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Learning from a simple model

Filed under: — gavin @ 10 April 2007

A lot of what gets discussed here in relation to the greenhouse effect is relatively simple, and yet can be confusing to the lay reader. A useful way of demonstrating that simplicity is to use a stripped down mathematical model that is complex enough to include some interesting physics, but simple enough so that you can just write down the answer. This is the staple of most textbooks on the subject, but there are questions that arise in discussions here that don’t ever get addressed in most textbooks. Yet simple models can be useful there too.

I’ll try and cover a few ‘greenhouse’ issues that come up in multiple contexts in the climate debate. Why does ‘radiative forcing’ work as method for comparing different physical impacts on the climate, and why you can’t calculate climate sensitivity just by looking at the surface energy budget. There will be mathematics, but hopefully it won’t be too painful.

So how simple can you make a model that contains the basic greenhouse physics? Pretty simple actually. You need to account for the solar radiation coming in (including the impact of albedo), the longwave radiation coming from the surface (which depends on the temperature) and some absorption/radiation (the ’emissivity’) of longwave radiation in the atmosphere (the basic greenhouse effect). Optionally, you can increase the realism by adding feedbacks (allowing the absorption or albedo to depend on temperature), and other processes – like convection – that link the surface and atmosphere more closely than radiation does. You can skip directly to the bottom-line points if you don’t want to see the gory details.

The Greenhouse Effect

The basic case is set up like so: Solar radiation coming in is S=(1-a) \mbox{TSI}/4, where a is the albedo, TSI the solar ‘constant’ and the factor 4 deals with the geometry (the ratio of the area of the disk to the area of the sphere). The surface emission is G=\sigma T_{s}^{4} where \sigma is the Stefan-Boltzmann constant, and  T_s is the surface temperature and the atmospheric radiative flux is written \lambda A=\lambda \sigma T_{a}^{4}, where \lambda is the emissivity – effectively the strength of the greenhouse effect. Note that this is just going to be a qualitative description and can’t be used to quantitatively estimate the real world values.

There are three equations that define this system – the energy balance at the surface, in the atmosphere and for the planet as a whole (only two of which are independent). We can write the equations in terms of the energy fluxes (instead of the temperatures) since it makes the algebra a little clearer.

Surface:  S + \lambda A = G
Atmosphere: \lambda G = 2 \lambda A
Planet: S  = \lambda A + (1-\lambda) G

The factor of two for A (the radiation emitted from the atmosphere) comes in because the atmosphere radiates both up and down. From those equations you can derive the surface temperature as a function of the incoming solar and the atmospheric emissivity as:

G=\sigma T_s^4= {S\over(1 - 0.5\lambda) }

If you want to put some vaguely realistic numbers to it, then with S=240 W/m2 and \lambda=0.769, you get a ground temperature of 288 K – roughly corresponding to Earth. So far, so good.

Point 1: It’s easy to see that the G (and hence T_s) increases from S to 2S as the emissivity goes from 0 (no greenhouse effect) to 1 (maximum greenhouse effect) i.e. increasing the greenhouse effect warms the surface.

This is an extremely robust result, and indeed has been known for over a century. One little subtlety, note that the atmospheric temperature is cooler than the surface – this is fundamental to there being a greenhouse effect at all. In this example it’s cooler because of the radiative balance, while in the real world it’s cooler because of adiabatic expansion (air cools as it expands under lower pressure) modified by convection.

Radiative Forcing

Now what happens if something changes – say the solar input increases, or the emissivity changes? It’s easy enough to put in the new values and see what happens – and this will define the sensitivity of system. We can also calculate the instantaneous change in the energy balance at the top of the atmosphere as \lambda or S changes while keeping the temperatures the same. This is the famed ‘radiative forcing’ you’ve heard so much about. That change (+ve going down) is:

F_{Top}= \Delta S + \Delta \lambda (G_0 - A_0) = \Delta S + {{0.5 \Delta \lambda S } \over { (1-0.5\lambda) }}

where \Delta S, \Delta \lambda are the small changes in solar and change in emissivity respectively. The subscripts indicate the previous equilibrium values We can calculate the resulting change in G as:

\Delta G \sim {\Delta S \over { (1-0.5\lambda) }} + {0.5 S \Delta \lambda \over { (1-0.5\lambda)^2 }} ={ F_{Top}\over { (1-0.5\lambda)}}

so there is a direct linear connection between the radiative forcing and the resulting temperature change. In more complex systems the radiative forcing is a more tightly defined concept (the stratosphere or presence of convection make it a little more complex), but the principle remains the same:

Point 2: Radiative forcing – whether from the sun or from greenhouse gases – has pretty much the same effect regardless of how it comes about.

Climate Sensitivity

The ratio of \Delta G/F_{Top} is the sensitivity of G to the forcing for this (simplified) system. To get the sensitivity of the temperature (which is the more usual definition of climate sensitivity, \Delta T_s/F_{Top}), you need to multiply by {0.25\over\sigma T_s^3} i.e. {0.25\over\sigma T_s^3 (1 - 0.5\lambda) }. For the numbers given above, it would be about 0.3 C/(W/m2). Again, I should stress that this is not an estimate for the real Earth!

As an aside, there have been a few claims (notably from Steve Milloy or Sherwood Idso) that you can estimate climate sensitivity by dividing the change in temperature due to the greenhouse effect by the downwelling longwave radiation. This is not even close, as you can see by working it through here. The effect on G due to the greenhouse effect (i.e. the difference between having \lambda=0 and its actual value) is { 0.5\lambda S\over(1 - 0.5\lambda) }, and the downward longwave radiation is just \lambda A, and dividing one by the other simply gives \lambda – which is not the same as the correct expression above – in this case implying around 0.2 C/(W/m2) – and indeed is always smaller. That might explain it’s appeal of course (and we haven’t even thought about feedbacks yet…).

Point 3: Climate sensitivity is a precisely defined quantity – you can’t get it just by dividing an energy flux by any old temperature.


Now we can make the model a little more realistic by adding in ‘feedbacks’ or amplifying factors. In this simple system, there are two possible mechanism – a feedback on the emissivity or on the albedo. For instance, making the emissivity a function of temperature is analogous to the water vapour feedback in the real world and making the albedo a function of temperature could be analogous to the ice-albedo or cloud-cover feedbacks. We can incorporate the first kind of physics by making \lambda=f(T_s) dependent on the temperature (or G for arithmetical convenience). Indeed, if we take a special linear form for the temperature dependence and write:

\lambda (G) =\lambda_0 + \lambda^\prime ({G\over G_0}-1)

then the result we had before is still a solution (i.e. \lambda_0=0.769, G_0={S\over (1-0.5\lambda_0)}=390). However, the sensitivity to changes (whether in the greenhouse effect or solar input) will be different and will depend on \lambda^\prime. The new sensitivity will be given by

\Delta G \sim { F_{Top}\over { (1-0.5(\lambda_0+\lambda^\prime))}}

So if \lambda^\prime is positive, there will be an amplification of any particular change, if it’s negative, a dampening i.e. if water vapour increases with temperature that that will increase the greenhouse effect and cause additional warming. For instance, \lambda^\prime=0.1, then the sensitivity increases to 0.33 C/(W/m2). We could do a similar analysis with a feedback on albedo and get larger sensitivities if we wanted. However, regardless of the value of the feedbacks, the fluxes before any change will be the same and that leads to another important point:

Point 4: Climate sensitivity can only be determined from changes to the system, not from the climatological fluxes.


While this is just a simple model that is not really very Earth-like (no convection, no clouds, only a single layer etc.), it does illustrate some relevant points which are just as qualitatively true for GCMs and the real world. You should think of these kinds of exercises as simple flim-flam detectors – if someone tries to convince you that they can do a simple calculation and prove everyone else wrong, think about what the same calculation would be in this more straightforward system and see whether the idea holds up. If it does, it might work in the real world (no guarantee though) – but if it doesn’t, then it’s most probably garbage.

N.B. This is a more pedagogical and math-heavy article than most of the ones we post, and we aren’t likely to switch over exclusively to this sort of thing. But let us know if you like it (or not) and we’ll think about doing similar pieces on other key topics.

296 Responses to “Learning from a simple model”

  1. 101
    Robin Johnson says:


    The geometry of the sphere works like this. If we draw a circle around the Earth where light transitions to dark and then project an imaginary cylinder back to the Sun, we can measure the solar output striking the Earth. Even though the Earth is a bulge at the bottom of that cylinder – the amount of solar light hitting the Earth (at whatever angles) is just the cross section of the cylinder – which is exactly the area of that circle we drew – which is, of course, given by Ï� * r2.

    It turns out that the surface area of a sphere is given by the formula 4 * � * r2. Even though the light is only striking half the Earth, we really want to know the total energy distributed across the entire surface area of the Earth since the acquired heat is not instantaneously radiated back into space. Given that, we divide the area of the sphere by the circle, we get really simple ratio of 4 to 1. So, we have to divide the solar output per meter squared by 4 to get the actual solar output per square meter over the Earth.

  2. 102
    DeWitt Payne says:

    [what is emissivity]

    An emissivity of zero means the object in question is either a perfect reflector or is perfectly transparent and as a result does not absorb and hence cannot emit radiant energy and has nothing to do with being surrounded by insulation. Gold metallized plastic film is a near perfect reflector and is used on spacecraft to prevent radiant heating from sunlight. A planet with an atmosphere of a noble gas like argon would have no greenhouse effect because argon is transparent in both the visible and infra-red wavelength region. If you have an infra-red kitchen thermometer and don’t use the emissivity correction, a stainless steel pot full of boiling water will apear to be at a much lower temperature than a black anodized aluminum pot full of boiling water. Stainless steel is a good reflector and thus has a lower emissivity than black anodized aluminum.

  3. 103
    DeWitt Payne says:

    If lambda can’t be greater than one, then isn’t the maximum surface temperature 303 K with this model, given a constant solar irradiance of 240 W/sq.m.? Yes, that’s fifteen degrees warmer than now, but it’s also an upper limit. I second the motion for something between this oversimplified model and a full bore planetary GCM. How about creating a multi-layer spreadsheet model or program of a one-dimensional, clear air, Earth normal nitrogen, oxygen and argon atmosphere containing only water vapor and carbon dioxide as ghg’s with graphs and charts of temperature and pressure with altitude. Of course, the more user adjustable parameters, the better. Including sensible and latent heat transfer from the surface and how they change with surface temperature and humidity would be nice too.

  4. 104
    The Wonderer says:

    Re: #84: Ike,

    I agree cross-field misunderstandings may be many and Iâ??m certainly not suggesting that engineers are all-knowledgeable or infallible, by any means. Climate science is certainly rich in underlying complexity, and I do not pretend to be an expert (unlike the engineer to whom you refer). Iâ??m just saying the misunderstandings are probably not due to the math itself. Linear algebra as used in this example should be in the basic toolkit of any engineer, and while climate feedbacks are multi-variable and non-linear, they are conceptually the same if not analogous to those found in other disciplines. Someone who believes that engineers â??know just enough math to get into troubleâ??, lives in the modern world, and is otherwise rational, should be very afraid to get out of bed in the morning. As far as history professors, wellâ?¦. theyâ??re interesting too.

  5. 105
    Pat says:

    Here’s how I might try to qualitatively introduce radiation transfer in a more complete manner:

    (LW = longwave = terrestrial = the kind of radiation emitted at typical atmospheric (except thermosphere, which, in terms of amount of energy, has a very small role to play in the totality of atmospheric radiation) and near-surface temperatures of the Earth – as opposed to SW = shortwave = solar (radiation))

    (optical depth, optical thickness, optical length, optical distance* = a measure of path length relative to the rate of absorption+scattering along that path – an opaque object has infinite optical depth, a transparent object has zero optical depth; the proportion of light starting at a point and moving along a path of optical thickness t which is neither scattered nor absorbed is exp(-t)
    * PS I’m assuming all these terms can be used, correct me if I’m wrong)

    (To the degree that objects are exchanging LW radiation, the net heat transfer via radiation will be from hot to cold)

    At any given level in the atmosphere (and at any given wavelength), radiation coming from any direction is originating from a distributed source along the pathlength, the distribution declining exponentially with optical distance; as the optical thickness increases along a given geometric distance, photons travel shorter paths, so the radiative energy exchange increases along shorter distances while decreasing along longer distances; hence, for LW radiation:

    increasing LW opacity (from increasing greenhouse gas concentration, or from clouds)
    => increased cooling to space from those parts of the atmosphere optically closest to space, decreased cooling from the surface and, if the starting opacity is high enough, lower atmosphere, to space, and if the starting LW opacity is high enough, decreased cooling from the lowest parts of the atmosphere to the colder upper parts, although – depending on the temperture profile and other details (wavelength dependencies) – increased cooling from the very top of the troposphere and the upper stratosphere to the base of the stratopshere, etc.

    Temperature rises or falls until, at any given level, emitted radiant power per unit atmosphere = absorbed radiant power per unit atmosphere + convergence of sensible heat transport by convection + net latent heat release per unit atmosphere

    net latent heat release = ((condensation + freezing) – (evaporation + melting))

  6. 106
    Harry Haymuss says:

    Re the response to #81, that sounds good until one remembers that Antarctica is over 13 million square km. Surely you’re not suggesting ghg warming of that much area is all damped by an ocean sometimes thousands of km away? Just doesn’t hold water…

  7. 107
    Harry Haymuss says:

    Re #71, your example of water vapor being a positive feedback is unrealistic. Clouds are a product of water vapor, and it is not known if that feedback is positive or negative. To say “water vapor is a positive feedback” is academic – and I thought we were trying to address reality here.

    You didn’t address #64, which states feedbacks *double or triple* the effect of CO2.

  8. 108

    Concerning G: Am I right that the given expression assumes that the emissivity of the ground is 1? There was a presentation at Santa Fe (2006) by Hartwig Volz from RWE showing that neglecting the emissivity of the large ocean surfaces (which is lower than 1) leads to climate sensitivities that are too high, and that this correction is absent from all (?) climate models used by the IPCC. What does Gavin think about this?
    BTW: I really appreciate some more theoretical contributions like this one. Please go on! Just a recommendation from someone who has taught physics since nearly 40 years: please don’t jump over intermediate calculating steps (like the derivative of G), short-cuts are a challenge for the reader but may put off someone used to more extended developments.

    [Response: This derivation assumed an emissivity of 1 at the surface. In the real world it’s slightly less (0.95 or so depending on various conditions), but that just divides through to the sensitivity and actually makes it slghtly larger (since dG/dTs is smaller => dTs/dFtop increases). This is important in various remote sensing applications, but a minor effect on the climate. And since it doesn’t change appreciably as a function of climate, it can’t be a dominant force in climate variability. -gavin]

  9. 109
    Dan says:

    re: 71. Goodness. You obviously did not read the link I provided:
    More specifically, the link contained in that link to Soden’s paper.

    Case closed. It is really not all that difficult to research it yourself and actually learn something instead of trolling with no basis in fact. Talk about reality. Sheesh.

  10. 110
    Dan says:

    re: 71. Lots more on water vapor being a positive feedback:

    Again, that took 10 seconds to research.

  11. 111
    Harry Haymuss says:

    Re 109 and 110:
    I’m not arguing that water vapor itself, with everything else remaining constant, is not a positive feedback.

    You are confusing that simple concept with the reality that increasing water vapor (including increasing the rate of convection) increases cloud cover, and we don’t know the overall effect of that.

    Look at the Pliocene. Much warmer, but indications are that the tropics were not much warmer than today and that may have been due to cloud cover.

  12. 112
    Dan says:

    Appropos to tropical clouds and feedbacks: Lin, Wielicki, et al., 2002, “The Iris Hypothesis: A Negative or Positive Cloud Feedback?” which found a weak positive feedback from tropical upper-tropospheric anvils. From

  13. 113

    [[There was a presentation at Santa Fe (2006) by Hartwig Volz from RWE showing that neglecting the emissivity of the large ocean surfaces (which is lower than 1) leads to climate sensitivities that are too high, and that this correction is absent from all (?) climate models used by the IPCC.]]

    I think you’re right that most (not all) GCMs set surface emissivity to one. I looked up emissivity figures for water and found these:

    0.93 — Transcat News,

    0.92-0.97 — Oke, T. R., (1987) Boundary layer climates. 2nd ed., Methuen, N.Y.

    0.95 — Jin Menglin and Liang Shunlin 2006. “An Improved Land Surface Emissivity Parameter for Land Surface Models Using Global Remote Sensing Observations.” J. Climate 19, 2867-2881.

    0.9637 at 3.7 microns, 0.9775 at 10.8 microns, 0.9431 at 12.0 microns, and 0.9967 at 8.5 microns — Surface Spectral Emissivity Derived from MODIS Data, Yan Chen, Sunny Sun-Mack, Patrick Minnis, David F. Young, William L. Smith, Jr. 2002. Extended Abstract for SPIE 3rd International Asia-Pacific Environmental Remote Sensing Symposium 2002: Remote Sensing of the Atmosphere, Ocean, Environment, and Space, Hangzhou, China, October 23-27, 2002

    [Response: I checked with our radiation people, and find that indeed, we do include longwave emissivity/albedo effects (which depend on spectral frequency and wind speed). It is a minor effect though. – gavin]

  14. 114

    Got cut off from that last one when my keyboard failed. Had to do a hard reboot to get it back again.

    The mean of the eight emissivities I found for water is 0.96, the standard deviation is 0.047. This is a significant difference; if applied over the whole Earth, it would mean we’re radiating at 374 watts per square meter instead of 390, a difference of 16 watts per square meter. Can any professionals comment? It looks like the skeptics might actually have a point here.

  15. 115

    My modest contribution to the discussion.

    I’d appreciate any feedback you think relevant, thx:-)

  16. 116

    [[I’m not arguing that water vapor itself, with everything else remaining constant, is not a positive feedback.
    You are confusing that simple concept with the reality that increasing water vapor (including increasing the rate of convection) increases cloud cover, and we don’t know the overall effect of that.

    I don’t think it’s clear that increased water vapor does increase cloud cover. I know Hart (1978, 1979) used such a relation in his model, but I think that was one of the parameterizations Schneider and Thompson (1981) said might not be valid. I’m not sure what the state of the question is these days. Does anyone know what determines cloud fraction, or it still an open question?

  17. 117
    Ike Solem says:

    RE the water vapor issue, see Robust Responses of the Hydrologic Cycle, Held&Soden 2006 for a discussion of the robust increase in lower tropospheric water vapor. The key number is the relationship of the saturation vapor pressure to temperature; for a 1K increase in temp there is a 7% increase in saturation vapor pressure. For a 3K equilibrium climate sensitivity, that’s about 20% increase in the amount of water that exists as vapor in the air (that’s derived from Clausius-Clapeyron). One of the robust responses is horizontal moisture transport from the equator to the poles (which also means latent heat transport). This means that warming at the poles should continue to accelerate, and a moister atmosphere should also encourage hurricane formation, as should warmer sea surface temps.

    What is less certain is the upper tropospheric water vapor, but the EOS MLS program is now measuring that. The greatest uncertianty wrt clouds seems to be the radiative effect of tropical clouds in the marine boundary layer… but is there any indication that cloudiness is increasing as the planet warms? In fact, the glocal record produced by the International Satellite Cloud Climatology Project shows a decreasing trend in global cloudiness, but the suitability of the data for trend analysis has been questioned (Evan et al 2007)… but it doesn’t seem that there is any evidence of increasing cloudiness.

    Incidentally, Lindzen has been given yet another media platform to promote his viewpoints: Newsweek ran “Why So Gloomy” in which Lindzen now claims that “A warmer climate could prove to be more beneficial than the one we have now.” and he also claims that “There is no evidence, for instance, that extreme weather events are increasing in any systematic way” (what does he mean by ‘systematic’? Is he ignoring the hurricane trend? The heat waves?) He also states that “Sea levels, for example, have been increasing since the end of the last ice age” while attributing the vastly accelerating rate of sea level rise to ‘short-term fluctuations’.

    Amazingly, he trots out this statement: “Many of the most alarming studies rely on long-range predictions using inherently untrustworthy climate models, similar to those that cannot accurately forecast the weather a week from now.” This is just nonsense.. see the real climate post Short and Simple Arguments . (this is why realclimate is such a valuable site!)

    He then goes on to make an economic argument, something he knows nothing about: “Moreover, actions taken thus far to reduce emissions have already had negative consequences without improving our ability to adapt to climate change”. This is even more nonsense, since renewable energy technology will result in economic expansion, new jobs, global energy security in an era of tight oil supplies – solar photovoltaics and wind technology coupled to innovative energy storage systems (hydrogen production, for example) are topics that Lindzen also ignores, while instead claiming that ethanol production will result in global starvation (and he also claims, like Sherwood Idso’s CO2science, that agricultural productivity will increase in a warmer world – ignoring the effects of temp extremes and drought…)

    What Lindzen also completely ignores is the well documented changes occuring in polar regions, and he makes no mention of the potential instability of the West Antarctic and Greenland Ice Sheets. The article reads like a list of fossil fuel lobby talking points, and has already been reposted all over the web. You can email Newsweek at and let them know what you think of this.

  18. 118
    Harry Haymuss says:

    Re 17:
    Aren’t you putting the cart before the horse when you say: “For a 3K equilibrium climate sensitivity, that’s about 20% increase in the amount of water that exists as vapor in the air ”

    Also, aren’t you being dogmatic when you deny even the *possibility* that warming may be good for us?

    Plus, I know I’m being a heretic here, but what exactly did you find wrong with Lindzen’s statement? Certainly not all of it…

  19. 119
    Eric says:

    Similar to comment #103, I checked the ratio of the temperature with full greenhouse effect (lambda=1) to no greenhouse effect (lambda=0). It is 2^0.25 = 1.189, so that going from no atmosphere to a fully absorbing atmosphere in this simple model only increases the surface temperature by 19% ! It seems to me that by definition no physical object can have emissivity greater than one, so that this simple model cannot explain the runaway greenhouse effect like on Venus. Could you elaborate on some of the other physical effects that can increase the temperature even more beyond this simple model for much higher concentrations of CO2 than exist on Earth now, like on Venus?

    [Response: You can make the strength of the effect as large as you like by adding layers. Each additional layer adds a factor of 1.189 to the temperature difference between the blackbody and the surface. However, in the real world the percentage difference between having a greenhouse effect and not (measured in Kelvin of course), is around 11 to 12% (i.e. 33 deg K out of 288 K). You are correct though, this model as is couldn’t cope with Venus… – gavin]

  20. 120

    Is not the solar constant even more divided, at high latitudes? Is their not an obliquity factor related to the latitude of the location? 288 Kelvin may be a number for the equator? It gets interesting at the Poles especially when there is no Solar constants during the long night, when S/(1-0.5*lambda) is equal to 0.

  21. 121
    Robin Johnson says:

    Wayne. See post #96.

  22. 122

    [[Aren’t you putting the cart before the horse when you say: “For a 3K equilibrium climate sensitivity, that’s about 20% increase in the amount of water that exists as vapor in the air “]]

    If you plug a 3 K increase in temperature, from 288 K to 291 K, into the Clausius-Clapeyron equation for saturation water vapor pressure, you get a 19% increase.

    [[Also, aren’t you being dogmatic when you deny even the *possibility* that warming may be good for us?]]

    It’s not really an open question at this point. We’re looking at narrowing of growing belts, increased drought in continental interiors, more violent weather on continental margins, and unpredictable changes at the local level. Bad seems more likely, on the whole.

    One doesn’t want to be dogmatic about it. No doubt some areas will benefit. But on average, it seems to be a bad prospect.

  23. 123
    James says:

    There is a critical flaw in the basic model.

    In the radiation field of a blackbody, a gas will obtain the Boltzmann distribution of states. This equilibrium is maintained by collision.

    I am aware of the LTE hypothesis.

    It is true that two black bodies radiating at each other will have the same radiation field at the same temperature, and thus absorption will equal emission.

    It is also true that two gas samples in the same state (composition, pressure, temperature, etc.) will have identical radiation properties and when in thermal equilibrium there will be no net transfer of radiation, absorption = emission.

    It is not true that a gas in the cavity of a black body will take on the radiation characteristics of the black body at thermal equilibrium of the entire system.

    Thermal equilibrium simply means that the average kinetic energy of the two systems is the same. A particle in a gas has considerably more translational freedom than a particle in a solid. Because of the large amount of continual random translational motion in a gas, it should be expected that its radiative power, when at the same temperature as a solid, will be considerably less than that of the solid.

    A solid at a given T will radiate as a function of T^4. A gas will not. A gas will radiate as a function of the T, which collisionally provides the Boltzmann distribution between excited and ground states,and the pressure, which must be used to account for the excited state lifetime. This provides for a considerably smaller radiation field associated with a gas than that of a solid.

    The basic assumption in the very first atmospheric layer that all of the energy the gas absorbs from the planet must be emitted is false. This assumption is then propogated up through the atmosphere layers. The net effect is that the model claims a T^4 power dependence for gas emission, because the surface (which is solid) has this power dependence.

    A much better model is to use the Boltzmann distribution for the gas excited state population in the radiation field of a black body – Atkins is a PChem book that provides a reasonable basic treatment.

    Using a basic two level system for the 15 micron absorption band provides an estimate that 4% of the CO2 molecules will be in the excited bending mode at any given time, when in the field of a black body at about 300K. This takes into account stimulated absorption and emission.

    Spontaneous emission is trivial at 1 atm pressure. It is present however, and a small radiation field will be maintained, and passed from atmospheric layer to atmospheric layer.

    It is simply incorrect to assume that all of the radiation CO2 absorbs from the planet’s surface will be subsequently reemitted, and it is this incorrect assumption that drives this theory.

    Consider however if it were true. In this model, only the CO2 is actually being heated, not the molecules around it. All excited states survive to emit, and the radiation field is maintained. Such a model actually does not lead to warming of the atmosphere, just the CO2 component.

    Gasses are not solid, and should not be treated as such. The Boltzmann distribution is well known, and should be used.

    It is not an accident that much of the basic theory used for global warming was devised before the introduction of quantum mechanics in 1900. The T^4 power dependence in these models is continuously scaled throughout a calculation because of the attempt to stuff a gas phase problem into a solution devised for solids. (Need we explain the band theory of solids and how it provides for continuum radiation whereas gas phases do not?)

    Well, I suppose that should rile a few folks and provide quite a bit of clucking. Sometimes it is fun to stir the hornets nest.

    [Response: No hornets here. I produced this to demonstrate pedagogically how the greenhouse effect works, not as a model for the real atmosphere (as I think I made clear). If you think that your explanation serves that purpose better, I would beg to disagree. Serious models (i.e. line-by-line codes or the radiative transfer models in GCMs) do this properly and certainly do not assume that all radiation absorbed is re-emitted. – gavin]

  24. 124
    jae says:

    89 and related responses to #22: I understand that more energy is “used” to form water vapor support photosynthesis in more humid/lush areas, compared to deserts. But this use of energy appears to lower the temperature, compared to a desert situation. My question was: does this not show that water vapor feedback exerts a NEGATIVE feedback to surface temperature?

  25. 125
    Richard Ordway says:

    re. 118 Troll. Please read this site before asking your question:

    [[Also, aren’t you being dogmatic when you deny even the *possibility* that warming may be good for us?]]

    If you did read this site, you would find that the negative effects far outweigh the good effects of warming.

    Warming can and does have some beneficial short-term effects by benefiting longer growing seasons in some areas, reducing heating bills, reducing freezing deaths, increasing some crop growth rates and opening up new navigation areas (already done)…but this is most likely minor compared to:

    The possible negative effects of ending/changing monsoons, water shortages, increasing storm damage, crop failures, flooding, sudden rising sea levels, abrupt climate change, wars, unrest, increased allergies, lung deaths (low-level ozone), mass human migration, species extermination (most likely already happened), coral destruction (oops, now many second and third world countries would lose alot of their fishing), increased diseases, ocean acidification (if you dump in co2) and possible wholesale evacuations of cities, islands and coastlines…

    and dare I say it…a threat to the integrity of our whole civilization itself through suddenly straining our federal system past its limits to adapt, fund and control too many combinations of sudden immigration pressures (tens of millions of “starving people ain’t goin to stay ‘down there’ when we ‘rich folks’ got food up here”), food shortages, mass evacuations, world economic collapse, civil unrest, and multiple simultaneous wide-spread disaster relief efforts…

    While suddenly trying to fight a rapidly increasing terrorist movement derived from a billion+ possible candidates from the second and third worlds who suddenly might not have food, a government or hope and lots of free A-bombs.

    Science always tries to be open to “possiblities”, so you do hear some of the possible good short term effects of warming.

    However, if you read the literature and talk to real scientists who do sustained peer-review research (that holds up under long-term scientific scrutiny), they will almost invaribly tell you that this current rapid average rate of warming is not good overall, is potentially very dangerous and far outweighs the possible good effects.

    …and by the way, Lindzen does not currently do peer-reviewed global warming research that stands up under scientific scrutiny…

  26. 126
    jae says:

    Question: Why isn’t the equation for surface flux: S + 1/2*lambda*A = G, since half the atmospheric flux goes up and half goes down?

  27. 127
    John Mashey says:

    re: #125

    There may be short-term benefits of warming (like Richard Selley’s amusing comments about the great future for wineries in Scotland after 2100), but there are plenty of concrete, specific issues that matter-of-fact state economic planners worry about … and this shows up rather strongly in which states are pushing for policies to deal with climate change. These people are not dealing with global theoretical concepts, they are trying to quantify direct effects in specific geographic areas they understand.

    Here in California:

    a) The state planners take the issues *very* seriously, because we already have serious water problems, and lower Sierra snowpacks don’t help one bit.

    b) 50% of the USA’s fruit and vegetables are grown here, sometimes in very limited areas where the soil & climate are “just right”, and a significant change of temperature in either direction will be disruptive.

    c) Good wineries won’t be helped either.

    d) We have a lot of coastline, and we have some farmlands that are already below sea level [Sacramento Delta].

    e) We have noticable travel industry, including ski resorts … and lower snowpack doesn’t help there either.

    f) Peak power usage is summer afternoons, i.e., air conditioning.

    f) In fact, for CA, it’s hard to identify economic benefits of warming, and easy to identify large downsides.

    g) All of this has something to do with CA’s persistence in pushing on energy efficiency, gas mileage improvements, clean air acts, etc.

    For others, there may be some mixed blessings: it’s probably milder in State College, PA, than when I was going to school there, but it’s not even an unmixed blessing for Northern states:

    – New Hampshire and Vermont worry about their ski resorts as well, for the same reason as California, but with less altitude. At least in NH, many southern resorts have closed for good, and tourism taxes are very important (to keep other taxes down).
    – They worry about the future of maple sugar … [sugar maples need cold]
    – They worry about the brilliant Fall Foliage (tourism) migrating 100-300 miles North.
    – Maine, Massachusetts, Rhode Island, Connecticut, New York, New Jersey have substantial low-lying seacoast. Most have at least some dense cities right on the coast, and have had experience with storm surges there.

    Southwest states worry about drought and cost of more air-conditioning; Pacific Northwest states worry about snowpack.
    Southeast has lots of coastline, water issues, and will want more air-conditioning.

    Anyway, I’ve long been curious about Lindzen’s current worldview, but I certainly trust state planners more to know whether warming benefits them or not.

  28. 128

    #121 Robin thanks double that Tamino! I was referring to what the equation speaks out loud, which is solar radiation spread out evenly on Earth as a sphere despite instantaneous solar radiation reaching only one side at any given time, half a sphere. There may be some deeper meaning to this; why is the radiation condidered spread out evenly when in reality it does not? The very reason for Climate/Weather is a differential in temperature , pressure driven by diurnal effects and the strength of solar radiation at every given location. A philosophical question so to speak, just exploring the meaning of this ‘spread the energy evenly about idea.’.. I would rather enjoy equations dealing with the dark side of the Earth and one with only the bright side. Applying 1/2 a sphere with T^4=S/(SBconstant *(1-0.5*lambda))gives interesting results,applying it to the dark side does not give any result. I really like simple models /equations as concepts explaining things, if well written they have great impact on spreading out knowledge of climate science. …..This is another great post!

  29. 129
    Alan D. McIntire says:

    Regarding no. 126.
    In the model, the 1/2 lambda going down to hit the earth will be re-radiated by the earth, giving an additional 1/4 lambda to space, 1/4 lambda back down to earth. That 1/4 lambda will be re radiated, of that, 1/8 will go up, 1/8 down.
    You wind up with a series 1/2 + 1/4 + 1/8 + 1/16…. that sums to 1.

  30. 130
    Robin Johnson says:

    Re #124 (jae)

    Well. Its tricky actually. And do you mean globally or locally?

    Locally, comparing a dry area [desert] with a wet area [ocean/forest] the main observable effect is that the daytime air temperature is lower but night time temperature is higher in the wet area. Since water vapor is a strong GHG, there is more heat rentention overnight plus the condensation effect – so more heat is retained by a wet atmosphere than a dry. But water vapor forms clouds which increases albedo reduces solar heating – a negative feedback. Also, water vapor is not as dense as “regular” oxygen-nitrogen air so it undergoes strong convection and lofts high in the atmosphere creating wind which transfers heat towards the polar regions [local negative feedback – but negative and positive global feedbacks]. The presence of plenty of non-frozen water means lower albedo – either its open water (very dark) or heavily covered in vegetation – so that’s a positive feedback assuming the desert is lighter in coloration (there are exceptions but that’s fairly true). Oceans however convect heat into the deep, convert heat into mechanical energy [currents] and plants consume energy to grow, retain water and sequester carbon. So those are negative feedbacks.

    Globally, warming the polar regions creates, you guessed it, more water vapor increasing the GHG effect – so that’s another set of positive and negative feedbacks. However, this convection process has taken the water vapor higher up where its longwave radiation leaves the atmosphere more easily (less absorption) creating a negative feedback. Also, heat escapes more readily from the polar regions – its drier and less hours of sun – also a negative feedback. Rain and snow have both negative and positive feedbacks.

    YET the global negative and positive feedbacks reach equilibrium at some point for a given solar input and other factors (atmosphere thickness, planet albedo, geography, other GHG).

    Locally, I as claimed earlier, the average temperature (total heat?) is probably going to be higher in the wet environment. But the day time high is going to be much higher in the dry desert than the wet forest. But that’s okay – its a DRY heat…

  31. 131
    Rod B. says:

    Interesting gas vs. solid discussion. When does a gas become dense enough to act like a liquid/solid? What is the surface of the Sun, e.g?

  32. 132
    Dr. M. Jorgensen-Petersen says:

    You may not want to answer this quesion, since it is politically tainted, but Laurie David and Sheryl Crow are currently touring the country lecturing to college students about global warming. One of their major points is that the current and recent unusually cold and snowy national weather is a sign of anthropogenic global warming, she is quoted here: “The things you mention like worse snowstorms and a chaotic start to baseball season are all symptoms of global warming. Global warming causes extreme weather in BOTH directions. For example, the reason the blizzards are getting worse in the Northeast is because the Great Lakes are no longer freezing over thus fueling the stronger snow storms that are topping the headlines and disrupting the start of the baseball season”. Many of my students know better and challenge this as unsceintific, but it seems this is becoming urban legend. I know this blog is usually politically sympathtic to the likes of Ms. David, but is there any science to back up her pronouncements? Thanks.

  33. 133

    #132, The last bit of cold in USA and Southern Canada now, was caused by the lack of mixing, a stable body of cold air in the North American Arctic in March. It is a small part of the world which is cooler than Normal, the big picture, planet Earth: is on the whole warmer at the same time. The two, cooler in North America and warmer rest of the world are linked by weather systems which exacerbate cooling by creating relatively small windows of clear air especially at locations in darkness or with low sun elevations which are surrounded by much larger above warmer than normal areas. The reverse can happen as well, when clouds and advection warm up the Polar regions, this translates into much warmer weather at lower latitudes, the latter has been happenning more often over the last few years. Being just a meteorological observer, I am recently seeing stranger and stranger weather, having very unfamiliar patterns. There is definitely Climate system Change going on on a world wide scale all of it linked by overall warming. So I do care about what the singers and poets are thinking, especially when they are right.

  34. 134
    theBhc says:

    I’m hoping you’ll notice this comment, because I found a seriously demented article that you all might be inclined toward responding to

    Is Climatology a Science?

    by the wonks at Real Clear Politics. If anyone ought to answer this question, I thought it would be this group. Robert Tracinksi amusingly cites Richard Lindzen, a man known to be in the employ of various petroleum interests.

  35. 135
    David B. Benson says:

    Re #132: Dr. M. Jorgensen-Petersen — Your question is about weather, not climate. Nonetheless, in an amateur way, I’ll attempt to answer part of it.

    It is my understanding that a frozen lake can produce no so-called lake effect.

    But some web trawling on your part can probably locate a more definitive and reliable answer to your question.

  36. 136
    Lynn Vincentnathan says:

    #132, from a scientific perspective, you cannot pin a single weather event on GW.

    However, from an environmentalist or victim’s perspective (or the perspective of those concerned about the world) of avoiding false negatives, you cannot prove beyond a shadow of a doubt that the weird weather we are experiencing is NOT caused or enhanced in some way or another by GW.

    At least one of my predictions (I’m a social scientist) is coming true — we are shifting from a period of underattribution of GW effects to overattribution (I even mentioned that on RC some years back), though it’s happening sooner than I had thought. So, if Cheshire Wogburton in Miami bumps his head on April 29th, it’s gotta be due to GW :)

    And I further predict that this trend in overattribution will continue and strengthen (not based on any good study, but just my general knowledge of humanity).

    Only problem is there is a serious lag between knowledge of and desire to solve environmental problems, and actual action to solve them (it’s like a disconnect) — and that IS based on a study.

  37. 137
    Doug Lowthian says:

    Re # 134: The cited article asks thes question at the end: Does climatology have a well-developed, thoroughly proven theoretical framework, derived from decades of observations and earlier discoveries? Does it have a proven set of laws to explain what factors drive the global climate on a scale of centuries? Does climatology have an established track record of being able to predict next week’s weather, much less the next century’s weather?

    Simple answers are Yes, Yes and No – but only because climatology is not concerned with predicting next weeks weather. More detailed answers are easy to find.

  38. 138
    Dennis Brown says:

    Greenhouse effect ??? Sorry, that only occurs in greenhouses – this specific effect does NOT occur on a global scale – this fact was proven around the turn of the previous century (1900s) by a German scientist; of course, there is global warming but greenhouses do not get warm because of heating due to absorption/re-radiation of longer wavelength photons- it is simple insulation effects by the glass only – the German built a greenhouse out of NaCl and it was the SAME temperature as the greenhouse made of glass – QED.

  39. 139

    In Comment 138, 14 Apr 2007 @ 5:38 pm, Dennis Brown says: “Greenhouse effect ??? Sorry, that only occurs in greenhouses – this specific effect does NOT occur on a global scale – this fact was proven around the turn of the previous century (1900s) by a German scientist; […]”

    Specifically who was the “German scientist”? Would you cite a particular report for further study?

  40. 140
    Chuck Booth says:

    Re #132 Dr. Jorgensen-Petersen,
    Regardless of whether the two singers have their facts straight on that specific point, or not, your students would be well advised to focus on the ladies’ music, and learn science from scientists.

    For what it is worth, the only Great Lake that reliably freezes over completely is Lake Erie, and there have always been years when it hasn’t froze completely, or when the ice lasts only a month or two (as opposed to 3-4, or even 5, months of ice in a cold winter). You can see the historical records of Great Lakes ice cover for yourself at:

  41. 141
    Harry Robertson says:

    I would like to recommend that Buffalo Bill be posthumously recognized as a top-notch environmentalist. He killed thousands of methane spewing critters that were about to cause global destruction. His quick actions saved the planet. By the way, shouldn’t folks get some kind of carbon credit or something for being descended from people who wiped out countless herds of greenhouse gas producing monsters, like buffalo, elephants, maybe even whales. Whales are big, do they flatulate alot? Everyone get your guns. It’s time to wipe out more species and save the planet from extinction!

  42. 142
    Harry Robertson says:

    Ummm… I didn’t seriously believe you would post that…

  43. 143
    Chuck Booth says:

    Re 139 greenhouse effect

    I don’t think a greenhouse, whether made of glass or NaCl (?) provides much insulation; glass (or plastic) has a low insulation value. The glass (or plastic) does greatly reduce convective heat loss, and this can play a significan role in keeping the interior warm, depending on the wind velocity, amount of solar heating, and other factors.

  44. 144
    Pat says:

    Re 141 – funny.

    Re 126, 129

    The equations in this model are set up to be solved without need of iteration. The atmosphere in this model is a ‘graybody’. A graybody emits as epsilon times blackbody emission. This emission is a flux – power per unit area – and is emitted from any surface; in the case of the atmosphere, the bottom surface and the top surface. The surface of the Earth can not radiate downward because it’s opaque; the radiation upward is unaffected by that.

    More technically, the emissivity is determined by the effect of optical depth or optical path length integrated over solid angle; for an isothermal atmosphere with top and bottom emitting surfaces, emission from either side will be the same; but for a complex shape, even if isothermal and homogeneous in other ways, emission from within the shape emerging from a given surface can be geometrically dependent (and refraction can play a role, too). You can visualize this by … well its simple and yet hard to explain clearly…

    Re 123,130 – The interaction of electron energy levels within solids generally gives rise to broader absorption/emission bands (over wavelength). Gasses tend to have more isolated absorption or emission peaks in their spectra, as far as I know. A single molecule would have a line spectrum. However, collisions of molecules and the range of velocities of molecules of a gas can spread the absorption/emission out from the lines due to the Doppler effect and what is called pressure broadenning. Higher up in the atmosphere the pressure broadenning is weaker.

    Emmissivity and absorptivity vary with wavelength in general. The black body radiation formula given for this model assumes emissivity and absorptivity are constant over wavelength (a ‘graybody’ approximation) and gives the total radiant power emitted over all wavelengths; this is fine for a simple model being used to illustrate the greenhouse effect as a general principle. In such a case, while a given mass path length (ie distance times density – is mass path length the right term?) through a gas may yield considerably lower emissivity than through a solid At a given wavelength, if the emissivity is independent of temperature, emitted radiant power per unit area will be proportional to the fourth power of temperature, even for a gas.

    To actually calculate the greenhouse effect of the Earth and it’s dependence on atmospheric composition, one must take wavelength into account. At a given wavelength, Thermal radiant emission can be expressed as a wavelength-dependent emissivity multiplied by the monochromatic blackbody radiation, which can be given as power per unit area per unit wavelength, and is a precise function of temperature and wavelength. Because of varying conditions with height, there will be differences in radiation intensity at different angles, so it is actually better to start with emissivity * spectral blackbody intensity, which is radiant power per unit area per unit wavelength per unit solid angle (this might also be called spectral radiance, whereas integration over a hemisphere solid angle weighted by the cosine of the angle from perpendicular to a plane gives the spectral (monochromatic) irradiance which is the spectral flux … sorry for all the jargon – and I may have been imprecise myself in some of the terms – someone correct me if my terms are mixed up.))

  45. 145
    Robin Johnson says:

    RE #128


    Well… The reason it makes sense to spread it across the whole surface is due to the latency and diffusion of the heat transfer mechanisms.

    For example, heat absorbed in the tropics is converted to mechanical energy in the form of winds, water vapor and currents that release the heat many days later mostly in the higher latitudes. The longwave radiation absorbed and re-radiated (in all directions) by the atmosphere sends a significant fraction of that re-radiated heat into the higher angle and dark parts of the planet in minutes. Also, obviously, the heat absorbed by objects during the day is radiated throughout the night while the Earth rotates around.

    So, for a “total average” energy budget it makes reasonable sense.

    In a more precise, complex model, you actually model the heat transfer mechanisms in blocks (100km across I believe is the current working standard in the GCMs) and account for the high angle effects, heat transfers in all their mathematical glory, etc. And use swags for cloud cover. The GCMs are currently fast enough to be used by the weather guys to give a global picture to inform their forecasts and more regional computer simulations.

    The ocean current guys have more trouble – water is much harder to model than air plus accurate initialization data is meager by comparison.

  46. 146
    Marcel Labonte says:

    Has anybody calculated how much each mode of vibration of the CO2 molecule is responsible for the retention of the heat (radiation)?
    The CO2 is a linear molecule that has many modes of vibrations,is one mode more responsible than the others?

    Another question is that these heat excitation are like photoexcitation, what is their quantum yields?

  47. 147
    Hank Roberts says:

    Search Google Scholar for “greenhouse effect” +misnomer

    The first hit will suffice:

    Clarification of Selected Misconceptions in Physical Geography
    by Burton D. Nelson, Robert H. Aron, and Mark A. Francek

    “… In this paper, a number of misconceptions relating to location, and to the earth’s hydrosphere, atmosphere; and lithosphere are discussed with appropriate correction and explanation…. it is intended principally for elementary and secondary level teachers who may have limited training and background in geography.”

  48. 148
    ChrisC says:

    Re 138

    A Greenhouse (made of glass) retains a hotter atmosphere within than the surrounding environment not because of any radiative insulating properties. A greenhose traps heat because it prevents convective heat transfer with the outside environment.

    Incident solar radiation penetrates the glass, causes the surface to warm, and re-radiate back some of this energy into the air contained within the greenhouse, which causes its temperature to rise. This air cannot mix with the air in the environment around it. Hence, it is unable to transfer heat, and becomes hotter than the surrounding environment.


    This is why a salt greenhouse should act similarly to a glass greenhouse. Provided it is transparent to solar radiation and inhibits convective mixing, the results shoulod be very similar.

    The term “greenhouse effect” is somewhat of a misnomer, as it is the absorption of thermal (infra-red) radiation by atmospheric constituents rather than mixing inhibition that results in the warming experienced on Earth.

  49. 149

    Wonderer (#21)

    First, no one’s saying there’s anything wrong with the average engineer.

    But frankly, engineers have given the world more than their share of cranks, including a small population that does know exactly enough math to get in trouble, i.e., they know that general relativity is a hoax (or global warming, or any number of other things).

    Take a look at this, for instance, by science fiction and alleged nonficion author James Hogan: | . Note in particular this bon mot: “Science really doesn’t exist. Scientific beliefs are either proved wrong, or else they quickly become engineering. Everything else is untested speculation.”

    I blame the Heinlein mystique partly. If you’ve ever read Ed Regis’ Great Mambo Chicken and the Transhuman Condition, look at the culture around Keith Henson (whom I know through Scientology critic circles) as an example.

    Also probably something to the EE/ME etc. divide. Advanced study in ME makes you an awesome seat of the pants calculator, but you don’t learn a very broad spectrum of math. In that sense, “rocket science” is a kind of misleading expression.

  50. 150
    William Astley says:

    Hi Marcel,

    In reply to your comment 146, the following is a link to Jack Barrett’s paper “Greenhouse molecules their spectra and function in the atmosphere.”

    See figure 9 for data from the Nimbus satellite that shows the reflected spectra from three climatic regions of the earth: Mediterranean, Sahara, and the Antarctic.