Slowing Down Part IV

Warning! This pushes the limits of every thermodynamic term or principle ever published.
The simple model of a steady state system in conditional equilibrium is pretty useful but a little bit limited because of the concerns of radiant versus other thermodynamic heat flows.  Since radiation has different spectra for different molecules and temperatures, it is somewhat more complicated.  Heat flow though is generally well understood since it is a vital part of so many aspects of life.  The resistance to heat flow or insulation value of a barrier can be describe in terms of R-value.  In SI units, R-value is in units meters square-K/Watts or simply degrees K/Wm-2. It is more often listed as degrees C, but degrees K are inter changeable.

With the heat of fusion of salt water as a lower limit and the 83.3Wm-2 established as the latent component of the system, we have part of the solution.  Using the increase in temperature at that temperature, 0.81C for 3.7Wm-2 we have a clue for the next step.  0.81/3.7=0.219 or the R-value associated with the internal system changes for the problem.  Since the resistance to energy flow from the system to ambient is the same, the R-value would be the same.  R-value is a general measure of heat flow and not specific to conductive, convective or radiant types of heat flow.  If we use the approximate initial conditions for the Earth, 288K and 390Wm-2 with the 271.5K 306.9Wm-2 determined with the simple model we can estimate an initial R-value.  (288-271.5)/(390-306.9)=. 16.5/83.1=0.198K/Wm-2.  This is slightly lower than the 0.219 estimate as it should be, but there is some uncertainty in how accurate both values are.  With Earth there is an external heat source, the sun which transfers some heat to the surface and some to the atmosphere with some reflected by both.  With the simple model we are only considering the internal balance to the external layer of the same temperature.  Since the internal energy calculation is based on the latent heat of fusion and evaporation, the R-value calculation would consider everything other than latent.  Convection, conduction and radiant combined would be equal to the latent heat transfer using this estimate.  That may be correct, perhaps not, but a reasonable initial estimate.  We need to start a new layer to find out.

306.9 is the new layer radiant energy.  The energy lost at the top of the atmosphere is estimated at 240Wm-2, the difference would be 66.9Wm-2.  This would be the total radiant heat flux from the 271.5K layer to space.  Using the initial R-value estimate 0.198*66.9=13.24K temperature drop from this layer to the next.  Using 0.219*66.9=14.65K temperature drop.  The difference 1.41K degrees would be the increase in temperature due to the 3.7Wm-2 of additional insulation.

This layer though is a transitional layer.  The 83.3Wm-2 of latent heat would transition to sensible heat which with the 66.9Wm-2 would total 150.2Wm-2.  0.198*150=29.7K temperature drop from the 271.5K layer or 241.8K degrees.  That would indicate a transitional layer from 241.8K to 258.3K for the initial conditions.  The final conditions would vary with the latent response and of course there is still uncertainty in the initial conditions.