The Atmospheric R Value

The surface of the Earth has an average temperature of 288K which would have a corresponding thermal energy flux of 390Wm-2, via Stefan-Boltzmann using perfect black body characteristics.

The Tropopause has a temperature on average of -55C or 218.15K which by the same S-B relationship would have a perfect black body equivalent thermal flux of 131.3Wm-2. The ratio of the change in temperature to the change in flux would be the R value of the atmosphere from the surface to the tropopause or (288-218.15)/(390-131.3)= 69.85/258.7=0.27K/Wm-2. The inverse of that value would be the thermal transmittance (U-value), 3.7Wm-2.k-1. Of the atmosphere from the surface to the tropopause, assuming the temperature of the tropopause, were somewhat stable.

The ratio of the effective thermal flux at the tropopause and at the surface, 131.3/390=0.337, a unitless value, is interesting. Why? It is the approximate conductivity or the value that determines the "Thermals" portion of the surface flux. The difference in effective flux ratio or an object emitting 390Wm-2 but receiving 131.3 Wm-2 from another object would have a net flux of 390-131.3 or 258.7 Wm-2. What is more interesting is that half of that 258.7Wm-2 or 129.3Wm-2 would be half of the change in flux. When combined with the 131.3 or minimum flux at the tropopause, we get a median flux of 260.65Wm-2 which would correspond to a temperature equivalent of 260.38K degrees. That is a rather convenient value for calculating change temperatures in the lower atmosphere. It is almost like there is a balance of forces.

If we were to consider the entire atmosphere, the R value would be (288-254.5)/(390-238)=0.22 which would equivalent to a transmittance of 4.53Wm-2.K-1.

So if our world only had radiant and conductive heat transfer, it would be very simple to use R-values and transmittance to determine changes in surface temperature and/or changes at the tropopause or TOA. We have water vapor and the latent heats of vaporization and fusion to contend with though.

If our surface is cooled by say, 79 Wm-2 of latent heat transfer, just to pull a number out of my hat, the effective flux from the surface would be reduced from 390Wm-2 to 311Wm-2, with a corresponding temperature of 272K degrees. Darn the bad luck! That would change our simple R values, now wouldn’t it?

Now the total energy transferred to the tropopause is from an effective temperature of 272K not 288K. (272-218)/(390-131.3)=0.209K/Wm-2 or in terms of transmittance, 4.77Wm-2.K-1. Because of that darn shift, the apparent net flux changed to 311Wm-2 – 131.3 = 179.7Wm-2 to the tropopause and 311-238=73Wm-2 to the TOA.

But what about the flux from the surface other than the latent? Good question! The surface flux minus the latent flux, 390-79=311 experiences a different R value, 69.85/(311-131.3)=0.39 with an equivalent transmittance of 2.57Wm-2.K-1.

So what does this all mean? Well, if we neglect latent heat, a 3.7Wm-2 increase at the surface or and improvement if atmospheric insulation that reduced flux by 3.7 Wm-2 at the tropopause would produce a 1K degree increase at the surface.

However, with latent removed from the surface considered, a 2.57Wm-2 increase at the surface or an improvement in atmospheric insulation that reduced flux at the tropopause by 2.57Wm-2 would produce a change of 1K degrees at the surface.

If we want to figure this out correctly, we consider the latent shift, then a 4.77Wm-2 increase in flux at the latent shift boundary or an improvement in the insulation of the atmosphere between this boundary and the tropopause would produce a 1K degree increase in temperature at the latent shift boundary.

From the surface to the latent shift boundary, 288K to 272K, there is a 390 to 311 change in flux or a 0.205 R value with equivalent transmittance or U value of 4.94 Wm-2.K-1.

So just for grins, if we improved the insulation of the atmosphere to retain 3.7Wm-2 at the tropopause, then 3.7/4.77=0.77K degrees increase would be felt at the latent boundary which would produce 4.77/4.94=0.743 degrees at the surface.

If you want to do it easy, 3.7Wm-2 at the tropopause using the surface R value corrected for latent shift, 2.57, 2.57/3.7=0.69 at the surface or 3.7/2.57=1.43 at the tropopause if the 1 K degree increase at the surface was caused due to other impacts.

Actually, the best way would be to consider the layers. From the surface to the latent shift, 311Wm-2 pass through 288K minus 272K differential temperature which would be an R value of 0.051 (U=19.4) seen by the combination of conductive/convective and radiant flux, before the shift latent heat is again added to the flow of energy.

So we have a surface layer, 288@390 to 272@311 R=0.054, U=19.4, Latent to Tropopause layer, 272K @311Wm-2 to 218.15K @ 131.3 R=0.209 U=4.77. R values are additive, add more insulation you improve the R value, so from the surface to the Tropopause the R value would be 0.054 + 0.209 = 0.263K/Wm-2 or an equivalent U of 3.80Wm-2.K-1.

So what happened to the 79Wm-2 of latent? Nothing. Its impact is spread over both the surface latent layer and the latent tropopause layer. The sensible portion of the latent, in this example ~24-21 or 3Wm-2 between the surface and latent boundary layers, has to be back calculated. Above the latent boundary, the sensible portion of latent would also have to be back calculated as varying air turbulence would change the sensible values with changing upper layer convection conditions.

Some may have noted that that neat 258 to 260 Wm-2 range calculated above is very close to what some consider to be the value of the Down Welling Longwave Radiation DWLR. That value is above the latent boundary layer. What would that value be at the surface? The apparent net flux above the latent boundary is 179Wm-2. The average apparent net flux would then be 218Wm-2, the effective DWLR, imagine that.

Those of you that have attempted to follow my calculations may have noticed that I am using the Tropopause here instead of the 249K 600mb reference level. Why? Because the tropopause changes, -55 C is an average, but it can drop to -90C like a rock! A better reference would be something a little more stable and more easily measurable.

I will clean this up and try to make a drawing to simplify communication, but the sensible portion of latent heat does not appear to be adequately considered in climate modeling. It is one of those confusing cloud feed back issues. Not very easy to directly measure it seems. They didn’t call it latent or hidden heat for nothing.

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