Radiant Layer Meet Golden Efficiency

The wind is blowing today so thought I would play some more with the Golden Efficiency concept.  The issue is that an ideal grey body, if one could exist, would have an emissivity of 50 percent.  Reading back through some of my stuff and a slightly failed post on realclimate.org, isothermal radiant layers can only emit what energy that they receive.  Because of the concept, an isothermal radiant layer would emit the exact energy up as it does down and being isothermal, the advection of energy in the plane of the isothermal layer could be neglected.  This is roughly the assumption required for the simple up/down radiant model.  

Ideal is a concept not a reality.  50 percent is purely based on the point where transmission from the surface equal emission at the top of the atmosphere.  The surface would have twice the energy available that is applied.  If the surface could store more energy, the system would run away creating second law issues.

The slightly failed realclimate post had a technical note that was botched then corrected.

[Technical digression: Imagine an atmosphere with multiple isothermal layers that only interact radiatively. At equilibrium each layer can only emit what it absorbs. If the amount of greenhouse gas (GHG) is low enough, each layer will basically only see the emission from the ground and so by Stefan-Boltzmann you get for the air temperature (Ta) and the ground temperature (Tg) that 2 Ta⁴ = Tg⁴ , i.e. Ta=0.84 Tg for all layers (i.e. an isothermal atmosphere). On the other hand, if the amount of GHGs was very high then each layer would only see the adjacent layers and you can show that the temperature in the top layer would approximate 0.84ⁿ Tg (n+1)^-1/4 Tg, (see note) where n is the number of layers - much colder than the low GHG case. Hence the increased GHG steepens the surface-to-top temperature gradient.]

If you use the corrected form, ideal isothermal radiant layer would be:

Layer	Emissivity
1	0.84090
2	0.75984
3	0.70711
4	0.66874
5	0.63894
6	0.61479
7	0.59460
8	0.57735
9	0.56234
10	0.54910
11	0.53728
12	0.52664
13	0.51697
14	0.50813
15	0.50000

I have used the label "Emissivity" instead of temperature since they should be interchangeable at these specific layers.  Based on this idealize model, 15 layers would produce the ideal grey body.

As the density of the atmosphere increases, the ideal isothermal radiant layer concept falls apart.  There will be advection and energy store as potential.  The advection or difference in rates of advection between adjacent radiant layers more specifically, is analogous to radiant scattering that is not easy to keep track of.  In the denser parts of the body, the Golden Efficiency may apply.  I am still not convinced, but there are quite a few coincidences.

Stage	Efficiency	Waste
-3	0.008131	0.991869
-2	0.021286	0.978714
-1	0.055728	0.944272
0	0.145898	0.854102
1	0.381966	0.618034
2	0.618034	0.381966
3	0.763932	0.236068
4	0.854102	0.145898
5	0.909830	0.090170
6	0.944272	0.055728
7	0.965558	0.034442
8	0.978714	0.021286
9	0.986844	0.013156
10	0.991869	0.008131

In the above table used in the Golden Efficiency link, I used stages noting that stage 2 was remarkably close to the TOA emissivity.  In the isothermal layer table, layer 6 is also close to TOA emissivity.  In a totally unscientific manner, I decided that could be more than just coincidence.  That could be a reasonable transition from a reasonably accurate isothermal radiant layer model to a golden efficiency model.  The weakness of the radiant model, advection, could be related to the stages of the Golden Efficiency.

Stage	Efficiency	Emissivity	Advection
-3	0.008131	0.53728
-2	0.021286	0.54910
-1	0.055728	0.56234
0	0.145898	0.57735
1	0.381966	0.59460
2	0.618034	0.61479	0.00325
3	0.763932	0.63894	0.12499
4	0.854102	0.66874	0.18536
5	0.909830	0.70711	0.20272
6	0.944272	0.75984	0.18444
7	0.965558	0.84090	0.12466
8	0.978714	1.00000
9	0.986844	--
10	0.991869	--

By combining the two tables, the only positive differences for "advection" for the using the rough TOA emissivity are between the TOA and surface for this fit.

Stage	Efficiency	Emissivity	Advection
-3	0.008131	0.53728
-2	0.021286	0.54910
-1	0.055728	0.56234
0	0.145898	0.57735
1	0.381966	0.61479
2	0.618034	0.63894
3	0.763932	0.66874	0.09519
4	0.854102	0.70711	0.14700
5	0.909830	0.75984	0.14999
6	0.944272	0.84090	0.10338
7	0.965558	1.00000
8	0.978714	--
9	0.986844	--
10	0.991869	--

A likely better fit is where the radiant emissivity is closer to the Golden Efficiency waste energy.  In this case the ideal black body emissivity of 1 is compared to the more realistic ocean surface emissivity of 0.9655.  The total "advection" for this case is 49.6%, so for a TOA of 240Wm-2, advection would be 118.6Wm-2 resulting in an "effective surface" energy of 358.6Wm-2.  Since the "true" surface energy is only approximated at 398 +/- 17 roughly with sensible convection and "surface" atmospheric window radiant on the order of 40Wm-2 total, this fit is rather comfortable.  That of course means that the other is more appropriate or both will be needed since the "surface", as far as the radiant model is concerned, is not a very clearly defined layer.

This may be just numerology, but there are plenty of coincidences that make the less than perfect fit perfectly logical.