The Atlantic meridional overturning circulation (AMOC) is sustained by changes in seawater density, but the link between those local density changes and the global-scale overturning circulation has not been fully resolved, in part due to ambiguity in the barotropic component of the flow. We define a new depth coordinate that compensates for the barotropic vortex stretching that balances planetary vorticity changes associated with meridional motion. In this framework, the divergence of barotropic flow is identically zero, and geostrophic divergence is determined solely by zonal density gradients. As a result, the geostrophic component of the Eulerian overturning circulation can be diagnosed directly from the density field without the need for a barotropic reference velocity or assumed level of no motion. We apply this framework to diagnose AMOC structure in a high-resolution numerical model. This application provides a direct link between density structure and overturning, clarifying how diapycnal processes at high latitudes influence Eulerian overturning at lower latitudes.