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## Simple Atmospheric Models – Part One

Most good textbooks introduce simple models to help the student gain a conceptual understanding.

In Elementary Climate Physics, Prof. F.W. Taylor does the same.

Now the atmosphere is mostly transparent to solar radiation (shortwave) which is centered around 0.5 μm, and quite opaque to terrestrial radiation (longwave) which is centered around 10 μm. Note that absorptivity is very wavelength dependent, especially for radiatively-active gases.

So in this first model, which is very common in introductory books on atmospheric physics, three things are assumed – and none of them are true:

• the atmosphere is isothermal – a slab of atmosphere all at the same temperature
• the atmosphere is completely transparent to solar radiation
• the atmosphere is completely opaque to terrestrial radiation

From Elementary Climate Physics, F.W. Taylor

Figure 1

As many people know, climate scientists introduce things that are not true in climate science books because either they have no idea what they are talking about, or because they are trying to deceive their readers.

Surprisingly, despite the incompetence and mendacity of these awful people the model is quite illuminating.

How do we calculate the surface temperature?

With apologies for the lengthy explanation that follows – necessary because of the confusion frequently spread on this subject. To grasp the essence of the simple model you don’t need to follow every point here.

When a body is a perfect absorber of radiation it is also a perfect emitter. If this is true at all wavelengths, the body is called a blackbody. In practice, no real bodies, or bodies of gases, are blackbodies but many come close. Especially, many come very close at certain wavelengths or bands of wavelengths.

If a layer of atmosphere has an optical thickness = 10 across a band, then its emissivity in that band = 1.0000.

This means, in this band it is.. still not actually a blackbody because its emissivity has not really reached 1, it is actually = 0.9999546. And if the optical thickness = 20 across a band, then its emissivity in that band = 0.9999999979 – still not a blackbody. For all practical purposes we can say it is a blackbody at these wavelengths because within the limits of accuracy we need, emissivity = 0.9999546 is the same as saying emissivity = 1. Nothing special or magical happens in the equations of heat transfer when we transition from 0.99 to 1.00. And assuming 0.9999546 = 1 introduces a 0.005% error.

The equation for the emission of thermal radiation is Planck’s law which describes how the intensity varies with wavelength for a perfect emitter (i.e., a blackbody), e.g.:

Figure 2

For any real surface (or body of gas) this Planck curve is multiplied by the emissivity curve (vs wavelength) to get the actual thermal emission vs wavelength.

To calculate the flux (W/m²) we can instead use the Stefan-Boltzmann equation (which is just the integral of the Planck curve over all wavelengths):

E = εσT4

where E = energy emitted in W/m², ε = emissivity, σ = 5.67 x 10-8, T = surface temperature

Because ε is a function of wavelength, and because increasing temperatures shift the emission to shorter wavelengths we need to use the value of emissivity for the temperature in question.

Notice, in figure 2, that the emission is very low below 4 μm.

Now, for our incorrect assumption that the atmosphere is completely opaque for all wavelengths greater than 4 μm (longwave) then the equation for emission from the atmosphere will be:

E = εσT4, and as ε=1 at these wavelengths,

E = σT4

Now we have that out of the way..

### Energy Balance

The (incorrectly assumed) optically thick atmosphere emits radiation to the surface at σTa4 and out to space at σTa4 (where Ta is the temperature of the atmosphere). The radiation from the earth’s surface is (incorrectly assumed) completely absorbed by the atmosphere.

In equilibrium, as the general rule:

Ein = Eout

Therefore, the absorbed solar radiation = energy emitted to space from the atmosphere.

Absorbed solar radiation = (1-0.3) x S/4, where S = solar constant of 1367 W/m². The “0.3” is the reflected radiation due to the albedo of the earth and climate system. So only 70% of solar radiation is actually absorbed on average. The term 1/4 appears because solar radiation is not directly overhead all points on the globe at all times. For the detailed explanation see The Earth’s Energy Budget – Part One.

Therefore, for the energy balance of the whole climate system:

(1-0.3) x S/4 = σTa4 [1]

And for the surface energy balance, where Ts= surface temperature (and refer to figure 1):

(1-0.3) x S/4 + σTa4 = σTs4 [2]

So, [1] -> [2]

2σTa4 = σTs4

Re-arranging, we get:   Ts = 21/4.Ta [3]

And from [1], Ta = (239/5.67 x 10-8)1/4 = 255 K

Therefore, Ts = 303 K

So we have a solution to the problem for our simple model with three totally incorrect assumptions. Compare the calculated value with the observed 288 K average surface temperature.

### Conclusion

As Taylor says:

This calculated greenhouse enhancement of 48 K is rather larger than the observed 33 K, not surprisingly in the light of the simplicity of the model.

This model helps us see how temperature of the atmosphere and the surface are related under the simplest of assumptions.

In practice, the atmosphere is not completely opaque to terrestrial radiation and therefore, does not emit like a blackbody. The atmosphere is not completely transparent to solar radiation, and therefore, the atmosphere is also warmed directly by the sun. The atmosphere is not isothermal and, therefore, emits differently to the surface compared with its emission to space.

And everyone in climate science knows this. Real climate models are slightly more sophisticated.

When you read examples like this and like the “multiple shell” model, they are for illumination and education. Simple models teach beginners more than complex models. Who can understand a GCM if they can’t understand this model?

When you read people writing that climate science assumes the atmosphere radiates as a blackbody you know they didn’t make much progress in their elementary climate science textbook. That is if they even picked one up.

Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations – the actual equations of radiative transfer (no blackbodies or Stefan-Boltzmann equations to be seen)

CO2 – An Insignificant Trace Gas? Part Five – the radiative-convective model with a couple of solutions

The Amazing Case of “Back Radiation” -Part One – for measurements of radiation from the atmosphere

### 42 Responses

1. Because ε is a function of wavelength, and because increasing temperatures…

My heat transfer book has only beam length and partial pressure concerning emissivity for gases. Please supply info.

2. SOD: Thanks for carefully reviewing this basic model and including all of the caveats. Unfortunately, one of your caveats – the atmosphere can be modeled by an isothermal slab – is a fundamental part of the model that is completely INCOMPATIBLE with atmospheric physics. (See below.) IMO, this model is somewhat analogous to a model of mechanics that postulates that velocity, not acceleration, is proportional to force, because all objects approach a terminal velocity due to rising friction. (It not a great analogy, but is the best I could devise.) When the basic physics is incompatible with the caveats/postulates of your model, the conceptual understanding obtained from the model will be wrong.

Summarizing the model: The earth’s surface receives 2*σTa^4; σTa^4 from the atmosphere and another σTa^4 (= (1-0.3) x S/4) from the sun. The earth radiates 2*σTa^4 to the atmosphere to maintain a constant temperature. The atmosphere radiates 2*σTa^4, half of it up and half of it down, also maintaining a constant temperature.

UNFORTUNATELY, Taylor’s drawing of the model doesn’t show the σTa^4 of energy that MUST be transferred THROUGH the slab atmosphere so it can escape to space. (It is also missing the 2*σTa^4 radiated upward from the earth’s surface.) How does this energy get through the slab?

Convection and conduct require a temperature gradient to transfer energy. An isothermal slab can’t use these processes.

Radiation might be capable of transferring energy without a temperature gradient*, but not through an opaque slab. Instead, let’s postulate that the slab is semi-transparent. Oops, a semi-transparent slab can’t have an emissivity of 1; if the absorptivity is less than 1, so is the emissivity. Therefore. a semi-transparent slab will emit LESS than the required σTa^4 from the top and bottom of the slab!

*What if we allow the energy to be transmitted through the slab by means of a chain of tens, hundreds or thousands of photons. For all practical purposes, such a slab would be opaque and have an emissivity close to 1. Unfortunately, photons travel in all directions, and NET transfer of energy by radiation requires a temperature gradient. When there is no vertical temperature gradient, equal numbers of photons travel up and down. Radiation can’t transmit the required σTa^4 of energy through an opaque slab without a temperature gradient.

(Unless I have made a gross error), the assumption that the atmosphere can be usefully modeled as an isothermal slab is incompatible with fundamental physics – there is no way to transfer the required σTa^4 of energy through an isothermal slab. This flawed model teaches the concept that temperature in the earth’s atmosphere and at the surface is controlled by radiative equilibrium. This concept is ALSO fundamentally flawed.

IMO, any model of the atmosphere built on isothermal slab atmosphere emitting blackbody radiation belongs in your “Debunking Flawed Science” section (and should be treated with the same disdain).

As you have written in other posts, a correct physical description of the atmosphere is based on radiative-convective equilibrium and REQUIRES a temperature gradient. From such a model, we learn that the earth’s surface temperature is mostly determined by radiative equilibrium high in the atmosphere AND the steepness and length of the temperature gradient that reaches from regions of radiative equilibrium to the earth’s surface. (“Mostly”; a complete model includes the roughly 10% of energy that escapes directly to space from the surface because the atmosphere isn’t completely opaque to LWR.)

3. Dear SoD.

I used to teach a similar example to my students. In my case, I supposed there are constant absorptions of shortwave and longwave radiations (as and al, different). Then, you can show that for the case as < al, the partial derivative of Ts with respect to al is positive. This has an obvious implication … This is a simple model, as you mention, nothing to do with real Earth, but … It is an interesting exercise.

4. on April 6, 2011 at 7:25 pm | Reply DeWitt Payne

mkelly,

Your heat transfer book leaves something to be desired. The emissivity of a gas is a function of the temperature as well as the mass path (partial pressure times the path length) and the absolute pressure because at different temperatures different bands absorb and emit and the line width varies as a function of absolute pressure.

The integrated emissivity function (after Hottel) for CO2 looks something like this:

The temperature is in degrees R which is Fahrenheit degrees referenced to absolute zero ( -459.7 F).

• It was my error in fast typing to forget to include temperature as one of the factors not the books. However, the Hottel charts do not include a frequency as part of the input. But none the less they show the emissivity of CO2 to be low and infact drop with an increase in temperature.

• What is the value of emissivity and what is:

– the path length
– partial pressure of CO2
– total pressure & temperature

The emissivity does drop as temperature increases. This is because individual lines narrow (as when pressure reduces).

The formula for line width can be seen in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Nine:

f(ν-ν0) = αL/π((ν-ν0)²+αL²)       [1]

where ν0 is the frequency of the absorption, ν = frequency and αL = half-width at half of the maximum

And the half-width at any given temperature and pressure is given by this formula:

αL = α0(p/p0).(T0/T)γ [2]
where α0 = lab measured half-width at pressure p0 and temperature T0, and γ is also a lab measured parameter, typically around 0.5 – 0.7.

So you can see that when T increases, T0/T reduces and so the line width reduces.

5. on April 6, 2011 at 8:46 pm | Reply DeWitt Payne

Frank,

UNFORTUNATELY, Taylor’s drawing of the model doesn’t show the σTa^4 of energy that MUST be transferred THROUGH the slab atmosphere so it can escape to space. (It is also missing the 2*σTa^4 radiated upward from the earth’s surface.) How does this energy get through the slab?

What is it about opaque and isothermal you don’t understand? There is no through. For heat transfer purposes, the slab has either infinite conductivity or zero thickness. It must or it wouldn’t be isothermal as defined in the initial conditions. No temperature gradient, no resistance to heat transfer. It absorbs 2*σTa^4 from the surface and emits 4*σTa^4, half up and half down. It transmits zero (in the IR). For short wavelengths, it transmits 0.7, reflects 0.3 and absorbs nothing. The surface absorbs everything and reflects and transmits nothing.

• DeWitt: Your suggestion that a slab atmosphere could have zero thickness and still be opaque seems unreasonable. The suggestion that the slab could have infinite conductivity has some merit. A thin layer of highly conduction metal foil would behave this way. Do you really want to propose that a thin metal foil surrounding the planet is a reasonable model for the greenhouse effect? :))

• This is not what making a simplified model is about. It’s “let’s try an approximation” and see what result we get.

Taylor is not trying to suggest that the atmosphere could be all at the same temperature.

He is saying “if we pretend it is, what result do we get?”

Given that the atmospheric temperature only varies by a factor of about 1.5 from coldest to hottest while other parameters vary by many orders of magnitude that is an interesting simplification for a model.

6. Frank:

Unfortunately, one of your caveats – the atmosphere can be modeled by an isothermal slab – is a fundamental part of the model that is completely INCOMPATIBLE with atmospheric physics..

I stated at the beginning:

..three things are assumed – and none of them are true:

– the atmosphere is isothermal – a slab of atmosphere all at the same temperature
..

It’s a simple model. This approach to teaching a subject isn’t confined to atmospheric physics.

Create the simplest model you can, and find the solution.
Add a refinement from reality, find the solution.
Add another refinement, find the solution.

It demonstrates how important the various elements of reality are to getting the right solution.

Like the pCO2 and pH2O model in Understanding Atmospheric Radiation and the “Greenhouse” Effect. These illuminate the subject because they are simpler.

• SOD: I would have had no complaints if your caveat had said, “A useful model of the earth’s surface and atmosphere should provide a sensible mechanism for energy absorbed by the earth’s surface to escape to space. Taylor models the earth’s atmosphere as a single, opaque, isothermal slab. Since the slab is isothermal and opaque, there is no mechanism by which energy can escape to space by conduction, convection or radiation. Despite this limitation, Taylor shows how this model can be used to calculate the Ta, Ts and a 44 degK greenhouse effect. These calculations misleadingly suggest that Ts is determined solely by radiative equilibrium. The real atmosphere has a steep temperate gradient. Its surface temperature is actually determined by a combination of radiative equilibrium high in the atmosphere and this temperature gradient.”

Is my version of the caveat scientifically accurate? Would anyone want to read about such a poor model?

Models of the atmosphere with one or more isothermal, optically-thick slabs are easy to use because because they emit equal amounts of blackbody radiation in two directions, no matter how much GHG is inside the slab. Unfortunately, the “top” of every atmosphere is not optically thick because the pressure is too low. IMO, one can’t build a useful model from such slabs, there are too many serious caveats.

One needs to work with optically thin slabs, but the emission from optically thin slabs also depends on how much GHG they contain. The best such slabs are infinitesimally thin with unit surface area; behavior described by the Schwartzschild equation. Your superb pCO2 and pH2O model in Understanding Atmospheric Radiation and the “Greenhouse” Effect uses the correct physics and the observed temperature gradient to calculate DLR and OLR. These post aren’t surpassed in elegance and utility by anything I have read at a blog, but their simplicity is not what makes them great. These posts used the correct physics (Schwartzschild), the observed lapse rate, and just enough complexity to illustrate many important aspects for atmospheric DLR and OLR: saturation, overlapping lines, the decrease in water vapor with height.

A useful model has all of the complexity needed to properly illuminate real phenomena, with nothing extra. A bad model uses compromised physics and produces misleading conclusions.

7. mkelly:

I will respond in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Ten, where you ask a similar question.

8. Frank on April 7, 2011 at 7:33 am:

I feel slightly bad arguing with someone who says my models are so wonderful.

But anyway, I’m only arguing with your caveat:

..Taylor models the earth’s atmosphere as a single, opaque, isothermal slab. Since the slab is isothermal and opaque, there is no mechanism by which energy can escape to space by conduction, convection or radiation..

Totally wrong.

There is a mechanism – radiation. Optically thick means that the emissivity of this slab = 1. This means it radiates very well – a perfect emitter. And this is the mechanism by which energy escapes to space.

..Despite this limitation, Taylor shows how this model can be used to calculate the Ta, Ts and a 44 degK greenhouse effect..

Ok

..These calculations misleadingly suggest that Ts is determined solely by radiative equilibrium. The real atmosphere has a steep temperate gradient. Its surface temperature is actually determined by a combination of radiative equilibrium high in the atmosphere and this temperature gradient.”

“Misleading” is what I take issue with.

These calculations demonstrate that when you use the simplest model possible you get a result which is inaccurate but in the right order of magnitude.

Misleading implies that someone is trying to trick you.

Is that what you think?

• SOD wrote:

“There is a mechanism – radiation. Optically thick means that the emissivity of this slab = 1. This means it radiates very well – a perfect emitter. And this is the mechanism by which energy escapes to space.”

I disagree. Do you agree that oTa^4 of radiative energy must pass THROUGH the slab for Taylor’s model to represent energy flow on the earth? If so, divide the optically-thick slab into a top half and a bottom half. The flux from bottom half to top half is oTa^4 and the flux from top half to bottom half is oTa^4. QED, the net vertical radiative flux through the center of an isothermal opaque slab is zero. Without any mechanism for energy flux through the slab, the model is internally inconsistent.

An isothermal, optically thick slab of atmosphere is an awkward concept reason. The model has infinitely steep temperature gradients at the interfaces between the slab and earth and between the slab and space. In the real world, energy flows to reduce steep gradients. The oTa^4 from the top to space for example must create a temperature gradient in the “isothermal” slab. You are correct in saying that an optically thick slab MUST emit oTa^4 from its top to space, but I am saying it CAN’T this for long without energy flux through the slab.

SOD also wrote: “Misleading implies that someone is trying to trick you”. I agree. The misconception that is being promoted by this model is that the earth’s surface and atmospheric temperature can be explained by radiative equilibrium alone. The model suggests that we can get reasonable answers – to a first approximation – by ignoring everything else. A look at the K-T energy budget shows how misleading Taylor’s model is.

First, let’s modify the K-T diagram to fit your caveats: The 40 W/m2 escaping directly to space and the 67 W/m2 absorbed by the atmosphere are assumed to be zero. These are reasonable “first approximations” given the larger fluxes involved elsewhere. When you ignore the 40 W/m2 escaping to space, the difference between DLR and upward radiation from the earth’s surface is only 26 W/m2. (Miskolczi apparently claims the difference is really only 12 W/m2). Taylor’s model indicates that the difference should be equal to OLR, but in reality the difference is 10-20X smaller. What is missing? Roughly 100 W/m2 of convection, which provides more than half of OLR.

Why does this distortion exist? Two of your caveats are sensible, the isothermal caveat is not. The temperature difference between the top (220? degK) and bottom (288 degK) of the troposphere is relatively small in an absolute sense (about 25%), but the difference is a factor of 4 (300% difference) in terms of oTa^4. The isothermal caveat is unreasonable in ANY first approximation. (You once wrote a post on how the non-linearity of oT^4 leads to major misconceptions.)

In summary, Taylor’s model is borders on scientific absurdity and gives very poor estimates for some energy fluxes. (IMO, it also promotes misconceptions.) It does happen to give a reasonable estimate of the greenhouse effect. Given the usual quality of the physics at your site, I think Taylor’s model deserves the type of careful debunking you often provide to skeptical ideas.

• Hi Frank,

I rather agree with you that an optically thick isothermal slab is an awkward concept.

The only way I can make sense of it is to say that the extreme optical thickness means that any photons that strike the inner layer are instantly thermalised, as are any photons emitted within the slab. In effect, there is no radiative transfer within the slab at all. The energy transfer from the inner to outer is purely by conduction. To meet the isothermal requirement we are then forced to imagine the existence of such a thing as a superconducting gas.

For myself, it is much easier to think that a thin metal shell would be a natural way to visualize a system that would have the specified thermodynamic properties. After all, we only need to slightly idealize common metallic characteristics to meet the requirements.

I think SoD is a bit trapped by trying to debunk some of the nonsense that is spouted by those who do not have a clue about physics and a desire to give a further education to those who want a deeper understanding of the complexities of radiation.

It seems to me that the Schwarzschild layer model needs to be the starting point for those of us that have grasped the basics. There is plenty of useful work to be done on showing how the properties of each layer can affect the OLR and I think SoD was making a good start.

9. […] Comments « Simple Atmospheric Models – Part One […]

10. Hi SoD,

That was an interesting post. When I first looked at the Taylor diagram I assumed he was going to reach a conclusion that I have seen before. That is to say, a very optically thick isothermal atmosphere would lead to the same OLR as an optically thin or non-existent one.

The discrepancy is clearly due to the fact that earlier ruminations on this had presumed that Ta would necessarily be the same as Ts. In this version there is clearly a thin vacuum layer separating the surface from the atmosphere or some assumtion that there can be no convection or conduction between the surface and the atmosphere.

I can see no difference now between this model and the single shell model alluded to by Frank. This atmosphere in your model is just a shell. It is superconducting, both inner and outer sufaces act like black body radiators and it is separated from the inner sphere by a vacuum.

If I am not mistaken all the equations are identical and lead to the same OLR, shell and inner sphere temperatures. The only thing that is an improvement on the shell model is that an “atmosphere” has a real mechanism to allow solar to pass through the shell because of the difference in wavelength.

Please don´t take this as being critical of the excellent work you are doing but this particular article does not seem to get us much further than the single shell model.

• Jorge: Thanks for the kind reply above. SOD does a great job debunking lousy climate science from skeptics, but criticizing the climate establishment here sometimes seems feels like spitting into the wind. (When I make scientific errors – sometimes from hurrying – it definitely seem like spitting into the wind, but I do get my physics straightened out.) I wasted a lot of time being confused optically thick slabs (more GHGs above the slab hinder OLR, but more GHGs in the slab don’t increase emission), so I equate optically thick slabs with Gore’s AIT.

The ultimate in optically thick slabs is the interior of the sun. It apparently takes thousands of years for energy generated inside the sun to diffuse through radiation (repeated cycles of emission and absorption) to the surface.

• on April 10, 2011 at 3:49 pm | Reply DeWitt Payne

this particular article does not seem to get us much further than the single shell model.

Nor was it intended to go further. Even a simple shell model seems to cause a lot of confusion. Did you want him to start with a line-by-line radiative transfer model of the whole atmosphere?

• Hi DeWitt,

“Even a simple shell model seems to cause a lot of confusion.”

This was a single shell model. If we are going to talk about such models it is much easier to visualize as a thin metal shell.

Yes, you do need some basic physics to follow how this leaves the inner sphere at a higher equilibrium temperature than it would be without the shell. I quite appreciate that a lot of folk seem to have real difficulty in grasping this point. If they could simply accept that material emits according to its temperature and has nothing to do with its surroundings or whether it is simultaneously absorbing radiation they would find it all slots into place.

If this were SoD´s first post then of course not, but I had hoped that some progress had been made in educating the readers to the point where continuing with this was a reasonable proposition.

• Jorge said:

If this were SoD´s first post then of course not, but I had hoped that some progress had been made in educating the readers to the point where continuing with this was a reasonable proposition.

I have new readers all the time.

Many people have misunderstood these simple models.

Many people have assumed that the assumptions in these simple models are the “best practice” of climate science.

Just recently an article on another blog claimed that the IPCC climate models worked from the basis that the atmosphere emitted as a blackbody.

And I read comments on blogs continually with assertions to this effect.

11. on April 10, 2011 at 3:44 pm | Reply DeWitt Payne

Frank,

I wasted a lot of time being confused optically thick slabs (more GHGs above the slab hinder OLR, but more GHGs in the slab don’t increase emission), so I equate optically thick slabs with Gore’s AIT.

Can you cite me an example of that? If it’s a single slab model, there are no GHGs above the slab. In the single slab model in Petty, for example (page 135(first edition) equation 6.25), increase the long wave absorptivity and emission from the slab increases. SoD’s slab is opaque, absorptivity = emissivity = 1, neither absorption nor emission can be increased any further. Are you sure you aren’t confusing short wave and long wave properties?

• DeWitt: I can’t remember all of the situations that confused me as I tried to learn the physics of radiation transfer. You must have seen many unsophisticated explanations of GW that rely on GHGs “trapping” OLR in the atmosphere.

I personally knew enough to recognize that GHGs both absorbed and emitted LWR and that more GHG must result in more emission as well as more absorption. (Since these processes are the time reverse of each other, the same QM constant applies to both processes.) However, in every model with an optically thick slab, emission would be unchanged by rising GHGs. None of the alarmists bothered to discuss the relationship between emissivity and absorptivity – they just let me assume that more GHGs would produce more absorption (trapping), but not more emission. I knew this wasn’t correct, but couldn’t figure out why.

Since emissivity is a constant for solids, liquids, and optically thick slabs, it is difficult (under self-taught circumstances) to realize that emissivity varies with the amount of GHG in the atmosphere. I knew that emissivity INTEGRATED over all wavelengths was <1, because there was no emission at some wavelengths. Integrated emissivity obscures the existence of emissivity at a single wavelength – we use the same word and symbol for both. It was hard to learn that emissivity at a single wavelength varied with GHG mixing ratio, especially when I encountered mostly optically-thick slabs. It didn't help that Professor Taylor omitted the red e from the Figure 3.5 in SOD's Part 2 on simple models, showing me an optically thin slab emitting blackbody radiation.

I had the most trouble trying to understand radiative forcing, which is calculated at the tropopause. I imagined myself at the tropopause, surrounded by 1X or 2X CO2. If there were twice as much CO2, why wasn't there twice as much emission as well as twice as much absorption? If my memory is correct, I pictured the something like the model in Part 2 of this series, with the troposphere being an optically thick slab, the stratosphere above being optically thin, and the tropopause (where radiative forcing is defined) as the top surface of the slab. Despite the presence of twice as many CO2's (which intuitively should be able to radiate away twice as much energy), emission was determined by blackbody considerations (after correcting for the wavelengths with no emission), but absorption "obviously" increased when 2X CO2 was present. I didn't worry about where the extra absorption took place.

Then SOD introduced me to the Schwartzschild equation; the clouds parted, the sun shown brightly, and I began to view optically thick slabs as the root of all evil.

For the self-taught, blackbody radiation can be a very confusing place to start when learning about atmospheric radiation. Blackbody radiation arises when nothing is changing; dI/ds = 0 = -noI_0 + no[B(T)] and therefore I_0 = B(T). The important physics is about how radiation leaving the earth's surface is CHANGED by its passage through the atmosphere. Radiative forcing is about this change. Both absorption and emission change with the amount of GHG (n), the same cross-section (o) applies to both absorption and emission, and everything makes sense intuitively. By eliminating all of this change, optically-thick slabs make computation easy and understanding change difficult.

12. One day, when we discuss the role of complexity and I introduce the double pendulum, I look forward to the unfolding discussion about why a point mass has been introduced.

Readers will no doubt ask how it is exactly that an object of infinite density can be constructed and why it is that the teachers of mechanical engineering have been so stupid as to miss the fact that point masses are a physical impossibility.

There is no chance that this would miss the reason for, and benefit of, the approximation.

• Hi SoD,

I don´t remember any mention of infinite density in my school books on statics and mechanics. So far as I remember it was simply an empirical fact that bodies acted just as they would if all the mass were concentrated at the centre of gravity. I don´t think there was ever any suggestion that a mass of infinite density was actually present at the centre of gravity.

Actually, I think Newton was quite worried about the mass of the earth being distributed when calculating the gravitational pull but eventually concluded that treating the mass as though it were concentrated at the centre gave the same result.

If you want to postulate a superconducting gas as a possible simple model for the atmosphere that is fine by me. My problem was that I don´t normally think of gas as anything but a rotten conductor of heat. This left me trying to figure out how the hell it could be isothermal if it was moving heat from one side to the other.

The other minor point that you did not mention was that this super conducting slab was not actually in contact with the earth surface which led to my initial reaction that it would behave as though it was not there.

13. Jorge:

You are still missing the point completely.

Making an approximation about the temperature of the atmosphere doesn’t mean that some mechanism for this needs to be found or assumed.

Take a look at all the atmospheric physics textbooks on the derivation of atmospheric pressure with height. They calculate a scale height – assuming a constant temperature.

Pressure varies by a factor of 100,000, temperature by a factor of 0.7, so it’s a convenient approximation to assume a constant temperature to get a nice analytical formula – as a first approximation.

This kind of “crude assumption” approach is used continually in physics and engineering. It is useful.

Point masses, point charges, bodies with infinite dimensions, perfect black bodies, infinite time for equilibrium, perfect insulators.. the list is long and if you don’t get the point now you never will.

My last comment on the isothermal atmospheric approximation.

14. Hi SoD,

This is not really on topic but this thing about imagining model worlds reminded me of an end of term A level physics class. Our teacher asked us to think about what would change if God decided to suspend the law of gravity.

It was all very interesting because the atheists said that was ridiculous because there was no god. Others were sure God had made the rules when the earth was created and couldn´t change them now. Some thought God could do this but would not cheat in this fashion.

The rest, including me, set to work reviewing all the equations we could think of that contained the symbol g to see what they they would imply with g set to zero.

I spent my working life with electronics and so, really, abstraction and approximation are second nature to me. I suspect that you and I are completely in harmony on this kind of thinking and I would not want you to feel that I am disparaging any of your efforts.

15. SOD: You asked an interesting question about what makes a model “misleading”. Is a model “misleading” only if the person who uses the model intends to “mislead” others. Or is a model “misleading” when a significant fraction of the audience for the model is “mislead”? What if someone keeps using it despite knowledge that some people will be mislead?

If one wanted to illustrate the principles of the greenhouse effect through a calculation of this type, a thin metal shell could be used in place of the slab atmosphere. Then one could explain in what way the real atmosphere is similar to a metal shell (DLR warms the surface) and how it is different (convection+radiation vs. conduction, significant temperature gradient in terms of oT^4, 25% of solar SWR is absorbed on the way down, 10% of surface LWR transmitted). Then one could compare the calculated fluxes for the metal shell to those of the real atmosphere.

• on April 13, 2011 at 2:58 pm | Reply DeWitt Payne

Isn’t that a distinction without a difference? Or is it so important that your shell has to be a real substance? Lots of toy models are constructed with properties that can’t exist in the real world like a reflective surface that is perfectly reflective at all frequencies including molecular collisions or a perfect blackbody. The calculation of 33 degree warming from the greenhouse effect assumes an isothermal sphere. You can’t have that unless either the heat capacity is infinite and there has been an infinite amount of time for equilibrium to be reached or the sphere is superconducting, neither of which is possible in the real world. I don’t see why you have such a problem accepting the conditions of the single layer, non-reflecting atmosphere model for the purposes of illustration.

In a metal shell model, you then have to introduce the complication of an internal heat source to represent the heat input from the sun. Either that or the metal shell has to be transparent at short wavelengths and opaque at long wavelengths and we’re back to a single layer, non-reflecting atmosphere again.

• on April 13, 2011 at 3:02 pm | Reply DeWitt Payne

You can’t model the real atmosphere with a single isothermal shell. Once you introduce a temperature gradient, the model is no longer simple and must be approximated by multiple shells, not to mention that the absorption/emission properties of the model atmosphere can’t be gray (constant at all frequencies) either.

• on April 13, 2011 at 6:40 pm | Reply DeWitt Payne

It looks like the minimum model you’re interested in discussing is a 1D radiative-convective model with at least a band model for the radiative transfer. That seems a lot like trying to run before you can crawl.

But of course that isn’t ‘real’ either because it leaves out the other two dimensions, not to mention the ocean. But you have to remember that no model is ‘real’. As Korzybski stated: “The map is not the territory.” Or as Box stated: “Essentially, all models are wrong, but some are useful.” The toy single layer, non-reflective atmosphere model is useful as a pedagogic tool, or at least all textbook writers that I know of think so.

• DeWitt: I agree with most of your points. I was picturing a thin, highly-conductive metal shell that let solar SWR pass through.

Models are often a trade-off between simple/unrealistic/potentially-misleading and complex/realistic. IMO, any model of the atmosphere that doesn’t include convection is potentially misleading, and I see too many of those. Forcing high in the atmosphere where radiation controls temperature must lead to warming there, but the environment lapse rate will determine how much of that warming (if any) is felt at the surface. (Other feedbacks, of course, are also important.) Alarmists want to oversimplify the situation so that the connection between absorption of OLR by GHGs and catastrophic warming appears more direct. For example, Taylor’s book follows these models to a no-feedbacks climate sensitivity of 16 degK for 2X CO2. (SOD may have undertaken this series of posts to clarify this situation.)

Unlike Taylor, SOD was very clear with his three caveat about how this model differs physically from the real atmosphere. He wasn’t clear about how much distortion those three caveats introduced into the result: no convection, absence of a temperature gradient big enough to produce a 4-fold difference between OLR and DLR, emissivity that varies with GHG concentration. When people see a model (even with caveats), they assume the model is being presented because it reflects important aspects of the real atmosphere. “All models are wrong, but some models are useful” – is problematic when we are told only the CAVEATS, but it can be true when we know what CONCLUSIONS are trustworthy and what conclusions are dubious or wrong. The advantage of the thin metal shell model – as opposed to the optically thick slab atmosphere – is that everyone can see how absorption and re-emission of OLR can warm the surface via DLR (the greenhouse effect), but no one will mistake the metal shell for a real atmosphere. (This is a disadvantage for alarmists.) An optically thick, isothermal slab atmosphere is actually a much better model for a thin metal shell transparent to SWR than for the real atmosphere.

16. on April 14, 2011 at 1:44 am | Reply DeWitt Payne

Frank,

You keep missing the point. A single slab, non-reflecting atmosphere model is used as a teaching tool to demonstrate the effects of different short wave vs long wave absorptivities on purely radiative heat transfer. Period. Low short wave and high long wave absorptivity increases surface temperature over the temperature of a surface with no ‘atmosphere’. The opposite decreases surface temperature. Thus we begin to understand why temperature declines with altitude in the troposphere and increases with altitude in the stratosphere. The other point that’s made is that this toy model with careful selection of short and long wave absorptivities gets you very close to the magnitude of the Earth’s greenhouse effect. But that’s all.

The next thing you might do is start looking at differences between the Earth and the toy model by looking at measured quantities like surface temperature and DLR and solar radiation. The first thing you see is that the surface temperature is too low so there must be another form of heat transfer that cools the surface. Then you can start talking about convection and work towards a more complex and realistic model.

Or you could start with classic physical meteorology which explains why the temperature profile in the troposphere exists but doesn’t explain why the surface temperature is what it is or why the temperature increases in the stratosphere. Then you introduce radiative heat transfer still with the toy model as a first pass and gradually get more complex.

• Dewitt wrote: “You keep missing the point. A single slab, non-reflecting atmosphere model IS USED as a teaching tool to demonstrate the effects of different short wave vs long wave absorptivities on purely radiative heat transfer. Period.” [My CAPS] Try replacing “is used” with “should be used only”? Aren’t multiple optically thick slabs used to explain why Venus is so hot? Isn’t the earth’s atmosphere broken up into multiple slabs (making the assumption that each slab is optically-thick increasingly untenable)? Taylor mentions both of these applications in his book immediately after the model.

DeWitt wrote: “The next thing you might do is start looking at differences between the Earth and the toy model by looking at measured quantities like surface temperature and DLR and solar radiation.”

I did precisely this (comment above on 4/8, 10:49) by comparing the predictions of Taylor’s model to the K-T energy budget for the earth. I found that the assumption that the slab atmosphere is isothermal leads to gross underestimates for DLR and overestimates for OLR. In the real troposphere there is a 4-fold difference between DLR and OLR, but the isothermal model requires them to be equal. A modest temperature between the top and bottom of the troposphere turns into a big change in oT^4.

Comparing the model to reality would have made an excellent addition to SOD’s post (and to Taylor’s book). Instead of specifics, we get meaningless generalities in SOD’s conclusion that don’t explain how good or bad the model really is:

“In practice, the atmosphere is not completely opaque to terrestrial radiation and therefore, does not emit like a blackbody.” How transparent is it? “The atmosphere is not completely transparent to solar radiation, and therefore, the atmosphere is also warmed directly by the sun.” How much energy gets absorbed there? “The atmosphere is not isothermal and, therefore, emits differently to the surface compared with its emission to space.” How different is it?

As Steven Schneider said: “as scientists we are ethically bound to the scientific method, in effect promising to tell the truth, the whole truth, and nothing but — which means that we must include all the doubts, the caveats, the ifs, ands, and buts.” I’m sure you know the rest of this quote about how to talk to the public. Are we being scientists or advocates here?

• Read the first sentence of this post. I’ll quote it again below.

Most good textbooks introduce simple models to help the student gain a conceptual understanding.

Then read the title of the post.

Simple Atmospheric Models – Part One

You’re spending a lot of time attacking a straw man of your own devising. If you can’t see this, then further comment from me or SoD is pointless.

17. Hello
In the book of “Global Physical Climatology” in chapter 3, the author use extend energy balance model contains two layer one at 0.5 km and the other at 2 km. The author say that the reason of that is in the book called “Atmospheres” chapter 3 ,for Goody , here is a link for the book. In the book of Atmosphere he say that he choose a layer at 0.5 km and 3 km(not 2 km) as the scale height of water is 2 km, is that logical???. Also in the same model in the book Global physical Climatology he used a layer high in the atmosphere and he called it skin layer, he put graph for the temperature of that layer, anybody knows how he knows it’s height ??

18. Ahmed,

The link didn’t get pasted in (if you paste text with a link only the text makes it through in the blog’s comment field).
Is this book by Hartmann?

19. Thanks for your note .

yes, the book “Global Physical Climatology” is for Hartman.

20. Ahmed,

Hartmann says (p.61):

..For an atmosphere with a large infrared optical depth, the radiative transfer process can be represented with a series of blackbodies arranged in vertical layers. Two layers centered at 0.5 & 2.0km altitudes provide a simple model for Earth’s atmosphere..

And references Goody & Walker (1972), who say (p.58):

..Experience suggests that a reasonable model consists of 2 layers, with the top layer centered at a height of about 3 km and the bottom layer centered at a height of about 0.5km. We can derive these heights by the method outlined earlier, if we remember that water vapor is the principal absorbing gas in the Earth’s atmosphere. Observations show that the scale height of water vapor is about 2km.

You are correct that there is a slight discrepancy between the two authors’ models.

The critical point to understand about these models is that they are all quite arbitrary. The real calculation of radiative transfer is much harder to grasp conceptually. So these simple models are intended to make a clearer link between concept and maths to derive a result that has some relevance without being perfect. Therefore, the teacher can pick one optically thick layer, or three, or five, and the choice of altitude (and therefore temperature) can be quite arbitrary because the result will not be correct, but it is intended tol be illuminating.

On your last point, the skin layer is the point at which the atmosphere becomes optically thin. Again, it is a conceptual idea which helps the student understand how radiative transfer works in the atmosphere for a very simple atmosphere.

21. Thank you for your answer. I now understood that now that the difference between height of layer between two author is not something wrong, But it due to different view or thought about the conceptual ideas that they try to proof or understand. About the two skin layers that are mentioned in Hartmann book, one at stratosphere and the other near the ground. He mention nothing about their height in spite of the two middle layer,that he mention their heights 0.5 and 2 km explicitly.
I just know the height of the two skin layer from the graph at p 63, the layer at stratosphere at 15 km and the other layer very near to the ground(it is not clear from the graph actually how is the height). I don’t know how to upload figures in the comment.