The first post in this series showed a theoretical proof that ‘runaway’ greenhouse effect is not possible, based on Miskolczi’s derivation of a constant (and maximum) greenhouse effect from energy conservation laws in a cloudy atmosphere. The virial theorem solves the semi-transparent (non-cloudy) case, where a fraction of the longwave radiation is transmitted directly from the surface to space, without absorption by the atmosphere.

The strategy we are using in this modeling process is to add an additional constraint (after energy conservation) in order to find the Eu term, the radiation emitted up from Earth’s absorbing atmosphere. Before, Eu was not defined and could have been zero, (in the case of a planet with no greenhouse effect). The virial theorem (and conservation of energy) allows the relationships between all the major radiation terms in a greenhouse system to be derived from first principles.

The virial theorem is a very general theorem used in cosmology (see Wikipedia entry) relating kinetic energy (KE) to potential energy (PE) in bound systems. A simple physical example is a small object orbiting around a larger object, bound by a force (or even an elastic string). The virial theorem states the KE (in the angular velocity) is half the PE (in the distance from the object):

PE = 2KE

Miskolczi relates without derivation the KE with the Eu, the longwave radiation emitted upward from the atmosphere, on the condition of hydrostatic equilibrium. The following was posted at the ClimateAudit forum.

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It’s actually introduced on page 6

Regarding the origin, 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.

Anyone have a problem with this? For the moment I’ll assume not. This balancing act is mediated by the virial theorem which may be generally expressed as where is the average kinetic energy (relation to temperature is well known) and is the total potential energy, so 2KE = PE. He takes the internal kinetic energy to be represented by (Is this valid? It is certainly related) now the potential energy I am thinking derives from the absorbed energy that is converted to potential energy as the air rises until hydrostatic balance is attained. The source of this energy is (IMO) but he takes it as being I’m not sure this is a fair thing since I would expect . Hence taking his representation of the internal kinetic energy and what I think is his original source of the potential energy we get .

At least we do have an idea where it comes from whether or not it’s right or valid I’m not yet ready to judge.

I hope this helps.

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The following is an example I thought of that has the flavor of the virial theorem in atmospheric absorption processes (but may not be relevant at all). Imagine an old-time dance hall where only couples of opposite sex can dance together (we are not in San Francisco now) and there are an equal number of males and females. There must be an equal number so that there is the ‘potential’ for all couples to be dancing simultaneously. Every few bars of the music, couples separate, turn to the nearest person, and if they are of the opposite sex, they dance. How many couples are dancing at any time?

Clearly, in a well mixed room (at hydrostatic equilibrium) as there is a 0.5 chance of a person encountering another of the opposite sex, half of the ‘potential’ couples will be dancing at any time. The ‘potential’ couples represent the PE, the actual couples dancing represents the KE. In the atmospheric case, photon-molecule interactions are the KE, and PE=2KE.

Back to the atmosphere, the potential energy is the longwave radiation up from the surface (Su) and the kinetic energy is the longwave radiation emitted from the atmosphere (Eu) (see figure). By virality, Su=2Eu. But from the previous post Su=2OLR/3. In the cloudy atmosphere, the outgoing longwave radiation is entirely due to the atmosphere Eu, i.e. OLR=Eu. But Su=2Eu and Su=2OLR/3 cannot be simultaneously equal if OLR=Eu.

A term is needed to allow
both the Su=2Eu and the Su=(3/2)OLR equations to be valid.
To obtain this Miskolczi uses St, the IR radiation from the ground transmitted
directly through the atmosphere. The outgoing
radiation is equal to the sum of the transmitted IR radiation
and the radiation up from the atmosphere OLR = St+Eu
(rather than just Eu as in the cloudy case). Then given
the equations:

Su = 3OLR/2 (energy conservation)
Su = 2Eu (Virial theorem)

The derivation of St/Su is easy, just basic algebraic substitution.

Su = 3(St+Eu)/2 as OLR = St+Eu
Su = 3(St+Su/2)/2 as Su=2Eu
Su = 3St/2 + 3Su/4
Su/4 = 3St/2
Su = 6St

As evidence Miskolczi presents the following figure relating Eu to Su from radiosonde observations.

The global average cloud cover of an Earth-like planet will organize itself so that one-sixth of the IR radiation is transmitted from the surface to space. Miskolczi states (but does not derive) an expected global average cloud cover of 0.6. He derives these relationships without any reference to the optical absorption of greenhouse gasses. Therefore they are not as important to temperatures as fundamental energy constraints on the system as a whole.

Miskolczi’s theory develops two types of stable atmosphere as possible solutions of the first principles analysis, Mars and Earth, and a third (Venus) is in development. These types are:

1. Planets with atmospheres largely transparent to IR, and very low greenhouse effect, that will not satisfy the virial theorem (hydrostatic equilibrium). Planets like Mars are of this type.
2. Planets with semi-transparent atmospheres, and transmittance from the surface around 1/6 of total IR emission, saturated greenhouse effect, with mixed cloud cover. Earth is a planet of this type.
3. Planets with semi-infinite atmospheres (no effective surface), permanently cloudy, with no appreciable transmittance.

So far this is a considerable achievement. There are a number of results that, if correct, render the some main themes of 10 years of climate modeling irrelevant: constant greenhouse effect, constant fractional cloud cover, to name only two. No wonder this paper sticks in their craw. The busywork of modelers is confirmed by the independent evaluation of models such as Douglass et al. 2007 and Koutsoyiannis et al. 2008, who conclude in an Assessment of the reliability of climate predictions based on comparisons with historical time series.

(M)odel outputs at annual and climatic (30‐year) scales are irrelevant with reality; also, they do not reproduce the natural overyear fluctuation and, generally, underestimate the variance and the Hurst coefficient of the observed series; none of the models proves to be systematically better than the others.

The huge negative values of coefficients of efficiency at those scales show that model predictions are much poorer that an elementary prediction based on the time average.

This makes future climate projections not credible.

Next I think I will tackle the existing models; what are the incorrect assumptions Miskolczi was referring to when he said that the “equations were totally wrong”, and exactly how does Miskolczi’s theory correct them. You will be surprised how simple they are, and learn more about the importance of assumptions in developing a model.