### 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 , where 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 where is the Stefan-Boltzmann constant, and is the surface temperature and the atmospheric radiative flux is written , where 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:
Atmosphere:
Planet:

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:

If you want to put some vaguely realistic numbers to it, then with S=240 W/m2 and =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 ) 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.

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 or 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:

where 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:

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 is the sensitivity of to the forcing for this (simplified) system. To get the sensitivity of the temperature (which is the more usual definition of climate sensitivity, ), you need to multiply by i.e. . 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 due to the greenhouse effect (i.e. the difference between having and its actual value) is , and the downward longwave radiation is just , and dividing one by the other simply gives – 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.

#### Feedbacks

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 dependent on the temperature (or for arithmetical convenience). Indeed, if we take a special linear form for the temperature dependence and write:

then the result we had before is still a solution (i.e. ). However, the sensitivity to changes (whether in the greenhouse effect or solar input) will be different and will depend on . The new sensitivity will be given by

So if 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, , 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.

#### Summary

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. 251
Pat says:

TO CLARIFY:

irradiance = power per unit area

Blackbody radiation as a function of temperature and wavelength is an idealized variation of radiance with wavelength (or wavenumber or frequency) and temperature. The thermally emitted radiation from an actual material in local thermodynamic equilibrium can be measured relative to blackbody radiation – at a given wavelength, or over some band of wavelengths, the emissivity is the emitted radiance as a fraction of blackbody radiance at the same temperature and wavelength. Emissivity can never be greater than 1 when in local thermodynamic equilibrium, or else it would be possible to build a perpetual motion machine and decrease the entropy of a closed system; this is because absorptivity – the fraction of radiance (or irradiance) absorbed – cannot be greater than 1, and emissivity must equal absorptivity when in local thermodynamic equilibrium, or else, again, it would be possible to reduce the entropy of a closed system.

(Interesting aside – because of total internal reflection, blackbody radiance is also proportional to the square of the real component of index of refraction. That is, a blackbody surface imbedded in glass with an index of refraction of 1.5 would emit into the glass 2.25 times the radiance that it could emit into a vacuum or into air (index of refraction is only slightly above 1). But looking at the blackbody from outside the glass, even with perfect antireflection coating, the radiance seen within the solid angle of visual field covered by the blackbody would be that of a blackbody in a vaccuum or into air, as one is typically more familiar with. The reason – again, if this were not the case, it would be possible with a system of lenses, etc, to break the second law of thermodynamics and decrease the entropy of a closed system.)

At a given wavelength, any substance which has a nonzero emissivity can be used to construct a nearly ideal blackbody (provided there is no limit of substance available and the amount used is isothermal). This is because:

Within a substance, if along pathlength of 1 mm, the radiance emitted is 0.001 times blackbody radiance, then along the next 1 mm, the same amount is emitted again, but also the same proportion of radiance from behind is absorbed (or otherwise blocked, if the microstructure has little surfaces mirrored on just one side, for example, with all mirrored sides facing the other way – but you get the idea). Thus after 2 mm, the emissivity along that direction is 0.001 + 0.001 * (1-0.001). Continued to infinity, the emissivity approaches 1; for radiation going in the reverse direction along the path of the same length, absorptivity will be the same as the emissivity.

Within a cavity, if the walls have an emissivity of 0.001 and thus a reflectivity of 0.999, then following a path from one wall, the radiance is 0.001 times that of a blackbody, but after reflection off another wall, there is an additional emission of 0.001, with an absorption of 0.001 times the radiance previously emitted, so the emissivity after this reflection is 0.001 + 0.001 * (1-0.001). Continued to infinity, the emissivity approaches 1, and going in the opposite direction, absorptivity also approaches 1. Thus the cavity will become filled with blackbody radiation. If a small hole is cut into a wall of the cavity, assuming it is quite small compared to the surface of the cavity, most paths going into the hole will rebound off the inner surfaces of the cavity walls many times before coming back out, so the hole will be like a blackbody.

2. 252
Pat says:

TO CLARIFY AGAIN:

So, ideal blackbody radiation (which is isotropic, which means the radiance at any angle is indepent of angle, so integrating over solid angle and factoring in the spreading out of radiation accross a surface at higher angles to the normal of the surface,

A steradian is a unit of solid angle; there are 2 pi steradians in a hemisphere.

—–

So, blackbody radiance or irradiance is a function of temperature and wavelength is not a property of any specific material but it sets a limit on what is possible for a material to thermally emit. As temperature rises, the size of energy transitions which may be involved in thermal emission increases; at a high enough temperature, the emission in visible and UV lines of a gas would become significant in thermal emission.

—–

Ozone is a greenhouse gas; it has an absorption band somewhere around 9 or 10 microns.

3. 253
Rod B. says:

Ray (240), great post… until you got to “The reason there is a net greenhouse effect is because sunlight peaks in the green, while, the peak of thermal emission from the surface is in the IR. Voila, an extra 30 degrees to make life possible.”

That made no sense to me. What difference does it make if insolation peaks in the green, yellow, blue or orange, or where in the IR band or even low visible band the earth’s radiation peaks?

4. 254
Rod B. says:

Fredrik (244), gasses are “greenhouse” gasses by definition if they absorb radiation from the earth. No greenhouse gasses = zero absorption. Both CO2 and H2O are greenhouse gasses.

5. 255

[[Ray (240), great post… until you got to “The reason there is a net greenhouse effect is because sunlight peaks in the green, while, the peak of thermal emission from the surface is in the IR. Voila, an extra 30 degrees to make life possible.”
That made no sense to me. What difference does it make if insolation peaks in the green, yellow, blue or orange, or where in the IR band or even low visible band the earth’s radiation peaks?
]]

Because sunlight, peaking at 0.5 microns, largely gets through the atmosphere to the ground, while terrestrial radiation, peaking at 10.6 microns, is largely absorbed by greenhouse gases. That’s how the greenhouse effect works, and it depends on the fact that carbon dioxide (for example) absorbs very little visual radiation but a great deal of infrared radiation.

6. 256

Re 253. Sorry for the lack of clarity, Rod. It’s not really the peak of the distribution that matters, but the distribution itself. The point is that there is a lot less energy in the IR in radiation coming in from the Sun (BB temperature ~6000 K) than there is in radiation heading back out to space from Earth (BB temperature ~300 K). I was pointing out that this is responsible for the net positive energy from the greenhouse effect.

7. 257
Alexander Harvey says:

I do not have a lot of time right now, but enough for one last go on gases.

Bascally, the simpler the structure of a gas molecule the more transparent it is in the 300K region.

The most transparent are the noble gases (He, Ne, Ar, Xe) next up are the symmetric diatoms (N2, O2, H2, etc.). More complicated molecules like CO2, H20, with their additional modes of vibration are far from transparent.

Now the noble gases are so transparent that they are used in spectroscopy as filler to increase the pressure in samples without colouring them. They are that transparent. I believe that Xenon is commonly used for this.

The reason they are transparent is because they lack a mechanism for absorbing or emitting radiation in that region. With out the vibrational-rotational modes of the more complex molecules they simply can not get excited by radiation in 300K region.

This is, I believe, the essence of why some gases are LW transparent (non-greenhouse) and some are opaque in some parts of the LW (greenhouse gas).

Now if a gas is transparent to radiation from a 300K source it neither absorbs nor radiates at those frequencies. One implication of this is that a body of that gas can not cool itself by radiating away its energy in comparison to CO2 H2O etc. which can and do.

If you look at the simplified model that commenced this thread you will see that much of the radiation that leaves earth is radiated into space by the atmospheric greenhouse gases.

Just to make this clear. I mean radiation produced by those gas molecules due to their temperature, not scattered radiation from the earth’s surface. A cloud of CO2 at 300K in empty space would quickly radiate heat away whereas a cloud of noble gas at 300K simply lacks the vibrational-rotational modes necessary to radiate in that fashion.

Precisely how low the emissivity of He or Xe is at 300K, I do not know but I suspect it is very, very low indeed. Mention of them is usually confined to a simple “they are transparent in the IR”. There are no 300K spectra for these gases that I can find which as they are used as transparent fillers when measuring other gases is hardly surprising.

I am not overly complicating things, I believe that it is detail at this sort of level and beyond that allows one to reason about behaviour of gases in a way that assuming they emit radiation by some unspecified mechanism does not. We are not talking blackbodies here, In gases radiation is confined to a small number of specific modes, the energy distribution (translational, rotational, etc.) follows the same laws but the spectrum is in origin a set of discrete lines, limited to the what is permitted by the mechanisms involved.

Given the importance of the lack of LW transparency of CO2 one might think that the mechanisms involved would be familiar to all. Sadly it seems that the science is widely considered as all too difficult.

Now you may feel that all of the above is nonsense or unnecessary, I do not. I brought it up precisely because it is counter-intuitive and has important consequences. The one I originally illustrated is that a non-greenhouse gas (one transparent to radiation produced by a 300K source) can neither be warmed by the upwelling radiation nor can it cool itself by turning its internal energy (kinetic) into LW radiation.

That alone I consider to be sufficiently important to have gotten myself into this wrangle. Hopefully we will all live and learn.

8. 258

Harvey, If what you are saying is that an atmosphere of noble gasses will not absorb in the IR, I agree. In that case, the planet would look from the outside like a blackbody except for some weak absorption in the UV, where there isn’t much radiation anyway. In fact, since the solar spectrum is richer in these wavelengths than the planetary spectrum, an “inert” gas atmosphere would a net (though slight) cooling effect.
Even the simplest model ain’t that simple.

9. 259
Alexander Harvey says:

To recap:

I speculated on the dynamics of a simplified model at each extreme of atmospheric emissivity to radiation from a 300K earth.

Zero emissivity leads to a very peculiar atmosphere indeed. Being totally transparent to a 300K raditaion source (by definition) it will lack the ability to either gain or lose heat through radiative processes at temperatures around 300K.

This implies that its only sink for heat will be its contact with the earth’s surface. Once in equillibrium there it would effective be dead thermodynamically and would have no apparent method of sustaining a temperature gradient.

This was met with some bald “all matter with a finite temperature radiates” statements. I merely pointed out that in order to radiate it must have internal energy and a mechanism. I.E. that matter does not “just radiate”, it radiates if it has internal energy and available transitions. I should perhaps have included that exceptions have to be made for ionised gases but I suspect that this really is not relevant for a noble gas at 300K.

Now in the 300K region the noble gases are considered to be transparent i.e. they have zero emissivity at the relevant frequencies. So the criticism that such an atmosphere is “unphysical” needed a little challenging. I requested anyone who knew that these gases radiated at 300K to state the relevant mechanism, produce the spectrum, or give the emissivity. I would have certainly have learned something new.

You alone seem to agree that the noble gases are “for all intents and purposes” transparent to 300K radiation. This requires that they have in effect zero emissivity (in the 300K band). So the model was not all that unphysical after all.

I only pursued that is point as I consider it important in itself to understand how and why some gases have high emissivities (greenhouse gases) and some low or zero emissivities (at 300K).

That a volume of noble gas (or a true non-greenhouse atmoshpere) can not cool itself effectively by radiating away its internal energy, seems counter-intuitive but it is I believe the case.

So a true non-greenhouse atmosphere (zero emissivity) in the original model could lead to some interesting thermodynamics, or should that be lack of them.

I was only pointing out that atmospheric emissivity is a two way street. It warms the earth surface but also allows the upper atmosphere to radiate into space thereby creating a sink for energy in the upper atmosphere that is a thermodynamic driver for the atmosphere.

10. 260

Alexander, I agree that blackbody radiation depends on the ability of the body to absorb radiation. To the degree that it is transparent or reflective, it is not a black body. I think where you got off on the wrong foot was that it was not clear whether you were talking about the entire planet + atmosphere system–which does radiate as a blackbody or just the atmosphere. I think I understand the point you were trying to make now is that the entire atmosphere would be an isotherm–unable to radiate away energy and insulated by the effective vacuum of space. It would not be an isobar, though, so if I’m not wrong it would be stratified and you wouldn’t have convection either. The only activity you would have would be at the boundary between day and night–midnight and noon would be still as death, but dawn and dusk quite active.
I think where I disagree with you is that you seem to be saying that the only way there’s a sink to space is via radiation of the atmosphere, and in a completely transparent atmosphere, that’s no the case. The radiation goes from surface to space unimpeded, so there’s still a sink, and the surface is in equilibrium with whatever radiation is incident on it. Would you not agree? Or am I misinterpreting what you had to say again?

11. 261
Pat says:

Re 260

I think that’s what Mr. Harvey was saying.

Re 258

Atmospheric SW absorption without any LW emission would tend to lead to a thermospheric atmosphere; the absorbed UV heat would still be transmitted by conduction to the surface (when in an equilibrium state).

12. 262
Rod B. says:

Barton (255), Ray (256) re my 253: O.K.

13. 263

If that’s all that Harvey was saying, then I agree, but it seemed to me — maybe I read him wrong — that he was portraying the atmosphere accumulating heat from conduction and convection and then being unable to radiate it, which would imply that it was heating up indefinitely. I also think there was some confusion between the atmosphere by itself and the planet-atmosphere system — even if the atmosphere doesn’t radiate, the planet must, or again, it will heat up indefinitely.

14. 264
Fredrik says:

“Once in equillibrium there it would effective be dead thermodynamically and would have no apparent method of sustaining a temperature gradient.”

I belive this is incorrect. The atmosphere is never in equillibrium due to night and day. The air close to the earth get colder due to conduction with the earth at night, creating on inversions. The sun then warms the air, creating thermals (convection), trying to get the atmospheric adiabatic.

I see no real difference if the greenhouse gases exist or not except in the radiation energy transfer with the earth. No more reason for a constant temperature atmosphere or an atmosphere without convection.

15. 265
Fredrik says:

A short question.

Does the greenhouse gases warm the atmosphere directly from radiation from the earth or sun, i.e. warming/coling not due to convection?

I belive the answer is no but I might be wrong.

16. 266

[[Does the greenhouse gases warm the atmosphere directly from radiation from the earth or sun, i.e. warming/coling not due to convection? ]]

Greenhouse gases work by allowing sunlight through to hit the ground, but absorbing infrared coming up from the ground (and from other locations in the atmosphere, e.g. if you model it as a number of horizontal layers). For an overview with a little math — nothing harder than algebra — try

http://members.aol.com/bpl1960/Climatology.html

17. 267
Lawrence Brown says:

Thank you for this introductory model. It’s good to know what some of the constants and variables are involved in climate studies. It’s especially gratifying to some of us amateurs who like to tool around with numbers.For example I’ve solved a few of your equations assuming that the Earth had no greenhouse gases in its atmosphere with what( to me,anyway) is an interesting result.
If lamda(L)( I have no greek letters on my keyboard)=0, also the^ sign stands for an exponent then:
(1-alpha)TSI/4= (1-L)G =sigmaTa^4 and
T=( TSI(1-alpha)/sigma)^1/4
From google: TSI= 1370 watts/m2 (the site shows a statue of James Watt standing on a square meter): alpha=.30 and sigma=5.67×10^-8.
Solving for T gives 254 kelvins or about -19 degrees C, which is about 0 Fahrenheit!! Cold yes but not too cold for life, such as polar bears or penguins, and perhaps many other species! Please do more of these, Gavin. Regards, Larry

18. 268
fredrik says:

Barton I dont think your site answer my question. My question is about convection and how the atmosphere is warming. My understanding from reading meteorology and some thinking (I might be completely wrong though) is that the atmosphere is only warmed by conduction from the ground. The green house gases radiate some energy back to earth and thus change the earth temperature but the radiation from the earth doesn’t change the temperatur of the atmosphere directly much. If that explaination make sense.

In other worlds. Look at a world with green house gases. Measure the earth temperatur over some time. Take away the greenhouse gases but install some warming of the eart surface such that the earth temperature is identical to before. My question is, do this result in the same atmosphere as before? I belive it does. This is related to Alexander’s thought experiment above.

19. 269

[[The green house gases radiate some energy back to earth and thus change the earth temperature but the radiation from the earth doesn’t change the temperatur of the atmosphere directly much. If that explaination make sense. ]]

It makes sense but it’s wrong. When greenhouse gases absorb energy from the ground, they heat up, and the atmosphere they’re mixed with heats up. If there were no greenhouse gases in the atmosphere, the atmosphere would be colder than it is.

20. 270
fredrik says:

yes, the atmosphere would be colder. No problem with that. I am interested in where the actual energy transfer happens.

“When greenhouse gases absorb energy from the ground, they heat up, and the atmosphere they’re mixed with heats up.”

The green house gases are less than 1% of the atmosphere. The green house gases warms by the radiation but the other >99% of the atmosphere must also be warmed. It needs quite a lot of energy to do that. My quess is that the warming due to the radiation is much much less compared to the warming (energy transfer) due to convection. How much difference in temperature would my thought experiment above result in?
Does the difference be significant? Measurable?

21. 271

Re 268 etc. Fredrik, think of it this way: IR from the surface excites vibrational energy in the CO2 molecule, raising its ehergy, and therefore its temperature. The vibration of the CO2 molecule is weakly dissipated by the surrounding molecules–that is, the rest of the atmosphere damps the vibrational motion, taking up the kinetic energy. You can kind of see this via the equipartition theorem–the energy that goes into any one degree of freedom–kinetic, vibrational, etc. eventually gets shared via all the degrees of freedom. So, whether CO2 is 0.0001% or 100%, you get warming of the entire atmosphere any time one component heats up. Does that help?

22. 272

[[The green house gases are less than 1% of the atmosphere. The green house gases warms by the radiation but the other >99% of the atmosphere must also be warmed. It needs quite a lot of energy to do that. My quess is that the warming due to the radiation is much much less compared to the warming (energy transfer) due to convection. How much difference in temperature would my thought experiment above result in?
Does the difference be significant? Measurable?
]]

Sensible heat (conduction and pure convection) heats the atmosphere at the expense of the surface at about 24 watts per square meter over the whole globe. Evaporation contributes another 78 W/m2. But the atmosphere picks up about 350 W/m2 in infrared radiation from the ground, so radiative effects beat other mechanisms by three to one.

23. 273
fredrik says:

Ray, the time it would take to warm the air should depend on the amount of the green house gases. My quess is that that the time is much longer compared to convection and other air motions. Definitely in the boundary layer, it might be a difference above the boundary layer. Do you have any numbers of the warming?

Barton, I fly paragliders and have thus a pretty good understanding of air movements in the boundary layer, night time inversions, thermals etc. Your numbers seems to suggest that the heat transfer from green house gases is much more compared to convection. I just dont belive that. I dont remember anything in the meteorology books that suggests that the air warms directly from the radiation. The energy comes from the the contact with the ground and evaporation.

I might be incorrect because I have limited knoweledge about the subject but your answers just doesn’t convience me.

24. 274
Pat says:

Re 273:

Lapse rate – the rate of temperature decline with height. Generally positive in the troposphere (with some exceptions – inversions near the surface in polar winters and calm clear nights, etc.), zero to negative in the stratosphere, generally positive in the mesosphere and negative in the lower thermosphere.

Air is gains (or loses in some cases) sensible heat by conduction at the surface; it can also gain by evaporation (or lose – dew and frost) latent heat, which is converted to sensible heat elsewhere by water phase changes, ultimately leading to precipitation.

Air rising cools adiabatically due to the drop in pressure and resulting expansion (it follows a dry adiabat). Latent heat release during condensation (and freezing) slows the cooling rate, so during such moist convection air follows a moist adiabat. If I understand correctly, the troposphere in general tends to follow a typical moist adiabat
(with much variation in time and space, but never exceeding a dry adiabatic lapse rate on a sizable scale (there often is such a higher lapse rate only in the immediate vicinity of the ground when the surface is being heated by solar radiation – convection is limited at the ground as it interupts vertical motion, but convection to and from the thin surface layer of air is important) that would lead to instant overturning until the lapse rate is reduced to dry adiabatic)
because moist convective processes tend to maintain such a lapse rate; an increase in the lapse rate will be more favorable for vertical motion, etc.

But radiation exchange among the sun, the surface, space, and multiple levels of atmosphere, is quite important. The atmosphere and surface both reflect some solar (SW) radiation back to space; the rest is ultimately absorbed, some in the atmosphere, but a majority at the surface (or within some distance below the surface in bodies of water). There is temperature and wavelength dependent emission of radiation, of the LW kind, by the surface and atmosphere; the limiting value is that of a blackbody at a given temperature – I think the surface is nearly (but not completely) like a blackbody for the wavelengths in the LW band; the atmosphere’s opacity in the LW band varies over wavelength and with the concentration of water vapor, clouds, and ozone (other important greenhouse gasses are much less spatially-temporally variable (in the short term)). For example, there is an atmospheric window between wavelengths of 8 and 12 microns (interupted by ozone around 9 or 10 microns) where a sizable fraction of radiation from the surface can escape directly to space, except where the water vapor concentration is very high.

Generally, the LW opacity (from absorption and emission – scattering may be a small factor but I think it’s much less important in LW radiation than in SW radiation, and for introductory purposes can be ignored in the LW) of the atmosphere blocks some fraction of radiation from the surface going to space – it absorbs it, and replaces it with it’s own space-bound emission – which is dimmer because the atmosphere is generally colder than the surface; hence less radiation is emitted to space then is emitted by the surface, enabling the surface to be at a higher temperature than otherwise.

More precisely, some radiation is absorbed and replaced with emitted radiation at each level in the atmosphere, going upward and downward. A higher opacity better enables the colder upper troposphere to block emission not just from the surface but the warmer lower troposphere from going to space – increasing the opacity makes the radiation emitted to space ‘colder’, allowing heat to build up until a new equilibrium is established. (At the same time, the radiation from the atmosphere that reaches the surface will be coming more from the lowest layers of atmosphere, which are warmer).

But in pure radiative equilibrium, the lapse rate of the lower atmosphere would be very high – convection does not allow such a situation to occur. Instead the climate tends to approach a radiative-convective equilibrium. Convection tends to keep the lapse rate of the troposphere near a moist adiabat; increasing the opacity of the atmosphere causes more of the radiation to space to come from higher up in the troposphere, where it is colder, so the surface and troposphere warm up together to compensate.

Of course, meanwhile the stratosphere tends to cool because it recieves even less radiation from below while emitting more strongly to space…

And as the radiation from the atmosphere to the surface is coming from closer to the surface, where it is typically warmer, the net radiative exchange is reduced…at least at first – this effect is couteracted by the nonlinear relationship between temperature and radiation, but is greatly strengthened by water vapor feedback … (See previous comments for more).

Increasing the greenhouse effect tends to reduce the diurnal temperature range by increasing the radiation from the atmosphere to the surface, which is much less variable over the course of a day than solar heating is.

25. 275
Pat says:

From
(Kiehl and Trenberth)
radiation from atmosphere to surface 324 W/m2
radiation from surface which heats the atmosphere 350 W/m2
NET: surface to atmosphere = 350 – 324 = 26 W/m2.

So getting back to the original point of contention:
the net heating of the atmosphere by the surface through radiation is smaller then that by convection, and I would expect it to be reduced in response to increased greenhouse forcing (while convective heating of the atmosphere from the surface would tend to increase); but it is not unimportant and is the difference between an upward and a downward radiative flux, each of which is quite large.

26. 276
fredrik says:

Looked through the article and especially figure 7.

Net radiation from earth to the atmosphere is 26 W/m2 as above but the total radiation to the atmosphere seems to be -102 W/m2. Thus the atmosphere is cooling if only radiation is included. Thus the warming of the atmosphere is only due to convection and latent heat. Ofcourse some of the heat due to convection is going to radiate out into space also.

Looking at the atmosphere without including the radiation transfer from the atmosphere in all directions much be wrong (or have I missunderstood something) and the numbers given by Pat and especially Barton is missleading.

“But in pure radiative equilibrium, the lapse rate of the lower atmosphere would be very high – convection does not allow such a situation to occur. Instead the climate tends to approach a radiative-convective equilibrium. ”

This and the numbers in the article seems to suggest that the atmosphere is actually to warm for radiation equilibrium. A green house atom actually emit more energy than it absorbs. It should be colder but convection/latent heat makes it warmer. Is this correct?

27. 277

[[Barton, I fly paragliders and have thus a pretty good understanding of air movements in the boundary layer, night time inversions, thermals etc. Your numbers seems to suggest that the heat transfer from green house gases is much more compared to convection. I just dont belive that. ]]

Every energy budget for the climate system gets numbers close to what I quoted. For a good review, and another set of estimates (the ones I used above), try:

You will note that the 12 other studies from 1975 to 1997 that K&T compare to their own all get similar numbers.

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[[A green house atom actually emit more energy than it absorbs]]

No. Energy is conserved.

29. 279

[[radiation from atmosphere to surface 324 W/m2
radiation from surface which heats the atmosphere 350 W/m2
NET: surface to atmosphere = 350 – 324 = 26 W/m2.
]]

And the latent and sensible heat fluxes don’t also get cooled? Let me clue you in on something, Pat — a parcel of atmosphere doesn’t know where its heat came from, and there’s no qualitative difference between heat from convection, heat from conduction, and heat from radiation. Heat is heat. Of the mechanisms which HEAT the atmosphere, radiation dominates. Which is what I said in the first place, and which Fredrik and now you seem to want to deny.

30. 280

Fredrik, The atmosphere has to emit more radiation than it absorbs, because that is the only way for the atmosphere to exchange radiation with the vacuum of space, and space is colder than the atmosphere. What greenhouse gases do is absorb radiation near the peak of Earth’s thermal emission spectrum, keeping the atmosphere warmer than it would be if it were a perfect black body. That absorbed radiation is largely why the atmosphere is warmer than a radiative equilibrium would make it. Within the atmosphere, heat is exchanged by conduction, convection and radiation. It is only at the boundary of space that radiation becomes dominant.

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fredrik says:

First I have to admitt that I dont really understand the whole system right now.

Most analysis and comments here assume that the atmosphere is in equlibrium, i.e. the net energy transfer for for example the ground is zero. This is almost never the case. The ground temperature almost always change. The same is true for most of the troposphere.

It is true that you cant just talk about radiation by itself as I did. That was a misstake. The atmosphere is going to radiate depending on it’s temperature independent on why the temperature is at that value.

My question could probably be posed better by noting that I am interested in the change around a mean value of tempereature in the atmosphere. The greenhouse gases make the atmosphere much warmer compared to an atmosphere without greenhouse gases but I was interested in the diurnal variations.

“[[A green house atom actually emit more energy than it absorbs]]

No. Energy is conserved.”

The energy is conserved but my statement is true if the atom isn’t at equlibrium and I should say that the atmosphere almost never is at equlibrium.

It is also true if the atom get some energy from some other source than from radiation, for example due to convection.

My question is something like this. Does the air outside the boundary layer (i.e. no convection) change temperature as a function of change in radiation from the earth (or sun) for a clear high pressure day and night?. If that is the case how much? The radiaten should be different for the day and night but I dont think I have seen anything about a change in temperature at for example 5000 meters altitude.

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Fredrik, Ever spend the night at 5000 meters altitude. If so, you would have your answer–it gets bloody cold. Even during the day, the air is chilly. I remember hiking to Everest base camp, being sick as a dog and curling up on a rock to warm up. During the day, sunlight radiates through the atmosphere and is absorbed by Earth. Earth warms and radiates in the IR, and both incoming and outgoing radaition pass through the atmosphere. Moreover, air in contact with the warm rock heats up and convects, taking that heat with it. Thus, despite being transparent to most of the radiation, the air is near equilibrium with the ground. At night, most of the radiation comes from the warmed Earth itself, and some of this is absorbed by ghg molecules. If you are talking about the atmosphere far from Earth, where convection is negligible, it is mostly transparent to radaition, except in certain bands–e.g. absorption bands for CO2, H2O, etc. Thus, at these altitudes, the atmosphere won’t be in thermal equilibrium with the radiation.

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fredrik says:

” If you are talking about the atmosphere far from Earth, where convection is negligible,”

Yes, outside the boundary layer.

“it is mostly transparent to radaition, except in certain bands–e.g. absorption bands for CO2, H2O, etc. Thus, at these altitudes, the atmosphere won’t be in thermal equilibrium with the radiation.”

Why want it be in thermal equilibrium? Thought you tried to explain to me in post 271 that the energy actually should be shared by all air molecules and equlibrium should thus be achived?

” The atmosphere has to emit more radiation than it absorbs, because that is the only way for the atmosphere to exchange radiation with the vacuum of space, and space is colder than the atmosphere.”

Guess we are talking mean values here.
Wouldn’t the atmosphere actually get warmer if this was true (if the convection/latent heat is included)? Why should the temperature of space matter?

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Fredrik, In order for it to be in thermal equilibrium, it would have to absorb the radiation (i.e. a black body), but it is transparent to most of the radiation. So, it can’t be in thermal equilibrium.

OK, think of the atmosphere as this thin layer of gas between a warm sphere of rock that is heated by the sun (on one side) and the inky blackness of space on the other. On the sun-facing side, we have a lot of radiation, but the atmosphere is mostly transparent to it. Likewise for the longwave radiation from Earth. The atmosphere only absorbs where it has absorption lines–due to vibration (e.g. CO2 in the IR), rotation, and electronic levels. The rotational and vibrational levels can exchange energy mechanically during collisions with other molecules. Even the absorption and emission of radiation involve a change in momentum, so to some extent there is exchange between the different atoms, but there is a lot of radiation the gas molecules just don’t see. Make sense?

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Pat says:

Re 279 – Barton Paul Levenson – I think you misunderstood my intent; I agree that radiation exchange between the atmosphere and surface is quite important. Perhaps I wasn’t clear, but I did my best to explain a lot in a small space.

Re – recent discusion:

True, at any given location and time, equilibrium may not been achieved. Much of this discussion has pertained to the behavior of a 1-dimensional column model under time-averaged solar forcing, which nonetheless is a good place to start.

Locally, any atmospheric column will often be out of equilibrium as cold air is advected here and warm air is advected there and the optical properties are altered as clouds and humidity and aerosols and ozone come and go and vary. But the resulting imbalances tend to drive the system toward equilibrium – a warmer layer of air will tend to cool by radiation to cooler layers of air above and below (and to the degree that the air above or below is not perfectly opaque, a warmer layer of air will cool more rapidly to the space, and to the surface if it is warmer then the surface, otherwise if it is still cooler than the surface, it will have a net radiative heat gain from the surface but it will not be warmed by the surface as rapidly as it would if it were cooler); in some conditions convection will adjust the temperature profile. It will be a dynamic equilibrium if ever achieved, and an equilibrium which is a diurnal oscillation (ie four dimensional equilibrium – if that is a valid concept (the idea being that the temperature distribution could settle into a daily cycle) – a climatic equilibrium surely must be considered this way over the course of a year…)

‘Locally’ above is distinctly different from ‘local’ in the context of local thermodynamic equilibrium. Local thermodynamic equilibrium is usually applied on a much smaller scale and describes a state in which the distribution of energy among states in populations of particles is in dynamic equilibrium – and consequently, for example, any subgroup of molecules of sufficient size will have the same temperature as any other such subgroup. The vast majority of the atmosphere (at least by mass) is quite close to local thermodynamic equilibrium.

The amount of radiant energy absorbed by a parcel or layer of air can certainly be different then the amount emitted – this occurs when radiative equilibrium is not achieved. But the absorptivity and emissivity – fractions of properties with respect to an ideal blackbody – are the same in local thermodynamic equilibrium.

The troposphere generally on average is in radiative-convective equilibrium – radiative transfer by itself is not in equilibrium even on average – more radiant energy is emitted then absorbed by radiation. In a single column model the stratosphere and above would tend to reach radiative equilibrium as convection would not penetrate to or through those levels.

Radiative solar heating per unit mass is generally greatest at the surface (on land, anyway) because radiation is absorbed over a small distance, and in the upper atmosphere because of the ozone layer and because the greatest amount of solar radiation available is at the top of the atmosphere, before any is absorbed (scattering complicates things but the distribution of solar heating is still generally as described). In the troposphere, water vapor and clouds absorb some solar radiation. As solar radiation varies between day and night, the most rapid temperature responses occur at the surface (and consequently near the surface) on land and in the upper atmosphere. A larger diurnal temperature range does not occur away from the surface in the troposphere, but a larger diurnal range can be expected in higher surface elevations because there is less atmosphere above and thus there will be less downward radiation from the atmosphere (which itself has a smaller diurnal temperature range) because it is colder because there is less of a greenhouse effect and also directly from the smaller greenhouse effect perhaps because of less pressure broadenning of the absorption/emission bands, etc… and depending on aerosols, humidity, and cloud cover, less solar radiation may be absorbed before reaching the surface in regions of high elevation, thus making the diurnal cycle of solar heating greater…

In pure radiative equilibrium the lowermost troposphere and surface would be warmer but the upper troposphere would be cooler than it is with convection.

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Timothy Chase says:

A Spreadsheet for the Simple Model
http://www.editgrid.com/user/timothychase/Greenhouse

To get a clearer idea of the principles behind the simple model, I created a spreadsheet which calculates the positive feedback as radiation is absorbed by the surface, emitted by the surface and either reaches space or is absorbed by the atmosphere, then emitted by the atmosphere either back to the surface or to space. The positive feedback is between radiation absorbed by the surface and emitted then absorbed by the atmosphere then emitted and absorbed by the surface. At each step in the process, the temperatures of both the surface and the atmosphere are calculated, and likewise the difference between radiation entering the system and radiation leaving the system are calculated. The positive feedback effectively comes to an end when this difference is reduced to zero.

Because this is a spreadsheet, the user can examine the formula, and assuming they export a copy as an Excel file(File->Export->Excel), modify the values within the input region.

One of the benefits of expressing the model in this manner is that the user can view this as steps, recognizing the positive feedback which lies at the foundation of the equality between radiation entering and leaving the system. Likewise, once the user recognizes the positive feedback, they will necessarily realize that such feedback does not necessarily lead to any form of runaway effect.

Let me know what you think…

http://www.editgrid.com/user/timothychase/Greenhouse

37. 287
Timothy Chase says:

PS

Here are the equations for the feedback…

1. In: Si = TSI*(1-a)/4

For each generation, Si=Si-1, that is, the same amount of thermal flux is entering the surface-atmosphere system from the sun. All other indexed variables are zero when the index is equal to 1.

2. Surface: Gi = Si-1+Ai-1
3. [Surface to Space]i: (1-Î»)Gi-1
4. [Surface to Atmos.]i: Î»Gi-1
5. [Atmos. to Surface]i: Ai = (1/2)[Surface to Atmos.]i-1
6. [Atmos. to Space]i: Ai = (1/2)[Surface to Atmos.]i-1
7. [Out to Space]i = [Surface to Space]i+[Atmos. to Space]i

Feedback effectively stops when [Out to Space]i = Si

… that is, there is no additional feedback once thermal flux entering the surface-atmosphere system is equal to thermal flux leaving the surface-atmosphere system.

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fredrik says:

Pat, good posts, thanks.

“Radiative solar heating per unit mass is generally greatest at the surface (on land, anyway) because radiation is absorbed over a small distance”

Isn’t the main warming close to the ground due to conduction with the ground and micro convection and not from radiation?

39. 289
Pat says:

Re 288 – “Isn’t the main warming close to the ground due to conduction with the ground and micro convection and not from radiation?”

Well, first, when I wrote “Radiative solar heating per unit mass is generally greatest at the surface (on land, anyway) because radiation is absorbed over a small distance”, I meant the radiative heating of the surface itself. On an asphalt surface, radiation is absorbed within a very short distance. Less so in a forest, where absorbtion is distributed from the canopy to the forest floor, but generally, for SW (solar) radiation absorption at least (and generally for LW too I think), absorption (and emission) at the surface is concentrated over a smaller mass than absorption and emission within most of the atmosphere (except for deep water surfaces in part of the SW spectrum), so radiative variation tends to cause greater temperature variation at the surface then elsewhere outside of the thermosphere (very very small mass of air that absorbs some of the shortest wavelengths of solar radiation). (Cloud boundaries would also be regions of relatively concentrated absorbtion or emission, at least relative to the surrounding air.) Air near the surface, via conduction and convection within the boundary layer, effectively adds some thermal mass to the surface but not enough to completely cancel the effect.

At any given level in the atmosphere, there is SW (solar) radiation going down and up (from backscattering/reflection), LW (emitted by the surface or atmosphere) radiation going down and up, and sensible and latent heat fluxes up and down. Subtracting upward and downward fluxes yields (on average) a positive net downward solar flux, positive net upward LW flux, and positive net upward sensible and latent heat fluxes. The rate of change with height of the flux represents either a flux convergence or divergence – meaning some net gain or loss. In the troposphere, on average there is a net gain from the convective fluxes (sensible and latent heat) which on average balances the net loss from radiant fluxes.

Upward and downward radiant fluxes may largely pass through a thin enough layer of atmosphere, so either absorption or emission from a thin enough layer may be quite small. In contrast, the sensible and latent heat is contained within the air and in a sense is entirely gained and lost simultaneously by a layer through which a a sensible and latent heat flux passes.

So in a thin layer near the surface, atmospheric heat gain from the surface is dominated by conduction and convection, not radiation, but most of that gain is lost to layers above by convective fluxes… and the net gain of sensible and latent heat will be, on average, equal to a net loss by emission of radiant energy, which itself will be the difference in absorption and emission, etc…

To make a long story short, both convection and radiation are important in the troposphere.

40. 290

[[Isn’t the main warming close to the ground due to conduction with the ground and micro convection and not from radiation? ]]

No matter how many ways you phrase this question, the correct answer will continue to be “no.”

41. 291
fredrik says:

It seems to be a difference between climatologists and meteorologists. The climatologist talk alot about radiation and meteorologist more or less “ignore” the influence of radiation in the atmosphere except the absorbation at the ground. I am not completely sure about this though and I am going to go to the library and look trough some meteorology books but that is atleast what I remember from reading before. I dont know if the difference is the time scale involved.

“So in a thin layer near the surface, atmospheric heat gain from the surface is dominated by conduction and convection, not radiation, but most of that gain is lost to layers above by convective fluxes…”

Yes, this is the mixing of the atmospehere. The air close to the ground heats by conduction and micro convection. Making a layer with a nice name that I dont remember. This layer became super adiabatic and thus unstable and give rise to thermals. This phenomenon continues during the day making the convection go higher as the earth warms. This makes the boundary layer dry adiabatic and moist adiabatic above the cloud base. Thus the air in the whole boundary layer warms by conduction by this mechanism. Some might also be warmed by radiation but the effect seems to be so small as meteorologist with expertise in the boundary layer ignore it (atleast in their explainations. The boundary layer called be several thousand meters thick so it is a lot of the atmosphere that is warmed.

The air close to the ground is cooled at night making a night inversion. I have never seen anyone claim that this is due to radiation but that it is due to conduction with the ground.

The temperature close to the ground change a lot between day and night. The temperature above the night inversion warms sligtly during the day due to convection. The air above the boundary layer is pretty constant.

I dont think this can be explained by radiation.

My scenario was for fair weather blue weather days or with small cumulus clouds.

“and the net gain of sensible and latent heat will be, on average, equal to a net loss by emission of radiant energy, which itself will be the difference in absorption and emission, etc…”

The energy transfer in the boundary layer seems to be dominant by convection so I am not sure this is the case.

42. 292
fredrik says:

“No matter how many ways you phrase this question, the correct answer will continue to be “no.” ”

I am not going to trust your no without any explaination anytime soon. I looked at your homepage and nothing on it seems to suggest that you are any authority on climate science or science in general. So why should I trust you?

“[[Isn’t the main warming close to the ground due to conduction with the ground and micro convection and not from radiation? ]]

Take a look in any meteorolgy book and this how they explain the warming of the air close to the ground. The ground is clearly warmed by raditation if that wasn’t clear but the air close to the ground isn’t mainly warmed by radiation. To be clear, I talk about the diurnal warming. Why is it much warmer 15.00 compared to 07.00. Not the mean value.

43. 293

Fredrik,
Conduction is not that effective for heat transfer without convection to cause circulation–likewise radiation. All three processes are important. Air in contact with the ground warms by conduction, but then rises due to its lower density, transporting its energy to the cool air above and cool air takes its place. Likewise, over the ocean, evaporation of water transports lots of energy upwards as latent heat. All these processes are necessary.
Conduction warms by contact. Without convection, it would not be effective. And radiation does transport energy to all levels of the atmosphere. It does not make sense to look at any of these mechanisms in isolation.

44. 294
Timothy Chase says:

Frederik wrote (#291):

It seems to be a difference between climatologists and meteorologists. The climatologist talk alot about radiation and meteorologist more or less “ignore” the influence of radiation in the atmosphere except the absorbation at the ground. I am not completely sure about this though and I am going to go to the library and look trough some meteorology books but that is atleast what I remember from reading before. I dont know if the difference is the time scale involved.

This isn’t a matter of disagreement, but a matter of context.

Meteorologists are concerned with what specifically happens in the next few days, weeks or months on the outside. They can take the total amount of thermal energy in the earth-atmosphere system more or less for granted, given the timescale of their concerns. Climatologists are concerned with the average behavior and variability of the system over years or decades rather than the specific behavior over the next few days. For climatologists, it matters whether or not the amount of thermal energy within the system is increasing over time. The more thermal energy, the higher the temperatures tend to be, and in terms of the trends for average behavior and variability, this has fairly predictable consequences.

When will a given iceshelf collapse? How soon will the next major outgassing of a methane hydrate take place? These are specifics at a lower scale of resolution than climatologists are generally capable of. It wasn’t that long ago that climate models reached the resolution at which they could handle hurricanes, and tornadoes are still at too small a scale. But climate models have improved a great deal within the past few decades, particularly since the NEC Earth Simulator, and they will continue to do so in the next few years. We are getting a much clearer picture of where this is headed.

45. 295
Hank Roberts says:

People keep coming in with this same question over and over.
I keep asking them where they’re getting their information and why they rely on it.
I haven’t ever gotten a clear answer to how this idea keeps being promoted, where they find it.
But it’s been such a steady flow of new names with the same, simple, wrong idea coming in here, saying how radiation can’t be important, so the whole theory about CO2 is wrong, it has to be conduction and convection.

I’m convinced it’s a talking point on one of the PR sites for Western Fuels, or one of the political sites.

Please, someone, where are you getting it? Who’s being so successful at fooling new readers that they come to RC believing this stuff and then spend large amounts of time insisting the science is missing their ‘fact’?

46. 296

[[“No matter how many ways you phrase this question, the correct answer will continue to be “no.” ”
I am not going to trust your no without any explaination anytime soon. I looked at your homepage and nothing on it seems to suggest that you are any authority on climate science or science in general. So why should I trust you?
]]

Don’t trust me. Do some basic reading in the field. Read the article at the link I provided, which is where I got my figures. Read John Houghton’s “The Physics of Atmospheres” (3rd edition 2002), or Grant W. Petty’s “A First Course in Atmospheric Radiation” (2006). I’m not saying anything that isn’t said by every climatologist I’ve ever come across in the literature. Your a priori commitment to convection being more important than radiation in heating the atmosphere is just making you into a crackpot. Educate yourself!