On the coupling of wave and three-dimensional circulation models: Choice of theoretical framework, practical implementation and adiabatic tests

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Many theoretical approaches and implementations have been proposed for the coupling of the three-dimensional ocean circulation with waves. The theoretical models are reviewed and it is shown that the formulation in terms of the quasi-Eulerian velocity circumvents the essential difficulty of alternative formulations for the Lagrangian mean velocity. Namely, models based on this Lagrangian velocity require an estimation of wave-induced motions to first order in the horizontal gradients of the wave field in order to estimate the vertical flux of wave pseudo-momentum. So far, only three-dimensional wave models have been able to provide these estimates, and all published theories based on the simpler Airy theory are not consistent at the leading order, because they ignore or incorrectly estimate the vertical momentum flux. With an adiabatic example on a sloping bottom it is shown that this inconsistency produces very large spurious velocities. These errors are independent of the slope for the inviscid case, and are still significant when a realistic vertical mixing is applied. A quick diagnostic of the potential accuracy of a theoretical model is the vertical profile of the wave-induced forcing terms: if it is not uniform over depth in adiabatic conditions then it will produce spurious artificial flow patterns in conditions with shoaling waves. Although conceptually more challenging, the quasi-Eulerian velocity theories only introduce minor modifications of the solution procedure for the standard primitive equations: a modification of the surface boundary condition for the mass conservation, the addition of the Stokes drift in the tracer advection equations, and sources of momentum and turbulent kinetic energy with associated surface and bottom fluxes. All the necessary modifications of primitive equation models are given in detail. This implementation is illustrated with the MARS3D model, which passes the test of the adiabatic shoaling waves.