Wave-forced barotropic currents

Waves rolling in to shallow seas will start to dissipate as a result of the bottom friction. The wave momentum will decrease from the dissipation process, and there is a transfer of momentum that accelerates an Eulerian bottom current. Water piles up toward the coast, thereby generating a return flow. When rotation is included, the return flows accelerate an alongshore current that moves to the left of the direction of the incoming wave field (Northern Hemisphere). With the assumption that the turbulent exchange can be mimicked by a constant exchange coefficient, there is a fairly simple analytical solution that relates the strength of the barotropic current to the incoming wave field. For deep water, that is, H ≫ 2υt/f, where f is the Coriolis force and νt is the turbulent exchange coefficient, the strength of the alongshore barotropic current becomes 3/2 of the Stokes drift near the bottom or 3ωka2/4 sinh(kH), where ω and k are the wave angular frequency and wavenumber, and a is the amplitude of the wave. Notably, the above expression is equal to the strength of the Eulerian streaming generated under a progressive wave by the wave-induced Reynold stresses in the viscous wave bottom boundary layer.