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The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thicknessweighted 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 equationsallowfor(i)finiteamplitudefluidmotions,(ii)thehorizontaldivergenceofcurrents,and(iii)aconcise treatment of both kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steadyandmonochromaticwaves anda uniformturbulentviscosity,theTWMmomentum equationsare used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface, which are implicit in previousstudiesofthewave-inducedmodification oftheclassicalEkmanspiralproblem. TheTWMapproach clarifies,inparticular,thesurfacemomentumfluxassociatedwiththeso-calledvirtualwavestressofLonguetHiggins.Overall,theTWMframeworkcanberegardedasanalternativetothethree-dimensionalLagrangian 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.

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