I have written before about what I perceive as the limit on Earth temperature increase due to an increase in the concentration of green house gases in the atmosphere. Here is an interesting discussion of the physics involved that I got from the blog 'Greenis Watch."

What is the maximum temperature of the Earth?

Questions and thought experiments around global warming and the greenhouse effect

Author: Ian Schumacher

The greenhouse effect is well known, although perhaps not well understood. Essentially greenhouse gases in the atmosphere act as a band-pass filter. High energy visible light can easily come in, heat up the ground and the atmosphere, and be converted to lower energy infrared light which the atmosphere absorbs and reflects. A question has not been asked of this process: How far can it go? What is the maximum possible temperature the greenhouse effect can produce? This paper attempts to answer this question and other related questions, and to discuss the postulates that arise as a result.

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:



A reasonably close approximation to a black body is a spherical cavity with a small hole for radiation to enter and exit. Many of us in our youth may have imagined such a cavity with a mirrored inner surface and a small hole in which to shine a light, mentally imagining how this might "trap" the light allowing it to build up and up to some bomb-like explosive level. However, now we know that the light energy will not build up to infinity but will be exactly balanced by outgoing thermal radiation once the temperature inside the sphere reaches a sufficient level.

Now imagine that we place a high-pass filter over the entrance of the cavity. High-energy high-frequency light will go in, hit the inner surface, bounce around, and be absorbed; low-energy, low-frequency thermal radiation will be produced which will hit our high-pass filter and be rejected. Have we changed anything? At first glance it might seem that we have, if we consider an idealized high-pass filter that does nothing but filter light. However, a real filter will have a temperature, and at thermodynamic equilibrium, by definition, the temperature of the filter will be the same as the temperature of the walls inside the sphere, and so at equilibrium the situation is exactly the same as before. Once we reach equilibrium the filter becomes irrelevant and the incoming and outgoing radiation will be equal as in the case where we don't have a filter.

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.

The total energy flux must be equal; however, can there be temperatures inside of the spherical cavity that are much higher than the average? No. At first it seems as though energy flux can be concentrated in some areas and spread out in others. A magnifying glass, or a parabolic mirror are obvious mechanisms to achieve such a concentration. However, unlike an external energy source such as the sun, the thermodynamic radiation inside the cavity at equilibrium will have no directional bias and therefore there are no mechanisms available to be able to focus this energy.

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.

Postulates 1 and 2, do not seem particularly revolutionary and to most people with a physics background they probably seem rather trivial and obvious. However, their statement up front is unfortunately necessary in order to overcome the common misinterpretation of the greenhouse effect that allows for conditions to violate postulates 1 and 2. When trying to determine the maximum temperature of the Earth, it is important to know which mechanisms limit this maximum. The parallels between our high-pass filter example and the greenhouse effect are obvious, so does this mean that the greenhouse effect does not exist? No, it does not mean any such thing. The greenhouse effect is real, however it does mean that the greenhouse effect can never produce a temperature that is higher than the temperature of a black body in the same environment.

Postulate 3: The greenhouse effect can never produce a temperature that is higher than the temperature of a black body in the same environment

For many readers this will cause a great pause and some reflection. It has become conventional wisdom that the greenhouse effect has essentially no limits, but this is clearly not true. The greenhouse effect works exactly as previously described. High energy visible light can easily come in, heat up the ground and the atmosphere, and be converted to lower energy infrared light which the atmosphere absorbs and reflects. High-energy high-frequency light enters through the atmosphere and is absorbed by the surface and atmosphere to produce low-energy low-frequency thermal radiation. This low frequency thermal radiation is more readily absorbed by the atmosphere and is radiated back to the surface and out to space. The result of the greenhouse effect is to raise the equivalent absorptivity of Earth closer and closer to unity (but never exceeding it). To those having trouble believing postulate 3, I recommend they work through postulated 1 & 2 in their mind until it becomes clear that this must be the case.

The sun, the moon, and the earth

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 390o K. Using the Stefan-Boltzmann equation described earlier the maximum flux on the moon is



which for our values gives a flux of . Already we have a problem. The flux on Earth from the sun as measured by satellites is widely reported to be around , 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 .

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 285o K. Again we have another inconsistency as this maximum temperature is below the widely reported global average temperature of 288o 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.

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

Ice ages and the runaway greenhouse effect

There is a surprising amount of debate about what causes ice ages and their ending. The core feature of ice ages is their remarkable periodicity. The figure below shows sample data for the last four ice ages.



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.

Of note in the above figure is the strong correlation between carbon dioxide and temperature. As the temperature increases, ice sheets recede, which increases the absorptivity of the earth, and more carbon dioxide, water vapor, methane, and other greenhouse gases are released. This increases the temperature further, which causes the ice sheets to recede further, and causes more greenhouse gases to be released, etc. This is a positive feedback loop and is the "runaway greenhouse effect" in action. The positive feedback also works in the opposite direction causing the earth to drastically fluctuate between these two metastable states. What causes this runaway greenhouse effect to end? The answer is that once the earth has achieved its maximum absorptivity (or very close to it), additional receding ice or greenhouse gases becomes irrelevant. The climate is "pinned" to the maximum possible value.

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

Climate variability/stability

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.

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.

(The paper above is an original publication. Ian Schumacher [ian.schumacher@gmail.com] has a degree in Engineering Physics and has done a master's program in physics and mathematical modeling. He used to work as a contract research scientist for the Canadian military, but has long since moved on and is now a programmer/software architect living in Vancouver, Canada)