Stefan's Law and AGW

An Eccentric Anomaly: Ed Davies's Blog

A while ago on the Green Building Forum, in the context of asserting that current theories of anthropogenic global warming are wrong, Tony wrote:

Finally any model that shows consistently rising temperatures or exponentionally rising temperatures must be ignoring Stefan’s Law which says that heat is lost in proportion to the fourth power of the absolute temperature, setting up a very strong force to stop or at least reduce any increases.

I think that's not right and I gave a quick response in the next post on that thread giving a “meta” reason why I thought so: that it's ridiculous to think that climate scientists aren't perfectly well aware of this law. Another quick near-meta argument I could have given is that feedback resulting from the supposed steep effects of Stefan's Law were not enough to prevent global temperatures dropping by around 6 °C during the last glacial maximum.

However, there's a physical reason, too, and the details are interesting enough that it seemed worth writing up my understanding of the situation.

Most of this stuff is basic enough that it should have been understood by anybody slightly interested in the subject about a decade ago. However, with the ridiculously poor level of coverage that the press gives to any science subject, and particularly this “controversial” one, it's not overly surprising that it's passed many by. Somebody (Tim, not the horse, silly) explained this to me ages ago but I didn't catch on to what he was saying until I'd read a bit more on the subject.

Note though, this is just my understanding pieced together from reading lots of individual bits of information over the last few years. I could easily have got something wrong here and would welcome any feedback by anybody who understands this better.

Stefan's Law

The Stefan–Boltzmann law says that the amount of power radiated by a body is proportional to the fourth power of its absolute temperature. (Actually, that's just for a black body; real objects will have a emissivity less than one so will emit less power. The emissivity is dependent on wavelength so may vary slightly as the temperature increases and the wavelengths reduce but over small temperature changes this will not make a lot of difference).

I don't think Stefan's Law is particularly relevant, as I'll try to show below, but it's still worth noting how much effect it would have if it did matter.

The average temperature of the Earth's surface is around 15 °C which is about 288 K (kelvins). Nobody's really clear how much harm different temperature rises would cause but there seems to be pretty widespread agreement that by the time we reach 6 °C (6 K) rise things will be getting serious - i.e., when the average gets to 21 °C (294 K).

In alternative universe A (I imagine that Tony imagines that the climate scientists imagine they live in this one) Stefan's law does not apply: radiation is simply proportional to temperature so raising the temperature like this would increase the heat radiation from the Earth by 294/288 times. The actual extra power would be 294/288 - 1 = 0.0208333 times the original.

In alternative universe B (the one I imagine Tony imagines he lives in) Stefan's law applies to this situation so the extra heat flux needed is 294⁴/288⁴ - 1 = 0.08597. This is about four times the extra required in alternative universe A.

So, even if it was relevant, Stefan's law only requires about four times more extra power flux to cause us serious temperature problems. To use this as convincing argument to eliminate the possibility of AGW would require a much more careful analysis of the amount of extra power available.

With that out of the way, let's concentrate on alternative universe C, the one that I imagine the climate scientists imagine they live in.

Heat fluxes

Sunlight arrives at the Earth. At the top of the atmosphere, in a plane perpendicular to the rays from the Sun the power in that sunlight (mostly visible light, short-wave infrared and ultraviolet) is about 1360 W/m². Some of the sunlight is returned straight back to space by reflection from clouds and the surface and by scattering in the atmosphere. Some, but not much, is absorbed in the atmosphere - in particular, UV is absorbed by ozone higher up. Somewhat less than 1000 W/m² (perpendicular to the sunlight) arrives at the surface.

The surface area of a sphere is four times the surface area of the disk it presents in any one direction so the energy arriving over the whole surface of the Earth averages to about 230 W/m². To remain in equilibrium the Earth has to get rid of that heat which it does by emitting long-wave (thermal) infrared to space.

Basically, Tony's assertion, as I understand it, is that if the surface of the planet warmed much then the resulting increased flux of IR would cool it back down again, restoring equilibrium and that the T⁴ part of Stefan's Law ensures that this is a powerful effect.

However, because the greenhouse gasses in the atmosphere absorb the long-wave IR this emission is not done directly from the surface. Instead the heat is transferred upwards through the atmosphere until it reaches a height at which the density of absorbing gases is low enough that the IR photons can escape to space.

Traditionally, three methods of heat transfer are listed: convection, conduction and radiation. Within the atmosphere conduction is not significant because air is a pretty good thermal insulator. Both convection and radiation are important for moving heat away from the surface.

Lapse rates

In meteorology a lapse rate is the rate at which some quantity reduces with increasing altitude. Usually, and throughout this essay, the quantity is temperature.

The temperature lapse rate varies from height to height and place to place. Sometimes it's negative: temperature increases with height in what is referred to as an inversion. However, overall the lapse rate is reasonably uniform.

Formally the SI units for lapse rate should be K/m (kelvins per metre) but that'd result in pretty small numbers. More commonly units like °C/km are used. Typical values in the troposphere are 6.5 °C/km - that is, for every 1000 metres of increased altitude the temperature drops by 6.5 °C.

Radiation

It's easiest to think of the transfer of heat up through the atmosphere by thermal (long-wave) infrared as proceeding layer by layer. Imagine two layers of air, say 1 metre thick, one above the other. Both will be radiating thermal IR in all directions depending on their respective temperatures in accordance with Stefan's law and depending on the amount of greenhouse gases in them.

Some radiation in each layer will be reabsorbed by a different molecule from the emitter, particularly for photons which are emitted sideways. Some from the upper layer will go upwards and some from the lower layer will go downwards. However, some radiation from the lower layer will be absorbed by the upper and vice-versa.

If we assume a typical temperature lapse rate then the lower layer will be a little bit warmer than the upper and so emit a little bit more radiation. The net effect of the radiation exchange will be to transfer heat upwards.

For a given amount of greenhouse gases in the atmosphere, in order to achieve a certain rate of energy transfer upwards (expressed in W/m²) you need a particular lapse rate. If the actual lapse rate is lower then the result will be a smaller heat transfer which will cause the lower atmosphere to warm until the required lapse rate is achieved. Similarly, if the lapse rate is higher more power will be transferred and the lower atmosphere will cool to balance.

If you increase the amount of greenhouse gases present then, for a given lapse rate, more power is transferred or for a given amount of power a lower lapse rate is required. If radiation was the whole story it would not be obvious what the net effect of increasing greenhouse gasses would be; the level at which radiation was lost to space would be higher tending to increase the surface temperature but the atmosphere would be more efficient at moving heat upwards tending to cool the surface. You'd have to do a bit of arithmetic to work out the net effect.

Convection

However, radiation is not the only way in which heat is transferred upwards in the atmosphere. Convection moves heat as parcels of air which are slightly warmer and so less dense than those around them rise and transfer heat. As a parcel of air rises it expands as the pressure decreases leading to cooling. If the air is dry enough that this cooling does not lead to condensation this cooling is at about 9.8 °C/km. If, on the other hand, the air contains enough water vapour that condensation (forming clouds) occurs this process releases heat which reduces the cooling rate to about 5 °C/km. These rates are called the dry and saturated adiabatic lapse rates (DALR and SALR) respectively.

Note that the DALR and SALR bracket the typical lapse rate mentioned above (6.5 °C/ km) as the actual lapse rate results from a mixture of dry and saturated convection (dry in clear air, saturated in clouds).

If the actual lapse rate of the atmosphere is less than the dry or saturated lapse rate as appropriate to a parcel of air then when that parcel rises it will cool to a lower temperature than the surrounding air and so be denser stopping the convection. This condition is referred to as stable. However, if the lapse rate is higher, then the rising parcel will be warmer and so less dense than its surroundings and so will rise higher feeding more convection resulting in an unstable condition.

Radiation and convection combined

Suppose we start with an isothermal (all at the same temperature) atmosphere. Because the lapse rate is zero the atmosphere is stable so there is no convection and are no clouds (which need convection to hold the water droplets up) so the sun shines on the ground warming it up.

By conduction to the air in direct contact and by radiation to the lowest layers the ground heats the air immediately above it. These layers in turn heat the layers above by radiation.

Gradually the lapse rate increases until the air becomes unstable and convection starts. As the upward flux of heat increases the lapse rate does not increase much more. Instead the amount of convection increases transporting more energy directly as sensible and, if condensation is happening at any level, latent heat.

(This is not a hypothetical situation. It's how a nice gliding day typically starts. Often there are other factors confusing the situation but it's not unknown for the story to unfold pretty much exactly as described here.)

Troposphere and stratosphere

There are two main layers of the atmosphere relevant to global warming. The lowest layer is the troposphere up to an altitude of 10 km or so (higher at the equator, maybe a bit less near the poles). This is the layer in which convection takes place. Above the troposphere is the stratosphere in which there is little or no convection. The boundary between the two is the tropopause.

The difference between these two layers is that in the troposphere there are enough greenhouse gases that the temperature lapse rate purely due to radiation would often be greater than the adiabatic lapse rate of the air, resulting in convection.

In the stratosphere the density of the air as a whole and therefore the density of the greenhouse gases is sufficiently low that radiation alone can transport heat upwards without the need for additional transport by convection.

Actually, because the stratosphere absorbs short-wave radiation from above its temperature is inverted; that is, it gets warmer with increasing altitude.

An interesting observation is that if the atmosphere was completely transparent (i.e., had no greenhouse gases to absorb and emit long-wave infrared and no molecules capable of absorbing incoming short-wave radiation) then it would be isothermal - the same temperature all the way up. No heat would be gained or lost at the top of the atmosphere so convection and conduction would rapidly see to it that the whole atmosphere was at about the same temperature as the surface (averaged out).

If it was just the greenhouse gasses which were missing then the whole atmosphere would be like the stratosphere with no convection and temperature increasing with height. Most of the interesting phenomena in the atmosphere result from convection which requires the greenhouse effect in order to happen at all.

(I came to realise this while trying to understand exactly where Hyperventilating on Venus goes so badly wrong.)

Earth's “photosphere”

In the core of the Sun mass is converted to energy (with only about the volumetric power density of a well-organized compost heap - but there's a lot of solar core at it so the total amount of power produced is not small) and emitted as gamma rays. These make their way (by repeated absorption and re-emission like long-wave radiation in Earth's atmosphere) over thousands of years until they reach a range of levels (the photosphere) where the gas is thin enough that they are radiated to space as sunlight.

A similar thing happens on Earth, only quite a bit quicker. The actual level at which radiation to space happens is quite smeared out because different wavelengths are absorbed and radiated differently and because of simple luck where some photons emitted upwards at a particular level escape all the way to space whereas others are reabsorbed.

Nevertheless, there is a level high in the troposphere where effectively radiation to space happens. As greenhouse gasses in the atmosphere increase this level gets higher - CO₂, etc, at the old level are now dense enough to re-absorb more photons than before.

As Tony correctly points out, the temperature of the surface emitting long-wave radiation to space is sensitively dependent on the power which needs to be got rid of. However, it's not the temperature at the surface which is dependent in this way but that at the effective emission level. The emission level with extra greenhouse gasses is higher in the troposphere but at the same temperature as the level without.

Surface Temperature

The important point, though, is that it's the applicable (dry or saturated) adiabatic lapse rate which typically controls the temperature profile of the atmosphere below the emission level. Stefan's law only applies to the lapse rate controlling radiation but that is a secondary matter; convection results in a lower lapse rate and takes up the slack in heat transfer.

It is the increased distance, from the surface to the emission level, for the relatively fixed adiabatic lapse rate to work over which results in higher temperatures at the surface. Neither of these factors is directly affected by Stefan's Law.

Update 2016-02-12

Rasmus E. Benestad of the Norwegian Meteorological Institute has published a similar description, A mental picture of the greenhouse effect: A pedagogic explanation, only with more graphs, numbers and citations.

Two things surprised me: that the rate of increase of the effective radiation level is only of the order of 23 metres/decade and that he says that not all his colleagues agree with this description and that it was a struggle to get it published, as he says on this shorter version on Real Climate).

Also, I suppose, that he doesn't mention Nils Ekholm's 1901 QJRMS paper which I haven't read but from the extract I have seen was the first (or, at least, a very early) exponent of the idea that surface warming is caused by the rise of the effective radiation level. OK, Norway and Sweden were having a bit of a tiff at that time but still…