Next is this series is Kirchhoff law. Kirchhoff is the third principle in Miskolczi’s atmospheric greenhouse theory, after energy conservation (part one) and the virial theorem (part two).

Miskolczi attributes Kirchhoff’s law (KL) to the significant findings of the paper. On page 6, two equations that are the direct consequence of KL and “may fundamentally change the general concept of greenhouse theories”. Elsewhere M states the validity of KL “is not trivial” and “we give a simple theoretical proof of the KL for atmospheres in radiative equilibrium”. So KL is important, and I wanted to understand that and flesh out the proof he refers to.

Firsly, KL is a law concerning thermal equilibrium, not to be confused with radiative equilibrium. Here, it concerns not only temperature but also black body radiation due to the temperature of bodies. Wikipedia states in Kirchhoff law the emissivity of a body (or surface) equals its absorptivity as at thermal equilibrium. A poor reflector is a good emitter, and a good reflector is a poor emitter. KL is why, for example, lightweight emergency thermal blankets are based on reflective metallic coatings: they lose little heat by radiation.

Miskolczi relates Kirchhoff law to the equality of IR absorbance of the atmosphere Aa (from the surface IR up Su) and the IR emission from the atmosphere down Ed. So the side of the atmosphere facing the earth acts like a body with equal absorptivity and emissivity of IR radiation. This formulation excludes the absorptions of SW by the atmosphere. Miskolczi also treats As=Ed as equivalent to surface/atmosphere thermal equilibrium.

Now the absorbed atmospheric flux Aa is equal to the surface IR Su minus the transmitted part Ta or Su(1-Ta), so I would assume that surface/atmosphere thermal equilibrium follows from Aa=Ed, though I am not going to pursue that. As evidence of Kirchhoff law Miskolczi presents the following linear regression that is the absorbed part of IR in the atmosphere Su(1-Ta) equals the IR down to the surface Ed.

Fig. Empirical evidence of the Kirchhoff law relationship in atmospheres provided by Miskolczi.

The first figure shows the relevant quantities at each point in the atmospheric profile: B the blackbody source function, Ed the IR down, Ta the flux transmittance (surface up Su transmission not absorbed in the atmosphere), and red dots are B(1-Ta). Evidence Kirchhoff law holds is shown by the correspondence of Ed with B(1-Ta). The second image is the linear regression of Su against Ed/(1-Ta) for selected radiosonde measurements.

Because the IRs are related, the temperatures of the atmosphere at the surface and the temperature of the ground must be closely related. Keep in mind that reasonable as this sounds it is not the basis of the current radiative models. Elsewhere pliny objects there is no reason to expect Aa=Ed. Nevertheless, this is the result that obtains from the proof I show later, which is why I suppose Miskolczi flags it as “non-trivial”.

Perhaps a flow-on effect of Aa!=Ed in current radiative models is a very large temperature discontinuity at the ground. Below is Fig 5 from Miskolczi’s paper.

His Fig 5 with original caption compares the new semi-transparent model, with a widely used reference atmosphere USST-76. Notice first the semi-transparent model almost fits exactly the observed lapse rate in the atmosphere. The USST-76 reference is a straight line approximation only. Note however the black dots on the USST-76 reference showing the temperature discontinuity at the ground. It is a whopping 125W/m² between the air and ground emission! I find this incredible! How it is derived and gets to be so large is subject of another post.

Elsewhere M explains how the discontinuity comes about through the semi-infinite atmosphere assumption. Calculations of the temperature profile as a linear function of the absorption path length, without reference to the surface temperature are referred to as the Schwarzschild-Milne approximation (M’s eqn 5). Warming of the atmosphere occurs in this model via path length increases due to increased GHG. Where the atmosphere is optically thin, Miskolczi claims the equation is invalid as it does not contain a lower boundary condition (the surface). He cites a number of papers to 2003 using results from the Schwartzchild-Milne approximation to derive surface and air temperatures with two (disconnected) equations.

t⁴_A=t⁴_E(1+τ_A)/2
t⁴_G=t⁴_E(2+τ_A)/2

At optical depth τ of 1.84, then t_A/t_G is 0.93 which is a temperature discontinuity at the surface of about 25 degrees C. A paper by Weaver and Ramanathan in 1995 suggest the Schwarzchild model was discarded 3 decades ago due to deficiencies, and replaced with “more sophisticated computer models”. It also suggests that addition of a spectral window, and unlinking optical depth and atmospheric pressure improves the Schwartzchild model. These are both features inherent in the Miskolczi model.

The temperature discontinuity at the surface are huge differences, and if due to a misspecified model, I can see the source of Miskolczi’s concerns. GCMs containing these models would need to be tuned to even approximate the correct radiation budget, and would demonstrate quite different behavior than the real atmosphere with even small deltas. If there is a basis for auditing the assumptions of the models, it is here. It is hard to believe they have been operating with such a large misspecification. However, it may explain the large systematic warming bias reported in almost all GCMs examined in Douglass et al. 2008.

Proof of Kirchhoff holding for atmosphere

We need to get to Aa=Ed to obtain Kirchhoff law in the atmosphere.

In section 4.2 M derives (21) a general solution temperature profiles for both the semi-infinite and semi-transparent case assuming radiative balance in a bounded atmosphere. When the optical path length is infinite (21) reduces to the semi-infinite case.

His eqn (28) Su=OLR/f derived from (21) for high emissivity and opaque areas is superficially similar to Su=2OLR/3 (8) derived earlier assuming energy balance and Kirchhoff. But Su=2OLR/3 is an overall energy balance requirement (not IR). In (28) f is a function of the optical path length τ:

f(τ)=2/(1 + τ + exp(-τ))

This function partitions the surface upward radiation into OLR by optical depth. For example, at the top of atmosphere, τ=0 and f(τ)=1. These two are not necessarily equal.

Miskolczi splits up outgoing LW radiation OLR to get an expression for Eu using two processes:

1. Su * f which is the part of surface radiation Su converted from solar SW radiation Fo to OLR
2. Su * Ta which is the transmitted part of the surface radiation

Then:

Su * f=Su * Ta+Eu as (OLR=Su* Ta + Eu)
Eu=Su * f – Su * Ta

Miskolczi argues that the difference above is the contribution to the OLR from energy processes not related to LW absorption, ie. from absorption of SW in the atmosphere F and from convection and other non-radiatiative surface processes K.

Therefore Eu = F + K.

He then substitutes the above into the surface balance equation.
Fo + Ed – F – K – Aa – St = 0 OK
St + Eu + Ed – F – K – Aa – St = 0 (as Fo = St + Eu)
St + F + K + Ed – F – K – Aa – St = 0 (as Eu = F + K)
and obtains Ed=Aa by cancellation.

The IR emitted down by the atmosphere equals the IR absorbed and Kirchoff law is obtained from radiative balance and energy balance at the surface. In real surfaces due to emissivity ε

t⁴S=t⁴E/(1+Ta(ε-1))/f

and given ε=0.95 and Ta=0.17 there will be a temperature discontinuity of 0.9% or 2.5deg.
This is much less than 25C obtained in the Schwartzchild equations and can be balanced by energy transport of non-radiative origin.

There is a view that uncertainty of future climate is either inherent in the system or or can be eliminated with more petaflops in computers. Miskolczi’s results argue that problems have been introduced due to an incorrent model of climate. Based on the types of errors reported by GCMS, especially those shown in Douglass et al 2008, I still find this to be a plausible explanation. In fact, the main alarmist claims:

• increasing greenhouse effect with larger τ,
• runaway greenhouse effects t unbounded with increasing τ, and
• positive feedback

have been shown by Miskolczi to result from misspecification of the atmospheric model. It is not the first time that errors propagated through the literature by mindless repetition of past claims. I will go into this further in the next post on radiative equilibrium.