The last installment of my review of Miskolczi’s theory of (almost) constant greenhouse effect examines his claim that attribution of global warming to greenhouse gases is due to an error in the equations. This part deals exclusively with equations of radiative equilibrium in the atmosphere. The other three parts dealt with various aspects of the overall energetic constraints on the atmospheric system: energy conservation (part one), the virial theorem (part two), and Kirchhoff’s law (part three).

I have tried to simplify the equations down to their essentials, to highlight the assumptions behind the different approaches using M’s overall balance balance equation (7). I look at three solutions to finding greenhouse effect on surface temperatures based on radiative equilibrium in the atmosphere: Willis Essenbach’s ‘steel greenhouse‘, Miskolczi’s semi-transparent atmosphere model, and finally the ‘received’ solution the semi-infinite atmosphere, radiative-convective solution.

The overall energy balance equation is from Miskolczi’s equation (7) where F is the solar insolation, Su is radiant heat from the Earth’s surface, Ed and Eu are radiant heats from the atmosphere, down from the lower atmosphere, and up from the top of atmosphere respectively:

F= Su-F + Ed-Eu (M7)

M describes it so:

In Eq. (6) represents two flux terms of equal
magnitude, propagating into opposite directions, while using the same F as energy sources. The first term (Su-F) heats the atmosphere and the second term (Ed-Eu) maintains the surface energy balance.

For conservation of energy, total energy input F, the total energy of the surface/atmosphere system, and also the total energy output OLR must all be balanced (illustrated above).

Steel Greenhouse

The ’steel greenhouse’ model is of the Earth surrounded by a steel (or some other material) shell, transparent to short-wave radiation from the sun but opaque to long-wave radiation up from the surface. In some respects this is a model of an ideal greenhouse.

There is a constraint on fluxes at the shell due to radiative equilibrium of Eu=Ed. Substitution this constraint into the overall energy balance equation gives:

F=Su-F
Su=2F

In the ideal greenhouse case, the surface flux is twice the shell flux. Translating fluxes into temperature using Stefan-Boltzmanns relationship, gives a greenhouse effect of 2^0.25 or 119% of temperature without the shell. If this model were correct the Earth average temperature would be 29C instead of 16C. A greenhouse effect of +48C is much higher than the accepted +33C figure.

Semi-transparent

In M’s theory of semi-transparent atmosphere, it is assumed that there is a radiative equilibrium at the earths surface (due to Kirchhoff’s law). Due to this assumption, Su=Ed. Substitution of this constraint into the overall balance equation gives:

F=Su-F+Su-F
3F=2Su

which is M’s overall energy balance equation. This gives a greenhouse effect of 1.5^0.25 or 110.6% of black body temperature, or a global average of 9C, much closer to the accepted value of 15C. (M adjusts this further with other aspects of the theory).

Semi-infinite theory

This is a simplified version of the received theory, called semi-infinite, based on a continuous atmosphere of optical depth τ. I found a detailed exposition here in some very nice lecture notes from Irina Sokolik at Georgia Institute of Technology. The semi-infinite theory is characterized by the assumption of radiative equilibrium at every level of the atmosphere itself. Once the parameters are worked out in the atmosphere, surface temperature is determined by substitution. This results in a discontinuity between surface and bottom of atmosphere temperatures (and fluxes).

Due to the radiative equilibrium assumption, the net flux is constant over the depth of the atmosphere or dF/dz=0. The rate of change of temperature B(τ) due to τ is constant dB/dτ=c indicating constant net radiative flux through the profile. Integration of this derivative gives the temperature increasing linearly with the gray-body optical depth.

B(τ)=πFτ+Bo

At B(0) or top of atmosphere Bo=πF, so that the whole equation is a line as shown by the triangle on the figure.

B(τ)=πF+πF τ

This would seem to show temperatures in the atmosphere increasing with increasing optical depth τ, caused say, by increases in greenhouse gases such as CO2.

To get surface flux Su by substitution it is then assumed that the
Su equals the sum of the downward fluxes: the shortwave flux from the Sun F and the longwave flux from the atmosphere at the surface B(τ)/π. Therefore (ignoring π for clarity):

Su=F+F+τF
Su=2F+Fτ

This solution suggests:

1. The temperature is dependent on the
   path length τ of absorbent gases in the atmosphere.
   Hence the greenhouse effect, that increase in IR absorbent gases will increase surface temperatures.
2. The temperature at bottom of atmosphere is B(τ)=F+τF but the temperature of the surface is Su=2F+Fτ. The temperature of the surface is discontinuous with the atmosphere.

This produces a few problems.

1. It gives a very high estimate for surface temperature of 36C
   some 20C higher than the actual global average temperature (see end of lecture notes).
2. The temperature must be adjusted with a
   convection model to bring it to more realistic level. The adjustments that
   together make a radiative-convective model match the observed lapsed rate are
   shown in the figure from the lecture notes below.

Consistency of the semi-infinite model

I noticed another issue with the semi-infinite model; that the equations have become largely energy inconsistent with a non-zero optical depth τ.
On substitution of the Su semi-infinite solutions into M’s energy balance equation:

F=2F+Fτ – F + Ed -F
Ed=F-Fτ

However, Ed=F+Fτ
so Ed=F-Fτ = F+Fτ and &tau=0

The semi-infinite model is only energy conservation consistent when τ=0. Ie. there is no optical thickness, the shell is isolated from the surface, and Su=2F. Thus the classic solution is really the (unrealistic) steel shell model in disguise. The lecture notes describe using the convective adjustment also to achieve heat balance (aka consistency).

If we make the surface temperature equal to the lower atmosphere temperature (Su=Ed) then
τ=0.5 and then Ed=Su=F+0.5F=3F/2, the same as the semi-infinite model. So the semi-infinite model reduces to the semi-transparent model when surface and air temperatures are assumed equal.
But then, surface temperatures should be bound to a constant relationship with incoming radiation
as the semi-transparent model implies.

Summary

Another way to look at it this: the semi-infinite models a maximum greenhouse effect of Su=2F (considerably above existing temperature) while the semi-transparent model gives a maximum greenhouse effect of Su=3F/2 (about current temperatures). Models of runaway greenhouse warming require the ‘headroom’ provided by the semi-infinite model. The semi-transparent models a system with no such headroom for temperature increases. This result is almost entirely due to the constraint of radiative equilibrium between the surface and the lower atmosphere.

Further comments

The central assumption, and distinguishing feature of the semi-transparent and semi-infinite models is the equilibrium state of surface and lower atmosphere temperatures Ed=Su.
The semi-transparent theory assumes radiative equilibrium, and the semi-infinite theory
does not. That the surface is in fact in radiant equilibrium with the atmosphere above is shown
by radiosonde observations and simulation in the
graphs supplied by Miskolczi below.

The semi-infinite case with the discontinuous surface atmosphere is
only consistent with the overall energy balance equation when τ=0,
or when surface temperature is adjusted to achieve thermal equilibrium
in a combined radiative-convective model.

But this equivalent to the semi-transparent case of constant greenhouse.

Unlike the semi-infinite case, the semi-transparent model does not require further adjustments to incorporate convection.
Convection is included in the Ed=Su constraint. Overall it is a much more parsimonious model
of the atmospheric greenhouse effect than the semi-infinite model. It produces a very
accurate atmospheric profile without adjustments, as shown in the figure 5 from M’s paper below.

The idea of a discontinuity
at the surface is obviously clumsy, and comes from trying to truncate a continuous model of atmospheres (the semi-infinite model), with a flawed boundary condition. It then becomes internally inconsistent, and inconsistent with conservation of energy, and requires adjustment with a convective model to make it ‘work’.

M’s assessment of the situation would seem to be correct:

As a consequence, Eq. (16) will underestimate A t ,
and Eq. (17) will largely overestimate G t (Miskolczi and Mlynczak, 2004).
There were several attempts to resolve the above deficiencies by
developing simple semi-empirical spectral models, see for example Weaver
and Ramanathan (1995), but the fundamental theoretical problem was never
resolved. The source of this inconsistency can be traced back to several
decades ago, when the semi-infinite solution was first used to solve bounded
atmosphere problems. About 80 years ago Milne stated: “Assumption of
infinite thickness involves little or no loss of generality”, and later, in the same
paper, he created the concept of a secondary (internal) boundary (Milne,
1922). He did not realize that the classic Eddington solution is not the general
solution of the bounded atmosphere problem and he did not re-compute the
appropriate integration constant. This is the reason why scientists have
problems with a mysterious surface temperature discontinuity and unphysical
solutions, as in Lorenz and McKay (2003). To accommodate the finite flux
optical depth of the atmosphere and the existence of the transmitted radiative
flux from the surface, the proper equations must be derived.

Weaver and Ramanathan (1995) concur:

Radiative equilibrium solutions are the starting point in our attempt to understand how the atmospheric composition governs the surface and atmospheric temperatures, and the greenhouse effect. The Schwarzschild analytical grey gas model (SGM) was the workhorse of such attempts. However, the solutions suffered from serious deficiencies when applied to Earth’s atmosphere and were abandoned about 3 decades ago in favor of more sophisticated computer models.

However it remains to be shown that the grand plans of ’sophisticated computer models’ (GCMs) have truly thrown off these conceptual deficiencies, or whether the poor, (or should I say abysmal) reproduction of atmospheric
temperature profiles is due to the persistence of the semi-infinite model assumptions in the structure of the code.

In his talk at Ohio State University, Steve McIntyre concludes:

Viewed from this perspective, a remarkable aspect of the climate debate has been the
seeming inability of the climate science community to narrow confidence intervals on
this estimate. In 1979, the Charney Report (National Research Council 1979) estimated
the impact at 3 deg C with a 1.5 degree range either way. In 2007, IPCC AR4 estimates
are virtually unchanged. With all the improvements in scientific knowledge and all the
efforts of climate scientists over the years, why has the improvement of these confidence
intervals proved so resistant? I don’t know, but it’s worth thinking about.

This analysis provides a possible answer. Because the semi-infinite structural equations rely on optical path length τ to constraint surface temperature, and τ is subject to many possible poorly understood influences (e.g. water vapor, sulphates, GHGs etc), this uncertainty is propagated through into the final results. However, when the semi-transparent conservation equations constrain (radiative equilibrium, virial theorem, Kirchhoff’s law which are known exactly), the CO2 sensitivity can be determined much more precisely (sensitivity of temperature to CO2 doubling of 0.24K). Thus the lack of progress could plausibly be due to a clumsy characterization of the problem.

A Miskolczi put it:

They (NASA scientists) did know very well, that – according to Von Neumann -
that climate prediction is a boundary condition problem and
unfortunately the mistake was there…

No matter the spatial and vertical resolution of a GCM,
if they do not put proper (greenhouse) physics into it
they will create useless climate predictions with higher
resolution.