Reality and Fiction of Stefan-Boltzmann's Radiation Law

Human accountant in charge of a non-physical fictional Stefan-Boltzmann Law.

In previous posts on radiative heat transfer I have compared two different formulations of Stefan-Boltzmann's Radiation Law (SB) for the radiative exchange of heat energy between two blackbodies of different temperatures:

1. one-way transfer from hot to cold,
2. two-way transfer with net transfer from hot to cold.

To see which formulation best represents physics, recall the wave equation model of a blackbody as a vibrating string with displacement $U$ subject to radiative damping:

• $U_{tt} - U_{xx} = f - (-\gamma U_{ttt})= f + \gamma U_{ttt}$,

which expresses a balance between the string force $U_{tt} - U_{xx}$ and the net force $f+\gamma U_{ttt}$ from the radiation pressure $-\gamma U_{ttt}$ and the exterior forcing $f$. For details see Mathematical Physics of Blackbody Radiation.

The essential aspect is now the interplay between the internal energy (density) $IE$ of the vibrating string

• $IE=\frac{1}{2}(U_t^2 + U_x^2)$

and the net forcing $f +\gamma U_{ttt}$, which is expressed in the following energy balance obtained by multiplying the force balance by $U_t$ and integrating in space and time to get

• $\int \frac{dIE}{dt}dxdt = \int (f +\gamma U_{ttt})U_tdxdt$.

We se that the rate of change $\frac{dIE}{dt}$ of internal energy $IE$ is balanced by a net force $f + \gamma U_{ttt}$ scaled with $U_t$. We can interpret $E=\int IE\, dx$ as an accumulator recording the net effect of the forcing and radiation, with $E$ proportional to $T_U^2$ with $T_U$ the temperature of the blackbody (with displacement) $U$.

In the case the forcing $f$ is delivered by another blackbody with displacement $V$ and temperature $T_V> T_U$, the energy balance takes the form

(1) $\int\frac{dE}{dt}dt= \int (\gamma V_{tt}^2 -\gamma U_{tt}^2)dxdt$,

where $\frac{dE}{dt}$ thus is the rate of heating of blackbody $U$ by the radiation from the hooter blackbody $V$ expressed as an integral of net forcing.

The right hand side of (1) can formally be rewritten as

$\int\frac{dE}{dt} dt= \int \gamma V_{tt}^2dx dt - \int\gamma U_{tt}^2dxdt$

from which follows by performing the integration with respect to $x$, and cancelling the integration with respect to $t$ (see Mathematical Physics of Blackbody Radiation for details):

(2) $\frac{dE}{dt} = \sigma T_V^4 - \sigma T_U^4$.

This is the version of Stefan-Boltzmann's Radiation Law (SB) cherished in climate science describing the heating $\frac{dE}{dt}$ as the difference of two gross flows of incoming radiation $\sigma T_V^2$ and outgoing radiation $\sigma T_U^4$.

We thus have two forms of SB:

1. (1) with one-way heat transfer from integration of net forcing,
2. (2) with two-way heat transfer as difference of integrated gross forcings.

I argue that (1) is physical since the internal energy $E$ acts as an accumulator of net forcing,

while (2) is unphysical because the accumulated quantities $\sigma T_V^4$ and $\sigma T_U^4$ lack physical realization.

The account of heat transfer expressed in (2) can formally be made by a human accountant on a piece of paper, but not by the blackbodies themselves and thus (2) lacks physical correspondence.

The conclusion is that one-way transfer of net flow is physics while two-way transfer of gross flows is fiction. In other words, DLR/backradiation is fiction.