Apparently, the radiative forcing contribution of different greenhouse gases is typically added to come up with the total contribution. For example, if radiative forcing between 1750 and 1998 is 1.46 W/m

^{2}for CO2 and 0.48 W/m

^{2}for methane, then the total for the two gases is 1.94 W/m

^{2}. This makes complete sense on the surface, does it not?

But if the effect of a change in the concentration of a GHG is non-linear, is it really correct to linearly add such effects? I was thinking this would be similar to trying to add decibels.

After some digging around, my impression was that the IPCC had addressed this issue in section 2.8.4 of 4AR WGI. If you read the papers cited, nevertheless, it appears that they refer to the equivalence of the addition of forcings and the addition of temperature changes. I don't doubt these operations are roughly equivalent for relatively small effects. But that's not what I'm talking about at all. I'm questioning the validity of

*either*operation, considering that

*both*forcings and temperature shifts are non-linear responses.

In order to illustrate the problem, I've come up with a thought experiment that I've dubbed

*the twin GHGs paradox*.

Imagine there are two GHGs that happen to be identical in their effects. If you have trouble picturing it, suppose one gas is industrially-produced CO2 and the other gas is naturally-produced CO2. Their effects are logarithmic.

Let's say the concentration of both gases has changed from 100 ppmv to 200 ppmv. Then their combined radiative forcing would be given by:

`ΔF = k·ln(200/100) + k·ln(200/100) = 2·k·ln(2)`

Since the gases are identical, the change just described should be equivalent to, say, that of one gas going from 190 ppmv to 390ppmv and the other gas remaining at 10 ppmv, not changing at all (i.e. the combined total is 200 ppmv initially and 400 ppmv later.) Therefore,

`ΔF = k·ln(390/190) + k·ln(10/10) = k·ln(2.05)`

Combining both equations, we have that:

`ln(2.05) = 2·ln(2)`

Evidently, we end up with reductio ad absurdum.

**To get more technical...**

When you're looking at the absorption bands of two GHGs, it seems clear that it matters whether the bands overlap or not. The net absorption calculation will depend on this.

## 2 comments:

The equations provided by the IPCC for N2O and CH4 forcing do take this twin GHG issue into account... (I think the equations appear in the radiative forcing chapter of the third assessment report, and are referred to off-hand in the 4th assessment report)

-Marcus

@Marcus: The equations you mention can be found here. They are non-linear for N20 and CH4 as well. Table 6.1 (section 6.3.3 of 3AR) suggests they just add the forcings linearly, but maybe I'm missing something.

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