This post comes about as an attempt to write down, slowly and carefully, a simple version of the “idealised GHE model“. This apparently simple concept causes lots of confusion, though mostly amongst people who want to believe there are fatal flaws at the heart of climate science.

Before I go on: this is an idealised model. No-one actually does any real calculations from it. Some of the concepts involved are used in GCMs, but anyone who says, for example, “aha! You’ve averaged the diurnal cycle in your model, therefore the GCMs are wrong” isn’t thinking.

Lets consider a very heavily idealised model of the greenhouse effect. There’s only one “atmospheric level”, and energy transport is by radiation only. Very importantly, this is an equilibrium model. In each layer (sfc, atmosphere, space) the fluxes balance; none of the layer’s temperatures are changing over time.

[SUN. SW to Earth] ^ [LW to space] (1) [S=1362 W/m2; 0.7*S/4=238 W/m2] | V ^ ------------ V ---[ATMOSPHERE Ta]-----+------------- (2) V ^ V V | | V ^ V V | | V ^ V [LW absorbed at sfc] /////////////////////[SURFACE Ts]//////////////////// (3)

The surface, at temperature Ts, emits thermal radiation given by (rho)Ts^4. The atmosphere at temperature Ta emits thermal radiation (upwards *and* downwards, this is important) according to the Stefan-Boltzmann law given by (rho)Ta^4; (rho) is the Stefan-Boltzmann_constant, 5.67/10^8 W/m^2/K^4. Or close enough for our purpose here.

[Update; note, although I've written (rho) here, and it all works of course because it is just an identifier, the usual symbol is the Greek "sigma", so (sigma) would have been clearer. Thanks to mz.]

You’ll see at once that this is not a strictly realistic model of the Sun-Earth-Atmosphere-Space system. Never mind, we’ll not worry about that much now. Lets just note the salient features of the model, without justifying them:

1. The Earth is assumed to be uniform and a perfect blackbody. Its also either perfectly super conducting, or some other appropriate set of assumptions to get a uniform surface temperature. It still spherical, though, if that helps.

2. Solar radiation, S, is 1361 W/m2. But because the Earth is a sphere, and the ratio of the area of a sphere (4.pi.r^2, Earth’s surface area, which emits thermal radiation) to a circle (pi.r^2, Earth’s cross-section to Solar radiation, which determines how much Solar we absorb) is 4, the average insolation per unit area of the Earth is 1361/4 at the top of the atmosphere. But since Earth’s albedo is ~0.3, the average insolation at the surface is 0.7*1361/4 = 238 W/m2 (if you do the approximations slightly differently you get 239; no-one cares about that difference at this level of accuracy).

3. In this version, the atmosphere is entirely transparent to Solar radiation (also known as SW, for Short Wave) and entirely opaque to Earth’s emitted thermal radiation (also known as LW, for Long Wave). And for the purposes of the model, SW and LW occupy non-overlapping bands; this last assumption is realistic, unless you’re being really picky.

4. Energy transport is by radiation only. This is not at all realistic for the totality of the Sfc-Atmos system, but it isn’t too implausible if you think of the model “surface” as representing some level of the atmosphere, perhaps 400 hPa.

5. There is only one “atmospheric” level, and only by convention is it called “atmosphere”: it could be a sheet of glass. It has no defined height above the surface.

We can now write down the energy balance of the three layers:

(1) Space. Solar radiation (0.7*1361/4 = 238 W/m2) absorbed by the Earth equals the LW radiation lost by the Earth. Hence,

238 = (rho)Ta^4

Woo, that’s good. We can immeadiately work out the temperature of this layer:

Ta = fourth_root(238*10^8/5.67) = 254 Kelvin.

(2) The Atmosphere. This one is a bit harder, but not much. The Solar SW radiation doesn’t appear, since the atmosphere is transparent to SW. So the terms are: the upwelling LW radiation from the Earth’s surface; balanced by *two* terms, from the upward (lost to space) and downward (absorbed by the Earth’s surface) thermal radiation emitted by the atmosphere. The crucial point here is that there are these *two* LW “loss” terms, and they must be equal, because they are both thermal emission from each side of an object, the atmosphere, with the same temperature, Ta. So:

(rho)Ts^4 = 2 * (rho)Ta^4.

From which the (rho) cancels, leaving:

Ts = fourth_root(2) * Ta.

Which is nice and simple. We now know the surface temperature, and its warmer than the atmosphere, by a factor of fourth_root(2) = 1.19. Ts = 302 Kelvin; roughly.

(3) The surface. Note that we’ve now run out of variables to find the value of, since we know the sfc and atmospheric temperatures, so we’d better hope the sfc budget balances. The terms are: incoming solar, 238. Outgoing thermal, (rho)Ts^4. And incoming atmospheric thermal, (rho)Ta^4. So:

238 + (rho)Ta^4 = (rho)Ts^4.

Happily, if you look back to the balance for layer (1), you’ll see we’ve already deduced that (rho)Ta^4 = 238, so this equation just says:

2 * (rho)Ta^4 = (rho)Ts^4

Which makes it a duplicate of the balance for layer (2), and thus it must balance, because we’ve already balanced that layer. So it all ends happily, phew (you may be tempted to see that as sleight of hand but it isn’t; it had to happen like that, or the physics would have been wrong).

### In summary

We end up with an energy balance that looks like this:

[SUN] ^ [LW to space] (1) [S=1362 W/m2; 0.7*S/4=238 W/m2] | [= 238 W/m2] V ^ ------------ V --[ATMOSPHERE 254 K]---+------------- (2) V ^ V V | | V ^ V V | | V (rho)302^4 ^ =238*2 V [LW absorbed at sfc=238 W/m2] /////////////////////[SURFACE 302 K]//////////////// (3)

### Having done that…

You’ve probably guessed that I’ve written all this down because someone else has got it wrong. That someone is “Derek CAVEMAN SCIENTIST”, and his version is a set of slides at www.globalwarmingskeptics.info/thread-2250.html (Update: actually he’s changed them. Sigh. Fortunately I took a webcite. As far as I can see from a quick scan he has removed some redundant pics, but his basic errors remain).

He starts off with some pics he’s found. Lets go through them.

A picture roughly corresponding to the above occurs in a Beeb page the greenhouse effect which appears to be part of some “GCSE bitesize” thingy. They don’t put any numbers on the arrows (I suppose S-B is beyond GCSE), (?hence?) they don’t explain the /4 geometry, and they do make it easy to mistake the thermal LW emitted by the Earth for “reflected heat” (though if you bother read their text they do get it right). They also don’t explain energy balance, or that the downwelling LW is part of it. Overall, not a very good effort I’d say.

Analogies with a real greenhouse are unhelpful. Real greenhouses work differently. Not everyone realises that, and some people get hung up on the terminology and forget that they’re trying to understand reality.

Here’s a pic that corresponds closely to mine. Its attributed to Lindzen, and that’s quite believeable, because about basic science he’s likely to be right.

This one is attributed to Gavin, and that’s plausible too, because this one is also correct. It reduces to the previous one, or to mine, if you set (lambda) to one. It isn’t clear to me if the source for my pix realises this. (lambda) represents the absoption of LW in the “atmosphere” layer. If (lambda)=1 all the LW is absorbed, as in my model; if (lambda)=0 the atmosphere is transparent to LW, and there is no greenhouse effect. This complexity adds nothing fundamentally interesting. Also the real atmosphere is opaque to LW when its of any thickness, so (lambda)=1 is reasonable at this level of detail.

Or, you can call the incoming solar Is*(1-(alpha)) where (alpha) is the planetary albedo of ~0.3; 1-0.3 is the 0.7 I’m using explicitly. This makes no difference; its just a matter of labels. You can also fold in a surface albedo, too; again, it makes no real difference.

But, what you can’t do is this: which is to say, write the wrong numbers onto the arrows. This is obviously broken, because the surface energy budget doesn’t balance, and nor does the atmospheric layer balance either. Not at all by coincidence, they are out of balance by the same amount, 239. This pic is described as “a fair and accurate representation of all the above model 2 diagrams”.

What’s gone wrong? Well, as we saw from my correct version, the upwelling LW from the surface needs to be twice the downwelling (and upwelling) from the atmosphere, in order to make this model balance. And that happens because the surface is warmer than the atmosphere. In the picture I’ve inlined, the author has the surface emitting 239 W/m2, which will only happen if Ts = Ta. So, this picture can be sort-of considered as an unstable state, before it equilibriates, if you want to be generous. You can make the picture correct by replacing the erroneous “239″ pointing to box 3 with the correct value, 2*239.

[Update: thinking about this, is it possible to explain the Author's confusion by him thinking this is a diagram of energy flowing round the system, in a step-by-step way? That is to say, he thinks the leftmost arrow happens *first*: SW enters and strikes the Sfc. *Then* he thinks the sfc emits LW, and naturally (on this incorrect view) he thinks that must be 239 too. *Then* he thinks the atmosphere emits thermal radiation. I think that really is what he's doing wrong. Oh dear. That isn't at all how the diagram should be interpreted: everything happens at once, and everything is in equilibrium. There is no step-by-step to it. The numbers on the arrows (1 to 4) do not represent a sequence in time. Uupdate: this is indeed how he is thinking; I notice that he actually explicitly says "One is supposed to follow the diagram from left to right. ie, arrow 1, arrow 2, arrow 3 and then arrow 4." Oh dear.]

What would happen, physically, in a world that really had those arrows on it? Well, instantaneously, to make the sfc upwelling LW consistent with the surface thermal radiation, Ts would have to be 254; ditto for Ta. From that picture, the surface would start to warm, because there’s a positive imbalance of 239 W/m2. The atmosphere would instantaneously start to cool (before warming back to 254 later as the surface warms), because it would have a negative balance of 239. The exact path that Ts and Ta take would depend on the heat capacities of the layers, but eventually it would equilibriate at my values.

Returning to the slideshow, our Author has got badly confused, because we next come across:

Ta has already been described by him as sunlight in (240W/m2) = Tg (240W/m2) = Ta (240W/m2)

This makes no sense at all: the incoming 240 W/m2 (or 238, or 239, whatever) is, errm, in W/m2. It can’t equal Tg (T_ground; my Ts) to Ta, because Tg and Ta are in Kelvin.

Our author then presents this pciture as “Professor Nathan Phillips also explains in this pdf Greenhouse effect “theory” model 3.” I don’t know why he calls this “model 3″; its the same as the previous ones.

### The Greenhouse effect “theory” as clearly depicted in the above diagrams has (at least) four fatal flaws, which are as follows.

This is the bold assertion our author makes. Lets go through it:

1) A 2 parallel plane (2PP) model used – Inappropriate, does not apply to earth.

This an idealised model, so its not expected to be entirely accurate. But this criticism isn’t a very good one: on Earth, the vertical scale is small compared to the horizontal, so representing the atmosphere as a plane is quite reasonable.

2) The power of sunlight received at the top of earth’s atmosphere is divided by four (P/4) – This is unphysical, and can not be applied to earth.

This is a failure of abstraction. I think our author understand the area-of-circle divided by area-of-sphere bit; what he hasn’t realised is that if you’re applying a time-independent view you can average over the day-night cycle, and over the Earth’s geographical area.

3) In the model 2 type, 240W/m2 is absorbed by earth’s atmosphere which becomes 480W/m2 radiated by the atmosphere. The atmosphere is depicted as emitting the same power or amount of energy both up and down, ie, twice what it receives. 240 W/m2 of energy is created. – Energy can not be created.

All we’re seeing here is the Author’s confusion. He has written the wrong numbers onto his “model type 2″ diagram, and then complained that the numbers are wrong. Um.

3 cont) In the model 3 type the atmosphere is depicted as radiating 240W/m2 before the surface warms it. This is double accounting, at best, and is by any other name, creating energy.

No, the picture is time-independent. Its a steady state once all layers have come to equilibrium. The surface has, indeed “warmed the atmosphere” and the atmosphere has “warmed the surface”. There is no double accounting, all fluxes balance, no energy is created or destroyed.

4) In the model 2 type earth’s surface receives 480W/m2, but is depicted as only radiating 240W/m2.

Only in our Author’s broken version. See above.

We now come on to a pile of pix of what our Author calls “model type 4″. these are no longer the idealised model; instead, they’re considerably more complicated diagrams dealing with a whole pile of extra processes that we deliberately abstracted out of the simple model to make it tractable. If you don’t understand the simple model, you’re unlikely to get far with this one.

That’s about the end of the slide show.

### Returning to the divide-by-4 bit, which seems to worry people

As I said earlier

Solar radiation, S, is 1361 W/m2. But because the Earth is a sphere, and the ratio of the area of a sphere (4.pi.r^2, Earth’s surface area, which emits thermal radiation) to a circle (pi.r^2, Earth’s cross-section to Solar radiation, which determines how much Solar we absorb) is 4, the average insolation per unit area of the Earth is 1361/4 at the top of the atmosphere.

This seems to worry some people as “unphysical”. Weeell, if it helps you, rest assured that GCMs don’t do this: they apply the diurnal cycle or radiation point-by-point, adjusting for latitude, and they include the effects of clouds and atmospheric scattering and surface albedo.

Another way of looking at it is, how could you rescue it so it was physical? One way is to render the Earth entirely uniform, and super-conducting to heat. In which case the “dark side” gets its share, irrespective of whether the sun is up or not. You have to make the “atmosphere” super-conducting too, of course.

But a better way is to realise that its an approximation: we know that in the real world, there is a day-night temperature cycle but (at least, say, over the oceans – we can imagine all this going on in an aqua-planet world if we like) its not very large. And then you’d need to fold in the latitudinal dependence too.

This all returns to my The New Aristotelians post: in order to make progress in science, you need to understand what to abstract, and what to keep.

### Another way of thinking about it…

…which is in fact exactly the same way of thinking about it, even further simplified.

If you don’t like the numbers of the maths, don’t despair, there is hope for you. Its possible to get a qualitative understanding with no numbers at all.

* Agree that the Earth is heated by the sun,

* and that it emits thermal radiation to balance the heat from the sun.

* In the absence of atmosphere, that’s it.

* With an atmosphere (that absorbs some or all LW, but no SW) the LW from the Earth warms up the atmosphere,

* which emits thermal radiation upwards and downwards.

* Therefore, the Earth is warmer in the presence of an atmosphere, because it is heated by two sources: the sun and the atmosphere.

Easy, no?

### Very late update

From March 10th: reading some comments at ATTP I think I begin to understand where the “its all the lapse rate” Doug-Cotton-types are going wrong (other than their a-priori insistence that it can’t possibly be GHG’s): they have in mind the black-body T, correct, and the lapse rate, correct, both of which they at least half understand; and the “effective emission height”, which they don’t understand at all. But they think of it as fixed. So they think of the sfc temperature as being from the temperature profile (as set by EEH and lapse rate) at the sfc. Or something like that – it can be hard to tell whether they think they’re starting from the EEH or the sfc. But all this is so obviously wrong (the EEH isn’t fixed) that its hard to know what else to say.

### Refs

Every man and his dog has their own pet explanation of the greenhouse effect. If you’re not a “skeptic”, and you can cope fairly easily with the maths and with S-B, then you probably want a more advanced version. Like the wiki one, or I’ll collect some here.

* Learning from a simple model – Gavin’s, at RC. Goes through the maths faster, and therefore gets beyond my very simple model to more interesting stuff.

The pic is Klee; found via the lost.

* Science of Doom Reference post – goes through some of the basics.