Climate Sensitivity Again

OK, here is a model (adapted from a wikipedia article that I can't find at the moment) that is simple enough that I can understand, but that reveals many of the key aspects of climate sensitivity. Define the following variables, all of which should be considered as planetary averages, numbers are from Lubos:

S – incoming solar flux at top of tropopause

A – albedo of the planet

e – emissivity of the planet

c – Stefan-Boltzmann constant = 5.67 * 10^-8 W* m^-2*K^-4

T – surface temperature of the planet = 288 K

Te – effective radiating (absolute) temperature of the planet = 256 K

R – radiated energy per unit area, (radiant flux)

In equilibrium, R has to equal the absorbed energy R = (1-A)*S
Te is defined by R = c Te^4. That is, Te is the temperature that a black body radiating R would have.

In the “grey body” approximation, we also have R= e c T^4. Note that the emissivity can be estimated from the last two equations e = (Te/T)^4 = .62

We would like to calculate climate sensitivity lambda = dT/dS, but it’s simpler to work with its inverse

lambda^-1 = dR/dT = d/dt (S*(1-A)) = d/dT(e c T^4)

= 4R/T - (R/(1-A))dA/dT +(R/e) de/dT

The solar input is being treated as constant. The 4 R/T term is what I called the grey body (inverse) climate sensitivity and is = 3.3 Adding CO2 to the atmosphere operates through the third term, decreasing the emissivity, so de/dT is negative, inverse lambda decreases, and lambda increases. Feedbacks due to increases in temperature put more water vapor in the atmosphere, also decreasing emissivity. Similarly, melting of polar ice decreases albedo, and lambda again increases. Other effects are more complex and controversial – clouds both decrease emissivity and increase albedo.

A small percentage change in either albedo or emissivity is needed to produce a K degree of change in temperature.

UPDATE: Well, I expected some commenter to have figured this out, but I guess I need to do it myself. In fact all the terms in the equation should have the same sign. Naturally its important to be a bit careful in ones definition. The radiative forcing should be defined as the difference between the equilibrium output = input = S(1-A) and the output with emissivity adjusted but not surface temperature = e c T^4 RF = e c T^4 - S(1-A)

d(RF)/dT = 4 R/T + (R/e)(de/dT) + (R/(1-A))(dA/dT)

Fairly modest feedbacks in the second two terms are adequate to produce the IPCC's lambda=0.8 value. (Black body = .18, "grey body" = .3)