The barotropic vorticity balance is fundamental to interpret the ocean
gyre circulation. However, it remains an overlooked diagnostic in ocean
circulation models. Here we propose a general theory of barotropic
vorticity balances for use in ocean models. We show that there exist
four distinct barotropic vorticity balances, each giving access to
diagnostic equations for the depth-integral ocean circulation, either
meridional, across geostrophic contours or its divergence. We then
formulate those balances in the Vorticity Balances in NEMO (VoBiN)
diagnostic package aimed at the NEMO ocean platform and more generally
C-grid ocean models. We show that space discretization of the equations
of motion have profound implications for those vorticity balances.
Finally, we diagnose the main balances of a global ocean climate
simulation. In all vorticity balances, topographic torques, both
physical and numerical, arise from interactions of the flow with
slanting topography. Contrary to Sverdrup theory, the wind stress curl,
although dominant in the interior Subtropics, becomes a minor player
anywhere significant bottom velocities prevail. Depth-dependent
vorticity dynamics show that the barotropic balance emerges from the
coupling of the surface (frictional) and bottom (topographic) boundary
layers to the geostrophic interior through the generation of vertical
motion. The vorticity balance for the depth-averaged momentum equations
highlights the limits of barotropic models of the ocean circulation.
Finally, the vorticity balance for the transport divergence highlights
the key role played by numerical torques for the oceanic mass balance.
This framework should encourage ocean modellers to diagnose more
routinely momentum and vorticity equations.