I don't have much time to go into the math of these simple models on realclimate.org, but I have had some thoughts after wading through the simplified math.
Nothing of this is new, since it all merely represents known physics. Although there some uncertainities I have with those models. First, there are these three energy levels, where Ground and Solar are independent:
I. System Earth
II. System Atmosphere
III. System Solar Planet
These are the three energy equations (equilibrium states?) and there are some basic formulae for G and S and :
So far, so nice, but then they start to derive the Ground Temperature via (V) in (I), (VI) in (I) and finally (II) in (I) :
Nice, idea, but what if I say that due to their connectivity, I set System (III) in (I):
We now have a dependence of on the Temperature of the atmosphere.
We can further say: and then
So, if we can measure the athmospheric temperature, we already know the Surface temperature, which is only slightly warmer?
Of course, this is a basic model, which is, as they stress, not applicable to Earth, but rather doable in a closed experiment in the lab.
For real models, we also have feedbacks and radiative forcings (Differentials). First of all, their is nothing short of only Greenhouse-Gases in this example. Then, we have multiple atmospheric levels, so we have energy emission on top and bottom of the different layers influencing each other. We have a lot of other influences on the emissivness, like clouds formation, radicals and so forth. This shows that lambda is no constant, but a dynamic variable:
So, lambda is certainly one of the decisive variables in the system, which have the biggest influence (along with the Solar-variable).
So, how do we determine lambda?