Why does the stratosphere cool while the troposphere warms daddy?

One of the strongest and most consistent predictions of Global Climate Models is that if greenhouse gas concentrations increase, then the troposphere will warm. Another is that the stratosphere will cool. Both cooling and warming have been observed in the right places, but before we start, let us say that most of the currently observed stratospheric cooling is due to ozone depletion.

To my knowledge no one has stated a simple first order explanation for greenhouse gas (ghg) stratospheric cooling. William at Stoat tried to do this, but his explanation is not simple. If you don't believe me go read. Talking over a carrot martini a simple explanation appeared between my ears.

The stratosphere is dry, dry, dry. Most stratospheric water vapor comes from oxidation of methane and it ain't much. The only stratospheric ghgs (e.g. ones that can absorb infrared [IR] radiation) are ozone and CO2. Most of the radiation that makes it into space is emitted from a tropospheric layer at about 6 km, often called the top of the atmosphere (TOA). Forcing is calculated in terms of its effects on the TOA. If you want to know more look at this NAS report or this article by Roger Pielke Sr.)

Thermal equilibrium requires that the TOTAL emission from the TOA (warning simplification) has to equal the solar energy flux into the atmosphere. Simply put, stratospheric cooling is driven by higher water vapor concentration at the TOA driven by tropospheric ghg warming. TOA IR emissions from water vapor are not absorbed in the bone dry stratosphere, IR emission from CO2 is very efficiently absorbed, and more so if the CO2 concentration increases. Ghg warming lead to higher water vapor concentrations. The proportion of the constant TOA IR emission from water vapor increases. Since none of this emission (warning simplification) can be absorbed in the stratosphere, the stratosphere cools. QED.