Commentary on 'The Three-Dimensional Current and Surface Wave Equations' by George Mellor

The lowest order sigma-transformed momentum equation given by Mellor (J. Phys. Oceangr. 2003) takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent due to inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure p and coordinate transformation function s to first order in parameters that define the large scale evolut ion of the wave field, such as the bottom slope. Unfortunately there is no analytical expression for p and s at that order. A numerical correction method is thus proposed and verified. Alternativ e coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inc onsistency nor require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directl y relevant to the mean flow evolution.