Over the last week or so, I’ve reported on my efforts to locate the provenance of the functional forms for the relationship between levels of CO2 and other greenhouse gases and temperature. Lubo has also chipped in on the topic from a different perspective proposing a derivation of a log formula from first principles.
We’ve noted that AR4 endorsed these particular TAR results (here), that Myhre et al 1998 was the primary source for these TAR results here and that Myhre et al 1998 specifically applied the IPCC 1990 forms (see here ); we noted that IPCC 1990 attributed the forms to Wigley 1987 and Hansen et al 1988 (see here for IPCC 1990 discussion) and that Hansen et al 1988 Appendix B simply stated results, attributed there to the Lacis et al 1981 radiative-convective model.
The other leg of their argument was Wigley 1987, published in Climate Monitor, a CRU house organ where Wigley was then employed. I doubt whether this was severely “peer reviewed”. However, the CRU authors are leaders in their field and I see no reason to disrespect Wigley 1987 merely because it appeared in a house organ. However, it has not been easy to locate. The University of Toronto did not have a copy; Wigley himself said that he did not have a copy. However a CA reader has located a copy and kindly emailed me a scanned version, enabling this source to be tracked down a bit further.
Once more there’s rather a dead end. Wigley 1987 simply stated his results, rather than deriving them, as shown below. Wigley also had some interesting comments about GCM performance in this article, which I’ve also excerpted at length below.
The Logarithmic Formula
Wigley simply states the results without deriving them:
On theoretical grounds it can be shown that the relationship between radiative forcing change at the top of the troposphere and concentration change is linear at low concentrations, square root at intermediate values and logarithmic at higher concentrations. Because of this, the results of detailed radiative transfer calculations for the various trace gases give a linear concentration dependence for CFCs, square root for CH4 and N2) and logarithmic for CO2.
For CO2 and CH4, I have used results from the Kiehl and Dickinson 1987 model, supplied by Jeff Kiehl.
For CO2 over the range 250 ppmv to 600 ppmv, the Kiehl-Dickinson model gives a change in radiative forcing ΔQ, resulting from a concentration change from C_0 to C which can be described by:
CO2: ΔQ= 6.333 ln (C/C_0) (14)
to within 0.01 wm-2. Note that the precision of this fit should not be confused with the accuracy of the implied ΔQ values. The equation is probably accurate to about +-10% with similar accuracy for the results for other trace gases given below. Equation (14) implies a ΔQ value of 4.39 wm-2 for a doubling of CO2 concentration.
Lubo also believes that the relationship is logarithmic and this idea is plausible. This may very well be, but I would be surprised if Wigley had precisely the same proof in mind. Hans Erren has written in saying that a logarithmic form was stated by Arrhenius but Arrhenius’ results were not derived “on theoretical grounds” within the terms of Wigley’s above assertion. I suspect that there is some folk history to the linear-square root-logarithm rule within the climate community of the 1980s – Ramanathan probably has something on it, but this is a dead end in terms of tracking IPCC references. There is more to be said on Lacis et al 1981 and Myhre et al 1998 methods, which I will get to.
Wigley on GCMS
Wigley 1987 contained an interesting discussion on a topic that continues to this day: the divergence between the warming predicted by GCMs and the historical record. Wigley:
The accepted range of equilibrium warming due to a doubling of CO2 concentration (or its radiative equivalent) is 1.5-4.5 deg C. Most recent GCM studies have given values of 4 deg C or more. A 4 deg C warming for doubled CO2 corresponds to a climate sensitivity of about 1 deg C/wm-2, i.e. an equilibrium warming of about 1.7 deg C for the 1880-1985 radiative forcing of 1.7 wm-2. This is very much larger than the observed global mean surface air temperature change. This discrepancy, which is partly accounted for by oceanic lag effects, has been noted earlier by other authors e.g. Gilliland and Schneider 1984; Wigley and Schlesinger 1985. It has a number of possible explanations: the magnitude of global warming may have been considerably underestimated; the damping or lag effect of the oceans may be much greater than is currently believed; large additional forcings may be operating on the century time scale; and/or the most recent GCM studies may have overestimated the climate sensitivity.
Uncertainties in the observational temperature record are discussed by Wigley et al 1985 and Jones et al 1986. Current opinion is that, if anything, the amount of warming has been overestimated, an opinion not shared by myself and my colleagues. I will not consider this option further, but, instead concentrate on the other three possibilities.
Wigley then goes on to consider oceanic lag as an explanation for the non-response, concluding against this on the basis that the “only way that one could obtain the observed warming would be for vertical ocean mixing to be much greater than could be obtained with a pure diffusion or upwelling-diffusion ocean model”.
He then discusses the possibility of an overlooked forcing, noting that one would have to reduce the 1880-1985 radiative forcing by 0.7 wm-2 or more, given GCM sensitivity. He canvassed the possibility of a decline in solar over the 20th century as being an explanation (something that all parties would now seem to agree is exactly opposite to what was going on):
This would occur if some other external forcing agent were operating on the century time scale. The obvious possibilities are solar irradiance changes and/or long time scale changes in the volcanogenic aerosol changes of the stratosphere. …Solar variations of this magnitude cannot be ruled out. A decline of 0.7 wm-2 would correspond to a 0.3% reduction in solar output, well within the uncertainty in historical (pre-satellite) measurements of the solar constant. Recent satellite data show a decline of about 0.1% in irradiance over the 1979-85 period (Kyle et al; Willson et al 1986) attesting to the feasibility of a 0.3% decline over the past century or so.
He dismissed the potential forcing from volcanic aerosols as being anything other than transient. His other suggestion was planetary albedo:
Another possibility is a long-term increase in planetary albedo, Since the incoming radiation is about 240 wm-2, an increase of only 0.002 in the planetary albedo i.e. about 0.7% would be sufficient to reduce the net radiation balance by 0.7 wm-2.
Notably and surprisingly missing from this inventory were manmade aerosols. I think that Charlson (Hansen) et al 1990 was seminal in putting these into the mix as an explanation for the divergence. As I noted in my comments on Ramanathan’s AGU presentation, while the “discovery” of manmade aerosols seems to be somewhat opportunistic, the aerosols themselves are real enough and the effect has to be considered in a historical context. (Of course, opportunism can creep in, as one notes by the inverse relationship between GCM sensitivity and aerosol history, so there’s a lot of softness in this topic.)
Wigley’s third alternative is that GCMs are too sensitive:
A third possibility is that he climate sensitivity of about 1 deg C/wm-2 implied by recent GCMs is too high. If one accounts for the ocean damping effect using either a PD or UD model, and, if one assumes that greenhouse gas forcing is dominant on the century time scale, then the climate sensitivity required to match model predictions is only about 0.4 deg C/wm-2. This corresponds to a temperature change of less than 2 deg C for a CO2 doubling. Is it possible that GCMs are this far out? The answer to this question must be yes. Feedbacks involving sea-ice and cloud variations are still relatively poorly handled by all climate models and the feedback due to changes in cloud optical properties (e.g. Somerville and Remer 1984) has not been included in any GCM studies. This latter factor alone could possibly reduce the climate sensitivity by a factor of two.
The model uncertainties described in (1) [oceanic lag] and (3) [sensitivity] are of course well known. Their existence is the reason that, in spite of recent GCM results, the equilibrium temperature change due to a CO2 doubling is still generally given as lying in the range 1.5-4.5 deg C. The lower limit is entirely compatible with observations.
It’s interesting to see once again the references to cloud feedback as a major uncertainty and the possibility that a particular cloud feedback could reduce climate sensitivity. I wonder how IPCC represented the uncertainties, as stated here by Wigley, in their contemporary reports. I’ll look at that some time.