[spunkets]

The eq I = A ln(C/C_o) Represents an increase in absorbance over some fixed concentration C_o. The factor A, must be evaluated at some fixed P_o and C_o. If either of those change, the relation no longer holds.

[spunkets]

JUst using numbers from memory. The Earth's avg temp is 18^oC. You gave a value of 1.2W/m² for the additional absorption due to doubling the CO₂, instead of the 3.7W/m² for a dry atmosphere. I assume the 1.2W/m² is due to absorption outside the band overlap? Then the temp rises 0.22^oC from CO₂. That would result in a 1.01% increase in avg water vapor concentration, and 0.062^oC of cooling, resulting in a net increase of 0.16^oC. What temp increase would the 1.01% increase in water, plus the overlap increase in CO₂ give?

[ancient_geezer]

The eq I = A ln(C/C_o) Represents an increase in absorbance over some fixed concentration C_o. The factor A, must be evaluated at some fixed P_o and C_o. If either of those change, the relation no longer holds.

Ahem, interesting that for logarithmic relationships, my spread sheet and slide rule demonstrates otherwise.

The change is a fixed amount for a given percentage change in concentration no matter what initial C_o one may select, the constant is a never mind where change in forcing is concerned, necessary only for when one needs to express the total cumulative change from 0 concentration, which is not what the functions convey, nor can they, as 0 concentration is undefined in a logarithmic relation as well as when chosen as a denominator in a fraction.

On a logarithmic curve, it doesn't matter what I choose as my C_o as long as it is not 0 and the log relationship holds in the range I wish to evaluate. The increment in the result remains fixed for a percentage change in C. That is an inherent characteristic of the log function and why it is the basis of the slide rule.

The change from any C_o remains constant for a doubling or indeed for any percentage one chooses to evaluate. It doesn't matter what your initial C_o is as long as the selected value of Co is such that it is within the empirical range of values for which the logarithmic relation holds.