Radiation stress and the drift in gravity waves with Rayleigh friction

By utilizing a Rayleigh friction formulation to model the dissipative processes in the ocean, a Lagrangian analysis for the mean drift due to surface gravity waves is performed. The waves have amplitudes that vary slowly in time and space, and can be forced by a prescribed wind-stress distribution normal to the free surface. The ocean depth is constant, and the analysis is valid for arbitrary wavelength/depth ratios. The derived equations for the horizontal Lagrangian mean drift contain depth-dependent forcing terms that are proportional to the Stokes drift, expressing the external forcing of the waves. There are no depth-varying radiation stress-like terms in these equations. For waves along the x-axis (1-axis), the mean drift equations contain an additional forcing from a depth-independent pressure gradient identical to the divergence of the radiation-stress tensor component 22 S of Longuet-Higgins and Stewart (1960). From the surfaceboundary condition for the wave field, a new relation is derived for the conservation of mean wave momentum. Applying this condition, and integrating the mean Lagrangian drift equations in the vertical from the bottom to the free surface, the only waveforcing term in the derived mass transport equations is the divergence of the radiation-stress tensor component 11 S of Longuet-Higgins and Stewart, as one would expect from an Eulerian analysis.