For nearly 100 years, Schwarzschild’s equation has been used to calculate the effect of outgoing longwave radiation (OLR) from the Earth’s atmosphere.  Here it is shown why that equation is inappropriate, and a novel method of calculating the effects of greenhouse gases is proposed.  This is of considerable importance for both meteorological and climate modelling.

In a classic paper1 read in 1913, Robert Emden proposed a model for the effects of radiation in the Earth’s atmosphere, which he viewed as a series of parallel slabs.  Each slab absorbs grey radiation from the slab below based on its greenhouse absorption spectrum.  The slab then re-emits black-body radiation upwards which can be calculated from the slab’s average temperature using Planck’s function.  This leads to Schwarzschild’s equation2:

dI = - Ikρ dz + B(T)kρ dz                                Equ. 1

where I is the intensity, k is the absorption coefficient, ρ is the density, z is the vertical coordinate and B(T) is Planck’s function at the temperature of the slab, T.

However, air does not behave as a black-body radiator.  Molecules of greenhouse gases emit line radiation at the same frequencies as those at which they absorb. No gas emits continuous black-body radiation.  Since gas molecules will emit on average equal amounts of radiation downwards and upwards, then the maximum change in radiation emitted to space at wavelengths where the greenhouse gases are absorbing is given by

dI = - Ikρ/2 dz.                                                Equ. 2

This means that the outgoing greenhouse radiation is halved as it passes through a slab which totally absorbs it.  McIlveen3  quotes a height of 30m for such a layer of the atmosphere near the surface.  Thus with a 50% attenuation every 30m of altitude, effectively no radiation in the bands absorbed by greenhouse gases will be emitted beyond the boundary layer.

Jack Barrett4 has already pointed out that greenhouse gases do not act as black-body radiators.  He was supported by Hug5 who added that the radiation absorbed by the greenhouse gases would be converted into kinetic energy of the air molecules, so heating the air near the surface, and preventing the OLR being cascaded to space.  However, both Barrett and Hug were mistaken in believing that increasing CO2 does not cause global warming.

An increase in the concentration of CO2 will mean that the same amount of radiation is trapped closer to the surface. Beer's Law states

A = ebc                                                Equ. 3

where A is absorbance, e is molar absorptivity, b is path length of the sample, and c is the concentration.  Thus if the concentration of CO2 is doubled, then for the same absorbance the path length will halve.  This means that the heat absorbed via the CO2 will be applied to a layer of air of half the thickness, raising the surface air temperature twice as much. See Figure 1.

Figure 1 – Diagram showing the way the lapse rate changes when the carbon dioxide concentration is doubled (not to scale) (a) the current Slab model adapted from reference 7 Fig. 1 (b) the new “Wine Press” model.

Currently it is believed that an increase in the concentration of CO2 will cause the forcing, and hence the surface temperature, to increase with the logarithm of the concentration of the greenhouse gases.  Using the new approach, the forcing of the air surface temperature is proportional to the change in concentration, and so the effect of anthropogenic CO2 is greater than is currently calculated.

It has been reported extensively6 that the measurements made by radiosondes and by microwave sounding units (MSUs) do not show a warming of the free troposphere, especially in the Southern Hemisphere where the air is free of the aerosols from the Asian Brown Cloud.  This is consistent with absorption of OLR being confined to the boundary layer.

Applying these principles to general circulation models are beyond the scope of this paper.  It is hoped that because of their urgency and importance experienced modellers will immediately develop these ideas.  It should be noted that the main greenhouse gas is water vapour.  Its concentration is controlled by the surface temperature through the Clausius-Clapeyron equation.  If that positive feedback resulted in a runaway event, it could explain how abrupt climate changes happen.

1.      Emden, R. ‘Über Stralungsgleichgewicht und atmosphärische Strahlung’ Sitzunsberichte der k. Bayerischen Akademie der Wissenschaften, Math.-phys. Klasse (1913).

2.      Houghton, J. The Physics of Atmospheres 2nd edn. p. 11, CambridgeUniversity Press, Cambridge, UK (2002).

1. McIlveen, R. Fundamentals of Weather and Climate 2nd edn. p. 250, Stanley Thornes (Publishers) Ltd, Cheltenham, UK (1992).
2. Barrett, J. “The roles of carbon dioxide and water vapour in warming and cooling the Earth’s troposphere” Spectrochimica Acta 51A, 415-417 (1995).
3. Hug, H. “A critical review of the hypothesis that climate change is caused by carbon dioxide” Energy and Environment 11, 631-638 (2000).
4. Tett, S. & Thorne, P. ‘Tropospheric temperature series from satellites’ Nature 42, doi:10.1038/nature03208 (2004)
5. Held, I.M. and Soden, B.J. “Water Vapour Feedback and Global Warming”, Annual Review of Energy and the Environment 25 441 – 475,  (2000)