Radiative Forcing from CO₂
Extra energy trapped by the atmosphere as carbon-dioxide concentration rises.
Formula
Variables
Example
Doubling CO₂: ΔF = 5.35·ln2 ≈ 3.7 W/m²
How it works
This relation estimates how much extra heat per square metre the Earth system retains when atmospheric CO₂ rises above a chosen baseline. It is logarithmic because the main infrared absorption bands of CO₂ become progressively less effective as they saturate, so each additional increase has a smaller incremental effect than the one before. The constant 5.35 is an empirically fitted coefficient used in climate physics to convert the concentration ratio into a forcing value.
When to use it
Use this formula when you need a quick estimate of the direct radiative impact of a CO₂ concentration change relative to pre-industrial conditions or another reference level. It is commonly used in climate modelling, greenhouse-gas comparisons, and back-of-the-envelope calculations of warming influence.
Worked example, step by step
Suppose atmospheric CO₂ rises from 278 ppm to 420 ppm. Step 1: Write the formula ΔF = 5.35 ln(C / C₀) Step 2: Substitute the values ΔF = 5.35 ln(420 / 278) Step 3: Compute the ratio 420 / 278 ≈ 1.5108 Step 4: Take the natural log ln(1.5108) ≈ 0.4129 Step 5: Multiply by 5.35 ΔF ≈ 5.35 × 0.4129 ≈ 2.21 W/m² Final answer: the radiative forcing is about 2.2 W/m².
Common mistakes to avoid
- Using log base 10 instead of the natural logarithm ln
- Forgetting to divide by the reference concentration C₀
- Treating the result as temperature change instead of energy forcing in W/m²
Did You Know?
This logarithmic law means each doubling of CO₂ adds roughly the same amount of warming.
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