In Part One I made the observation:

If the atmosphere has an invariant optical thickness then surely all molecules should be included?

Meaning all ‘radiatively-active’ gases. Then I cited some results from Collins (2006) on the ‘radiative forcing’ for other gases, and added:

..So if total optical thickness from CO2 and water vapor has stayed constant over 60 years then surely total optical thickness must have increased?

In response, Miskolczi supporter Miklos Zagoni said:

Optical thickness was calculated over 60 years for CO2 and water vapor and other 9 IR-active molecular species (O3, N2O, CH4, NO, SO2, NO2, CCl4, F11 and F12), and turned out to be strictly fluctuating around a theoretically predicted equilibrium value

I asked for more details (concentrations of each of these gases over time which were used for the calculations) which weren’t forthcoming.

Later Miskolczi supporter Ken Gregory said:

Only the H2O and CO2 gases were changed. Other minor GHG were held constant.

So, working with this data I thought it would be interesting to see what changes **had** taken place in optical thickness due to these minor “greenhouse” gases.

I should point out that there are substantial problems identified with Miskolczi’s theory and experimental work and this is a very minor issue – it is more of an interesting aside.

A little while ago I managed to recreate the CO2 transmittance in the atmosphere – as shown in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Nine. This was done using the HITRAN database in a MATLAB model I created.

The question about changes in optical thickness over time from other gases was a good motivator to update my MATLAB model to bring in other molecules. It was something I wanted to do anyway.

*Note that radiative forcing or (surface emission – OLR) is a much more useful value than total optical thickness (as explained in Part One).*

Extracting the HITRAN data proved to be the most tedious and challenging part of the project. It turns out that the “minor gases” like CFC-11 and CFC-12 are stored in a totally different format from gases like CO2, N2O, CH4 etc. These minor gases have a dataset for each temperature and pressure, with different sizes of dataset at various temperature/pressures. Nothing mathematically or conceptually challenging, just very tedious.

Another challenge was working out what concentrations to use for 1948 – the start date that Miskolczi uses. From Collins (2006) it seemed that the main “greenhouse” gases to evaluate were N2O (nitrous oxide), CH4 (methane) plus CFC11 (CCl3F) and CFC12 (CCl2F2). There are other halocarbons to include but time is limited.

Here are the values used:

……………………1948 2008

CO2 311 ppmv 386 ppmv

N2O 289 ppbv 319 ppbv

CH4 1250 ppbv 1775 ppbv

CFC11 0 267 pptv

CFC12 0 535 pptv

The later CO2 value is from 2008 from Miskolczi’s spreadsheet while the other values are from 2005.

**ppmv **= parts per million by volume, **ppbv **= parts per billion (10^{9}) by volume, **pptv **= parts per trillion (10^{12}) by volume.

Earlier values of N2O and CH4 are taken from various papers, I can provide citations if anyone is interested – but pre-1980 values are thin on the ground.

In any case, my calculations of total optical thickness are rudimentary and provided as a starting point.

### The Model

I used a 5 layer model up to 200mbar, with a surface temperature of 289K. The diffusivity approximation was used to estimate total hemispherical transmittance (see Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations*). *The wavenumber step, Δν = 1 cm^{-1}. The calculations were done from 100 cm to 2500 cm (4μm – 100 μm) and the “Planck weighted” transmittance (at 289K) was calculated. This transmittance was converted back to an optical thickness, which is the same approach that Miskolczi uses (see comment).

Water vapor was assumed to be 10g/kg at the surface with a straight line reduction (vs pressure) to zero at 200mbar. Previously I carried out calculations where water vapor was varied from 5g/kg to 15g/kg and the effect on the transmittance change due to other gases was quite small.

Water vapor absorption lines are included from the HITRAN database but the water vapor continuum is not. This is next in my wishlist to include.

### Changes in Water Vapor

The model deliberately did not try to follow Miskolczi’s water vapor values. The point of this article is to demonstrate that if (and only if) CO2 optical thickness is canceled out by water vapor changes, then significant increases in optical thickness from other gases impact negatively on his hypothesis.

If his calculations show:

optical thickness (CO2 + water vapor) = constant

then this article demonstrates that:

optical thickness (CO2 + other gases + water vapor) = increasing

*Many people might not realize that there are a number of water vapor datasets. The one Miskolczi uses is not the only one. Others show different trends.*

### Results

Note that water vapor is included, but at unchanged concentration.

- The change in optical thickness, Δτ, for CO2 only changing = 0.0167
- The change in optical thickness, Δτ, for CO2+N2O+CH4+CFC11+CFC12 = 0.0238

The % increase (over CO2) due to the nominated “minor gases” = is 43%.

The* total optical thickness* is not so important in this analysis. If the number of layers is changed, the total optical thickness changes, but **percent changes** due to “greenhouse” gas increases are roughly similar.

### Conclusion

If (and only if) water vapor has canceled out CO2 increases, then the increase in optical thickness due to these other gases (methane, nitrous oxide plus halocarbons) has destroyed the idea that optical thickness can be considered to be constant.

Of course, my calculations are rudimentary. My model is much less exact than the HARTCODE model used by Miskoczi and it would be interesting to see his results reproduced in full with the correct concentrations of all of the GHGs from 1948 – 2008.

As I commented earlier – this is one of the least important of the criticisms of Ferenc Miskolczi’s papers.

Now I have updated the model I can produce results like these:

*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 Three – Kinetic Energy - why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)*

*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*

### References

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

*Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4)*, Collins et al, *JGR* (2006)

## The Mystery of Tau – Miskolczi – Part Four – Emissivity

Posted in Basic Science, Commentary on May 1, 2011 | 293 Comments »

In Part Two we looked at the claimed relationship E

_{D}=A_{A}in Miskolczi’s 2007 paper.I showed that they could not be exactly equal. Ferenc Miskolczi himself has just joined the discussion and confirmed:

That is, there is not a theoretical basis for equating A

_{A}=E_{D}as an identity.There is a world of difference between demonstrating a thermodynamic identity and an approximate experimental relationship. In the latter situation, it is customary to make some assessment of how close the values are and the determining factors in the relationship.

But in reviewing the 2007 paper again I noticed something very interesting:

From Miskolczi (2007)

Figure 1Now the point I made in Part Two was that A

_{A}≠ E_{D}because the atmosphere is a little bit cooler than the surface – at the average height of emission of the atmosphere. So we would expect E_{D}to be aa little lessthan A_{A}.Please review the full explanation in Part Two to understand this point.

Now take a look at the graph above. The straight line drawn on is the relationship E

_{D}=A_{A}.The black circles are for

an assumptionthat the surface emissivity, ε_{G}= 1. (This is reasonably close to the actual emissivity of the surface, which varies with surface type. The oceans, for example, have an emissivity around 0.96).In these calculated results you can see that Downwards Emittance, E

_{D}is a little less than A_{A}. In fact, it looks to be about 5% less on average. (And note that is E)_{D }= Absorbed Downwards EmittanceOf course in practice, ε

_{G}< 1. What happens then?Well, in the graph above, with ε

_{G}= 0.96 the points appear to lie very close to the line of E_{D}=A_{A}.I think there is a calculation error in Miskolczi’s paper – and if this is true it is quite fundamental. Let me explain..

Here is the graphic for explaining Miskolczi’s terms:

From Miskolczi (2007)

Figure 2When the surface is a blackbody (ε

_{G}=1), S_{U}= S_{G}– that is, the upwards radiation from the surface = the emitted radiation from the ground.The terms and equations in his 2007 are derived with reference to the surface emitting as a blackbody.

When ε

_{G}< 1, some care is needed in rewriting the equations. It looks like this care has not been taken and the open circles in his Fig 2 (my figure 1) closely matching the E_{D}=A_{A}line are an artifact of incorrectly rewriting the equations when ε_{G }< 1.That’s how it looks anyway.

Here is my graphic for the terms needed for this problem:

Figure 3As much as possible I have reused Miskolczi’s terms. Because the surface is not a blackbody, the downward radiation emitted by the atmosphere is not completely absorbed. So I created the term E

_{DA}for the emission of radiation by the atmosphere. Then some of this, Er, is reflected and added to S_{G}to create the total upward surface radiation, S_{U}.Note as well that the relationship emissivity = absorptivity is only true for the same wavelengths. See note 4 in Part Two.## Some Maths

Now for some necessary maths – it is very simple. All we are doing is balancing energy to calculate the two terms we need. (

Updated note – some of the equations are approximations – the real equation for emission of radiation is a complex term needing all of the data, code and a powerful computer – but the approximate result should indicate that there is an issue in the paper that needs addressing – see comment).And

the objectiveis to get a formula for the ratio E_{D}/A_{A}– if E_{D}=A_{A}, this ratio = 1. And remember that in Figure 1, the relationship E_{D}/A_{A}=1 is shown as the straight line.First, instead of having the term for atmospheric temperature, let’s replace it with:

T

_{A}= T_{S}– ΔT [1]where ΔT represents the idea of a small change in temperature.

Second, the emitted atmospheric downward radiation comes from the Stefan-Boltzmann law:

E

_{DA}= ε_{A}σ(T_{S}– ΔT)^{4}[2]Third, downward

atmospheric radiationabsorbedby the surface:E

_{D}= ε_{G}E_{DA}[3]Fourth, the upward surface radiation is the emitted radiation plus the

reflectedatmospheric radiation. Emitted radiation is from the Stefan-Boltzmann law:S

_{U}= ε_{G}σT_{S}^{4}+ (1-ε_{G}) E_{DA}[4]Fifth, the

absorbed surface radiationis the upward surface radiation multiplied by the absorptivity of the atmosphere (= emissivity at similar temperatures):A

_{A}= ε_{A}S_{U}[5]So if we put [2] -> [3], we get:

E

_{D}= ε_{G}ε_{A}σ(T_{S}– ΔT)^{4}[6]And if we put [4] -> [5], we get:

A

_{A}= ε_{G}ε_{A}σT_{S}^{4}+ E_{D}ε_{A}(1-ε_{G})/ε_{G}[7]We are almost there. Remember that we wanted to find the ratio E

_{D}/A_{A}. Unfortunately, the A_{A}term includes E_{D}and we can’t eliminate it (unless I missed something).So let’s create the ratio and see what happens. This is equation 6 divided by equation 7 and we can eliminate ε

_{A}that appears in each term:E

_{D}/A_{A}= [ ε_{G}σ(T_{S}– ΔT)^{4}] / [ ε_{G}σT_{S}^{4}+ E_{D}(1-ε_{G})/ε_{G}] [8]And just to make it possibly a little clearer, we will divide top and bottom by ε

_{G}and color code each part:E

_{D}/A_{A}= [ σ(T_{S}– ΔT)^{4}] / [ σT_{S}^{4}+ E_{D}(1-ε_{G})/ε_{G}^{2}] [8a]And so the ratio = blackbody radiation at the atmospheric temperature divided by

( blackbody surface radiation plus a factor of downward atmospheric radiation that increases as ε

_{G }reduces )We didn’t make a blackbody assumption, it is just that most of the emissivity terms canceled out.

## What Does the Maths Mean?

Take a look at the green term – if ε

_{G}= 1 this term is zero (1-1=0) and the equation simplifies down to:E

_{D}/A_{A}= (T_{S}– ΔT)^{4}/ T_{S}^{4}Which is very simple. If ΔT = 0 then E

_{D}/A_{A}= 1.Let’s plot E

_{D}vs A_{A}for a few different values of ΔT and for T_{S}= 288K:Figure 4Compare this with figure 1 (Miskolczi’s fig 2).

Note: I could have just cited the ratios of E

_{D}/A_{A}, which – in this graph – are constant for each value of ΔT.And we can easily see that as ΔT →0, E

_{D}/A_{A}→1. This is “obvious” from the maths for people more comfortable with equations.That’s the simplest stuff out of the way. Now we want to see

what happens when ε. This is the interesting part, and when you see the graph, please note that the axes are not the same as figure 4. In figure 4, the graph is of E_{G}< 1_{D}vs A_{A}, but now we will plotthe ratioof E_{D}/A_{A}as other factors change.Take a look back at equation 8a. To calculate the ratio we need a value of Ed, which we don’t have. So I use some typical values from Miskolczi – and it’s clear that the value of Ed chosen doesn’t affect the conclusion.

Figure 5You can see that when ε

_{G}= 1 the ratio is almost at 0.99. This is the slope of the top line (ΔT=1) in figure 4.This is clear from equation 8a – as ε

_{G}reduces below 1, the second term in the denominator of equation 8a increases from zero. As this increases, the ratio must reduce.In Miskolczi’s graph, as ε

_{G}changed from 1.0 → 0.96 thecalculated ratio increased. I believe this is impossible.Here is another version with a different value of ΔT:

Figure 6## Conclusion

Perhaps I made a mistake in the maths. It’s pretty simple – and out there in the open, so surely someone can quickly spot the mistake.

Of course I wouldn’t have published the article if I thought it had a mistake..

On conceptual grounds we can see that as the emissivity of the surface reduces, it absorbs

lessenergy from the atmosphere and reflectsmoreradiation back to the atmosphere.This must reduce the value of E

_{D}and increase the value of A_{A}. Thisreducesthe ratio E_{D}/A_{A}.In Miskolczi’s 2007 paper he shows that as emissivity is reduced from a blackbody to a more realistic value for the surface, the ratio goes in the other direction.

If my equations are correct then the equations of energy balance (for his paper) cannot have been correctly written for the case ε

_{G}<1.This one should be simple to clear up.

Update May 31st - Ken Gregory, a Miskolczi supporter appears to agree – and calculates E_{D}/A_{A}=0.94 for a real world surface emissivity.Other articles in the seriesThe 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&EPart Two – Kirchhoff - why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paperPart Three – Kinetic Energy - why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)Part Five – Equation Soufflé - explaining why the “theory” in the 2007 paper is a complete dog’s breakfastPart 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.Read Full Post »