Stratospheric Cooling

An Eccentric Anomaly: Ed Davies's Blog

Last autumn I wrote a post on the Navitron forum about a reason for cooling of the stratosphere with increased carbon dioxide in the atmosphere. I later posted a reference to it in a comment on Kaitlin Alexander's blog (Climate Sight) which lead to further discussion which showed that what I described is only one of the mechanisms which tends to cause stratospheric cooling. In particular, Martin Vermeer's comment explains the other, perhaps the main, mechanism well.

Nevertheless, I'm sufficiently smug about my own explanation that I'm going to put a slightly-edited copy here.

It all started with an article in the Independent from which the first quote originally came.

It is thought global warming could be responsible for some of the damage because greenhouse gases trap energy at lower altitude, heating up the atmosphere nearer the ground but cooling the stratosphere, creating conditions for the formation of chemicals that break apart oxygen molecules of ozone.

I used to think that but somebody who'd gone into the maths and physics of global warming in a lot more depth than I have said no, it's totally wrong. I didn't fully understand his explanation but went looking for something written for a bit more study. I've never found an explanation I both understood and which made physical sense to me but putting together the half understood bits I've come up with this description for my own use:

Think of a cube of stratosphere, say 1 m³. It contains oxygen, nitrogen, carbon dioxide and so on in pretty much the same proportions as the lower atmosphere (because, on a timescale of many years, the atmosphere is pretty well mixed even across the tropopause for all but very short lived gasses like water vapour).

Because the stratosphere is stratified (hence its name) most of the energy flows into and out of this cube are by radiation rather than convection or conduction. It absorbs long-wave infra-red from all around but slightly more from below than other directions (because the surface of the planet and lower part of the atmosphere is warmer) and radiates it in all directions. The amount it does this is proportional to the amount of CO₂ in the cube.

It also absorbs short-wave radiation from the Sun (UV, visible light and short-wave IR) as the result of the action of other molecules, particularly ozone absorbing UV. Because the air is cold it doesn't emit any noticeable amount of short-wave radiation (as is true of most natural things on or above the planet's surface other than fires or volcanoes).

The energy absorbed as short-wave radiation therefore goes to slightly warm the air in the cube causing it to radiate a little more long-wave IR. If the power of the short-wave from the Sun is swi (short-wave in), the long-wave IR from the surroundings in lwi and the long-wave radiated out is lwo then, because the cube is in energy balance, swi + lwi = lwo.

However, the amount of long-wave emitted and absorbed is a function of the amount of CO₂ in the cube. Suppose we call that factor C. Changing the meaning of our variables a bit we can write the balance equation as swi + C·lwi = C·lwo.

Note that the absorption of short-wave radiation is not affected by the amount of CO₂. Therefore if the amount of CO₂ is increased then the equation doesn't balance any more - the long-wave output increases more than enough to balance the short-wave input with the result that the cube cools. Cooling the cube reduces the amount of long-wave out without affecting the inputs thereby bringing the cube back into balance.

In symbols: using T for the factor by which temperature affects the long-wave output (it'll actually be the fourth power of the absolute temperature) we can mess with the equation a bit more to get:

swi + C·lwi = T·C·lwo

Keeping the swi, lwi and lwo factors fixed, if there's an increase in the CO₂ in the cube (C) then for the energy flows to continue to balance T must decrease.

This is all a bit beside the point for most discussions but for having any understanding of the interaction between global warming and the ozone holes I think it's fairly important to be a little clearer on why the stratosphere cools in a warming world.

Also, there are various memes around about warming not being caused by extra CO₂ but by the Sun or whatever. This observed stratospheric cooling is a good example of why this doesn't make sense; if it was extra solar radiation directly causing warming (yes, I know there are other proposed mechanisms) then the swi part of the equation would be increasing and it'd need a higher temperature in the cube to increase the long-wave radiation out to keep things in balance.