The Schwarzschild equation is the basis of understanding radiative transfer; the passage of radiation from the surface to space. This is a brief introduction to the equation and how it may be interpreted. A model is presented that serves to explain the consequences of the Schwarzschild equation for an optically thick atmosphere.

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**The Schwarzschild Equation and Radiative Transfer**

The Schwarzschild Equation for the transmission of radiation of a particular frequency through an absorbing medium is:

d*I* = -*Ik*r* *d*z* + *Bk*rd*z*

The first term is called the sink function [radiation is being transferred into molecular energy, rotations and vibrations and thence to the thermal reservoir, i.e., the bulk atmosphere], the second is the source function which allows for emission of thermal radiation along the optical path. The emission is caused by the transfer of thermal energy of the bulk atmosphere to GHGs, producing vibrational/rotational excited states which decay radiatively. The negative sign of the first term indicates that the intensity of radiation decreases as the path length increases and the opposite is the case for the second term.

*I* is the radiation intensity at any given height

*k* is the absorption coefficient [values are well known for the GHGs] and is used in the source function for emission intensity assuming that Kirchhoff's equation is valid.

*r* is the density of the absorbing/emitting gas

*B* is the Planck function:

*B* = 2p*hc ^{2}v^{3}/*(

*e*- 1)

^{hcv/kT}

n is the wavenumber of the line being considered, *T* is the absolute temperature, *c* is the speed of light and *h* is Planck's constant

The equation can be simplified for a beam of monochromatic radiation originating at the Earth's surface and directed outwards to space along the *z* axis. There is an assumption of zero scattering which is reasonable for IR radiation.

The integrated form of the equation becomes:

ln(*I*-*B*) = -*k*r*z* + constant

When *z* = 0, *I *= *I*_{0}

So that ln(*I*_{0}-*B*) = constant

*I - B = I*_{0}(exp(-*k*r*z*)) - *B*exp(-*k*r*z*)

* *

*I* = *I*_{0}(exp(-*k*r*z*)) + *B*[1 - exp(-*k*r*z*)]

*I* is the intensity of radiation passing through the absorbing layer of thickness *z*. The first term is an expression of the Beer-Lambert law of light absorption. *I*_{0} is the intensity of the incident radiation.

To be applied to any particular line *k *must be known and is known for all the greenhouse gases. The data are contained in the HITRAN database.

The equation suffers from having the density as a constant, which for the ever dwindling atmosphere is not so. Another problem with the equation is that the temperature of the atmosphere varies very much with altitude and for a thick slab of atmosphere the initial absorption of radiation occurs at a higher temperature than that of the part where the emission takes place. The integreation was made with these problems in mind, but suffers from being mathematically impure! The results are to be considered as an indication of the issues involved, rather than giving an accurate description of the Schwarzchild equation.

A more appropriate equation would be:

*I* = [*I*_{0}(exp(-*k*r*z*))]_{T}_{absorption} + {*B*[1 - exp(-*k*r*'z*)]}_{T}_{emission}

*T*_{absorption} is the temperature of the slab where absorption occurs and *T*_{emission} is the temperature where emission from the slab occurs. The different densities at the two altitudes are also indicated.

To get an accurate answer requires more mathematics and computation, but that can be done and is done by programmes such as MODTRAN. Much more detailed calculations are incorporated into General Circulation Models, but the input files are MODTRAN results suitably parameterized to save computing time.

At some frequencies the absorption of radiation by some of the greenhouse gases is so great to allow the absorption to be 'saturated'. This means that any further addition of the gas will not change the amount of absorption and will not contribute further to warming of the atmosphere. For real saturation to occur the product *k*r*z* has to be large enough to make the factor exp(-*k*r*z*) equal to zero. In absolute terms this can only be achieved if *k*r*z* has the value of infinity. So, saturation can never be achieved, but for practical purposes may be regarded as being achieved if the factor exp(-*k*r*z*) has a value of ~0.001, i.e., only a tenth of a percent of the terrestrial radiation escapes to space. In the saturated limit the equation becomes:

*I* = [*B*]_{T}_{emission}

Thus for any line at any concentration of an absorbing substance *I* may be calculated. Sufficient data are available for the equations to be solved for atmospheres of any composition. The equation also indicates the possible heating that occurs in a slab of the atmosphere that is not in radiative equilibrium, e.g., when the sun shines in the morning and the air warms up. The difference between the two terms gives the absorbed radiation intensity and this can easily be transformed into a temperature difference when the heat capacity of the slab is included. Of course, this is tempered by the non-radiative heat transfer that occurs between a heated surface and the atmosphere and if the surface is ocean the evaporation of water is a dominant mechanism for cooling the surface and warming the atmosphere with the release of energy when water vapour condenses.

When the Earth/atmosphere system is in quasi-equilibrium the value of the rate of change in the escaping flux is zero:

d*I* = -*Ik*rd*z* + *Bk*rd*z* = 0

The two terms are equal in magnitude under such conditions. An increase in forcing causes the first term to be larger, more absorption occurs in a smaller altitude increment. To re-establish the quasi-equilibrium the second term must also become larger. With more radiation being absorbed in a smaller altitude increment means a higher warming rate and this must be countered by a higher cooling rate. Since warming and cooling are at different temperatures, this means that the system warms up. Both *T*_{absorption} and *T*_{emission} must increase. Just how much these terms increase depend on the extent of the forcing and on the non-radiative contributions to the cooling of the surface.

If the atmosphere were to become devoid of all the GHGs the integrated equation would reduce as shown for all *k*'s = 0.

*I* = [*I*_{0}(exp(-*k*r*z*))]_{T}_{absorption} + {*B*[1 - exp(-*k*r'*z*)]}_{T}_{emission}

In such a case the value of *I *would be given by the Stefan-Boltzmann equation with appropriate value for the emissivity of the surface:

*I* = es*T*_{surface}^{4}

In this case the emission from the surface would be equal to the intensity of insolation.

A crude approach to the problem of the Earth's surface temperature is to use the radiative balance between insolation and IR escape:

S/4(1 - albedo) = ets*T*_{surface}^{4}

S = solar constant = 1368 W m^{-2}

albedo = 0.31

e = emissivity = 0.95

t = infrared transmission

s = Stefan-Boltzmann constant = 5.67 x 10^{-8} W m^{-2} K^{-4}

For instance, if the estimate of t = 235/390 = 0.603 [from the K/T budget], using the above figures gives:

*T*_{surface} = 291.9 K

If a forcing of 3.7 W m^{-2} is caused by a doubling of CO_{2} concentration then t = 231.3/390 = 0.593 and this gives *T*_{surface} = 293.2 K, an increase of 1.3 K.

This is not far removed from the results from the GCMs for an instant doubling of CO_{2} concentration. Our main criticism of the GCMs is the amplification of their results by the use of positive feedbacks which are not fully understood and fraught with uncertainty, and the possible downplaying of the negative feedback from the evaporative water thermostat.

**A note about saturation**

Although some parts of the CO_{2} spectrum show saturation in the Earth's atmosphere and others that are near saturation there are two absorption bands which are very far from being saturated. These are centred at 961 and 1064 cm^{-1} and are very weak absorptions. They do contribute considerably to the greenhouse effect in the atmosphere of Venus.

**Acknowledgement**

The integrated form of the Schwarzschild equation was brought to our attention by Hartwig Volz and we thank him for that.

A Model based on the Schwarzschild equation for an optically thick atmosphere

This model uses specific atmospheric layers that are just physically thick enough to ensure that no radiation from other than their immediate neighbours is transmitted. They are optically thick over the frequency range of terrestrial emissions. Because of the pressure gradient the layers are thinner at lower altitudes. These layers represent one of the limits of the Schwarzschild equation in which the sink function is large enough to allow all the incident radiation to be absorbed and none transmitted. The source function is entirely that given by the Planck emission equation. s is the Stefan-Boltzmann constant (normally represented by the Greek sigma).

Radiative equilibrium: sT_{1}^{4} = sT_{e}^{4}

Layer 1: sT_{2}^{4} = 2sT_{e}^{4}

Layer 2: 2sT_{2}^{4} [=4sT_{e}^{4}] = sT_{1}^{4} [=sT_{e}^{4}] + sT_{3}^{4} sT_{3}^{4} = 3sT_{e}^{4}

Layer 3: 2sT_{3}^{4} [=6sT_{e}^{4}] = sT_{2}^{4} [= 2sT_{e}^{4}] + sT_{4}^{4} sT_{4}^{4} = 4sT_{e}^{4}

Layer 4: 2sT_{4}^{4} [=8sT_{e}^{4}] = sT_{3}^{4} [= 3sT_{e}^{4}] + sT_{g}^{4} sT_{g}^{4} = 5sT_{e}^{4}

Note that the ground temperature increases with the number of optically thick layers. By induction there is a general equation:

sT_{g}^{4} = (1 + optical thickness).sT_{e}^{4}

A crude attempt to use this equation for the Earth's surface temperature in a purely radiative world would put the optical thickness as 235/390 = 0.6, based on the emissions from the surface and to space.

The Earth's atmosphere is not optically thick over the whole frequency range of terrestrial radiation. A fraction of the CO_{2} spectrum confers optical thickness to the atmosphere, some other portions being somewhat less than that needed for optical thickness. An increase in the concentration of CO_{2} increases the optical thickness of a specific part of the terrestrial spectrum and should lead to some further warming.

The observations since 1900 show that, although the CO_{2} concentration has increased continually there have been periods of cooling and warming. The warming from the increasing CO_{2} concentration must therefore be small enough for it to be enhanced or cancelled out by the other factors that affect the atmosphere's temperature. It is not the major reason for the temperature variations.

The model deals with the case for an atmosphere without absorbers where the optical thickness is zero.

sT_{g}^{4} = (1 + optical thickness).sT_{e}^{4}

sT_{g}^{4} = sT_{e}^{4}

It would be cold.