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The 2nd law of thermodynamics and the greenhouse effect

Posted on 22 October 2010 by TonyWildish

Skeptics sometimes claim that the explanation for global warming contradicts the second law of thermodynamics. But does it? To answer that, first, we need to know how global warming works. Then, we need to know what the second law of thermodynamics is, and how it applies to global warming. Global warming, in a nutshell, works like this:

The sun warms the Earth. The Earth and its atmosphere radiate heat away into space. They radiate most of the heat that is received from the sun, so the average temperature of the Earth stays more or less constant. Greenhouse gases trap some of the escaping heat closer to the Earth's surface, making it harder for it to shed that heat, so the Earth warms up in order to radiate the heat more effectively. So the greenhouse gases make the Earth warmer - like a blanket conserving body heat - and voila, you have global warming. See What is Global Warming and the Greenhouse Effect for a more detailed explanation.

The second law of thermodynamics has been stated in many ways. For us, Rudolf Clausius said it best:

"Heat generally cannot flow spontaneously from a material at lower temperature to a material at higher temperature."

So if you put something hot next to something cold, the hot thing won't get hotter, and the cold thing won't get colder. That's so obvious that it hardly needs a scientist to say it, we know this from our daily lives. If you put an ice-cube into your drink, the drink doesn't boil!

The skeptic tells us that, because the air, including the greenhouse gasses, is cooler than the surface of the Earth, it cannot warm the Earth. If it did, they say, that means heat would have to flow from cold to hot, in apparent violation of the second law of thermodynamics.

So have climate scientists made an elementary mistake? Of course not! The skeptic is ignoring the fact that the Earth is being warmed by the sun, which makes all the difference.

To see why, consider that blanket that keeps you warm. If your skin feels cold, wrapping yourself in a blanket can make you warmer. Why? Because your body is generating heat, and that heat is escaping from your body into the environment. When you wrap yourself in a blanket, the loss of heat is reduced, some is retained at the surface of your body, and you warm up. You get warmer because the heat that your body is generating cannot escape as fast as before.

If you put the blanket on a tailors dummy, which does not generate heat, it will have no effect. The dummy will not spontaneously get warmer. That's obvious too!

Is using a blanket an accurate model for global warming by greenhouse gases? Certainly there are differences in how the heat is created and lost, and our body can produce varying amounts of heat, unlike the near-constant heat we receive from the sun. But as far as the second law of thermodynamics goes, where we are only talking about the flow of heat, the comparison is good. The second law says nothing about how the heat is produced, only about how it flows between things.

To summarise: Heat from the sun warms the Earth, as heat from your body keeps you warm. The Earth loses heat to space, and your body loses heat to the environment. Greenhouse gases slow down the rate of heat-loss from the surface of the Earth, like a blanket that slows down the rate at which your body loses heat. The result is the same in both cases, the surface of the Earth, or of your body, gets warmer.

So global warming does not violate the second law of thermodynamics. And if someone tells you otherwise, just remember that you're a warm human being, and certainly nobody's dummy.

This post is the Basic Version (written by Tony Wildish) of the skeptic argument "The 2nd law of thermodynamics contradicts greenhouse theory".

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Comments 1 to 50 out of 110:

  1. Cracking closing line Tony - nice work all round.
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  2. Would it be too complicated for this bit to say that the person in the "blanket analogy" also gets warmed because the blanket is like a one way valve. Light and short wave radiation passes through and warms ones skin but the blanket acts to slow down the re-radiation of long wave heat - put another blanket on and it slows down even more
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  3. I don't know if this gets beyond a "basic" version, but when you are talking about heat exchange by radiation, the 2nd law really states that there can be no spontaneous NET flow of heat from a cold place to a hot. The cold air in the atmosphere helps keep us warm because it emits infrared radiation back to ground. It's less than what the ground radiates up, but more than would be radiated down from space alone.

    The blanket analogy is really that you have a heat flow:

    warm body -> colder blanket -> cold room

    On earth you have a heat flow:

    warm surface -> colder air -> really cold space
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  4. Putting a blanket on a tailor's dummy will cause it to heat up more than its surroundings when there is a nearby heat source.
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  5. Probably so obvious I'm foolish to say it, but I think where a lot of folks go off the rails on this has to do w/the conversion of wavelength of energy in versus energy out. We're looking at a situation where the ease of input is staying relatively the same versus the ease of output, which is increasing slightly. It stands to reason that Earth will have to become slightly warmer to account for this.

    Easy unless, that is, some folks go to a lot of trouble to confuse the issue. Just a few more hours of science education in middle school and we'd be able to skip this entire impediment.
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  6. Bibiovermis (01:44 AM on 23 October, 2010),
    Umm, no. Not unless the energy source is giving off radiation that is more transparent to the blanket than it is to the dummy.

    I use the term energy source because it is more generic; I suppose 'heat source' might refer to an object emitting energy primarily in the infrared spectrum, but microwave through, IR, visible, UV, and beyond, is all electromagnetic energy.
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  7. Oh, Doug, I think you mean ease of output is _decreasing_ slightly.
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  8. Another source of confusion ist "heat" and "radiation".

    The Second Law of Thermodynamics states, that HEAT can't go from a colder atmosphere to a warmer earth.

    But of course RADIATION is still possible. Heat is the net flow of radiations. You think, that's clear? Then listen zu extreme-sceptics ;-)
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  9. Right you are, Chris. Thanks!

    Amazing that people actually -read- these comments, heh!
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  10. Chris,

    Yes, my apologies for not explicitly stating that assumption. The blanket analogy also fails in that it prevents convection, like real greenhouses and unlike the greenhouse effect.
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  11. I would suggest that those who haven't seen the discussion on Spencer's site about this take the time (and you'll need a lot of it) to look it up and wade through it. Spencer and a very small number of commenters (as I recall) do a heroic job of convincing the hard core deniers how reality works. It's about as perfect an example as I can imagine of how difficult communication can be.
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  12. "If you put the blanket on a tailors dummy, which does not generate heat, it will have no effect. The dummy will not spontaneously get warmer. That's obvious too!"

    Don't be so sure about it being obvious. I once went hiking with a fairly intelligent but non-scientist friend, and having no good "thermos" (vacuum-insulated) bottle for our cold drinking water, took a plastic bottle, filled with ice water, and wrapped it in a sweater. She asked "What are you doing?". She assumed that the insulation would warm it up, not keep it cold. I think this is fairly typical thinking among non-scientists.
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  13. Argumentação muito fraca frente a um processo extremamente complexo como o aquecimento da atmosfera.
    Vejamos. O sol envia para a terra 1368 W/m2 de energia radiante. Ao penetrar na atmosfera (troposfera) esta energia encontra nuvens e as converte em vapor de agua e reflete algo de volta ao espaço. Chegam ao solo algo em torno de 965,6 W/m2.
    No solo aquecido encontramos;
    1- Calor irradiado (IR) 595,3 W/m2
    2- Calor transferido ao ar por contato 616,3 W/m2. (convecção)
    3- Calor absorvido pelo solo 246,3 W/m2
    Somatorio da energia no solo 1457,9 W/m2 assim distribuidos 492,3 W/m2 irradiados pela atmosfera e 965,6 W/m2 provenientes do sol.
    Este 492,3 W/m2 é que representa o efeito estufa aqui em Manaus.
    Dum total de 1457,9 W/m2 apenas 616,3 (item 2) estão sujeitos à segunda lei da termodinamica que é respeitada em sua integra.
    A segunda lei da termodinamoca refere-se a transferecia de calor entre os corpos pelo processo de condução e neste caso só é aplicavel à convecção (item 2) pois o restante segue a lei da irradiação.
    Este exemplo é de condições local entre as 14 e 15 horas, pois a transferencia por convecção requer que o solo esteja mais quente que o ar ambiente. O que não ocorre à noite.

    Efeito estufa em Manaus...



    Very weak argument against an extremely complex process like the heating of the atmosphere.
    Let's see. The sun sends to earth 1368 W/m2 of radiant energy. By penetrating the atmosphere (troposphere) this energy convert the clouds into water vapor and reflects something back to space. Reach the ground somewhere around 965.6 W/m2.
    In the soil heated encounter;
    1 - Heat irradiated (IR) 595.3 W/m2
    2 - Heat transferred to the air by contact 616.3 W/m2. (Convection)
    3 - Heat absorbed by the soil 246.3 W/m2
    Sum of the energy in the soil 1457.9 W/m2 distributed, 492,3 W/m2 irradiated by the atmosphere and 965.6 W/m2 from the sun.
    This 492.3 W/m2 represents the greenhouse effect here in Manaus.
    Of a total of 1457.9 W/m2 only 616.3 W/m2(item 2) are subject to the second law of thermodynamics and is respected in its entirety.
    The second law of thermodynamics concerns of transfer of heat between the bodies and the process of conducting this case is only applicable to convection (item 2) because the remainder follows the law of irradiation.
    This example is from local conditions between 14 and 15 hours, for transfer by convection requires that the soil is warmer than the ambient air. What does not occur at night.
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  14. I like the description of the blanket analogy in #3 better than in the post. It seems to me important to point out that the blanket makes you warmer despite the fact that the blanket is colder than you are. By denialist logic blankets also violate the 2nd Law.
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  15. Here's a nice little analogy, rigorously tested and free of convection and other distractions:

    The equipment compartments are insulated by multi-layered blankets of aluminized plastic. Temperature-responsive louvers at the bottom of the equipment compartment, opened by bi-metallic springs, allow controlled escape of excess heat. Other equipment has individual thermal insulation and is warmed by electric heaters and 12 one-watt radioisotope heaters, fueled with plutonium-238.

    The Pioneer Jupiter Spacecraft

    That's a technique often used for thermal control of spacecraft:

    The thermal control of the spacecraft was intended to be as passive or automatic as possible. The greatest part of the heat load came from the Sun and a lesser amount from the onboard electronics equipment, the latter also being among the most heat-sensitive components. For passive control, materials with different absorption and emission properties were used to radiatively balance the heat within the spacecraft. In addition, one of the six [25] boxes around the hexagonal structure was fitted with louvers activated by a temperature-sensitive bimetallic element. If temperatures within the box rose too high, the louvers opened to radiate the heat to black space; when temperatures were too low, the louvers closed to keep in the heat generated by electronic components.

    SP-480 Far Travelers: The Exploring Machines: Creating an Exploring Machine

    Were these louvers cooler than the spacecraft? Were they heating the spacecraft?

    CO2, water vapor. Louvers.
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  16. Tarcisio, I have not the faintest idea of what you are trying to say
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  17. Sceptics have been arguing this one for a while. Surely it's about net radiation. Does a photon know which way it's going?
    And you can take a pyrgeometer (longwave radiometer) - into the backyard – point upwards on a cloudless night and measure the downward radiation with a pyrgeometer. So what's the back radiation doing?
    In fact a net radiometer is a pair of pyranometers and pyrgeometers. http://en.wikipedia.org/wiki/Net_radiometer So all those people doing energy balance mustn’t know what they’re doing according to the sceptics?
    Eli Rabbet did a nice demonstration of net radiation with his alfoil light bulb. http://rabett.blogspot.com/2008_09_01_archive.html
    Do radiation shields show that cooler can increase warmer?
    And it's even done commercially be Osram and GE
    The IRC technology used in the installation lamp (burner) contains the routing of the infrared (IR) radiation of the lamp back to the filament by means of a suitable reflecting layer mounted outside on the lamp (burner).
    http://www.osram.com.au/_regional_APAC/pdf/Consumer/General_Lighting/IRC_brochure_APR_lowres.pdf
    Although my sceptic debater didn't like the fact that it was a reflector. So I guess warming from brick walls doesn't count here then http://wattsupwiththat.com/2008/07/02/sydneys-weather-station-150-meters-makes-all-the-difference/
    An alleged refutation experiment by some is Roy Spencer's solar box - http://www.drroyspencer.com/2010/07/first-results-from-the-box-investigating-the-effects-of-infrared-sky-radiation-on-air-temperature/ but I think the relatively small cooling actually proves the point - and note here the impact of clouds here http://www.drroyspencer.com/2010/08/help-back-radiation-has-invaded-my-backyard/
    A more formal treatment of the subject at http://scienceofdoom.com/2010/10/07/amazing-things-we-find-in-textbooks-the-real-second-law-of-thermodynamics/
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  18. This argument assumes that carbon-gases trap heat from both the Earth and the sun, but there's no evidence to support this; on the contrary, atmospheric oxygen and nitrogen are more absoptive than carbon-gases at certain wavelenghts; carbon-gases are not a "blanket" in any sense, they simply have different resonant-frequencies than non-carbon ones.
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    Moderator Response: You are very incorrect. See CO2 effect is weak.
  19. Tony

    Several comments have already mentioned this, and I understand that this is a basic post, but the distinction between

    'Heat cannot flow from a cold source to a hot source' (NOT a correct statement of the 2nd Law)

    and

    'NET Heat cannot flow from a cold source to a hot source' (A more correct statement - we wont mention entropy at this level)

    is fundamental to identifying the fallacy in this sceptic argument. So somehow you need to weave in the NET element to avoid giving sceptics rope to hang themselves with. (On the other hand....)
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  20. Glenn Tamblyn
    it shouldn't be necessary to add "net" to heat, it is already the net energy exchange (or flux). Though, you're right that there's a lot of confusion on this concept, we all often call heat the energy emitted by warm bodies (the Stefan-Boltzman law).
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  21. Visitei o site sugerido por LukeW, scienceofdoom.com e achei um absurdo aplicar a segunda lei aos corpos sujeitos à energia radiante. É como se um corpo ao receber energia radiante mandasse a energia parar e responder se veio de um corpo mais quente ou mais frio. Se a resposta for mais quente, pode entrar, caso contrario será bloqueada.

    Quanto a questão da atmosfera ser mais fria que o solo, podemos diser a mesma coisa. O solo recebe milhões de emissões das particulas do ar e não tem a propriedade de escolher se esta emissões provem de um corpo mais quente ou mais frio. Ele simplesmente recebe esta energia.


    "Google tranlated"

    I visited the site suggested by LukeW, scienceofdoom.com and thought absurd to apply the second law for bodies subjected to radiant energy. It's like a body to receive radiant energy to send power to stop and answer came from a body warmer or colder. If the answer is hotter, you can get, otherwise it will be blocked.

    Regarding the question of the atmosphere is colder than the ground, we see the same thing. The soil receives millions of emission of particles from the air and has no property to choose whether this emission comes from a body warmer or colder. He just gets this energy.
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  22. Bibliovermis, the blanket analogy is precisely that, an analogy. It is not a model that represents every aspect of global warming. Nor does it need to, it shows the point that a warm body can be made warmer by something cooler, without violating the second law. So it's enough to disprove the skeptics' claim.

    Tarcisio, are you implying that the second law does not apply to radiation? If that were true then this rebuttal would not need to exist. As it is, the second law applies equally to all forms of heat-transfer - you will not find a textbook anywhere that defines the second law as behaving differently for conduction, convection, or radiation.

    KirkSkywalker, nothing in this rebuttal is specific to a certain gas, any GHG will exhibit the same effect. CO2 is important because there's so much of it, and adding more is like adding another blanket, and another, and another. As others have pointed out (#3, #14) I didn't emphasise this point enough in the post, but adding more blankets will make you warmer, even though the outermost blanket may itself be quite cold.
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  23. At least once I agree with SkS. This is some point where some skeptics are horribly wrong. Not including me.
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  24. protestant
    virtually all the skeptic scientists agree, just some blogger/blog commentrs don't. Definitely you're not alone :)
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  25. Be very careful with the blanket analogy. Increased GHGs keep the surface warmer due to changes in radiative transfer and a blanket keeps the body warmer by suppressing convection.

    They both keep it warmer but for very different reasons. You did note this in your third to last paragraph but it may get lost in translation if somebody carries the message from here. A person, after reading this post, might tell somebody that GHGs act like a blanket to keep us warm.

    "What do you mean I am wrong, I read this over at Skeptical Science."
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  26. Tarcisio José D'Avila, the value 1368 W/m2 is the energy flux coming from the sun to the unit area of a plane surface perpendicular to the line between the sun and the center of the earth. To get the value of solar energy flux per unit area of the earth's surface, it should be multiplied by the cosine of the solar zenith angle (in other words, the sine of the solar elevation angle), which depends on latitude, season, and time of day. (We are sure that the global and annual average of it is 1/4). There are many textbooks, both meteorological and astronomical, to explain how to calculate it. As for a conceptual reminder, here is an introduction from NASA.

    http://edmall.gsfc.nasa.gov/inv99Project.Site/Pages/science-briefs/ed-stickler/ed-irradiance.html
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  27. Help please! I do not find good textbooks of thermodynamics which contain how to interpret radiative transfer as heat exchange. Some books do contain Planck's theory of blackbody radiation, but that is not enough. Even if we are content with the blackbody spectrum, we need to handle different energy flux densities from the thermal equilibrium. We also need to evaluate the effective tempearture of radiation after partial absorption.
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  28. I think that the blanket analogy is less troublesome than greenhouse (glass house) analogy, for a blanket is obviously different from the atmosphere.

    By the way, I remember some planetary scientists use "blanket effect" as a technical term distinguished from "greenhouse effect". But, I do not remember the precise context. Probably, it is "greenhouse effect" if energy comes primarily from the sun, and it is "blanket effect" if it comes from the interior of the planet (geothermal).
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  29. Re: Kooiti Masuda (27)

    I pulled these references off of Barton Paul Levenson's Climatology pages (he specializes in atmospheric physics):
    Goode, Philip 1998. "Earthshine Measurements of Global Atmospheric Properties." http://www.bbso.njit.edu/Research/Earthshine/earthshine-proposal.html, accessed 10/13/1998.

    Houghton, J.T. 2002 (1977). The Physics of Atmospheres. Cambridge: Cambridge University Press.

    Lorenz, R.D. and C.P. McKay 2003. "A Simple Expression for Vertical Convective Fluxes in Planetary Atmospheres." Icarus 165, 407-413.

    Trenberth, K.E., J.T. Fasullo and J. Kiehl 2009. "Earth's Global Energy Budget." Bull. Amer. Meteorol. Soc., preprint.
    BPL also has an online planetary temperature calulator here.

    Not exactly what you may have been looking for, but I hope it helps!

    The Yooper
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  30. TonyWildish

    Suponha uma antena trnsmitindo 1W e outra recebendo este sinal num ponto tal que seja induzidos 1 dBm (1mW). Divida então este 1W de sinal e aplique em 1000 antenas. Em nossa antena receptora teremos 1w de sinal. Como ? dirias e a segunda lei da termodinamica ?.
    No caso da atmosfera ocorre a mesma coisa. São 6.022 E23 antenas por moll de ar transmitindo IR (ondas eletromagneticas).

    Translated by google

    Kooiti Matsuda

    Voce está correto, mas apresentei apenas um exemplo local sem o intuito de apresentar avoles absolutos. Mostrando que o fluxo energetico que chega ao solo é maior que a energia enviada pelo sol.


    TonyWildish

    Suppose an antenna trnsmitindo 1W and the other receiving this
    signal at a point that is induced 1 dBm (1mW).
    Then divide this signal and apply 1W in 1000 antennas.
    In our receiving antenna will 1w signal. How?
    would you say and the second law of thermodynamics?.
    In the case of the atmosphere is the same thing. 6022 E23 St
    antennas for transmitting IR moll air (waves
    electromagnetic).


    Kooiti Matsuda

    You're right, but has just one local example
    without the intention of presenting absolute avola. Showing
    the energetic flow that reaches the ground is greater than the
    energy sent by the sun.
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  31. 1. A blanket is opaque to IR just as much as to light. If a blanket warmed by absorption and emission, rather than by blocking convection, then the first blanket -- blocking 100% of the IR radiation -- would do all the warming that could occur and a second blanket could cause no further warming (contrary to reality).

    2. If greenhouse is a blanket that sends some heat back the way it came then this must be true of the radiation coming in from the sun as well as going out. Notice these two diagrams show that water vapour absorbs far more frequencies of radiation, especially incoming from the sun, than CO2. Therefore water vapour would provide a negative feedback, cooling the earth.

    3. Even though there's radiation from the cooler atmosphere to the warmer ground, heat does not flow from the cooler to the warmer. An object passively warmed can't make the warming object warmer than it otherwise would be unless you believe you can get energy for free. Yet, remarkably, this fallacy is the basis of the greenhouse effect.
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    Moderator Response: You should read the Intermediate version of 2nd law of thermodynamics contradicts greenhouse theory. Then read CO2 effect is weak, both the Intermediate and Advanced versions.
  32. #25 ProfMandia at 08:26 AM on 24 October, 2010
    "Increased GHGs keep the surface warmer due to changes in radiative transfer and a blanket keeps the body warmer by suppressing convection."

    Indeed the primary reason a green house is warmer is because of the suppressed convection. So the mechanism of "greenhouse" gases starts off with a basic confusion.

    The easiest way I've found to explain the GHG effect is to ask someone to compare a dry cloudless night with a humid cloud-covered night. Using this as the example, it isn't very hard to get across the concept of the surface radiating upward and that energy getting reflected either just a little (clear, dry night) or a lot (humid, cloudy night).
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  33. Riccardo @20

    "it shouldn't be necessary to add "net" to heat, it is already the net energy exchange (or flux)" Of course it is, but that is when speaking to a moderatly technically literate audience. I have had sceptics tell me that down-welling longwave radiation can't exist because heat cannot radiate from a cold atmosphere to the warmer Earth's surface.

    This is the point, particulary with the basic posts. They have to provide information to the undecided who for example have never heard of this "Stefan Boltzman guy" (I have encountered this) in simple form without providing fodder for deliberate or ignorant mis-representation along the lines of:

    "I was over at this warmist site and they didn't even understand that heat can't flow from a cold source to a hot source. These people are trying to con us all with this junk science ... blah ... blah ..."
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  34. Stylo exhibits the common confusion that has so succesfully been exploited in this non debate. Even G&T do the same thing: confuse heat (in the thermodynamic sense) and energy, or radiation.
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  35. Glenn Tamblyn,
    I understand your point (and ProfMandia). It's difficult to judge what should and what should not be said in a basic level explanation.
    If a user has no previous kowledge, the simple blanket analogy is easy enough. A blanket reduces your body heat losses, the GHGs do the same with the earth. No details, no conduction, radiation or Plank, but essentially it shows that putting something less cold than the ambient near a warm body helps keep it warm by reducing heat losses.
    The 2nd law is not addressed explicitly; the analogy just shows that what may seem to violate the 2nd law, in reality does not. In my view, it is an intermediate level explanation that should explicitly mention the mechanism to explain why the second law is not violated.

    In the end, it all depends on the audience one immagine to have when writing a post. I myself wrote a few posts here and the hardest part was answering to the question "to whom am I talking?". The problem is that there's no single answer or, better, the real answer is "to everyone".
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  36. #31, Stylo:

    1. On blankets: This relates to an analogy rather than physics, so I'll leave it aside.

    2. "Doesn't water vapor block radiation coming from the sun? Shouldn't increased water vapor act as a negative feedback?"

    Yes, water vapor does block some infrared (IR) radiation at high frequencies. However, if you look at these absorption bands in the context of the two blackbody-radiation graphs, you find:
    - Your first graph shows the spectrum of incoming radiation from the sun.
    - Your second graph shows the spectra from both the incoming radiation and the outgoing radiation (from the earth, headed into space).

    - The water-vapor lines have taken a bit out of the solar-radiation emission spectrum (which is mostly in the visible and higher-frequency IR), but nearly wipe out about 2/3 of the earth-radiation spectrum: This can be seen in your 2nd graph.

    - So if you add more water vapor, you cut back on radiant input (solar) a bit, but you cut back on radiant output (into space) a lot.

    - Since the water vapor has a huge percentage impact on the earth-radiation spectrum but just a minor effect on the solar-radiation spectrum (and since the total power under both these graphs have to be very very nearly the same (SEE NOTE)), the overall impact of additional water vapor will be to increase the warming effect.

    3. "Heat does not flow from cooler to warmer."

    - True, if you put a cold object in direct physical contact with a warm object, heat will be conducted only from the warmer to the cooler, not the other way.
    - But we are not talking about heat conduction, we are talking about radiant energy transfer. There is no 2nd-law problem with SOME radiated power from the cooler object (which still gives off radiation, remember: Power proportional to (temperature)^4 ) being absorbed by the warmer object. This doesn't violate the 2nd law because EVEN MORE power radiated from the warmer object is absorbed by the colder object. So the net transfer of heat is going to be from warmer to hotter. There is no problem with this.
    - Perhaps you are getting confused by the old-fashioned language of heat engines, etc. A slightly more abstract but simpler, and equivalent, approach is to think of it in terms of entropy: The 2nd law states that the entropy of the universe can only increase; so a process that would result in a decrease of entropy would violate the 2nd law. Example: Heat (dQ) flows from hot object 1 to cold object 2 (T_1 > T_2): the entropy change is:
    dS = (1/T_1)*(-dQ) + (1/T_2)*(dQ)
    = (1/T_2 - 1/T_1)*dQ
    = {(T_1-T_2)/(T_1*T_2)}*dQ

    Since T_1 > T_2, dS > 0, and this is OK by the 2nd law. But if you try having dQ be negative (heat flowing from cold object at T_2 to hot object at T_1), you will get dS < 0. WARNING! That means it's impossible. So heat conduction will ALWAYS and ONLY be from the warmer to the colder body.

    But now let's consider radiant transfer: Both bodies at T_1 and T_2 are radiating energy away, and absorbing whatever radiant energy falls upon them.

    The energy that is radiated from object 1 and absorbed by object 2 is proportional to the intensity of the radiation from object 1, I_1. Correspondingly, the energy radiated from object 2 and absorbed by object 1, in time dt, is proportional to the intensity of radiation from object 2, I_2. Since T_1 > T_2, I_1 > I_2 (Stefan-Boltzmann law). (The constants of proportionality are the same, because they are determined by geometrical considerations which are reciprocal between the two objects.)

    Hence, the entropy increment associated with the radiant energy flow from 1 to 2 is:
    dS_a = -(1/T_1)*(I_1 * dt) + (1/T_2)*(I_1 * dt)
    .....= {(T_1 - T_2)/(T_1*T_2)}*(I_1 * dt)

    Likewise, the entropy increment associated with the radiant energy flow from 2 to 1 is:

    dS_b = (1/T_1)*(I_2 * dt) - (1/T_2)*(I_2 * dt)
    .....= {(T_2 - T_1)/(T_1*T_2)}*(I_2 * dt)

    Now, if we look at dS_a, we see that dS_a > 0 because T_1 > T_2: no problem. But if we look at dS_b, we see that dS_b < 0. Does that mean that process b (radiant flow from object 2 to object 1) is impossible? NO, not if it is done as part & parcel with process a (and it WILL be part & parcel: as soon as radiation can go from 1 to 2, it can also go from 2 to 1): In that case,
    dS_total = dS_a + dS_b
    .........= {(T_1 - T_2)/(T_1*T_2)} * (I_1 - I_2)*dt

    and this is > 0, because T_1 > T_2 and I_1 > I_2.

    Hence, when you put processes a and b together, it's completely legitimate with regard to the 2nd-law.

    You might ask, "Is it legal to do that? To combine a process that would seem to be illegal alone with another so that the total is legal?" Yes, it is: Consider the Carnot cycle, run in refrigerator mode: It takes heat from a cooler object and deposits in a hotter object (horrors!). It's all legal, because the work you have to put in to make it balance ensures that the total entropy change is non-negative. That means it's fine from a 2nd-law perspective.

    And that's a good thing for those bottles of beer you have in your 'fridge.


    NOTE: Someone might wonder, "Why is the total power under the solar-radiation input curve the same as the total power under the earth-radiation output curve? Isn't the sun much hotter, and so doesn't the Stefan-Boltzman law state that the sun must be radiating much more energy?" The answer is Yes, the sun is radiating much more energy in total; but we're just receiving a fraction of it, as we are at a distance 1 AU away from the sun, and so the intensity of the radiation we receive is reduced by factor (1/distance^2). Whereas we are sitting on the earth, and get the full benefit of the blackbody emission spectrum from it. Since the earth is only warming slowly, the total influx must very nearly balance the total outflux; therefore the integrated powers have to be very nearly the same.
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  37. Tarcisio, #30. I'm sorry, I do not understand what you are trying to say.

    Stylo, #31.
    1. your first blanket would radiate away heat, but slower than your body alone would do so because it is cooler. That's why you warm up. A second blanket would slow down the heat-loss from the first blanket for the same reason, so you would still get warmer. sure, there is a physical limit in this analogy, eventually you would die of heat-stroke and stop emitting heat. Then you and your blankets would all cool down.

    3. What do you mean by 'passively warmed'? The earth is being actively warmed by the sun. Energy is being pumped into it by solar radiation. Nothing requires heat to flow from the atmosphere to the earth, we need only slow the rate of loss of heat by the earth, and it will get warmer. That's what GHGs do.
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  38. #36 nealjking, re: the graphs: the incoming from the sun is 1366 W/m2 but the outgoing is only, what, 342 W/m2? So a little out of the sun's is worth a lot out of the earth's energy flow in terms of greenhouse blocking.

    #37 TonyW,

    1. "your first blanket would radiate away heat because it is cooler."

    It would be cooler but, without convection-cooling from the air the blanket would just get to the same temperature as the body and radiate away just as much. So, radiation blocking is not really a mechanism of heat trapping.

    3. By passively warmed I just mean it doesn't have its own power source and so can't add to the temperature like, say, a lowered emissivity could.
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  39. #38, Stylo:

    According to the wiki on the Earth's energy budget (http://en.wikipedia.org/wiki/Earth%27s_energy_budget ), 1366 (W/m^2) is the instantaneous intensity of the sunlight at 1 AU. However, to compare that with the radiated power from the Earth, you have to convert that into a surface-area average. That gives a factor of 4.

    More explicitly:
    - The solar constant S = 1366 (W/m^2). The cross-sectional area of the Earth is pi*R^2, so the total power absorbed from the sun is
    P_in = S * (pi*R^2) = pi * R^2 * S
    (The point is that, at any one time, the sunlight only shines on the daytime half of the Earth; and you have to take the tilting of the surface relative to the rays. Both issues are taken care of by using the cross-sectional area to calculate total absorbed power.)

    - The time-averaged power radiated away through heat radiation, per unit area, is B (W/m^2). Therefore, the total heat radiation lost to space is:
    P_out = B * 4*pi*R^2 = 4 * pi * R^2 * B
    (This is emitted day and night, so over the entire area of the Earth.)

    - Since P_out is, on the average, almost perfectly equal to P_in,
    4 * pi * R^2 * B = P_out = P_in = pi * R^2 * S

    so:
    B = S/4 = 1366/4 = 341.5 (which is close enough to 342 for government work).

    So the match is perfect, to within the precision of these numbers.

    (If there were a 1366 - 342 - 1024 (W/m^2) difference between input power and output power, we would not be talking about "global warming": We would comparing recipes for "global roasting"!)
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  40. #39, cont'd:

    Stylo:

    So the point is that a little bit from solar input is worth just a little bit from the heat-radiation output: When compared on an apples-to-apples basis, total solar input to the Earth is essentially equal to total Terran heat-radiation output.

    So the point still stands: Additional water vapor will do much more to reduce the heat loss than it does for blocking incoming solar radiation.
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  41. stylo, #38.
    The incoming solar and outgoing terrestrial radiation spectra are different (different temperatures of sun and earth). GHGs selectively block frequencies nearer the peak of the earth's spectrum, so they stop only a small fraction of the incoming energy but a larger fraction of the outgoing energy.

    So a portion of the spectrum becomes essentially unavailable for the earth to use to radiate away energy, and it warms up until it emits enough energy at other frequencies to maintain the balance.

    If it helps, you can think of it as effectively lowering the emissivity of the earth at certain frequencies, by lowering the transparency of the atmosphere to those frequencies.
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  42. TonyWildish

    Pense em "ganho de antena". Só é possivel porque as ondas eletromagnéticas não seguem a 2nd lei da termodinamica. Elas seguem a lei da sobreposição linear das ondas mecanicas.
    Na atmosfera;
    Um mol de N2 colocando-se uma molécula ao lado de outra, cobriremos 92,7 m2/mol ou 6,62 m2/gr e cada m2 representa um elemento radiante. Como o ganho de antena é igual ao numero de elementos radiantes teremos 6,62 vezes a radiação de um m2 para cada grama de N2.


    Think of "antenna gain" . I's possible only because the
    Electromagnetics waves do not follow the 2nd law of thermodynamics.
    They follow the law of linear superposition of mechanics waves.
    In the atmosphere;
    One mole of N2 placing one next to another molecule,
    will cover 92.7 m2/mol or 6.62 m2/gr and each m2 represents
    a radiating element. Because the antenna gain is equal to
    number of radiating elements will have 6.62 times the
    radiation from an m2 per gram of N2.
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  43. Em tempo:
    A lei de Stefan-Boltzmann tem apenas dois parâmetros; area e temperatura.

    At time:
    The Stefan-Boltzmann law has only two parameters, area and temperature.
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  44. Tarcisio,

    Google-Translate is not providing a translation of your Portuguese adequate to the task of explaining what you are saying. Perhaps you can find a colleague who can clarify the output and render it into English that we can understand?

    I often use Google-Translate for German/English, and it takes some effort to make the English comprehensible. Quite often it misses an important word, "nicht", thus giving the exact opposite meaning of the original!
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  45. This article is great! I am particularly pleased to see how the many discussions of the blanket analogy in this forum have taken such excellent fruit in this article.

    But I do have to wonder: how exact is that translation of Clausius's original words? I wonder because the 'generally' does not sound right. The word I remember (from Pauli's formulation of Clausius's original words) is 'spontaneously'.

    This makes much more sense, and lends itself to precise interpretation much more easily. But precise interpretation is exactly what is so sorely missing from so much discussion of the 3 laws of thermo, especially when rebutting this ridiculous fallacy.
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  46. This is my very dump stupid explanation than even an engineer can understand:

    Without atmosphere the temperature of the sky would be roughly 3K or -270 C. The presence of the atmosphere make the sky much hotter.
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  47. Tarcisio #42,

    I speak Portuguese and I have a background in Electronics. Sorry, you cannot just magically multiply energy by antenna gain.

    If that was the case, I would very gladly prescribe re-transmitting the energy of our big Brazilian hydroelectrical powerstation over and over through the world as a definitive solution to the world's carbon-free energy problem.
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  48. #41 TonyWildish at 02:07 AM on 25 October, 2010
    you can think of it as effectively lowering the emissivity of the earth at certain frequencies, by lowering the transparency of the atmosphere to those frequencies

    No, you can't. In fact by lowering atmospheric transparency, you increase emissivity. That's Korchoff's law of thermal radiation.

    If the atmosphere would be in thermodynamic equilibrium, that is, if it were isothermal, lowering its transparency would increase not only emissivity, but also its emittance, so adding a small amount of GHG to a perfectly transparent atmosphere (let's say made of Nitrogen and Argon) would decrease its temperature (something like that is happening to Earth's stratosphere).

    However, the atmosphere is not in equilibrium, just in a more or less steady state with a continuous flow of energy going through it, associated with a steady rate of entropy production.

    If there's already some GHG content in such an atmosphere, the upper layers tend to cool faster (because they are more "visible" from space), which is why they are kept at a lower temperature. If you increase GHG content (as we have seen, also increasing emissivity by the very same act), the "photosphere" (the layer from which IR photons have a chance to escape to space unimpeded) goes higher, that is, to a colder place. Emittance is proportional to the product of emissivity (a dimensionless number) and the fourth power of absolute temperature. In some cases the latter one decreases more than the increase in the first term, so their product also decreases.

    But by decreasing the emittance, entropy production rate also decreases (while internal entropy contents of the system increases). Now, open systems with many degrees of freedom tend to assume a state of maximum entropy production (with minimum entropy contents, that is, a tendency to maintain internal order). So to prove increasing GHG contents of the atmosphere decreases overall IR emittance indeed, it is not enough to show photosphere rises to a higher layer in a narrow IR band, you also have to demonstrate the climate system has no way to reconfigure itself (even with its gazillion degrees of freedom) to a state which restores entropy production rate to its former value or at least makes it decrease less than indicated by first order GHG calculations.

    If you fail to do so, a negative feedback is likely (which makes AGW scare moot).
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  49. I agree to the 2nd to 5th paragraphs of #48 by Berényi Péter (#48), but I disagree to the last two paragraphs. Excuse me, I cannot fully write my reasoning right now.

    For entropy production in the climate system, see Ozawa (2003) (Unfortunately I do not find a free copy). The authors suggest that the principle of maximum entropy production seems to be applicable to fluid motion processes, but not to the whole climate system including radiative processes.
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  50. #49 Kooiti Masuda at 22:42 PM on 25 October, 2010
    For entropy production in the climate system, see Ozawa (2003) (Unfortunately I do not find a free copy)

    Dear prof Masuda, here is a copy of the paper you are referring to.

    Reviews of Geophysics, 41, 4 / 1018 2003
    doi:10.1029/2002RG000113
    THE SECOND LAW OF THERMODYNAMICS AND THE GLOBAL CLIMATE SYSTEM: A REVIEW OF THE MAXIMUM ENTROPY PRODUCTION PRINCIPLE
    Hisashi Ozawa, Atsumu Ohmura, Ralph D. Lorenz & Toni Pujol

    Unfortunately their claim "On the contrary, absorption of radiation is essentially a linear process; its rate is given by the flux of radiation multiplied by the absorptivity of the material under consideration. There can be no feedback mechanism for the strength of the flux or the absorptivity in this process. Radiation can therefore be seen to be just an energy source for the climate system." is not a valid one, as absorption of radiation clearly depends on things like clouds, snow cover, foliage and airborne dust. Looks like there can be feedback mechanisms after all, as distribution of many absorbers are determined by climate.

    Their omission of the factor 4/3 from the standard expression of radiation entropy is strange.

    Also, total entropy production is the difference between entropy of outgoing (reflected/dispersed short wave and emitted thermal) radiation and that of incoming sunlight. The MEP, being a pretty universal principle covering all steady state open thermodynamic systems with sufficient degrees of freedom, should not differentiate between internal modes of energy transfer in a way they claim.
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