I’m in the process of writing a couple more in-depth articles but have been much distracted by “First Life” in recent weeks.. sad and unfortunate, because writing Science of Doom articles is much more interesting..
While writing a new article – What’s the Palaver? – Kiehl and Trenberth 1997 – I thought that I should separately explain a few things which related to the earlier article: Do Trenberth and Kiehl understand the First Law of Thermodynamics? Part Three.
I know that many readers already get the point. But clearly some people find the model – and real life – so controversial that they will find many ways to claim “real life” wrong. Stefan-Boltzmann, who was he? Pyrgeometers– clearly a fake product that should be investigated by the Justice Department? And so on.
One of the problems is that radiant heat transfer is not something in accord with everyday life and so – as we all do – people draw on their own experience. But people also draw on confused ideas about the First Law of Thermodynamics to make their case.
In this article, two ideas.
First, is the Atmosphere Made of PVC?
In the original article – Do Trenberth and Kiehl understand the First Law of Thermodynamics? – I used a simple heat conduction problem to demonstrate that temperatures can be much higher inside a system than outside a system, when the system is heated from within.
One commenter explained the link between this and the atmosphere, although perhaps my attempts at humor had slightly back-fired. I had disclaimed any relationship between PVC spheres and the atmosphere..
Well, I confirm the atmosphere is not made of PVC, and that conduction is not important for heat transfer through the atmosphere.
But there is relevance for the atmosphere. Where is the relevance?
Solar radiation heats the climate system “from within”. The atmosphere is mostly transparent to solar radiation so the solar energy initially heats the surface of the earth. Then the surface of the earth heats the atmosphere. Finally the atmosphere radiates energy back out to space.
If it were true that the first law of thermodynamics – the conservation of energy – was violated by a simple “lagged pipe” model – well, that would be the end of an important branch of thermodynamics.
The model showed that the temperature of an inner surface can be higher than an outer surface – and, therefore, radiation from an inner surfaces can be higher than the radiation to space from the outer surface.
The reason for providing the model of the PVC sphere – much simpler than the atmosphere – was to demonstrate that simple point.
Second, What if the Radiation from an Inner Surface CANNOT be Higher than from an Outer Surface
Many people write entertainingly inaccurate articles about this subject (see Interesting Refutation of Some Basics for one example). Apparently, if the radiation from the atmosphere/surface into space is 239 W/m² then the radiation from the inner surface itself cannot be more than 239 W/m². A confusion about the First Law of Thermodynamics.
To be specific, the actual claim from the believers in the Imaginary First Law of Thermodynamics (IFTL) is that the total radiation from the earth’s surface cannot be higher than the total radiation from the climate system into space.
This, according to the IFLT, is not allowed.
Let’s consider the consequences and calculate the results. All we need to connect the two values – when we have the W/m² for both surfaces – is the ratio of surface areas.
If E1 is the radiation in W/m² from inner surface A1, and E2 is the radiation from the outer surface, of area A2:
E1A1 = E2A2
The area of a sphere is proportional to its radius squared (A = 4πr²), so the above equation becomes:
E1r1² = E2r2²
a) The Earth and Climate System
In the case of the Earth and climate system, the radius of the earth and the radius of the “climate system” are almost identical..
The radius of the earth, r1 = 6,380 km or 6.38 x 106 m.
The radiation to space takes place from an average height of around 6km from the surface, so the radius of “the climate system”, r2 = 6.39 x 106 m (at most).
The total radiation to space, E2 = 239 W/m² (measured by satellite).
If the IFTL believers are correct then E1r1² = E2r2²
Therefore, E1 = 239 x (6.39 x 106)² / (6.38 x 106)² = 240 W/m²
Unsurprisingly, this surface radiation value is almost the same as the radiation into space because the two areas are almost identical.
The Stefan-Boltzmann law says that radiation from a surface, E = εσT4
The “currently believed” average value from the earth’s surface is 396 W/m². This is due to the emissivity of the earth’s surface being very close to 1.
So there are three simple choices for why the “believed value” of 396 W/m² is so much higher than the believers in the IFLT appear to claim:
- The Stefan-Boltzmann law is wrong
- The emissivity of the earth’s surface, for the wavelengths under question, is an average of 0.61
- The surface temperature has been massively over-estimated and the “average” temperature of the earth’s surface is actually around -18°C (see note 1).
The 3rd choice should not be ruled out. Perhaps Antartica is a lot larger than measured, or a lot colder. How many temperature stations are there on Antarctica anyway? Maybe there is some cartographical error in estimating the area of this continent from when planes have flown over Antarctica and satellites have crossed the poles.
Perhaps the Gobi desert is a lot colder than people think. No one really makes an effort to measure this stuff, climate scientists just take it all for granted, sitting in their nice warm comfortable offices looking over the results of supercomputer climate models. No one does any field research.
Quite plausible really. It’s not too hard to make the case that the average temperature of the earth is much much much colder than is generally claimed.
b) The PVC Sphere
Let’s review the very simple hollow PVC sphere model. In the original article, the inner radius was 10m and the outer radius was 13m.
Let’s look at what happens as the inner radius is increased up to 10,000m while the wall thickness stays at 3m.
Instead of keeping the internal energy source of 30,000W constant, we will keep the internal energy source per unit area of inner surface constant. In the original example, this value was 23.9 W/m².
Real First Law
With the equations provided in the maths section of Part One, and an energy source of 23.9W/m², here is the temperature difference from inner to outer surface as the inner radius increases:
Note that the x-axis is a log scale. The initial value, 10¹ (=10) was the value from the original example, and the temperature difference was 290K.
As the sphere becomes much larger (and the wall thickness stays constant) the temperature difference tends towards 377K.
Now that is a very interesting number that we can check.
When the wall thickness becomes very thin in comparison to the sphere it is really approximating a planar wall. The equation for heat conduction (per unit area) through a planar wall is:
q = k . ΔT/Δx
where q = W/m², k = conductivity (0.19 W/m.K), ΔT = temperature difference, Δx = wall thickness, m
So for a 3m thick planar PVC wall conducting 23.9 W/m², let’s re-arrange and plug the numbers into the equation:
ΔT = 23.9 x 3 / 0.19 = 377 K
Correct.
So this is a very simple test. There is no other way to link heat conduction and temperature difference. The simple equations that anyone can check support the PVC sphere model results.
Imaginary First Law
Let’s find out what happens under the imaginary first law. It will be quite surprising for the supporters of the theory.
I couldn’t check the imaginary first law in any textbooks, because it’s.. anyway, as far as I can determine, here are the steps:
1. The radiation from the inner surface must be 23.9 W/m². This means (for an emissivity, ε = 0.8 that has already been prescribed for this model) that the inner surface temperature, T1 = 151.5K (E = εσT4)
2. The inner and outer surface radiation values are related by the equations provided earlier:
E1r1² = E2r2²
3. Therefore, we can calculate the outer surface temperature and therefore the temperature difference.
Here is the graph of temperature difference as the radius increases:
Note the important point that as the radius increases the temperature difference reduces to almost nothing – this is the inevitable consequence of the (flawed) argument that inner surface radiation has to equal outer surface radiation.
Because when r1=10,000m, r2=10,003m, therefore, the areas are almost identical.
Therefore, the radiation values are almost identical, therefore the temperatures are almost identical.
Ouch. This means that somehow 23.9 W/m² is driven by heat conduction across 3m of PVC with no temperature difference.
How can this happen? Well – it can’t. To get 23.9 W/m² across a planar PVC wall 3m thick requires a temperature difference of 377 K.
When r1 = 10,000 m, ΔT = 0.02 K according to my IFTL calculations – and so the conducted heat per unit area, q = 0.0013 W/m². The heat can’t get out, which means the temperature inside increases.. and keeps increasing until the temperature differential is high enough to drive 23.9 W/m² though the wall.
Hopefully, this makes it clear to anyone who hasn’t already made a total nana of themselves that the imaginary first law of thermodynamics, is .. imaginary.
Notes
Note 1 – The concept of an average temperature is not really needed to actually do this calculation. Averaging temperatures across different surface materials like oceans, rocks, deserts clearly has some problems – see for example, Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored.
All that is really required is to calculate the average radiation value instead. Just find the temperature at each location and calculate the emitted radiation. Then average up all the numbers (area-weighted).
As a note to the note.. To get 240 W/m² with an emissivity close to 1, the “average temperature” can be at most -18°C. With a wider day/night and seasonal variation than we actually experience on earth the “average temperature” would then be lower than -18°C.
The Science of Doom 
Please do not take this amiss but I really can´t see the point you are making. The physics of the PVC shell with an internal energy source, reaching a steady state by radiation from the outer suface seems a fairly trivial exercise.
First the outer temperature is fixed by the emissivity, the radiating area and the total energy to be dissipated. This energy is, of course, the same as the internal energy source if a steady state is to be reached. Then the temperature difference between the internal and external temperature is fixed by the conductivity, geometry and the number of Watts to be conducted.
It makes no difference whether the internal energy source is a radiant heater, a fan heater or a resistive layer glued to the internal suface. In fact, it could be any combination of these as long as the total Watts supplied is the same.
I just cannot imagine why anyone would think that the internal surface emissivity had anything to do with the internal surface temperature. You seem to imply that someone would seek to fix this temperature by using this emissivity together with the Watts/m2 and the Stefan-Boltzmann law. In the case of an internally heated PVC shell this would only be attempted by someone with a very poor grasp of physics and the correct calculation is easily shown.
If this PVC shell is supposed to have something to do with the Earth and its atmosphere you need to be clearer on how the analogy is supposed to work. As far as I can see, the internal surface of the shell is also the external surface of the Earth and as neither of the two materials support the transmission of radiation any concept of the shell radiating to the Earth is quite impossible in this model.
All I´m getting from this model is that the outside temperature is fixed by radiation to space and the inside temperature is then derived by the temperature difference due to the thermal resistance. This is not dissimilar to what happens in our atmospheric gas shell except that the concept of an outside surface is blurry and the temperature drop across the shell is fixed and not dependant on the actual heat flow.
I don´t suppose this is quite the message you intend as you appear to set great store in the concept of back radiation ´warming´the earth. Personally, I think back radiation is very relevant to the energy balance at the Earth´s surface but it is the overall system balance together with the lapse rate that actually sets the surface temperature. At the least, I would argue that the sums are easier if we concentrate on the fate of upwelling radiation rather than trying to mix up/downwelling radiation with convection/evaporation at the surface.
This comment is not meant to be adversarial and simply reflects my possibly flawed understanding of the physics.
Jorge,
I might be wrong in my interpretation, but I see the conductivity of the shell as somewhat analogous to the greenhouse gas ‘impedance’ to flow of radiation from the surface to space.
This isn’t what we observe with Earth’s atmosphere because convection is a parallel process with the Earth’s atmosphere that effective shorts out the greenhouse gas impedance when it exceeds the adiabatic lapse rate.
Mike,
I understand what you mean about ´impedance´ but I have trouble with making a connection to the electrical version in terms of temperature and heat flow when talking about radiation making its way through a thick layer of gas. Not only is there the fact that things depend on the fourth power of temperature, there is also the difficulty of the emission and absorbtion all varying along the length of the resistor.
I suppose we could simply define the impedance as the temperature difference divided by the heat flow and hope this relationship is linear for small changes in temperature or heat flow.
In any event, I agree that what is observed in the real atmosphere is a linear relationship between temperature and altitude that is not a function of the actual heat flow. I would simulate this by a Zener diode in parallel with a resistive wire. Once the current/heat flow exceeds a certain point the Zener will impose a fixed voltage/ temperature drop. Obviously we can look at the voltage along the wire to mimic the change in temperature.
I think the resistive wire is akin to your gas ´impedance´ and the Zener is like the convective short circuit you mentioned.
My apologies to all who don´t follow this but electronics is about the only trade I know something about.
Once again, it isn’t whether S-B is wrong but whether you use the correct version of it, ie the difference form that takes into account the ambient temperature, which is therefore very much affected by heat flux from convection. There’s little point in being simplistic if doing so removes any link to reality. You may think you can logically separate out the different feedbacks but nature doesn’t. As i said before, it’s perforce an iterative calculation.
An interesting discussion of the greenhouse effect is in place on J Curry’s site. You’d learn something from a comment by “Nullius in Verba” about how it is well known in climate science circles (really?) that the back radiation concept has actually been superceded as an explanation of the “Greenhouse” effect by a more modern convective model. Apparently despite this, too many “teachers” still teach the old, discredited model which is inherently nonsensical and wrong. Well this was news to me. No wonder we think you climateers are all confused about real life atmospheric heat transfer: You patently are!
JamesG: Back-radiation is still part of the physics– it doesn’t go away just because more complex models are being used to integrate it with convection. No-one working on these models was ever ‘confused’ about this– the older models are not “nonsensical and wrong”. Giving separate treatments of convection and radiation and then combining the results to model the two together is perfectly legitimate; shifting to a model that integrates the two more fully is just a step up in model sophistication, not a transformation of physical understanding that invalidates previous work.
I would like to question your statement that the integrated radiation convective model is more complex.
I went through many iterations in trying to understand the greenhouse effect. It was extremely difficult to see how to calculate the OLR based on the surface temperature when radiation and convection were treated as separate things. Conversely, it was hard to work out the surface temperature that would lead to a given OLR.
It was not until I read the met notes by Rodrigo Caballero that I finally understood how to do the calculations. The critical point is that the lapse rate gives you the temperatures that are essential to working out how outgoing radiation is generated as you go up from the surface to the top. It is not a trivial calculation but I can see that it is doable.
There is no complication in handling back radiation or worrying about how much energy is lost by convection. The lapse rate neatly sidesteps all these issues.
As I said above, you need to worry about convection and back radiation if you are interested in working out a surface energy balance, but if the important thing is the relationship between surface temperature and OLR they are a side issue.
To me the the integrated model seems much simpler rather than more complex as an explanation of why the surface temperature is higher than it would be in an atmosphere without GHGs.
Not least, the lapse rate/radiation explanation will put a stop to the interminable questioning about whether back radiation can or does warm the surface of the earth!
Jorge:
The model is not an exact copy of climate system processes and is intended not to be exact.
The model has ONE aim.
To demonstrate that total internal radiation in a system can be higher than the total energy supplied to the inside of the system.
Of course, this is uncontroversial, except for people who somehow don’t understand the subject. They claim “climate science doesn’t understand the first law of thermodynamics because the radiation at the earth’s surface CANNOT be higher than the radiation into space..“.
Does this explain the point of the model?
I set “great store” by accuracy and scientific understanding.
Many people claim – and have claimed on this blog:
– that back radiation doesn’t exist
– if it does exist it isn’t caused by CO2 and other “greenhouse gases”
– it isn’t absorbed by the surface
– if it is absorbed it can’t affect the temperature of the surface
Of course, articles to explain “back radiation” therefore prove I am some champion of a different explanation of the “greenhouse effect”..
See The Earth’s Energy Budget – Part Two
And Part Three for explanation about how the energy balance of the climate system works with no reference at all to “back radiation”.
Thanks SoD,
Now I am confused again, Surely the nett energy supply from the surface of the earth is the input to the shell and it is this value that has to turn into OLR at the outside of the shell. I think it is the phrase total internal radiation that I don´t really understand. Partly because I don´t follow what this internal space consists of. Is it not the earth itself? If it is just a vacuum then the amount of radiation bouncing from internal surface to internal surface is not very interesting. I think it was established on a previous thread that we can make this an enormous amount simply by making the emissivity of the internal surface very low.
I just read the part three explanation and agree totally with it. It seems that we could avoid all the back radiation stuff if we just stuck to that model. Mind you, if we did that you would be redundant!
S.o.D:
Many people claim – and have claimed on this blog:
– that back radiation doesn’t exist
– if it does exist it isn’t caused by CO2 and other “greenhouse gases”
– it isn’t absorbed by the surface
– if it is absorbed it can’t affect the temperature of the surface
You missed mine out.
– if it is absorbed by the ocean surface it is promptly re-emitted and can’t affect the temperature of the vast bulk of the ocean to any significant extent.
It’s had 11,000 years to try since the end of the last ice age and the deeps are still at 1-2C. Surely we won’t have to carry on paying modelers while we wait that long again for greenhouse?
JamesG:
Here is the comment I made on Judith Curry’s site:
“I would like to comment on “when they try to explain it“.
In atmospheric physics text books and papers on the subject, when climate scientists “try to explain it” the explanation is correct.
Perhaps someone did it incorrectly once, but I haven’t found it yet in any technical publications discussing this subject.
Many websites set up to explain to the general public in media-friendly sound-bites probably do explain it badly. Or non-technically. Or completely wrongly.
But I believe it is important to differentiate between the two worlds.
To criticize how climate scientists describe atmospheric physics based on what NASA or the Met Office explain it via their marketing departments on their public websites seems somehow unfair to me.
Or even to criticize how someone technically competent tries to explain a complex technical subject to a non-technical audience by saying “you made it too simple”.. Equally you could say “you made it too complex”..
In any case, the key word in your statement is “Apparently”.
Note that no one has actually demonstrated any flawed teaching. It’s just a claim floating around which is made by people who haven’t read any atmospheric physics textbooks.
I wonder why tallbloke is commenting on this blog, after accusing me of dishonesty on his own.
Those readers interested in that kind of stuff can do a search on this blog for his comments and my responses over a few articles and draw their own conclusion about our respective characters.
On another blog I wrote:
The author had apparently written:
You can read the rest at that blog.