An ocean-climate model that shows high fluxes of anthropogenic carbon dioxide into the Southern Ocean, but very low storage of anthropogenic carbon there, agrees with observation-based estimates of ocean storage of anthropogenic carbon dioxide. This low simulated storage indicates a subordinate role for deep convection in the present-day Southern Ocean. The primary mechanism transporting anthropogenic carbon out of the Southern Ocean is isopycnal transport. These results imply that if global climate change reduces the density of surface waters in the Southern Ocean, isopycnal surfaces that now outcrop may become isolated from the atmosphere, tending to diminish Southern Ocean carbon uptake.