As a non-climate specialist, this topic is on the edge of my understanding of meteorology, but here are some papers that I found potentially relevant to your topic.
Here is a quote from the summary of Limpasuvan and Hartmann, 1999:
The planetary wave propagation characteristics can be used to link
changes in the tropospheric annular mode to changes in the strength of
the stratospheric polar vortex during winter. Planetary waves are
strongly attracted to polar regions during the low phase of the
annular mode when the tropospheric westerlies are displaced
equatorward. We should expect a weaker stratospheric polar vortex
then, since planetary waves weaken the polar vortex by generating
And here is a quote from Shepherd and McLandress, 2011
In particular, the tropospheric warming that is robustly expected from
climate change leads to a strengthening of the upper flanks of the
subtropical jets, which causes the critical layers on the equatorward
side of the jets to move upward, and with them the Rossby wave drag,
allowing more Rossby wave activity to penetrate into the subtropical
lower stratosphere (Figs. 4, 5). Because the subtropics represent the
critical region for wave driving of the Brewer–Dobson circulation, the
circulation is thereby strengthened.
So my conclusion from these two papers (there is obviously a lot more detail in the full text) is that polar vortex will weaken, subtropical jets and Brewer-Dobson circulation will strengthen, and Rossby waves will penetrate farther south. Of note, neither paper seemed (to me) to indicate that a cooler stratosphere was predicted.