The positive trend of the Southern Annular Mode (SAM) will impact the Southern Ocean’s role in Earth’s climate, however the details of the Southern Ocean’s response remain uncertain. We introduce a methodology to examine the influence of SAM on the Southern Ocean, and apply this method to a global ocean–sea-ice model run at three resolutions (1$^{\circ}$, 1/4$^{\circ}$ and 1/10$^{\circ}$). Our methodology drives perturbation simulations with realistic atmospheric forcing of extreme SAM conditions. The thermal response agrees with previous studies; positive SAM perturbations warm the upper ocean north of the windspeed maximum and cool it to the south, with the opposite response for negative SAM. The overturning circulation exhibits a rapid response that increases/decreases for positive/negative SAM perturbations and is insensitive to model resolution. The longer term adjustment of the overturning circulation, however, depends on the representation of eddies, and is faster at higher resolutions.