In Part One we looked at the calculation of total atmospheric optical thickness.
In Part Two we looked at the claim that the surface and atmosphere exchanged exactly equal amounts of energy by radiation. A thermodynamics revolution if it is true, as the atmosphere is slightly colder than the surface. This claim is not necessary to calculate optical thickness but is a foundation for Miskolczi’s theory about why optical thickness should be constant.
In this article we will look at another part of Miskolczi’s foundational theory from his 2007 paper, Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service.
For reference of the terms he uses, the diagram from the 2007 paper:
On pages 6-7, we find this claim:
Regarding the origin, EU is more closely related to the total internal kinetic energy of the atmosphere, which – according to the virial theorem – in hydrostatic equilibrium balances the total gravitational potential energy. To identify EU as the total internal kinetic energy of the atmosphere, the EU = SU / 2 equation must hold.
Many people have puzzled over the introduction of the virial theorem (note 1), which relates total kinetic energy of the atmosphere to total potential energy of the atmosphere. Generally, there is a relationship between potential energy and kinetic energy of an atmosphere so I don’t propose to question it, we will accept it as a given.
By the way, on the diagram SU = SG, i.e. SU = upwards radiation from the surface. And EU = upwards radiation from the atmosphere (cooling to space).
Kinetic Energy of a Gas
For people who don’t like seeing equations, skip to the statement in bold at the end of this section.
Here is the equation of an ideal gas:
pV = nkT (also written as pV = NRT) 
where p = pressure, V = volume, n = number of molecules, k = 1.38 x 10-23 J/K = Boltzmann’s constant, T = temperature in K
This equation was worked out via experimental results a long time ago. Our atmosphere is a very close approximation to an ideal gas.
If we now take a thought experiment of some molecules “bouncing around” inside a container we can derive an equation for the pressure on a wall in terms of the velocities of the molecules:
pV = Nm<vx²> 
where m = mass of a molecule, <vx²> = average of vx², where vx = velocity in the x direction
Combining  and  we get:
kT = m<vx²>, or
m<vx²>/2 = kT/2 
The same considerations apply to the y and z direction, so
m<v²>/2 = 3KT/2 
This equation tells us the temperature of a gas is equal to the average kinetic energy of molecules in that gas divided by a constant.
For beginners, the kinetic energy of a body is given by mv²/2 = mass x velocity squared divided by two.
So temperature of a gas is a direct measure of the kinetic energy.
The Kinetic Error
So where on earth does this identity come from?
..To identify EU as the total internal kinetic energy of the atmosphere..
EU is the upwards radiation from the atmosphere to space.
To calculate this value, you need to solve the radiative transfer equations, shown in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. These equations have no “analytic” solution but are readily solvable using numerical methods.
However, there is no doubt at all about this:
EU ≠ 3kTA/2 
where TA = temperature of the atmosphere
that is, EU ≠ kinetic energy of the atmosphere
As an example of the form we might expect, if we had a very opaque atmosphere (in longwave), then EU = σTA4 (the Stefan-Boltzmann equation for thermal radiation). As the emissivity of the atmosphere reduces then the equation won’t stay exactly proportional to the 4th power of temperature. But it can never be linearly proportional to temperature.
A Mystery Equation
Many people have puzzled over the equations in Miskolczi’s 2007 paper.
The direct consequences of the Kirchhoff law are the next two equations:
EU = F + K + P (M5)
SU − (F0 + P0 ) = ED − EU (M6)
Note that I have added a prefix to the equation numbers to identify they as Miskolczi’s. As previously commented, the P term (geothermal energy) is so small that it is not worth including. We will set it to zero and eliminate it, to make it a little easier to see the problems. Anyone wondering if this can be done – just set F’ = F0 + P0 and replace F0 with F’ in the following equations.
EU = F + K (M5a)
SU − F0 = ED − EU (M6a)
Please review figure 1 for explanation of the terms.
If we accept the premise that AA = ED then these equations are correct (the premise is not correct, as shown in Part Two).
M5a is simple to see. Taking the incorrect premise that surface radiation absorbed in the atmosphere is completely re-emitted to the surface: therefore, the upward radiation from the atmosphere, EU must be supplied by the only other terms shown in the diagram – convective energy plus solar radiation absorbed by the atmosphere.
What about equation M6a? Physically, what is the downward energy emitted by the atmosphere minus the upward energy emitted by the atmosphere? What is the surface upward radiation minus the total solar radiation?
Well, doesn’t matter if we can’t figure out what these terms might mean. Instead we will just do some maths, using the fact that the surface energy must balance and the atmospheric energy must balance.
First let’s write down the atmospheric energy balance:
AA + K + F = EU + ED  - I’m jumping the numbering to my equation 10 to avoid referencing confusion
This just says that Surface radiation absorbed in the atmosphere + convection from the surface to the atmosphere + absorbed solar radiation in the atmosphere = energy radiated by the atmosphere from the top and bottom.
Given the (incorrect) premise that AA = ED, we can rewrite equation 10:
K + F = EU [10a]
We can see that this matches M5a, which is correct, as already stated.
So first, let’s write down the surface energy balance:
F0 – F + ED = SU + K 
This just says that Solar radiation absorbed at the surface + downward atmospheric radiation = surface upward radiation + convection from the surface to the atmosphere.
Please review Figure 1 to confirm this equation.
Now let’s rewrite equation 11:
SU – F0 = ED – F – K [11a]
and inserting eq 10a, we get:
SU – F0 = ED -EU [11b]
Which agrees with M6a.
And as an aside only for people who have spent too long staring at these equations – re-arrange the terms in 11b:
Su – Ed = F0 – Eu; The left side is surface radiation – absorbed surface radiation in the atmosphere (accepting the flawed premise) = transmitted radiation. The right side is total absorbed solar radiation – upward emitted atmospheric radiation. As solar radiation is balanced by OLR, the right side is OLR – upward emitted atmospheric radiation = transmitted radiation.
Now, let’s see the mystery step :
In Eq. (6) SU − (F0 + P0 ) and ED − EU represent two flux terms of equal magnitude, propagating into opposite directions, while using the same F0 and P0 as energy sources. The first term heats the atmosphere and the second term maintains the surface energy balance. The principle of conservation of energy dictates that:
SU − (F0) + ED − EU = F0 = OLR (M7)
This equation M7 makes no sense. Note that again I have removed the tiny P0 term.
Let’s take [11b], already demonstrated (by accepting the premise) and add (ED -EU) to both sides:
SU – F0 + (ED – EU) = ED – EU+ (ED -EU) = 2(ED -EU) 
So now the left side of eq 12 matches the left side of M7.
The M7 equation can only be correct if the right side of eq 12 matches the right side of M7:
2(ED -EU) = F0  – to be confirmed or denied
In concept, this claim is that downward radiation from the atmosphere minus upward radiation from the atmosphere = half the total planetary absorbed solar radiation.
I can’t see where this has been demonstrated.
It is not apparent from energy balance considerations – we wrote down those two equations in  and .
We can say that energy into the climate system = energy out, therefore:
F0 = OLR = EU + ST  (atmospheric upward radiation plus transmitted radiation through the atmosphere)
Which doesn’t move us any closer to the demonstration we are looking for.
Perhaps someone from the large fan club can prove equation 7. So many people have embraced Miskolczi’s conclusion that there must be a lot of people who understand this step.
I’m confused about equation 7 of Miskolczi.
Running with the odds, I expect that no one will be able to prove it and instead I will be encouraged to take it on faith. However, I’m prepared to accept that someone might be able to prove that it is true (with the caveat about accepting the premise already discussed).
The more important point is equating the kinetic energy of the atmosphere with the upward atmospheric radiation.
It’s a revolutionary claim.
But as it comes with no evidence or derivation and would overturn lots of thermodynamics the obvious conclusion is that it is not true.
To demonstrate it is true takes more than a claim. Currently, it just looks like confusion on the part of the author.
Perhaps the author should write a whole paper devoted to explaining how the upwards atmospheric flux can be equated with the kinetic energy – along with dealing with the inevitable consequences for current thermodynamics.
Update 31st May: The author confirmed in the ensuing discussion that equation 7 was not developed from theoretical considerations.
Other Articles in the Series:
The Mystery of Tau – Miskolczi - introduction to some of the issues around the calculation of optical thickness of the atmosphere, by Miskolczi, from his 2010 paper in E&E
Part Two – Kirchhoff - why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper
Part Four - a minor digression into another error that seems to have crept into the Aa=Ed relationship
Part Five – Equation Soufflé - explaining why the “theory” in the 2007 paper is a complete dog’s breakfast
Part Six – Minor GHG’s - a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.
New Theory Proves AGW Wrong! – a guide to the steady stream of new “disproofs” of the “greenhouse” effect or of AGW. And why you can usually only be a fan of – at most – one of these theories.
Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi, Quarterly Journal of the Hungarian Meteorological Service (2007)
Note 1 – A good paper on the virial theorem is on arXiv: The Virial Theorem and Planetary Atmospheres, Victor Toth (2010)