We must now look at Roy Spencer’s simple model of the greenhouse effect to see what he’s actually doing with it and what he actually thinks about it.
I have updated his current model but also have a copy from some time he did this before using a sphere with concentric shell as an example – they’re both updated with “reality” sheets and “Spencer” sheets.
We now get to see exactly what Roy thinks about backradiation, independent of all of the usual obfuscatory language, because we can refer directly to his mathematics. Here is the relevant Excel equation from his most recent model:
B12 = B11 + ($C$2-C11+$C$5*0.0000000567*E11*E11*E11*E11)*$C$3/$C$4
Now this is an iterative equation and has Excel formatting, so I will rewrite it to look more like real math:
TSurf(i+1) = TSurf(i) + (FSun – FSurf(i) + ασTAtmo(i)4)*dt/Cp
We can simplify the appearance of this a bit with the atmospheric flux being FAtmo = ασTAtmo(i)4, so that
TSurf(i+1) = TSurf(i) + (FSun – FSurf(i) + FAtmo(i))*dt/Cp
And therefore, we now clearly see, that the atmospheric flux is being directly added in as heat. It is precisely the same as the consensus radiative greenhouse effect diagrams such as this one.
Note what happens inside the brackets. He takes the difference between the solar input and the surface output because, as is correct, the heat input from the Sun on the surface is the difference between the temperature of the solar radiation and the temperature of the surface, i.e., the difference between the solar input flux and the surface output flux.
But then, he goes on to add in directly the entire flux from the atmosphere, irrespective of the surface’s temperature. In other words, the input flux is now the flux from the Sun and the atmosphere combined. A slight rearrangement of terms shows this more clearly:
TSurf(i+1) = TSurf(i) + (FSun + FAtmo(i) – FSurf(i) )*dt/Cp
And so there you have it: the atmospheric flux is being added back in as input energy, as heat (or as whatever else you want to call it).
And that most certainly is in violation of the Laws of Thermodynamics and hence all of physics.
The approach isn’t even self-consistent. If the heat input from the sun is given by the difference in flux between the solar input and surface output, FSun – FSurf(i), then why wouldn’t the heat input from the atmosphere be given by the difference between the atmospheric flux and surface flux, as in FAtmo(i) – FSurf(i)? Why is the atmospheric flux being added back in directly irrespective of the temperature of the surface? You see how the logic automatically begins to break down when you start violating the laws of existence.
So, let’s say we add in heat from the atmosphere in this “more appropriate way”. We then have:
TSurf(i+1) = TSurf(i) + [(FSun – FSurf(i)) + (FAtmo(i) – FSurf(i))]*dt/Cp
= TSurf(i) + [FSun + FAtmo(i) – 2*FSurf(i)]*dt/Cp
If you modify the Excel sheet to match that, then the final surface temperature is -47°C! Why so cold? Because none of it makes sense. Because the entire concept is not based in reality.
It’s not logically self-consistent, and it can’t be self consistent because it isn’t logically consistent with existence in the first place with the entry postulate that backradiation can cause further temperature increase than what the source initially provided.
That’s Ontological Mathematics philosophy right there for you.