Nine hydrographic sections are combined in an inverse box model of the Southern Ocean south of ;128S. The inverse model has two novel features: the inclusion of independent diapycnal flux unknowns for each property and the explicit inclusion of air‐sea fluxes (heat, freshwater, and momentum) and the water mass transformation they drive. Transformation of 34 3 106 m3 s21 of Antarctic Surface Water by air‐sea buoyancy fluxes, and cooling and freshening where Subantarctic Mode Water outcrops, renews cold, fresh Antarctic Intermediate Water of the southeast Pacific and southwest Atlantic. Relatively cold, fresh mode and intermediate water enter the subtropical gyres, are modified by air‐sea fluxes and interior mixing, and return poleward as warmer, saltier mode and intermediate water. While the zonally integrated meridional transport in these layers is small, the gross exchange is approximately 80 3 106 m3 s21. The air‐sea transformation of Antarctic surface water to intermediate water is compensated in the Southern Ocean by an interior diapycnal flux of 32 3 106 m3 s21 of intermediate water to upper deep water. The small property differences between slightly warmer, saltier intermediate water and cold, fresh Antarctic Surface Water results in a poleward transfer of heat and salt across the Polar Front zone. Mode and intermediate water are crucial participants in the North Atlantic Deep Water overturning and Indonesian Throughflow circulation cells. The North Atlantic Deep Water overturning is closed by cold, fresh intermediate water that is modified to warm, salty varieties by air‐sea fluxes and interior mixing in the Atlantic and southwest Indian Oceans. The Indonesian Throughflow is part of a circum-Australia circulation. In the Indian Ocean, surface water is converted to denser thermocline and mode water by air‐sea fluxes and interior mixing, excess mode water flows eastward south of Australia, and air‐sea fluxes convert mode water to thermocline water in the Pacific.