## Wednesday, April 11, 2007

### Real Climate and Math

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. $S+ \lambda * A = G$ System Earth
II. $\lambda * G = 2 * \lambda * A$ System Atmosphere
III. $S = \lambda * A + (1-\lambda ) * G$ System Solar Planet

These are the three energy equations (equilibrium states?) and there are some basic formulae for G and S and $\lambda * A$:

(IV) $\lambda * A = \lambda * \sigma * T^4_a$
(V) $G = \sigma * T^4_s$
(VI) $S = (1 - a)*\frac {TSI}{4}$

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) :

$G = \sigma * T^4_s = \frac {S}{1 - 0.5*\lambda} = \frac{(1-a) * \frac {TSI}{4}}{1-0.5*\lambda}$

Nice, idea, but what if I say that due to their connectivity, I set System (III) in (I):

$G = 2*\lambda * A + (1-\lambda)* G \rightarrow G = -2 * \sigma * T^4_a$

We now have a dependence of $G(T_s)$ on the Temperature of the atmosphere.

We can further say: $T^4_s = 2 * T^4_a$ and then $T_s = T_a * \sqrt[4]{2}$

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 $\lambda$ 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:

$\lambda = f(T, c(CO_2), c(Aerosols), c(Watervapour), Cloudformation)$

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?