Sunday, December 10, 2006

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 [] 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)


Eppur si muove -- "and yet it moves" -- was supposedly Galileo's final statement after being forced by the Church to retract his revolutionary cosmological theories. He had run up against the overwhelming consensus of his time -- that the Earth was the center of the universe and that saying otherwise was detrimental to the public good, not to mention Galileo's health. For centuries, the scientific method has been an antidote to such persecution. Right or wrong, scientists should be free to advance their theories without the threat of extra-scientific censure, except perhaps when national security is at stake. Science alone should judge scientific validity.

Yet today, there appears to be a band of scientists and agitators who are willing to use the methods of Galileo's persecutors to protect their own cherished theories. In the field of climate science, some people want to declare the scientific debate closed, allowing only those public statements that advance the approved idea that global warming is occurring, that man is responsible for it, and that it will probably be catastrophic unless greenhouse gas emissions are drastically curbed.

In its most extreme form, this phenomenon has involved calls for scientific versions of the Nuremberg Trials (from a writer at the environmental magazine Grist) and the equation of "climate change denial" with Holocaust denial. Others have branded as criminal those who question restrictions on greenhouse gas emissions. In September 2006, on CNBC's Global Players program, Jeremy Leggett, CEO of a solar power company, called for fellow guest Fred Smith, president of the Competitive Enterprise Institute (the think tank where I work), to be locked up for expressing his views. James Hansen, head of NASA's Goddard Institute for Space Studies, who has vociferously complained in the media of being silenced by the Bush administration for his research on global warming, suggested -- without a hint of irony -- that Mr. Smith should not even have been given such a platform for his views.

So just what is the nature of the "denial" that these scientists and environmentalists want to eradicate? First, there is the proposition that the Earth may not be warming at all. The truth is that there are not many scientists who publicly express this view nowadays. While there are many who question the reliability of surface temperature records, there are few who dispute the evidence from satellite records showing that the Earth has warmed 0.13 degrees Celsius per decade since the start of the data in the 1970s. These records, however, also show virtually no warming in the Southern Hemisphere (global warming isn't very global). There is ongoing scientific debate about the calibration of the data, but essentially this debate is over: The Earth has warmed since the 1970s.

Yet that isn't a very long time at all, certainly not long enough to establish whether or not the warming is so unprecedented that civilization and the biosphere have not had to deal with similar warming before. So the second target of the "denial" charge is those who dispute that the current warming is unprecedented. Yet here there is clearly ongoing scientific debate, with developments in just the past few months. A small group of paleoclimatologists issued a series of temperature reconstructions finding that global temperature was mostly stable for the past thousand years until a precipitous recent rise. Questions, however, were raised about the quality of the data and the statistical methods used to achieve this result. A team of eminent statisticians charged by the House Energy and Commerce Committee to investigate the scientists' methodology confirmed that the methods they had used virtually guaranteed the result they obtained. Meanwhile, the National Research Council (NRC) found that the quality of the historic data meant that nothing more could be said with certainty than that the current warm period is warmer than at any time since the 1600s, which the NRC agreed was part of the "Little Ice Age" -- something that the paleoclimatologists' reconstruction suggested had not occurred. The NRC found that the suggestion that the current warm period was the warmest for a thousand years was merely plausible, but both unprovable and unfalsifiable given the current state of the historic data. The NRC also upheld the methodological criticisms. It is therefore somewhat of a stretch to claim that science has proven beyond reasonable doubt that the current warm period is unprecedented.

Third, the "denial" charge is aimed at those who purportedly suggest that mankind has nothing to do with the current warming. This represents a considerable oversimplification of the issue. Such "contrarian" scientists -- such as S. Fred Singer, Patrick J. Michaels, and Richard S. Lindzen -- have affirmed time and again that mankind is responsible for some of the warming. Basic physics indicates that the more greenhouse gases in the atmosphere, the more heat will be trapped there. Yet there are far more climate "forcings" than just greenhouse gases. The Intergovernmental Panel on Climate Change (IPCC) and the NRC both admit that our current understanding of these other forcings is low. Until we know much more about land-use change, aerosols, and solar activity, to name but a few, we cannot be certain that greenhouse gases have been driving the recent warming trend. That is why the NRC concluded that, "Because of the large and still uncertain level of natural variability inherent in the climate record and the uncertainties in the time histories of the various forcing agents (and particularly aerosols), a causal linkage between the buildup of greenhouse gases in the atmosphere and the observed climate changes during the twentieth century cannot be unequivocally established."

The final charge against "deniers" is that they fail to acknowledge that global warming will be catastrophic. Most deniers would happily cop to this accusation, and they have plenty of evidence to back up their stance. When Al Gore talks about twenty feet of sea-level rise from the melting of the West Antarctic Ice Sheet (WAIS), he is failing to acknowledge the science. The IPCC estimates less than a meter of sea-level rise this century and considers catastrophic destabilization of ice sheets unlikely. Even if the WAIS were to melt, research indicates it would take several thousand years to do so, more than enough time for people to get out of the way.

The facts are similar with regard to most other supposed "catastrophic" impacts. Even if the theories linking increased sea surface temperature to more intense hurricanes are correct -- and the hypothesis is the subject of intense ongoing debate -- hurricanes will only be about 5 percent stronger by 2100 in a worst-case scenario. Polar bear biologists dispute whether or not the Arctic ursines are under any real threat from the melting of Arctic ice -- 15 of 17 populations do not even appear to be affected. Some evidence from nineteenth-century Arctic explorers even suggests that ice sheet extents were as receded then as they are today. The same holds true for claims about air quality, heat waves, and precipitation. There is no uncontested, compelling scientific evidence that the effects of global warming will be catastrophic to health and welfare.

So if that's all that the climate change denial charge can mean, why is it being made with such enthusiasm? The answer seems to be the chilling effect it has on the scientific debate. It makes public profession of opposing views unpalatable. Thus, Richard Lindzen of M.I.T. argues, "Scientists who dissent from the alarmism have seen their grant funds disappear, their work derided, and themselves libeled as industry stooges, scientific hacks, or worse. Consequently, lies about climate change gain credence even when they fly in the face of the science that supposedly is their basis."

There is another, even more worrying result of the denial charge. It enables alarmists to portray the science as dispositive. The only way to solve the problem, science supposedly shows, is to reduce greenhouse gas emissions radically. Anyone who argues against this conclusion is deemed a denier.

Yet in public policy, science is not dispositive. Economic, political, and moral considerations also need to be taken into account. Practical tradeoffs and competing priorities need to be considered. By tarring those opposed to climate-change policy with the charge of denial, the alarmists have elided the economic, political, and moral debate to their great advantage.

Even worse, the denial charge obscures the many uncertainties that surround our understanding of climate change and its implications. Global warming is a serious enough subject that it needs to be debated fully, submitting every hypothesis to rigorous testing and hard-headed analysis. When the alarmists say the debate is over, responsible scientists and policymakers must reply, like Galileo, "And yet it moves."



Investors are pouring billions of dollars into "renewable" energy sources such as ethanol, biodiesel and solar power that promise to reduce the world's reliance on petroleum. But exploiting these alternatives may produce unintended environmental and economic consequences -- fallout that could offset many of the expected benefits.

Here on the island of Borneo, a thick haze often encloses this city of 500,000 people. The cause: forest fires that have blazed across the island, some of which were set to clear land to produce palm oil -- a key ingredient in biodiesel, a clean-burning diesel fuel alternative.

The bluish smoke is at times so dense that it leaves the city dark and gloomy even at midday. The haze has sometimes closed Pontianak's airport and prompted local volunteers to pass out face-masks on city streets. From July through mid-October, Indonesian health officials reported 28,762 smog-related cases of respiratory illness across the country.

"I feel it in my breath when I breathe," said Imanuel Patasik, a 26-year-old delivery man, as he sat in one of Pontianak's many open-air coffee shops on a recent evening. When the smoke is really bad, he wears a mask to work, but still wakes up feeling sick. "It's part of life here," he said with a sigh.

Seasonal rains have helped quell the fires over the past few weeks. Even so, the miasma of smoke from Borneo and the island of Sumatra -- an annual phenomenon that blankets large parts of Southeast Asia in smog, including Singapore and Kuala Lumpur -- underscores a troubling dark side of the world's alternative-energy boom. Among other problems, the fires set to clear forest land spew millions of tons of carbon dioxide and other greenhouse gases into the atmosphere, experts say. In doing so, they exacerbate the very global-warming concerns biofuels are meant to alleviate.

Such side-effects aren't an isolated problem. In Indonesia and Malaysia, forests are being slashed for new energy-yielding crops or other unconventional fuels. In India, environmental activists say, water tables are dropping as farmers try to boost production of ethanol-yielding sugar.

"Let's be brutally frank: (The push for alternative fuels) is going to cause significant changes for the environment," says Sean Darby, an equities analyst and expert on alternative-energy companies at Nomura International in Hong Kong. He is most worried about the strain on water resources caused by accelerated crop production. Water, he says, is "just as precious" as oil.

Some experts are also concerned that crops for biofuels will compete with other farmland, possibly driving up global costs of basic food production.

It isn't clear how serious these problems will become -- or whether they eventually will be resolved through new technologies and stricter environmental measures. Proponents of alternative energy, including palm-oil industry executives, say the dangers are exaggerated and are outweighed by the benefits new fuels promise.



They LOATHE air travel -- except for themselves of course

A big expansion of Heathrow, including a new runway, will be endorsed by the Government next week after a decision by ministers that the economic benefits vastly outweigh the environmental damage.

BAA, which owns Heathrow, is secretly developing a pollution charging scheme targeted at lorries to overcome the problem of poor air quality around the airport in West London. An internal BAA document states that the best way to resolve the problem of nitrogen dioxide pollution is to reduce the number of older, more polluting lorries passing the airport on the M4 and nearby roads. Lorries emit about 15 times more nitrogen dioxide than cars. BAA believes lorries could be deterred either by road tolls or by penalties for the most polluting trucks.

The new runway would allow an additional 500 flights a day to pass over London and create a new flight path over Acton, Chiswick and Fulham. Up to 700 homes, including eight Grade II listed buildings, would have to be demolished to make way for the runway and a terminal.

Gordon Brown, the Chancellor, has accepted the argument put by Sir Rod Eddington in his report last week on the future of Britain’s transport system, that expansion of Heathrow is crucial to the economy and to maintaining London’s position as Europe’s financial centre. This week’s Pre-Budget Report said: “To avoid the economic consequences of constraining aviation growth, further expansion of UK airport capacity is needed. Heathrow plays a unique role in the UK as a hub airport and demand for capacity already significantly exceeds supply, leading to less competition, greater congestion, reduced choice and higher prices for passengers.”

The report made no mention of Stansted, which the Government had promised, in its aviation White Paper of December 2003, would receive a new runway before Heathrow. Next week’s progress report on the White Paper will continue to support a second runway at Stansted but the Government no longer believes Heathrow should have to wait for the Essex airport to be expanded. None of the leading airlines support BAA’s plans for a new runway at Stansted. Ferrovial, the Spanish company which recently bought BAA, agrees with British Airways that the priority should be expanding Heathrow because demand for flights is far greater there.

The Department for Transport has been monitoring air pollution levels around Heathrow and is confident they can be reduced to comply with European limits by the time a third runway opens. BAA hopes to submit a planning application for a third runway at Heathrow in 2008 and to open it by 2017. A BAA source said: “We are completely committed to expanding Heathrow.”

Ministers are also keen to allow more take-offs and landings on the existing two runways while the third runway goes through the planning process. A consultation paper will be published in the new year proposing the ending of runway alternation, which gives residents under the flight path respite from aircraft noise for half the day. The two runways could accommodate an extra 60,000 flights a year if each was used for take-offs and landings throughout the day.

John Stewart, chairman of ClearSkies, which opposes the expansion of Heathrow, said: “The Government is gearing up to allow a new runway at Heathrow before Stansted but they won’t be honest about it. “There will be the mother of all battles over Heathrow because the environmental movement sees it as a cause celebre. It will be the Newbury bypass of the skies.” In October 40 people living near Heathrow attended training sessions on direct action techniques. One idea is for a convoy of cars to stop inside the road tunnels under the northern runway, causing chaos for people trying to reach the terminals.



Many people would like to be kind to others so Leftists exploit that with their nonsense about equality. Most people want a clean, green environment so Greenies exploit that by inventing all sorts of far-fetched threats to the environment. But for both, the real motive is to promote themselves as wiser and better than everyone else, truth regardless.

Global warming has taken the place of Communism as an absurdity that "liberals" will defend to the death regardless of the evidence showing its folly. Evidence never has mattered to real Leftists

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Anonymous said...

really nice!

Anonymous said...

Oh, yes, and I just thought of this...

If we can't affect the amplitude, then all we can do, at most, is shift the phase a bit, or maybe increase the frequency. But we will probably never be able to measure the difference we make, especially on the time scales involved.

M. Simon said...

Nice bit.

I have turned it into a post:

Al Be Doh!