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In a recent study, the authors hypothesize that the Clausius-Clapeyron relation provides a strong constraint on the temperature of the extratropical tropopause and hence the depth of mixing by extratropical eddies. The hypothesis is a generalization of the fixed-anvil temperature hypothesis to the global atmospheric circulation. It posits that the depth of robust mixing by extratropical eddies is limited by radiative cooling by water vaporand hence saturation vapor pressuresin areas of sinking motion. The hypothesis implies that 1) radiative cooling by water vapor constrains the vertical structure and amplitude of extratropical dynamics and 2) the extratropical tropopause should remain at roughly the same temperature and lift under global warming. Here the authors test the hypothesis in numerical simulations run on an aquaplanet general circulation model (GCM) and a coupled atmosphere-ocean GCM (AOGCM). The extratropical cloud-top height, wave driving, and lapse-rate tropopause all shift upward but remain at roughly the same temperature when the aquaplanet GCM is forced by uniform surface warming of +4 K and when the AOGCM is forced by RCP8.5 scenario emissions. Locking simulations run on the aquaplanet GCM further reveal that 1) holding the water vapor concentrations input into the radiation code fixed while increasing surface temperatures strongly constrains the rise in the extratropical tropopause, whereas 2) increasing the water vapor concentrations input into the radiation code while holding surface temperatures fixed leads to robust rises in the extratropical tropopause. Together, the results suggest that roughly invariant extratropical tropopause temperatures constitutes an additional robust response of the climate system to global warming.