In several recent posts, we’ve been reviewing the provenance of the radiative forcing estimate of about 4 wm-2 and the logarithmic form for estimating climate response to increased CO2 levels. This has led on the one hand to several primary references, including Lacis et al 1981 and, on the other hand, to references to realclimate collaborator David Archer’s MODTRAN calculator.
I want to focus today on the structure of these calculations – or, at least, to do so to the extent possible given that the methodologies are poorly described. In our discussions, many readers have volunteered observations relating to formulas for simple absorbance of radiation by gas. However, what these observations fail to address is that the relevant calculation here requires a detailed specification of the atmosphere, which includes not just CO2 levels, but a temperature varying from surface to atmosphere exit, other radiative gases including water vapor, clouds, etc.
Given this atmospheric profile, the calculations yield downwelling (or upwelling) infrared radiation in wm-2. The Archer calculator provides Java results for a variety of conditions, permitting the calculation of upwelling and downwelling radiation at specified altitudes. (It would be nice to have this algorithm available in a Matlab or R function and probably it wouldn’t be too hard to convert.)
What catches my eye about these calculations is that atmospheric conditions are held rigid – as though it were a coincidence that the CO2 radiation-to-space maximum and tropopause are at the same elevation. My own intuition is that the atmospheric profile is itself affected by presence of CO2. In Lubo short note on the matter, he took a similar perspective, hypothesizing that additional CO2 would raise the tropopause – a change in atmospheric profile that doesn’t occur in the rigid calculations of Myhre et al and Archer, where the entire impact is concentrated on downwelling wm-2. The form of the problem strikes me as an interesting type of calculus of variations problem and it would be interesting to see what this type of mathematician could do with it.
On to a review of the sources.
Myhre et al 1998
I’m going to start my review with Myhre et al 1998 online here, as that is relatively recent and was relied upon by IPCC TAR. Unfortunately,they do not provide a detailed or even cursory description of their algorithm, saying only that radiative forcing is calculated as a “difference between irradiances”:
In this study, radiative forcing is calculated as the difference between irradiances in the pre-industrial and present day atmosphere due to changes in concentrations of WMGG as described in IPCC 1995. The full definition of radiative forcing includes stratospheric temperature adjustment (IPCC 1995). However radiative forcing prior to this adjustment is often used in the intercomparison of radiation schemes since it is less computationally expensive. We refer to these forcings as “adjusted” and “instantaneous” respectively.
They state that they only consider direct forcing (i.e. no feedbacks):
Only the direct forcing due to a change in well-mixed greenhouse gas (WMGG) concentration is considered here.
Whereas Lacis et al 1981, as shown below, appear to have used only one vertical atmospheric profile, Myhre et al say that more than one vertical profile must be studied. In this article, they use three vertical vertical profiles (SH, Tropic, NH):
Myhre and Stordhal 1997 and Freckleton et al 1998 have shown that for global radiative forcing calculations, it is not sufficient to use a single vertical profile. Freckleton et al 1998 have shown that three vertical profiles ( a tropical profile and northern and southern hemisphere extratropical profiles) can represent global calculations sufficiently. These profiles are used for all calculations in this study except for the adjusted BBM forcing calculations ….
They also compare results from line-by-line and more parameterized models:
Three radiative transfer schemes are used, a line-by-line (LBL) model [Edwards, 1992], a narrow-band model (NBM) [Shine 1991] and a broad-band model (BBM) [Myhre and Stordhal 1997] …
They report closely similar results for CO2 forcing from pre-industrial to the present for the LBL (1/76 wm-2), the Shine NBM model (1.79 wm-2) and the Myhre and Stordal 1997 BBM (1.80 wm-2). These values are consistent with the forcing values (1880-1985: 1.7 wm-2) discussed in Wigley 1987, which caused Wigley to reflect on the discrepancy between these results and GCM output).
The cursory description does not suggest any iteration in the process, so I take it that the atmospheric profile is held rigid through the addition of CO2.
Freckleton et al 1998
Freckleton et al is cited in Myhre et al 1998, and, to me, was a higher quality article, in the sense that it is actually a bit more than an abstract. They state that their results are based on the Shine NBM model (which is a parameterization of LBL models), asserting that the parameterization is an effective interpolation as follows:
We use the 10 cm-1 narrow-band radiative transfer scheme of Shine (1991) with the HITRAN 1992 (Rothman et al. 1992) spectral-band data, except where otherwise stated. In a number of publications (Freckleton et al. 1996; Christidis et al. 1997; Pinnock and Shine 1998) it has been shown that this scheme can reproduce both irradiances and forcings to within a few percent of line-by-line calculations for a wide range of gases.
While computer advances seem to make the need for NBM models increasingly unnecessary (especially in 1-D calculations), I don;t see the use of NBM radiation models as being problematic and, while I’ve not cross-checked the above claims, they don’t seem implausible to me.
Data for the three vertical profiles used in Muhre et al 1998 is available in print form in Freckleton et al 1998 (which I have transcribed from the pdf.) The plot below shows the temperature by altitude for the three profiles.
Freckleton also parameterizes clouds in the 3 profiles in terms of “high” (85 kPa), “medium” (50 kPa) and “low” (Tropic-10 kPa, Extratropic – 25 kPa). High cloud decreases quite noticeably from SH (31.8%) through the tropics (25.5%) to the NH ( 20.4%) – an interesting asymmetry, given the many statements about how uncertain cloud effects and feedbacks are. Mid-level clouds are higher in the extratropics (SH-23%, NH-26%) than in the tropics (13%), while low clouds are higher in the tropics (15%) and NH(14%) than in the SH.
Although both Myhre et al and Freckleton et al say that several vertical profiles must be considered, they don’t report actual results from these profiles.
Lacis et al 1981
Let’s go back and review the methodology of Lacis et al 1981 in light of the later methods. This early (1981) 1-D radiative-convective model was said to have been used in deriving the parameters and forms cited in Hansen et al 1988, which were cited in IPCC (1990). The parameters (though not the forms) were re-estimated in Myhre et al 1998, which was cited in IPCC TAR and IPCC TAR results were re-asserted without modification in AR4. So it still lives on a little bit. They said:
The 1-D model uses a time-marching procedure to compute the vertical atmospheric temperature profile from the net radiative and convective energy fluxes. The radiative flux is obtained by integrating the radiative transfer equation over all frequencies, using the temperature profile of the previous time step and an assumed atmospheric composition, The convective flux is the energy transport needed to prevent the temperature gradient from exceeding a preassigned limit (6.5 deg C/km) which parameterized effects of vertical mixing and large scale dynamics.
The radiative calculations were made with a method (Lacis et al 1979) which groups absorption coefficients by strength for efficiency. Pressure and temperature dependent absorption coefficients are from line-by-line calculations for H2), CO2, O3, N2) and CH4 (McClatchey et al 1973) including continuum H2) absorption (Roberts et al 1976). Transmission of thermal radiation by these gases is shown in Fig 1. Climatological cloud cover (50%) and aerosol properties (Toon and Pollack 1976) are used, with appropriate fractions of low (0.3), middle (0.1) and high (0.1) clouds. Wavelength dependence of cloud and aerosol properties is obtained from Mie scattering theory. Multiple scattering and overlap of gas absorption bands are included.
Model approximations and uncertainties are discussed by Hansen et al 1981. The models equilibrium sensitivity is ~3 deg C for doubled CO2. The model includes major feedback effects believed to operate in the climate system. The sensitivity is similar to the global mean sensitivity of 3-D climate models (NAS, 1979. It is widely believed that this equilibrium sensitivity is correct to within a factor of 2.
A detailed description of the radiative calculations will be given in a separate paper. …
Unfortunately, Lacis et al have here provided nothing more than an abstract of how their model works. I’ve been unable to locate the promised “detailed description of the radiative calculations” in any contemporary articles by Lacis, so perhaps this promise never materialized.
I was unable to locate any specification of the atmospheric parameters used in the Lacis et al calculation. I presume that they had to specifiy an atmospheric profile, as was done in Myhre et al? Did they use a U.S. mid-latitude atmosphere, a tropical atmosphere,…? Impossible to tell.
Thirdly, unlike Myhre et al, Lacis (Hansen) et al state that they use a type of iteration (“time-marching procedure”) to calculate the “vertical atmospheric temperature profile”. It would be very interesting to know how they performed this calculation. It sounds like an empirical solution to the calculus of variations problem and well worth seeing how they did it. It’s too bad that climate science documentation is so horrendous. Yes, I realize that this is from 1981, but 1981 papers in other disciplines can still be de-ciphered.
I think that it makes sense to view David Archer’s popular Modtran Java implementation in the same context as Myhre et al 1998.
Archer’s script appears to be structurally similar to the Myhre et al 1998 calculation in the sense that he uses an atmospheric profile (temperatures by altitude, gas composition, clouds etc) to calculate upwelling and downwelling radiation in wm-2. Archer provides for a wider range of profiles (tropical, mid-latitude summer and winter, subarctic summer and winter), each with a variety of cloud conditions and permits the calculation of both upwelling and downwelling radiation at different altitudes. It’s a very nice tool. Archer’s webpage says that his
This is an old model (1990’s), illustrative but not necessarily in strict quantitative agreement with current state of the art line-by-line models.
It would have been nice to know the major areas of difference and whether there are any material differences in the calculations. It seems to me that it wouldn’t be very hard to turn this calculator into an R- or Matlab function, which would definitely be handy as well.
I’ve experimented a bit with Archer’s program and will discuss these results on another occasion. In today’s context, the main issue that I wish to draw attention to is that Archer, like Myhre et al 1998, has a rigid atmospheric profile. Thus, the entire impact of additional CO2 goes to downwelling wm-2.
As someone who took some microeconomic courses and inhaled that approach to comparing different “equilibria” (or steady states), the rigid-atmospheric profile approach seems reminiscent of an economist doing supply-demand diagrams with completely inelastic demand and then reporting that price is very sensitive to changes in supply. I’ll discuss this analogy more on another occasion.
If CO2 levels are, in some way, connected to the atmospheric temperature profile, as seems to be the case, then the atmospheric profile should change with CO2 levels. I’m not sure that this should necessarily be termed a “feedback” if the atmospheric temperature profile (especially the tropopause location) is a type of calculus of variations problem. In this case, the new atmospheric temperature profile with additional CO2 should be calculated in the same step. As noted above, it would be interesting to examine the “time-marching procedure” of Lacis et al to see how they did it.
It also seems to me that one could derive some interesting mathematical properties of toy models that more precisely capture the mechanisms than the simple absorption models that preoccupy too many readers. The sort of toy model that I have in mind would specify an atmospheric analytically and likewise specify the radiation model analytically (in terms of simple functions) and then see what happened. If I remembered how to do calculus of variations, I’d try to do it.