The impacts of Stratospheric Aerosol Injection (SAI) on the atmosphere and surface climate depend on when and where the sulfate aerosol precursors are injected, as well as on how much surface cooling is to be achieved. We use a set of CESM2(WACCM6) SAI simulations achieving three different levels of global mean surface cooling and demonstrate that unlike some direct surface climate impacts driven by the reflection of solar radiation by sulfate aerosols, the SAI-induced changes in the high latitude circulation and ozone are more complex and could be non-linear. This manifests in our simulations by disproportionally larger Antarctic springtime ozone loss, significantly larger intra-ensemble spread of the Arctic stratospheric jet and ozone responses, and non-linear impacts on the extratropical modes of surface climate variability under the strongest-cooling SAI scenario compared to the weakest one. These potential non-linearities may add to uncertainties in projections of regional surface impacts under SAI.