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 ?.) 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 surface value. The equations were solved numerically, using a constant wind stress and monochromatic wave field simultaneously applied in the same direction at time t = 0. The eddy viscosity ? was assumed to be proportional to depth, using Madsen's relation (? = ?kKu*?, where kK is von Kármán's constant, u* = (? long-term average values ranged from 2.2% to 2.8% of the wind speed at the 10 m level, and were directed between 12° and 17° to the right of the wind and wave direction (in the Northern Hemisphere). The deviation of the current from the wind direction is closer to observed drift current observations than the corresponding results for a constant eddy viscosity. The Lagrangian mean current is surprisingly close to the current obtained from Madsen's theory, even though Madsen does not account explicitly for the effect of surface waves. The theory can easily take account of random sea states. There are good prospects for coupling it with the output of a numerical model for surface gravity waves, using the wave model's input and dissipation source terms.