Global Warming?

  • Climate change should be seen as the greatest challenge to face man and treated as a much bigger priority in the United Kingdom. (Prince Charles)
  • The issue of climate change is one that we ignore at our own peril…What we can be scientifically certain of is that our continued use of fossil fuels is pushing us to a point of no return. And unless we free ourselves from a dependence on these fossil fuels and chart a new course on energy in this country, we are condemning future generations to global catastrophe. (Barack Obama)
  • I want to testify today about what I believe is a planetary emergency – a crisis that threatens the survival of our civilization and the habitability of the Earth. (Al Gore)
  • All across the world, in every kind of environment and region known to man, increasingly dangerous weather patterns and devastating storms are abruptly putting an end to the long-running debate over whether or not climate change is real. Not only is it real, it’s here, and its effects are giving rise to a frighteningly new global phenomenon: the man-made natural disaster. (Barack Obama)

Climate Sensitivity?

Once Icarus and Daedalus have escaped from the Labyrinth of Ignorance, they wiil be ready to take on problems.

What is the biggest problem facing humanity today? Is it Global Warming because of increasing CO_2 in the atmosphere from burning fossil fuels like oil and coal, and from humans breathing and cows letting out? Can we all go on breathing or will it be reserved for the rich?

The key question is climate sensitivity, which is how much global mean temperature will increase if the concentration of CO_2 doubles from the present 0.038%. The United Nations Intergovernmentak Panel on Climate Change IPCC tells us that the increase can be about 3 degrees Celcius C, give and take 1.5 C, that is up to 4.5 C, which if true would end human civilization as we know it.

Jurrasic Park would be back. But from where does the 4.5 C or more of potentially catastrophical global warming, come? Can the climate sensitivity really be so catastropically large? And if so, then why?

To start our studies, let’s try our hands on this problem. As always we ask

  • What are the physical laws?
  • What are the numbers?

Once we know the answers to these basic questions, we can use mathematics to produce some answer.

OK, so what do we have here? Well, the Earth with atmosphere is heated by the Sun through incomingradiation of all wavelengths, short to long, and the Earth with atmosphere: troposphere plus stratosphere radiates longwave infrared light to outer space. The reason the outgoing radiation is infrared is that the Earth is not nearly as hot as the Sun, and a colder object tends to transform incoming shortwave to outgoing longwave. It acts like a transformer transforming high voltage to low voltage current.

We don’t need to know the details of the absorption-emission process, only Stefan-Boltzmann’s Radiation Law of blackbody radiation stating that energy from a surface of a blackbody (like the Sun or the Earth with atmosphere) is proportional to the fourth power of the surface temperature in degrees Kelvin K. We recall that 273 Kelvin = 0 Celcius and 373 K = 100 C, so that temperature increase measured in Kelvin K or Celcius C is the same. Stefan-Boltmann’s law applies to any isolated bodies, in particular the Earth with its atmosphere if the temprerature is that of the upper atmosphere.

This means that doubling the temperature increases the radiated energy by the factor 2^4=16.

That is the physical law. Now to the numbers:

  • The surface temperature of the Sun can be estimated to 5700 K.
  • The radius of the Sun is 695.000 kilometers about 2.3 light seconds.
  • The distance between the Earth and the Sun is 500 light seconds.
  • The ratio is about 220, and since the area of a sphere scales like the radius squared, the intensity of the incoming Sun light is decreased by a factor 220^2.
  • We assume that the incoming light is distributed evenly over the surface of the Earth (with atmosphere), which is 4 times bigger than the disc area as seen from the Sun, and thus the radition from the Sun is diluted by the factor 4\times 220^2.

Now we start the mathematics combining laws with numbers (assuming that the Sun and the Earth with atmosphere can be viewed as blackbodies):

  • Whatever the Earth (with atmosphere) absorbs from the Sun has to be emitted according to Stefan-Boltzmann’s law.
  • What ratio r of the temperatures of Earth atmosphere and of the Sun, then gives r^4=4\times 220^2\approx 194 000? Your pocket calculator gives r\approx 21.
  • We conclude that the temperature on the top of the Earth atmosphere must be \frac{5700}{21}\approx 273 K or 0 C. This agrees pretty well the observed temperature at the top of the stratosphere.
  • If the Earth did not have an insulating atmosphere this would also be the surface temperature of the Earth (assuming no albedo or reflection).
  • Without insulating atmopshere the Earth could be covered by ice at 0 C. This agrees with the observation that the night temperature in Sahara (with clear sky an very dry air) is about 0 C.
  • Luckily, the Earth has an atmosphere, which increases the mean Earth surface temperature to 15 C, by acting like an insultaing window connecting to the top of the stratosphere at 0 C.
  • Climate sensitivity measures change of temperature vs change of radiative forcing. We know the effective total radiative forcing from the Sun to be about 280 Watts (per square meter), from the Stefan-Boltzmann Law and also from measurements.
  • We can then say that total climate sensitivity equals \frac{15}{280}\approx 0.06 K (squaremeter) per Watt.
  • Now IPCC informs us that the extra radiative forcing from doubling of CO_2 should be something like 2-4 Watts per square meter.
  • If we used the total climate sensivity just computed, we can estimate the corresponding global warming to 0.12-0.24  C.
  • We see that IPCC’s 4.5 C is 20-40 time bigger than our result, and thus IPCC must assume very large positive feed back, if started from the basic computation we have made.

So there we stand now: We have using basic physics and mathematics estimated the crucial climate sensitivity to be less than 0.5 C, including a safety factor of 2, which is not alarming at all. Pooh! But is our calculation correct?


IPCC sends out an alarm by suggesting that the climate sensitivity can be 10 times bigger: 4.5 C, apparently assuming very large positive feed-back. What is the truth? Your further studies will help to find the answer, from physical laws and numbers.

Is global warming a real threat or only imagined? Only science and mathematics can give an answer. Politics and religion cannot.

We have used a simple rational argument based on basic physics and math. We don’t know if our model describes the reality, only comparison with more precise models can tell. But we have subjected the IPCC alarm to a first rationality test, and the IPCC  alarm did not pass. We have done what seems reasonable, namely to ask when confronted with a message that the World is going to burn up from fossil fuels, if the message makes sense.

We have not found that it makes sense, and we have thus been warned that the message may be incorrect. We cannot say that it is incorrect, but we can say that we have found a reason to dig deeper.

A More Familiar Example

Suppose you are a poor student living in one-room student dwelling heated by a 280 Watt lamp maintaining 15 C inside at an outdoor temperature of 0 C. A common situation for a student in Sweden.

Suppose now by chance you have acquired an additional heat source of $3$ Watts. What increase of the temperature can you expect? Can you see the similarity with the above? Yes, the answer is the same \frac{15}{280}\times 3 =0.16 C.

From 15.0 C to less than 15.2 C say. Impossible to detect. Do you see the implication?

U-factor of a window measures heat transfer per square meter and degree Kelvin. The U-factor of the atmosphere is thus \frac{280}{15}. Doubling CO_2 with radiative forcing of 2.8 W would correspond to a decrease of U by 1\% and a corresponding global warming of 1\% of 15 C, that is 0.15 C.


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