Virial Paper 6_12_2010 submitted by Adolf J. Giger.

Allow me to make some more comments on the Virial Theorem (VT) as used by Ferenc Miskolczi (FM) for the atmosphere.

As I said on this blog back in February, a very fundamental derivation of the VT was made by H. Goldstein in Section 3-4 of “Classical Mechanics”, 1980, Ref.[1] : PE= 2*KE (potential energy=2 x kinetic energy). Then, he also derives the Ideal Gas Law (IGL), P*V = N*k*T as a consequence of the VT, and shows that PE=3*P*V and KE=(3/2)*N*k*T. The two laws, IGL and VT, therefore are two ways to describe the same physical phenomenon. Despite its seemingly restrictive name, we know that the IGL is a good approximation for many gases, monatomic, biatomic, polyatomic and even water vapor, as long as they remain very dilute. Goldstein’s derivations are made for an enclosure of volume V with constant gas pressure P and temperature T in a central force field like the Earth’s gravitational field. They also hold for an open volume V anywhere in the atmosphere. As to FM, he points out that the VT reflects the fact that the atmosphere is gravitationally bounded.

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Ferenc Miskolczi in his papers [2,3] relates the total potential energy of the atmosphere, PEtot, to the total IR upward radiation Su at the surface. This relationship has to be considered a proportionality rather than an exact equality, or Su=const* PEtot. We see that this linkage makes sense since Su determines the surface temperature Ts through the Stefan-Boltzmann law, Su = (5.6703/10^8)*Ts^4 , and finally the IGL ties together Ts, P(z=0) and PEtot.

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FM then assigns the kinetic IR energy KE (temperature) in the atmosphere to the upward atmospheric IR emittance Eu, or Eu=const*KE. The flux Eu is made up of two terms F + K , where F is due to thermalized absorption of short wave solar radiation in atmospheric water vapor, and K due to heat transfers from the Earth’s surface to air masses and clouds through evaporation and convection. Neither F or K are directly radiated from the Earth’s surface. They represent radiation from the atmosphere itself. There is an obvious limitation for such an assignment mainly because for the VT , or the IGL in general, the temperature (the KE) has to be measured with a thermometer, whereas Eu represents the radiative temperature (flux) that has to be measured with a radiometer, and these two measurements can give vastly different results as we see for the two following extreme cases:

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In between these two extremes we have the Earth where FM’s version of the VT , Su = 2 * Eu applies reasonably well. We will see next in a discussion of FM’s exact solution how close, and for what types of atmospheres FM’s VT ( Eu/Su=0.5) holds, but we can say already that no physical principle is violated if it doesn’t. The VT that always holds for gases is not being violated, it is simply not fully recognized by FM’s fluxes that have to be measured by radiometers. This may be an indication that the VT is less important for FM’s theory than normally assumed.

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On the other hand, the IPCC assumes a positive water vapor feedback and arrives at very imprecise predictions for the Climate Sensitivity ranging from 1.5 to 5K (and even more). It is clear that this wide range of numbers is caused by the assumed positive feedback system, which apparently is close to instability (or singing, as the electrical engineer would call it in an unstable microphone-loudspeaker system). With such large uncertainties in their outputs true scientists should be reluctant to publish their results.

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