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It is not surprising that the people most confused about basic physics are the ones who can’t write down an equation for their idea.

The same people are the most passionate defenders of their beliefs and I have no doubts about their sincerity.

I’ll meander into what it is I want to explain..

I found an amazing resource recently – iTunes U short for iTunes University. Now I confess that I have been a little confused about angular momentum. I always knew what it was, but in the small discussion that followed The Coriolis Effect and Geostrophic Motion I found myself wondering whether conservation of angular momentum was something independent of, or a consequence of, linear momentum or some aspect of Newton’s laws of motion.

It seemed as if conservation of angular momentum was an orphan of Newton’s three laws of motion. How could that be? Perhaps this conservation is just another expression of these laws in a way that I hadn’t appreciated? (Knowledgeable readers please explain).

Just around this time I found iTunes U and searched for “mechanics” and found the amazing series of lectures from MIT by Prof. Walter Lewin. A series of videos. I recommend them to anyone interested in learning some basics about forces, motion and energy. Lewin has a gift, along with an engaging style. It’s nice to see chalk boards and overhead projectors because they are probably no more in use (? young people please advise).

These lectures are not just for iPhone and iTunes people – here is the weblink.

The gift of teaching science is not in accuracy – that’s a given – the gift is in showing the principle via experiment and matching it with a theoretical derivation, and “why this should be so” and thereby producing a conceptual idea in the student.

I haven’t got to Lecture 20: Angular Momentum yet, I’m at about lecture 11. It’s basic stuff but so easy to forget (yes, quite a lot of it has been forgotten). Especially easy to forget how different principles link together and which principle is used to derive the next principle.

What caught my attention for the purposes of this article was how every principle had an equation.

For example, in deriving the work done on an object, Lewin integrates force over the distance traveled and comes up with the equation for kinetic energy.

While investigating the oscillation of a mass on a spring, the equation for its harmonic motion is derived.

Every principle has an equation that can be written down.

Over the last few days, as at many times over the past two years, people have arrived on this blog to explain how radiation from the atmosphere can’t affect the surface temperature because of blah blah blah. Where blah blah blah sounds like it might be some kind of physics but is never accompanied by an equation.

Here’s the equation I find in textbooks.

Energy absorbed from the atmosphere by the surface, Ea:

Ea = αRL↓ ….[eqn 1]

where α = absorptivity of the surface at these wavelengths, RL↓ = downward radiation from the atmosphere

And this energy absorbed, once absorbed, is indistinguishable from the energy absorbed from the sun. 1 W/m² absorbed from the atmosphere is identical to 1 W/m² absorbed from the sun.

That’s my equation. I have provided six textbooks to explain this idea in a slightly different way in Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics.

It’s also produced by Kramm & Dlugi, who think the greenhouse effect is some unproven idea:

Now the equation shown is a pretty simple equation. The equation reproduced in the graphic above from Kramm & Dlugi looks a little more daunting but is simply adding up a number of fluxes at the surface.

Here’s what it says:

Solar radiation absorbed + longwave radiation absorbed – thermal radiation emitted – latent heat emitted – sensible heat emitted + geothermal energy supplied = 0

Or another way of thinking about it is energy in = energy out (written as “energy in – energy out = 0“)

Now one thing is not amazing to me -  of the tens (hundreds?) of concerned citizens commenting on the many articles on this subject who have tried to point out my “basic mistake” and tell me that the atmosphere can’t blah blah blah, not a single one has produced an equation.

The equation might look something like this:

Ea = f(α,Tatm-Tsur).RL↓ ….[eqn 2]
where Tatm = temperature of the atmosphere, Tsur = temperature of the surface

With the function f being defined like this:

f(α,Tatm-Tsur) = α, when Tatm ≥ Tsur and

f(α,Tatm-Tsur) = 0, when Tatm < Tsur

In English, it says something like energy from the atmosphere absorbed by the surface = 0 when the temperature of the atmosphere is less than the temperature of the surface.

I’m filling in the blanks here. No one has written down such ridiculous unphysical nonsense because it would look like ridiculous unphysical nonsense. Or perhaps I’m being unkind. Another possibility is that no one has written down such ridiculous unphysical nonsense because the proponents have no idea what an equation is, or how one can be constructed.

My Prediction

No one will produce an equation which shows how no atmospheric energy can be absorbed by the surface. Or how atmospheric energy absorbed cannot affect internal energy.

This is because my next questions will be:

  1. Please supply a textbook or paper with this equation
  2. Please explain from fundamental physics how this can take place

My Challenge

Here’s my challenge to the many people concerned about the “dangerous nonsense” of the atmospheric radiation affecting surface temperature -

Supply an equation.

If you can’t, it is because you don’t understand the subject.

It won’t stop you talking, but everyone who is wondering and reads this article will be able to join the dots together.

The Usual Caveat

If there were only two bodies – the warmer earth and the colder atmosphere (no sun available) – then of course the earth’s temperature would decrease towards that of the atmosphere and the atmosphere’s temperature would increase towards that of the earth until both were at the same temperature – somewhere between the two starting temperatures.

However, the sun does actually exist and the question is simply whether the presence of the (colder) atmosphere affects the surface temperature compared with if no atmosphere existed. It is The Three Body Problem.

My Second Prediction

The people not supplying the equation, the passionate believers in blah blah blah, will not explain why an equation is not necessary or not available. Instead, continue to blah blah blah.

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The Rotational Effect

Climate scientists think that the rotation of the earth is responsible for a lot of the atmospheric and ocean effects that we see. In fact, most climate scientists think it is easy to prove. (Although not as simple as proving that radiatively-active gases affect the climate).

Now suppose the earth’s rotation speed was reducing by X% per year as a result of some important human activity (just suppose, for the sake of this mental exercise) and had been for 100 years or so.

Then atmospheric physics papers and textbooks would comment on the effect of the current speed of rotation of the planet – quantifying its effect by analyzing what climate would be like without rotation. This would be just as an introduction to the effect of rotation on climate. Let’s say that the mean annual equator-arctic temperature differential is currently 35°C (I haven’t checked the exact value) but without rotation it might be thought to be 45°C. So we will describe the rotational effect as being responsible for a 10°C arctic-equatorial temperature differential.

More specifically the rotational effect might be quantified as the number of petawatts of equatorial to polar heat transported vs the value calculated for a “no rotational” earth. But by way of introduction the temperature differential is an easier value to grasp than the change in petawatts.

Various researchers would attempt to calculate the much smaller changes likely to occur in the climate as a result of the rotational changes that might take place over the next 10-20 years. They would use GCMs and other models that would be exactly like the current ones.

And of course there would be many justifiable questions about how accurate the models are – like now.

And many from the general public, not understanding how to follow the equations of motion in rotational frames, or the thermal wind equation, or Ekman pumping, or baroclinic instability, or pretty much anything relating to atmospheric & ocean dynamics might start saying:

The rotational effect doesn’t exist

Many of these people would be skeptical about the small changes to climate that could result from an impercetible change in the rotation rate.

Many blogs would spring up with people using hand-waving arguments about the climatic effects of rotation being vastly overstated.

Other blogs would write that climate science makes massively simplistic assumptions in its calculations and uses the geostrophic balance as its complete formula for climate dynamics. Many other people unencumbered with any knowledge from climate science textbooks, or any desire to read one, would curiously label themselves as skeptics and happily repeat these “facts” without ever checking them.

People with some scientific qualifications, but without solid understanding of the complete field of oceanic or atmospheric dynamics, would write poor quality papers explaining how the rotational effect was much less than climate science calculated and produce some incomplete or incorrectly derived equations to demonstrate this.

These scientists and their new work would be lauded by many blogs as being free from the simplistic assumptions that has dogged climate science and yes, finally, accurate and high quality work has been done!

Other blogs would claim that climate science was ignoring the huge effects of absorption and emission of radiation on the climate.

Then some more serious scientists would come along and write lengthy papers to argue that the rotational effect as defined by climate science does not exist because the “no rotation” result is incorrectly defined, or is not possible to accurately calculate.

Papers of incalculable value.

Read Full Post »

In Kramm & Dlugi On Illuminating the Confusion of the Unclear I pointed out that the authors of Scrutinizing the atmospheric greenhouse effect and its climatic impact are in agreement with climate science on the subject of “back radiation” from the atmosphere contributing to the surface temperature.

No surprise to people familiar with the basics of radiative heat transfer. However, Kramm & Dlugi are apparently “in support of” Gerlich & Tscheuschner, who famously proposed that radiation from the atmosphere affecting the temperature of the ground was a violation of the second law of thermodynamics. A perpetual motion machine or something. (Or they were having a big laugh). For more on the exciting adventures of Gerlich & Tscheuschner, read On the Miseducation of the Uninformed..

The first article on the Kramm & Dlugi paper was short, highlighting that one essential point.

Given the enthusiasm that new papers which “cast doubt” on the inappropriately-named “greenhouse” effect are lapped up by the blogosphere, I thought it was worth explaining a few things from their complete paper.

If I sum it up in simple terms, it is a paper which will annoy climate scientists and add confusion to scientifically less clear folk who wonder about the “greenhouse” effect.

And mostly, I have to say, without actually being wrong – or not technically wrong (note 1). This is its genius. Let’s see how they “dodge the bullet” of apparently slaying the “greenhouse” effect without actually contradicting anything of real significance in climate science.

Goody & Yung’s Big Mistake

Regular readers of this blog will know that I have a huge respect for Richard M. Goody, who wrote the seminal Atmospheric Radiation: Theoretical Basis in 1964. (The 2nd edition from 1989 is coauthored by Goody & Yung).

However, they have a mistake in a graph on p.4:

Kramm & Dlugi say:

..This figure also shows the atmospheric absorption spectrum for a solar beam reaching the ground level (b) and the same for a beam reaching the temperate tropopause (c) adopted from Goody and Yung [30]. Part (a) of Figure 5 completely differs from the original twin-peak diagram of Goody and Yung. We share the argument of Gerlich and Tscheuschner [2,4] that the original one is physically misleading..

I have the same argument about this one graph from Goody & Yung’s textbook. You can see my equivalent graph in 4th & 5th figures of The Sun and Max Planck Agree – Part Two.

There is nothing in the development of theory by Goody & Yung that depends on this graph. Kramm & Dlugi don’t demonstrate anything else in error from Goody & Yung. However, I’m sure that someone who wants to devote enough time to the subject will probably find another error in their book, or at least, an incautious statement that could imply that they have carelessly tossed away their knowledge of basic physics. This is left as an exercise for the interested reader..

To clarify the idea for readers – the energy emitted by the climate system to space is approximately equal to the energy absorbed from the sun by the climate system. This is not in dispute.

Kramm & Dlugi point out that one should be careful when attempting to plot equal areas on logarithmic graphs. Nice point.

Kepler & Milankovitch

Kramm & Dugli spend some time deriving the equations of planetary motion. These had been lost by climate science so it is good to see them recovered.

They also comment on Milankovitch’s theory in terms that are interesting:

Thus, on long-term scales of many thousands of years (expressed in kyr) we have to pay attention to Milankovitch’s [33] astronomical theory of climatic variations that ranks as the most important achievement in the theory of climate in the 20th century [10].

The theory definitely has a lot of mainstream support as being the explanation for the ice ages. However, as a comment to be developed one day when I understand enough to write about it, there isn’t one Milankovitch theory, there are many, and of necessity they contradict each other.

Interesting as well to suggest it as the most important achievement in the theory of climate last century – as the consequence of accepting Milankovitch’s theory is that climate is very sensitive to small peturbations in radiative changes in particular regions at particular times. In essence, the Milankovitch theory appears to rely on quite a high climate sensitivity.

Anyway, I’m not criticizing Kramm & Dugli or saying they are wrong. It’s just an interesting comment. And excellent that Kepler’s theories are no longer lost to the world of climate science.

Energy Conversion in the Atmosphere & at the Surface

The authors devote some time to this study (with no apparent differences to standard climate science) with the conclusion:

..Note that the local flux quantities like Q(θ, φ), H(θ, φ), G(θ, φ) and RL↑(θ, φ) are required to calculate global averages of these fluxes, but not global averages of respective values of temperature and humidity.

An important point.

They also confirm – as noted in Kramm & Dlugi On Illuminating the Confusion of the Unclear - that the energy balance at the surface is affected by the energy radiated by the atmosphere. Just helping out the many blog writers and blog commenters – be sure to strike Kramm & Dlugi off your list of advocates of the imaginary second law of thermodynamics.

The Gulags for Everyone? – Climatology Loses Its Rational Basis

The authors cite this extract from the WMO website about the “greenhouse” effect:

In the atmosphere, not all radiation emitted by the Earth surface reaches the outer space. Part of it is reflected back to the Earth surface by the atmosphere (greenhouse effect) leading to a global average temperature of about 14°C well above –19°C which would have been felt without this effect.

This website statement is incorrect as the radiation emitted by the Earth’s surface is absorbed and re-emitted by the atmosphere – not reflected. This is a very basic error.

Kramm & Dlugi say:

Note that the argument that “part of it is reflected back to the Earth surface by the atmosphere” is completely irrational from a physical point of view. Such an argument also indicates that the discipline of climatology has lost its rational basis. Thus, the explanation of the WMO is rejected..

[Emphasis added]

Well, we could argue that if one person writing a website for one body writes one thing that is not technically correct then that whole discipline has lost its rational basis. We could.

Seems uncharitable to me. Although I have to confess that on occasion I am a little bit uncharitable. I wrote that Gerlich & Tscheuschner had lost their marbles, or were having a big laugh, with their many ridiculous and unfounded statements. We all have our off days.

I think if we want to uphold high standards of defendable technical accuracy we would say that the person that wrote this website and the person that reviewed this website are not technically sound as far as the specifics of radiative physics go. I’m hard pressed to think it is justified to cast stones at say Prof. Richard M Goody for this particular travesty. Or Prof. R. Lindzen. Or Prof. V. Ramanathan. Or Prof. F.W. Taylor. Otherwise it might be a bit like Stalin with the Gulag. Everyone and their mother gets tarred with the sins of the fellow down the road and 30 million people wind up digging rocks out of the ground in a very cold place..

But let’s stay on topic. If indeed there is one.

The Main Point

Now that we have found a graph in Goody that is wrong, a website that has a mistake and have rediscovered Kepler’s equations of motion, we turn to the main course.

Kramm & Dlugi turn to perhaps their main point, about the surface temperature of the earth with and without radiatively-active gases.

As a clarification for newcomers, average temperature has many problems. Due to the non-linearity of radiative physics, if we calculate the average radiation from the average temperature we will get a different answer compared with calculating the radiation from the temperature at each location/time and then taking the average.

For more on this basic topic see under the subheading How to Average in Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored

First citing Lacis et al:

The difference between the nominal global mean surface temperature (TS = 288 K) and the global mean effective temperature (TE = 255 K) is a common measure of the terrestrial greenhouse effect (GT = TS – TE = 33 K).

The authors develop some maths, of which this is just a sample:

Using Eq. 3.8 and ignoring G(θ,φ) will lead to:

<Ts> = 23/2Te/5 ≈ 144K (3.9)

for a non-rotating Earth in the absence of its atmosphere, if S = 1367 W/m² , α (Θ0, θ, φ) = αE = 0.30 and ε(θ, φ) = ε = 1 are assumed [2]

Ts = 153 K if αE = 0.12 and Ts = 155 K if αE = 0.07

It might surprise readers that these particular points are not something novel or in contradiction to the “greenhouse” effect. In fact, you can see similar points in two articles (at least) on this blog:

- In The Hoover Incident we had a look at what would happen to the climate if all the radiatively-active gases (= “greenhouse” gases) were removed from the atmosphere. Here is an extract:

..And depending on the ice sheet extent and whether any clouds still existed the value of outgoing radiation might be around 1.0 – 1.5 x 1017 W. This upper value would depend on the ice sheets not growing and all the clouds disappearing which seems impossible, but it’s just for illustration.

Remember that nothing in all this time can stop the emitted radiation from the surface making it to space. So the only changes in the energy balance can come from changes to the earth’s albedo (affecting absorbed solar radiation).

And given that when objects emit more energy than they absorb they cool down, the earth will certainly cool. The atmosphere cannot emit any radiation so any atmospheric changes will only change the distribution of energy around the climate system.

What would the temperature of the earth be? I have no idea..

Notice the heresy that without “greenhouse” gases we can’t say for sure what the surface temperature would be.. (It’s definitely going to be significantly lower though).

- In Atmospheric Radiation and the “Greenhouse” Effect – Part One:

..The average for 2009 [of outgoing longwave radiation] is 239 W/m². This average includes days, nights and weekends. The average can be converted to the total energy emitted from the climate system over a year like this:

Total energy radiated by the climate system into space in one year = 239 x number of seconds in a year x area of the earth in meters squared..

ETOA= 3.8 x 1024 J

The reason for calculating the total energy in 2009 is because many people have realized that there is a problem with average temperatures and imagine that this problem is carried over to average radiation. Not true. We can take average radiation and convert it into total energy with no problem..

[Emphasis added]

The point here is that the total emitted top of atmosphere radiation is much lower than the total surface emitted radiation. It can be calculated. In that article I haven’t actually attempted to do it accurately – it would require some work (spatial and temporal temperature across a year and the longwave emissivity of the surface around the globe) – it is a straightforward yet tedious calculation. (See note 2).

A note in passing that this difference between the top of atmosphere radiation and the surface radiation is also derided by the internet imaginary second law advocates as being a physical impossibility because it “creates energy”.

Now I am not in any way a “representative of climate science” despite the many claims to this effect, it’s just that the basics are.. the basics. And radiative transfer in the atmosphere is a technical yet simple subject which can be easily solved with the aid of some decent computing power. So I have no quarrel with anything of substance that I have so far read in textbooks or papers on radiative physics. Yet I appear to have stated similar points to Kramm & Dlugi.

Perhaps Kramm & Dlugi have not yet stated anything controversial on the inappropriately-named “greenhouse” effect.

They take issue with what I would call the “introduction to the greenhouse effect” where a simple comparison is drawn. This is where the “greenhouse” effect is highlighted as “effective temperature”.

It could more accurately be highlighted as “difference in average flux between surface and TOA” or “difference in total flux between surface and TOA”

Is it of consequence to anything in climate science if we agreed that the difference between the TOA radiation to space and the upward surface radiation is a better measure of the “greenhouse” effect?

Kramm & Dlugi comment on a paper by Ramanathan et al:

“At a surface temperature of 288 K the long-wave emission by the surface is about 390 W/m², whereas the outgoing long-wave radiation at the top of the atmosphere is only 236 W/m² (see Figure 2 [here presented as Figure 17]). Thus the intervening atmosphere causes a significant reduction in the long-wave emission to space. This reduction in the long-wave emission to space is referred to as the greenhouse effect”

As discussed before, applying the power law of Stefan and Boltzmann to a globally averaged temperature cannot be justified by physical and mathematical reasons.

Thus, the argument that at a surface temperature of 288 K the long-wave emission by the surface is about 390 W/m² is meaningless.

Just for interest here is how Ramanathan et al described their paper:

The two primary objectives of this review paper are (1) to describe the new scientific challenges posed by the trace gas climate problem and to summarize current strategies for meeting these challenges and (2) to make an assessment 0f the trace gas effects on troposphere-stratosphere temperature trends for the period covering the pre-industrial era to the present and for the next several decades. We will rely heavily on the numerous reports..

We could assume they don’t understand science basics, despite their many excellent papers demonstrating otherwise. Or we could assume that someone writing their 100th paper in the field of climate science doesn’t need to demonstrate that something called the “greenhouse” effect exists, or quantify it accurately in some specific way unless that is necessary for the specific purpose of the paper.

However, this is the genius of Kramm & Dlugi’s paper..

Dodging the Bullet

Casual readers of this paper (and people who rely on the statements of others about this paper) might think that they had demonstrated that the “greenhouse” effect doesn’t exist. They make a claim in their conclusion, of course, but they haven’t proven anything of the sort.

Instead they have written a paper explaining what everyone in climate science already knows.

So, to clarify matters, what is the emission of radiation from the top of atmosphere to space in one year?

ETOA= 3.8 x 1024 J

What is the emission of radiation from the surface in one year?

Esurface = ?

My questions to Kramm & Dlugi:

Is  Esurface significantly greater than ETOA ?

Obviously I believe Kramm & Dlugi will answer “Yes” to this question. This confirms the existence of the greenhouse effect, which they haven’t actually disputed except in their few words at the conclusion of their paper.

Hopefully, the authors will show up and confirm these important points.

Conclusion

The authors have shown us:

  • that a graph in the seminal Goody & Yung textbook is wrong
  • Kepler’s laws of planetary motion
  • that a website describes the “greenhouse” effect inaccurately
  • that without any “greenhouse” gases the effective albedo of the earth would be different
  • the average temperature of the earth’s surface can’t be used to calculate the average upward surface radiation

However, the important calculations of “radiative forcing” and various effects of increasing concentrations of radiatively-active gases are all done without using the “33K greenhouse effect”.

Without using the 33K “greenhouse” effect, we can derive all the equations of radiative transfer, solve them using the data for atmospheric temperature profiles, concentration of “greenhouse” gases, spectral line data from the HITRAN database and get:

  • the correct flux and spectral intensity at top of atmosphere
  • the correct flux and spectral intensity of downward radiation at the surface

We can also do this for changes in concentrations of various gases and find out the changes in top of atmosphere and downward surface flux. (Feedback and natural climate variations are the tricky part).

The discussions about average temperature are an amusing sideshow.

They are of no consequence for deriving the “greenhouse” effect or for determining the changes that might take place in the climate from increases or decreases in these gases.

Notes

Note 1: I didn’t check everything, so there could be mistakes. As the full article makes clear, not much need to check. I don’t endorse their last paragraph, as my conclusion – and article – makes clear.

Note 2: The calculation in that article for total annual global surface radiation doesn’t take into account surface emissivity. The value of ocean emissivity is incorrectly stated (see Emissivity of the Ocean). There are probably numerous other errors which I will fix one day if someone points them out.

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Many people are confused about science basics when it comes to the inappropriately-named “greenhouse” effect.

This can be easily demonstrated in many blogs around the internet where commenters, and even blog owners, embrace multiple theories that contradict each other but are somehow against the “greenhouse” effect.

Recently a new paper: Scrutinizing the atmospheric greenhouse effect and its climatic impact by Gerhard Kramm & Ralph Dlugi was published in the journal Natural Science.

Because of their favorable comments about Gerlich & Tscheuschner and the fact that they are sort of against something called the “greenhouse” effect I thought it might be useful for many readers to find out what was actually in the paper and what Kramm & Dlugi actually do believe about the “greenhouse” effect.

Much of the comments on blogs about the “greenhouse” effect are centered around the idea that this effect cannot be true because it would somehow violate the second law of thermodynamics. If there was a scientific idea in Gerlich & Tscheuschner, this was probably the main one. Or at least the most celebrated.

So it might surprise readers who haven’t opened up this paper that the authors are thoroughly 100% with mainstream climate science (and heat transfer basics) on this topic.

It didn’t surprise me because before reading this paper I read another paper by Kramm - A case study on wintertime inversions in Interior Alaska with WRF, Mölders & Kramm, Atmospheric Research (2010).

This 2010 paper is very interesting and evaluates models vs observations of the temperature inversions that take place in polar climates (where the temperature at the ground in wintertime is cooler than the atmosphere above). Nothing revolutionary (as with 99.99% of papers) and so of course the model used includes a radiation scheme from CAM3 (=Community Atmospheric Model) that is well used in standard climate science modeling.

Here is an important equation from Kramm & Dlugi’s recent paper for the energy balance at the earth’s surface.

Lots of blogs “against the greenhouse effect” don’t believe this equation:

Figure 1

The highlighted term is the downward radiation from the atmosphere multiplied by the absorptivity of the earth’s surface (its ability to absorb the radiation). This downward radiation (DLR) has also become known as “back radiation”.

In simple terms, the energy balance of Kramm & Dlugi adds up the absorbed portions of the solar radiation and atmospheric longwave radiation and equates them to the emitted longwave radiation plus the latent and sensible heat.

So the temperature of the surface is determined by solar radiation and “back radiation” and both are treated equally. It is also determined of course by the latent and sensible heat flux. (And see note 1).

As so many people on blogs around the internet believe this idea violates the second law of thermodynamics I thought it would be helpful to these readers to let them know to put Kramm & Dlugi 2011 on their “wrong about the 2nd law” list.

Of course, many people “against the greenhouse thing” also – or alternatively – believe that “back radiation” is negligible. Yet Kramm & Dlugi reproduce the standard diagram from Trenberth, Fasullo & Kiehl (2009) and don’t make any claim about “back radiation” being different in value from this paper.

“Back radiation” is real, measurable and affects the temperature of the surface – clearly Kramm & Dlugi are AGW wolves in sheeps’ clothing!

I look forward to the forthcoming rebuttal by Gerlich & Tscheuschner.

In the followup article, Kramm & Dlugi On Dodging the “Greenhouse” Bullet, I will attempt to point out the actual items of consequence from their paper.

Further reading -  Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part One and New Theory Proves AGW Wrong!

Note 1 – The surface energy balance isn’t what ultimately determines the surface temperature. The actual inappropriately-named “greenhouse” effect is determined by:

  • the effective emission height to space of outgoing longwave radiation which is determined by the opacity of the atmosphere (for example, due to increases in CO2 or water vapor)
  • the temperature difference between the surface and the effective emission height which is determined by the lapse rate

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A long time ago I started writing this article. I haven’t yet finished it.

I realized that trying to write it was difficult because the audience criticism was so diverse. Come to me you huddled masses.. This paper, so simple in concept, has become somehow the draw card for “everyone against AGW”. The reasons why are not clear, since the paper is nothing to do with that.

As I review the “critiques” around the blogosphere, I don’t find any consistent objection. That makes it very hard to write about.

So, the reason for posting a half-finished article is for readers to say what they don’t agree with and maybe – if there is a consistent message/question – I will finish the article, or maybe answer the questions here. If readers think that the ideas in the paper somehow violate the first or second law of thermodynamics, please see note 1 and comment in those referenced articles. Not here.

==== part written article ===

In 1997, J. T. Kiehl and Kevin Trenberth’s paper was published, Earth’s Annual Global Mean Energy Budget. (Referred to as KT97 for the rest of this article).

For some reason it has become a very unpopular paper, widely criticized, and apparently viewed as “the AGW paper”.

This is strange as it is a paper which says nothing about AGW, or even possible pre-feedback temperature changes from increases in the inappropriately-named “greenhouse” gases.

KT97 is a paper which attempts to quantify the global average numbers for energy fluxes at the surface and the top of atmosphere. And to quantify the uncertainty in these values.

Of course, many people criticizing the paper believe the values violates the first or second law of thermodynamics. I won’t comment in the main article on the basic thermodynamics laws – for this, check out the links in note 1.

In this article I will try and explain the paper a little.  There are many updates from various researchers to the data in KT97, including Trenberth & Kiehl themselves (Trenberth, Fasullo and Kiehl 2009), with later and more accurate figures.

We are looking at this earlier paper because it has somehow become such a focus of attention.

Most people have seen the energy budget diagram as it appears in the IPCC TAR report (2001), but here it is reproduced for reference:

From Kiehl & Trenberth (1997)

From Kiehl & Trenberth (1997)

History and Utility

Many people have suggested that the KT97 energy budget is some “new invention of climate science”. And at the other end of the spectrum at least one commenter I read was angered by the fact that KT97 had somehow claimed this idea for themselves when many earlier attempts had been made long before KT97.

The paper states:

There is a long history of attempts to construct a global annual mean surface–atmosphere energy budget for the earth. The first such budget was provided by Dines (1917).

Compared with “imagining stuff”, reading a paper is occasionally helpful. KT97 is simply updating the field with the latest data and more analysis.

What is an energy budget?

It is an attempt to identify the relative and absolute values of all of the heat transfer components in the system under consideration. In the case of the earth’s energy budget, the main areas of interest are the surface and the “top of atmosphere”.

Why is this useful?

Well, it won’t tell you the likely temperature in Phoenix next month, whether it will rain more next year, or whether the sea level will change in 100 years.. but it helps us understand the relative importance of the different heat transfer mechanisms in the climate, and the areas and magnitude of uncertainty.

For example, the % of reflected solar radiation is now known to be quite close to 30%. That equates to around 103 W/m² of solar radiation (see note 2) that is not absorbed by the climate system. Compared with the emission of radiation from the earth’s climate system into space – 239 W/m² – this is significant. So we might ask – how much does this reflected % change? How much has it changed in the past? See The Earth’s Energy Budget – Part Four – Albedo.

In a similar way, the measurements of absorbed solar radiation and emitted thermal radiation into space are of great interest – do they balance? Is the climate system warming or cooling? How much uncertainty do we have about these measurements.

The subject of the earth’s energy budget tries to address these kind of questions and therefore it is a very useful analysis.

However, it is just one tiny piece of the jigsaw puzzle called climate.

Uncertainty

It might surprise many people that KT97 also say:

Despite these important improvements in our understanding, a number of key terms in the energy budget remain uncertain, in particular, the net absorbed shortwave and longwave surface fluxes.

And in their conclusion:

The purpose of this paper is not so much to present definitive values, but to discuss how they were obtained and give some sense of the uncertainties and issues in determining the numbers.

It’s true. There are uncertainties and measurement difficulties. Amazing that they would actually say that. Probably didn’t think people would read the paper..

AGW – “Nil points”

What does this paper say about AGW?

Nothing.

What does it say about feedback from water vapor, ice melting and other mechanisms?

Nothing.

What does it say about the changes in surface temperature from doubling of CO2 prior to feedback?

Nothing.

Top of Atmosphere

Since satellites started measuring:

  • incoming solar (shortware) radiation
  • reflected solar radiation
  • outgoing terrestrial (longwave) radiation

- it has become much easier to understand – and put boundaries around – the top of atmosphere (TOA) energy budget.

The main challenge is the instrument uncertainty. So KT97 consider the satellite measurements. The most accurate results available (at that time) were from five years of ERBE data (1985-1989).

From those results, the outgoing longwave radiation (OLR) from ERBE averaged 235 W/m² while the absorbed solar radiation averaged 238 W/m². Some dull discussion of error estimates from earlier various papers follows. The main result being that the error estimates are in the order of 5W/m², so it isn’t possible to pin down the satellite results any closer than that.

KT97 concludes:

Based on these error estimates, we assume that the bulk of the bias in the ERBE imbalance is in the shortwave absorbed flux at the top of the atmosphere, since the retrieval of shortwave flux is more sensitive than the retrieval of longwave flux to the sampling and modeling of the diurnal cycle, surface and cloud inhomogeneities.

Therefore, we use the ERBE outgoing longwave flux of 235 W/m² to define the absorbed solar flux.

What are they saying? That – based on the measurements and error estimates – a useful working assumption is that the earth (over this time period) is in energy balance and so “pick the best number” to represent that. Reflected solar radiation is the hardest to measure accurately (because it can be reflected in any direction) so we assume that the OLR is the best value to work from.

If the absorbed solar radiation and the OLR had been, say, 25 W/m² apart then the error estimates couldn’t have bridged this gap. And the choices would have been:

  • the first law of thermodynamics was wrong (150 years of work proven wrong)
  • the earth was cooling (warming) – depending on the sign of the imbalance
  • a mystery source of heating/cooling hadn’t been detected
  • one or both of the satellites was plain wrong (or the error estimates had major mistakes)

So all the paper is explaining about the TOA results is that the measurement results don’t justify concluding that the earth is out of energy balance and therefore they pick the best number to represent the TOA fluxes. That’s it. This shouldn’t be very controversial.

And also note that during this time period the ocean heat content (OHC) didn’t record any significant increase, so an assumption of energy balance during this period is reasonable.

And, as with any review paper, KT97 also include the results from previous studies, explaining where they agree and where they differ and possible/probable reasons for the differences.

In their later update of their paper (2009) they use the results of a climate model for the TOA imbalance. This comes to 0.9 W/m². In the context of the uncertainties they discuss this is not so significant. It is simply a matter of whether the TOA fluxes balance or not. This is something that is fundamentally unknown over a given 5-year or decadal time period.

As an exercise for the interested student, if you review KT97 with the working assumption that the TOA fluxes are out of balance by 1W/m², what changes of note take place to the various values in the 1997 paper?

Surface Fluxes

This is the more challenging energy balance. At TOA we have satellites measuring the radiation quite comprehensively – and we have only radiation as the heat transfer mechanism for incoming and outgoing energy.

At the surface the measurement systems are less complete. Why is that?

Firstly, we have movement of heat from the surface via latent heat and sensible heat – as well as radiation.

Secondly, satellites can only measure only a small fraction of the upward emitted surface radiation and none of the downward radiation at the surface.

Surface Fluxes – Radiation

To calculate the surface radiation, upward and downward, we need to rely on theory, on models.

You mean made up stuff that no one has checked?

Well, that’s what you might think if you read a lot of blogs that have KT97 on their hit list. It’s easy to make claims.

In fact, if we want to know on a global annual average basis what the upward and downward longwave fluxes are, and if we want to know the solar (shortwave) fluxes that reach the surface (vs absorbed in the atmosphere), we need to rely on models. This is simply because we don’t have 1,000′s of high quality radiation-measuring stations.

Instead we do have a small network of high-quality monitoring stations for measuring downward radiation – the BSRN (baseline surface radiation network) was established by the World Climate Research Programme (WCRP) in the early 1990′s. See The Amazing Case of “Back Radiation”.

The important point is that, for the surface values of downward solar and downward longwave radiation we can check the results of theory against measurements in the places where measurements are available. This tells us whether models are accurate or not.

To calculate the values of surface fluxes with the resolution to calculate the global annual average we need to rely on models. For many people, their instinctive response is that obviously this is not accurate. Instinctive responses are not science, though.

Digression – Many Types of Models

There are many different types of models. For example, if we want to know the value of the DLR (downward longwave radiation) at the surface on Nov 1st, 2210 we need to be sure that some important parameters are well-known for this date. We would need to know the temperature of the atmosphere as a function of height through the atmosphere – and also the concentration of CO2, water vapor, methane – and so on. We would need to predict all of these values successfully for Nov 1st, 2210.

The burden of proof is quite high for this “prediction”.

However, if we want to know the average value of DLR for 2009 we need to have a record of these parameters at lots of locations and times and we can do a proven calculation for DLR at these locations and times.

An Analogy – It isn’t much different from calculating how long the water will take to boil on the stove – we need to know how much water, the initial temperature of the water, the atmospheric temperature and what level you turned the heat to. If we want to predict this value for the future we will need to know what these values will be in the future. But to calculate the past is easy – if we already have a record of these parameters.

See Theory and Experiment – Atmospheric Radiation for examples of verifying theory against experiment.

End of Digression

And if we want to know the upward fluxes we need to know the reflected portion.

Related Articles

Kiehl & Trenberth and the Atmospheric Window

The Earth’s Energy Budget – Part One – a few climate basics.

The Earth’s Energy Budget – Part Two –  the important concept of energy balance at top of atmosphere.

References

Earth’s Annual Global Mean Energy Budget, Kiehl & Trenberth, Bulletin of the American Meteorological Society (1997) – free paper

Earth’s Global Energy Budget, Trenberth, Fasullo & Kiehl, Bulletin of the American Meteorological Society (2009) – free paper

Notes

Note 1 – The First Law of Thermodynamics is about the conservation of energy. Many people believe that because the temperature is higher at the surface than the top of atmosphere this somehow violates this first law. Check out Do Trenberth and Kiehl understand the First Law of Thermodynamics? as well as the follow-on articles.

The Second Law of Thermodynamics is about entropy increasing, due to heat flowing from hotter to colder. Many have created an imaginary law which apparently stops energy from radiation from a colder body being absorbed by a hotter body. Check out these articles:

Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics

The Three Body Problem

Absorption of Radiation from Different Temperature Sources

The Amazing Case of “Back Radiation” – Part Three and Part One and Part Two

Note 2 – When comparing solar radiation with radiation emitted by the climate system there is a “comparison issue” that has to be taken into account. Solar radiation is “captured” by an area of πr² (the area of a disc) because the solar radiation comes from a point source a long way away. But terrestrial radiation is emitted over the whole surface of the earth, an area of 4πr². So if we are talking about W/m² either we need to multiply terrestrial radiation by a factor of 4 to equate the two, or divide solar radiation by a factor of 4 to equate the two. The latter is conventionally chosen.

More about this in The Earth’s Energy Budget – Part One

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In Part Two we looked at the claimed relationship ED=AA in Miskolczi’s 2007 paper.

  • Ed = downward atmospheric radiation absorbed by the surface
  • Aa = surface radiation absorbed in the atmosphere

I showed that they could not be exactly equal. Ferenc Miskolczi himself has just joined the discussion and confirmed:

I think I was the first who showed the AA≈ED relationship with reasonable quantitative accuracy.

That is, there is not a theoretical basis for equating AA=ED as an identity.

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

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

From Miskolczi (2007)

Figure 1

Now the point I made in Part Two was that AA ≠ ED because the atmosphere is a little bit cooler than the surface – at the average height of emission of the atmosphere. So we would expect ED to be a a little less than AA.

Please review the full explanation in Part Two to understand this point.

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

The black circles are for an assumption that the surface emissivity, εG = 1. (This is reasonably close to the actual emissivity of the surface, which varies with surface type. The oceans, for example, have an emissivity around 0.96).

In these calculated results you can see that Downwards Emittance, ED is a little less than AA. In fact, it looks to be about 5% less on average. (And note that is  ED = Absorbed Downwards Emittance)

Of course in practice, εG < 1. What happens then?

Well, in the graph above, with εG = 0.96 the points appear to lie very close to the line of ED=AA.

I think there is a calculation error in Miskolczi’s paper – and if this is true it is quite fundamental. Let me explain..

Here is the graphic for explaining Miskolczi’s terms:

From Miskolczi (2007)

Figure 2

When the surface is a blackbody (εG =1), SU = SG  – that is, the upwards radiation from the surface = the emitted radiation from the ground.

The terms and equations in his 2007 are derived with reference to the surface emitting as a blackbody.

When εG < 1, some care is needed in rewriting the equations. It looks like this care has not been taken and the open circles in his Fig 2 (my figure 1) closely matching the ED=AA line are an artifact of incorrectly rewriting the equations when ε< 1.

That’s how it looks anyway.

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

Figure 3

As much as possible I have reused Miskolczi’s terms. Because the surface is not a blackbody, the downward radiation emitted by the atmosphere is not completely absorbed. So I created the term EDA for the emission of radiation by the atmosphere. Then some of this, Er, is reflected and added to SG to create the total upward surface radiation, SU.

Note as well that the relationship emissivity = absorptivity is only true for the same wavelengths. See note 4 in Part Two.

Some Maths

Now for some necessary maths – it is very simple. All we are doing is balancing energy to calculate the two terms we need. (Updated note – some of the equations are approximations – the real equation for emission of radiation is a complex term needing all of the data, code and a powerful computer – but the approximate result should indicate that there is an issue in the paper that needs addressing – see comment). 

And the objective is to get a formula for the ratio ED/AA – if ED=AA, this ratio = 1. And remember that in Figure 1, the relationship ED/AA=1 is shown as the straight line.

First, instead of having the term for atmospheric temperature, let’s replace it with:

TA = TS – ΔT      [1]

where ΔT represents the idea of a small change in temperature.

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

EDA = εAσ(TS – ΔT)4      [2]

Third, downward atmospheric radiation absorbed by the surface:

ED = εGEDA      [3]

Fourth, the upward surface radiation is the emitted radiation plus the reflected atmospheric radiation. Emitted radiation is from the Stefan-Boltzmann law:

SU = εGσTS4 + (1-εG) EDA    [4]

Fifth, the absorbed surface radiation is the upward surface radiation multiplied by the absorptivity of the atmosphere (= emissivity at similar temperatures):

AA = εASU      [5]

So if we put [2] -> [3], we get:

ED = εGεAσ(TS – ΔT)4    [6]

And if we put [4] -> [5], we get:

AA = εGεAσTS4 + EDεA(1-εG)/εG   [7]

We are almost there. Remember that we wanted to find the ratio ED/AA. Unfortunately, the AA term includes ED and we can’t eliminate it (unless I missed something).

So let’s create the ratio and see what happens. This is equation 6 divided by equation 7 and we can eliminate εA that appears in each term:

ED/AA = [ εGσ(TS - ΔT)4 ] / [ εGσTS4 + ED(1-εG)/εG ]    [8]

And just to make it possibly a little clearer, we will divide top and bottom by εG and color code each part:

ED/AA = [ σ(TS - ΔT)4 ] / [ σTS4 + ED(1-εG)/εG2 ]      [8a]

And so the ratio = blackbody radiation at the atmospheric temperature divided by

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

We didn’t make a blackbody assumption, it is just that most of the emissivity terms canceled out.

What Does the Maths Mean?

Take a look at the green term – if εG = 1 this term is zero (1-1=0) and the equation simplifies down to:

ED/AA = (TS – ΔT)4  /  TS4

Which is very simple. If ΔT = 0 then ED/AA = 1.

Let’s plot ED vs AA for a few different values of ΔT and for TS = 288K:

Figure 4

Compare this with figure 1 (Miskolczi’s fig 2).

Note: I could have just cited the ratios of ED/AA, which – in this graph – are constant for each value of ΔT.

And we can easily see that as ΔT →0, ED/AA →1. This is “obvious” from the maths for people more comfortable with equations.

That’s the simplest stuff out of the way. Now we want to see what happens when εG < 1. This is the interesting part, and when you see the graph, please note that the axes are not the same as figure 4. In figure 4, the graph is of ED vs AA, but now we will plot the ratio of ED/AA as other factors change.

Take a look back at equation 8a. To calculate the ratio we need a value of Ed, which we don’t have. So I use some typical values from Miskolczi – and it’s clear that the value of Ed chosen doesn’t affect the conclusion.

Figure 5

You can see that when  εG = 1 the ratio is almost at 0.99. This is the slope of the top line (ΔT=1) in figure 4.

But as surface emissivity reduces, ED/AA reduces

This is clear from equation 8a – as εG  reduces below 1, the second term in the denominator of equation 8a increases from zero. As this increases, the ratio must reduce.

In Miskolczi’s graph, as εG changed from 1.0 → 0.96 the calculated ratio increased. I believe this is impossible.

Here is another version with a different value of ΔT:

Figure 6

Conclusion

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

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

On conceptual grounds we can see that as the emissivity of the surface reduces, it absorbs less energy from the atmosphere and reflects more radiation back to the atmosphere.

This must reduce the value of ED and increase the value of AA. This reduces the ratio ED/AA.

In Miskolczi’s 2007 paper he shows that as emissivity is reduced from a blackbody to a more realistic value for the surface, the ratio goes in the other direction.

If my equations are correct then the equations of energy balance (for his paper) cannot have been correctly written for the case εG <1.

This one should be simple to clear up.

Update May 31st - Ken Gregory, a Miskolczi supporter appears to agree – and calculates ED/AA=0.94 for a real world surface emissivity.

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 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.

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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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Many people have requested an analysis of Miskolczi’s theories.

I start with his more recent paper:  The Stable Stationary Value of the Earth’s Global Average Atmospheric Planck-Weighted Greenhouse-Gas Optical Thickness, Energy & Environment (2010).

It’s an interesting paper and clearly Miskolczi has put a lot of time and effort into it. I recommend people read the paper for themselves, and the link above provides free access.

The essence of the claim is that the optical thickness of the earth’s atmosphere is a constant – at least over the last 60 years – where water vapor cancels out any change from CO2. So if more CO2 increases the optical thickness, then the optical thickness from water vapor will reduce.

In his paper he make this statement:

Unfortunately no computational results of EU, ST, A, TA and τA can be found in the literature, and therefore our main purpose is to give realistic estimates of their global mean values, and investigate their dependence on the atmospheric CO2 concentration.

Among the terms noted in this quote, τA is the optical thickness of the atmosphere.

As we delve into the paper, hopefully the reasons why this value isn’t calculated in any papers will become clear. In fact, the first question people should be asking themselves is this:

If the result is of significant importance why has no one else calculated this parameter before?

There are thousands of papers about radiative transfer, CO2 and water vapor.

Why has no one (apparently) published their calculations of the globally averaged optical thickness of the atmosphere and how it has changed over time?

There is a reason..

What is Optical Thickness?

You can find a more complete explanation of optical thickness in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations, which I definitely recommend reading even though it has many equations. (Actually, because it has many equations..)

Because optical thickness isn’t an obvious parameter, let’s start with a simpler property called transmittance.

Transmittance is the proportion of radiation which is transmitted through a body (in this case, the atmosphere). We will use the letter “t” to refer to it.

t has a value between 0 and 1. Slightly more formally, we can write 0 ≤ t≤ 1.

For t = 1, the body is totally transparent to incident radiation.

For t = 0, the body is totally opaque and absorbs all incident radiation.

For non-scattering atmospheres (note 1), absorptance, a = 1- t, which means that whatever is not absorbed gets transmitted. This is simple enough, and everyone would expect this from the First Law of Thermodynamics.

Now for optical thickness. We will use τ for this parameter. τ is the Greek letter “tau”.

The Beer-Lambert law says that the transmittance of a beam of radiation:

t = exp(-τ)

The “exp” is a mathematical convention for “e to the power of”. So this can alternatively be written as:

t = e

Which means that when τ = 1, t = 0.36;   when τ = 2, t = 0.14; and when τ = 10, t = 0.000045.

Optical thickness is tedious to calculate because the properties of each gas vary strongly with wavelength.

In brief, for each molecule at each wavelength, the total optical thickness is equal to the total number of molecules in the path x the absorption coefficient (which is a function of wavelength).

So optical thickness is a very handy parameter. Calculating it does take some work and a pre-requisite is a database of all the spectroscopic values for each molecule – as well as knowing the total amount of each gas in the path we want to calculate.

Absorption and Emission

The atmosphere absorbs and also emits.

Absorption, as we have just seen, is a function of the total amount of each gas (in a path) as well as the properties of each gas.

And, in case it is not obvious, the total radiation absorbed is also a function of the intensity of radiation travelling through the body that we want to calculate. This is because absorption = incident radiation x absorptance.

What about emission?

Emission of radiation is a function of the temperature of the atmosphere, as well as its emissivity, ε. This parameter emissivity is equal to the absorptivity or absorptance, of a body at any given wavelength – or across a range of wavelengths. This is known as Kirchhoff’s law.

Emission = ε . σT4 in W/m², where T is the temperature of the atmosphere at that point.

If we want to calculate the radiative transfer through the atmosphere we need both terms.

Here is a simple example of why. Readers who followed the series Understanding Atmospheric Radiation and the “Greenhouse” Effect will remember that I introduced a simple atmosphere with two molecules, pCO2 and pH2O. These had a passing resemblance to the real molecules, but had properties that were much simpler, for the purposes of demonstrating some important aspects of how radiation interacts with the atmosphere.

This following example has three scenarios. Each scenario has the same total amount of water vapor through the atmosphere, but a different profile vs height. These are shown in the graph:

Figure 1

The bottom graph shows the top of atmosphere (TOA) flux from each of the three scenarios.

If we calculated the total transmittance through the atmosphere it would be the same in each scenario (update: correction – see Ken Gregory’s point below). Because the optical thickness is the same. The optical thickness is the same because the total number of pH2O molecules in the path is the same.

Yet the TOA flux is very different.

This is because where the atmosphere emits from is very important in calculations of flux. For example, in the case of the 3rd scenario, the TOA flux is lower because more of the water vapor is at colder temperatures, and less is at hotter temperatures.

From Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations:

dIλ/dτ = Iλ – Bλ(T)     [12]

which is also known as Schwarzschild’s Equation – and is the fundamental description of changes in radiation as it passes through an absorbing (and non-scattering) atmosphere. Bλ(T) = the Planck function, which is a function of temperature. And the subscript λ in each term identifies the wavelength dependence of this equation.

For the mathematically minded, it will be clear reviewing the above equation that total optical thickness tells you less than you need. As the location of optical thickness varies, if temperature varies (which it does in the atmosphere) then you can get different results for the same optical thickness.

That is, the simulations above demonstrate what is clear, and easily provable, from the form of the fundamental equation.

This is why papers on total optical thickness of the atmosphere over time are hard to come by. It is of curiosity value only.

What About Methane, Nitrous Oxide and Halocarbons?

The total optical thickness of the atmosphere is not just determined by water vapor and CO2. If the atmosphere has an invariant optical thickness then surely all molecules should be included?

According to WM Collins and his co-authors (2006):

The increased concentrations of CO2, CH4, and N2O between 1750 and 1998 have produced forcings of +1.48, +0.48, and +0.15 W m, respectively [IPCC, 2001]. The introduction of halocarbons in the mid-20th century has contributed an additional +0.34 Wm for a total forcing by WMGHGs of +2.45Wm with a 15% margin of uncertainty.

I’m sure someone with enough determination can find some results for the changes in the radiative forcing from CH4 and N2O between 1950 and 2010. But this at least demonstrates that there is some significant absorption characteristics for other molecules. After all, halocarbons have added a quarter of the longer term CO2 increase in radiative forcing from CO2 (from 1750 to the present day) in just half a century.

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

This is not mentioned in the paper and seems to be a major blow to the not-particularly-useful result calculated.

Update, 31st May: Ken Gregory, a Miskolczi supporter armed with the spreadsheet of calculations, says that minor gases were kept constant. So Part Six demonstrates my basic calculations of optical thickness changes due to CO2 and some minor gases.

Cloudy Thinking

Miskolczi says:

In all calculations of A, TA, tA, and of the radiative flux components, the presence or absence of clouds was ignored; the calculations refer only to the greenhouse gas components of the atmosphere registered in the radiosonde data; we call this the quasi-all-sky protocol. It is assumed, however, that the atmospheric vertical thermal and water vapor structures are implicitly affected by the actual cloud cover, and that the atmosphere is at a stable steady state of cloud cover.

Assumed but not demonstrated.

Clouds have a huge impact on the radiative (and convective) heat transfers in the atmosphere. From Clouds and Water Vapor – Part One:

Clouds reflect solar radiation by 48 W/m² but reduce the outgoing longwave radiation (OLR) by 30 W/m², therefore the average net effect of clouds – over this period at least – is to cool the climate by 18 W/m².

Are they constant?

Here is a snapshot from Vardavas & Taylor (2007):

From Vardavas & Taylor (2007)

Figure 2

Another important point – given the non-linearity of the equations of radiative transfer, even if the cloud cover stayed at a constant global percentage but the geographical distribution changed, the optical thickness of the atmosphere cannot be assumed constant.

Here are some values of cloud emissivity from Hartmann (1994):

From Hartmann (1994)

Figure 3

Just for some perspective, as emissivity reaches 0.8, τ =  1.6; with emissivity = 0.9, τ = 2.3. And Miskolczi calculates the global average optical thickness of the atmosphere – without clouds – at 1.87.

At the end of his paper, Miskolczi concludes:

Apparently, the global average cloud cover must not have a dramatic effect on the global average clear-sky optical thickness..

I can’t understand, from the paper, where this confidence comes from.

Conclusion

There is more in the paper, including some very suspect assumptions about radiative exchange. However, six out of the 19 references in the paper are to Miskolczi himself and the fundamental equations brought up for energy balance (where radiative exchange is referenced) rely on his more lengthy 2007 paper, Greenhouse effect in semi-transparent planetary atmospheres.

I will try to read this paper before commenting on these energy balance equations.

However, the key points are:

  • optical thickness of the total atmosphere is not a very useful number
  • the useful headline number has to be changes in TOA flux, or radiative forcing, or some value which expresses the overall radiative balance of the climate system (update: see this comment for the correct measure)
  • optical thickness calculated as constant over 60 years for CO2 and water vapor appears to prove that total optical thickness is not constant due to increases in other well-mixed “greenhouse” gases
  • clouds are not included in the calculation, but surely overwhelm the optical thickness calculations and cannot be assumed to be constant

Other Articles in the Series:

Part Two – Kirchhoff - why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper

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

Part Four - a minor digression into another error that seems to have crept into the Aa=Ed relationship

Part Five – Equation Soufflé - explaining why the “theory” in the 2007 paper is a complete dog’s breakfast

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

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

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

Radiation and Climate, Vardavas & Taylor, Oxford University Press (2007)

Global Physical Climatology, Hartmann, Academic Press (1994) – reviewed here

Notes

Note 1 – For longwave radiation (>4 μm), scattering is negligible in the atmosphere.

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Reading one good text book on climate science can save 100′s of hours of reading rubbish on the internet. And there is a lot (of rubbish). Well-meaning people without the baggage of any knowledge of the subject writing rubbish, then repeated by other well-meaning people.

Text books cost money. But depending on which country you live in and whether you have an income, the “payback” means that not buying it is like working for $1/hr. That assumes reading rubbish isn’t a hobby for you..

And depending on where you live you can often join a university library as an “outsider” for anything ranging from $100/year up – and borrow as many books as you like.

Learning can be like a drug. In which case, other justifications aren’t necessary, you have to feed the habit regardless. So pawn family jewelery, sell your furniture, etc. Well, as an addict you already know the drill..

Just some ideas.

Global Physical Climatology – by Dennis Hartmann

Academic Press (1994)

Amazon for $88 (reduced from $118, the price at the normally amazing bookdepository.co.uk).

Why am I recommending such an old book? This covers the basics very thoroughly. When someone covers a lot of subjects there is inevitably a compromise. To cover each of the subjects “properly” would be 4,000 pages or 40,000 pages – not 400 pages. What I like about Hartmann:

a) very readable

b) very thorough

c) enough detail to feel like you understand the basics without drowning in maths or detail.

Maths is the language of science, and inevitably there is some maths. But without any maths you can still learn a lot.

Now, a few samples..

From Chapter 4:

From chapter 11:

As you can see, there is some maths, but if you are maths averse you can mostly “punch through” and still get 80% instead of the full 100%.

Elementary Climate Physics by Prof. F.W. Taylor

Oxford University Press (2005)

bookdepository.co.uk for $44 with FREE shipping lots of places in the world, unbelievable but true.

Amazon has it for $60 plus shipping.

This is an excellent book with more radiative physics than Hartmann, but also more maths generally. For example, in the derivation of the lapse rate there is some assumed knowledge. That’s par for the course with textbooks. They are written with an audience in mind. The audience in mind here is people who already have a decent knowledge of physics, but not of climate.

However, even with a tenuous grasp of physics you will get a lot out of this book. Here’s the downside though – quite some maths:

Well, he is teaching physics.

A First Course in Atmospheric Radiation – Grant Petty

Sundog Publishing 2006

Amazon from $48

Thanks to DeWitt Payne for recommending this book, which is excellent. This is the best place to start understanding radiation in the atmosphere. Goody & Yung 1989 is comprehensive and detailed – but not the right starting point.

Radiative physics is no walk in the park. There is no way to make it astoundingly simple. But Petty does a great job of making it five times easier than it should be:

Now onto “not climate science”:

An Introduction to Thermal Physics – Daniel Schroeder

Published by different companies in different countries.

Amazon from $45 plus shipping and Bookdepository for $56 free shipping.

A book that is nothing to do with climate science, but quite brilliant in explaining very hard stuff – heat and statistical thermodynamics – so it sounds really easy. Not many people can explain hard subjects so they sound easy. Most textbooks writers make slightly difficult stuff sound incomprehensible until after you understand it – at which point you don’t need the textbook.

It wasn’t until I read this book that I realized that Statistical Thermodynamics was actually interesting and useful.

The Inerrancy of Textbooks?

Are textbooks without error and without flaw?

No

So what’s the point then?

The people who write textbooks usually have 20+ years of study in that field behind them. And until such time as E&E start a line of textbooks, the publishers of textbooks, with their own reputation to protect, only ask people who have a solid background in that field to write a textbook.

So even if you are intent on demonstrating that climate science has no idea about basic physics – how are you going to do this?

You could follow the path of many other brave bloggers and commenters who write about the “paltry understanding” of climate science without actually knowing anything about climate science.

But if you choose to do it the old-fashioned way then you should at least find out what climate science says.

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