Prove that radiative forcing grows logarithmically withCO 2 concentration under the circumstances illustrated in Figure 34.13 Specifically,

Question:

Prove that radiative forcing grows logarithmically withCO₂ concentration under the circumstances illustrated in Figure 34.13

Specifically, assume that for wavelengths in the vicinity of the 15 ?m absorption peak, theCO₂ absorption coefficient ?(?) exceeds the value ?^? at which absorption saturates. Further, assume that ?(?)falls exponentially ?(?) = ?₀C _CO2 e^?c(???₀⁾, beyond some ?₀ in the right wing of the 15 ?m absorption peak. Here ?₀ is a constant and C_CO2 is the concentration of atmospheric CO₂. Show that the range of wavelengths over which absorption is saturated grows proportional to the logarithm of C_CO2 .