Back Radiation: Algebraic Fiction or Physical Reality? 2

This is a continuation of the previous post with focus on the unphysical nature of back radiation as expressed in Schwarzschild's two-stream model of radiative heat transfer taking the following form in a setting of a horisontal atmospheric layer connecting the Earth surface to outer space with $z$ a vertical coordinate representing optical distance:

• $\frac{dU}{dz}=-U+B,$
• $\frac{dD}{dz}= D-B,$

where $U$ is upwelling radiation and $D$ is downwelling radiation and $B(T)$ is upward/downward emission of radiation according Stefan-Boltzmann's law with $T$ temperature depending on $z$. Subtracting the equations, we have

• $\frac{d(D-U)}{dz}=D-2B+U.$  (0)

We see a balance at a certain level $z$ between U as incoming radiation from below, D as incoming radiation from above and -2B as emission upward and downward. Discretizing $z$ into discrete levels/nodes $z_i=i*h$ with $h$ a mesh size and $i=0,1,2...$, the balance of upward and downward radiation involves  

• $B_{i+1}-2B_i+B_{i-1}$    (1)

with $B_i=B(T(z_i))$, if we set $B_{i+1}=D(z_{i+1})$ and $B_{i-1}=U(z_{i-1})$. 

We will now compare (0)-(1) with the following discretised heat equation for a temperature $T(z,t)$ with $t$ a time coordinate:

• $dT_i/dt = (T_{i+1}-2T_i+T_{i-1})*\frac{1}{pow(h,2)}$   (2)

where $T_i=T(z_i,t)$, which is a discretisation of $\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial x^2}$. We know that the heat equation is a stable physical model subject to perturbations $p$ of (2) of the form

• $dT_i/dt = (T_{i+1}-2T_i+T_{i-1})*\frac{1}{pow(h,2)}+p$.    (3)

while perturbations $P$ of (1) would have the form

• $B_{i+1}-2B_i+B_{i-1}+P$.    (4)

We see that introducing a perturbation $P$ of the form (4) in (2) would correspond to a perturbation $p=\frac{P}{pow(h,2)}$ which would be big since $h$ is small. We thus see a fundamental difference as concerns stability between a heat equation of the form (2), which is stable, and a two-stream model of radiative transfer of the form (0). 

Note that the idea of two-stream radiative heat transfer with upwelling/outgoing (longwave) radiation OLR and downwelling (long wave) DLR is firmly implanted in the discussion of the GHE through the two-stream Schwarzschild model. In particular, the existence of DLR is supposedly being demonstrated experimentally with the help of a pyrgeometer, which however is a ghost detector.