Climate variability and change in the Southern Hemisphere (SH) are influenced by the Southern Annular Mode (SAM) and are closely related to changes in the kinematic properties of the SH surface zonal winds. The SAM and SH surface zonal winds have strong effects on the atmospheric and oceanic circulation system. In this study we investigate the variability and trend in the SAM and position and strength of the surface zonal wind stress (TAUX), using two ensembles of simulations covering the historical record from the Energy Exascale Earth System Model (E3SM‐HIST and Atmospheric Model Intercomparison Project) for 1979–2014. In addition, performance of two CO forcing simulations from the E3SM (E3SM‐1pctCO2 and 4xCO2) is assessed to examine the sensitivity of the variability and changes in the SAM and SH surface TAUX to climate forcing. In general, all E3SM simulations tend to capture the dominant feature of the SAM pattern reasonably well. The annual SAM index in the E3SM‐HIST simulation shows a significant increasing trend. These features are similar to the trends in the strength (along with poleward shift in the position) of the annual surface TAUX. For the climatological surface TAUX position and strength, the two CO forcing simulations show slightly poleward movement and stronger intensity, while the E3SM‐HIST is equatorward and weaker than observations. In the relationship between the SAM and surface TAUX, we show that the SAM index exhibits a positive (negative) relationship with the strength (position) of the surface TAUX in the variability for all seasons and annual mean.