Guest Post by Willis Eschenbach
[Update: I have found the problems in my calculations. The main one was I was measuring a different system than Kiehl et al. My thanks to all who wrote in, much appreciated.]
The IPCC puts the central value for the climate sensitivity at 3°C per doubling of CO2, with lower and upper limits of 2° and 4.5°.
I’ve been investigating the implications of the canonical climate equation illustrated in Figure 1. I find it much easier to understand an equation describing the real world if I can draw a picture of it, so I made Figure 1 below.
Be clear that Figure 1 is not representing my equation. It is representing the central climate equation of mainstream climate science (see e.g. Kiehl ). Let us accept, for the purpose of this discussion, that the canonical equation shown at the bottom left of Figure 1 is a true representation of the average system over some suitably long period of time. If it is true, then what can we deduce from it?
Figure 1. A diagram of the energy flowing through the climate system, as per the current climate paradigm. I is insolation, the incoming solar radiation, and it is equal to the outgoing energy. L, the system loss, is shown symbolically as lifting over the greenhouse gases and on to space. Q is the total downwelling radiation at the top of the atmosphere. It is composed of what is a constant (in a long-term sense) amount of solar energy I plus T/S, the amount of radiation coming from the sadly misnamed “greenhouse effect”. T ≈ 288 K, I ≈ 342 W m-2. Units of energy are watts per square metre (W m-2) or zetta-joules (10^21 joules) per year (ZJ yr-1). These two units are directly inter-convertible, with one watt per square metre of constant forcing = 16.13 ZJ per year.
In the process of looking into the implications this equation, I’ve discovered something interesting that bears on this question of sensitivity.
Let me reiterate something first. There are a host of losses and feedbacks that are not individually represented in Figure 1. Per the assumptions made by Kiehl and the other scientists he cites, these losses and feedbacks average out over time, and thus they are all subsumed into the “climate sensitivity” factor. That is the assumption made by the mainstream climate scientists for this situation. So please, no comments about how I’ve forgotten the biosphere or something. This is their equation, I haven’t forgotten those kind of things. I’m simply exploring the implications of their equation.
This equation is the basis of the oft-repeated claim that if the TOA energy goes out of balance, the only way to re-establish the balance is to change the temperature. And indeed, for the system described in Figure 1, that is the only way to re-establish the balance.
What I had never realized until I drew up Figure 1 was that L, the system loss, is equal to the incoming solar I minus T/S. And it took even longer to realize the significance of my find. Why is this relationship so important?
First, it’s important because (I – Losses)/ I is the system efficiency E. Efficiency measures how much bang for the buck the greenhouse system is giving us. Figure 1 lets us relate efficiency and sensitivity as E = (T/I) / S, where T/I is a constant equal to 0.84. This means that as sensitivity increases, efficiency decreases proportionately. I had never realized they were related that way, that the efficiency E of the whole system varies as 0.84 / S, the sensitivity. I’m quite sure I don’t yet understand all the implications of that relationship.
And more to the point of this essay, what happens to the system loss L is important because the system loss can never be less than zero. As Bob Dylan said, “When you got nothin’, you got nothin’ to lose.”
And this leads to a crucial mathematical inequality. This is that T/S, temperature divided by sensitivity, can never be greater than the incoming solar I. When T/S equals I, the system is running with no losses at all, and you can’t do better than that. This is an important and, as far as I know, unremarked inequality:
I > T/S
or
Incoming Solar I (W m-2) > Temperature T (K) / Sensitivity S (K (W m-2)-1)
Rearranging terms, we see that
S > T/I
or
Sensitivity > Temperature / Incoming Solar
Now, here is the interesting part. We know the temperature T, 288 K. We know the incoming solar I, 342 W m-2. This means that to make Figure 1 system above physically possible on Earth, the climate sensitivity S must be greater than T/I = 288/342 = 0.84 degrees C temperature rise for each additional watt per square metre of forcing.
And in more familiar units, this inequality is saying that the sensitivity must be greater than 3° per doubling of CO2. This is a very curious result. This canonical climate science equation says that given Earth’s insolation I and surface temperature T, climate sensitivity could be more, but it cannot be less than three degrees C for a doubling of CO2 … but the IPCC gives the range as 2°C to 4.5°C for a doubling.
But wait, there’s more. Remember, I just calculated the minimum sensitivity (3°C per doubling of CO2). As such, it represents a system running at 100% efficiency (no losses at all). But we know that there are lots of losses in the whole natural system. For starters there is about 100 W m-2 lost to albedo reflection from clouds and the surface. Then there is the 40 W m-2 loss through the “atmospheric window”. Then there are the losses through sensible and latent heat, they total another 50 W m-2 net loss. Losses through absorption of incoming sunlight about 35 W m-2. That totals 225 W m-2 of losses. So we’re at an efficiency of E = (I – L) / I = (342-225)/342 = 33%. (This is not an atypical efficiency for a natural heat engine). Using the formula above that relates efficiency and sensitivity S = 0.84/E, if we reduce efficiency to one-third of its value, the sensitivity triples. That gives us 9°C as a reasonable climate sensitivity figure for the doubling of CO2. And that’s way out of the ballpark as far as other estimates go.
So that’s the puzzle, and I certainly don’t have the answer. As far as I can understand it, Figure 1 is an accurate representation of the canonical equation Q = T/S + ∆H. It leads to the mathematically demonstrable conclusion that given the amount of solar energy entering the system and the temperature attained by the system, the climate sensitivity must be greater than 3°C for a doubling of CO2, and is likely on the order of 9°C per doubling. This is far above the overwhelming majority of scientific studies and climate model results.
So, what’s wrong with this picture? Problems with the equation? It seems to be working fine, all necessary energy balances are satisfied, as is the canonical equation — Q does indeed equal T/S plus ∆H. It’s just that, because of this heretofore un-noticed inequality, it gives unreasonable results in the real world. Am I leaving something out? Problems with the diagram? If so, I don’t see them. What am I missing?
All answers gratefully considered. Once again, all other effects are assumed to equal out, please don’t say it’s plankton or volcanoes.
Best wishes for the New Year,
w.
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It seems at this juncture that the paradox Willis proposes is resolved by the difference of differential versus absolute measures, plus a few other issues, but I would like to add a clarification of something first stated by
Solar spectrum contains a very large amount of thermal IR, it’s true, but in the atmosphere the Earth’s surface subtends a solid angle of about a hemisphere while the sun is only a little-bitty disk a half degree in diameter, and as a result the thermal (wavelength longer than 10 micrometers, say) radiation is dominated by that emitted from the Earth’s surface, and internal to the atmosphere. That coming from the sun is pretty insignificant. Solar IR of wavelengths from 0.7 to 10 micrometers is much more significant, but CO2 doesn’t absorb any.
Zero hits searching this web page for Boltzmann or Wein …
One hit for Planck.
Two hits for Atmospheric Window however!
Until consideration is given to these facets of LWIR from the earth discussion on this topic is pretty well separated from that physical processes that take place in reality.
But, maybe you have to start somewhere before graduating to the stage where one comprehends radiative energy flow from a ‘black body’ is proportional to temperature to the fourth power …
.
For those folks who are bringing up the subject of day/night and Earth not being flat etc.
My understanding is that there is 1368Wm2 arriving at the top of atmosphere. This is divided by 4 to simulate the day/night sphere conundrum you are raising.
So I believe all that has been accounted for. How accurately I couldn’t say.
I would have thought the energy transferred to the surface and ultimately to the oceans by falling precipitation to be a factor in the energy budget. Maybe I missed it. It is a nearly constant movement of energy when averaged. But I’m in Seattle where rain is never far from our thoughts.
Looking at the diagram: why is energy “re-radiated equally upwards and downwards”? The Earth being a sphere, would there not be a slightly larger amount going “up”? That is, any re-radiation horizontally, or even slightly downwards, would tend to leave the Earth. Or is the difference too small to matter?
Kai says, “Yep, quite a few of other commenters have spotted the problem with your analysis, the Sensibility relationship is established for T difference (difference from an equilibrium situation, atmosphere with pre-industrial CO2 concentration for example).”
Kai, could I ask you further clarify what you mean by “an equilibrium situation”?
Moreover, is there a reason to believe that an “an equilibrium situation” of some kind existed within the earth’s climate system at pre-industrial CO2 concentrations?
If so, what are the characteristics and dimensions of that “equilibrium situation”?
Sorry, again I haven’t spent a lot of time thinking about this. But it looks to me like a “one-shell” pure radiative model. Climate scientists might conceivably use it in explanatory thought experiments, but I don’t think anyone uses it as an actual model of the atmosphere.
The “shells model” is a way of calculating a pure radiative greenhouse effect by treating the atmosphere as a series of concentric LW-opaque shells, each one optical depth thick. If the world absorbs W units of power, then the outermost shell must radiate W units upwards for the whole Earth to be in equilibrium. It radiates W units downwards, because radiation is isotropic and only radiative heat transfer is being considered here. Because it is radiating 2W units in total, it must be absorbing 2W units at equilibrium, which can only come from below. The next shell down is therefore radiating 2W units up, 2W units down, and absorbing W units from above. It must therefore absorb 3W units from below to maintain equilibrium. And so on.
The first shell radiate W units up/down, the second 2W units, the third 3W units, and so on, with radiated power being proportional to optical depth measured from the top of the atmosphere, and hence the temperature proportional to the fourth root of optical depth. With sufficiently many shells, there is no limit to the temperature that may theoretically be achieved. (In practice, it would stop when the surface got hot enough to radiate in the visible spectrum where the atmosphere is no longer opaque.) As the density of the atmosphere, and hence optical depth, varies exponentially with altitude, so would the temperature. If you run the numbers for Earth, the surface temperature would be something like 45 C or even 60 C (I’ve seen various estimates) and you wouldn’t get the straight line lapse rate that is actually observed. I imagine you would also get a higher temperature sensitivity, although I haven’t calculated it.
It gives completely the wrong answer, because the atmosphere isn’t pure radiative. Convection short-circuits the process once the gradient exceeds a certain fixed level, and thereafter controls the temperature profile of atmosphere (and hence the surface) totally. (I’ve written about that extensively elsewhere.) It’s simply not possible to calculate realistic greenhouse warming without incorporating this non-linear adiabatic lapse rate effect. Shell models don’t work.
Your calculation looks like a one-shell (or “up to one shell”) pure radiative model, and therefore has no chance of correctly calculating, or setting limits on, greenhouse warming. But it is an interesting demonstration that the simplistic one-shell picture you often see in popular presentations of the greenhouse effect cannot be correct. Understanding how to do this sort of calculation is essential to the debate about greenhouse mechanisms.
Willis,
As Dan Kirk-Davidoff says:
The main issue here is that that Kiehl used (delta T) and (delta Q).
The sensitivity in your formula is not constant over temperature because the “upwelling radiation” is proportional to T^4. The sensitivity is much larger at lower temperatures. I think the sensitivity you arrive at is some kind of average from zero K to 288 K, not the sensitivity at 288 K.
The result reminds me of mathematical proof by contradiction. Either the model is incorrect, or one of the formulas underlying the model are not correct, or you have made a mistake in the calculations.
This could be an important result because it presents a possible falsification of the GHG model. Also, it shifts the argument from Climatology to Physics.
On a side note, as an old sea dog with 20 years experience at sea in the tropics, I would say the model does not adequately represent the scale of the heat exchange between the oceans and atmosphere. The idea that the sun mostly heats the air which heats the water is from my observations false, at least in the tropics.
A large portion of the energy from the sun passes through the atmosphere, where it is absorbed by the oceans. This is released though evaporation, convection and condensation back into the atmosphere, often quite violently.
KE of molecules in a coordinated direction is wind or currents. When uncoordinated, i.e. random in direction, it’s temperature. Eventually KEcoord turns into heat since turbulence eventually leads to uncoordinated motion. If KEcoord were to increase without devolving into heat, wind speeds would not be limited. Obviously, that doesn’t happen. Thus, KEcoord can be thought of as an intermediate (potentially ingnored), but also as a carrier of energy from one place to another in a more refined model. Eventually, it all turns into heat and radiates.
Hi , as a complete layman , I was just wondering wether anyone has actually doubled the amount of CO2 gas in an actual greenhouse and measured the temperature increase this creates ? Would be interesting to know how much warmer it gets .
New important invention:
Heatballs with 95% efficiency.
http://heatball.de/en/
Charles Duncan says:
January 4, 2011 at 5:50 am
I’m not sure it’s fair to treat the atmosphere as a single thin layer,…
Willis,
If this treatment is correct. ( ideal gas )
The equations must be adjusted to two dimensions.
(k= Boltzmann)
E = 3 / 2 kT
1/2kT (i) + 1 / 2 kt (j) + 1 / 2 kt (z)
In this case the result of 33% indicates the removal of two planes.
This is the result of the first reading.
A Holmes suggest it would be interesting to double the CO2 in a greenhouse and measure the effect. It might be, but it would bear little resemblance to the atmosphere where warm air can freely rise…
Depending on whose model you use convection is either trivial or the major heat transfer mechanism.
I would be interested to know what data there is that links evaporation rate (of sea, puddles etc) and hence latent heat transfer with temperature. My back-of-an-envelope calculation suggests it is represents around 75W/M2 at 13°C and rises steeply with temperature.
“I ≈ 342 W m-2.”
Where do we draw our box? What “area” are we talking about? It can’t be the earth’s surface, because the energy balance begins at TOA. However, warming the atmosphere causes it to expand (and thin!) and increases the surface area at “TOA.” My own thoughts are that this expansion would not have much influence on incoming radiation, but would increase the atmosphere’s heat loss through a larger surface.
There seem to be multiple problems here:
The upward and downward radiation from the atmosphere are the same in this Model. This implies that this is a single shell model. However there are two observations that suggest a multi-shell model:
*A lot of energy is being absorbed and re-radiated multiple times within the atmosphere before it reaches space.
*The absorption through CO2 is mostly saturated, so there are hardly any changes possible in a single shell model
In a multi-shell model, however, the downward forcing would be higher than the upward forcing. So you would need two separate sensitivities for radiation onto the ground and radiation into space.
The second problem is that T and S are not linear and the change in forcing is deltaT/S
So the total downward forcing would be the integral(deltaT/S * dT) from 0 to T.
A Holmes says:
January 4, 2011 at 9:59 am
“Hi , as a complete layman , I was just wondering wether anyone has actually doubled the amount of CO2 gas in an actual greenhouse and measured the temperature increase this creates ? Would be interesting to know how much warmer it gets .”
A real green house operates by limiting convection. Normally, in the open, warm air near the ground surface tends to depart in small (or large) vortices at that circulate it upward and away from sun warmed surfaces. Cooler air moves in replacing the warmed air. In a green house the formation of vortices that carry away the heat is not permitted physically by the walls and roof of the enclosure. This is also why the phrase the “so-called green house effect” is often encountered when reading about climate. The energy absorption and release of the atmosphere have little in common with a real green house. Doubling the CO2 would make the plants happier though since they fix light energy into carbohydrates using CO2 and water.
@ur momisugly Dave Springer
The fly in the ointment is that thermal IR cannot heat the ocean. It only heats the land. IR is absorbed by water in the first few micrometers at the surface. This doesn’t heat the body of water at all but instead increases the evaporation rate. The thermal energy is thus instantly carried away from the surface in latent heat of vaporization.
That seems to be nonsense to me. Shallow pools of water heated by the sun are demonstrably warmer than deeper pools with the same surface area. For particularly deep bodies, with not a lot of turbulance, you can end up with a scenario where the top couple of feet are much warmer than the lower areas.
The IR doesn’t vaporize water. IR increases molecule speed (heat.) With more heat, more water molecules evaporate certainly, _but not all particles that gain energy evaporate_. The additional heat increases molecular motion and a large amount gets distributed throughout the water body via conduction and convection.
Charles Duncan says:
January 4, 2011 at 10:49 am
Would Pan Evaporation data be useful to you? If so, google it, there are numerous papers on the subject.
_Jim says:
January 4, 2011 at 8:40 am
Zero hits searching this web page for Boltzmann or Wein …
One hit for Planck.
Two hits for Atmospheric Window however!
Until consideration is given to these facets of LWIR from the earth discussion on this topic is pretty well separated from that physical processes that take place in reality.
But, maybe you have to start somewhere before graduating to the stage where one comprehends radiative energy flow from a ‘black body’ is proportional to temperature to the fourth power …
———————-
Yes, even after your post there’s still no mention of WIEN.
There were some variations on the S-B theme.
I’m not sure if there’s another point you wanted to make.
I found it interesting when the geniuses at RC basically told me this: not only does the mythical ‘greenhouse effect’ not work in a greenhouse, but it can’t be made to work (i.e. demonstrated).
Perhaps you have discovered why the true believers are so convinced. Their equation shows catastrophic warming is inevitable, and is even worse than the models suggest. They are so sure it is fundamentally correct that it never occurs to them that observations could disprove their theory. Therefore they can go through life fudging away and ignoring inconveniet data because they “know” they are right.
Offhand, the amount of Watts/m2 (whatever it is) of “forcing” produced by doubling CO2 is vastly overestimated. By your sensitivity estimate (which seems reasonable), the actual W/m2 of CO2 forcing should be about 1/9th of what the IPCC says (assuming a “real” 1C rise).
“T/S” is a troubling term. It is not derived using mathematics and physics, but is declared valid by definition. Theoretical worlds are built through definitions (complex mathematics based upon the square root of -1 comes to mind), but does the world they are building have any validity to the real world. S is supposed to represent all of those noisy and complicated systems that haven’t been properly defined nor quantified, yet they assume that every one of these systems is largely static except for the impact from CO2. Is there any proof or even data that supports this? I haven’t seen any.