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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I am quite sure to have solved Willis’ problem at 12:57 a.m. and further explained it a few minutes ago. Yet the post seems to have escaped Willis’ notice.
342 W/m^2 is for directly perpendicular to the sun, not at the polls. Winter in the Arctic get’s nothing. Thus the 3d aspect to the atmosphere is important, as warmed parts of the world is trying to heat the cold parts not getting heated by the sun.
Plus, he only part of the planet that is directly perpendicular to the sun and getting 342 W/m^2 is at noon, hence only a small diameter of surface gets that input, the rest of the planet gets less. Soon as night comes, there is no solar input. So again, those parts of the planet getting sun will have its heat moved into colder areas. This is why there are winds (fiction loss as well as redistrubution of heat).
Bottom line is this model is wrong because it is far too simplistic. It is modeling a flat surface equally getting the same solar input. The real world is a sphere, which has only a small percent of its surface getting solar input. Hence wind moves heat around with frontal systems. A large portion of your upwelling must be supplying this redistribution.
Take this into account and CO2 sensitivity drops to zero.
Willis, another excellent and thought-provoking article. As a Physical chemist, all these equations worry me. I remember back in graduate school when we were collecting vibrational structure of electronically excited diatomic molecules; my advisor would say “Why bother figuring out the theoretical quantum mechanical equations when you can just shine a laser on the molecule and nature solves the equation for you”. He is an excellent experimental chemical physicist. Most of our energy was spent on the experiments and then we’d use theory to check our results. If theory agreed that was a bonus. But if it didn’t agree, then that was usually the theory was incomplete.
It really bothers me that AGW is based primarily on observational (historical) data and computer models. Where are the experiments? I checked the literature and there doesn’t seem to be much. How hard would it be to set up a real Greenhouse modelling the major parameter of the earth and just measure what’s going on? All major factors can be controlled and manipulated. Sure we can’t control any weather patterns in the greenhouse, but we aren’t interested in weather..we are interested in the climate.
I tried this in my backyard with some glass bottles. (A similar experiment was also found on Youtube demonstrated by a fourth grader so I’m not saying this is amazing science: just the first step in thinking about what a real experiment would look like). One bottle had air, one had air and 50g water, one had 50g water and 100% CO2 (from the beer meister!) and each had a type K thermocouple attached to data logger. The air temp was 1.7C. Air alone was 7.2C. Air plus water was 12.8 and CO2 plus water was 15.3. Difference by adding water was 5.6C. Difference by switching air with 100% CO2 was only 2.5C after 2 hours of monitoring (water’s a better greenhouse gas than CO2….who would have known????) These were then moved to the shade and each cooled to 5.6C with the same rate constant (it’s the cooling of the 50g water and glass bottle so that makes sense).
My point is: why isn’t someone tapping into all that AGW research money and doing real experiments? Throw the equations and computer models away and start doing the experiments!!!! It doesn’t have to be hard…and they’ll be a lot closer to the truth than coming up with a bunch of partial and/or faulty equations.
Willis,
I know this is not your model, but it is not any sort of a model of the Earth. There seems to be confusion between power and thermal energy, at least in the choice of variable names. The L eqn is wrong. the assumption that T is linear is wrong, the assumption that the power flow up and down is the same amount is wrong. There is not a conservation of energy at work.
To support my comments, lets check out some simple common numbers.
the toa is about 342 w/m^2 but the albedo is roughly 0.30 meaning that only about 235 w/m^2 actually enters the atmosphere to be absorbed by atmosphere and surface. the surface averages around 288K and radiates by stefan’s law about 391 w/m^2. Total power balance means that a total of around 391 w/m^2 must be hitting the Earth’s surface. we can make the crude assumption that most of the 235 w/m^2 incoming power reaches the surface. That means we are missing 391 -235 = ~155 w/m^2 which must be coming back to the surface from the blocking of the atmosphere. Note that ghgs are good for about 120 w/m^2 blocking in clear skies, leaving around 35 w/m^2 to be accounted for by our typical cloud cover IR blocking / reemission for about 60% coverage. With these, the surface is in balance. For the outgoing LWR power balance with incoming power means that what leaves the atmosphere must be a total of 235 w/m^2. For clear skies what escapes is 391 – 120 (ghgs only) = ~270 w/m^2 – which is too much were all of the sky to be clear. What balances this is the cloudy sky component that emits somewhat less than 235 w/m^2 due to the lower temperatures of the cloud tops and the weighted average comes out to 235 w/m^2 for balance.
Note that while the atmosphere in a single shell must have both conservation of energy (and hence conservation of power) AND must radiate upward and downward at the same rate, the flow of power downward will not be the same as the upward flow. For one thing, the cloud component has a different temperature bottom and top – which are radiating surfaces for thick clouds.
The atmosphere must radiate (on average) downward to the surface at a rate of ~150 w/m^2 to maintain surface T and power balance. Clear skies (~40%) permit the surface to radiate about 270w/m^2. If we take what escapes for this, 0.4 x 270 = 108w/m^2 averaged over the whole. Add in the whole average for outbound, ASSUMING the 150 w/m^2 downward provides us with 258 w/m^2 plus the top of cloud emissions for the cloudy sky 60% portion. NOTE that we’re already above the 235 w/m^2 balance point by 23 w/m^2 and we have not accounted for the cloud radiation which will have to be positive and significant.
Therefore, the assumption of T/S outbound being equal to T/S downward is absolutely wrong in the real world. The back of the envelope numbers don’t even get close.
I suppose one could say it’s what easily happens when one tries to use nonphysical models on physical situations. It has me wondering whether whoever invented this travesty ever successfully passed university physics. I’m pretty sure they never read a book on feedback control systems either. Maybe that’s why they call themselves climatologists.
The only thing useful to do with this model (kheil’s???) is to gut it completely and do something physical, such as along the lines of kheil & trenberth’s 1997 paper – while avoiding their grossly erroneous numbers that they failed to correct in their paper a decade later.
I don’t have time to track down all the issues here, but the main one is that you’ve taken an equation from Kiehl that’s expressed in terms of changes of temperature and radiative fluxe, and converted it to an equation expressed in terms of absolute temperature and radiative flux. In Kiehl, the equation is that = /sensitivity + . That equation itself is based on linearization of a differential equation expressing energy conservation. It’s no longer valid if you get rid of the &Delta’s! The equation is being used (by Kiehl) to describe a system that is not in equilibrium (it’s warming over time), so it’s unlikely that you can meaningfully derive limits based on thermodynamic efficiency in that framework. That’s why, by the way, your picture has the weird quality of having a fixed T, despite the fact that energy is leaking out of it into the lower boundary! In Kiehl’s formulation, he’s partitioning the imbalance of energy due to CO2 at the TOA (&Delta Q) into a portion that’s going into warming the system (&Delta T/S) and a portion that’s going into warming of the ocean (H).
For a climate realist, if the observed differences are this far from theoretical, the equations must be incorrect.
For a global warmist, if the observed differences are this far from theoretical, the earth must be incorrect.
Sorry, tried to use too much html. I meant to say, in Kiehl, the equation is (Change in Flux at TOA due to doubling CO2) = (Change in global average surface temperature)/(Climate Sensitivty) + (Flux into the Ocean).
3 Cheers for the model! Finally, a model that describes the difference in thermal balance between land and ocean. But we still have to address the inherent heat emanating from the Earth’s core (which I know is much lower than solar heating). It is there as a basic starting point on top of which the atmospheric and ocean thermal balance rides. We also need to know how to deal with hot spots in the ocean, which can create a regional hot spot and throw the balance point of your model off. There are clearly regions of some decent size where the water (or air to a lesser degree) are being warmed from the magma coming to the surface. While not a driver, it is a factor that (IMHO) is much greater than CO2’s GHG effects when integrated over the entire planet.
Hence the Monty Python-esk search for missing heat by the AGW crowd. And the absurd amounts of flung funds being used to do it.
Reminds me much of the wild Alchemy chase to find the “turn base metal into gold” formula. We all know how that turned out.
http://en.wikipedia.org/wiki/Alchemy
These discussions always confirm my long-held impression that the science of the CO2 greenhouse effect on Earth is, at best, in what biologists call a “pre-embryonic stage,” and most likely not even that far. The egg seems to be there; it hasn’t yet been implanted in the uterus; we are not quite sure if it has even been truly fertilized, and the potential father could be one among many. But the boldest among us see no impediment in predicting the precise behavior and features of the adult it might one day turn into. Medieval astrology was probably more solid.
The basic assumption in the construction of CAGW models is that OLR is sensitive to the atmospheric concentration of CO2 and that sensitivity controls the “greenhouse effect” of other atmospheric components. The problem with these models is that it is more likely that water in the processes of evaporation/condensation and freezing/thawing is controlling the rate of energy lost to space and those processes are also controlling the atmospheric concentration of CO2. Think about how much more energy is exchanged in evaporation compared with raising atmospheric temperature.
Temperature sensitivity to CO2 and positive feedbacks are “fudge factors” that tend to make the models appear to work. Read my presentation http://www.kidswincom.net/CO2OLR.pdf and let me know if you think I have made mistakes.
I is presented as a constant, without any supporting evidence. The earth is a sphere, not a flat plane. The sun has variations in energy. In addition, the constant I would appear to be valid only at a point on the earth that is receiving sunlight. What happens at night to the constant I?
S is presented as some magical number that is a constant. What the hell is “S”? Sensitivity? Sensitivity to what? I have found no definition, no table of various values, supporting the notion of a universal sensitivity that can be reduced to a single simple constant.
In surveying, we were taught that if you don’t know where you started, then you don’t know where you are going. Any real science or engineering always begins with a complete list of real known quantities and measurements. As a thought experiment, this is interesting. If it was turned in as a first-year engineering homework, it would get an “F” as being irrelevant to any real problems.
Willis
“For starters there is about 100 W m-2 lost to albedo reflection from clouds and the surface.”
This is not a loss to the system, it’s an efficent part of the process of getting from ‘in’ to ‘out’.
Sorry for getting excited in my first comment, but overly simplistic models simply produce crude and inaccurate results (I have had too much orbital mechanics I guess).
What you have shown in your calculation is another one of those back of the envelope checks that shows the model above to be incomplete and wrong. What it means is that energy loss and sensitivity are more complex in reality than the model allows. What astounds me in these equations is how energy loss never seems to account for energy used.
Someone once commented that ocean waves, by running laterally, never expend energy. Explain that to battered coast lines. The movement of masses of air and water is energy consuming. Rising masses against gravity are energy consuming. Electrical discharge, liquefying and evaporating, etc all transfer energy. How much of this is thermal based (vs wind or gravity)? And even though you did not want it said, biology consumes a lot of energy. Biology is the act of capturing energy and holding it in a living structure before it continues on its path of entropy.
Since turning the equation around produces nonsense, it means the model is nonsense. There is so much more at work than a bunch of heated air.
Willis, a couple of quick thoughts on your diagram:
1) As drawn, Figure 1 implies that T/S Up (upper right) is nominally the same as T/S down. They are not the same, nor should they be. The T/S down should contain only downwelling IR, mostly from mid to lower atmosphere. The T/S Up should represent radiant emissions from the upper atmosphere and includes the reflected radiant energy term, which is 107 of 342 total per K&T. I have not worked through your equations to see the implication, but I suggest that is a place to start.
2) The “U” Upwelling Radiation term is a misnomer – Energy losses from the surface are dominated by Convection and Latent Heat terms, with radiant energy no more than 40% (per K&T diagram) and possibly as low as 8% (per Chilingar et al “According to our estimates, convection accounts for 67%, water vapor condensation in troposphere accounts for 25%, and radiation accounts for about 8% of the total heat transfer from the Earth’s surface to troposphere.”) Again, as it relates back to my first comment, T/S Up as drawn in Fig1 includes the energy transported by Convection and Latent Heat, so would not be equal to T/S down.
3) The T/S Downwelling is dominated by IR from water vapor as noted by MostlyHarmless and others. The Evans and Puckrin (2006) measurements of IR back radiation show CO2 to be no more than 20 to 25% in low moisture winter clear conditions, and in summer with higher humidity, both the numeric value and the percentage contribution of CO2 back radiation drop significantly. Again, not sure how that plays through your math, but we should be careful to keep total “GHG” sensitivity separate from CO2 sensitivity.
I always enjoy your articles. They make me review what I think I know, and I dig back through my saved information to compare with your presentations.
While working on this, I see there are additional comments that I have not yet read through – my apology if this is a repeat of others views.
thanks,
Dave
Slightly OT:
It looks like NASA has been upgrading its website and has added new features. IMO, this feature is a step in the right direction.
NASA JPL
Eyes on the Earth 3D
http://climate.nasa.gov/Eyes/eyes.html
Select Aqua and then select CO2 monthly — tilt to view to show different angles — then do the same with Water vapor last 3 days. Note: it looks like they are still working out some bugs and loading data as some of the data maps are missing.
The data indicates, distribution of CO2 and water vapor in the atmosphere are not uniform (no news there). The equation implies a uniform effect which doesn’t exist. I realize you asked to keep this focused on the supplied equation and simple but thought you’d find it interesting as it relates to sensitivity.
I’m bothered in principle by the zero dimensional approach, IN vs OUT vs STORED. Its useful for thought experiments such as this, but not real world modeling. The Earth system is a 3D system and includes transport terms/factor. What comes in and goes out don’t have to be related to each other directly – especially the outgoing part which will vary a lot location to location.
Maybe the inequality is an expression of this?
Can I tell you that climate sensitivity is zero? Ferenc Miskolczi determined that the optical density of the atmosphere in the infrared band where CO2 absorbs had not changed for 61 years despite constant addition of carbon dioxide to air. This means that the greenhouse absorption signature of that added carbon dioxide is missing. No absorption, no greenhouse effect, case closed. I have come to the same conclusion because the Hansen warming he announced in 1988 does not exist in satellite records. During the Hansen warming of the eighties and nineties satellite record shows only climate oscillations due to alternating El Nino and La Nina periods but no warming whatsoever for twenty years. This too means that CO2 sensitivity = zero, independently of Miskolczi’s work.
The “work” that world heat engine might produce, might be referred to as “weather”
Yep, quite a few of other commenters have spoted 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).
Willis, you have made very very interesting contributions, I like your atmospheric engine models, with an equatorial iris a LOT, I really think you are on the right path and that H20 have a huge stabilising effect on average T, and completely control tropical average T above the oceans.
But for this post, you made a glaring error, which is a pity because it will feed your opponents at RC for example and may distract from your more interresting theories.
I would kindly suggest you post an errata or drop the subject asap, before it turn round the blogosphere as an example of skeptic science….
Another “storage” mechanism is the evaporation/melting condensation/freezing of water. These store and dump energy w/o a temperature change, & vary a lot around the globe. The simple T/I does not hold, And even including radiated heat I-Losses) gets messed up.
Let’s look at the nature of the losses due to ‘inefficiency.’ For example, a butterfly on a tree shades the trunk as the sun warms its wings and the lizard that eats it spends less energy metabolizing the warmed butterfly and uses the unspent energy to ascend to a new home higher in the rocks, ultimately raising the habitat of that specie of lizard by a few feet. So, the heat is not really lost: it has been converted into lizard elevation. But, how do we quantify this ‘loss’ to the system. And, if we can’t, is it really lost?
Dave Springer says, “The so-called amplification by water vapor is fictional.”
I have been asking the following question for three years on this and other forums, “What specific research activities, experimental and observational, conducted within the operative climate system itself, are necessary to prove or disprove current AGW theories?”
No one, AGW skeptic or AGW warmer alike, has ever given me anything in the way of an answer to my question, or even the most minimal starting point for a more comprehensive answer to my question.
cba says:
January 4, 2011 at 6:35 am
“…the surface averages around 288K and radiates by stefan’s law about 391 w/m^2. Total power balance means that a total of around 391 w/m^2 must be hitting the Earth’s surface. we can make the crude assumption that most of the 235 w/m^2 incoming power reaches the surface. That means we are missing 391 -235 = ~155 w/m^2 which must be coming back to the surface from the blocking of the atmosphere. ”
cba, I don’t claim expertise in this field, I’m just a learner, but to calculate the total energy hitting the surface don’t you have to include the non-radiative heat loss from the surface? Which would make the total rather more than 391W/m^2. (An additional 102W/m^2 according to Kiehl & Trenberth (1997).) Also, K & T have 168 W/m^2 solar being absorbed at the surface, rather than 235 W/m^2. Would that not add 77 + 102 = 179 to your required 155 downward longwave? hr
crosspatch says:
January 3, 2011 at 11:34 pm
“CO2 absorbs very little in the visible frequencies.”
“I wasn’t meaning visible frequencies. Solar radiation outside the atmosphere includes a lot of IR in addition to visible. This IR will be absorbed by the GHGs on the way in.”
Good point crosspatch. It seems sensible that the incoming IR would saturate the “adsorptive” capacity of all CO2 in the atmosphere and returning IR from the surface would simply pass on through unaffected by the already satiated CO2. This has got to be too simple for all those astrophysicists to have missed if so. Willis?