Discussion of Gerlich and Tscheuschner

This is a copy of a comment I've posted at Andy Revkin's "DotEarth" - somewhat off the main subject area of this site, but I didn't have another place to continue this if needed. References to comment numbers are to the AGU discussion that has now reached over 1000 comments! 

There's an awful lot to respond to, and for one I am definitely getting exhausted by it all. However, there's one very small point I'd like to focus on with Drs. Kramm, Gerlich and Tscheuschner (and G and T's comments here in #974 are much appreciated for their willingness to get involved).

The detail I want to get into here is what exactly G and T proved in their paper on the issue of a planet with no "greenhouse" absorption. That is, the bulk of the discussion in section 3.7.

Gerlich and Tscheuschner claim (in #974) "In our paper, we clearly show that the standard calculation giving the 33 Celsius degrees for the greenhouse effect is wrong." Dr. Kramm seems to echo this in earlier comments on this thread - first that "one should analyze what is right and what is wrong in this article" (#180), and "We consider the Earth without an atmosphere and calculate an temperature on the basis of a radiative equilibrium [...] Then we obtain nearly 255 K and state that the difference between this value and the mean global temperature amounts to 33 K. Unfortunately, this uniform temperature of the radiative equilibrium has nothing to do with the mean global temperature derived from observations [...] " (#457) and

"This simple scheme of the planetary radiation balance for the Earth without an atmosphere (in principle, a thought experiment) is, indeed, highly unsuitable from a physical point of view. The reasons are: First, a uniform temperature which is deduced from the planetary radiation balance does not exist, and second, it has nothing to do with a globally averaged near-surface temperature. [...] Our Moon, for instance, nearly fulfills the requirement of this though experiment of an Earth without atmosphere. It is well known that the Moon has no uniform temperature. There is not only a variation of the temperature from the lunar day to the lunar night, but also from the Moon’s equator to its poles. To calculate the temperature difference between lunar day and lunar night would not only require recognizing its rotation, but also the penetration of energy into the near-surface layers of the ground. [...] If we accept that there is no uniform temperature, then we have to ask what the correct procedure of averaging is. Colleagues engaged in large-scale and meso-scale meteorological modeling know this problem that also occurs in the parameterization of subgrid scale processes. In their manuscript Gerlich & Tscheuschner demonstrate that a correct averaging would lead to highly awkward results." (#770)

and in #851, Kramm responds to my comments: "[...]it is physically inappropriate because a uniform Earth’s surface temperature does not exist. Therefore, we have to consider that the Earth’s surface temperature depends on longitude and latitude in quantifying the greenhouse effect. Consequently, the simple radiation balance model using a uniform surface temperature does not satisfy any requirement of a true radiation balance for an Earth without an atmosphere.".

If I may restate the point you have been repeating here:

Is that the gist of the argument?

I hope I've got it. Here's the problem with it: G&T actually proved something quite important in their section 3.7.4 - the average temperature is always lower than the "effective" temperature. This follows from a simple inequality: ((T1+T2)/2)^4 is less than or equal to (T1^4 + T2^4)/2; ie. the fourth power of the average temperature is always less than the average fourth-power of temperatures.

Now, the fourth-power is important for the surface because the surface, composed of solid or liquid materials, absorbs radiation across the thermal region essentially completely in the continuum, and so is a very good "black body". So for radiation directly from the surface, the Stefan-Boltzmann law holds almost everywhere. That means the total radiated energy from the surface can be calculated as the integral over Earth's surface of the fourth power of the local temperature multiplied by the appropriate constant.

Furthermore, this integral can be divided by Earth's surface area to get an average of the fourth power of the temperature. From what G&T prove, that average is always greater than or equal to the fourth power of the average temperature. Therefore, the radiated energy from Earth's surface, when the average temperature is calculated as it is now at about 15 degrees C, is always greater than what would be radiated by a planet at a uniform temperature of 15 degrees C.

In addition to knowing the average temperature of the planet at 15 degrees C or 288 K, we also know the average incoming solar radiation: as you point out and as G&T calculate, that is equivalent to the radiation emitted by an atmosphere-free planet at a uniform temperature of 255 K. Given that the temperatures on our and any planet are not uniform, we can use the same relation to show that the average temperature on such an atmosphere-free planet that emits at that average radiation level (i.e. has that same average fourth-power of temperature) has to be *less* than 255 K.

So we have 2 input facts:

and therefore from the inequalities we have two more facts we can deduce, without question:

If you combine (2) and (3), you see there is a discrepancy of *at least* 33 degrees between the uniform-planet temperatures corresponding to the levels of outgoing and incoming radiation for Earth. If you combine (1) and (4) similarly, you see there is a discrepancy of *at least* 33 degrees between the actual average temperature of Earth as it is and the average temperature that a greenhouse-free Earth would have based on just incoming solar energy.

So from your premises as I summarized above (A, B, C); (A) I think we all agree with; the average temperature of the Earth's surface reflects a wide range of local temperatures and day/night variations.

But for (B), this lack of uniformity does not remove the relationship between temperature and radiation entirely, but simply puts a lower bound on the outgoing energy flow for a given average temperature, or an upper bound on average temperature for a given energy flow.

And therefore, contrary to your conclusion (C), there remains a discrepancy of *at least* 33 K that needs to be accounted for. And the so-called "greenhouse" effect is in fact the way to understand it.

As Ray Pierrehumbert has repeatedly asserted here, if your argument were correct, there would be no greenhouse effect, and by your own calculations, Earth's surface would have an average temperature of less than 255 K. There is no getting around these inequalities.

Do you agree with this logic? If not, please state clearly what the issue is. Atmospheric absorption has nothing to do with the discrepancy at issue here. Thanks.

Created: 2008-02-11 22:34:54 by Arthur Smith
Modified: 2008-02-11 22:49:29 by Arthur Smith