More Fun with Three Disc Radiant Models
With the three disc models, I was attempting to simulate what that disc would sense. That disc is limited to energy bands or lines that can interact with the material of the disc. When I use a CO2 disc, I should use only the CO2 spectrum to determine what interactions that disc would have with the energy incident to its surface. I cheated and used a percentage of the spectrum based on the difference in temperature between the source of the energy and the sink, or disc receiving the energy. That is not a major cheat, it gives a ballpark which is all I was looking for at the time.
I can adjust my little model to get more information on just CO2, with just minor tweak, by making all the discs CO2 discs. That way a warmer disc would interact with all of the energy incident on its surface from a cooler disc and two equal temperature disc would not have any energy pass through. The two equal temperature discs with perfect absorption and return is fun to think about.
Disc 1 emits X towards disc 2 which returns x/2 to disc 1 which returns x/4 to disc 2 and so on. Since disc 1 started at X on the face toward disc 2, it effective emission due to interaction with disc 2 would approach 2X. Because of conservation of energy, the other side of disc 1 would approach 0 as the opposite face approached 2X. The same thing would happen at disc 2 since it started at the same temperature. We would end up with two discs emitting and receiving 2X on the common faces and emitting 0 on the opposite faces. There is no perfection, so this would be impossible, but where the discs would try to be headed.
Without some other source of CO2 friendly photons, nothing else would happen. If we inserted another CO2 disc between the two, all the faces would approach the same value of X. If the inserted disc had initial flux values of 2X, nothing would change with the other two discs which were in equilibrium at the perfect 2X flux prior to insertion. If we inserted a disc with X flux on each face, it would approach a higher value and the outer discs would have to find a new equilibrium. But what would happen on the other faces of the original discs?
Since the new disc had energy equal to half the total system energy, the outside faces of the original discs can emit a portion of that energy, so at perfection, the opposite faces could emit X each, or half of the addition 2X per face of the new inserted disc. That poses a small problem though. With three discs we have six faces, two at X and four at 2X. That would make a total of ten X when we started with eight X, two faces at 2X for the original disc and two faces at 2X for the inserted disc. So is this model wrong?
Not really, the Xs are energy flux or energy flow, which I am not allowing to flow. This model is similar to charging a capacitor which can only hold so much energy but can appear to have a greater potential energy. If I allow some of the energy in the model to flow for a short amount of time, the X value of all the faces would decrease proportionally. Since the disc in the middle has to interact emitting its energy with the outer discs, it would reach zero energy last by some fraction of a second.
If instead of letting energy flow, I inserted another disc at 2X per face, nothing would change, just the time required for the inside discs to lose their energy. If I inserted an infinite amount of discs at 2X per face, only the time to discharge would increase because more energy is stored.
This should be an example of saturation. The CO2 discs cannot manufacture energy, only delay its transfer. We have a stack of disc swapping photons with no net energy flow.
Now let’s imagine I take a stack of discs more massive than the stack above, capable of emitting X plus something and put it up to one of the faces. The more massive stack contains more energy, so there can be flow from it to the less massive stack of discs. The adjacent faces would approach 2X plus 2 something and the outside face of the less massive stack would approach X plus half of something. If the more massive stack could maintain its energy, the outer face of the less massive stack would emit X plus half of something as long as there was energy available.
This would be steady state energy transfer at saturation. As long as the more massive stack can maintain X plus something, the face between the more and less massive stacks can maintain the 2X plus twice something, while the opposite face of the less massive stack maintains emission of X plus half of something.
Adding more discs doesn’t change anything unless the space between the faces of the discs is not at 2X plus twice something. If the energy available at the face changes then the flux interaction at the faces change by twice that change, but that only applies to the CO2 portion of the change.
For example: 3.7Wm-2 of additional CO2 forcing would produce 7.4Wm-2 net effect. If the surface temperature were 288K it would increase to 289.3K or 1.3K increase in temperature. If the same forcing were at the Antarctic surface with -20C or 253K @ 232Wm-2, then the result would be 255K or 2 degrees of warming. Since the Antarctic is not warming significantly, odds are that any forcing due to CO2 change is not near the surface.
This estimate does not attempt to determine the impact of the change in temperature due to more CO2, but instead changes in forcing that are impacted by CO2 at saturation. So it may be useful for locating the effective radiant layer of CO2 to determine the feedback of water vapor on the altitude of the effect radiant layer. An atmospheric layer that shows nearly twice the flux change of another layer could hold clues to the magnitude of the CO2 impact.
I have no clue if this is actually measurable because it is not a true net flux, but it could be an indication of location of the radiant layer because local temperature should change even though there is no real energy transfer, just a backup or flux charging, if you will.
Note: This is just a musing post. The increase in Antarctic 600mb temperature by 0.7C per decade with no apparent impact on the surface got me wondering if that oddity may be of any use. I haven't checked it out and may never. Just in case anyone would like to, it may be worth a few minutes.
I know this is a simple model but that can be a good thing. Spectral broadening and a few other things change the basic relationship. An almost two times impact though is a lot easier to spot than a, “I have no clue”, impact. Since it really doesn’t matter where in the stack the hot disc is placed, the Antarctic troposphere might have an odd hot spot or two worth checking out.