Here is a neat way to sum up a range of models of greenhouse effect using the overall energy balance equation of Miskolczi (M7). The energy balance equation represents two flux terms of equal magnitude, propagating into opposite directions, while using the same solar energy F as an energy source. The first term (Su-F) heats the atmosphere and the second term (Ed-Eu) maintains the surface energy balance.

F — Solar flux in
Su — Surface flux up
Eu — Atmospheric flux up
Ed — Atmospheric flux down

They can be represented as equations of linear algebra:

1.1 F = Su – F + Ed – Eu — overall energy balance equation
1.2 0 = F – Eu — energy balance at top of atmosphere

The following are different three constraints:

2.1 0 = Ed – Eu — the steel greenhouse, top of atmosphere constraint.
2.2 0 = Su – Ed — the Kirchhoff’s law, IR radiative equilibrium between surface and atmosphere
2.3 0 = Su – F — the third option, for completeness.

By substituting each of 2.1, 2.2 and 2.3 into 1.1 and 1.2 we get three different solutions for surface temperature with three decreasing levels of greenhouse effects.

3.1 Su=2F
3.2 Su=3F/2
3.3 Su=F

The three models of greenhouse effect are shown in the figure below, ordered by increasing surface temperature. Below the diagrams are representation of the modeled and equilibrium lapse rates, the increase in air temperature with altitude for each of the models.


Here are a few points of interest that argue that the middle semi-transparent model is the correct model:

  1. In the left-hand model the model lapse rate increases faster than the equilibrium lapse rate — a quasi-stable atmospheric condition called an inversion. In the right-hand model the lapse rate increases more slowly than the equilibrium value — an atmospheric situation where large volumes of air rise through the profile. In each of these situations the equilibrium is eventually reestablished to the middle model, where the lapse rate is ‘just right’.
  2. Note that the models on the left and right side also have a discontinuity between the surface and the lower atmosphere. The center does not. Only the center model minimizes energy and maximizes entropy. Temperature discontinuities are not consistent with maximizing entropy.
  3. The three options could also represent zonal difference, from high to tropical latitudes.

In a previous post it was noted that the widely regarded semi-infinite model of greenhouse effect follows the ‘steel greenhouse’ solution. However, as noted above, this solution is one extreme that is unphysical due to the temperature discontinuity between the surface and the lower atmosphere.

Note that this simple model represents only the overall conservation of energy constraints on the system, and a number of other constraint and processes are in play (more discussed in the category Miskolczi above left). However, the central Kirchhoff law model is the only plausible solution with radiative balance throughout the whole atmosphere. However, this model suggests that all of the processes that contribute to the greenhouse effect are already contributing their maximum warming effect, as they cannot increase beyond the limits set by energy conservation. Miskolczi concludes that global warming must therefore be due to other mechanisms and not greenhouse gases.

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