Thickness-Weighted Mean Theory for the Effect of Surface Gravity Waves on Mean Flows in the Upper Ocean

Abstract

Abstract The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thickness-weighted mean (TWM) theory applied in a vertically Lagrangian and horizontally Eulerian coordinate system. Depth-dependent equations for the conservation of volume, momentum, and energy are derived. These equations allow for (i) finite amplitude fluid motions, (ii) the horizontal divergence of currents, and (iii) a concise treatment of both kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steady and monochromatic waves and a uniform turbulent viscosity, the TWM momentum equations are used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface, which are implicit in previous studies of the wave-induced modification of the classical Ekman spiral problem. The TWM approach clarifies, in particular, the surface momentum flux associated with the so-called virtual wave stress of Longuet-Higgins. Overall, the TWM framework can be regarded as an alternative to the three-dimensional Lagrangian mean framework of Pierson. Moreover, the TWM framework can be used to include the residual effect of surface waves in large-scale circulation models. In specific models that carry the TWM velocity appropriate for advecting tracers as their velocity variable, the turbulent viscosity term should be modified so that the viscosity acts only on the Eulerian mean velocity.