A very interesting theory of global warming proposed by the Hungarian mathematician
Ferenc Miskolczi contains a simple proof that the greenhouse effect
is bound to a fixed value and cannot ‘runaway’, or even increase. In order to understand, or audit, parts of the theory I step through a simplified version of the derivation of his result below.
The first step in modeling a system’s dynamics is representing the main
constant relationships, usually based on conservation of energy.
The ‘big picture’ view of the flow of atmospheric energy consists of three
linked components: the Sun’s energy in, flux circulation within the atmosphere/surface system, and radiant heat out.
This view of the system as a linear converter of shortwave (SW) into longwave (LW) radiation is shown in diagram with the relevant symbols. Energy comes in
as SW energy (Fo), is circulated and transformed by surface (S) and atmospheric (E) fluxes, and goes out as LW energy (OLR).
Fo -> S,E -> OLR (1)
For these components to be in balance, the energy passing through each of the three components must be equal. We ignore geothermal energy at this stage (Po).
The internal energy of the surface/atmosphere system is also composed of a number of fluxes. Miskolczi represents the energy of the system by two main net fluxes. The first net flux is the difference between the surface up flux (Su) and the radiant energy down flux Fo. This net flux (Su-Fo) warms the atmosphere. If the atmosphere was non-absorbing then these would be equal and the net term zero.
The other internal energy term is the net flux down from
the warmed atmosphere. This is composed of up and down atmospheric components, Eu and Ed. The net flux (Ed-Eu) produces the increase in land surface temperature known as the greenhouse effect. If this net flux was zero, the land surface would be much colder than it is.
For conservation of energy, total energy input Fo, the total energy of the surface/atmosphere system, and also the total energy output OLR must all be balanced (illustrated in the Figure 1):
Su-Fo + Ed – Eu = Fo = OLR (2)
This is the Miskolczi energy conservation equation 7.
Here it is assumed that the atmosphere is able to use all of the absorbed solar
flux density (Fo=OLR) to warm the system.
The factor missing is the transmittance (T) directly from the surface
to space without absorbance in the atmosphere. This is through
frequencies called the atmospheric window, and is why the
Miskolczi atmosphere theory is “semi-transparent” and not “infinitely thick”.
I will deal with transmittance in a later post, however for now we assume the
system is able to utilize St (surface radiation that would be transmitted
in a clear sky) and return it to the surface from the cloud bottom.
(2) is a unique equation for cloudy atmosphere. In the Earth you always have some transmitted flux density, meaning that OLR>Eu.
The expression (2) can be simplified using the Kirchoff law equilibrium
relationship: the energy balance at the ground Ed=Su and
the energy balance at the top of the atmosphere Eu=Fo=OLR. Miskolczi
explains that thermal equilibrium between the surface and the atmosphere is not self-evident, and in fact is one of the major innovations of his approach.
But I will return to his discussion of that issue in a later post.
Substitution of these into (2) gives SU-OLR+SU-OLR = OLR or
Miskolczi offers the following simple linear regression model as evidence of the
postulated relationship between Su and OLR based on actual radiosonde measurements (Figure 2 below).
The relevant information on the figure is the a global average OLR at 61km of 250W/m2 and Su of 375 W/m2, confirming the 2/3 relationship. The spread of results about these values are due to latitudinal variation I presume (figure supplied by Ferenc). The comparable IPCC AR4 estimates (see FAQ and chapter 1) based on Kiehl and Trenberth are 235W/m2 and 390W/m2.
An easy way to get the greenhouse effect in terms of temperature is take the fourth root of the ratio of W/m2 (square root twice). This gives 1.106682 for the ratio of the increase in temperature due to the greenhouse effect. If the Earth’s temperature would be 257.7K without the greenhouse effect, with gives an expected increase of temperature of 27.5K due to greenhouse effect, which is comparable to estimates (see wikipedia).
So it seems that Miskolczi derives upper limits to the greenhouse effect in
a fairly straightforward way from the conservation of energy. Based on the rough calculations above the effect is also at a maximum, leaving no possibility of increases in temperature due to ‘enhanced greenhouse effect’ without violation of principles of conservation of energy.
Implications of Increasing CO2
The 2/3 value is the amplification of IR at the surface causing Su to be greater than OLR. This is related to the greenhouse effect caused by the absorption or optical depth of the atmosphere, called f. Greenhouse gases that increase the optical depth of the
atmosphere would change f, expressed as a positive derivative df/f.
The results of most general circulation models (GCMs) indicate such an increase df/f>0, would increase the temperature of the atmosphere
and hence the atmospheric flux down Ed, warming the surface. This is shown in the elevated curve in the
figure from Douglass et al. 2007 marked with a red arrow.
To examine changes in the system, Miskolczi takes a derivative form of (3) where f is the coefficient for 2/3:
df/f = dOLR/OLR – dSu/Su (4)
If df/f were positive, due to a sudden increase in a GHG say, then the dOLR/OLR must be
greater than dSu/Su. That is, (4) indicates that atmospheric temperatures
would increase faster than surface temperatrues, as shown in the
experimental results from GCMs reported in Douglas et al 2007
and reviewed here.
However, (3) shows that the value of Su is locked to OLR which in turn is
locked to incoming radiation Fo. So such increases would be
temporary and the equilibrium would be reestablished with
f going back to its original value. This necessitates a fall in other
greenhouse elements, such as a decrease in water vapor in
the air, restoring the original optical depth.
The energy of
the surface/atmosphere system cannot continue increasing
as energy inputs and outputs must remain the same
to keep the energy of the system in balance. So the
temperature of the atmosphere must remain fairly constant.
This model provides an explanation for lack of troposphere warming despite increasing CO2.
as shown in the observed trends for the troposphere shown on the Douglass figure above (blue lines).
Some outstanding issues need to be addressed, particularly the cloudy sky and Kirchoff equilibrium assumptions. In future posts, I want to get to claims such as the following:
The observed 5 % increase in CO2 can be compensated with ~0.005 prcm decrease in the global H2O content. This amount is so
small it cannot be measured or monitored.
This means that in the long run the Earth has a saturated greenhouse
effect with fixed optical depth, with profound consequences
for global warming. As long as we have an atmosphere with a virtually infinite
water reservoir, neither nature nor humans can influence the
The only levers to play with that can alter surface temperatures
are the solar inputs Fo, geothermal
energy Po, and the distribution of heat within the system such as changes in albedo or
Next: The Virial Theorem in Miskolczi’s Atmosphere Theory