Wind and Wave Induced Currents in a Rotating Sea with Depth-varying Eddy Viscosity

Abstract A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of a version of the Navier-Stokes equations in Lagrangian coordinates. The Coriolis effect and the effect of a depth-dependent eddy viscosity are included. (The eddy viscosity is taken to depend on the Lagrangian vertical coordinate ĉ.) Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. The vertical variation of the eddy viscosity causes an extra source term to appear in the equation for the evolution of the current profile. This additional source of momentum within the water column is exactly balanced by an extra term in the surface boundary condition, which in turn represents the contribution to wave dissipation caused by the eddy viscosity within the water column being different from its s...