This post examines how radiative forcing depends on CO2 concentrations in the atmosphere. In CO2 greenhouse demystified, we calculated the effective emission height where “thermal” photons escape to space . This height depends on the lapse rate temperature and defines the outgoing radiative flux for a given wavelength. By integrating over all lines in the CO2 15 micron band we were able to derive by how much the radiative flux reduced for a doubling of CO2. This “instantaneous” energy imbalance is usually called radiative forcing. Now we study how in detail how this radiative forcing depends on CO2 concentration. Figure 1 shows CO2 induced radiative forcing varies as a function of fractional concentration (C/C0), assuming a constant surface temperature of 288 K and a constant lapse rate.

Figure 1: Radiative forcing versus the Fractional inncrease in CO2 concentration (C/C0) where C0 = 100ppm

The increase shows an approximately logarithmic dependency. A best fit to the data for CO2 concentrations from 100ppm(as reference) up to 1000 ppm(C/C0=10), assuming a fixed surface temperature and stable atmosphere gives

R.F = 6.6 log (C/C0) , where C is CO2 .

We have derived the often quoted formula in climate science for the radiative forcing for a CO2 increase from concentration Co to C. This canonical equation is given by ~5.4 log (C/C0)[1] ! OK so our result is about 20% higher – but that is pretty good IMHO, since the formula is never explained without reference to results from various GCM black boxes. Now we see here how it can be approximately derived just from changes to the emission height with increasing CO2.

Figure 2 shows the Planck spectra for a range of concentrations showing how OLR reduces with increasing CO2 to produce this dependence.

Fig 2. Change in outgoing IR spectra for a range of CO2 concentrations. Each increasing spectra has been offset by 5 mmW/m2sr-1cm-1 to better visualise the differences.

Finally we can also make an estimate for the net change in surface temperature due to CO2.

$S = \epsilon\sigma T^4$
$DS = 4\epsilon\sigma T^3 DT$
$DT = \frac{DS}{4\epsilon\sigma T^3}$

Averaging over clouds($\epsilon = 0.5$), oceans and land($\epsilon = 0.95$) we then get a global averaged $\epsilon = 0.65$

If we assume that each increment in forcing DS is offset by the same increase in Black Body radiation due to a small surface temperature rise DT, then we can iterate through long term increases in CO2 concentrations. The result is shown in Figure 3.

Fig 3: Surface temperature change induced by a gradual increase in CO2 concentrations from a starting temperature of 284K

Summary: By calculating effective emission heights for Co2 we have shown that:

1. Modern levels of 300 ppm of CO2 have resulted in an average surface temperature ~ 4 deg.C higher than an atmosphere free of CO2.

2. A doubling of CO2 from 300 – 600 ppm results in a further ~1.5C increase in surface temperatures.

Published results from more sophisticated GCMs reduce these figures by ~ 20%.

References

[1] Myhre et al. New estimates of radiative forcing due to well mixed greenhouse gasses Phys.Rev.Lett., 25, 2715-2718,1998

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