A Simple Model of the Atmosphere

Some 30+ years ago I used to be a university researcher into heat flow in buildings. We constructed models of heat flow through walls, ceilings, floors, windows.

But we didn’t consider the environment outside the buildings. We didn’t build models of the atmosphere.

Nevertheless, ever since climate models became a source of contention over global warming (or the lack of it), I’ve always wanted to extend my heat flow models to include the atmosphere.

And I’ve had several tries at doing so, which were mostly unsuccessful. Because I’ve yet to come across an explanation of how to do it. Although I have come across a few people who have been building very complex models.

So it all rather ground to a halt. Until a few weeks ago, that is, when I decided it might be worth building a very simple atmospheric model from first principles, ignoring everything I knew (or rather, didn’t know) about Earth’s atmosphere.

I’ve recently been building simulation models of rock clouds, and so I thought that maybe I could think of an atmosphere as being made up of lots and lots of little rocks in orbit around the Earth. There’d be no ‘air’, no ‘wind’, no ‘air pressure’, no ‘adiabatic lapse rate’ or any of the other stuff meteorologists talk about. There’d just be lots of rocks in empty space. There’d be a dust cloud.

And the dust would be found in layers above the surface of the Earth, and would be heated by sunlight filtering down through the layers to the surface of the Earth. At each layer, sunlight would either pass through the layer, or else would strike dust particles. And when sunlight struck a particle, it would either be reflected back up, or absorbed by the particle, warming it up. And as the dust warmed up, it would radiate heat upwards and downwards.

Now there would be a lot of sunlight and radiated heat bouncing around inside this dust cloud. There’d be multiple reflections between layers. But once a ray of sunshine was admitted at the top of the cloud, it would all end up somewhere or other. It would either be absorbed by the dust cloud, or by the surface of the Earth, or reflected or re-radiated back out into outer space. Once you knew the transparency of each layer, and the absorptivity of the dust, and a few other things about it (emissivity, thermal capacity, etc), it was possible to calculate where all  sunlight would end up.

So I wrote a computer program to do all these calculations for a cloud with a number of layers, and constant sunlight falling on it. And underneath the dust cloud ‘atmosphere’ I put a few conductive layers of earth.

And then I set the model running, so that the dust cloud and earth gradually warmed up from an initial temperature of absolute zero, 0° K, under a constant solar flux of 340 Watts/square metre (which is about what the Earth receives every day on average).

And I found that the rock cloud and earth layers gradually rose and reached equilibrium at 278° K  (about 5° C), just above the melting point of ice. And this is almost exactly what it’s supposed to be, as derived analytically.

However, the actual temperature profile of the atmosphere is quite different. It’s warm near the surface, very cold 10 km up, warmer at 50 km, cold at 100 km above the surface, and very hot at the highest altitudes of the thermosphere, where temperatures can reach 1000° K .

This profile was explained by absorption by oxygen of UV light in the thermosphere, and absorption by ozone in the stratosphere, and absorption by water vapour and carbon dioxide in the troposphere. The regions of the atmosphere that were coldest were those that did not absorb much sunlight.

I tried changing the absorptivity of the particles in my dust cloud, but whatever value I set this at, the equilibrium temperature remained the same. It just heated up more quickly if it was strongly absorptive.

My model doesn’t include the conduction or convection of heat from the Earth’s surface. But even if this was included, it would only have an effect in the troposphere or stratosphere.

However, in my model I had set absorptivity equal to emissivity, according to Kirchhoff’s Law of Radiation. If my dust particles could readily absorb sunlight, they would equally readily radiate heat. And it turned out that this was why my dust cloud always reached equilibrium at 278 ° K.

So I tried using values of absorptivity different from those of emissivity. I used high absorptivity and low emissivity at the top of the dust cloud, and low absorptivity and high emissivity at the bottom. And with that there came a dramatic new result. The dust cloud temperatures rose to 877° K at the top, and fell to 132° K at the bottom.

And with judicious choice of different absorptivities and emissivities in different layers, I could – in rough outline – reproduce the earth’s atmospheric temperature profile.

Most likely Kirchhoff’s Law of Radiation is valid, although there seem to be some question marks about it. What’s probably happening is that while the dust cloud is absorbing shortwave solar radiation from the hot sun (5,778° K), it is radiating at a much longer wavelength because of its lower temperature (160 – 877° K). My current model doesn’t consider shortwave or longwave radiation.

So maybe the next thing I should consider is the wavelength of radiation. But at least now I have a atmospheric model which is behaving a little bit like the real one.  Which is a big improvement on the earlier models.