Sunday, May 20, 2007

Al Be Doh!

Terrible pun. About what you ask? Al Gore and the Climate Menace. It is my way of saying - Al is not too smart, but saying it in a way that segues into the topic at hand. Which is: what is the maximum possible temperature of the earth and are we near it? Which is all tied up with Maxwellian thermal distributions, the Stephan-Boltzman Law, Planck's Law, and albedo (get it? al be do). Which will all be explained shortly.

My friend Eric sent me this interesting piece that asks an important question. What is the maximum temperature of the earth? Which is why the above science comes into play.

Black bodies

A black body absorbs all of the light that reaches it. It has an absorptivity of 1. Thermodynamics states that objects at thermodynamic equilibrium radiate as much energy as they receive. The Stefan-Boltzmann equation describes the energy flux as it relates to temperature for a body in thermodynamic equilibrium:

S= σ T4
Simplified the energy radiated by a perfect radiating body goes up as the fourth power of the temperature of the body. An interesting and important point is that a perfect radiator is also a perfect absorber. An absorbtivity of 1 is equal to an albedo of 0. Conversely an absorbtivity of 0 is equal to an albedo of 1. (1 - absorbtivity = albedo)

Which brings us to the next point:
Postulate 1: The average temperature of a body in thermodynamic equilibrium with an external energy source can never exceed the temperature of a black body in the same environment.
This is true for a number of reasons. One of which is the conservation of energy. If you have two black bodies in the same environment and one was hotter than the other you could get energy out of such a system by absorbing heat at the higher temperature and rejecting it at the lower temperature. Now if you took that energy and re-injected it into the black body you should be able to make the temperature of the black body rise thus getting even more energy out of the system. Perpetual motion. i.e. it can't happen. So a black body in thermal equilibrium with a source is going to have a temperature defined by the source and its distance from the black body.
Postulate 2: The maximum temperature of a body in thermodynamic equilibrium with an external energy source can never exceed the temperature of black body in the same environment.
So neither the average nor the maximum can exceed the black body temperature. Other wise you could get perpetual motion from black bodies. So you think a perfect reflector would help? Nope. Perfect reflectors do not radiate energy. You can't pump energy into a perfect reflector in thermal equilibrium. Because if such a thing was possible the temperature would rise without limit. Then you could extract thermal energy from it by rejecting the heat to a black body. You could take the energy extracted and use it to raise the temperature of the white body making even more energy available. i.e. perpetual motion. Not going to happen.

One other important point. A perfect white body couldn't absorb ANY heat because its temperature would become infinite if you continued to pump heat into it. Another little stumbling block.
Postulate 3: The greenhouse effect can never produce a temperature that is higher than the temperature of a black body in the same environment
If it could it would violate thermodynamics principles and we could in theory have a perpetual motion machine. Not going to happen.

So there is a maximum temperature that the earth can reach no matter how many zillions of tons of green house gasses are pumped into the atmosphere.

So the question is what is that temperature?
It should now be clear that the maximum temperature of Earth can be no higher than the maximum temperature of an equivalent black body. We will now try to evaluate what that maximum is. For simplicity, all values and graphs have been obtained from Wikipedia unless otherwise stated.

The moon is quite close to a black body. It is estimated to have an absorptivity of 0.88. Conveniently the moon is nearly in the same environment in space as the Earth. The maximum temperature found on the moon is approximately 390° K. Using the Stefan-Boltzmann equation described earlier the maximum flux on the moon is

αS = σ T4

which for our values gives a flux of 1491 w/m2. Already we have a problem. The flux on Earth from the sun as measured by satellites is widely reported to be around 1366 w/m2, or significantly lower. Why the discrepancy? It is interesting to note that even with only these three elements, moon data, sun data, and the Stefan-Boltzmann equation, we end up with slightly inconsistent results, which may give us some insight into the level of uncertainty in the data that still remains in this area. Since we are interested in the maximum temperature we will take the maximum value of 1491 w/m2.

The earth is approximately spherical and receives light from the sun on a cross-sectional area of a circle, but radiates thermal energy from the area of a sphere. The ratio of the spherical area to the circular area is 4. Dividing the incoming energy flux by 4 gives the Earth an approximate maximum temperature of 285° K. Again we have another inconsistency as this maximum temperature is below the widely reported global average temperature of 288° K. Also the earth has an uneven distribution of temperatures and therefore an uneven distribution of flux, the end result of which would be to lower the average temperature even more. Still the result is quite close and it suggests that the Earth is behaving very closely to a black body and is operating very close to its maximum possible temperature.
Which leads to a restatement of the last bit as a postulate:
Postulate 4: The earth is operating very close to its maximum possible temperature.

Again, this will cause many to pause as it goes against the conventional wisdom. However we will attempt to provide two pieces of evidence to support this case:

- ice ages and the runaway greenhouse effect

- climate variability/stability
So let us look at the ice age data. Specifically the interglacial periods (like now) when the earth warms up after an ice age. What is postulated is that when an ice age ends the earth's temperature rises rapidly to a maximum (due to positive feed backs) and stays there with very little fluctuation in termperature while during the ice age phases the fluctuations are significant. Which would mean that the earth's albedo (reflectivity) varies a lot during ice ages, and not very much during warm periods.
The most likely cause of the ice ages is due to fluctuations in the intensity and the distribution of solar radiation caused by changes in the tilt in the Earth's axis. This theory was first described by the Serbian scientist, Milutin Milankovitch, in 1938. There are three major cyclical components of the Earth's orbit about the sun that contribute to these fluctuations: the procession (tilt of the Earth's axis), as well as Earth's orbital eccentricity and orbital tilt. The exact cause and effect relationship between orbital forcing and ice ages is still a matter of great debate, however the match of glacial/interglacial frequencies to the Milankovitch orbital forcing periods is so close that orbital forcing is generally accepted. Other theories include greenhouse gas forcing, changes in the Earth's plate tectonics, changes in solar variation, and changes in absorptivity due to dust and gases spewed by volcanoes.

The exact cause of the ice ages is not critical to our discussion other than to note that the Earth appears to have two metastable states: an ice age period and a warm period.
I refer to the metastable states as "strange attractors" from chaos theory. What that says is that you have a local maximum or minimum in an unstable system and once you are far enough from the "strange attractor" the system will tend to rapidly switch states. When a system oscillates between two such states it is said to be bifructed. Which is just a fancy way of saying two stable states. In electronics we have a circuit that does that. It has positive feedback in both directions once you are far enough from a stable input. The circuit is called a Schmidt trigger. Once the input gets out of the stable region it switches rapidly to the alternate stable state. Positive feedback all the way.
Postulate 5: The transition from Ice Age to warm period and back to Ice Age is achieved through a runaway greenhouse effect and its opposite

Another remarkable feature is the relative stability of the climate at the peak of the warming cycle. The variability of temperatures during an ice age is relatively high compared to periods of warming. However this makes perfect sense if one considers the climate as being "pinned" to the upper limit during the warm periods and therefore remaining stable due to strong positive feedback. At the upper limit, the major driver of upper temperatures becomes solar input as this is the only thing remaining that can effectively increase temperatures.
Once your effective albedo is close to zero the temperature is only determined by black body considerations. No amount of additional radiation capture within the body is going to change the temperature. There is no kind of heat trap we can devise which will increase the temperature above the black body limit. In fact we can only raise the temperature locally on such a black body by concentrating the energy from a given area on a smaller area. However that will increase the radiation from the hotter area and the average temperature of the body will in fact decline, because radiation goes up as the fourth power of temperature. You can't beat mother nature. In fact here is a good point to give the three laws of thermodynamics in laymans terms:

1. You can't win - there is no way to beat the system, energy is conserved
2. You can't break even - there will always be losses
3. You can't get out of the game - the rules always apply

When it comes to thermal systems there can never be any such thing as perpetual motion. There is always a maximum of work that can be extracted from two bodies at different temperatures. Sadi Carnot figured that one out. The work out can only be equal to the heat energy in if the cold body that the heat is rejected to is at absolute zero. Otherwise there will be a certain amount of heat that must flow into the cold body. That heat is unavailable for work.
Postulate 6: The runaway greenhouse effect ends when the Earth has achieved a effective absorptivity as close to unity as it can get after which the earth becomes insensitive to further positive feedback changes.

Can there be a tipping point or a runaway greenhouse effect from a sudden injection of CO2/methane or the melting of ice?

No there can not. The Earth has already experienced a runaway greenhouse effect thousands of times during its lifetime. Each time it is run to the maximum possible level that it can, bringing us the much more habitable climate that we have today. It is not possible for there to be a tipping point to spiral us into a third metastable climate state that has not been shown to exist during the entire history of Earth. Barring a sudden change in input from the sun, changes in climate upwards can only occur in a smooth, slow and limited fashion. A tipping point is possible, however, towards another ice age as has happened thousands of times before.
So there you have it. The green house gasses (mainly water vapor) have done their job in changing the albedo of the earth to close to zero. The albedo can never go below zero no matter how much CO2 is pumped into the atmosphere.

Ian Schumacher, the author of the bits quoted above, follows a little different argument than I do on the matter of thermodynamics. The results are the same. Which means you should compare what I said to what Ian has written. In other words - read the whole thing.

H/T Eric of Classical Values

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