The concept of perfection is on occasion misinterpreted, not only by the lay public, but the scientists using the concept. The Carnot Engine, is a classic example of perfection as a tool. A Carnot engine perfectly converts energy into work. The efficiency of an engine is how close it performs with respect to the Carnot engine as a percentage. The amount of energy not converted to useful work, entropy, is the difference between perfection and reality. Studying the climate of Earth, perfection has to be considered but there is another puzzle piece that also needs to be considered, perfect imperfection, or maximum entropy.
Maximum entropy may be considered as the point of worst performance where the system does not destroy itself. For an object radiating electromagnetic energy to space, that may be the ideal gray body.
All objects in space will emit the same amount of energy they absorb over sufficient time. A perfect black body, described by the Stefan-Boltzmann equation E=e*sigma*(T)^4, where sigma is the Stefan-Boltzmann constant. E is energy in joules/sec, T is degrees Kelvin and e is the emissivity or the amount not absorbed or emitted by the less than ideal gray body. For the purposes of comparison, a gray body with an emissivity of 50% could be considered an ideal gray body.
Using the Stefan-Boltzmann equation, the chart about shows the relationship of emission and transmission. In this case the surface energy is represented by the E-value or the energy emitted at the surface and the T-value is the energy transmitted to space from some point above the surface. If there were no atmosphere, the E-value would be constant. With an atmosphere, the energy emitted from the surface interacts with the atmosphere transferring energy into heating the atmosphere. The point where the E-value and the T-value equal is what may be considered a perfect gray body.
Also on the chart is the R-value, or the energy one would expect to be transferred from a source at one temperature to a sink at another temperature. The R-valueis equal to the difference in temperature between the source of the energy flux and the sink of the energy flow divided by the energy flux through the barrier or layers of barriers.
Since the atmosphere of the Earth has a Tropopause, a layer in the lower atmosphere with a minimum temperature, the R-value is based on that approximate temperature, 182K at 62Wm-2 equal to approximately -91 degrees C. This value is slightly lower than the minimum temperature measured on the surface of the Earth and slightly higher than the minimum temperature measured in the Tropopause. If the R-value intersected the crossover of the E and T values, I would consider that to be a perfect gray body adjusted for thermal conductivity.
In this chart, the E and T values have been shift upwards so that all three curves cross at the same point, approximately 0.60, a value very close to the measure emissivity of the Earth viewed from space. Flowing the E-value to the right, at 288K degrees, the surface emissivity is approximately 0.9996 at 335Wm-2 at 277K degrees. So for this simple model, since the value of an ideal black body cannot exceed 1, the "effective" surface emission temperature is approximately 277K degrees. The estimated actual surface temperature of the Earth is 288K with emission of approximately 390Wm-2, the difference, 11K and 55Wm-2 could be considered energy flux from the surface that does not interact with the atmosphere.
This model provides some interesting potential. First, the cross over is at 0.6 instead of 0.609 the actual measured average emissivity of the Earth. The temperature of the cross over is 208K with a flux of 102Wm-2 approximately, indicating an imbalance of approximately 0.9 Wm-2, the modeled value of the energy imbalance use by Kiehl and Trenberth in their Earth energy budget.
By adjusting the the curves so that they intersect at 0.609, there is no energy imbalance. This required changing the tropopause temperature by 0.2 degrees K and changing the T and E offset value from 0.185 to 0.20, which is Planck parameter or Planck response made somewhat famous in the Monckton Lucia dust up.
To indicate what relatively small changes in the troposphere temperature can have on the Planck response and energy imbalance:
In the above chart the Tropopause temperature used is 190K
In this chart the tropopause temperature is shift to 173K or 100C.
In the last two charts a %I curve was added to illustrate the relative changes in thermal conductivity of the atmosphere with tropopause temperature.
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