The proof consists of using Stefan-Boltzmann's Radiation Law Q = cT^4 as a model of the Earth in radiative equilbrium with an atmospheric layer absorbing a factor f of the radiation E from the Earth surface and re-emitting f/2 E back again and f/2 E to outer space together with the non-absorbed (1-f) E, to give the total outgoing
(1 - f/2) E
suggesting a global warming with the factor (1 - f/2)^1/4 (fourth root of (1 - f/2)) by Stefan-Boltzmann. (If you iterate with n layers transmitting a factor (1-f/2n), then the outgoing radiation decrease to approximately exp(-f/2)E with global warming factor exp(-f/8)).
This supposedly generic (trivial) model of the greenhouse effect is presented also by Richard Lindzen in Greenhouse Effect: A Scientific Analysis, with the result that with f = 1 the Earth surface temperature comes out to be 303 K, about 15 K too high. Definitely a major greenhouse effect, much more than asked for, but not convincing scientifically since it does not fit at all with observation. Of course, adjusting f can give any desired result with less warming:
For example, if you say that the Earth temperature would be 255 K without atmosphere, then
you would get the observed 288 K if you arbitrarily choose f = 0.68. Any model with a free parameter at your disposal can be made to fit this data, but any model is not a correct model.
A scientist has two ways to handle a theory which does not fit with observation: (i) Dismiss the theory as incorrect and start anew with some different theory. (ii) Or simply add an ad hoc assumption, for example that you can divide your model results by 2, or fiddle with f, and you have it! Perfect match, but the trouble is that the corresponding physical model is missing, and without a physical model it is not science. What physics divides by 2?
The evidence is overwhelming that you cannot prove any greenhouse effect simply by waving at the Stefan-Boltzmann's Radiation Law. Science does not work that way. Either your model gives results consistent with observations, and then the model may be correct, or the model results do not agree at all with observations, and then the model is useless.
The above argument is developed in some more detail in the new article Greenhouse Gases without Greenhouse Effect.
A climate model only based on radiation cannot explain the observed temperture profile starting at 15 C at the Earth surface, dropping linearly to -55 C at the top of troposphere (tropopause) and climbing back to 0 C at the top of the stratosphere (stratopause).
Why? Because black body radiation from the tropopause as compared to that from the Earth surface is reduced with a factor (218/288)^4 = o.33, and compared to 0 C the drop is a factor 0.4. The cold tropopause would then act as a barrier to radiation and in order to reach radiation balance the Earth surface temperature would have to increase to 15 + 55 = 70 C. You could then speak about a terrible greenhouse effect as a result of a cold tropopause! (without any greenhouse gas effect at all). Fortunately there is also convection-evaporation/condensation, which changes the model.
In any case, a model without convection-evaporation/condensation only based on radiation
does not describe physics correctly, and is useless for prediction. Nevertheless such a model
is the starting point for IPCC global warming alarmism.
Inlägg
Looking at that paper I see that in his example he uses a flat surface. I've long wondered what such equations would look like if you considered spherical coordinates along with the angle of the rotational axis of earth would look like? Add to this that during the northern hemisphere summer the planet the furthest away from the sun. My math is too rusty to set up the calculations I'm afraid.
SvaraRaderaIt just seems to me that all calculations I see with how much the sun heats the surface gives 280 W/m^2 on average, but never how it's applied so to speak.