Radiative Forcing of Carbon Dioxide (Simplified Expression)

Radiative forcing is a change in the balance of energy, counted per square metre, in watts per square metre. It asks a narrow question: if the concentration of carbon dioxide moves from a reference value $C_0$ to a new value $C$, how much more energy is the climate system left holding, with the carbon dioxide alone changed? The answer $\Delta F$ is positive when $C$ is above $C_0$ and negative when it is below. It is a bookkeeping quantity, a way to put the change in one gas on a common scale before anyone asks what happens to temperature. This page uses the simplified expression in Table 6.2 of the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), whose coefficient 5.35 comes from the radiative transfer calculations of Myhre and others (1998).

The logarithm means the formula responds to the ratio $C/C_0$, not to the difference. Equal ratios give equal forcing: every doubling adds $5.35 \ln 2 = 3.71$ W/m², which the report states as 3.7, so going from 400 to 800 ppm adds exactly as much as going from 200 to 400 (both give 3.70834). The practical reading is that a given number of extra parts per million adds less forcing when the concentration is already high than when it is low.

The page's example is the report's own test of the expression. From 278 ppm in 1750 to 365 ppm in 1998, $5.35 \ln(365/278)$ gives 1.4567 W/m², and Table 6.1 of the report prints 1.46. Because the relation can be rearranged, you can also run it backwards: a forcing of 3.708337416 W/m² from a starting value of 278 ppm points to 556 ppm, which is a doubling. That direction answers the question of what concentration a stated forcing implies.

Know what the number is not. It is the forcing at the tropopause after the stratosphere has adjusted, for carbon dioxide alone; the other gases have their own expressions in the same table. It is not a temperature: how far the surface warms depends on the climate sensitivity, which this formula does not contain. And it is the Third Assessment Report's simplified expression, not the updated figures of the later IPCC assessments, which differ by a few percent. The concentrations are mixing ratios of carbon dioxide in dry air, in parts per million.