The Tropopause and the Second Law of Thermodynamics Paradox
Most of the relationships developed for radiant physics are based on disks. Flat surfaces with one side analyzed. This started with an original problem of an oven with an observation hole. The inside of the oven was at a constant temperature and the observer determined the characteristics of the radiation leaving the hole. This resulted in a vast improvement in our knowledge of electromagnetic radiation.
Today, we are dealing with radiant physics problem using that same hole in an oven. Not that that data is wrong in any way, we just may not have the same oven or the same hole. When two objects in ideal space are used for example, typically we use spheres. Then relate the hole or disk to a sphere, use the classic relationships and call it good. That may not provide all the accuracy needed.
Consider two disks in a vacuum:
Disk one is at some temperature and disk two is at another. The disks have a common face, or one is facing the other, and an opposite face, facing away from the other. The warmer disk is at a temperature 2T and the cooler at a temperature T as observed from between the two disks.
We insert a disk between these two. The new disk is at absolute zero with an area equal the two original disks and less mass than either of the original disks. Now we observe what happens.
Since the new disk is at absolute zero, initially it receives energy proportional to 2T on one side and T on the other. Since it initially has no energy, i.e. is at absolute zero, the new disk does not emit any radiation.
As the new disk slowly gains energy, it begins to return energy to the original disks in some proportion to the energy it acquires. Eventually, the new disk will reach equilibrium with the original disks.
Since the new disk does not have an energy source, it cannot add energy to the system, only impact the distribution of energy between the two original disks. Since the 2T disk has more energy, the impact on that disk will not be the same as the impact on the lower energy disk originally at temperature T.
The new disk will absorb at 2T and return half or T to that disk. It will absorb at T and return half or T/2 to that disk. The new disk will pass on the opposite face, T to the cooler disk and T/2 to the warmer disk. The energy passing through the new disk is now 3/2T, its apparent temperature.
The original warmer disk sees a warmer disk than before and the cooler disk sees a disk cooler than before. So the original disks apparent temperature will have to adjust to this new condition.
If the total energy of the system is conserved, then the cooler disk which is receiving 1/2T worth less energy , will reduce by 1/2 T. So it will emit 1/4 T less on both faces. The warmer disk which is being returned 1/2T more. would emit 1/4 T more on each face.
Now we have a paradox. While energy is conserved, the opposite face of the cooler disk may not be able to emit less energy or the opposite face of the warmer disk may not be able to emit more energy. There may exist constraints outside of the three disk system.
If the system of disks are in a vacuum at absolute zero, the cooler face of the cooler disk was at equilibrium with zero energy. It cannot absorb or gain energy from a source with no energy. That means than the warmer disk cannot release more energy, since if the cooler disk cannot emit less, energy is not conserved.
In order to conserve energy the opposite face of the warmer disk must emit less energy. That implies that the warmer disks opposite face can absorb more energy or the apparent temperature of all disk face would have to decrease proportionally to conserve the energy of the system.
An interesting situation, it is kinda like the convective rule of radiant energy.