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Archive for the ‘Debunking Flawed “Science”’ Category

In The “Greenhouse” Effect Explained in Simple Terms I list, and briefly explain, the main items that create the “greenhouse” effect. I also explain why more CO2 (and other GHGs) will, all other things remaining equal, increase the surface temperature. I recommend that article as the place to go for the straightforward explanation of the “greenhouse” effect. It also highlights that the radiative balance higher up in the troposphere is the most important component of the “greenhouse” effect.

However, someone recently commented on my first Kramm & Dlugi article and said I was “plainly wrong”. Kramm & Dlugi were in complete agreement with Gerlich and Tscheuschner because they both claim the “purported greenhouse effect simply doesn’t exist in the real world”.

If it’s just about flying a flag or wearing a football jersey then I couldn’t agree more. However, science does rely on tedious detail and “facts” rather than football jerseys. As I pointed out in New Theory Proves AGW Wrong! two contradictory theories don’t add up to two theories making the same case..

In the case of the first Kramm & Dlugi article I highlighted one point only. It wasn’t their main point. It wasn’t their minor point. They weren’t even making a point of it at all.

Many people believe the “greenhouse” effect violates the second law of thermodynamics, these are herein called “the illuminati”.

Kramm & Dlugi’s equation demonstrates that the illuminati are wrong. I thought this was worth pointing out.

The “illuminati” don’t understand entropy, can’t provide an equation for entropy, or even demonstrate the flaw in the simplest example of why the greenhouse effect is not in violation of the second law of thermodynamics. Therefore, it is necessary to highlight the (published) disagreement between celebrated champions of the illuminati – even if their demonstration of the disagreement was unintentional.

Let’s take a look.

Here is the one of the most popular G&T graphics in the blogosphere:

From Gerlich & Tscheuschner

From Gerlich & Tscheuschner

Figure 1

It’s difficult to know how to criticize an imaginary diagram. We could, for example, point out that it is imaginary. But that would be picky.

We could say that no one draws this diagram in atmospheric physics. That should be sufficient. But as so many of the illuminati have learnt their application of the second law of thermodynamics to the atmosphere from this fictitious diagram I feel the need to press forward a little.

Here is an extract from a widely-used undergraduate textbook on heat transfer, with a little annotation (red & blue):

From Incropera & DeWitt (2007)

From “Fundamentals of Heat and Mass Transfer” by Incropera & DeWitt (2007)

Figure 2

This is the actual textbook, before the Gerlich manoeuvre as I would like to describe it. We can see in the diagram and in the text that radiation travels both ways and there is a net transfer which is from the hotter to the colder. The term “net” is not really capable of being confused. It means one minus the other, “x-y”. Not “x”. (For extracts from six heat transfer textbooks and their equations read Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics).

Now let’s apply the Gerlich manoeuvre (compare fig. 2):

Fundamentals-of-heat-and-mass-transfer-post-G&T

Not from “Fundamentals of Heat and Mass Transfer”, or from any textbook ever

Figure 3

So hopefully that’s clear. Proof by parody. This is “now” a perpetual motion machine and so heat transfer textbooks are wrong. All of them. Somehow.

Just for comparison, we can review the globally annually averaged values of energy transfer in the atmosphere, including radiation, from Kiehl & Trenberth (I use the 1997 version because it is so familiar even though values were updated more recently):

From Kiehl & Trenberth (1997)

From Kiehl & Trenberth (1997)

Figure 4

It should be clear that the radiation from the hotter surface is higher than the radiation from the colder atmosphere. If anyone wants this explained, please ask.

I could apply the Gerlich manoeuvre to this diagram but they’ve already done that in their paper (as shown above in figure 1).

So lastly, we return to Kramm & Dlugi, and their “not even tiny point”, which nevertheless makes a useful point. They don’t provide a diagram, they provide an equation for energy balance at the surface – and I highlight each term in the equation to assist the less mathematically inclined:

Kramm-Dlugi-2011-eqn-highlight

 

Figure 5

The equation says, the sum of all fluxes – at one point on the surface = 0. This is an application of the famous first law of thermodynamics, that is, energy cannot be created or destroyed.

The red term – absorbed atmospheric radiation – is the radiation from the colder atmosphere absorbed by the hotter surface. This is also known as “DLR” or “downward longwave radiation, and as “back-radiation”.

Now, let’s assume that the atmospheric radiation increases in intensity over a small period. What happens?

The only way this equation can continue to be true is for one or more of the last 4 terms to increase.

  • The emitted surface radiation – can only increase if the surface temperature increases
  • The latent heat transfer - can only increase if there is an increase in wind speed or in the humidity differential between the surface and the atmosphere just above
  • The sensible heat transfer – can only increase if there is an increase in wind speed or in the temperature differential between the surface and the atmosphere just above
  • The heat transfer into the ground – can only increase if the surface temperature increases or the temperature below ground spontaneously cools

So, when atmospheric radiation increases the surface temperature must increase (or amazingly the humidity differential spontaneously increases to balance, but without a surface temperature change). According to G&T and the illuminati this surface temperature increase is impossible. According to Kramm & Dlugi, this is inevitable.

I would love it for Gerlich or Tscheuschner to show up and confirm (or deny?):

  • yes the atmosphere does emit thermal radiation
  • yes the surface of the earth does absorb atmospheric thermal radiation
  • yes this energy does not disappear (1st law of thermodynamics)
  • yes this energy must increase the temperature of the earth’s surface above what it would be if this radiation did not exist (1st law of thermodynamics)

Or even, which one of the above is wrong. That would be outstanding.

Of course, I know they won’t do that – even though I’m certain they believe all of the above points. (Likewise, Kramm & Dlugi won’t answer the question I have posed of them).

Well, we all know why

Hopefully, the illuminati can contact Kramm & Dlugi and explain to them where they went wrong. I have my doubts that any of the illuminati have grasped the first law of thermodynamics or the equation for temperature change and heat capacity, but who could say.

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Over the last few years I’ve written lots of articles relating to the inappropriately-named “greenhouse” effect and covered some topics in great depth. I’ve also seen lots of comments and questions which has helped me understand common confusion and misunderstandings.

This article, with huge apologies to regular long-suffering readers, covers familiar ground in simple terms. It’s a reference article. I’ve referenced other articles and series as places to go to understand a particular topic in more detail.

One of the challenges of writing a short simple explanation is it opens you up to the criticism of having omitted important technical details that you left out in order to keep it short. Remember this is the simple version..

Preamble

First of all, the “greenhouse” effect is not AGW. In maths, physics, engineering and other hard sciences, one block is built upon another block. AGW is built upon the “greenhouse” effect. If AGW is wrong, it doesn’t invalidate the greenhouse effect. If the greenhouse effect is wrong, it does invalidate AGW.

The greenhouse effect is built on very basic physics, proven for 100 years or so, that is not in any dispute in scientific circles. Fantasy climate blogs of course do dispute it.

Second, common experience of linearity in everyday life cause many people to question how a tiny proportion of “radiatively-active” molecules can have such a profound effect. Common experience is not a useful guide. Non-linearity is the norm in real science. Since the enlightenment at least, scientists have measured things rather than just assumed consequences based on everyday experience.

The Elements of the “Greenhouse” Effect

Atmospheric Absorption

1. The “radiatively-active” gases in the atmosphere:

  • water vapor
  • CO2
  • CH4
  • N2O
  • O3
  • and others

absorb radiation from the surface and transfer this energy via collision to the local atmosphere. Oxygen and nitrogen absorb such a tiny amount of terrestrial radiation that even though they constitute an overwhelming proportion of the atmosphere their radiative influence is insignificant (note 1).

How do we know all this? It’s basic spectroscopy, as detailed in exciting journals like the Journal of Quantitative Spectroscopy and Radiative Transfer over many decades. Shine radiation of a specific wavelength through a gas and measure the absorption. Simple stuff and irrefutable.

Atmospheric Emission

2. The “radiatively-active” gases in the atmosphere also emit radiation. Gases that absorb at a wavelength also emit at that wavelength. Gases that don’t absorb at that wavelength don’t emit at that wavelength. This is a consequence of Kirchhoff’s law.

The intensity of emission of radiation from a local portion of the atmosphere is set by the atmospheric emissivity and the temperature.

Convection

3. The transfer of heat within the troposphere is mostly by convection. The sun heats the surface of the earth through the (mostly) transparent atmosphere (note 2). The temperature profile, known as the “lapse rate”, is around 6K/km in the tropics. The lapse rate is principally determined by non-radiative factors – as a parcel of air ascends it expands into the lower pressure and cools during that expansion (note 3).

The important point is that the atmosphere is cooler the higher you go (within the troposphere).

Energy Balance

4. The overall energy in the climate system is determined by the absorbed solar radiation and the emitted radiation from the climate system. The absorbed solar radiation – globally annually averaged – is approximately 240 W/m² (note 4). Unsurprisingly, the emitted radiation from the climate system is also (globally annually averaged) approximately 240 W/m². Any change in this and the climate is cooling or warming.

Emission to Space

5. Most of the emission of radiation to space by the climate system is from the atmosphere, not from the surface of the earth. This is a key element of the “greenhouse” effect. The intensity of emission depends on the local atmosphere. So the temperature of the atmosphere from which the emission originates determines the amount of radiation.

If the place of emission of radiation – on average – moves upward for some reason then the intensity decreases. Why? Because it is cooler the higher up you go in the troposphere. Likewise, if the place of emission – on average – moves downward for some reason, then the intensity increases (note 5).

More GHGs

6. If we add more radiatively-active gases (like water vapor and CO2) then the atmosphere becomes more “opaque” to terrestrial radiation and the consequence is the emission to space from the atmosphere moves higher up (on average). Higher up is colder. See note 6.

So this reduces the intensity of emission of radiation, which reduces the outgoing radiation, which therefore adds energy into the climate system. And so the climate system warms (see note 7).

That’s it!

It’s as simple as that. The end.

A Few Common Questions

CO2 is Already Saturated

There are almost 315,000 individual absorption lines for CO2 recorded in the HITRAN database. Some absorption lines are stronger than others. At the strongest point of absorption – 14.98 μm (667.5 cm-1), 95% of radiation is absorbed in only 1m of the atmosphere (at standard temperature and pressure at the surface). That’s pretty impressive.

By contrast, from 570 – 600 cm-1 (16.7 – 17.5 μm) and 730 – 770 cm-1 (13.0 – 13.7 μm) the CO2 absorption through the atmosphere is nowhere near “saturated”. It’s more like 30% absorbed through a 1km path.

You can see the complexity of these results in many graphs in Atmospheric Radiation and the “Greenhouse” Effect – Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database, and see also Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book

The complete result combining absorption and emission is calculated in Visualizing Atmospheric Radiation – Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

CO2 Can’t Absorb Anything of Note Because it is Only .04% of the Atmosphere

See the point above. Many spectroscopy professionals have measured the absorptivity of CO2. It has a huge variability in absorption, but the most impressive is that 95% of 14.98 μm radiation is absorbed in just 1m. How can that happen? Are spectroscopy professionals charlatans? You need evidence, not incredulity. Science involves measuring things and this has definitely been done. See the HITRAN database.

Water Vapor Overwhelms CO2

This is an interesting point, although not correct when we consider energy balance for the climate. See Visualizing Atmospheric Radiation – Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed.

The key point behind all the detail is that the top of atmosphere radiation change (as CO2 changes) is the important one. The surface change (forcing) from increasing CO2 is not important, is definitely much weaker and is often insignificant. Surface radiation changes from CO2 will, in many cases, be overwhelmed by water vapor.

Water vapor does not overwhelm CO2 high up in the atmosphere because there is very little water vapor there – and the radiative effect of water vapor is dramatically impacted by its concentration, due to the “water vapor continuum”.

The Calculation of the “Greenhouse” Effect is based on “Average Surface Temperature” and there is No Such Thing

Simplified calculations of the “greenhouse” effect use some averages to make some points. They help to create a conceptual model.

Real calculations, using the equations of radiative transfer, don’t use an “average” surface temperature and don’t rely on a 33K “greenhouse” effect. Would the temperature decrease 33K if all of the GHGs were removed from the atmosphere? Almost certainly not. Because of feedbacks. We don’t know the effect of all of the feedbacks. But would the climate be colder? Definitely.

See The Rotational Effect - why the rotation of the earth has absolutely no effect on climate, or so a parody article explains..

The Second Law of Thermodynamics Prohibits the Greenhouse Effect, or so some Physicists Demonstrated..

See The Three Body Problem – a simple example with three bodies to demonstrate how a “with atmosphere” earth vs a “without atmosphere earth” will generate different equilibrium temperatures. Please review the entropy calculations and explain (you will be the first) where they are wrong or perhaps, or perhaps explain why entropy doesn’t matter (and revolutionize the field).

See Gerlich & Tscheuschner for the switch and bait routine by this operatic duo.

And see Kramm & Dlugi On Dodging the “Greenhouse” Bullet - Kramm & Dlugi demonstrate that the “greenhouse” effect doesn’t exist by writing a few words in a conclusion but carefully dodging the actual main point throughout their entire paper. However, they do recover Kepler’s laws and point out a few errors in a few websites. And note that one of the authors kindly showed up to comment on this article but never answered the important question asked of him. Probably just too busy.. Kramm & Dlugi also helpfully (unintentionally) explain that G&T were wrong, see Kramm & Dlugi On Illuminating the Confusion of the Unclear - Kramm & Dlugi step up as skeptics of the “greenhouse” effect, fans of Gerlich & Tscheuschner and yet clarify that colder atmospheric radiation is absorbed by the warmer earth..

And for more on that exciting subject, see Confusion over the Basics under the sub-heading The Second Law of Thermodynamics.

Feedbacks overwhelm the Greenhouse Effect

This is a totally different question. The “greenhouse” effect is the “greenhouse” effect. If the effect of more CO2 is totally countered by some feedback then that will be wonderful. But that is actually nothing to do with the “greenhouse” effect. It would be a consequence of increasing temperature.

As noted in the preamble, it is important to separate out the different building blocks in understanding climate.

Miskolczi proved that the Greenhouse Effect has no Effect

Miskolczi claimed that the greenhouse effect was true. He also claimed that more CO2 was balanced out by a corresponding decrease in water vapor. See the Miskolczi series for a tedious refutation of his paper that was based on imaginary laws of thermodynamics and questionable experimental evidence.

Once again, it is important to be able to separate out two ideas. Is the greenhouse effect false? Or is the greenhouse effect true but wiped out by a feedback?

If you don’t care, so long as you get the right result you will be in ‘good’ company (well, you will join an extremely large company of people). But this blog is about science. Not wishful thinking. Don’t mix the two up..

Convection “Short-Circuits” the Greenhouse Effect

Let’s assume that regardless of the amount of energy arriving at the earth’s surface, that the lapse rate stays constant and so the more heat arriving, the more heat leaves. That is, the temperature profile stays constant. (It’s a questionable assumption that also impacts the AGW question).

It doesn’t change the fact that with more GHGs, the radiation to space will be from a higher altitude. A higher altitude will be colder. Less radiation to space and so the climate warms – even with this “short-circuit”.

In a climate without convection, the surface temperature will start off higher, and the GHG effect from doubling CO2 will be higher. See Radiative Atmospheres with no Convection.

In summary, this isn’t an argument against the greenhouse effect, this is possibly an argument about feedbacks. The issue about feedbacks is a critical question in AGW, not a critical question for the “greenhouse” effect. Who can say whether the lapse rate will be constant in a warmer world?

Notes

Note 1 - An important exception is O2 absorbing solar radiation high up above the troposphere (lower atmosphere). But O2 does not absorb significant amounts of terrestrial radiation.

Note 2 – 99% of solar radiation has a wavelength <4μm. In these wavelengths, actually about 1/3 of solar radiation is absorbed in the atmosphere. By contrast, most of the terrestrial radiation, with a wavelength >4μm, is absorbed in the atmosphere.

Note 3 – see:

Density, Stability and Motion in Fluids - some basics about instability
Potential Temperature - explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate - lots more about the temperature profile in the atmosphere

Note 4 – see Earth’s Energy Budget - a series on the basics of the energy budget

Note 5 – the “place of emission” is a useful conceptual tool but in reality the emission of radiation takes place from everywhere between the surface and the stratosphere. See Visualizing Atmospheric Radiation – Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.

Also, take a look at the complete series: Visualizing Atmospheric Radiation.

Note 6 – the balance between emission and absorption are found in the equations of radiative transfer. These are derived from fundamental physics – see Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies. The fundamental physics is not just proven in the lab, spectral measurements at top of atmosphere and the surface match the calculated values using the radiative transfer equations – see Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.

Also, take a look at the complete series: Atmospheric Radiation and the “Greenhouse” Effect

Note 7 – this calculation is under the assumption of “all other things being equal”. Of course, in the real climate system, all other things are not equal. However, to understand an effect “pre-feedback” we need to separate it from the responses to the system.

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As a friend of mine in Florida says:

You can’t kill stupid, but you can dull it with a 4×2

Some ideas are so comically stupid that I thought there was no point writing about them. And yet, one after another, people who can type are putting forward these ideas on this blog.. At first I wondered if I was the object of a practical joke. Some kind of parody. Perhaps the joke is on me. But, just in case I was wrong about the practical joke..

 

If you pick up a textbook on heat transfer that includes a treatment of radiative heat transfer you find no mention of Arrhenius.

If you pick up a textbook on atmospheric physics none of the equations come from Arrhenius.

Yet there is a steady stream of entertaining “papers” which describe “where Arrhenius went wrong”, “Arrhenius and his debates with Fourier”. Who cares?

Likewise, if you study equations of motion in a rotating frame there is no discussion of where Newton went wrong, or where he got it right, or debates he got right or wrong with contemporaries. Who knows? Who cares?

History is fascinating. But if you want to study physics you can study it pretty well without reading about obscure debates between people who were in the formulation stages of the field.

Here are the building blocks of atmospheric radiation:

  • The emission of radiation – described by Nobel prize winner Max Planck’s equation and modified by the material property called emissivity (this is wavelength dependent)
  • The absorption of radiation by a surface – described by the material property called absorptivity (this is wavelength dependent and equal at the same wavelength and direction to emissivity)
  • The Beer-Lambert law of absorption of radiation by a gas
  • The spectral absorption characteristics of gases – currently contained in the HITRAN database – and based on work carried out over many decades and written up in journals like Journal of Quantitative Spectroscopy and Radiative Transfer
  • The theory of radiative transfer – the Schwarzschild equation – which was well documented by Nobel prize winner Subrahmanyan Chandrasekhar in his 1952 book Radiative Transfer (and by many physicists since)

The steady stream of stupidity will undoubtedly continue, but if you are interested in learning about science then you can rule out blogs that promote papers which earnestly explain “where Arrhenius went wrong”.

Hit them with a 4 by 2.

Or, ask the writer where Subrahmanyan Chandrasekhar went wrong in his 1952 work Radiative Transfer. Ask the writer where Richard M. Goody went wrong. He wrote the seminal Atmospheric Radiation: Theoretical Basis in 1964.

They won’t even know these books exist and will have never read them. These books contain equations that are thoroughly proven over the last 100 years. There is no debate about them in the world of physics. In the world of fantasy blogs, maybe.

There is also a steady stream of people who believe an idea yet more amazing. Somehow basic atmospheric physics is proven wrong because of the last 15 years of temperature history.

The idea seems to be:

More CO2 is believed to have some radiative effect in the climate because of the last 100 years of temperature history, climate scientists saw some link and tried to explain it using CO2, but now there has been no significant temperature increase for the last x years this obviously demonstrates the original idea was false..

If you think this, please go and find a piece of 4×2 and ask a friend to hit you across the forehead with it. Repeat. I can’t account for this level of stupidity but I have seen that it exists.

An alternative idea, that I will put forward, one that has evidence, is that scientists discovered that they can reliably predict:

  • emission of radiation from a surface
  • emission of radiation from a gas
  • absorption of radiation by a surface
  • absorption of radiation by a gas
  • how to add up, subtract, divide and multiply, raise numbers to the power of, and other ninja mathematics

The question I have for the people with these comical ideas:

Do you think that decades of spectroscopy professionals have just failed to measure absorption? Their experiments were some kind of farce? No one noticed they made up all the results?

Do you think Max Planck was wrong?

It is possible that climate is slightly complicated and temperature history relies upon more than one variable?

Did someone teach you that the absorption and emission of radiation was only “developed” by someone analyzing temperature vs CO2 since 1970 and not a single scientist thought to do any other measurements? Why did you believe them?

Bring out the 4×2.

Note – this article is a placeholder so I don’t have to bother typing out a subset of these points for the next entertaining commenter..

Update July 10th with the story of Fred the Charlatan

Let’s take the analogy of a small boat crossing the Atlantic.

Analogies don’t prove anything, they are for illustration. For proof, please review Theory and Experiment – Atmospheric Radiation.

We’ve done a few crossings and it’s taken 45 days, 42 days and 46 days (I have no idea what the right time is, I’m not a nautical person).

We measure the engine output – the torque of the propellors. We want to get across quicker. So Fred the engine guy makes a few adjustments and we remeasure the torque at 5% higher. We also do Fred’s standardized test, which is to zip across a local sheltered bay with no currents, no waves and no wind – the time taken for Fred’s standarized test is 4% faster. Nice.

So we all set out on our journey across the Atlantic. Winds, rain, waves, ocean currents. We have our books to read, Belgian beer and red wine and the time flies. Oh no, when we get to our final destination, it’s actually taken 47 days.

Clearly Fred is some kind of charlatan! No need to check his measurements or review the time across the bay. We didn’t make it across the Atlantic in less time and clearly the ONLY variable involved in that expedition was the output of the propellor.

Well, there’s no point trying to use more powerful engines to get across the Atlantic (or any ocean) faster. Torque has no relationship to speed. Case closed.

Analogy over.

 

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In previous articles in this series we looked at a number of issues first in Miskolczi’s 2010 paper and then in the 2007 paper.

The author himself has shown up and commented on some of these points, although not all, and sadly decided that we are not too bright and a little bit too critical and better pastures await him elsewhere.

Encouraged by one of our commenters I pressed on into the paper: Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service (2007), and now everything is a lot clearer.

The 2007 paper by Ferenc Miskolczi is a soufflé of confusion piled on confusion. Sorry to be blunt. If I was writing a paper I would say “..some clarity is needed in important sections..” but many readers unfamiliar with the actual meaning of this phrase might think that some clarity was needed in important sections rather than the real truth that the paper is a shambles.

I’ll refer to this paper as M2007. And to the equations in M2007 with an M prefix – so, for example, equation 15 will be [M15].

Some background is needed so first we need to take a look at something called The Semi-Gray Model. Regular readers will find a lot of this to be familiar ground, but it is necessary as there are always many new readers.

The SGM – Semi-Grey Model or Schwarzschild Grey Model

I’ll introduce this by quoting from an excellent paper referenced by M2007. This is a 1995 paper by Weaver and the great Ramanathan (free link in References):

Simple models of complex systems have great heuristic value, in that their results illustrate fundamental principles without being obscured by details. In particular, there exists a long history of simple climate models. Of these, radiative and radiative-convective equilibrium models have received great attention..

One of the simplest radiative equilibrium models involves the assumption of a so-called grey atmosphere, where the absorption coefficient is assumed to be independent of wavelength. This was first discussed by Schwarzschild [1906] in the context of stellar interiors. The grey gas model was adapted to studies of the Earth by assuming the atmosphere to be transparent to solar radiation and grey for thermal radiation. We will refer to this latter class as semigrey models.

And in the abstract they say:

Radiative equilibrium solutions are the starting point in our attempt to understand how the atmospheric composition governs the surface and atmospheric temperatures, and the greenhouse effect. The Schwarzschild analytical grey gas model (SGM) was the workhorse of such attempts. However, the solution suffered from serious deficiencies when applied to Earth’s atmosphere and were abandoned about 3 decades ago in favor of more sophisticated computer models..

[Emphasis added]

And they go on to present a slightly improved SGM as a useful illustrative tool.

Some clarity on a bit of terminology for new readers – a blackbody is a perfect emitter and absorber of radiation. In practice there are no blackbodies but some bodies come very close. A blackbody has an emissivity = 1 and absorptivity = 1.

In our atmosphere, the gases which absorb and emit significant radiation have very wavelength dependent properties, e.g.:

From spectralcalc.com

Figure 1

So the emissivity and absorptivity vary hugely from one wavelength to the next (note 1). However, as an educational tool, we can calculate the results for a grey atmosphere – this means that the emissivity is assumed to be constant across all wavelengths.

The term semi-grey means that the atmosphere is considered transparent for shortwave = solar wavelengths (<4 μm) and constant but not zero for longwave = terrestrial wavelengths (>4 μm).

Constructing the SGM

This model is very simple – and is not used to really calculate anything of significance for our climate. See Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations for the real equations.

We assume that the atmosphere is in radiative equilibrium - that is, convection does not exist and so only radiation moves heat around.

Here is a graphic showing the main elements of the model:

Figure 2

Once each layer in the atmosphere is in equilibrium, there is no heating or cooling – this is the definition of equilibrium. This means energy into the layer = energy out of the layer. So we can use simple calculus to write some equations of radiative transfer.

We define the TOA (top of atmosphere) to be where optical thickness, τ=0, and it increases through the atmosphere to reach a maximum of τ=τA at the surface. This is conventional.

We also know two boundary conditions, because at TOA (top of atmosphere) the downward longwave flux, F↓(τ=0) = 0 and the upwards longwave flux, F↑(τ=0) = F0, where F0 = absorbed solar radiation ≈ 240 W/m². This is because energy leaving the planet must be balanced by energy being absorbed by the planet from the sun.

We also have to consider the fact that energy is not just going directly up and down but is going up and down at every angle. We can deal with this via the dffusivity approximation which sums up the contributions from every angle and tells us that if we use τ*= τ . 5/3  (where τ is defined in the vertical direction) we get the complete contribution from all of the different directions. (Note 2). For those following M2007 I have used τ* to be his τ with a ˜ on top, and τ to be his τ with a ¯ on top.

With these conditions we can get a solution for the SGM (see derivation in the comments):

B(τ) = F0/2π . (τ+1)   [1]   cf eqn [M15]

where B is the spectrally integrated Planck function, and remember F0 is a constant.

And also:

F↑(τ) = F0/2 . (τ+2)    [2]

F↓(τ) = F0/2 . τ    [3]

A quick graphic might explain this a little more (with an arbitrary total optical thickness, τA* = 3):

Figure 3

Notice that the upward longwave flux at TOA is 240 W/m² – this balances the absorbed solar radiation. And the downward longwave flux at TOA is zero, because there is no atmosphere above from which to radiate. This graph also demonstrates that the difference between F↑ and F↓ is a constant as we move through the atmosphere, meaning that the heating rate is zero. The increase in downward flux, F↓, is matched by the decrease in upward flux, F↑.

It’s a very simple model.

By contrast, here are the heating/cooling rates from a comprehensive (= “standard”) radiative-convective model, plotted against height instead of optical thickness.

Heating from solar radiation, because the atmosphere is not completely transparent to solar radiation:

From Grant Petty (2006)

Figure 4

Cooling rates due to various “greenhouse” gases:

From Petty (2006)

Figure 5

And the heating and cooling rates won’t match up because convection moves heat from the surface up into the atmosphere.

Note that if we plotted the heating rate vs altitude for the SGM it would be a vertical line on 0.0°C/day.

Let’s take a look at the atmospheric temperature profile implied by the semi-grey model:

Figure 6

Now a lot of readers are probably wondering what the τ really means, or more specifically, what the graph looks like as a function of height in the atmosphere. In this model it very much depends on the concentration of the absorbing gas and its absorption coefficient. Remember it is a fictitious (or “idealized”) atmosphere. But if we assume that the gas is well-mixed (like CO2 for example, but totally unlike water vapor), and the fact that pressure increases with depth then we can produce a graph vs height:

Figure 7

Important note – the values chosen here are not intended to represent our climate system. 

Figure 6 & 7, along with figure 3, are just to help readers “see” what a semi-grey model looks like. If we increase the total optical depth of the atmosphere the atmospheric temperature at the surface increases.

Note as well that once the temperature reduction vs height is too large a value, the atmosphere will become unstable to convection. E.g. for a typical adiabatic lapse rate of 6.5 K/km, if the radiative equilibrium implies a lapse rate > 6.5 K/km then convection will move heat to reduce the lapse rate.

Curious Comments on the SGM

Some comments from M2007:

p 11:

Note, that in obtaining B0 , the fact of the semi-infinite integration domain over the optical depth in the formal solution is widely used. For finite or optically thin atmosphere Eq. (15) is not valid. In other words, this equation does not contain the necessary boundary condition parameters for the finite atmosphere problem.

The B0 he is referring to is the constant in [M15]. This constant is H/2π – where H = F0 (absorbed solar radiation) in my earlier notation. This constant B0 later takes on magical properties.

p 12:

Eq. (15) assumes that at the lower boundary the total flux optical depth is infinite. Therefore, in cases, where a significant amount of surface transmitted radiative flux is present in the OLR , Eqs. (16) and (17) are inherently incorrect. In stellar atmospheres, where, within a relatively short distance from the surface of a star the optical depth grows tremendously, this could be a reasonable assumption, and Eq. (15) has great practical value in astrophysical applications. The semi-infinite solution is useful, because there is no need to specify any explicit lower boundary temperature or radiative flux parameter (Eddington, 1916).

[Emphasis added]

The equations can easily be derived without any requirement for the total optical depth being infinite. There is no semi-infinite assumption in the derivation. Whether or not some early derivations included it, I cannot say. But you can find the SGM derivation in many introductions to atmospheric physics and no assumption of infinite optical thickness exists.

When considering the clear-sky greenhouse effect in the Earth’s atmosphere or in optically thin planetary atmospheres, Eq. (16) is physically meaningless, since we know that the OLR is dependent on the surface temperature, which conflicts with the semi-infinite assumption that τA =∞..

..There were several attempts to resolve the above deficiencies by developing simple semi-empirical spectral models, see for example Weaver and Ramanathan (1995), but the fundamental theoretical problem was never resolved..

This is the reason why scientists have problems with a mysterious surface temperature discontinuity and unphysical solutions, as in Lorenz and McKay (2003). To accommodate the finite flux optical depth of the atmosphere and the existence of the transmitted radiative flux from the surface, the proper equations must be derived.

The deficiencies noted include the result in the semi-gray model of a surface air temperature less than the ground temperature. If you read Weaver and Ramanathan (1995) you can see that this isn’t an attempt to solve some “fundamental problem“, but simply an attempt to make a simple model slightly more useful without getting too complex.

The mysterious surface temperature discontinuity exists because the model is not “full bottle”. The model does not include any convection. This discontinuity is not a mystery and is not crying out for a solution. The solution exists. It is called the radiative-convective model and has been around for over 40 years.

Miskolczi makes some further comments on this, which I encourage people to read in the actual paper.

We now move into Appendix B to develop the equations further. The results from the appendix are the equations M20 and M21 on page 14.

Making Equation Soufflé

The highlighted equation is the general solution to the Schwzarschild equation. It is developed in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations - the equation reproduced here from that article with explanation:

Iλ(0) = Iλm)em + ∫ Bλ(T)e 

The intensity at the top of atmosphere equals..

The surface radiation attenuated by the transmittance of the atmosphere, plus..

The sum of all the contributions of atmospheric radiation – each contribution attenuated by the transmittance from that location to the top of atmosphere

For those wanting to understand the maths a little bit, the 3/2 factor that appears everywhere in Miskolczi’s equation B1 is the diffusivity approximation already mentioned (and see note 2) where we need to sum the radiances over all directions to get flux.

Now this equation is the complete equation of radiative transfer. If we combine it with a simple convective model it is very effective at calculating the flux and spectral intensity through the atmosphere – see Theory and Experiment – Atmospheric Radiation.

So equation B1 in M2007 cannot be solved analytically. This means we have to solve it numerically. This is “simple” in concept but computationally expensive because in the HITRAN database there are 2.7 million individual absorption lines, each one with a different absorption coefficient and a different line width.

However, it can be solved and that is what everyone does. Once you have the database of absorption coefficients and the temperature profile of the atmosphere you can calculate the solution. And band models exist to speed up the process.

And now the rabbit..

The author now takes the equation for the “source function” (B) from the simple model and inserts it into the “complete” solution.

The “source function” in the complete solution can already be calculated – that’s the whole point of the equation B1. But now instead, the source function from the simple model is inserted in its place. This equation assumes that the atmosphere has no convection, has no variation in emissivity with wavelength, has no solar absorption in the atmosphere, and where the heating rate at each level in the atmosphere = zero.

The origin of equation B3 is the equation you see above it:

B(τ) = 3H(τ)/4π + B0   [M13]

Actually, if you check equation M13 on p.11 it is:

B(τ) = 3H.τ/4π + B0   [M13]

This appears to be one of the sources of confusion for Miskolczi, for later comment.

Equation M13 is derived for zero heating rates throughout the atmosphere, and therefore constant H. With this simple assumption – and only for this simple assumption – the equation M13 is a valid solution to “the source function”, ie the atmospheric temperature and radiance.

If you have the complete solution you get one result. If you have the simple model you get a different result. If you take the result from one and stick it in the other how can you expect to get an accurate outcome?

If you want to see how various atmospheric factors are affected by changing τ, then just change τ in the general equation and see what happens. You have to do this via numerical analysis but it can easily be done..

As we continue on working through the appendix, B6 has a sign error in the 2nd term on the right hand side, which is fixed by B7.

This B0 is the constant in the semi-gray solution. The constant appears because we had a differential equation that we integrated. And the value of the constant was obtained via the boundary condition: upward flux from the climate system must balance solar radiation.

So we know what B0 is.. and we know it is a constant..

Yet now the author differentiates the constant with respect to τ. If you differentiate a constant it is always zero. Yet the explanation is something that sounds like it might be thermodynamics, but isn’t:

If someone wants to explain what thermodynamic principle create the first statement – I would be delighted. Without any explanation it is a jumble of words that doesn’t represent any thermodynamic principle.

Anyway B0 is a constant and is equal to approximately 240 W/m². Therefore, if we differentiate it, yes the value dB0/dτ=0.

Unfortunately, the result in B10 is wrong.

If we differentiate a variable we can’t assume it is a constant. The variable in question is BG. This is the “source function” for the ground, which gives us the radiance and surface temperature. Clearly the surface temperature is a function of many factors especially including optical thickness. Of course, if somewhere else we have proven that BG is a constant then dBG/dτ=0.

It has to be proven.

[And thanks to DeWitt Payne for originally highlighting this issue with BG, as well as explaining my calculus mistakes in an email].

A quick digression on basic calculus for the many readers who don’t like maths – just so you can see what I am getting at.. (you are the ones who should read it)

Digression

We will consider just the last term in equation [B9]. This term = BG/(eτ-1). I have dropped the π from the term to make it simpler to read what is happening.

Generally, if you differentiate two terms multiplied together, this is what happens:

d(fg)/dx = g.df/dx + f.dg/dx   [4]

This assumes that f and g are both functions of x. If, for example, f is not a function of x, then df/dx=0 (this just means that f does not change as x changes). And so the result reduces to d(fg)/dx = f.dg/dx.

So, using [4] :

d/dτ [BG/(eτ-1)] = [1/(eτ-1)] . dBG/dτ + BG . d [1/(eτ-1)]/dτ  [5]

We can look up:

d [1/(eτ-1)]/dτ = -eτ/(eτ-1)²  [6]

So substituting [6] into [5], last term in [B9]:

= [1/(eτ-1)] . dBG/dτ – eτ.BG /(eτ-1)²   [7]

You can see the 2nd half of this expression as the first term in [B10], multiplied by π of course.

But the term for how the surface radiance changes with optical thickness of the atmosphere has vanished.

end of digression

Soufflé Continued

So the equation should read:

Where the red text is my correction (see eqn 7 in the digression).

Perhaps the idea is that if we assume that surface temperature doesn’t change with optical thickness then we can prove that surface temperature doesn’t change with optical thickness.

This (flawed) equation is now used to prove B11:

Well, we can see that B11 isn’t true. In fact, even apart from the missing term in B10, the equation has been derived by combining two equations which were derived under different conditions.

As we head back into the body of the paper from the appendix, equations B7 and B8 are rewritten as equations [M20] and [M21].

Miskolczi adds:

We could not find any references to the above equations in the meteorological literature or in basic astrophysical monographs, however, the importance of this equation is obvious and its application in modeling the greenhouse effect in planetary atmospheres may have far reaching consequences.

Readers who have made it this far might realize why he is the first with this derivation.

Continuing on, more statements are made which reveal some of the author’s confusion with one part of his derivation. The SGM model is derived by integrating a simple differential equation, which produces a constant. The boundary conditions tell us the constant.

Equation [M13] is written:

B(τ) = 3H/4π + B0   [M13]

Then [M14] is written:

H(τ) = π (I+ – I-)    [M14]

So now H is a function of optical depth in the atmosphere?

In [M15]:

B(τ*) = H (1 + τ*)/2π    [M15]

Refer to my equation 1 and you will see they are the same. The only way this equation can be derived is with H as a constant, because the atmosphere is in radiative equilibrium. If H isn’t constant you have a different equation – M13 and 15 are no longer valid.

..The fact that the new B0 (skin temperature) changes with the surface temperature and total optical depth, can seriously alter the convective flux estimates of previous radiative-convective model computations. Mathematical details on obtaining equations 20 and 21 are summarized in appendix B.

Miskolczi has confused himself (and his readers).

Conclusion

There is an equation of radiative transfer and it is equation B1 in the appendix of M2007. This equation is successfully used to calculate flux and spectral intensity in the atmosphere.

There is a very simple equation of radiative transfer which is used to illustrate the subject at a basic level and it is called the semi-grey model (or the Schwarzschild grey model). With the last few decades of ever increasing computing power the simple models have less and less practical use, although they still have educational value.

Miskolczi has inserted the simple result into the general model, which means, at best, it can only be applied to a “grey” atmosphere in radiative equilibrium, and at worst he has just created an equation soufflé.

The constant in the simple model has become a variable. Without any proof, or re-derivation of the simple model.

One of the important variables in the simple model has become a constant and therefore vanished from an equation where it should still reside.

Many flawed thermodynamic concepts are presented in the paper, some of which we have already seen in earlier articles.

M2007 tells us that Ed=Aa due to Kirchhoff’s law. (See Part Two). His 2010 paper revised this claim as to due to Prevost.. However, the author himself recently stated:

I think I was the first who showed the Aa~=Ed relationship with reasonable quantitative accuracy.

And doesn’t understand why I think it is important to differentiate between fundamental thermodynamic identities and approximate experimental results in the theory section of a paper. “My experiments back up my experiments..”

M2007 introduces equation [M7] with:

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

Note the pseudo-thermodynamic explanation. The author himself recently said:

Eq. 7 simply states, that the sum of the Su-OLR and Ed-Eu terms – in ideal greenhause case – must be equal to Fo. I assume that the complex dynamics of the system may support this assumption, and will explain the Su=3OLR/2 (global average) observed relationship.

[Emphasis added]

And later entertainingly commented:

You are right, I should have told that, and in my new article I shall pay more attantion to the full explanations. However, some scientists figured it out without any problem.

Party people who got the joke right off the bat..

M07 also introduces the idea that kinetic energy can be equated with the flux from the atmosphere to space. See Part Three. Introductory books on statistical thermodynamics tell us that flux is proportional to the 4th power of temperature, while kinetic energy is linearly proportional to temperature. We have no comment from the author on this basic contradiction.

This pattern indicates an obvious problem.

In summary – this paper does not contain a theory. Just because someone writes lots of equations down in attempt to back up some experimental work, it is not theory.

If the author has some experimental work and no theory, that is what he should present – look what I have found, I have a few ideas but can someone help develop a theory to explain these results.

Obviously the author believes he does have a theory. But it’s just equation soufflé.

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 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.

Further reading:

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.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi, Quarterly Journal of the Hungarian Meteorological Service (2007)

Deductions from a simple climate model: factors governing surface temperature and atmospheric thermal structure, Weaver & Ramanathan, JGR (1995)

Notes

Note 1 – emissivity = absorptivity for the same wavelength or range of wavelengths

Note 2 – this diffusivity approximation is explained further in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. In M2007 he uses a different factor, τ* = τ . 3/2 – this differences are not large but they exist. The problems in M2007 are so great that finding the changes that result from using different values of τ* is not really interesting.

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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:

From Miskolczi (2007)

Figure 1

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)   [1]

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>     [2]

where m = mass of a molecule, <vx²> = average of vx², where vx = velocity in the x direction

Combining [1] and [2] we get:

kT = m<vx²>, or

m<vx²>/2 = kT/2     [3]

The same considerations apply to the y and z direction, so

m<v²>/2 = 3KT/2      [4]

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   [5]

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.

On p6:

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.

So:

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   [10]   -  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    [11]

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)   [12]

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      [13] – 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 [10] and [11].

We can say that energy into the climate system = energy out, therefore:

F0 = OLR = EU + ST    [14]   (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.

Conclusion

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.

Further Reading:

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.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi, Quarterly Journal of the Hungarian Meteorological Service (2007)

Notes

Note 1 – A good paper on the virial theorem is on arXivThe Virial Theorem and Planetary Atmospheres, Victor Toth (2010)

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In Part One we looked at the usefulness of “tau” = optical thickness of the atmosphere.

Miskolczi  has done a calculation (under cloudless skies) of the total optical thickness of the atmosphere. The reason he is apparently the first to have done this in a paper is explained in Part One.

The 2010 paper referenced the 2007 paper, Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service.

The 2010 paper suggested an elementary flaw, but referenced the 2007 paper. The 2007 paper backed up the approach with the same apparently flawed claim.

The flaw that I will explain doesn’t affect the calculation of optical thickness, τ. But it does appear to affect the theoretical basis for why optical thickness should be a constant.

First, the graphic explaining the terms is here:

From Miskolczi (2007)

Figure 1

The 2010 paper said:

One of the first and most interesting discoveries was the relationship between the absorbed surface radiation and the downward atmospheric emittance. According to Ref. 4, for each radiosonde ascent the
ED = AA = SU – ST = SU(1− exp(−τA)) = SU(1− TA ) = SU.A             (5)
relationships are closely satisfied. The concept of radiative exchange was the discovery of Prevost [17]. It will be convenient here to define the term radiative exchange equilibrium between two specified regions of space (or bodies) as meaning that for the two regions (or bodies) A and B, the rate of flow of radiation emitted by A and absorbed by B is equal to the rate of flow the other way, regardless of other forms of transport that may be occurring.

Ref. 4 is the 2007 paper, which said:

According to the Kirchhoff law, two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts, therefore, the thermal energy of either system can not be changed. In case the atmosphere is in thermal equilibrium with the surface, we may write that..

What is “thermal equilibrium“?

It is when two bodies are in a closed system and have reached equilibrium. This means they are at the same temperature and no radiation can enter or leave the system. In this condition, energy emitted from body A and absorbed by body B = energy emitted from body B and absorbed by body A.

Kirchhoff showed this radiative exchange must be equal under the restrictive condition of thermal equilibrium. And he didn’t show it for any other condition. (Note 2).

However, the earth’s surface and the atmosphere are not in thermal equilibrium. And, therefore, energy exchanged between the surface and the atmosphere via radiation is not proven to be equal.

Dr. Roy Spencer has a good explanation of the fallacy and the real situation on his blog. One alleged Miskolczi  supporter took him to task for misinterpreting something – here:

With respect, Dr Spencer, it is not reasonable, indeed it verges on the mischievous, to write an allegation that Miskolczi means that radiative exchange is independent of temperature. Miskolczi means no such thing. To make such an allegation is to ignore the fact that Miskolczi uses the proper laws of physics in his calculations. Of course radiative exchange depends on temperature, and of course Miskolczi is fully aware of that.

and here:

..Planck uses the term for a system in thermodynamic equilibrium, and the present system is far from thermodynamic equilibrium, but the definition of the term still carries over..

I couldn’t tell whether the claimed “misinterpretation” by Spencer was of the real law or the Miskolczi interpretation. And this article will demonstrate that the proper laws of physics have been ignored.

And I have no idea whether the Miskolczi supporter represented the real Miskolczi. However, a person of the same name is noted by Miskolczi for his valuable comments in producing the 2010 paper.

Generally when people claim to overturn decades of research in a field you expect them to take a bit of time to explain why everyone else got it wrong, but apparently Dr. Spencer was deliberately misinterpreting something.. and that “something” is very clear only to Miskolczi supporters.

After all, the premise in the referenced 2007 paper was:

According to the Kirchhoff law, two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts, therefore, the thermal energy of either system can not be changed. In case the atmosphere is in thermal equilibrium with the surface, we may write that..

Emphasis added.

So if the atmosphere is not in thermal equilibrium with the surface, we can’t write the above statement.

And as a result the whole paper falls down. Perhaps there are other gems which stand independently of this flaw and I look forward to a future paper from the author when he explains some new insights which don’t rely on thermodynamic equilibrium being applied to a world without thermodynamic equilibrium.

Thermodynamic Equilibrium and the Second Law of Thermodynamics

If you put two bodies, A & B, at two different temperatures, TA and TB, into a closed system then over time they will reach the same temperature.

Let’s suppose that TA > TB. Therefore, A will radiate more energy towards B than the reverse. These bodies will reach equilibrium when TA = TB (note 1).

At this time, and not before, we can say that ” ..two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts”. (Note 2).

Obviously, before equilibrium is reached more energy is flowing from A to B than the reverse.

Non-Equilibrium

Let’s consider a case like the sun and the earth. The earth absorbs around 240 W/m² from the sun. The sun absorbs a lot less from the earth.

Let’s just say it is a lot less than 1 W/m². Someone with a calculator and a few minutes spare can do the sums and write the result in the comments.

No one (including of course the author of the paper) would suggest that the sun and earth exchange equal amounts of radiation.

However, they are in the condition of “radiative exchange”.

The Earth’s Surface and the Atmosphere

The earth’s surface and the bottom of the atmosphere are at similar temperatures. Why is this?

It is temperature difference that drives heat flow. The larger the temperature difference the greater the heat flow (all other things remaining equal). So any closed system tends towards thermal equilibrium. If the earth and the atmosphere were left in a closed system, eventually both would be at the same temperature.

However, in the real world where the climate system is open to radiation, the sun is the source of energy that prevents thermal equilibrium being reached.

The bottom millimeter of the atmosphere will usually be at the same temperature as the earth’s surface directly below. If the bottom millimeter is stationary then it will be warmed by conduction until it reaches almost the surface temperature. But 10 meters up the temperature will probably reduce just a little. At 1 km above the surface the temperature will be between 4 K and 10 K cooler than the surface.

Note: Turbulent heat exchange near the surface is very complex. This doesn’t mean that there is confusion about the average temperature profile vs height through the atmosphere. On average, temperature reduces with height in a reasonably predictable manner.

Energy Exchanges between the Earth’s Surface and the Atmosphere

According to Miskolczi:

AA = ED   [4]

Referring to the diagram, AA is energy absorbed by the atmosphere from the surface, and ED is energy radiated from the atmosphere to the surface.

Why should this equality hold?

The energy from the surface to the atmosphere = AA+ K (note 3), where K is convection.

The energy absorbed in total by the atmosphere = AA + K + F, where F is absorbed solar radiation in the atmosphere.

The energy emitted by the atmosphere = ED + EU , where EU is the energy radiated from the top of the atmosphere.

Therefore, using the First Law of Thermodynamics for the atmosphere:

AA + K + F = ED + EU + energy retained

i.e., energy absorbed = energy lost – energy retained

No other equality relating to the atmospheric fluxes can be deduced from the fundamental laws of thermodynamics.

In general, because the atmosphere and the earth’s surface are very close in temperature, AA will be very close to ED.

It is important to understand that absorptivity for longwave radiation will be equal to emissivity for longwave radiation (see Planck, Stefan-Boltzmann, Kirchhoff and LTE), therefore, if the surface and the atmosphere are at the same temperature then the exchange of radiation will be equal.

Where does the atmosphere radiate from, on average? Well, not from the bottom meter. It depends on the emissivity of the atmosphere. This varies with the amount of water vapor in the atmosphere.

The atmospheric temperature reduces with height- by an average of around 6.5 K/km – and unless the atmospheric radiation was from the bottom few meters, the radiation from the atmosphere to the surface must be lower than the radiation absorbed from the surface by the atmosphere.

If radiation was emitted from an average of 100 m above the surface then the effective temperature of atmospheric radiation would be 0.7 K below the surface temperature. If radiation was emitted from an average of 200 m above the surface then the effective temperature of atmospheric radiation would be 1.3 K below the surface temperature.

Mathematical Proof

For people still thinking about this subject, a simple mathematical proof.

Temperature of the atmosphere, from the average height of emission, Ta

Temperature of the surface, Ts

Emissivity of the atmosphere = εa

Absorptivity of the atmosphere for surface radiation = αa

If Ta is similar to Ts then εa ≈ αa (note 4).

(In the paper, the emissivity (and therefore absorptivity) of the earth’s surface is assumed = 1).

Surface radiation absorbed by the atmosphere, AA = αaσTs4 .

Atmospheric radiation absorbed by the surface, ED = εaσTa4 .

Therefore, unless Ta = Ts, AA ≠ ED .

If Roy Spencer’s experience is anything to go by, I may now be accused of deliberately misunderstanding something.

Well, words can be confused – even though they seem plain enough in the extract shown. But the paper also asserts the mathematical identity:

AA = ED   [4]

I have demonstrated that:

AA ≠ ED   [4]

I don’t think there is much to be misunderstood.

Two bodies at different temperatures will NOT exchange exactly equal amounts of radiation. It is impossible unless the current laws of thermodynamics are wrong.

As a more technical side note.. because εa ≈ αa and not necessarily an exact equality, it is possible for the proposed equation to be asserted in the following way:

AA = ED if, and only if, the following identity is always true, αa(Ts)σTs4 = εa(Ta)σTa4 .

Therefore:

Ts/Ta = (εa(Ta)/αa(Ts))1/4  [Equation B]

- must always be true for equation 4 of Miskolczi (2007) to be correct. Or must be true over whatever time period and surface area his identity is claimed to be true.

Another quote from the 2007 paper:

The popular explanation of the greenhouse effect as the result of the LW atmospheric absorption of the surface radiation and the surface heating by the atmospheric downward radiation is incorrect, since the involved flux terms (AA and ED) are always equal.

Emphasis added.

Note in Equation B that I have made explicit the dependence of emissivity on the temperature of the atmosphere at that time, and the dependence of absorptivity on the temperature of the surface.

Emissivity vs wavelength is a material property and doesn’t change with temperature. But because the emission wavelengths change with temperature the calculation of εa(Ta) is the measured value of εa at each wavelength weighted by the Planck function at Ta.

It is left as an exercise for the interested student to prove that this identity, Equation B, cannot always be correct.

The “Almost” Identity

In Fig. 2 we present large scale simulation results of AA and ED for two measured diverse planetary atmospheric profile sets. Details of the simulation exercise above were reported in Miskolczi and Mlynczak (2004). This figure is a proof that the Kirchhoff law is in effect in real atmospheres. The direct consequences of the Kirchhoff law are the next two equations:

EU = F + K + P (5)
SU − (F0 + P0 ) = ED − EU (6)

The physical interpretations of these two equations may fundamentally change the general concept of greenhouse theories.

From Miskolczi (2007)

Figure 2

This is not a proof of Kirchhoff’s law, which is already proven and is not a law that radiative exchanges are equal when temperatures are not equal.

Instead, this is a demonstration that the atmosphere and earth’s surface are very close in temperature.

Here is a simple calculation of the ratio of AA:ED for different downward emitting heights (note 5), and lapse rates (temperature profile of the atmosphere):

Figure 3

Essentially this graph is calculated from the formula in the maths section and a calculation of the atmospheric temperature, Ta, from the height of average downward radiation and the lapse rate.

Oh No He’s Not Claiming This is Based on Kirchoff..

Reading the claims by the supporters of Miskolczi at Roy Spencer’s blog, you read that:

  1. Miskolczi is not claiming that AA = ED by asserting (incorrectly) Kirchhoff’s law
  2. Miskolczi is claiming that AA = ED by experimental fact

So the supporters claim.

Read the paper, that’s my recommendation. The 2010 paper references the 2007 paper for equation 4. The 2007 paper says (see larger citation above):

..This figure is a proof that the Kirchhoff law is in effect in real atmospheres..

In fact, this is the important point:

Anyone who didn’t believe that it was a necessary consequence of Kirchhoff would be writing the equations in the maths section above (which come from well-proven radiation theory) and realizing that it is impossible for AA = ED.

And they wouldn’t be claiming that it demonstrated Kirchhoff’s law. (After all, Kirchhoff’s law is well-proven and foundational thermodynamics).

However, it is certain that on average ED < AA but very close to AA.

Hence the Atmospheric Window Cooling to Space Thing

From time to time, Miskolczi fans have appeared on this blog and written interesting comments. Why the continued fascination with the exact amount of radiation transmitted from the surface through the atmospheric window?

I have no idea whether this point is of interest to anyone else..

One of the comments highlighted the particular claim and intrigued me.

Yes, indeed, that’s right: Simpson discovered the atmospheric window in 1928. It was not till the work of Miskolczi in 2004 and 2007 that it was discovered that practically all the radiative cooling of the land-sea surface is by radiation direct to space.

Apart from the (unintentional?) humor inherent in the Messianic-style claim, the reason why this claim is a foundational point for Miskolczi-ism  is now clear to me.

If exactly all of the radiation absorbed by the atmosphere is re-radiated to the surface and absorbed by the surface (AA = ED) then these points follow for certain:

  1. radiation emitted by the atmosphere to space = convective heat from the surface into the atmosphere + solar radiation absorbed by the atmosphere
  2. total radiative cooling to space = radiation transmitted through the atmospheric window + convective heat plus solar radiation absorbed by the atmosphere

A curiosity only.

Changing the Fundamental View of the World

Miskolczi claims:

The physical interpretations of these two equations may fundamentally change the general concept of greenhouse theories.

He is being too modest.

If it turns out that AA = ED then it will overturn general radiative theory as well.

Or demonstrate that the atmosphere is much more opaque than has currently been calculated (for all of the downward atmospheric radiation to take place from within a few tens of meters of the surface).

This in turn will require the overturning of some parts of general radiative theory, or at least, a few decades of spectroscopic experiments, which consequently will surely require the overturning of..

Conclusion

How is it possible to claim that AA = ED and not work through the basic consequences (e.g., the equations in the maths section above) to deal with the inevitable questions on thermodynamics basics?

Why claim that it has fundamentally changed the the general concept of the inappropriately-named “greenhouse” theory when it – if true – has overturned generally accepted radiation theory?

  • Perhaps α(λ) ≠ ε(λ) and Kirchhoff’s law is wrong? This is a possible consequence. (In words, the equation says that absorptivity at wavelength λ is not equal to emissivity at wavelength λ, see note 4).
  • Or perhaps the well-proven Stefan-Boltzmann law is wrong? This is another possible consequence.

Interested observers might wonder about the size of the error bars in Figure 2. (And for newcomers, the values in Figure 2 are not measured values of radiation, they are calculated absorption and emission).

As already suggested, perhaps there are useful gems somewhere in the 40 pages of the 2007 paper, but when someone is so clear about a foundational point for their paper that is so at odds with foundational thermodynamic theory and the author doesn’t think to deal with that.. well, it doesn’t generate hope.

Update 31st May – the author comments in the ensuing discussion that Aa=Ed is an “experimental” conclusion. In Part Four I show that the “approximate equality” must be an error for real (non-black) surfaces, and Ken Gregory, armed with the Miskolczi spreadsheet, later confirms this.

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 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

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.

Further Reading:

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.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi , Quarterly Journal of the Hungarian Meteorological Service (2007)

The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness, Miskolczi, Energy & Environment(2010)

The Theory of Heat Radiation, Max Planck, P. Blakiston’s Son & Co (1914) : a translation of Waermestrahlung (1913) by Max Planck.

Notes

Note 1 – Of course, in reality equilibrium is never actually reached. As the two temperatures approach each other, the difference in energy exchanged is continually reduced. However, at some point the two temperatures will be indistinguishable. Perhaps when the temperature difference is less than 0.1°C, or when it is less than 0.0000001°C..

Therefore, it is conventional to talk about “reaching equilibrium” and no one in thermodynamics is confused about the reality of the above point.

Note 2 – Max Planck introduces thermodynamic equilibrium:

Note 3 - Geothermal energy is included in the diagram (P0). Given that it is less than 0.1 W/m² – below the noise level of most instruments measuring other fluxes in the climate – there is little point in cluttering up the equations here with this parameter.

Note 4 - Emissivity and absorptivity are wavelength dependent parameters. For example, snow is highly reflective for solar radiation but highly absorbing (and therefore emitting) for terrestrial radiation.

At the same wavelength, emissivity = absorptivity. This is the result of Kirchhoff’s law.

If the temperature of the source radiation for which we need to know the absorptivity is different from the temperature of the emitting body then we cannot assume that emissivity = absorptivity.

However, when the temperature of source body for the radiation being absorbed is within a few Kelvin of the emitting body then to a quite accurate assumption, absorptivity = emissivity.

For example, the radiation from a source of 288K is centered on 10.06 μm, while for 287 K it is centered on 10.10 μm. Around this temperature, the central wavelength decreases by about 0.035 μm for each 1 K change in temperature.

An example of when it is a totally incorrect assumption is for solar radiation absorbed by the earth. The solar radiation is from a source of about 5800 K and centered on 0.5 μm, whereas the terrestrial radiation is from a source of around 288 K and centered on 10 μm. Therefore, to assume that the absorptivity of the earth’s surface for solar radiation is equal to the emissivity of the earth’s surface is a huge mistake.

This would be the same as saying that absorptivity at 0.5 μm = emissivity at 10 μm. And, therefore, totally wrong.

Note 5: What exactly is meant by average emitting height? Emitted radiation varies as the 4th power of temperature and as a function of emissivity, which itself is a very non-linear function of quantity of absorbers. Average emitting height is more of a conceptual approach to illustrate the problem.

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With apologies to my many readers who understand the basics of heat transfer in the atmosphere and really want to hear more about feedback, uncertainty, real science..

Clearing up basic misconceptions is also necessary. It turns out that many people read this blog and comment on it elsewhere and a common claim about climate science generally (and about this site) is that climate science (and this site) doesn’t understand/ignores convection.

The Anti-World Where Convection Is Misunderstood

Suppose – for a minute – that convection was a totally misunderstood subject. Suppose basic results from convective heat transfer were ridiculed and many dodgy papers were written that claimed that convection moved 1/10 of the heat from the surface or 100x the heat from the surface. Suppose as well that everyone was pretty much “on the money” on radiation because it was taught from kindergarten up.

It would be a strange world – although no stranger than the one we live in where many champions of convection decry the sad state of climate science because it ignores convection, and anyway doesn’t understand radiation..

In this strange world, people like myself would open up shop writing about convection, picking up on misconceptions from readers and other blogs, and generally trying to explain what convection was all about.

No doubt, in that strange world, commenters and bloggers would decry the resulting over-emphasis on convection..

First Misconception – Radiation Results are All Wrong Because Convection Dominates

There are three mechanisms of heat transfer:

  • radiation
  • convection
  • conduction

Often in climate science, people add:

  • latent heat

In more general heat transfer this last one is often included within convection, which is the movement of heat by mass transfer. Although sometimes in general heat transfer, heat transfer via “phase change” of a substance is separately treated – it’s not important where “the lines are drawn”.

Update note from Dec 9th – I leave my poorly worded introduction above so that readers comments make sense. But I should have written

In fact in atmospheric physics we almost always see the breakdown like this:

-radiation
-latent heat
-sensible heat

Latent heat being the movement of heat via evaporation – convection – condensation. Sensible heat being the movement of heat via convection with no phase change. Conduction is actually also included in sensible heat, but is negligible in atmospheric physics.

Therefore when convection is written about it is both sensible and latent heat. That is, heat transfer in the atmosphere is via either convection or via radiation.

End of update note

Let’s look at conduction, safe from criticism because it is largely irrelevant as a form of heat transfer within the atmosphere. Conduction is also the easiest to understand and closest to people’s everyday knowledge.

The basic equation of heat conduction is:

q =- kA . ΔT/Δx

where ΔT is the temperature difference, Δx is the thickness of the material, A is the area, k is the conductivity (the property of the material) and q is the heat flow. (See Heat Transfer Basics – Part Zero for more on this subject).

Notice the terms in this equation:

  • the material property (k)
  • the thickness of the material (Δx)
  • the temperature difference (ΔT)
  • the area (A)

Where are the convective and the radiative terms?

Interestingly, conduction is independent of convection and radiation. This is a very important point to understand – but it is also easy to misunderstand if you aren’t used to this concept.

It doesn’t mean that we can calculate a change in equilibrium condition – or a dynamic result – only using one mechanism of heat transfer.

Let’s suppose we have a problem where we know the temperature at time = 0 for two surfaces. We know the heating conditions at both surfaces (for example,  zero heat input). We want to know how the temperature changes with time, and we want to know the final equilibrium condition.

The way this problem is solved is usually numerical. This means that we have to work out the heat flow from each mechanism (conduction, convection, radiation) for a small time step, calculate the resulting change in temperature, and then go through the next time step using the new temperatures.

For many people, this is probably a fuzzy concept and, unfortunately, I can’t think of an easy analogy that will crystallize it.

But what it means in simple terms is that each heat transfer mechanism works independently, but each affects the other mechanisms via the temperature change (if I come up with a useful analogy or example, I will post it as a comment).

So if, for example, convection has changed the temperature profile of the atmosphere to something that would not happen without convection – the calculation of conduction through the atmosphere is still:

q = -kA . ΔT/Δx

And likewise the more complex equation of radiative transfer (see Theory and Experiment – Atmospheric Radiation) will also rely on the temperature profile established from convection.

So – an ocean surface with an emissivity of 0.99 and a surface temperature of 15°C will still radiate 386 W/m², regardless of whether the convection + latent heat term = zero or 10 W/m² or 100 W/m² or 500 W/m².

Second Misconception – Atmospheric Physics Ignores Convection

This is a common claim. It’s simple to demonstrate that the claim is not true.

Let’s take a look at a few atmospheric physics text books.

From Elementary Climate Physics, Prof F.W. Taylor, Oxford University Press (2005):

Extract from Elementary Climate Physics, F.W. Taylor (2005)

Extract from Elementary Climate Physics, F.W. Taylor (2005)

From Handbook of Atmospheric Science, Hewitt & Jackson (2003):

From Handbook of Atmospheric Science, Hewitt and Jackson (2003)

From Handbook of Atmospheric Science, Hewitt and Jackson (2003)

From An introduction to atmospheric physics, David Andrews, Cambridge University Press (2000):

Davies, Atmospheric Physics

David Andrews, Atmospheric Physics (2000)

In fact, you will find some kind of derivation like this in almost every atmospheric physics textbook.

Also note that it is nothing new –  from Atmospheres, by R.M. Goody & J.C.G. Walker (1972):

Both convection and radiation are important in heat transfer in the troposphere.

Lindzen (1990) said:

The surface of the earth does not cool primarily by infrared radiation. It cools mainly through evaporation. Most of the evaporated moisture ends up in convective clouds.. where the moisture condenses into rain..

..It is worth noting that, in the absence of convection, pure greenhouse warming would lead to a globally averaged surface temperature of 72°C given current conditions

Note the important point that convection acts to reduce the surface temperature. If radiation was the dominant mechanism for heat transfer the surface temperature would be much higher.

Convection lowers the surface temperature. However, it only acts to reduce the effect of the inappropriately-named “greenhouse” gases. And convection can’t move heat into space, only radiation can do that, which is why radiation is extremely important.

The idea that climate science ignores or misunderstands convection is a myth. This is something you can easily demonstrate for yourself by checking the articles that claim it.

Where is their proof?

Do they cite atmospheric physics textbooks? Do they cite formative papers that explained the temperature profile in the lower atmosphere?

No. Ignorance is bliss..

Third Misconception – Convection is the Explanation for the “33°C Greenhouse Effect”

Perhaps in a later article I might explain this in more detail. It is already covered to some extent in On Missing the Point by Chilingar et al (2008).

As a sample of the basic misunderstanding involved in this claim, take a look at Politics and the Greenhouse Effect by Hans Jelbring, which includes a section Atmospheric Temperature Distribution in a Gravitational Field by William C. Gilbert.

If you read the first section by Jelbring (ignoring the snipes) it is nothing different from what you find in an atmospheric physics textbook. No one in atmospheric physics disputes the adiabatic lapse rate, or its derivation, or its total lack of dependence on radiation.

Clearly, however, Jelbring hasn’t got very far in atmospheric physics text books, otherwise he would know that his statement (updated Dec 9th with longer quotation on request):

T is proportional to P and P is known from observation to decrease with increasing altitude. It follows that the average T has to decrease with altitude. This decrease from the surface to the average infrared emission altitude around 4000 m is 33 oC. It will be about the same even if we increase greenhouse gases by 100%.

- was very incomplete. How is it possible not to know the most important point about the inappropriately-named “greenhouse” effect with a PhD in Climatology? Or even no PhD and just a slight interest in the field?

What determines the average emission altitude?

The “opacity” of the atmosphere. See The Earth’s Energy Budget – Part Three. Clearly Jelbring doesn’t know about it, otherwise he would have brought it up – and explained his theory of how doubling CO2 doesn’t change the opacity of the atmosphere – or the average altitude of radiative cooling to space.

Gilbert adds in his section:

I was immediately amazed at the paltry level of scientific competence that I found, especially in the basic areas of heat and mass transfer. Even the relatively simple analysis of atmospheric temperature distributions were misunderstood completely.

Where is Gilbert’s evidence for his amazing claim?

Gilbert also derives the equation for the lapse rate and comments:

It is remarkable that this very simple derivation is totally ignored in the field of Climate Science simply because it refutes the radiation heat transfer model as the dominant cause of the GE. Hence, that community is relying on an inadequate model to blame CO2 and innocent citizens for global warming in order to generate funding and to gain attention. If this is what“science” has become today, I, as a scientist, am ashamed.

I’m amazed. Hopefully, everyone reading this article is amazed.

The derivation of the lapse rate is in every single atmospheric physics textbook. And no one believes that radiative heat transfer determines the lapse rate.

And the important point – the Climate Science 101 point – is that the altitude of the radiative cooling to space is affected by the concentration of “greenhouse” gases.

Actually understanding a subject is a pre-requisite for “debunking” it.

Conclusion

Many people read blog articles and comments on blog articles and then repeat them elsewhere.

That doesn’t make them true.

Science is about what can be tested.

What would be a worthwhile “debunking” is for someone to take a well-established atmospheric physics textbook and point out all the mistakes. If they can find any.

It would be more valuable than just “making stuff up”.

References

Elementary Climate Physics, Prof F.W. Taylor, Oxford University Press (2005)

Handbook of Atmospheric Science, Hewitt & Jackson, Blackwell (2003)

An introduction to atmospheric physics, David Andrews, Cambridge University Press (2000)

Atmospheres, R.M. Goody & J.C.G. Walker, Prentice-Hall (1972)

Some Coolness Regarding Global Warming, Lindzen, Bulletin of the American Meteorological Society (1990)

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I’m in the process of writing a couple more in-depth articles but have been much distracted by “First Life” in recent weeks.. sad and unfortunate, because writing Science of Doom articles is much more interesting..

While writing a new article – What’s the Palaver? – Kiehl and Trenberth 1997 -  I thought that I should separately explain a few things which related to the earlier article: Do Trenberth and Kiehl understand the First Law of Thermodynamics? Part Three.

I know that many readers already get the point. But clearly some people find the model – and real life – so controversial that they will find many ways to claim “real life” wrong. Stefan-Boltzmann, who was he? Pyrgeometers- clearly a fake product that should be investigated by the Justice Department? And so on.

One of the problems is that radiant heat transfer is not something in accord with everyday life and so – as we all do – people draw on their own experience. But people also draw on confused ideas about the First Law of Thermodynamics to make their case.

In this article, two ideas.

First, is the Atmosphere Made of PVC?

In the original article – Do Trenberth and Kiehl understand the First Law of Thermodynamics? – I used a simple heat conduction problem to demonstrate that temperatures can be much higher inside a system than outside a system, when the system is heated from within.

One commenter explained the link between this and the atmosphere, although perhaps my attempts at humor had slightly back-fired. I had disclaimed any relationship between PVC spheres and the atmosphere..

Well, I confirm the atmosphere is not made of PVC, and that conduction is not important for heat transfer through the atmosphere.

But there is relevance for the atmosphere. Where is the relevance?

Solar radiation heats the climate system “from within”. The atmosphere is mostly transparent to solar radiation so the solar energy initially heats the surface of the earth. Then the surface of the earth heats the atmosphere. Finally the atmosphere radiates energy back out to space.

If it were true that the first law of thermodynamics – the conservation of energy – was violated by a simple “lagged pipe” model – well, that would be the end of an important branch of thermodynamics.

The model showed that the temperature of an inner surface can be higher than an outer surface – and, therefore, radiation from an inner surfaces can be higher than the radiation to space from the outer surface.

The reason for providing the model of the PVC sphere – much simpler than the atmosphere – was to demonstrate that simple point.

Second, What if the Radiation from an Inner Surface CANNOT be Higher than from an Outer Surface

Many people write entertainingly inaccurate articles about this subject (see Interesting Refutation of Some Basics for one example). Apparently, if the radiation from the atmosphere/surface into space is 239 W/m² then the radiation from the inner surface itself cannot be more than 239 W/m². A confusion about the First Law of Thermodynamics.

To be specific, the actual claim from the believers in the Imaginary First Law of Thermodynamics (IFTL) is that the total radiation from the earth’s surface cannot be higher than the total radiation from the climate system into space.

This, according to the IFLT, is not allowed.

Let’s consider the consequences and calculate the results. All we need to connect the two values - when we have the W/m² for both surfaces -  is the ratio of surface areas.

If E1 is the radiation in W/m² from inner surface A1, and E2 is the radiation from the outer surface, of area A2:

E1A1 = E2A2

The area of a sphere is proportional to its radius squared (A = 4πr²), so the above equation becomes:

E1r1² = E2r2²

a) The Earth and Climate System

In the case of the Earth and climate system, the radius of the earth and the radius of the “climate system” are almost identical..

The radius of the earth, r1 = 6,380 km or 6.38 x 106 m.

The radiation to space takes place from an average height of around 6km from the surface, so the radius of “the climate system”, r2 = 6.39 x 106 m (at most).

The total radiation to space, E2 = 239 W/m² (measured by satellite).

If the IFTL believers are correct then E1r1² = E2r2²

Therefore, E1 = 239 x (6.39 x 106)² / (6.38 x 106)² = 240 W/m²

Unsurprisingly, this surface radiation value is almost the same as the radiation into space because the two areas are almost identical.

The Stefan-Boltzmann law says that radiation from a surface, E = εσT4

The “currently believed” average value from the earth’s surface is 396 W/m². This is due to the emissivity of the earth’s surface being very close to 1.

So there are three simple choices for why the “believed value” of 396 W/m² is so much higher than the believers in the IFLT appear to claim:

  1. The Stefan-Boltzmann law is wrong
  2. The emissivity of the earth’s surface, for the wavelengths under question, is an average of 0.61
  3. The surface temperature has been massively over-estimated and the “average” temperature of the earth’s surface is actually around -18°C (see note 1).

The 3rd choice should not be ruled out. Perhaps Antartica is a lot larger than measured, or a lot colder. How many temperature stations are there on Antarctica anyway? Maybe there is some cartographical error in estimating the area of this continent from when planes have flown over Antarctica and satellites have crossed the poles.

Perhaps the Gobi desert is a lot colder than people think. No one really makes an effort to measure this stuff, climate scientists just take it all for granted, sitting in their nice warm comfortable offices looking over the results of supercomputer climate models. No one does any field research.

Quite plausible really. It’s not too hard to make the case that the average temperature of the earth is much much much colder than is generally claimed.

b) The PVC Sphere

Let’s review the very simple hollow PVC sphere model. In the original article, the inner radius was 10m and the outer radius was 13m.

Let’s look at what happens as the inner radius is increased up to 10,000m while the wall thickness stays at 3m.

Instead of keeping the internal energy source of 30,000W constant, we will keep the internal energy source per unit area of inner surface constant. In the original example, this value was 23.9 W/m².

Real First Law

With the equations provided in the maths section of Part One, and an energy source of 23.9W/m², here is the temperature difference from inner to outer surface as the inner radius increases:

Note that the x-axis is a log scale. The initial value, 10¹ (=10) was the value from the original example, and the temperature difference was 290K.

As the sphere becomes much larger (and the wall thickness stays constant) the temperature difference tends towards 377K.

Now that is a very interesting number that we can check.

When the wall thickness becomes very thin in comparison to the sphere it is really approximating a planar wall. The equation for heat conduction (per unit area) through a planar wall is:

q = k . ΔT/Δx

where q = W/m², k = conductivity (0.19 W/m.K), ΔT = temperature difference, Δx = wall thickness, m

So for a 3m thick planar PVC wall conducting 23.9 W/m², let’s re-arrange and plug the numbers into the equation:

ΔT = 23.9 x 3 / 0.19 = 377 K

Correct.

So this is a very simple test. There is no other way to link heat conduction and temperature difference. The simple equations that anyone can check support the PVC sphere model results.

Imaginary First Law

Let’s find out what happens under the imaginary first law. It will be quite surprising for the supporters of the theory.

I couldn’t check the imaginary first law in any textbooks, because it’s.. anyway, as far as I can determine, here are the steps:

1. The radiation from the inner surface must be 23.9 W/m². This means (for an emissivity, ε = 0.8 that has already been prescribed for this model) that the inner surface temperature, T1 = 151.5K (E = εσT4)

2. The inner and outer surface radiation values are related by the equations provided earlier:

E1r1² = E2r2²

3. Therefore, we can calculate the outer surface temperature and therefore the temperature difference.

Here is the graph of temperature difference as the radius increases:

Note the important point that as the radius increases the temperature difference reduces to almost nothing – this is the inevitable consequence of the (flawed) argument that inner surface radiation has to equal outer surface radiation.

Because when r1=10,000m, r2=10,003m, therefore, the areas are almost identical.

Therefore, the radiation values are almost identical, therefore the temperatures are almost identical.

Ouch. This means that somehow 23.9 W/m² is driven by heat conduction across 3m of PVC with no temperature difference.

How can this happen? Well – it can’t. To get 23.9 W/m² across a planar PVC wall 3m thick requires a temperature difference of 377 K.

When r1 = 10,000 m,  ΔT = 0.02 K according to my IFTL calculations – and so the conducted heat per unit area, q = 0.0013 W/m². The heat can’t get out, which means the temperature inside increases.. and keeps increasing until the temperature differential is high enough to drive 23.9 W/m² though the wall.

Hopefully, this makes it clear to anyone who hasn’t already made a total nana of themselves that the imaginary first law of thermodynamics, is .. imaginary.

Notes

Note 1 – The concept of an average temperature is not really needed to actually do this calculation. Averaging temperatures across different surface materials like oceans, rocks, deserts clearly has some problems – see for example, Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored.

All that is really required is to calculate the average radiation value instead. Just find the temperature at each location and calculate the emitted radiation. Then average up all the numbers (area-weighted).

As a note to the note.. To get 240 W/m² with an emissivity close to 1, the “average temperature” can be at most -18°C. With a wider day/night and seasonal variation than we actually experience on earth the “average temperature” would then be lower than -18°C.

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When I wrote Do Trenberth and Kiehl understand the First Law of Thermodynamics? I imagined that (almost) no one would have a problem with the model created. Instead, I thought perhaps some might question its relevance to climate.

It was a deliberate choice to use conduction to demonstrate the point – the reason is that radiation is less familiar to most people, while conduction is more straightforward and easier to understand.

Here is the model from that article – a heat source in a hollow PVC sphere, located in the depths of space:

Many people have experienced a lagged hot water pipe. The more lagging (insulation), the higher the temperature rises. It seems straightforward.

However, the conceptual barrier that some people have is so large that anything – literally – will be put forward to make the model fit their conceptual idea. In case the case of one blog, claiming that energy can be destroyed in an effort to get the “right” result. A delicious irony that the first law of thermodynamics is cast aside to protect.. the first law of thermodynamics.

The reason this PVC sphere model appears so wrong to many people is for similar reasons that the famous Kiehl & Trenberth diagram seems wrong – the radiation “internally” (earth surface) is higher than the external radiation to space. (Note that the radiation values in the K&T diagram can be measured).

Explaining How the Result is Calculated

..in simple terms.

Solving the maths for the model above is straightforward (refer to the first article for the actual maths). Here is the solution in simple terms:

For the steady state condition the energy radiated from the outer surface must equal the energy source in the center (30,000 W). Otherwise the system will keep accumulating energy.

Given the surface area and the stated emissivity the outer surface temperature (T2) must be 133K (to radiate 30,000 W).

The only way that heat can be transferred from the inner surface to the outer surface is through conduction. This means 30,000 W is conducted through the PVC.

Given the (low) thermal conductivity of PVC and the dimensions, the temperature difference must be 290K, making T1 = 423K.

If the temperature differential is any lower then less than 30,000W will be conducted through the wall. And if that was the case then heat would be accumulated at the inner surface – increasing its temperature until eventually 30,000W did flow through.

Conversely, if the temperature was higher than 423K then more than 30,000W would be conducted through the sphere. This would start to reduce the temperature until only 30,000W was conducted.

Simple really. However, when the result doesn’t seem right, people begin their mental gyrations to get the “right” result.

This article is not written to convince people who have their minds made up. It’s written to help those who are asking the legitimate question:

Haven’t you just created energy? And can’t I use that to run a small power station?

Good question.

This article is not about proving what has already been demonstrated, it’s about helping with mental models.

The equations of heat transfer have already been clearly explained in Part One. So far, the arguments against that have been put forward consist of:

  • the argument from incredulity
  • 3m of PVC can transmit radiation straight through (no it can’t)
  • energy disappears under the right circumstances (that was just the first of many flaws in that person’s argument..)

Of course, if someone comes up with yet another alternative calculation of the heat transfer I will be happy to look at it.

In the meantime, let’s create a mental model..

The Power Station

A few people have jubilantly claimed that the model I created, if correct, can run a power station of 1.8 MW, from a source of only 30,000 W.

That’s what it might seem like on the surface. But strangely, the model results were derived by conserving energy. That is, no energy was created or destroyed..

In the steady state condition:

  • 30,000 W is produced from the internal source
  • 30,000 W is conducted through the PVC “wall”
  • 30,000 W is radiated from the outer surface

Energy is not being created or destroyed. Where is the energy accumulation in this model? Where is the usable energy being stockpiled?

  • If you want to understand the subject, this point is the one to focus on and think about
  • If you don’t want to understand the subject say “he’s created 1.8 MW of energy from 30,000W – ridiculous”, and move on (it sounds good)

The inner surface of the sphere has an area of 1,257 m² (4πr²). Consider one square meter of internal surface, we’ll call it “A” – these kind of models always have catchy names for different components of the model.

  • Each second, A receives 23.9 W/m² from the internal heat source (30,000W / 1,257 m²).
  • Each second, A conducts 23.9 W/m² through the wall.
  • Each second, A absorbs 1,452 W/m² radiated from the rest of the inner wall.
  • Each second, A re-radiates 1,452 W/m².

This is another way of saying that no energy is being created or destroyed. Where is the energy to run this power station?

All that happens if we start drawing power out of this system is the temperature internally reduces very quickly.

How Does the Sphere Heat Up?

In my efforts to understand the conceptual problems people have, I believe that this might help. I can’t be certain – this article is about mental models.

Let’s picture the scene when the PVC sphere is “started up”.

Outside it is 0K. Inside it is 0K. Chilly. Very chilly.

Now the 30,000 W heat source is fired up. 30,000 J every second gets radiated out from this source. Every second, 80% of this 30,000 J gets absorbed by the inner surface (with 20% reflected).

At this stage almost no energy is conducted through the PVC sphere. It can’t – because the temperature differential is not nearly high enough. Conduction requires a heat differential. So instead, the energy goes into heating up the inner surface of the sphere.

As the inner surface heats up it begins to conduct heat through to the outer surface – but most of the energy still goes into heating the inner surface.

A necessary consequence of the inner surface being heated up is that it radiates. All of this radiation is absorbed by the rest of the inner surface AND THEN re-radiated. Energy is not being created. This energy can’t be “tapped off” to do anything useful.

A small supply of energy is simply being “bounced around” (not really “bounced” but it might be a useful way to think about it)

This energy is simply the energy that has been accumulated by the inner surface during the initial heating process. It keeps being accumulated until finally the temperature is high enough to conduct the full 30,000 W through to the outer surface.

Now we have reached equilibrium! On our journey to equilibrium, while the inner surface was heating up, it accumulated heat, and this accumulated heat is now radiated, absorbed, re-radiated, absorbed…

You can connect it to a power station and very quickly you will draw down this accumulation of energy. The maximum you can draw out long term will be 30,000 W.

Conclusion

This article is all about mental models – explaining why the actual results for this model don’t violate the First Law of Thermodynamics. The results were calculated from the very simple and standard heat transfer equations.

Analysis of this model, with the results that I have presented (in part one), demonstrates that energy is conserved.

At first glance it might not seem like it to many people – because the inner surface radiation is so high. But the energy is just re-radiated from the energy absorbed. It’s like a small stockpile of energy that is being “bounced around” from wall to wall.

There is only one (legitimate) way to solve the heat transfer equations for this model. Other approaches invent /destroy physics in an attempt to get a low enough value for the radiation emitted from the inner wall.

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With apologies to my many patient readers who want to cover more challenging subjects.

Many people trying to understand climate science have a conceptual problem.

I have written (too) many articles about the second law of thermodynamics – the real and the imaginary version. Resulting comments on this blog and elsewhere about those articles frequently contain comments of this form:

So if we take bucket A full of water at 80°C and bucket B full of water at 10°C, Science of Doom is saying that bucket A will heat up because of bucket B? Right! That’s ridiculous and climate science is absurd!

Yes, if anyone was saying that it would be ridiculous. I agree. To take one example from many, in The Real Second Law of Thermodynamics I said:

Put a hold and cold body together and they tend to come to the same temperature, not move apart in temperature.

Of course, it could be that I am inconsistent in my application of this principle.

One observation on the many contrary claims resulting from my articles – not a single person has provided a mathematical summary to demonstrate that the examples provided contradict the first or second law of thermodynamics.

It should be so easy to do – after all if one of the many systems I have outlined contravenes one of these laws, surely someone can write down the equations for energy conservation (1st law of thermodynamics) or for change in entropy (2nd law of thermodynamics) and prove me wrong. We aren’t talking complex maths here with double integrals or partial differentiation. Just equations of the form a + b = 0.

And here’s the reason why – the problem that people have is conceptual. It seems wrong so they keep explaining why it seems wrong.

Conceptual problems are the hardest to get around. At least, that’s what I have always found. Until a subject “clicks”, all the mathematical proof in the world is just a jumble of letters.

So with that introduction, I offer a conceptual model to help those many people who don’t understand how a cold atmosphere can lead to a warmer surface than would occur without the cold atmosphere.

And if you are one of those people in the “firmly convinced” camp, let me suggest this reason for making the effort to understand this conceptual model. If you understand why others are wrong you can help explain it to them. But if you just don’t understand the argument of people on “the other side” you can’t offer them any useful assistance.

Model 2 – Two bodies – The Boring One that Everyone Really Does Agree With

Very quickly, to “warm everyone up”, and to once again state the basics – if we have two bodies in a closed system, and body A is at temperature 80°C and body B is at 10°C, then over a period of time both will end up at the same temperature somewhere between 10°C and 80°C. It is impossible, for example, for body A to end up at 100°C and body B at 0°C.

Everyone is in agreement on this point.

Note that the “period of time” might be anything between seconds and many times the age of the universe – dependent upon the circumstances of the two bodies.

Model 3A – Three Bodies with the Third Body Being Quite Cold

Where’s Body 1? This picture is the view from Body 1, also known as “Chilly Earth”, which is a spherical solid planet.

To make the problem much easier to solve we will state that the heat capacities of Body 2 and Body 3 are extremely high. This means that whether they gain or lose energy, their temperature will stay almost exactly the same. Body 1, “Chilly Earth”, has a much lower heat capacity and will therefore adjust quickly to a temperature which balances the absorption and emission of radiation.

“Chilly Earth” doesn’t have an atmosphere.

However, for the purposes of helping the conceptual model, “Chilly Earth” reflects 30% of shortwave radiation from the Sun but at longer wavelengths absorbs 100% (reflects 0%). This means its emissivity at longwave is also 100%.

“Chilly Earth” has a very high conductivity for heat, and therefore the whole planet is at the same surface temperature. (See note 1).

“Sun” is 150M km away from “Chilly Earth”, and “Chilly Earth” has a radius of 1,000 km (a little different from the planet we call home).

Let’s calculate the approximate equilibrium temperature of “Chilly Earth”, T1

How do we do this? By calculating the energy absorbed from Body 2 and from Body 3, and calculating the temperature of a surface that will radiate that same energy back out.

The method is simple – see below.

Energy Absorbed from Body 2, “Sun”

Radiation from “Sun” at 5780K = 6.3 x 107 W/m² – near the surface of the sun. By the time the sun’s radiation reaches earth, because of the inverse square law (the radiation has “spread out”), it is reduced to 1,369 W/m². Remember that 30% is reflected, so the absorbed radiation = 958 W/m².

The surface area that “captures” this radiation = πr² = 3.14 x 106 m².

Energy absorbed from body 2, Er2 = 958 x 3.14 x 106 = 3.01 x 109 W.

Energy Absorbed from Body 3, “Space”

Radiation from “Space” at 3K = 4.59 x 10-6 W/m². Apart from the very tiny angle in the sky for “Sun”, the entire rest of the sky is radiating towards the earth from all directions in the sky.

The surface area that “captures” this radiation = 4πr² = 1.26 x 107 m².

Energy absorbed from body 3, Er3 = 4.59 x 10-6 x 1.26 x 107 = 57.7W.

So energy from body 3 can be neglected which is not really surprising.

Energy Radiated from Body 1, “Chilly Earth”

For thermal equilibrium (energy in = energy out), “Chilly Earth” must radiate out 3.01 x 109 W, from its entire surface area of 1.26 x 107 m².

This equates to 239 W/m², which for a body with an emissivity of 1 (a blackbody) means T1 = -18°C.

So we have calculated the equilibrium temperature of “Chilly Earth”.

Now, if we change the model conditions – the reflected portion of solar radiation, the emissivity of the earth at longwave, or the conductivity of the planet’s surface – any of these factors would affect the result. They wouldn’t invalidate the analysis, they would simply lead to a different number, one that was slightly more difficult to work out.

But hopefully everyone can agree that with these conditions there is nothing wrong with the method. (I realize that a few people will not agree..)

Model 3B – Three Bodies with the Third Body Being Somewhat Warmer

So now we are going to perform the same analysis with our new Body 1, “Warmer Earth” (a wild stab at an appropriate name).

The only thing that has really changed about the environment is that Body 3, “Crazy Background Radiation”, is now at 250K instead of 3K.

Note that the temperature of Body 3 is higher than before but lower than the equilibrium temperature of 255K calculated for “Chilly Earth” in the last model. As before, body 3 has an emissivity of 1 for longer wavelengths.

Body 1, “Warmer Earth”, still reflects 30% of solar radiation and is the same in every way as “Chilly Earth”.

What are we going to find?

We will do the same analysis as last time. Repeated in full to help those unfamiliar with this kind of problem.

Energy Absorbed from Body 2, “Sun”

Radiation from “Sun” at 5780K = 6.3 x 107 W/m² – near the surface of the sun. By the time the sun’s radiation reaches earth, because of the inverse square law (the radiation has “spread out”), it is reduced to 1,369 W/m². Remember that 30% is reflected, so the absorbed radiation = 958 W/m².

The surface area that “captures” this radiation = πr² = 3.14 x 106 m².

Energy absorbed from body 2, Er2 = 958 x 3.14 x 106 = 3.01 x 109 W.

Energy Absorbed from Body 3, “Crazy Background Radiation”

Radiation from “Crazy Background Radiation” at 250K = 221 W/m². Apart from the very tiny angle in the sky for “Sun”, the entire rest of the sky is radiating towards the earth from all directions in the sky.

The surface area that “captures” this radiation = 4πr² = 1.26 x 107 m². (See note 2).

Energy absorbed from body 3, Er3 = 221 x 1.26 x 107 = 2.78 x 109 W.

In this case, energy from body 3 is comparable with body 2.

Energy Radiated from Body 1, “Warmer Earth”

Body 1 absorbs Etot= Er2 + Er3 = 5.79 x 109 W

For thermal equilibrium (energy in = energy out). “Warmer Earth” must radiate out 5.79 x 109 W, from its entire surface area of 1.26 x 107 m².

This equates to 460 W/m², which for a body with an emissivity of 1 (a blackbody) means T1 = +27°C.

Discussion

Our two cases have revealed something very interesting.

A very very cold sky led to a surface temperature on our slightly different earth of -18°C, while a cold sky (colder than the original experiment’s planetary surface temperature) led to a surface temperature of 27°C.

Well, and here’s the thing, strictly speaking the temperature is actually caused primarily by the bright object in the middle of the picture, “Sun”. The energy absorbed from the sky just changes the outcome a little.

In both cases we calculated the equilibrium temperature by using the first law of thermodynamics (energy in = energy out).

If we do the calculation of entropy change we will find something interesting.. but first, let’s consider the conceptual model and what exactly is going on.

It’s very simple.

In a 3-body problem the temperature of the coldest body still has an effect on the equilibrium temperature of the body being heated by a hotter body.

I could make it more catchy, more media-friendly, but that would go against everything I stand for. I will call this Doom’s Law.

Entropy

The second law of thermodynamics says that entropy can’t reduce. The many cries of anguish that will now arise will claim that Model 3B has broken the Second Law of Thermodynamics. But it hasn’t.

See The Real Second Law of Thermodynamics for more on how to do this calculation. And even clearer, the article by Nick Stokes:

Change in entropy, δS = δQ / T

where δQ = change in energy, T = temperature

We will consider both models over 1 second.

Model 3A

Body 2, “Sun”, δS2 = -3.85 x 1026 / 5780 = -6.66 x 1022 J/K

Body 3, “Space”, δS3 = 3.85 x 1026 / 3 = +1.28 x 1026 J/K

And finally, Body 1, “Chilly Earth”, δS1 = 0 / 255 = 0 J/K

Total Entropy Change = δS1 + δS2 + δS3 = +1.28 x 1026 J/K  :a net increase in entropy.

Model 3B

Body 2, “Sun”, δS2 = -3.85 x 1026 / 5780 = -6.66 x 1022 J/K

Body 3, “Crazy Background Radiation”, δS3 = 3.85 x 1026 / 250 = +1.54 x 1024 J/K

And finally, Body 1, “Warmer Earth”, δS1 = 0 / 255 = 0 J/K

Total Entropy Change = δS1 + δS2 + δS3 = +1.47 x 1024 J/K   :a net increase in entropy.

Important points to note about the entropy calculation

Both scenarios increase entropy – by transferring heat from a high temperature source, “Sun”, to a low temperature source, “Space” in 3A, and “Crazy Background Radiation” in 3B (which is really also Space at a higher temperature).

The earth-like planet is sitting in the middle and doesn’t have a significant effect on the entropy of the universe.

In both cases the entropy of the system increases, so both are in accordance with the second law of thermodynamics.

The earth cools to space, but just at a slower rate when the background temperature of “space” is higher.

If we replaced “crazy background radiation” by an atmosphere that was mostly transparent to solar radiation, the analysis would be a little more complex but the result wouldn’t be much different.

Reasons Why It Might be Wrong

Just to be clear, these aren’t true..

1. The hotter body can’t absorb radiation from the colder body

a) see Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics for six textbooks on heat transfer which all say, yes it does. Actually, seven textbooks, thanks to commenter Bryan identifying his “non-cherrypicked” textbook by “real physicists” which also agreed.

b) see The Amazing Case of “Back Radiation” – Part Three which includes the EBEX experiment as well as a brief explanation of fundamental physics

c) see Absorption of Radiation from Different Temperature Sources – clearing up a few misconceptions on this idea

2. It’s not a real situation because the atmosphere isn’t a black body

It is true that the atmosphere is not a blackbody. But look back at model 3B. It doesn’t matter. Body 3 in this model could be a 250K body with an emissivity of 0.1 and the temperature would still increase over model 3A.

In fact, if the claim is that a colder body can never increase the temperature of a warmer body – all we need is one counter-example to falsify this theory. Now, if you want to modify your theory to something different we can examine this new theory instead.

Reasons Why It Has to be Right

1. The First Law of Thermodynamics. This neglected little jewel is quite important. Energy can’t disappear (or be created) or be quarantined into a mental box.

There is a reason why all the people disputing these basic analyses never explain where the energy goes (if it “can’t” go into changing the temperature of the hotter body that might have absorbed it). The reason – they don’t know.

2. The Second Law of Thermodynamics. This law says that in a closed system entropy cannot decrease. Despite angry claims about “no such thing as a closed system” – that’s what the second law says. Entropy is often simple to calculate.

If a solution uses simple radiation of energy (Stefan-Boltzmann’s law) and satisfies the first and second law of thermodynamics, and some people don’t like it, it suggests that the problem is with their conceptual model.

Conclusion

This is a conceptual model that is very simple.

The sun warms up the earth, and the earth cools to space. The colder “space” is, the faster the rate of net heat transfer. The warmer “space” is, the slower the rate of net heat transfer. And because the sun “pumps in” heat at the same rate, if you slow the rate of heat loss the equilibrium temperature has to increase.

The first law of thermodynamics is the key to understanding this problem. It is simple to verify that model 3A & 3B both satisfy the first law of thermodynamics. In fact, more importantly, a different result would contradict the first law of thermodynamics.

It is also easy to verify that in both 3A & 3B entropy increases.

Just to be clear on a tedious point, the earth and space do not have to radiate as a blackbody to have these conclusions. They just make the model simpler to explain, and the maths easier to understand. We could easily change the emissivity of the planet to 0.9 and the emissivity of space to 0.5 in both models and we would still find that Model 3B had a warmer planetary surface than Model 3A.

Many people will be unhappy, but this blog is not about bringing happiness. Clarity is the objective.

One more hopeless note of despair – this article uses simple theory to prove a point, which is actually a very valuable exercise. Next, some will say – “I don’t want that pointless over-theoretical theory, these people need to prove it with some experiments“.

And so I offer the series, The Amazing Case of “Back Radiation” as proof, especially Part Three. Result of Part Three was – “well, that can’t happen because it goes against theory“..

And so the circle is complete.

Notes

Note 1 – These strange conditions that don’t relate to the real world are to make the conceptual model simpler (and the maths easy). This is the staple of physics (and other sciences) – compare simple models first, then make them more complex and more realistic. If you can prove a theory with a simple model you have saved a lot of work and more people can understand it.

Note 2 – Solar radiation is from a tiny “angle” in the sky, and so the radiation is effectively “captured” by the earth as a flat disk in space. This area is the area of a disk = πr². By contrast, radiation from the sky is from all around the planet, and so the radiation is effectively captured by the surface area of the sphere. This area = 4πr². See The Earth’s Energy Budget – Part One for more explanation of this.

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