It would be exceedingly bad news for planetary habitability if real greenhouse gases were grey gases (see Exercise 4.3.1). Greenhouse gas concentrations would have to be tuned exceedingly accurately to maintain a planet in a habitable temperature range, and there would be little margin for error. Thus, it is of central importance that, for real gases, OLR varies much more gradually with greenhouse gas concentration than it would for an idealized grey gas 2. This is another area in which the quantum nature of the Universe directly intervenes in macroscopic phenomena governing planetary climate.
Infrared Radiative transfer is a very deep and complex subject, and mastery of the material in this section will still not leave the reader prepared to write stateoftheart radiation codes. Nor will we cover the myriad engineering tricks large and small which are needed to make a radiation code fast enough to embed in a general circulation model, where it will need to be invoked a dozen times per model day at each of several thousand grid points. We do aspire to provide enough of the basic physics to allow the student to understand why OLR is less sensitive to the concentration of a typical real gas than to a grey gas, and to help the student develop some intuition about the full possible range of behaviors of greenhouse gases on Earth and other planets, now and in the distant past or future. Such an understanding should extend even to greenhouse gases that are not at present commonly considered in the context of climate, or implemented in standard "off the shelf" radiation models. What would you do, for example, if you found yourself wondering whether SO2 or H2S significantly affected the climate of Early Earth or Mars? The grey gas model does not provide an adequate first attack on such problems. We thus aspire to provide enough of the basic algorithmic equipment to allow the student to build simplified radiative models from scratch, that get the OLR and infrared heating profiles roughly correct.
Even though we will have recourse to a "professionally" written radiation code in Section 4.5, we'd like to at least draw back the curtain a little bit, so that the reader will not be left with the alltoocommon notion that radiation routines are black boxes, the internal workings of which can only be understood by the high priesthood of radiative transfer. Hopefully, this will also open the door to entice more people into innovative work on the subject.
Since the main point is to understand how the wavenumber dependence of absorptivity affects the sensitivity of OLR to greenhouse gas concentration, we'll begin with a discussion of the spectrum of outgoing longwave radiation in an idealized case. Let's consider a planet whose surface radiates like an ideal blackbody in the infrared, having an atmosphere whose air temperature at the surface Tsa is equal to the ground temperature Tg. The temperature T(p) is monotonically decreasing with height in the troposphere, and is patched continuously to an isothermal stratosphere having temperature Tstrat. The atmosphere consists mostly of infraredtransparent N2 and O2 with a surface pressure of 105Pa, like Earth. Unlike Earth, the only greenhouse gas is a mythical substance (call it Oobleck), which is a bit like CO2, but much simpler to think about. It has the same molecular weight as CO2, but it's absorption coefficient KOob(v) has an absorption band centered on wavenumber v0 = 600cm1. Within 100cm1 of v0, KOob has the constant value k0. Outside of this limited range of wavenumbers, Oobleck is transparent to infrared, i.e. KOob = 0. To make life even simpler for the atmospheric physicists of this planet, KOob is independent of both temperature and pressure. Like real CO2, the specific concentration of Oobleck (qOob) is constant throughout the depth of the atmosphere.
What does the spectrum of OLR look like for this planet? The answer is shown in the left panel of Figure 4.5. In this figure, we have assumed that the Oobleck molecule has an absorptivity of 1m2/kg. Then, with a molar concentration of 300ppmv (like CO2 in the 1960's), the specific concentration is 4.6 ■ 104 and the optical thickness KoqOobps/g cos 6 is 9.4 within the absorption band. Since the atmosphere is optically thick in this wavenumber region, infrared radiation in this part of the spectrum exits the atmosphere with the temperature of the stratosphere. This is exactly what we see in the graph. Outside the absorption band, the atmosphere is transparent,
2Lest there be any misunderstanding, we must emphasize at this point that "less sensitive" does not mean
"insensitive." If CO2 were a grey gas, then doubling it's concentration, as we are poised to do within the century, would be unquestionably lethal. Because CO2 is not in fact a grey gas, the results may be merely catastrophic.
Wavenumber (cm1) Wavenumber (cm1)
Figure 4.5: The OLR spectrum for a hypothetical gas which has a piecewise constant absorption coefficient. The dasheddotted lower curve is the blackbody spectrum corresponding to the stratospheric temperature Tstrai while the dashed upper curve is the blackbody spectrum corresponding to the surface temperature Ts. The calculation was carried out for Ts = 280K and TSirai = 200K, and with a greenhouse gas concentration sufficient to make the optical thickness « 10 in the central absorption band. Left panel: The gas has an absorption coefficient of 1m2/kg within a single absorption band extending from 500 to 700 cm1. Right Panel: The gas has additional weak absorption bands from 300 to 500 cm1 and 700 to 900 cm1, within which the absorption coefficient is .125m2/kg.
and hence infrared leaves the top of the atmosphere at the much higher temperature of the ground. The overall appearance of the OLR spectrum is that the greenhouse gas has "dug a ditch" in the spectrum of OLR, or perhaps "taken a bite" out of it. The ditch in the spectrum reduces the total OLR of the planet, but not so much so as if the absorption were strong throughout the spectrum, as would be the case for a grey gas. This is the typical way that real greenhouse gases work: they make the atmosphere optically thick in a limited part of the spectrum, while leaving it fairly transparent elsewhere. The strength of the greenhouse effect is not so much a matter of how deep the ditch, but how wide.
Oobleck is a very contrived substance, but the above exercise gives us a fair idea of what to look for when interpreting real observations of the spectrum of OLR. Figure 4.6, giving the OLR spectrum of Mars observed at two times of day by the TES instrument on Global Surveyor, is a case in point. Mars has an essentially pure CO2 atmosphere complicated only be optically thin ice clouds and dust clouds (which can be very thin between major dust storms). The planet thus provides perhaps the purest illustration of the CO2 radiative effect available in the Solar system. In Figure 4.6 a CO2 "ditch" centered on about 650cm1 is evident both in the afternoon and sunset spectra. At the trough of this ditch, the radiation exits the atmosphere with a radiating temperature of about 170 K both in the afternoon and sunset cases. This temperature is similar to the coldest temperature encountered in the upper atmosphere of Mars in the Summer (see Fig 2.2), and is compatible with the strong decrease of temperature with height seen in the soundings. Away from CO2 ditch, the atmosphere appears transparent, and the emission resembles the blackbody emission from a land surface having temperature 265K in the afternoon case and 212K in the sunset case. These numbers are compatible with the observed range of ground temperature on Mars, crosschecked by nearsurface data from landers.
Afternoon limb observation 
1 1  
Sunset nadir ofcserv.ati.oii  
 
Nightside observation 
—  
/ 
l C0fi gas 
—  
y 
^ \ Amio spheric dust \\ 
  

 
200 400 BOO BOO lOi/i^. 1200 1400 1000
200 400 BOO BOO lOi/i^. 1200 1400 1000
Figure 4.6: Some representative OLR spectra for Mars, observed by the Thermal Emission Spectrometer on Mars Global Surveyor at various times of day.
In a situation like that shown in the left panel of Figure 4.5, the OLR is as low as it is going to get, provided the stratospheric temperature is held fixed. Increasing the greenhouse gas concentration qG cannot lower the OLR further since, in the spectral region where the gas is radiatively active the atmosphere is already radiating at the coldest available temperature 3. Suppose, however, that instead of the gas being transparent outside the central absorption band, there is a set of weaker absorption bands waiting in the wings on either side of the primary band a gas we may call "TwoBand Oobleck." In this case, illustrated in the right panel of Figure 4.5, the effect of the weaker bands on OLR is not yet saturated, and increases in qG will cause the OLR to go down until these bands, too, are saturated. But what if there are yetweaker absorption bands waiting a bit farther out? Then further increases of qG will yield additional decreases in the OLR. One can imagine making the process continuous by making the width of the bands smaller, and the jump in absorption coefficients between adjacent bands smaller. Real greenhouse gases act very much like this, as they almost invariably have absorption whose overall strength decays strongly with distance in wavenumber from a central peak. The rate at which the absorption decays with distance determines the rate at which the OLR decreases as the greenhouse gas concentration is made larger.
From Eq. 4.11, 4.12 or 4.13, if we know the transmission function, we can carry out the integral needed to obtain the radiative fluxes. As we shall see shortly, in most cases the dependence of k on wavenumber is so intricate that solving the problem by doing a bruteforce integral over wavenumber is prohibitive if one aims to do the calculation enough times to gain some insight from modelling a climate (even in a single dimension). In any event, doing the calculation with enough spectral resolution to directly resolve all the wiggles in k(v) provides much more information about
3This example is somewhat contrived, since increasing the concentration of a greenhouse gas generally cools the stratosphere. However, it serves to illustrate the way additional weak absorption bands influence the OLR.The additional OLR reduction from cooling of the stratosphere as qG increases is a secondary effect. Since the temperature there is already so low, it wouldn't throw off the result very much to simply replace the OLR at the depths of the ditch to zero.
spectral variability than is needed in most cases. What we really want is to understand something about the properties of the transmission function averaged over a finite sized spectral region of width A, centered on a given frequency v. Specifically, let's choose A to be small enough that the Planck function B and its derivative dB/dT are both approximately constant over the spectral interval of width A. In that case, when the solution for the flux given in Eq. 4.12 or its alternate forms is averaged over A, B can be treated as nearly independent of v and taken outside the average. In consequence, the resulting bandaveraged equations have precisely the same form as the original ones, save that the fluxes are replaced by average fluxes like
and the transmission function is replaced by i rv+A/2
We need to learn how to derive properties of Tv (p,p'). The essential challenge is that the nonlinear exponential function stands between the statistics of kv and the statistics of Tv.
The transmission function satisfies the multiplicative property, that
if p' is between p1 and p2. The multiplicative property means that the transmission along a path through the atmosphere can be obtained by taking the product of the transmissions along any number of constitutent parts of the path. The bandaverage transmission loses this valuable property, because for two general functions f and g, f f (v)g(v)dv = (/ f (v)dv)(/g(v)dv). The equality holds only if the two functions are uncorrelated, which is not generally the case for the transmission in two successive parts of a path. In the first part of the path, the strongly absorbed frequencies are used up first, and are no longer available for absorption in the second part of the path. The system has memory, and one can think of the light as becoming "tired," or depleted more and more in the easily absorbed frequencies the longer it travels, with the result that the absorption in the latter parts of the path are weaker than they would be if fresh light were being absorbed.
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