Physical Principals for the Cartoon Sensitivity Calculation?

In my rambling post on the Cartoon, I wandered a bit.

lucia (Comment #82813)
October 2nd, 2011 at 6:20 am

Dallas–
Your explanation isn’t very clear because the text mostly just throws out numbers without mentioning names or physical principle. I also notice for some reason, your first step seems to be to create a figure with entirely different numbers:

You have some sort of discussion of:

At the surface we can use the averages of solar absorbed, 168 plus thermals, 24 plus latent 78 plus radiant, 26 calculated as the net, and 40 lost to space for a total of 168 in and 24+78+26+40=168 out. If the 26 changes by 3.7, then the total radiative impact would be (26+3.7)/(168+3.7)=0.173 or 17.3% if

Could you turn those into some type of formula, and mention a physical principle of some sort? The formula, and physical principles would help me understand why you think you assembling the numbers as you have results in the climate sensitivity.

For example: I don’t know why you think this:

The 3.7Wm-2 change would produce a 17.3-15.4=1.9% change in the energy balance or a 0.019*33K=0.63K change in surface temperature.

Physical principles could help here.

For the some sort of discussion, for the surface energy balance, the total solar absorbed by the surface Es, 168wm-2 should balance the energy leaving the surface, Latent, El + Thermals, Et + Radiant absorbed atmosphere, Era +Radiant to space Ers. The radiant to space and radiant absorbed are separated because the energy lost directly to space would not feedback to the surface. Assuming that the surface over time will have a balance, then Es=El+Et+Era+Ers. Should one of the surface energies out change, the surface would attempt to re-balance requiring a higher effective temperature.

The atmosphere's energy inputs are partially dependent on the surface energy fluxes. In the Atmosphere, Esun + El +Et + Era total to equal the energy out as the atmosphere attempts to obtain equilibrium, at a time different than the surface.

Both the surface and the atmosphere would be sensitive to a common perturbation, albeit at differing amounts and time frames.

A 3.7 Wm-2 change in Era is equivalent to a 1K change in surface temperature per generally accepted theory assuming a 33K atmospheric effect on surface temperature. So for an instantaneous perturbation of 3.7Wm-2 impacting Era initially, the change in the energy imbalance would result in a temperature change proportional to the relative impact of the energy flux changed, (Era2-Era1)/(Es2-Es1). {Era2/(El+Et+Ers+Era2)}-{Era1/(El+Et+Ers+Era1)}, Era2=Era1+3.7Wm-2 for a unit change in surface temperature. {(26+3.7)/(24+78+40+(26+3.7))}-{26/(24+78+40+26)}= 0.173 - 0.155 = 1.8 percent change in the atmospheric effect, 0.018*33= 0.6 K which would the surface temperature transient sensitivity to radiative forcing, based on a Cartoon.

The same logic in the atmosphere yields atmospheric transient sensitivity {Era2/(Esun+Et+El+Era2} - {Era1/(Esun+Et+El+Era1)} yielding 1.7% or 0.54K transient sensitivity(TS). Since the change in the surface TSs is related to the change in TSa (atmsophere), the overall Ts of the Earth would lay within the range of the two.

lucia (Comment #82820)
October 2nd, 2011 at 8:38 am

Dallas–
I do agree that energy balance holds. I agree that if energy input (or OLR as a function of surface temperature) change, the surface temperature will adjust to balance.

I’m still not reading a physical principle why you think the ratios you are using are appropriate to estimating climate sensitivity. Why use 33K? Why not 288K? Explaining your notion more fully in a more conventional way might help.


I am not sure how to best describe the principle. At the surface Energy out, Eo is the sum of Et+El+Er which would be proportional to partial fraction of each with respect to the variables of dependence, time, temperature, radiative emissivity, radiative absorption, lapse rate and available moisture. I could get a pretty messy equation if I don't simplify by using the more important relationships. Et is most dependent on temperature differential which is dependent on density. El is most dependent on moisture availability and temperature difference which are both dependent on density. Et is most dependent on temperature and radiative emissivity, which is dependent on radiative absorptivity of the atmosphere. Two thirds of the Er is at wave lengths where the radiative aspects of the atmosphere are minimal. Only the Era, radiative emissions in the spectrum absorbed by the atmosphere is most dependent on emissivity and absorption. So a better answer than , "it is what it is", Would be a partial differential equation showing all the changes with respect other changes. Simplicity allows for estimations, where delta Et/delta T is sufficiently large, delta El/delta T is sufficiently large and delta Es/delta emissivity is sufficiently small, that delta Era/delta emissivity is most impacted by a change in emissivity. In the atmosphere, density decrease with height so the Et and El terms approach zero. As they decrease the Era impact increases. For the above calculation, the average of all, Et, El and Era are assumed from the K&T cartoon. An integration of the changes of all three should result in approximately those values.

This is still hand waving since I haven't set up the partial differential equation, but the physical justification will be in there. :)