The barotropic vorticity (BV) balance is fundamental when interpreting the ocean gyre circulation. Here we propose an intercomparison of vorticity equations for the depth‐integrated flow applied to ocean models. We review four distinct variants of the BV balances, each giving access to diagnostic equations for the depth‐integrated 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 spatial 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 arise from interactions of the flow with slanting topography. We identify significant spurious topographic torques related to the model's C‐grid discretizations, and we suggest ways to address them. In the depth‐integrated and BV balances, bottom vortex stretching and bottom pressure torque drive the flow interaction with topography, respectively. Contrary to Sverdrup theory, the wind stress curl, although dominant in the interior Subtropics, becomes a minor player anywhere significant bottom velocities prevail. The geostrophic contour vorticity balance highlights the limits of barotropic models of the ocean circulation through the so‐called JEBAR term. Finally, the transport divergence vorticity balance stresses the limitations of Ekman plus geostrophic dynamics for the mass balance closure in ocean models. This framework should encourage ocean modellers to diagnose more routinely momentum and vorticity equations. Plain Language Summary Ocean gyre theories involve the key role played by the wind variations across latitudes to force an interior flow. However, recent work has put forward the role played by bottom topography as a guide, or an obstacle, to the gyre circulation. The general framework employed in those theories is the so‐called barotropic vorticity equation involving the balancing of the spin induced by the Earth rotation in an ocean in motion. This work proposes a review on oceanic vorticity balances that synthesizes the informations that they provide about the ocean circulation. We then apply this framework to an ocean model used for climate projections. Our analysis confirms the key role played by interactions with topography in driving the gyre circulation. The ocean bottom topography is a geological constant, so that it constitutes a long‐lasting constraint for the circulation. Finally, we also stress the large effect of the ocean circulation formulation in the computer programs that make up an ocean model and should be designed carefully. Key Points The large‐scale ocean circulation includes four distinct vorticity balances for the depth‐integrated flow Those vorticity balances are discretized in the VoBiN diagnostic module designed for C‐grid ocean models such as the NEMO platform The barotropic vorticity equation of a global NEMO climate simulation is dominated by Sverdrup and topographic balances