Frank Lansner has presented an interesting derivation of climate response to a doubling of CO2. He uses current values for Greenhouse warming and MODTRAN radiative forcing calculations to derive the temperature response. While his approach is interesting, his presentation is confusing and poorly presented. Here I drop most of his discussion to draw out the basic math underlying his main insight – using current Greenhouse conditions to derive a temperature climate response to CO2 doubling.
Derivation of Climate Reponse to CO2 Doubling
The Greenhouse effect accounts for about 33C warming.
CO2 accounts for about 18% of the Greenhouse effect.
The warming due to CO2 is dT_total_CO2 = 33C * 0.18 = 6C.
Starting at 5ppm CO2, there are 6 doublings to 320ppm CO2.
Using MODTRAN, at 0ppm CO2, the total forcing is:
RF0 = 318.396
Using MODTRAN, at 5ppm CO2, the total forcing is:
RF5 = 308.631
Using MODTRAN, at 320ppm CO2, the total forcing is:
RF320 = 288.597
The change in forcing from 0ppm to 320ppm is:
RF0 – RF320 = 29.799
The change in forcing from 5ppm to 320ppm is:
RF5 – RF320 = 20.034
The change in forcing from 5ppm to 320 ppm is xx%:
(RF5-RF320)/(RF0-RF320) = q = 0.6723044
Assuming that the change in temperature is
directly proportion to the change in forcings,
then the change in temperature over the last 6 doublings is
dT_6doubles_CO2 = dT_total_CO2 * q = 6C * 0.67 = 4C
Assuming that the change in temp is roughly the same
between each doubling (and the MODTRAN model supports this)
then the change in temperature due to a single doubling
is simply the change in temp over 6 doublings divided by 6
dT_6doubles_CO2/6 = dT_1double_CO2 = 4C / 6 = .67C
This is the final answer:
each doubling of CO2 results in a change of temperature of .67C
Lansner’s presentation is confusing. The 9.25 factor thingee doesn’t seem to make much sense. It appears to simply be the inverse of the fraction of CO2 greenhouse warming over a single doubling within a range which includes an intial warming + 6 doublings. But we could easily take an additional doubling or drop one (starting with 2.5ppmCO2 or 10ppmCO2) and this factor will change by a large degree. It is an arbitrary factor, a magic number based on starting the doubling counts at 5ppm.
But what about this version of the derivation? Is it similiarly arbritrary? Let’s test it.
First, we check just the previous 5 doublings.
RF10 = 305.522
(RF10-RF320)/(RF0-RF320) = q = .57
dT_5doubles_CO2 = dT_total_CO2 * q = 6C * .57 = 3.4C
dT_5doubles_CO2/5 = dT_1double_CO2 = 0.68C
Next, we check the previous 7 doublings.
RF2.5 = 311.237
(RF2.5-RF320)/(RF0-RF320) = q = .76
dT_7doubles_CO2 = dT_total_CO2 * q = 6C * .76 = 4.6C
dT_7doubles_CO2/7 = dT_1double_CO2 = 0.65C
So this derivation is fairly robust even when the number of doublings changes. It is not tied to an arbitrary starting point. Lansner had an interesting insight and I find this form of the derivation charming in its simplicity.
Note that my derived climate response is 0.67C while Lansner’s is 0.54C. That’s largely due to our different values for the amount of current warming due to CO2 – I have used 6C and he used 5C. We can nearly recover his climate response simply by multiplying mine by 5/6ths: 0.67C * 5 / 6 = .56C.
All forcings were calculated with MODTRAN web interface set to the default values except for the CO2. That includes alt = 70km, sensor facing down.
A ppt presentation addressing some of these issues: