Models don’t over-estimate warming?

At the reblogged article, I made reference to both a recommendation by Judith Curry for an article in Physics Today by Ray Pierrehumbert on infrared radiation and the original article. That article is available online.

More is available in Professor Pierrehumbert’s book, Principles of Planetary Climate (“PoPC“).

As to the elementary model of radiative balance to which I referred, I reproduce it below. The matter is more complicated than that because (a) atmosphere is actually layered, something Professor Pierrehumbert takes on in his Chapter 4. This doesn’t really matter in principle, but if someone wants to calculate what the Outgoing Longwave Radiation (“OLR”) is for a given carbon dioxide concentration of atmosphere, it does matter, since infrared radiation is emitted by successive layers of atmosphere, not just at the top nor just at the bottom. The matter is (b) then more complicated when treating additional effects, such as scattering and thermodynamics, also available in PoPC.

The basic equation of radiative transfer is a balancing act. For Earth (or any planet) not to overheat or freeze, incoming solar radiation needs to be balanced by outgoing blackbody radiation. That incoming radiation is $F \pi a^{2} (1 - \alpha)$ where $F$ is the solar flux, $a$ is the Earth’s radius, and $\alpha$ is the albedo or the amount of solar flux reflected back into space, about 0.3. That is balanced by the outgoing blackbody radiation from the Stefan-Bolzmann law, $4 \pi a^{2} \sigma T^{4}_{s}$ , where $\sigma$ is the Stefan-Boltzmann constant and $T_{s}$ is the resulting blackbody radiation at the surface. (This is for an Earth without an atmosphere. That will be important as will be seen.) Equating gives

$F \pi a^{2} (1 - \alpha) = 4 \pi a^{2} \sigma T^{4}_{s}$

which simplifies to

$F (1 - \alpha) = 4 \sigma T^{4}_{s}$ .

Isolating for the surface temperature gives $T_{s} = \sqrt[4]{\frac{F (1 - \alpha)}{4 \sigma}}$ . Now, $\sigma = 5.67x10^{-8} W/(m^{2} (^{\circ}K)^{4})$ . $F = 1360 W/m^{2}$ , an average over Earth’s average distance from the Sun and over wavelengths. Plugging in gives a $T_{s}$ of $254.5^{\circ}K$ . Actual surface temperature averages at around $288^{\circ}K$ . The difference is because there is, in fact, an atmosphere, and the principle component of the difference is atmospheric concentration of carbon dioxide.

One way of looking at this is to consider CO₂ an albedo reducer. Rewriting

$F (1 - \alpha) = 4 \sigma T^{4}_{s}$

as

$\alpha = 1 - \frac{4 \sigma T^{4}_{s}}{F}$

then, substituting, $\alpha_{\text{effective}} \approx \mathbf{-0.15}$ if $T_{s} = 288^{\circ}K$ . In other words, having an atmosphere with greenhouse gases produces what is effectively a negative or absorptive albedo.

...and Then There's Physics

I thought I might write about the new paper by Jochem Marotzke and Piers Forster called Forcing, feedback and internal variability in global temperature trends. It’s already been discussed in a Carbon Brief post called claims that climate models overestimate warming are unfounded.

Having read the paper, I’m not sure I quite agree with the Carbon Brief title. I think (although happy to be convinced otherwise) that a fairer assessment would be that the paper shows that internal variability can explain the discrepancy between forced model trends and observed trends for periods of 15 years. It also shows that for longer period (62 years) the impact of internal variability is small and the forced model trends are a good match to observations.

I’ll briefly try to explain. They use a very basic energy balance-like model

$latex Delta T = dfrac{Delta F}{alpha + kappa} + epsilon,$

where $latex alpha$…

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2 Responses to Models don’t over-estimate warming?

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Willard says:

3 February 2015 at 11:27

Thanks for the calculation, Hyper!