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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:
dI = -Ik(dz + Bk(dz
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 Kirchhoffs equation is valid.
( is the density of the absorbing/emitting gas
B is the Planck function:
EMBED Equation.DSMT4
( is the wavenumber of the line being considered, T is the absolute temperature, c is the speed of light and h is Plancks constant
The equation can be simplified for a beam of monochromatic radiation originating at the Earths 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:
EMBED Equation.DSMT4
i.e., ln(I-B) = -k(z + constant
When z = 0, I = I0
So that ln(I0-B) = constant
EMBED Equation.DSMT4
EMBED Equation.DSMT4
I - B = I0(exp(-k(z)) - Bexp(-k(z)
I = I0(exp(-k(z)) + B[1 exp(-k(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. I0 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.
A more appropriate equation would be:
I = [I0(exp(-k(z))]Tabsorption + {B[1 exp(-k(z)]}Temission
Tabsorption is the temperature of the slab where absorption occurs and Temission 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 and in much more detail in General Circulation Models.
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(z has to be large enough to make the factor exp(-k(z) equal to zero. In absolute terms this can only be achieved if k(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(z) has a value of ~0.999, i.e., only a tenth of a percent of the terrestrial radiation escapes to space. In the saturated limit the equation becomes:
I = [B]Temission
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:
dI = -Ik(dz + Bk(dz = 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 Tabsorption and Temission 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 ks = 0.
I = [I0(exp(-k(z))]Tabsorption + {B[1 exp(-k(z)]}Temission
I = I0
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 = ((Tsurface4
In this case the emission from the surface would be equal to the intensity of insolation.
A crude approach to the problem of the Earths surface temperature is to use the radiative balance between insolation and IR escape:
S/4(1 albedo) = (((Tsurface4
S = solar constant = 1368 W m-2
albedo = 0.31
( = emissivity = 0.95
( = infrared transmission
( = Stefan-Boltzmann constant = 5.67 ( 10-8 W m-2 K-4
For instance, if the estimate of ( = 235/390 = 0.603 [from the K/T budget], using the above figures gives:
Tsurface = 291.9 K
If a forcing of 3.7 W m-2 is caused by a doubling of CO2 concentration then ( = 231.3/390 = 0.593 and this gives Tsurface = 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 CO2 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 CO2 spectrum show saturation in the Earths 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.
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