Models of Greenhouse Effect

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.

Greenhouse Heat Engine

In reviewing the points of controversy raised here in Miskolczi’s controversial theory of (almost constant) greenhouse effect and the impossibility of runaway global warming, I thought about the role of convection.

Convection is a heat engine. A heat engine is defined as a device that converts heat energy into mechanical energy. In this model, the circulation of the air is analogous to a Stirling or other simple heat engine, producing work as the result of temperature differential between the earths surface and the edge of the atmosphere. The diurnal cycle plays a large (but not complete) role in the operation of the engine, creating a cycle of heating and cooling air packets.

Keep in mind convection is not the only thermal process in the atmosphere. While convection is necessary, the following does not represent the totality of the energy conservation relationships governing the atmospheric system. As Miskolczi says:

If you like to put that way, the su=3olr/2, su=2eu, su=ed/a and
su=olr/f relations are for convective-radiative-hydrostatic equilibrium
global average atmosphere which is in total energy balance.

Treating convection as a heat engine is not a new idea and a research field has developed in the area of thermodynamics of natural convections (e.g. Renno). I don’t think it has been discussed in relation to Miskolczi’s theory though, and below I go through some of the ways it could potentially apply. I refer to comments by Nick Stokes and Pat Cassen in previous simple descriptions of parts of the theory.

Below is a schematic of natural convection as a heat engine. The source of heat at the surface, cooling and work output is shown. Wikipedia has a good article on different types of heat engines, and natural convection is probably most similar to the Brayton cycle (adiabatic/isobaric/adiabatic/isobaric).


Figure: The model of convection as a heat engine, much like a Stirling or Brayton engine, work is done by across a temperature potential. In the atmosphere, absorbed heat performs atmospheric work of raising packets of air to the colder upper atmosphere Eu.

Points of comparison with Miskolczi’s theory are as follows:

Conservation of Energy

M’s equation (6) is F = (Su-F) + (Ed-Eu) where F is the solar isolation on the surface, Ed and Eu are down and up atmospheric radiation, and Su is the radiation up from the surface. This equation asserts the existence of an Su-OLR flux term that heats the
atmosphere, and the existence of the Ed-Eu flux term that heats the

If Ed-Eu can be equated with work done, and Su-F with internal energy, and F with heat added to the system, then the equation describes the first law of thermodynamics.

ΔQ = ΔU + ΔW

Pat Cassen expresses a concern here

Equation 7 expresses the balance of energy of…what? I don’t know.

M’s equation could describe the first law of thermodynamics relating conversions of energy from one form to another via mechanical work.

The Virial Theorem

The virial theorem mentioned by Miskolczi relates kinetic and potential energy in gravitationally bounded systems to the proportion 2KE=PE. The natural convection engine could be analyzed for kinetic and potential energies, where the moving parts are the KE, and the potential energy is the gravitational field.

Here Pat Cassen also was concerned as:

I cannot figure out how Miskolczi is applying the virial theorem, or why it is necessary for any planetary atmosphere.

The virial theorem is applicable to convection — the atmospheric heat engine would not work if there was no gravitation. Air parcels could not rise without gravity.

Kirchhoff’s law

Nick Stokes was concerned that he should:

use Kirchhoff properly, which he doesn’t do (no mention of gas emissivity).

From Wikipedia, Kirchhoff’s law states the emissivity of a body (or surface) equals its absorptivity as at thermal equilibrium. However, the origin of the law is in the description of what was originally a mysterious process where an object in side a cavity achieved thermal equilibrium. Study of this leads to theories of black body radiation, and eventually to Planck’s and Einsteins treatment of radiation as quantized energy.

Local Thermodynamic Equilibrium (LTE) in the atmosphere requires the
equality of the absorbed and emitted radiation, that Su=Sa=Sg (on the average) and the simulation results in the related figures demonstrate this relationship to hold.

Kirchhoff’s law can be related to thermal equilibrium via radiative (black body) emission, and this is the context I think M intends here. In the atmospheric heat engine, the temperature of the gas inside the engine at the lowest point in the cycle is equal to the temperature of the surface (i.e. isobaric).

Optimal optical path

Pat Cassen also expressed concern at M’s radiative equilibrium equations showing that the system acts to optimize the conversion of solar energy into heat at the edge of the atmosphere Bo. Because of the cloudiness – the Earth-atmosphere system may convert Fo to OLR in such a way that the effective absorption coefficient is 1.

A heat engine is a also converter of solar energy into heat, which through M’s equations may be self-regulating. A self-regulating engine will not run faster (in this case due to solar energy constraints), or run slower (in order to utilize available energy). The Earth’s convection engine is currently at maximum greenhouse effect, and cannot be increased (the engine can’t run faster) or decreased (the engine can’t run slower) except through changes to the overall energy input to the system.


This just sketches out a model for natural convection in the atmosphere. Contrasting the heat engine model with the ‘steel shell‘ model of Willis Eschenbach, and another model of greenhouse warming applied to ice beads called the solid-state greenhouse effect, would demonstrate different types of greenhouse effect.

Pat and Nick seem to be concerned with lack of good motivation for these relationships in the paper. At this stage, I can’t see that they constitute errors that undermine the theory.

It might be argued that it is the greenhouse effect that drives the atmospheric heat engine and not the other way around. Perhaps IR greenhouse helps get the engine started. If there was no atmospheric heat engine driving warm air packets into the upper atmosphere, the atmosphere would like as a stable layer on the surface. Heat would transfer by thermal conduction, and temperature would be driven by the coefficient of conductivity of the air. This is describing an inversion condition, an occasional but not widespread phenomenon.

The predictions of GCMs due to increased greenhouse gases shows increased heating in the troposphere, kind of like a temperature profile of inversion conditions. The measurements of actual air temperature are as predicted by Miskolczi’s theory: Douglass et al 2007 show increased surface temperatures, but little increase in tropospheric temperatures. I wonder if anyone has made the connection between the profile of GCM’s and inversion conditions. This suggests a major source of lack of realism in GCM’s is inadequate representation of convection processes.

Chaitén Going Ultra-Plinian?

I have been watching the eruption of Mt Chaitén in Chile, South America very closely for the last few weeks. It appears as if it may produce a major eruption soon. The last known eruption was in 7420 BC ± 75. The progress of the 2008 eruption has been as follows.

April 30: A significant earthquake preceded the first explosions;
May 2: More earthquakes arrayed radially around the caldera implying a very large magma chamber;
May 15: Near steady-state explosive eruptions (Plinian) releasing about two cubic kilometers of ejecta of 4-5km in altitude.

Eruptions are rated by the Volcanic Explosivity Index (VEI). The following are examples of recent eruptions and their VEI.

VEI=5 Plinian, ejecta > 1 km³, e.g. St. Helens (1980)
VEI=6 Plinian/Ultra-Plinian, ejects > 10 km³, e.g. Mount Pinatubo (1991)
VEI=7 Plinian/Ultra-Plinian, ejecta > 100 km³, e.g. Tambora (1815)
VEI=8 Ultra-Plinian, ejecta > 1,000 km³, e.g. Toba (73,000 BP)

Mt Chaitén eruptions have ejected around 2 km³ of ash, giving it a solid VEI=5 on a par with Mt. St. Helens.
Mt Chaitén eruptions are not yet of sufficient energy to reach the 10km to the stratosphere (Ultra-Plinian), needed to affect weather on the global scale, although emissions reached an altitude of 8 km on the 12th May.

Some of the stunning lighning images being taken of the Mt Chaitén eruption.

The initial fuselage of shallow focus earthquakes (10 km depth), larger than magnitude 4, delineate a potentially very large magma chamber, accompanied the start of the Plinian phase on May 2.

    4.4     April 30, 2008 at 11:52 PM     17km E
    5.3     May 02, 2008 at 01:51 AM     30km NE
    4.9     May 02, 2008 at 07:13 AM     13km NW
    4.1     May 02, 2008 at 06:13 PM     16km SW
    5.0     May 02, 2008 at 10:36 PM     30 km NE

Note the epicenters arrayed radially around the caldera. Their locations imply a very large magma chamber might be released.

Today the Volcanism blog reports changes in seismic activity indicating fragmentation and instability around the volcano’s central conduit and beneath the lava dome. The volcano may go ultra-plinian if the magma conduit breaks up, and the capping lava dome explodes away. The consequences of large-scale injections of ash and gases into the stratosphere are disruptions of global climate, particularly initial cooling for a few years, depending on the amount of material released. A Mt Tambora scale eruption would cause enough disruption of normal climate to cause widespread crop losses.

Below are some resources that I will update over the next few days.

Some Resouces

Volcanism blog.– Latest reports from Chile

Satelite Image – of cloud-top height

Volcano Live – Reports by Jon Seach

Greenhouse Effect Physics

Before delving into the fourth and final installment of Miskolczi’s controversial theory of the greenhouse effect, below is a slide depicting the relationships covered so-far. The last part, on radiative equilibrium, binds warming as a function of optical depth (or concentration of greenhouse gases) and will be a bit technical.


Figure: The major relationships between fluxes in the description of Miskolczi’s atmospheric theory.

Each of the installments has dealt with a fundamental principle of physics. Below are the conroversial points that have been raised.

Conservation of energy. Here the magnitude and theoretical maximum to surface radiation (and hence surface temperatures) is shown through energy conservation equations.

Controversy: Is Fo=Su-Fo+Ed-Eu=OLR a valid conservation of energy equation?

The virial theorem. — Here the fraction of infra-red radiation absorbed by the atmosphere is shown to be 5/6 of the overall infra-red radiation from the surface.

Controversy: Is the identification of KE=Eu and PE=Su valid?

Kirchhoff’s law — Here the surface is shown to be in thermal equilibrium with the atmosphere, binding atmospheric absorbed and transmitted infra-red radiation.

Controversy: Can Ed=Aa be attributed to Kirchhoff’s law, or not?

Radiative equilibrium — To be done, quantifying the (small) effect of changes in optical depth such as the doubling of CO2 on surface temperatures. Interestingly, the theory describes is a niche model, a dynamic system bound in an energetic minimum defined by an optimal value of optical path length.

Controversy: Is Bo(τ), the shell temperature as a function of optical depth unimodal?

From the figure below, its clear that the main magnitudes of fluxes are defined by the global constraints introduced, except for the solar input Fo and the geothermal input Po. This is why Miskolczi states that current warming could not have been due to greenhouse gases, and if driven by anything, must be due to variations in solar input.

Keep in mind that my goal to this date has been to understand his theory and not to defend it. It seems like many people on the web are keen to attack without understanding first. While there have been some insightful doubts expressed about the theory, and about the way the theory is exposed, I have yet to see an objection that shows any of the relationships Miskolczi describes is wrong.

If and when a flaw is identified that materially affects the results, I will describe it in detail.


2002. Simulation of uplooking and downlooking
high-resolution radiance spectra with two
different radiative transfer models

Rolando Rizzi, Marco Matricardi, and Ferenc Miskolczi

2005. An inter-comparison of far-infrared line-by-line radiative
transfer models

David P. Kratza;∗, Martin G. Mlynczaka, Christopher J. Mertensa, Helen Brindleyb,
Larry L. Gordleyc, Javier Martin-Torresd, Ferenc M. Miskolczid, David D. Turnere

Greenhouse effect in semi-transparent planetary

Ferenc M. Miskolczi

Modeling Global Warming (Miskolczi Part 1)

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

Su=3OLR/2 (3).

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
greenhouse effect.

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

Euromodels run hot then cool then hot again

“NASA just released their new report on global warming or, as President Bush, calls it — Spring.” –Jay Leno

The recent Science has an editorial by Richard Kerr, Mother Nature Cools the Greenhouse, But Hotter Times Still Lie Ahead. He reports on the results of short-term model runs at the Hadley Center in England showing masking of warming due to natural changes in ocean currents. This is why temperatures have not been increasing as noted here and here.

So it is natural to ask, if natural cycles are masking global warming, why hasn’t the same natural cycle also been enhancing the appearance of warming, so that global warming appears greater than it is? A natural question, but one that doesn’t occur to Kerr to address in his article. More warming bias I suppose.

Kerr also acknowledges the climate stability:

As climate-change skeptics like to point
out, worldwide temperatures haven’t risen
much in the past decade. If global warming
is such hot stuff, they ask, why hasn’t it
soared beyond the El Niño–driven global
warmth of 1998?

Actually, pointing out recent cooling, branded as ‘devilish’ by Jim Hansen on NASA in
his newsletter, is simply being objective about the numbers.

However, while Kerr claims putting numbers on the cycle is supposed to be a rebuttal,
the numbers only have force if the counter-argument is contained,
that past warming could have been due to the other phase of the cycle.
Otherwise this is a case of numbers being misused to provide false authority.

Mainstream climate
researchers reply that greenhouse warming
isn’t the only factor at work. And in a new
paper, they put some numbers on that rebuttal.

Here are a couple of link to very informative, up-to-date information on the issue of ocean cycles and climate.

Is the Earth getting warmer or cooler?

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The Relationship of the PDO to El Nino and La Nina Frequency