Much ink is spilled on this site, both in the articles themselves and in the threads that follow, on CO2 and its impact on long wave infrared radiation (LWIR). As a site visited by many skeptics of the CO2/global warming nexus, the bulk of these arguments seek to exonerate CO2 as a cause of a warming climate.
One common theme is that the relatively small effect a doubling of CO2 has on LWIR at the top of atmosphere (TOA) seems too small to produce the much larger effect needed for a 3K warming at the surface. Several articles from last summer called into question the claims of the IPCC versus the Stefan-Boltzmann law; and many comments reflect this sentiment also. There appears to be a common misconception that was initiated and is perpetuated to the present time by the climate research community themselves. Let’s examine the problem of infrared transport in the atmosphere through some simple modeling. Some of these results may surprise folks.
Essence of a misconception
If the concentration of CO2 is doubled specifically from its present value of 400ppm to 800ppm, what results is a reduction in outgoing LWIR of about 3.7 W/m2 at TOA. Almost everyone refers to this decline as a “forcing”. The impression made by this unfortunate phrase is that the 3.7 W/m2 at TOA combines with solar radiation heading to the surface and directly drives surface temperature higher. It suggests something like a heat flow vector, if you will, pointing toward the surface.
This is not what happens. The deficit of 3.7 W/m2 doesn’t beam toward the surface from TOA at all. It represents a deficit of outgoing radiation caused by the higher concentration of CO2 at every level of the atmosphere which absorbs more radiation and emits radiation more effectively into all directions. The figure of 3.7 W/m2 at TOA depends on the assumed temperature and composition of the atmosphere.
Near the Earth’s surface something similar occurs. LWIR is absorbed and emitted differently in this atmosphere because of our new CO2 concentration. Because the atmosphere is relatively opaque to LWIR (not completely opaque though; various models are 12% to 35% transparent over the column from surface to TOA), what goes on at TOA has limited effect on what happens at the surface. What matters at the surface is mainly absorption and emission of LWIR in the warmest and wettest part of the atmosphere near the ground. Climate forcing is the disturbance that CO2 causes on local LWIR transport; it is not the 3.7 W/m2 decline in LWIR that we might happen to observe at TOA because of this disturbance.
Radiation energy transfer is a complex problem. People often use the Stefan-Boltzmann equation alone as the basis of their radiation computations. This works only when the radiation is exchanged between surfaces with no IR active medium between them. In other words, depending on the Stefan-Boltzmann law alone severely restricts the problems one may legitimately tackle.
When a physical situation involves boundaries with fixed temperature, or fixed radiant flux, and a medium which can participate in the transfer through emission, absorption or scattering, the problem becomes hugely more difficult and requires a real solver of radiation transport. The only tool at my disposal is MODTRAN which can calculate transport between two points in space as long as temperature and composition are specified. It will not solve for unknown temperature because it isn’t a complete solver of the transport equation.
What the radiation transport problem entails is this: Radiation incident at any point within a space has traversed space on beams from other points along which there has been absorption, emission and scattering. Meanwhile some amount of emitted radiation originates at every point because of its temperature and this is propagated away into all directions. MODTRAN calculates the accumulated changes of radiant intensity along any specified direction of travel from all these factors.
This problem is nearly identical to that of calculating the flux of neutrons in a nuclear reactor delivering heat, transmuting elements, and emitting more neutrons along the way. Without a transport equation one would never understand what goes on in a reactor and the same is true of radiant heat transport in an active atmosphere.
Limitations of MODTRAN
The version of MODTRAN available at the University of Chicago consists of a legacy FORTRAN program written in the 1980s which a person may access in a limited way through a “wrapper” written in some other language. The wrapper provides a graphical interface (GUI) of edit boxes where one may change boundary temperatures, choose from a limited set of atmosphere models and set levels of greenhouse gasses or cloud cover. It is handy but prevents using the full power of the legacy FORTRAN code.
For one thing, one cannot choose at any viewing angle. Only vertical views are allowed. In order to transform from radiant intensity (I) along a vertical path to a true irradiance (flux) on a surface (G), one usually assumes isotropic radiation which makes the conversion as simple as G= pI. This isn’t especially important except to say it is part of a list of issues  that prevent accuracy of MODTRAN being better than 5-10 W/m2. However, differences between models with only a parameter or two of distinction between them are probably more accurate than this.
Second, when a situation involves an infrared (IR) active atmosphere a temperature discontinuity occurs at bounding surfaces. This makes a concept like surface temperature ambiguous. It complicates mixing convective heat transfer into a problem.
Third, after running many models, I have noticed an error in the legacy FORTRAN code. The error manifests itself in separate runs with different parameters which result in exactly the same solutions – and I mean exactly. This is an error. Forty years ago I was paid to rewrite legacy FORTRAN codes, of the same vintage as MODTRAN, to port them between machines using different word lengths, different languages and varying compilers. Just like MODTRAN, these codes came from Federal contracts or Federal agencies. Most contained some number of errors. In the case of MODTRAN the code seems to use single precision arithmetic in places it should use double precision.
The impact of CO2 concentration changes
MODTRAN offers several models of the atmosphere. Let’s take the tropical model as an example for no reason other than a huge proportion of solar input to the Earth takes place in the tropics. Furthermore, let’s look at differences between 260ppm (preindustrial) to 420ppm (nearly current) concentration of CO2. As our top of atmosphere (TOA) we will assume a height of 18km above the 1013mb surface. Our reference model is 260ppm. There are no clouds and we neglect any effects from convection. I focus solely on radiant transport.
As Figure 2 shows, MODTRAN calculates radiant transport differences caused by our chosen increase in CO2 as leading to a decline at TOA of 3 W/m2. The new atmosphere absorbs more LWIR. This decline in LWIR would be measured by satellites, except that it has taken place over 400 years and amounts to only a tenth watt per square meter per decade. It is immeasurable, in other words, within the resolving power of any instrument and noise contributed by the Earth..
The graph of OLR (Figure 3) in this recent WUWT essay shows that climate disturbances present far larger impacts. For example, two recent El Ninos and Mt. Pinatubo show up with far greater effect in this graph than our slow increase of CO2 ever could. As a consequence the graph doesn’t prove anything about the present influence of CO2.
Meanwhile, near the tropical surface this increase in CO2 boosts downward directed flux by 1W/m2. If we assume that the tropical atmosphere and surface were in energy balance in our reference model, then the changes in LWIR flux as calculated at 420ppm will increase temperature of the atmosphere and surface until the outgoing LWIR at TOA reaches the reference value of 301.6 W/m2 again. To figure out how much temperature rise is needed to restore balance, we can use MODTRAN through a number of iterations.
Using first the Stefan-Boltzmann law we can calculate that a 1 W/m2 increase in downward longwave at the surface would boost surface boundary temperature by only 0.18K. This is far too small to balance energy at TOA, but even this small adjustment of surface temperature increases not only OLR but downward LWIR at the surface significantly. Repeated MODTRAN calculations with refined guesses of surface temperature reveal that energy balance at TOA is restored with a temperature rise of 0.7K. Figure 4 shows this. However, a 0.7K increase at the surface which is nearly black (e=0.97), is not possible to maintain without an increase of emitted power (calculated with the Stefan-Boltzmann law) of 4.2 W/m2. A much larger value than we began with.
How does this happen? A small increase in irradiance at the ground surface increases its temperature. This, in turn, increases emitted power from the surface. The increased emitted power is absorbed in the atmosphere; near the surface particularly because this is the moist part of the atmosphere. Some of this absorbed radiation is re-emitted to ground. This, again, raises the surface temperature slightly. The process repeats ad infinitum. It’s an infinite series of re-radiation which eventually converges when there is a return to energy balance. Infinite series like this are common in problems of radiation exchange between surfaces with or without an active medium in between – the resonant cavity of a laser is an extreme example of the former.
Our experiment shows that radiant transport to the bounding surfaces of an IR active medium mainly involve re-radiant exchanges between the surfaces and nearby medium. Exchanges taking place in one part of the medium might have minor influence on other parts. The more that parts of the medium are isolated from one another, the slower the return to equilibrium through re-radiation and temperature adjustments.
In our test case, increasing CO2 concentration from preindustrial estimated values to the present time manifests itself first as a decrease of 3 W/m2 at TOA and an increase of 1 W/m2 at the surface. The disturbance from increased CO2 throughout the atmosphere passes through a chain of events that ends in energy balance restored at TOA and a temperature rise at the surface appearing to require an increase of 4.2 W/m2. To some people this looks inconsistent with the Stefan-Boltzmann law. Yet, it is not. Such apparent inconsistencies occur commonly when an active medium is involved in transport.
1.- A serious question. In this case the 3.7 W/m2 at TOA was compared to energy density required to maintain an increase in surface temperature of only 1K, calculated with the Stefan-Boltzmann equation which is 7 W/m2. The apparent contradiction is easy enough to see.
2.- Hayden quoted in the most recent SEPP newsletter.
3.- This 3.7 W/m2 value of “climate forcing” is quoted in many places, but can be demonstrated by using MODTRAN with any of its summer atmospheric models by doubling CO2.
4.- Mostly from radiation not being isotropic. As an example, from a height of 16km above the Earth’s surface a view of the lower hemisphere will have 0.28 steradians filled with a background of cold space rather than the warm Earth surface of the remaining 6.00 steradians. This is plainly visible out the porthole of an airliner. Because of this change in background the flux value calculated by assuming G= pI will be biased too large.
5.- This long list mainly involves problems with the isotropic assumption in various contexts. Interestingly the wrapper doesn’t use p but the constant 3.14 instead. There is no time dependence in any of these calculations. MODTRAN calculations are instantaneous, but the true changes are delayed and take time.
6.- See for instance: Houghton, J.T., 1986, Physics of Atmospheres, 2nd Ed., Cambridge. P. 13. Or, M. Necati Ozisik, 1973, Radiative Transfer, Wiley-Interscience. p. 320.
7.- A person could almost say that the Earth’s surface temperature causes of much of its own greenhouse effect.
8.-The return to equilibrium can take a long time to achieve as it may involve changes at the Earth’s surface like melting ice or changing vegetation.
9.- Other apparent inconsistencies include that a rising concentration of CO2 can apparently act to cool the Earth rather than warm it especially when the IR active gasses in the atmosphere are concentrated in a thin layer near the ground and the remaining atmosphere above is especially transparent, such as in the Antarctic or winter-time Arctic, or at the ground surface of elevated regions.