  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 evolution 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. Alternative
coordinate transforms that allow a separation of wave and mean flow momenta do
not suffer from this inconsistency 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 directly
relevant to the mean flow evolution.
