Radiative Forcing of Methane (Simplified Expression with Nitrous Oxide Overlap)

This is the methane counterpart of the carbon dioxide expression: the extra energy per square metre, in W/m², that methane traps when its concentration rises from $M_0$ to $M$. It has two parts. The first, $0.036(\sqrt{M} - \sqrt{M_0})$, is the direct effect of the methane. The second, the bracketed difference $f(M, N_0) - f(M_0, N_0)$, is subtracted, because methane and nitrous oxide absorb at some of the same wavelengths and the two gases partly compete for the same radiation. The function $f$ carries that overlap, and it is evaluated at the unperturbed nitrous oxide concentration $N_0$, which is 270 ppb in 1750.

The square root, where carbon dioxide has a logarithm, comes from the strength of the methane absorption: its bands are already strongly absorbing, so more methane still raises the forcing but progressively more slowly than the concentration itself. The coefficient $0.036$ and the function $f$, with its constants 0.47, $2.01 \times 10^{-5}$ and $5.31 \times 10^{-15}$ and exponents 0.75 and 1.52, are printed in Table 6.2 of the Third Assessment Report of the Intergovernmental Panel on Climate Change, with $M$ in ppb of methane, $N$ in ppb of nitrous oxide, and the subscript 0 marking the unperturbed value. Use the constants as printed.

The page's example is the report's own case. Methane rose from 700 ppb in 1750 to 1745 ppb in 1998, and with $N_0 = 270$ ppb the expression gives 0.4838 W/m², where Table 6.1 of the report prints 0.48. The square-root term alone is 0.5514, so the overlap correction takes some off it, and leaving the correction out would overstate the forcing. Run backwards, 0.4838206548 W/m² from 700 ppb returns 1745 ppb, which shows that the arithmetic closes in both directions.

Be clear about what is included. This is the forcing of the methane itself. The report adds the effects of methane on ozone and on stratospheric water vapour separately, as indirect effects, and they are not in this expression. Later assessments update the numbers, so treat the result as the Third Assessment Report's figure. Both concentrations are mixing ratios in dry air, in parts per billion, and the nitrous oxide value is held at its unperturbed level: the page does not follow a changing nitrous oxide concentration.