model based on ROMS/SWAN to study dynamics in coastal waters.## References

Garrett, C., 1976, "Generation of Langmuir circulations by surface waves-A feedback Mechanism, J. Mar. Res., 34, 116-130.

Kemp, P. H. & Simons, R. R., 1982, "The interaction between waves and a turbulent current: waves propagating with the current", J. Fluid Mech., 116, 227-250.http://dx.doi.org/10.1017/S0022112082000445

Kirby, J. T. & Chen, T-M., 1989, "Surface waves on vertically sheared ï¬‚ows: approximate dispersion relations", J. Geophys. Res., 94, 1013-1027.http://dx.doi.org/10.1029/JC094iC01p01013

Longuet-Higgins, M. S. & Stewart, R. W., 1960, "Changes in the form of short gravity waves on long waves and tidal currents". J. Fluid Mech., 8, 565-583.http://dx.doi.org/10.1017/S0022112060000803

Longuet-Higgins, M. S. & Stewart, R. W., 1961, "The changes in amplitudeof short gravity waves on steady non-uniform currents", J. Fluid Mech., 10, 529-549.http://dx.doi.org/10.1017/S0022112061000342

Longuet-Higgins, M. S. & Stewart, R. W., 1962, "Radiation stress and mass transport in gravity waves, with applications to 'surf-beats'", J. Fluid Mech., 13, 481-504.http://dx.doi.org/10.1017/S0022112062000877

Longuet-Higgins, M. S. & Stewart, R. W., 1964, "Radiation stress in water waves: a physical discussion, with applications", Deep-Sea Res. Oceanogr. Abstr., 11, 529-562.http://dx.doi.org/10.1016/0011-7471(64)90001-4

Longuet-Higgins, M. S., 1970, "Longshore currents generated by obliquely incident sea waves I.", J. Geophys. Res., 75, 6778-6789.http://dx.doi.org/10.1029/JC075i033p06778

McWilliams, J. C., Restrepo, J. M. and Lane, E. M., 2004, "An asymptotic theory for the interaction of waves and currents in coastal waters", J. Fluid Mech., 511, 135-178.http://dx.doi.org/10.1017/S0022112004009358

Mellor, G., 2003, "The three-dimensional current and surface wave equations", J. Phys. Oceanogr., 33, 1978-1989.http://dx.doi.org/10.1175/1520-0485(2003)033<1978:TTCASW>2.0.CO;2

Mellor G., 2008, "The depth dependent current and wave interaction equations: a revision", J. Phys. Oceanogr., 38, 2587-2596.http://dx.doi.org/10.1175/2008JPO3971.1

Uchiyama, Y., McWlliams, J. C., Shchepetkin, A. F., 2010, "Wave-current interaction in an oceanic circulation model with a vortex-force formalism: application to the surf zone", Ocean Modeling, 34, 16-35.http://dx.doi.org/10.1016/j.ocemod.2010.04.002

Warner, J. C., Sherwood, C. R., Signell, R. P., Harris, C. K., Arango, H. G., 2008, "Development of a three-dimensional, regional, coupled wave, current, and sediment-transport model", Computers & Geosciences, 34, 1284-1306.http://dx.doi.org/10.1016/j.cageo.2008.02.012