The Second Law and its Criminal Misuse
UPDATE: (See comments for discussion of changes below)
FW?IW the idiocy du jour is that thermal energy is not heat. Thermal energy is heat. Joule showed that about 150 years ago
Last changed 9:30 PM EST
This is probably time to move on to the next section, the mystical revolving planet
The following might be inserted into the introduction (from Robert P):
Gerlich and Tscheuschner  assert that Clausius' statement of the second law of thermodynamics forbids transfer of energy from a colder atmosphere to a warmer surface. As shown in Section (3.9), the second law requires consideration of all heat flows in a process, so one must also include the transfer of thermal energy from the surface to the atmosphere. Ref. 1 does not consider this second part of the process and thus errs. When done properly, there is no contradiction
The fundamental equations of radiative transfer have the Second Law of Thermodynamics built into them, via Kirchoff's Law, which can be derived directly from the 2nd Law. Thus when solved numerically the solutions perforce obey the 2nd law. This applies equally well to simple models described below, and to the most elaborate line-by-line calculations. All show that the presence of greenhouse gases in the atmosphere results in a warmer surface than in their absence.
(As part of Rabett Run's Gerlich and Tscheuschner project, Eli has started drafting parts of a response, which we will gift wrap in Bozo paper and send to some unsuspecting journal, but certainly arXiv. This first part comes almost completely from >pliny but with contributions, in no particular order Eli (it is Rabett Run and don't try and push in line), Barton, Joel, Arthur, Jochen, taavi, Robert and others who have all sharpened the arguments. Anyone who wants on or off the list should write to the comments. Admittedly most of what is below belongs to pliny, so the Kopywrong Kops will have to get in touch, but words have been changed to shelter the bunnies in the meantime. Suggestions for changes and additions are welcome)
Gerlich and Tscheuschner  make fundamental mistakes in their arguments about the thermodynamics of the greenhouse effect which are profoundly revealing. They invoke Clausius' classic statement of the Second Law of Thermodynamics, no process is possible whose result is the transfer of heat from a cooler to a hotter body, to claim that thermal radiative energy from the colder atmosphere cannot warm the hotter surface (principally section 3.9 of Ref. 1).
When following how energy moves between the sun, the Earth's surface and atmosphere, and space, the increases in entropy through every step of the process are simple and obvious, and the net energy flows are always from hotter to colder, as they must be. Estimation of the greenhouse effect contrasts cases when there is no atmosphere, or an atmosphere with no greenhouse gases to cases where there are varying amounts of greenhouse gases. The simplest calculations require significant simplifications but capture the essence of the situation. Radiative transfer models provide detailed information at the cost of complexity. In all cases surface temperatures are found to be higher for higher greenhouse gas concentrations.
It is important to understand Gerlich and Tscheuschner's objection. A clear statement can be found in Fig. 32 on page 340
Fig. 32. A machine which transfers heat from a low temperature reservoir (e.g., stratosphere) to a high temperature reservoir (e.g., atmosphere) without external work applied, cannot exist — even if it is radiatively coupled to an environment, to which it is radiatively balanced. A modern climate model is supposed to be such a variant of a perpetuum mobile of the second kind.
Their view of the second law is both clear and clearly wrong. The simplest explanation of why it wrong is that the Clausius statement refers to an entire process, not a single part of it. By isolating transfer from the colder atmosphere to the warmer surface they are neglecting heat transfer in the reverse direction. Radiative transfer is discussed below using simplified examples to appreciate how the greenhouse effect is a result of basic physics, consistent with all the laws of thermodynamics, and to show how Ref. 1 errs.
There appears to be confusion about whether the Clausius statement applies to net heat flow or simply any flows of heat. Qualitatively one can make a simple argument about interchange of thermal energy between two bodies. Consider two perfectly absorbing disks in a vacuum at temperatures TA and TB, with TA > TB. If B is isolated, it will emit thermal energy at a rate given by the Stefan-Boltzmann Law. If the Clausius statement referred to any flow of heat when the two disks were placed opposite each other B would have to stop radiating towards A because if it did not, heat would be transferred between a body at lower temperature to a body at higher temperature. This is obviously absurd. The ability of either disk to radiate does not depend on the presence of another disk that absorbs the emitted radiation. Further it is not necessary to restrict the heat transfer mechanism to radiation, the same argument holds when energy is transferred by molecular motion, or electrons. Thus, the Clausius statement clearly must apply only to net heat flow, and one must consider all heat flows when applying the second law and not just selected flows in isolation from the others.
Using Fig. 32 and in other places in Ref. 1., Gerlich and Tscheuschner repeatedly apply the second law to the isolated heat flow between the atmosphere and the surface and from this conclude that the greenhouse effect is impossible because it would be a perpetual motion machine of the second kind. We have shown that this is an absurd argument and thus the most basic part of their thesis fails.
One can illustrate this quantitatively in a simplified manner with an idealized example. Again we use two infinite, flat and parallel plates. In this case we will treat the two plates as infinite heat sinks. For the sake of argument Face A is at 300K, face B at 260 K, somewhat the temperatures of the surface and the level of the atmosphere at which greenhouse gases radiate to space. Using the Stefan- Boltzmann law we can calculate the thermal energy and entropy exchanges between the two plates as shown in Fig. 1 which is similar to that of Fig. 32 of Ref. 1 except that includes heat transfer in both directions, which, as was discussed above, must be the case.
Only heat is transferred, energy is conserved and, the net entropy increase of the entire system is positive as the second law requires, but equally clearly, the colder body radiates thermal energy that the hotter body absorbs. The argument of Ref. 1, which considers only part of the process is unphysical and wrong. The Clausius statement is about a complete process, not what happens to individual steps. The example makes clear that there is an interchange of heat by radiation between the colder and the warmer surface. Such an interchange occurs because the net entropy change for the process is positive.
In the idealized example the disks were considered infinite. If they were finite, they would eventually reach a common temperature, however, the argument would be essentially the same for the process with minor changes to account for the changing temperatures of the disks. The point with respect to Ref. 1 is not the details of the process, but the fact that there must be constant heat exchange from the colder to the hotter disk, as well as a larger one from the hotter to the colder.
These simplest examples can be expanded upon. Consider a spherical body whose temperature is maintained at T. Around it place two concentric shells A and B, each infinitesimally larger than the other. Surrounding all this is empty space at absolute zero . For convenience treat everything as perfect blackbodies.
First remove shell B. At equilibrium, the amount of thermal radiative energy leaving shell A will balance that impinging on it
σT4 - 2 σ TA4 = 0 so TA = T/21/4 = 0.84 TNext insert shell B. The equilibrium conditions for both shells are
Shell A: σT4 + σTB4 -2 σTA4 = 0which can be solved to yield
Shell B: σTA4 - 2σTB4 = 0
TA = (2/3)1/4 T = 0.90 TThe net energy flow from A to B is (1/3)σT4 The assumptions that the spheres are perfect blackbodies and the radii of the shells are only slightly larger than the radius of the sphere could be relaxed at the expense of making the solution more complex.
TB = (1/3)1/4 T = 0.76 T
Thus, the addition of the Shell B has caused the temperature of Shell A to be higher than it would be in the absence of Shell B (~0.90 T instead of ~0.84 T), yet Shell B is at a lower temperature than Shell A. This is exactly the situation that Gerlich and Tscheuschner claim would violate the Second Law of Thermodynamics, i.e., that we have warmed an object (Shell A) to a higher temperature than it would have an the absence of the “back-radiation” from a cooler object (Shell B).
Of course, as one can see, the net heat flow is from Shell A to Shell B and thus the 2nd law is not in fact violated, just as is true of the earth / atmosphere case where the net flow of heat is from the earth to the atmosphere and yet the presence of the IR-absorbing atmosphere still results in the surface being warmer than it would be without greenhouse gases.
UPDATE: The next part will probably go in the final version, but I am leaving it in for now for interest. If it is to be included, it has to be made a) obvious and b) bulletproof
The entropy flux of the Earth is interesting. Suppose the Earth had no greenhouse gases. Ideally, it would, as discussed elsewhere, receive at its surface 235 W/m2 at the surface at 255K, and radiate it back as IR. The influx creates 235/255=0.92 W/m2/K entropy, but exactly the same amount is radiated out. What is not usually noted is that this allows for no creation of negative entropy on Earth except for biologically and chemically driven processes. No winds, no heat conduction. For these to happen, the Earth (which is not an isolated system) export net entropy. In other words having absorbed thermal energy from the sun, some portion of this must be transformed into free energy, capable of creating physical work to drive circulation.
Since the outflux equals the influx of radiant heat, that means that at least some of the outgoing radiation must be emitted at a temperature lower than that at which the incoming was thermalized (so Q/T is higher). Due to the greenhouse effect, this happens. A substantial part of the IR leaves from the top of the atmosphere (TOA) at a much cooler temperature. In crude terms, if the greenhouse effect raises the surface temp from 255K to 288 K, the net entropy exported is 235/255-235/288 =0.106 W/m2/K. This is the entropy created by the wind. So the greenhouse effect does more than just keep us warm. It excretes our entropy garbage.
The atmosphere can be treated as a huge heat engine, and the net entropy export is the driver.