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Why does each doubling of atmospheric carbon dioxide produce a similar increase in warming potential?

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World map shaded to show annual carbon dioxide emissions by country in 2022, with China, the United States, and India highlighted as top emitters in dark red. A bar chart below lists the top ten emitting countries by million metric tons. — Greenhouse gas Map of annual carbon dioxide emissions by country in 2014.

The relationship between atmospheric carbon dioxide (CO₂) concentration and its warming potential is approximately logarithmic rather than linear. This occurs due to a saturation effect within the atmosphere.1

As CO₂ concentrations rise, it becomes increasingly difficult for additional CO₂ molecules to further influence the "infrared window"—a narrow band of wavelengths in the infrared region that is not typically absorbed by atmospheric gases. Because the gas becomes less efficient at trapping additional radiation as its concentration increases, the radiative forcing does not increase proportionally with the amount of gas added.1

Consequently, this logarithmic relationship predicts that the surface warming potential will rise by roughly the same amount for each doubling of CO₂ concentration. For instance, a doubling of CO₂ levels is estimated to produce an increase of approximately 4 watts per square metre of radiative forcing, leading to a projected warming of 2 to 5 °C (3.6 to 9 °F) over preindustrial temperatures.1