Reader DAV raised the following interesting question:
The strange thing about 6..3.5 Simplified Equations that gets me is why should CO2, CH4 and N2O have different equational forms? And what would be the physical basis for raising something to the 0.75 or 1.52 power? The whole thing looks ad hoc as if someone was insistently forcing a linear regression fit.
This was raised in the context of a discussion of the logarithmic form of the CO2 relationship, reader DAV observing that other structural forms were reported for other GHGs. So where did these other relationships come from originally?
IPCC TAR reported their “simplified expressions” in their Table 6.2 shown below here:
The provenance of the various CO2 expressions is provided, but the provenance for some of the other ones is not as clear as one would like. However, it’s not hard to determine that they come from Myhre et al 1998, the corresponding table being shown below here. Myhre states explicitly that the functional forms, with all their peculiarities, are derived from IPCC AR1 (1990); indeed, some of the parameters remain unchanged (CH4 for example.)
Moving back to IPCC (1990), it cited Hansen et al 1988, mentioning that the functional forms were adopted from Wigley 1987, a publication in the CRU house organ, Climate Monitor. And sure enough if one continues back to Hansen et al 1988, one finds the functional forms in Appendix B, as shown below:
As previously noted, there is no derivation of the functional forms in Hansen et al 1988; it cites Lacis et al 1981, where the matter is not discussed at all. Possibly Wigley 1987 will shed some light on this. The CRU library is mailing me a copy.
On a more positive note, whatever the reasons for the original derivation, Myhre et al 1998 re-estimated the functional forms from their radiative-convective model, showing the following relationship generated from their radiative-convective model. From this one can see that, at levels that interest us, not much really turns on whether the relationship is modeled as a log relationship or a square root relationship.
What is perhaps more instructive here is that the results in Myhre et al 1998, as in Lacis et al 1981, were calculated from a 1-D radiative-convective model, which should be much more accessible analytically than a 3-D model. Myhre et al 1998 summarize their model 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.
The forcing is defined as: the change in irradiance at the tropopause following adjustment of stratospheric temperatures, following the trace gas perturbation. This so-called “adjusted forcing” is a better indicator of climatic impact than the so-called “instantaneous forcing” in which stratospheric temperatures are kept constant (see IPCC 1994, 1995; Hansen et al. 1997). We use the fixed-dynamical-heating approximation (see e.g. Forster et al. 1997) to calculate the temperature changes.
At this point, I’m reasonably content that the narrow-band radiative transfer scheme relied on here does do what it’s said to do here. I haven’t confirmed this, but I don’t plan to pursue this at this time. My browse of the radiative transfer literature leaves me with a pretty high comfort level that the authors are not Mannian.
I’m mainly interested in the apparent assumption of an unchanging atmospheric profile with additional CO2. It hardly seems a very unlikely coincidence to me that the tropopause just “happens” to occur at a level of pretty much maximum CO2 impact; it seems far more likely to me that tropopause height is interrelated to CO2 levels. This complicates the math significantly. It looks to me like a calculus of variations problem and not necessarily a very easy one.
One of the problems with articles like Myhre et al 1998 and Lacis et al 1981 is that they deal with the matter in purely parametric terms. They leave atmosphere unchanged and derive a sharp response. My guess is that concurrent changes in the atmospheric profile would reduce the response, perhaps even substantially. Because neither Lacis et al nor Myhre et al took a mathematical approach to the problem, they don’t really provide much insight into the relationships. Lubo” has posted on this aspect of the topic and I’m going to examine his logic.
Hansen, J., M. Sato, A. Lacis, R. Ruedy, I. Tegen, and E. Matthews, 1998: Climate forcings in the Industrial Era. Proc. Natl. Acad. Sci., 95, 12753-12758.
Myhre et al 1998 online here