Schwarzschild, Chandrasekhar, Back Radiation and Ghost Detection

Non-physical two-way back-and-forth energy transfer with "back radiation". Notice that with N layers the "back radiation" to the surface increases like N x F into an absurdity for many layers. Nevertheless (or because of that), pretending to understand "back radiation" has come to be viewed as the sign of deep scientific knowledge acknowledged by all CO2 alarmists and quite a few (so called) skeptics, which differs the expert from the man in the street, who cannot understand how such a back-and-forth process can be real. See also posts under category "radiative heat transfer" connecting to the old controversy between Pictet and Prevost.

The notion of "back radiation" can be traced back to the Schwarzschild equations for radiative transfer from 1906, which formally mathematically decompose net radiative heat energy transfer into the difference of opposite two-way energy transfers, in order to faciliate symbolic solution by analytical mathematics. This method is also used by Chandrasekhar in his book on radiative transfer, and there correctly referred to as "formal solution".

Both Schwarzschild and Chandrasekhar thus use a formal mathematical decomposition similar to formally rewriting a physical Stefan-Boltzmann law

• $Q=\sigma (T_1^4-T_2^4)$                         (1)

for one-way net energy transfer $Q$ from a body with temperature $T_1$ to a body of lower temperature $T_2$,  into a non-physical form:

• $Q =\sigma T_1^4 - \sigma T_2^4$,                      (2)

formally expressing the net $Q=Q_1-Q_2$ as the difference of two opposite transfers with $Q_1=\sigma T_1^4$ from warm-to-cold and $Q_2=\sigma T_2^4$ from cold-to-warm, as if emitted to a background at 0 K (which is not the case).

Compare with Fourier's law of heat conduction $Q=\sigma (T_1-T_2)$ (or in differential form $Q=\sigma dT/dx$), which nobody would even think of splitting into $Q=\sigma T_1- \sigma T_2$ (or in differential form $Q=\sigma T_1/dx-\sigma T_2/dx$), not even a first year student, since a system acting like that with "back conduction", would be unstable and thus could not exist. "Back radiation" is as un-physical and unstable as "back conduction". Fourier would turn in his grave at the mere thought of such a horrendous concept.

Confusion arises if the non-physical form (2) is interpreted as a describing actual physics including the transfer $Q_2$ from cold-to-warm in violation of the 2nd law of thermodynamics. The confusion can lead to serious errors (as any violation of the 2nd law), with errors resulting from working with differences of gross quantities subject to perturbations (like all physical quantities), which easily may be unstable, instead of working with net quantities as being more stable. 

In the setting of CO2 global warming alarm, gross two-way energy transfer between Earth surface and atmosphere is postulated to be about 350 W/m2, while net transfer is about 35 W/m2. As acknowledged by Henning Rodhe in the discussion we had, the difference is a factor 10 reduction of alarm from 3 C to 0.3 C = no alarm. 

The lesson is: Do not confuse manipulations of symbols on a piece of paper with actual physical processes. Do not confuse the correct physical form of Stefan-Boltzmann's law (1) with the incorrect non-physical form (2). If you follow this lesson, then CO2 global warming alarm collapses to zero.

For an illuminating comparison of one-way and two-way equations for radiative heat transfer, see this article by Joseph Reynen.

Notice that another way of formally rewriting (1) is:

• $Q =\sigma T_1^4+GHOST_1 - (\sigma T_2^4 +GHOST_2)$,         

where $GHOST_1=GHOST_2=GHOST$are equal (odorless, weightless) "ghost quantities" emitted by the bodies. By measuring $Q$ as the net energy absorbed by body 2 and postulating that body 2 is emitting $\sigma T_2^4 + GHOST$, one can argue that the measurement gives instrumental evidence of the existence of the quantity $\sigma T_1^4+GHOST$ and thus instrumental evidence of the existence of the quantity $GHOST$, which as you understand can be anything (odorless and weightless). The instrument is thus a perfect "ghost-detector" and as such potentially has a huge commercial market. Right? Anyway here is a handy instrument affordable to anyone in need of "ghost detection":

This is the way a pyrgeometer measuring DLR as atmospheric "back radiation" functions.

PS1 David Andrews states on p 84 of An Introduction to Atmospheric Physics, about two-way radaitive heat transfer:

• We find F-up and F-down by a sequence of tricks.

Yes, it is a "sequence of tricks" with no physical correspondence.

PS2 The fundamental error is clearly exposed in the above book:

• If the Earth is assumed to emit as a black body: $Q=\sigma T^4$...(page 5). 
• The ground is assumed to emit as a black body (page 6).

But the ground does not emit as a black body emitting into a backgound at 0 K. The Earth-atmosphere system emits into a background at 3 K, and the ground emits into an atmosphere of about 255 K, thus vastly different form 0 K. The fact that physicists of today do not react to this fundamental violation of basic physics, can only be understood as a degeneracy of modern physics away from scientific principles into black magic. The day of reckoning is approaching and the verdict in the history of science to be written will be harsh. Assuming the Earth is flat is a small error in comparison.