  According to the Snowball Earth hypothesis, Earth has experienced periods of
low-latitude glaciation in its deep past. Prior studies have used general
circulation models (GCMs) to examine the effects such an extreme climate state
might have on the structure and dynamics of Earth's troposphere, but the
behavior of the stratosphere has not been studied in detail. Understanding the
snowball stratosphere is important for developing an accurate account of the
Earth's radiative and chemical properties during these episodes. Here we
conduct the first analysis of the stratospheric circulation of the Snowball
Earth using ECHAM6 general circulation model simulations. In order to
understand the factors contributing to the stratospheric circulation, we extend
the Statistical Transformed Eulerian Mean framework. We find that the
stratosphere during a snowball with prescribed modern ozone levels exhibits a
weaker meridional overturning circulation, reduced wave activity, stronger
zonal jets, and is extremely cold relative to modern conditions. Notably, the
snowball stratosphere displays no sudden stratospheric warmings. Without ozone,
the stratosphere displays slightly weaker circulation, a complete lack of polar
vortex, and even colder temperatures. We also explicitly quantify for the first
time the cross-tropopause mass exchange rate and stratospheric mixing
efficiency during the snowball and show that our values do not change the
constraints on CO$_2$ inferred from geochemical proxies during the Marinoan
glaciation ($\sim$635 Ma), unless the O$_2$ concentration during the snowball
was orders of magnitude less than the CO$_2$ concentration.
