Eulerian versus Lagrangian Approaches to the Wave-Induced Transport in the Upper Ocean

Abstract

It is demonstrated that the Eulerian and the Lagrangian descriptions of fluid motion yield the same form for the mean wave-induced volume fluxes in the surface layer of a viscous rotating ocean. In the Eulerian case, the volume fluxes are obtained in the familiar way by integrating the horizontal components of the Navier?Stokes equation in the vertical direction, as seen, for example, in the book by Phillips. In the direct Lagrangian approach, the perturbation equations for the second-order mean drift are integrated in the vertical direction. This yields the advantage that the form drag, which is a source term for the wave-induced transports, can be related to the virtual wave stress that acts to transfer dissipated mean wave momentum into mean currents. In particular, for waves that are periodic in space and time, comparisons between empirical and theoretical relations for the form drag yield an estimate for the wave-induced bulk turbulent eddy viscosity in the surface layer. A simplistic approach extends this analysis to account for wave breaking. By a generalization from a wave component to a wave spectrum, a set of equations for the wave-induced transport in the surface layer is derived for a fully developed sea. Solutions are discussed for an idealized spectral formulation. The problem is formulated such that a numerical wave prediction model can be used to generate the wave-forcing terms in a numerical barotropic ocean surge model. Results from the numerical simulations with a wave-influenced surge model are discussed and compared with similar results from forcing the surge model only by the traditional mean horizontal wind stress computed from the 10-m wind speed. For the simulations presented here, the wave-induced stress constitutes about 50% of the total atmospheric stress for moderate to strong winds.