Back Radiation: Algebraic Fiction or Physical Reality?

The physics of a "greenhouse effect" (GHE) resulting from "back radiation" from a colder atmosphere with "greenhouse gases" (GH-gases) to a warmer Earth surface, is still a subject of active discussion. A recent contribution is Verification of the Greenhouse Effect in the Laboratory by Hermann Harde and Michael Schnell published in Science of Climate Change, presenting the following main points:

• The impact of the different greenhouse gases (GH-gases) on our climate is not really well understood and even under experts this repeatedly leads to doubts about its existence and its influence on our climate.
• This article summarizes the theoretical background of the GHE and presents first quantitative measurements of this effect with an advanced experimental set-up. 
• For our studies we use an experimental set-up, which consists of two plates in a closed housing, one plate heated to 30 C, the other cooled to -11.4 C. 
• ...the set-up...uses a heated plate as radiation source and simultaneously as sensitive detector for the back-radiation from GH-gases. We measure the increasing temperature of this plate or, alternatively at stabilized temperature, the energy saving due to the back-radiation.
• We measure the additional warming of a pre-heated plate due to back-radiation of the greenhouse gases carbon dioxide, methane and nitrous oxide as a function of the gas concentration, and we derive from the observed warming the radiative forcing of these gases.
• Our studies also demonstrate that contrary to the often-misinterpreted 2nd law of thermodynamics a warmer body can further be heated by absorbing the radiation from a colder body, here the radiation from the cooled plate and a GH-gas.
• In addition and independent of the temperature measurements is the back radiations of the GH gases directly recorded as reduced electrical heating of the upper plate.
• These measurements clearly demonstrate that contrary to the often misinterpreted 2nd law of thermodynamics a warmer body can further be heated by absorbing the radiation from a colder body, here the radiation from the cooled plate and a GH-gas.
• The presented measurements and calculations clearly confirm the existence of an atmospheric GHE, but they also demonstrate the only small impact on global warming with increasing GH-gas concentrations, which in any way are apperently dominated by natural emissions. So, there is no reason for panic and climate emergency.

The article claims to give the first verification of GHE in a laboratory setting with measurements supported by theory in the form of Schwarzschild's two-stream equation for radiative heat transfer. GHE is viewed to be an effect of back radiation with a warmer plate absorbing radiation from a colder plate, yet without violation of the 2nd law of thermodynamics. 

I have agued (see also video) that back radiation with two-way radiative heat transfer is unphysical and should be replaced by one-way transfer from warm to cold, with recent detailed follow up by Joseph Reynen. The presence of back radiation is directly connected to Schwarzschild two-stream model, which is described as follows in standard treatise Atmospheric Radiation: Theoretical Basis by Goody and Yung: 

• While it sets the pattern of the formalism used in radiative transfer problems, its physical content is very slight.
• The Schwarzschild-Schuster approximation is now of historical interest only.

The key issue concerning back radiation and Schwarzschild's two-stream equation is as follows: Stefan-Boltzmann's equation (SB) for one-way radiative heat transfer $Q$ between two bodies $B_1$ and $B_2$ of temperatures $T_1$ and $T_2$ with $T_2>T_1$ takes the form

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

with $\sigma$ Stefan-Boltzmann's constant and $0<\epsilon \le1$ a coefficient depending on the nature of the bodies with $\epsilon =1$ for black bodies. Algebraically the SB equation can formally be written 

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

expressing two-way transfer with $\epsilon\sigma T_2^4$ heat transfer from $B_2$ to $B_1$ and $\epsilon\sigma T_1^4$ heat transfer from $B_1$ to $B_2$.  We see that (1) expresses $Q$ as net transfer from warm to cold, which in (2) is formally written as the difference between two gross transfers back and forth between warm and cold. 

Implicit in (2) is Kirchhoff's Law expressing that absorptivity is equal to emissivity since in (2) the coefficient $\epsilon$ has both the role of emissivity and absorptivity depending on the interpretation of (2) as both heat balance for $B_1$ absorbing heat from $B_2$ while emitting heat to $B_2$, and heat balance for $B_2$ absorbing heat from $B_1$ while emitting heat to $B_1$. 

Algebraically (1) and (2) are formally the same but the physics is different with one-way net heat transfer in (1) and two-way difference of gross heat transfer in (2). Now comes the key observation: From stability point of view (1) and (2) are not the same: A percentagewise small perturbation of net transfer will result in a small change of net transfer, while a small percentagewise perturbation of gross flow may result in a big perturbation of net transfer. 

One can now argue that physics must be stable to persist over time and so two-way radiative heat transfer as expressed by Schwarzschild's two-stream equations is unphysical and that is also what Goody and Yung says. 

It thus remains to find a mathematical model for one-way radiative heat transfer and here the stack model studied by Reynen is a first step. In a next post I will exhibit the crucial aspect of stability in a simple example comparing from stability point of view Schwarzschild's unphysical two-stream equations with a physical heat equation.