lördag 19 mars 2011

Simple Model for Radiative Heat Transfer: Cooling

Loschmidt against Maxwell and Boltzmann in a match about the lapse rate: Who won? Or who had the longest beard?

Anders of Skeptic's Guide to the Greenhouse Effect has posed the following question:
  • What is the effect of more atmospheric greenhouse gases: Warming or cooling of the Earth surface temperature?
To seek an answer consider the following simple model for radiative heat transfer through the atmosphere with X a height coordinate in say the interval (0,1), may take the form of the transport equation
  • A U + B dU/dX - C d^2U/dX^2 = F for X in (0,1),
  • U(X) is heat energy/temperature at height X,
  • A U(X) represents energy radiated to outer space from height X,
  • B dU/dX represents heat energy radiated as convection,
  • - C d^2U/dX^2 represents heat energy radiated as diffusion,
with the transport equation complemented by boundary conditions at X = 0 (ground) and
X = 1 (top of the atmosphere) and A, B, C are nonnegative coefficients, and to start with F = 0.

In the case C = 0, B = 1 the solution is given (with proper boundary conditions) as the exponential
  • U(X) = exp (- A X),
and in the case B = 0, C = 1,
  • U(X) = exp (- a X) with a^2 = A .
In both cases we see that the temperature profile flattens when A decreases, decreasing lapse rate, which means cooling.

The model thus predicts
  • cooling effect of more greenhouse gases
  • opposite of basic postulate of climate alarmism.
The related question of the lapse rate in an atmosphere under gravitation, was hotly debated at the end of the 19th century with Boltzmann and Maxwell on one side and Johann Josef Loschmidt on the other side. Boltzmann and Maxwell claimed that the lapse rate would be zero, while Loschmidt claimed that gravitation alone would cause a positive lapse rate = decreasing temperature with height. Who was right?

We know today that Loschmidt was right. Nevertheless, climate alarmism seeks its roots in the incorrect theory of Boltzmann and Maxwell in its basic postulate that the lapse rate is largely determined by radiation (and not gravitation).

The story is told in the upcoming scientific thriller: Dr Faustus of Modern Physics.

PS1 As suggested by Anders, the model can be augumented by an integral term modeling atmospheric absorption of the form
  • - D W (X) with dW/dX = U
where D is a positive coefficient which is now increasing with increasing absorption. The analysis is now different, and will be addressed in a upcoming post.

PS2 Adding a non-zero force F = -1 models a constant lapse rate. The same conclusion as above in the case C = 0 and B = 1: decreasing lapse rate as A tends to zero corresponding to increasing absorption = cooling.

PS3: Consider the case B=1, C=0, F constant: We have the heat balance
  • Q = U0 - U1 + sigma U1^4 = Q
where Uo temperature at X = 0, U1 temp at X = 1, and U0 - U1 the total heat energy lost towards space from the atmosphere, sigma U1^4 the heat energy radiated at x = 1 to space and Q is the given forcing. Differentiation with respect to A, assuming Q is constant, gives
  • 0 = dU0/dA - dU1/dA + 4 sigma U1^4/U1 dU1/dA ~ dU0 + 3 dU1/dA
since sigma U1^4/U1 ~ 240/255 ~ 1. We conclude that
  • dU0/dA ~ - 3dU1/dA
Increasing A (from 0 say ) means that U1 decreases, that is dU1/dA is negative, that is U0 is increasing. Increasing A means decreasing opacity, and thus increasing opacity means decreasing U0. In other words,
  • More of absorbing greenhouse gases mean cooling. The greenhouse effect is a cooling effect.

46 kommentarer:

  1. so, according to this model the temperature on the moon should be higher then on the earth?

  2. i wrote directly after commenting that i took my comment back because i did not consider the two other contributions.
    i think you have a very strange policy in choosing what and when publishing comments.

    but maybe i can take the chance to ask you once more to answer to the question i have been asking for a long time but that you always avoid answering: a source at temperature T1 will emit radiation (waves) towards a source at temperature T2 if T2 > T1 or not?

  3. Yes waves but not any heat transfer by the waves. I have stated this over and over and I do it once more.

  4. it's the first time i understand your answer.
    so let me see if my graceful state of mind allows me to understand the rest of your model:
    -the earth emits radiation (waves) towards the atmosphere, this radiation generates heat in the atmosphere since the atmosphere is at lower T than the earth
    -the atmosphere radiates waves back to the earth, but the earth is totally transparent to this waves because the earth temperature is higher than the temperature of the atmosphere. the waves coming back from the atmosphere do not transfer any energy at all to the earth.

    is this correct?

  5. Yes, basically: The Earth is not transparent but rather reflective, and thus does not get heated by the incoming radiation/waves. Again, there is a distinction between two-way wave propagation and one-way heat transfer
    by waves.

  6. How can the earth reflect your waves without being affected by them itself? This seems to violate the basic principles of physics. There can be no action without reaction.

    Maybe a more precise question. What are your waves made of and how do they know if they are going to impart energy into a object or not? Heat is an abstract property, a mean energy of motion for matter, so heat is not something that can be transferred. Only energy can.

  7. Claes said:" The Earth is not transparent but rather reflective."
    For a surface the emissivity = absorbtivity for a certain frequency spectrum. The outgoing radiation spectrum from earth is definitively overlapping the frequency spectrum of the backradiation from the atmosphere (e.g. CO2) so the earth can´t be reflective to all radiation from the atmosphere. Simple logic!?

  8. "We know today that Loschmidt was right. Nevertheless, climate alarmism seeks its roots in the incorrect theory of Boltzmann and Maxewell in its basic postulate that the lapse rate is largely determined by radiation (and not gravitation)."

    Claes, why not read a few climate science textbooks of the last few decades?

    The lapse rate in the troposphere - agreed by all - is determined by the adiabatic cooling as the pressure decreases. (Equally you can rewrite this equation as the conversion of potential energy to kinetic energy - i.e., the result of gravitation).

    Nothing to do with radiation.

    So who are you arguing with?

  9. So you say that climate alarmists do not claim a connection between lapse rate and greenhouse gases?

  10. I look at it more simply, based on the evidence I have uncovered. Heat transfer is diffusive, not directed. As a commenter (Alberto Miatello) on my site recently noted, the dark side of Venus is just as hot as the lighted side (due, I replied, to its massive atmosphere of IR absorbing and emitting CO2). The fundamental directive force in the atmosphere is gravity, which establishes a basic temperature lapse rate structure due to increasing pressure with depth (at least for a sufficiently massive atmosphere, such as on Earth or Venus, but not so clearly on Mars with a surface pressure of only about 6.5 mb). As most simply explained by chemical engineer William C. Gilbert (in "Politics and the Greenhouse Effect" by Ph.D. climatologist Hans Jelbring): U = CpT + gh (for unit mass), or dU = CpdT + gdh, thus dU/dh = -g/Cp (the simplest case, with "no input/output of energy or mass"). Radiative transfer of IR (heat) in the presence of matter (atmospheric molecules here) is just the radiative component of diffusive heat transfer through a medium. The lapse rate is simpler, more fundamental, and appropriate, than radiative transfer theory, as the proper comparison of atmospheric temperatures in Venus and Earth, as done on my site, easily confirms, and shows there is no greenhouse effect, either warming or cooling, just enhanced heat transfer, with increased IR absorption and emission in the atmosphere (which is not to say the planetary atmosphere is not warmed by IR absorption, to its capacity to hold the Sun's heat -- but the lapse rate does not care how the atmosphere is heated). Fundamentally, in my understanding, the effective specific heat Cp and local gravity determine the temperature profile, and the mass of the atmosphere determines the surface temperature.

    And the atmospheres of Venus and Earth are both warmed by direct absorption of the same portion of the incident solar IR radiation, not from the surface; the IR emitted by the surface during the day is basically matched by IR from the atmosphere next to the surface to the surface, both quickly brought to the same temperature. The large loop of radiation between surface and adjoining atmosphere, in the "greenhouse effect consensus" view, is just the thermal background for the daytime near-surface temperature; it has no overall effect but temperature equilibration between surface and atmosphere.

  11. That is right. I just wanted to make a comment on a question raised by Anders, leaving here out gravitation from the discussion, about the effect of increased absorption: cooling or warming?

  12. See PS2 update of the post to include a lapse rate determined by gravitation.

  13. what about the question raised by anonymous: How can the earth reflect your waves without being affected by them itself? This seems to violate the basic principles of physics.

    would you be so kind to answer that?

  14. It is not reflection: What the model describes is absorption by the blackbody of incoming frequencies combined with radiation from the blackbody of the same frequencies up to a cut-off determined by the blackbody temperature, with the effect that incoming frequencies above cut-off are stored in the blackbody and thus cause heating. So the blackbody is not at all unaffected by the incoming waves.

  15. the earth is at higher temperature than the atmosphere.
    the atmosphere radiates waves towards the earth.
    this waves you said are reflected but do not exchange heat (that is do not exchange energy). how does the reflection mechanism work?

    and another thing: microwaves have longer wavelength than temperature radiation at 300 K, but they warm up water to its boiling point and higher. isn't this in contrast with what you just said?

  16. The "reflection mechanism" is analyzed in the model I consider in the form of a wave equation with radiation.

    In the microwave the amplitude of the incoming waves is larger than that of corresponding blackbody radiation, which causes heating. There are two effects involved: frequency and amplitude of incoming/outgoing waves.

  17. what you analyse is a mathematical solution to a mathematical equation. what is its physical interpretation?

    how can the amplitude of the wave influence its *capability* of interacting with matter? (again i am asking a physical interpretation, not a mathematical solution)
    what is the amplitude of a microwave in a oven?
    and what is the amplitude of the EM-waves that the atmosphere radiates back to the earth?
    it should be very easy to do an experiment by tuning microwaves amplitude under your predicted amplitude cut-off and see whether or not you can warm up a cup of water. or is there a catch?

  18. There is no catch: a blackbody subject to radiating from another blackbody re-emits all incident radiation below cut-off. A blackbody subject to higher amplitude incident radiation, like a beef in a microwave owen, re-emits less than absorbed in a transient phase of heating, but in stationary state the beef gets overheated
    and burns. Try it!

  19. Re 21 mars 2011 06:42:

    You frequently write "climate alarmists" to mean anyone who believes that the inappropriately-named "greenhouse" effect exists.

    So this takes in all of climate science by your inappropriate naming.

    And yes, everyone believes the lapse rate is nothing to do with radiation.

    Because it isn't anything to do with radiation.

    Take a look at Things Climate Science has Totally Missed? – Convection.

    Even better, find out about climate science by reading a few textbooks.

  20. The lapse rate essentially determines the Earth surface temperature. Do you really mean that all those who speak about a "greenhouse (gas) effect", whatever it is, say that radiation has no influence on the lapse rate?

  21. cont'd: Lindzen states that without convection (and gravitation I guess) the lapse rate would be much bigger than observed and the role of convection is to reduce it to the observed. Is this what SciDoom is saying?

  22. Claes Johnson on 21 mars 2011 14:11:

    The lapse rate is determined by adiabatic expansion.

    The lapse rate by itself does not determine the Earth's surface temperature.

    The average height of radiative cooling to space is determined by the opacity of the atmosphere. The more opaque the atmosphere, the higher this average emission takes place.

    The higher the emission, the colder the emission temperature (because of the lapse rate) and so the radiative cooling is reduced.

    Therefore, with a fixed lapse rate a more opaque atmosphere leads - all other things being equal - to a warmer surface.

    Simple to understand.

    Basic atmospheric physics textbook stuff.

    So why am I explaining it to you? You have written a book about climate science. Surely you read up on the subject first?

  23. Claes Johnson on 21 mars 2011 14:33:

    Yes, everyone who understands atmospheric physics agrees with Lindzen on this fundamental point.He describes it well, although of course he was not the person who discovered this basic fact.

  24. There are several simplistic arguments floating around, and Lindzen's is one of them. A simplistic argument may carry some truth or just be off the point, usually the latter. Climate comes out as a complex combination of thermodynamics forced by radiation, which needs to be modeled and simulated to open to understanding a posteriori. A priori understanding by some simplistic model, like the one I consider or Lindzen's model, should be viewed as some form of mental preparation for a more realistic analysis, nothing connected to reality in itself.

  25. It is a simple claim.

    That doesn't make it "simplistic".

    As another relevant example, that the lapse rate is not determined by radiative physics is a simple claim. It is true or false. It is not simplistic.

    A simplistic claim is one which overly simplifies a problem.

    The lapse rate is easily determined from thermodynamics basics.

    The lapse rate without convection - is easily determined by Schwarzschild’s Equation.

    That one is less than the other is either true or false and not an over-simplification.

    Of course, it is a thought experiment because convection exists.

    And I realize that you have some non-standard perspectives because you claim to have over-turned "standard physics".

    There is nothing wrong with bold claims, but what would be helpful to the punters out there is to actually explain to them when you are attacking a theory:

    a) this result is consistent with the last 100 years of physics that I, Claes Johnson, actually don't agree with


    b) this result is inconsistent with the last 100 years of physics that I, Claes Johnson, don't agree with ANYWAY

    The people who happily repeat your claims don't actually realize that you have such a novel understanding of physics.

    However, as you asked such basic questions about the derivation of the lapse rate maybe you can't choose between a) or b).

  26. I don't know what "standard physics" I am supposed to confess to or deny.
    Again: if you know the solution to a complex problem, then you may find a simple way of describing the essence of the solution. But if you don't know the solution, then a simple argument may just be too simple or simplistic, right?

  27. I insist upon being connected to reality at all times, whereas current climate science (and ScienceofDoom's "simple understanding" of it) does not. It is theoretical nonsense, that is definitively denied by the Venus/Earth data, which undeniably and unambiguously shows there is no greenhouse effect, period. You have been disproved, ScienceofDoom. After nearly 20 years of having that data, climate scientists refuse to confront and accept the truth. That which you insist is "simple to understand" is indeed simple to understand -- but it is proved wrong, by the even simpler physical facts of two detailed atmospheres, and it is incompetent of you (and Lindzen, and Spencer, and Curry, and "97% of all climate scientists") to continue to uphold it. I don't care how pretty you think your understanding is, or how large the consensus behind you -- the real world Venus/Earth data shows you obviously don't know what you are talking about. You have been miseducated by your textbooks, and are miseducating others in turn. It is obscene, and as a competent physical scientist, I deny you and your climate science utterly. Your science is a cartoon science, and like a cartoon character you have run off a cliff and now stand upon thin air, held up only by your being unaware you have run off the cliff. Eh, what's up, Doc? You're going to have to look down at the hard facts sooner or later, and then you're done for. But you'll have 97% of climate scientists, and all those who trusted in your "consensus", right alongside you, when you hit the dirt. And I will have no sympathy for you, who have done so much evil to true science through your incompetence.

  28. SoD

    "The lapse rate in the troposphere - agreed by all - is determined by the adiabatic cooling as the pressure decreases. (Equally you can rewrite this equation as the conversion of potential energy to kinetic energy - i.e., the result of gravitation)."

    Your dishonesty and incompetence is excruciating. Read my blog.

  29. this sounds more like a tv-debate with attacks on the opponent instead of on their arguments than a scientific one.
    i do not understand the meaning of this kind of behaviour.
    why not just present convincing arguments instead of judgements?

  30. the argument about venus data is interesting. i have a question though:
    the radius of venus is around 0,95 of the earth radius, the solid angle has therefore not to be corrected by the factor 1,176 but by 0,95^2*1,176 ~ 1,1. this mens that the temperature of venus at the "earth sea level" should be around 306 K, considerably less than the observed one.

    or am i wrong somewhere?

  31. I think there is a debate with almost religious signs here. So, the climate science have several non consensus riddles still to solve. Are there any climate questions where all of you agree?
    I think lorenzo raises many logical, good questions.
    Why is the lapse rate issue so hard to solve?

  32. I do not think it is so hard: just thermodynamics with radiative forcing under gravitation.But requires solution of Navier-Stokes and thus is nontrivial.

  33. ...who has the longest beard?

    Boltzmann, I reckon (unless he was a midget).

    The main problem with physics these days is that physicists no longer grow beards, and so nobody knows whose beard is the longest, and therefore who is the wisest. 130 years ago, you could tell at a glance...

  34. The fatal flaw is in assuming (or demanding) you should be able to tell at a glance who is giving you the objective truth, and who is misleading you. When times get tough, the tough-minded think for themselves (or learn to do so). Lorenzo's question about my Venus/Earth analysis shows that he is willing to think, but he has been misled so much in the past that he makes a hash of it. He supposes there is a "solid angle correction" to be made, but that is a false supposition, which I would have thought anyone could see from following my simple analysis. So he makes a false supposition, follows it through and gets an obviously wrong answer, and then wonders why. He needs to study what I wrote more simply, without trying to drag in false complications. The question for me who knows the answer is, how do I get across to him, without triggering resentment (which occurs all too often on the internet), that he has tried to inject a false complication into the analysis, because he has seen such complications again and again in the "explanations" given by the promulgators of the incompetent climate consensus. To put it bluntly, he has been miseducated, and this one example should be multiplied millions of times to appreciate the harm that has been done to the education of a whole generation, by the incompetent consensus (whole textbooks have been written, that must be largely drivel because they assume the "science is settled" on many points, when in fact it is not only unsettled, it is clearly wrong). It is the miseducation of the whole world that is at stake in the uncovering of a false consensus belief (a belief which is being wielded tyrannously, even ruinously, by politicians worldwide, and by scientists in positions of authority, on every level and in every scientific institution -- think of the careers, of all those whose search for truth have led them outside of the fold, that have been cut short or blighted by such entrenched tyranny). On my blog, I try to write only about the most basic, most obvious points, that will allow even a layperson to understand where the truth lies. But too many come to any article they read not with a fresh, unprejudiced determination to find the truth, which shines with its own internal consistency and explanatory power, but merely to judge it according to what they think they already know. It is what people think they already know that is tripping them up, and making a hash of their reasoning ability. It is called false dogma, and the climate consensus is riddled with it, and people like ScienceofDoom chant it continually and ritually, allowing no deviation from it. That is what is going on; it is not just a genial academic debate, and you should not see it as one, nor try to make it one.

  35. The fatal flaw is in assuming (or demanding) you should be able to tell at a glance who is giving you the objective truth, and who is misleading you.


    Well, if not length of beard, then what about the number of letters after their names? Like FRS and WTF and LBW? Surely that's a good indicator of competence?

    I mean, if you were to hire a lawyer, you'd surely hire someone who had a proper framed Doctor of Law certificate hanging on their wall, rather than somebody you met in a bar somewhere, no? Same with physics. Or climate science.

    Now take this Claes Johnson chap. He's not a climate scientist. He's a professor of mathematics. I bet you'll find lots of nice framed certificates on his wall showing that he's a world expert in vector algebra or complex numbers and stuff like that. But I bet you won't find a single qualification in geology or dendrochronology or climate science. Not one. So when he gives a lecture on, say, the solution of second order differential equations, he knows what he's talking about, and you should pay attention. But if he's talking about climate science or cooking or cinematography, his opinions are no more valuable than any other Tom, Dick, or Harry, and you're perfectly entitled to just gaze out the classroom window at the girls walking by. Particularly Natasha, leastways.

    How else is academia to be organised, but that people teach what they are skilled and trained in, and not teach things they are not trained in? Or are we to have plumbers teaching law, or lawyers teaching plumbing? Would that not result in chaos, and nobody knowing anything about anything, least of all about law or plumbing?

  36. harry,

    actually i think that he best way to get across to me would have been to remind me that the emission and absorption depends on the same way on the radius of the planet and therefore the equilibrium temperature is independent on it (or that it is sufficient to consider the power absorbed per unit area because...)
    your long monologue has a good probability of triggering resentment.

    so my advice to you is: stick to the point and cut the crap, you'll see an incredible lowering of resentment, believe me!

    in particular let me ask you

  37. Frank: You argue that authority should set the truth. It used to be that way, but it is no longer. Now it is (or should be) more about the weight of the argument, less about the length of the beard of the scientist.

  38. You argue that authority should set the truth. It used to be that way, but it is no longer.

    It's more that climate alarmists regularly argue this way - from authority. They say that the climate scientists are the 'acknowledged experts'. That there is a 'consensus' among these experts. That they publish 'peer-reviewed' papers - i.e. reviewed by other acknowledged experts.

    And this essentially boils down to the measuring the length of their beards.

    But what is the alternative? Don't we all spend a lot of our time measuring the length of beards? e.g. when we go to a three-star Michelin restaurant, or watch an Academy Award winning movie, or read a book by a Nobel prize-winning scientist, or buy a chart-topping hit record. We're always looking for clues that signify good quality, like Le Creuset on the bottom of a frying pan, or Stradivarius on a violin. Is this a terrible thing to do?

    Or, at what point does it become mistaken?

  39. It is true that the battle between authority and reason goes on everywhere all the time, but somehow you have to find your way through this swamp.
    My idea is that reason, perspective and a healthy disrespect of authority
    is helpful, combined with respect for true knowledge and mastery.

  40. Claes said:
    "Yes, basically: The Earth is not transparent but rather reflective, and thus does not get heated by the incoming radiation/waves."
    I agree completely in this statement. The earth will not get heated this way by backradiation, because emissivity=absorptivity. But there is another way of seeing things. The atmosphere will as known be heated by this radiation from earth, by the part of the energy which is not radiated up into space. And with more "greenhouse" gases, more energy will be stored in atmosphere with higher temperature as a consequence. And with higher surrounding temperature the earth will have less net radiation acc to the SB twobody law. With constant incoming sun radiation this will mean a warmer earth surface to get a new energy balance. Right? (This has nothing to do with lapse rates and thermodynamic issues)

  41. Simplistic argument with only radiation. It is necessary to include thermodynamics to be able to say something meaningful.

  42. The back radiation mechanism is independent of other climate mechanisms so therefore it can be treated by itself. To mix up all climate mechanisms into one model is very difficult, so I can´t. Perhaps you can, which your answer indicates.

  43. To understand walking on two legs, it is not sufficíent to study jumping on one.

  44. This blog is really interesting to read with all different ideas and you fighting in the middle. As an example I think this discussion with SoD gets 5+: http://claesjohnson.blogspot.com/2010/08/blackbody-transformer-of-radiation.html
    Hoping for new thrilling exchanges of ideas to come!

  45. Claes, about this back-radiation business a moment. My understanding of it is that a photon is emitted by the the warm surface of the earth, and is captured by a CO2 molecule, which then releases it shortly afterwards, either upwards or downwards. So that there's a 50% chance of it going up or down.

    And most of what I've read about the atmosphere suggest that something like 50% of photons captured by CO2 are "back-radiated" to the surface of the earth.

    But this would seem to assume that the atmosphere is a single layer. But I believe that the mean distance a photon travels before encountering a CO2 molecule is only a few tens of metres. If so, the atmosphere is made up of several layers, and a photon emitted from the ground will be captured in the lowermost CO2 layer, and then emitted either up or down. If it goes up, it will be captured by the next CO2 layer up, and so on.

    I wrote a little computer programme to work out what fraction of a stream of photons coming up from the ground would end up getting through 20 or 30 such layers, and found that hardly any got through all the layers to outer space, and 99+% of them found their way back to the earth's surface. And this suggested that several layers of photon-trapping-and-re-emitting CO2 layers would behave like an almost perfect reflector, and that almost ALL the heat radiated from the ground would be returned to it.

    The same argument would apply with other greenhouses gases, like water vapour. Multiple layers of those gases would also act like a perfect reflector.

    The same argument would also apply for light coming from outer space. Most of that would be reflected back into space.

    Do climate alarmists treat the atmosphere as a perfect reflector in some wavelengths? If not, why not?