Fixing the basic AGW calcs II

This comment was left in the thread of Judith Curry’s blog.  It is an explanation of the problems introduced when the simple black body sphere estimate for basic forcing is not used.  While Tomas makes the point that the solution is nonsense, these things can still be estimated and do have uses so I’m not as dismissive as he of the idea, however on the complexity of the problem we agree.  So when people say 33 C from global warming gasses, you have to wonder where they got their facts or if they understand them.  The 33C value is just an estimate which is likely to be a poor quality one.

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Tomas Milanovic | December 14, 2010 at 7:23 am | Reply

Judith just look at how bungled this “sensitivity” concept is .

(1) F = ε.σ.T⁴(definition of emissivity)
dF = 4.ε.σ.T³.dT + σ.T⁴. dε => dT = (1/4. ε.σ.T³).(dF – σ.T⁴. dε)

Ta = 1/S . ∫ T.dS (definition of the average temperature at time t over a surface S)

Now if we differentiate under the integral sign even if it is mathematically illegal because the temperature field is not continuous we get :

dTa = 1/S . ∫ dT.dS

Substituting for dT
dTa = 1/S . ∫ [(1/ 4.ε.σ.T³).(dF – σ.T⁴. dε)] . dS

Now this is a sum of 2 terms :

dTa = { 1/S . ∫ [(T/ 4.ε) . dε].dS } + { 1/S . ∫ [dF/4. ε.σ.T³].dS }

The first term is due to the spatial variation of emissivity.
It can of course not be neglected because even if the liquid and solid water has an emissivity reasonably constant and near to 1 , this is not the case for rocks , sable , vegetation etc . It is then necessary to compute the integral which depends on the temperature distribution.
E.g same emissivity distribution , different temperature distributions give different values of the integral and different “sensitivities”.

The second term is more problematic.
Indeed the causality in the differentiated relation goes from T to F . If we change the temperature by dT , the emitted radiation changes by dF .
However what we want to know is what happens with T when the incident radiation changes by dF.
This is a dynamical question whose answer can not be given by the Stefan Boltzmann law but by Navier Stokes (for convection) , the heat equation (for conduction) and the thermodynamics for phase changes and bio-chemical energy.

OK as we can’t answer this one , let’s just consider the final state postulated as being an equilibrium.
Not enough , we must also postulate that the initial and final “equilibrium” states have EXACTLY the same energy repartition in the conduction , convection , phase and chemical energy modes.
In other words the radiation mode must be completely decoupled from other energy transfer modes.
Under those (clearly unrealistic) assumptions we will have in the final state dF emitted = dF absorbed.

Now comes the even harder part . The fundamental equation (1) is only valid for a solid or some liquids so the temperatures and fluxes considered are necessarily evaluated at the Earth surface.
If we took any other surface (sphere) going through the atmosphere , all of the above would be gibberish.

Unfortunately the only place we know something about the fluxes is the TOA because it is there that we will postulate that radiation in = radiation out.
This is also wrong (just look at the difference between the night half, the day half and the sum of both) but this is the basic assumption of all and any climate models sofar.
So what we postulate at some height R where the atmosphere is supposed to “stop” is :
FTOA = g(R,θ,φ) with g some function.
From there via radiative transfer model and assuming known lapse rate, we’ll get to the surface and obtain F = h(R,θ,φ)] with h some other function depending on g (note that h , so F depends also on the choice of R e.g the choice where the atmosphere “stops”)
Last step is just to differentiate F because we need dF in the second integral.
dF = ∂h/∂θ.dθ + ∂h/∂φ.dφ

Now substitute dF and compute the second integral . The sum of both gives dTa , e.g the variation of the average surface temperature.
We can also define the average surface flux variation :
dFa = 1/S . ∫ dF.dS

It appears obvious that {∫ [(1/ 4.ε.σ.T³).(dF – σ.T⁴. dε)] . dS} / ∫ dF.dS
(e.g dTa/dFa) will depend on the spatial distribution of the temperatures and emissivities on the surface as well as on the particular form of the h function which transforms TOA fluxes in surface fluxes.
It will of course also change with time but this dynamical question has been evacuated by considering only initial and final equilibrium states even if there actually never is equilibrium.

A careful reader will have noted and concluded by now that it is impossible to evaluate these 2 integrals because they necessitate the knowledge of the surface temperature field which is precisely the unknown we want to identify.
The parameter dTa/dFa is a nonsense which can only have a limited use for black bodies in radiative equilibriums without other energy transfer modes.
The Earth is neither the former nor the latter.