Abstract During northern winter the stratospheric polar vortex is highly variable — even breaking down completely during sudden stratospheric warmings. The strength of the stratospheric polar vortex is observed to be coupled to surface weather, primarily through swings in the phase of the Northern Annular Mode — which is nearly uuivalent to polar‐cap sea‐level pressure. Pressure changes (as a function of height) are largest at the surface, which is difficult to reconcile with the remote effects of stratospheric variability. This conundrum requires some mechanism(s) to amplify the polar sea‐level pressure anomalies. A simple, intuitive explanation has remained elusive. We use Japanese 55‐year Reanalysis (JRA‐55) reanalyses to investigate the reason that stratospheric variability causes large polar sea‐level pressure changes. The main result of this paper is that vertical movement of the polar tropopause is essential in understanding how surface pressure changes are generated. The strength of the polar vortex is strongly coupled to the height of the polar tropopause, resulting in stretching/compression of the polar tropospheric air column. This causes pressure changes that are largest at the surface. For example, the lowering of the polar tropopause during a sudden stratospheric warming leads to a compression and spin down of the tropospheric column and is therefore associated with higher polar sea‐level pressure. This mechanism can be compared to the textbook example of conservation of potential vorticity as an air column moves over a mountain range — except in this case the column depth changes because the stratosphere is changing the height of the tropopause. The observed vertical profile of pressure changes matches closely what would be expected from the stretching/compression mechanism. The higher polar sea‐level pressure associated with a sudden stratospheric warming is due mainly to mass moving into the polar cap within the troposphere, rather than rearranged mass in the stratosphere.