The shallow overturning circulation of the oceans transports heat from the tropics to the mid-latitudes. This overturning also influences the uptake and storage of anthropogenic carbon (Cant). We demonstrate this by quantifying the relative importance of ocean thermodynamics, circulation and biogeochemistry in a global biochemistry and circulation model. Almost 2/3 of the Cant ocean uptake enters via gas exchange in waters that are lighter than the base of the ventilated thermocline. However, almost 2/3 of the excess Cant is stored below the thermocline. Our analysis shows that subtropical waters are a dominant component in the formation of subpolar waters and that these water masses essentially form a common Cant reservoir. This new method developed and presented here is intrinsically Lagrangian, as it by construction only considers the velocity or transport of waters across isopycnals. More generally, our approach provides an integral framework for linking ocean thermodynamics with biogeochemistry.