The characteristics of the wintertime Arctic Oscillation (AO) and North Atlantic Oscillation (NAO) and their impacts on climate variability over the Northern Hemisphere are important metrics for evaluating a climate system model. Observational analyses reveal that the horizontal and vertical structures in the AO and NAO exhibit a meridional dipole and a large-scale barotropic pattern between the Arctic and mid-latitudes. Historical model simulations from the Energy Exascale Earth System Model (E3SM-HIST) are used to identify how well it captures these major climate modes. It is found that the simulated AO and NAO modes have spatial structures similar to the observed features. In addition, the observed frequency bands in the AO and NAO-related time variability are captured well in the E3SM-HIST simulation. Associated with the positive phase in wintertime AO and NAO, zonal flow and warm advection in mid-latitude continents are enhanced, along with stronger cold flow from enhanced northerly winds over high latitudes. These features are linked to the atmospheric circulation pattern reflected by lower SLP anomalies over the Arctic and higher SLP anomalies over the mid‐latitudes. In E3SM-HIST, these spatial associations and main structural features are analogous to those in observations. In the time-height evolution related to winter AO and NAO modes, it can also be seen that the simulations reproduce the downward propagating patterns in observations. Nevertheless, the vertical structures associated with AO and NAO in E3SM-HIST exhibit substantial biases in the lower stratosphere. The cause of these stratospheric biases is investigated using the strength of climatological stratospheric polar vortex (SPV) and wave activity fluxes. The results herein suggest that E3SM-HIST has a reasonable skill in reproducing the observed characteristics related to the winter AO and NAO, although there exist systematic biases in the associated climate variability.