  With a new unifying model for layered rotating shallow-water (RSW) and
quasi-geostrophic (QG) equations, this paper sheds light on the relation
between these two sets of equations. We propose here a new formulation of the
quasi-geostrophic equations as a projection of the rotating shallow-water
equations. This QG formulation uses the same prognostic variables as RSW,
namely velocity and layer thickness, restoring the proximity of these two sets
of equations. It allows direct access to the ageostrophic velocities hidden in
geostrophic velocities resolved by the QG equations. It opens the path of
studying the difference between QG and RSW using the same underlying numerical
discretization. We illustrate this new possibility on a vortex shear
instability and a double-gyre configuration.
