Volume transport in the Southern Ocean is investigated using a high-resolution, fully turbulence-resolving direct numerical simulation. The response of zonal transport, abyssal overturning and upper overturning to varying wind stress and turbulence is quantified. We find that zonal transport is fully eddy saturated when all scales of motion are resolved. Therefore, zonal transport is insensitive to strengthening winds, primarily due to a concomitant increase in fluctuating kinetic energy which results in no net change in the meridional stratification. The abyssal overturning strengthens significantly with increasing turbulence, while the upper cell shrinks in magnitude and spatial extent. Fluctuating flow features play a dominant role in driving the abyssal overturning, while mean flow (specifically, mean Ekman transport due to surface winds) drives the upper overturning. Therefore, we observe a linearly increasing response of the upper overturning to increasing winds. The abyssal cell also strengthens with increasing winds, driven by enhanced diapycnal mixing and fluctuating flow features. Therefore, it appears that the sensitivity of the abyssal overturning to winds is highly dependent on the degree to which turbulence is characterised. These findings indicate that resolving turbulence is critically important to investigating volume transport in the Southern Ocean.