A logarithmic relationship between CO2, radiative forcing and direct impact is reported by IPCC and widely relied on. While this may well be a plausible relationship (Lubo, for one, endorses it), it is not easy finding a proof of the relationship. In a recent post, I noted this in connection with IPCC AR1 (1990), where I reported their references. Today, I’m going to discuss the handling of the logarithm formula in AR4, AR3 and then report on today’s search for the source of the Nile.
AR4 stated that the logarithmic formula of TAR remained valid as follows:
(2.3.1) The simple formulae for radiative forcing of the LLGHG [long-lived greenhouse gases] quoted in Ramaswamy et al. (2001) [IPCC TAR] are still valid. These formulae are based on global radiative forcing calculations where clouds, stratospheric adjustment and solar absorption are included, and give an RF of +3.7 W m2 for a doubling in the CO2 mixing ratio. (The formula used for the CO2 RF calculation in this chapter is the IPCC (1990) expression as revised in the TAR. Note that for CO2, radiative forcing increases logarithmically with mixing ratio.)
Section 1.3.1 of TAR stated:
If the amount of carbon dioxide were doubled instantaneously, with everything else remaining the same, the outgoing infrared radiation would be reduced by about 4 Wm-2. In other words, the radiative forcing corresponding to a doubling of the CO2 concentration would be 4 Wm-2. To counteract this imbalance, the temperature of the surface-troposphere system would have to increase by 1.2°C (with an accuracy of ±10%), in the absence of other changes. In reality, due to feedbacks, the response of the climate system is much more complex. It is believed that the overall effect of the feedbacks amplifies the temperature increase to 1.5 to 4.5°C. A significant part of this uncertainty range arises from our limited knowledge of clouds and their interactions with radiation. …
It has been suggested that the absorption by CO2 is already saturated so that an increase would have no effect. This, however, is not the case. Carbon dioxide absorbs infrared radiation in the middle of its 15 mm band to the extent that radiation in the middle of this band cannot escape unimpeded: this absorption is saturated. This, however, is not the case for the bands wings. It is because of these effects of partial saturation that the radiative forcing is not proportional to the increase in the carbon dioxide concentration but shows a logarithmic dependence. Every further doubling adds an additional 4 Wm-2 to the radiative forcing.
I have not been able to locate any mention of the logarithmic relationship in SAR (or for that matter any use of the quantity 4.37 wm-2 for doubled CO2). Nor in IPCC (1994).
I discussed this aspect IPCC AR1 (1990) here giving an extended quotation. I observed that IPCC (1990) cited two references for the logarithmic relationship:
1) Wigley Climate Monitor 1987. Climate Monitor is a house publication of CRU, where Wigley was then employed. It is not carried at U of Toronto and Wigley said that he no longer had a copy.
2) Hansen et al 1988. As noted in previous post, Appendix B of Hansen et al 1988 stated the logarithmic relationship referring to Lacis et al 1981 as authority:
Radiative forcing of the climate system can be specified by the global surface air temperature change ΔT0 that would be required to maintain energy balance with space if no climate feedbacks occurred (paper 2). Radiative forcings for a variety of changes of climate boundary conditions are compared in Figure B1, based on calculations with a one-dimensional radiative-convective model (Lacis et al, 1981). The following formulas approximate the ΔT0 from the 1D RC model within about 1% for the indicated range of composition. The absolute accuracy of these forcings is of the order of 10% because of uncertainties in the absorption coefficients and approximations in the 1D calculations:
where x_0=315 ppmv; X<1000 ppmv.
Lacis et al GRL 1981
I obtained Lacis et al 1981 today. It did not mention the logarithmic relationship anywhere. It contained a very sketchy description of their 1-D radiative-convective model.
TAR 6.3.5 Simplified Expressions
As noted in comment #3 below, TAR 6.3.5 also contains the following statement:
IPCC (1990) used simplified analytical expressions for the well-mixed greenhouse gases based in part on Hansen et al. (1988). With updates of the radiative forcing, the simplified expressions need to be reconsidered, especially for CO2 and N2O. Shi (1992) investigated simplified expressions for the well-mixed greenhouse gases and Hansen et al. (1988, 1998) presented a simplified expression for CO2. Myhre et al. (1998b) used the previous IPCC expressions with new constants, finding good agreement (within 5%) with high spectral resolution radiative transfer calculations. The already well established and simple functional forms of the expressions used in IPCC (1990), and their excellent agreement with explicit radiative transfer calculations, are strong bases for their continued usage, albeit with revised values of the constants, as listed in Table 6.2. Shi (1992) has suggested more physically based and accurate expressions which account for (i) additional absorption bands that could yield a separate functional form besides the one in IPCC (1990), and (ii) a better treatment of the overlap between gases. WMO (1999) used a simplified expression for CO2 based on Hansen et al. (1988) and this simplified expression is used in the calculations of GWP in Section 6.12. For CO2 the simplified expressions from Shi (1992) and Hansen et al. (1988) are also listed alongside the IPCC (1990)-like expression for CO2 in Table 6.2. Compared to IPCC (1990) and the SAR and for similar changes in the concentrations of well-mixed greenhouse gases, the improved simplified expressions result in a 15% decrease in the estimate of the radiative forcing by CO2 (first row in Table 6.2), a 15% decrease in the case of N2O, an increase of 10 to 15% in the case of CFC-11 and CFC-12, and no change in the case of CH4.
Their Table 6.2 entitled “Simplified expressions for calculation of radiative forcing due to CO2, CH4, N2O, and halocarbons” contains the following expressions described as follows”
The first row for CO2 lists an expression with a form similar to IPCC (1990) but with newer values of the constants. The second row for CO2 is a more complete and updated expression similar in form to that of Shi (1992). The third row expression for CO2 is from WMO (1999), based in turn on Hansen et al. (1988).
F = 5.35 ln(C/C0)
F= 4.841 ln(C/C0) + 0.0906 (C – C0)
F= 3.35 *( ln(1+1.2C+0.005C2 +1.4 x 10-6C3) – ln(1+1.2C0)+0.005C0)2 +1.4 x 10-6C0)3))
The third row expression appeared previously in Appendix B of Hansen et al 1988 (See above). So that doesn’t help.
The IPCC 1990 form and the WMO (1999 – Ozone Depletion) both derive from Hansen et al 1988, which cites Lacis et al 1981, which turns out to be a dead end with no mention of logarithms. I haven’t located Shi (1992) yet and again will update this when I do so. It’s quite possible, perhaps even plausible, that there should be a logarithmic relationship between CO2 and direct temperature impact, but it’s certainly not easy to locate a clear derivation.
Shi, G., 1992: Radiative forcing and greenhouse effect due to the atmospheric trace gases. Science in China (Series B), 35, 217-229.
WMO, 1999: Scientific Assessment of Ozone Depletion: 1998, Global Ozone Research and Monitoring Project, World Meteorological Organization, Report No. 44, Geneva, Switzerland.