Unphysical Schwarzschild vs Physical Model for Radiative Transfer

The basic model for radiative heat transfer in a horisontal slab atmosphere, with vertical coordinate $x$ with $x=0$ at the Earth surface and $x=X$ at the top of the atmosphere, was formulated by Schwarzschild in 1906 as a two-stream  gross-flow model in terms of a gross upward heat flux $F^+(x)$ and a gross downward heat flux  $F^-(x)$ satisfying the following advection-absorption equations for $0\lt x\lt X$:

• $\frac{dF^+}{dx} + F^+ = Q$               (1)
• $-\frac{dF^-}{dx} + F^- = Q$               (2) 

where $Q(x) =\sigma T(x)^4$ is supposed to express Stefan-Boltzmann's law with $T(x)$ the temperature at $x$ and $\sigma$ Stefan-Boltzmann's constant, and $x$ serves as an optical coordinate normalizing absorption. The atmosphere is supposed to be heated from below at $x=0$ by a heat source $H$, and the heat is radiatively transported to the top of the atmosphere from where it is radiated into outer space at 0 K. Conservation of heat energy gives the additional equation

• $F^+-F^- = H$,                                      (3)

from which follows by adding/subtracting (2) from (1) that $F^+ + F^-=2Q$ and $\frac{d(F^++F^-)}{dx}=-H$ and thus:

• $2Q(x) = H(X-x)+H$,                          (4)
• $F^+ =\frac{H}{2}(X-x)+H$
• $F^-=\frac{H}{2}(X-x)$                        

which determines the temperature profile $T(x)$. We note that Schwarzschild's model with linear $Q(x)$, is very simplistic. Only a model with $Q(x)$ constant could be more simplistic.

Schwarzschild's equations (1-2) are supposed to express conservation of upward and downward heat fluxes through a thin atmospheric layer radiating both upward and downward according to Stefan-Boltzmann in the form $Q(x) =\sigma T(x)^4$.

Schwarzschild's two-stream model is unphysical in the sense that the gross downward flux $F^-$ is directed against the temperature gradient and thus violates the 2nd law of thermodynamics. Further, the two sided radiation up/down according to $Q(x) =\sigma T(x)^4$ in (1-2) is also unphysical, because  $Q(x) =\sigma T(x)^4$ is the blackbody radiation into a background at 0 K, which is not the case. Scwarzschild's model is thus doubly unphysical and we shall now see that the unphysical aspects do not cancel to give a physical model.

Let us thus compare Schwarschild's unphysical two-stream gross-flow model with a one-stream net-flow model based on Stefan-Boltzmann's law in its correct physical form

• $Q=\sigma (T_2^4-T_1^4)$                     (5)

as the net radiative heat transfer radiative between two blackbodies of temperature $T_1$ and $T_2$ with $T_2>T_1$, in accordance with the 2nd law with heat transfer from warm to cold (but not the other way).

We then start from the following balance equation expressing that the total inflow of heat at a thin layer at level $0\lt x\lt X$ from layers at levels $y$ with $0\le y\lt x$, equals the total outflow into levels $x\lt y\le X$:

• $\int_0^x\sigma (T(y)^4-T(x)^4)\exp(-\alpha (x-y))dy = \int_x^X\sigma (T(x)^4-T(y)^4)\exp(-\alpha (y-x))dy$
• $+H\exp(-\alpha x)$                                                         (6)

with the exponential factor $\exp(-\alpha\vert x-y\vert )$ accounting for the absorption between levels $x$ and $y$ with a non-negative absorption coefficient $\alpha$, and the term $H\exp(-\alpha x)$ accounts for the effect of the heat source $H$ at $x=0$, and $\sigma T(X)^4 = H$.

Reshuffling terms in (6), we obtain the following integral equation in $Q(x)=\sigma T(x)^4$ for $0\le x\le X$

• $ \int_0^xQ(y)\exp(-\alpha (x-y))dy+ \int_x^XQ(y)\exp(-\alpha (y-x))dy$
• $= Q(x)(\int_0^x\exp(-\alpha (x-y))dy+ \int_x^X\exp(-\alpha (y-x))dy)+H\exp(-\alpha x)$,

which can be written

•  $Q(x)-c(x)\int_0^XQ(y)\exp(-\alpha\vert x-y\vert )dy =-c(x)H\exp(-\alpha x)$   (7)

for $0\le x\le X$ with $c(x) = 1/(2-\exp(-\alpha x)-\exp(-\alpha (X-x)),$ which together with the outflow condition $Q(1) = H$ determines $Q(x)$ uniquely as the solution of a Fredholm integral equation of the 2nd kind.

To compare with the two-stream model, let us formally split Stefan-Boltzmann's equation (5) algebraically into the (unphysical) form

• $Q=\sigma T_2^4-\sigma T_1^4$                   (8)  

and rewrite (6) accordingly collecting positive terms and writing $Q(x)=\sigma T(x)^4$ as above, into

• $A+B = C+D+E$                                                  (9)
• $A(x)= \int_0^xQ(y)\exp(-(x-y))dy$
• $B(x)= \int_x^XQ(y)\exp(-(y-x))dy$
• $C(x)=Q(x)(1-\exp(-x))$
• $D(x)=Q(x)(1-\exp(-(X-x))$
• $E(x) = H\exp(-\alpha x)$. 

To compare the models we observe that by the definitions of $A$ and $B$:

• $\frac{dA}{dx}+A = Q$
• $-\frac{dB}{dx}+B = Q$

which shows that $A$ corresponds to $F^+$ and $B$ to $F^-$.  We recall that

• $F^+ + F^-=2Q$ 

which we we compare with (9):

• $A+B=C+D+E=Q(2-\exp(-x)-\exp(X-x))+H\exp(-\alpha x)$,

to conclude that the two models are not identical, and so the models give different temperature distributions. 

The lesson is that radiative heat transfer should better be modeled using the physical one-stream net-flow correct form of Stefan-Boltzmann's law (5). Using the unphysical two-stream gross-flow form (1-2) can lead to unphysical results.

It is possible that an unphysical gross-flow model can give a physically meaningful result by happy cancellation of unphysical gross-flow aspects. But to rely on an unphysical model to derive conclusions about real physics is risky because the happy cancellation may not be granted, in particular not if the question concerns the effect of perturbations, as is the case in global warming modeling.

The unphysical aspects of Schwarzschild's model are:

1. The downward flux $F^-$ violates the 2nd law.
2. The downward flux $F^-$ generates unphysical absorption in (2).
3. The basic equations (1) and (2) do not represent correct physics. 

Despite these shortcomings, Schwarzschild's model has come to serve as the basic model of atmospheric radiation supporting CO2 alarmism.  The unphysical of nature of this basic model gives one reason to view also CO2 alarmism to be unphysical.

In a following post we will solve the integral equation (7) and compare with Schwarzschild's solution. To start with we note that for $\alpha$ large (7) approaches $\frac{d^2Q}{dx^2}=0$ resulting in a linear $Q(x)$ as in Schwarzschilds model. More generally, the physical model (7) is close to Schwrzschilds model in the trivial cases of a nearly opaque and transparent atmosphere, but not so in the more relevant case in between.

The integral equation (7) can by differentiation be turned into a (diffusion-advection-absorption) ordinary linear differential equation.