We apply residual mean theory of tracer transport, in which eddy transfer plays a fundamental role, to develop scalings and idealized numerical models of the Southern Ocean uptake of transient tracers. The streamline-averaged numerical model, which represents transport in the meridional plane, captures the observed distributions of CFC-11, bomb-Δ14C, and anthropogenic CO2. The model reproduces the observed relationship between CFC-11 and bomb-Δ14C and suggests that the upper branch of the residual overturning flow in the Southern Ocean is about 14 Sv, supporting previous inferences based on the observed buoyancy distribution and air-sea buoyancy fluxes. Scale analysis suggests that the limit of fast air-sea gas exchange is applicable to CFC-11, for which the surface concentration is close to equilibrium and cumulative ocean uptake is largely determined by physical transport processes. In the slow gas exchange limit, applicable to bomb-Δ14C, the surface concentration is far from equilibrium and the cumulative uptake is most sensitive to the parameterization of the gas transfer coefficient. Anthropogenic CO2 falls between those two limit cases and is sensitive to both transport processes and the gas transfer coefficient. Sensitivity studies using the streamline-averaged model suggest that uncertainties in air-sea buoyancy fluxes in current climatologies result in significant uncertainty in estimates of Southern Ocean uptake of anthropogenic CO2 based on circulation and biogeochemistry models driven by, or brought into consistency with, the climatological fluxes. This uncertainty is sufficient to explain a significant amount of the spread in a recent model comparison study.