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’m with Nylo and later AndrewG on this one. Surely the T/S term must be the temperature due to greenhous gasses (~33K) and thats not the whole temperature (~288K).
If you dont think so, then could you please explain the “meaning” behind the term T/S if thats at all possible?
Willis,
O/T, but were you Long by Chron. or Marcq St. Hilaire?
My head hurts, thnx Willis.
Houston we have a problem.
Q = T/S + SOLAR
Q= T/S + ΔH
SOLAR = ΔH
WUWT??
I also think T/S may need to be halved but I haven’t got my head around that yet.
A couple of observations.
1. The atmosphere is heated by two mechanisms:
A. Energy absorbed from sunlight. This occurs mainly in the upper atmosphere.
B. Energy absorbed from the surface. There are 3 components to this:
(i) Direct conduction (about one fifth). This enters the atmosphere at the surface.
(ii) Net Radiation (about the same as conduction – about one fifth). This enters the atmosphere at a highish level – it is a measure of the inefficiency of the back-radiation. ( For wavenumber 670, John Nicol has calculated that all the back radiation must be coming from the lowest 5m of atmosphere – as this will be nearly the same temperature as the surface there will be little NET absorption at this frequency. For other frequencies, where the average height of emission of the back radiation is much higher, the temperature is less and so the back radiation cannot balance the surface emissions, and there is a NET transfer of energy.)
iii) Evaporated water condensing in the atmosphere as fogs, mists, frosts and clouds (about three fifths). This is all over by around the cloud tops.
2. The energy from the surface is converted to atmospheric kinetic energy no matter what its original form or entry height. It modifies the Adiabatic Lapse Rate (9.8DegC/km) to some other value, on average taken to be 6.5DegC/km.
3. The Surface anchors the atmosphere: the atmosphere cannot heat up and maintain the lapse rate without the surface driving it. (or to put it another way, if the conditions are not sufficient to maintain the surface at a new, elevated temperature, it will cool down, dragging the atmosphere with it.) The critical sensitivity is therefore that of the Surface. (This also satisfies commonsense – we want to know the relationship of surface temperature with changes in GHG concentration, so we shouldn’t be calculating what is occurring high in the atmosphere – we need to calculate for the surface.)
4. When the surface heats up, the energy emitted as radiation increases , as does the energy emitted as evaporated water. (In contrast, if you heat the atmosphere, all you get is an increase in radiation). So the sensitivity of the surface to a change in conditions (either a change in back-radiation, or a change in insolation) is much less than the sensitivity of the atmosphere. [The change in evaporation rate is one of the big unknowns. The climate modellers use a very low figure – around 2%/DegC. It could be much higher than this, near the theoretical maximum of 7%/DegC.] If you plug in the simple numbers from the Kiehl & Trenbeth diagrams you get between 0.095 and 0.15 DegC/W/m^2 for the surface sensitivity, or between 0.35 and 0.55 DegC for a doubling of CO2. Let’s say 0.5DegC.
4. To get any higher than this AT THE SURFACE you need to make rabbits come out of hats. Specifically the static increase in Forcing at the Surface has to be much higher than the 3.7W/m^2 at the Tropopause. For example, for a 3DegC increase in surface temperature the increase in surface forcing has to be between 21 and 31W/m^2. About 15W/m^2 of this is due to increased atmospheric temperature (therefore increased back-radiation). The remaining 6 to 16 W/m^2 also has to come from increased back radiation – and there’s only one way to do that – by increasing the GHG concentrations sufficiently to lower the height of emission of the back-radiation, thus increasing its intensity.
5. Such a change could not be due to the changed CO2 concentration. Virtually all back-radiation from CO2 is coming from below 200m, the vast majority from below 20m, so in the CO2 frequencies the surface and back radiation are nearly balanced. The only way would be to greatly increase the water vapour concentration, so that the inefficient absorption bands in the water spectrum are greatly improved.
6. I guess this long ramble has merely shown that to get a high sensitivity you MUST have positive water vapour feedback, and it must be quite fierce. This is the area where I think the CO2-causes -lots-of-warming hypothesis is weak. The detail of the effect of water vapour concentration on the SURFACE conditions is in my view central to that hypothesis, but is mostly vague, lost in the models. In particular, the modellers need as high a surface sensitivity as possible (ie as low an increase in evaporation rate as possible) but the largest possible increase in atmospheric water vapour. So they have a 2%/DegC increase in evaporation rate but a 7%/DegC increase in atmospheric water vapour.
7. I think that area needs more explanation to satisfy us ancient sea-dogs. It is relevant that measurements suggest evaporation rate changes around 5%/degC.
The elephant in the room has to be ocean heat – the rest is round-off error in comparison. One buck of that bronco (to mix metaphors) and all the delicate little atmospheric calculations come crashing down.
TomVonk:
I completely agree with your post at January 4, 2011 at 2:22 am . It summarises to your statements that say;
“The real Earth is not a homogeneous isothermal body in radiative equilibrium and the 0 dimensional “model” has no relevance for its dynamics .
You just trivially found out that Q=T/S with S and T constant is wrong and can’t even begin to describe what happens when the spatially highly non homogeneous temperature and radiation fields vary .”
Energy lost by radiation from a surface is proportional to T^4. Therefore, small changes to temperature distribution over the Earth’s surface have large effect on total radiated energy loss from the total surface.
The cartoon K-T budget that Willis is assessing is so simplified that it is plain wrong to the degree of being meaningless. And all Willis’ analysis does is to illustrate that the K-T budget is plain wrong. The cartoon is a useful illustration of the GHG but it is not an analytical tool, and this is why GCMs are constructed.
Nobody is more AGW-skeptic than me, but I see no point in assessing a cartoon used for illustration purposes.
Richard
The system equation completely ignores cloud and dust cover, which reduce I. The diagram shows that ALL exiting radiation comes from the surface or was absorbed from below and re-radiated in the atmosphere. There is no factor for incoming energy that “bounces off” the atmosphere and back into space, or was absorbed before it got to the surface and reradiated into space.
So either the incoming arrow is drawn incorrectly, and I is a measured value at the surface, not from space, or it would seem there’s a huge factor missing.
here is a model of the atmosphere:-
http://upload.wikimedia.org/wikipedia/commons/3/3c/Crude_Oil_Distillation.png
you note that the higher up the gravity well you go the lower the molecular weight of the molecules. Also note that the ‘lapse rate’ is also present.
With regard to the atmosphere, the hotter it gets then the distribution of molecules at the top of the atmosphere changes; you get more low molecular weight molecules. The molecule with the lowest molecular weight is water. Heat the atmosphere and gravity fractionation increases the number of water molecules at the top; where they can radiate heat into space.
Willis,
the original equation in Kiehl is differential. delta Q = lambda * delta T
You are not using differential values?
The missing factors!
Whats the equation that calculates the CHANGES of T/I T/S and E? The T/I is a varible factor that effects S (sensitivity). What were the T/S and E and S during MWP? LIA? Who decided to make them constants ? Were beeing fooled by the lack of time perspective and the redicoulous assumption of co2 as main driver of climate.When you add and find the true”forcings” into the histrorical changes of T/S E and S you will find the keys to complete the equation of the radiation budget of the planet. S changes! We have relations between the factors that are not linear.
Whats the forcing behind KAOS?
The atmosphere heats the oceans by conduction alone. Back-radiation is never an issue because the emissivity of CO2 is 1/1000th that of the liquid/solid surface.
It’s elementary.
Another thoughtful piece Willis, thanks. I have had the nagging feeling the important factor is the amount of radiation hitting the earth’s surface, and believe crosspatch is correct wrt IR composing a large fraction of TSI. Someone else posted a reference above, here’s one I keep. http://i55.tinypic.com/zn1yt3.jpg
If CO2 is as interactive with IR as the CAGW people profess, it impacts incoming as well as outgoing, ie. blocking IR coming in. This would make CO2 self-regulating/limiting, more= more atmospheric retention of heat, but less incoming IR or heat to retain. The main problem I see with this is that most of the incoming IR is absorbed by H2O, so CO2 is hardly a player. Regardless of formulas and how things work out on paper, CO2 never lives up to high or even significant sensitivity.
Intellectually, under ideal circumstances, communism seems like a good idea. The only flaw is not taking human nature into account.
crosspatch says:
January 3, 2011 at 10:38 pm
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. Water vapor being lighter than air it carries the latent heat upward until adiabatic lapse rate causes condensation. So the IR energy never registers on a thermometer at the ocean surface but rather only registers at the cloud level. That’s why the downwelling thermal IR from CO2 doesn’t raise the surface temperature anywhere near as much as AGW boffins think it must.
Over land the downwelling IR does indeed heat the surface but since only 30% of the earth’s surface is land only 30% of the downwelling IR has any effect on surface temperature. The other 70% only shows up in the cloud layer. Then since clouds reflect up to 85% of visible sunlight the clouds radically limit daytime heating of the ocean which then lowers the evaporation rate which results in fewer clouds and it just keeps going round and round in that manner.
The bottom line appears to be, near as I can tell, is that CO2 doublings beginning from a base level of 150ppm or more (where the logarithmic curve is well established) can increase surface temperature by 1.0C and that’s it. The so-called amplification by water vapor is fictional. Moreover the surface temperature increase can only be observed over land with more of it at the higher latitudes in the winter when there is little evaporation occuring because the surface is frozen. Because there is a greater concentration of land surface in the northern hemisphere the NH will exhibit more surface temperature increase than the southern hemisphere.
There is not a single observation that contradicts anything written above. Observation only supports it.
crosspatch:“so, what impact would increased CO2 have on absorbing *incoming* solar IR and re-radiating half of that into space?”
JohnH“Do any of these models take into account the sun only shines on 1/2 the globe at any one time, if a Greenhouse gas absorbs radiation could it also radiate it more with no input.”
The other thing missing from this kind of diagrammatic explanation is what happens at night when the sun is not shining. It would be instructive to start explaining the ‘greenhouse effect’ from that viewpoint, as it would show the other part of the IR-active gas dynamic, their cooling properties.
Willis,
I think you’ve established that the Kevin Trenberth zero dimensional cartoon version of the “greenhouse effect” leads to absurd results. But that’s about it.
Fortunately you haven’t made the same terrible mistakes as your previous thermodynamics forays (eg “The Steel Greenhouse”) which comes as something of a relief.
I agree with crosspatch and Dave Springer.
Tom Vonk made a post some while back about
the M-B kinetic energy distribution curves for gases.
To claim re-radiation of half up half down of all
absorbed energy is nonsense and invalidates
all calculations.
Re: Dave Springer @ur momisugly January 4, 2011 at 5:17 am
So maybe it’s:
+CO2 => +Downwelling IR on ocean => +Evaporation => +Clouds => Global Cooling!
but then
-Temp => +Ocean CO2 absorption => -Clouds => Negative feedback => Stability
@Willis Eschenbach
I agree with what you say (i.e. the model is for all types of forcing) but the IPCC is obsessed with CO2 increases, which seems daft since that absorption is already saturated. So it seems a bit irrelevant. Or did I miss something?
Also, CO2 is good for life on the planet. So again, it seems a bit daft to worry about having too much of it.
Willis, previously you said (in comments on your thermostat hypothesis thread)
“In the tropics, mid-day surface insolation averages about a kilowatt per square metre, and the amount reflected by clouds is on the order of 340 W/m2.
This gives us a cloud effect about five times as strong as the global average. In the tropics there is a change of 5 W/m2 in reflected energy for each 1% change in cloud cover. This allows for large swings, as I showed in my paper.”
Looking at the graph of low level tropical cloud cover http://www.climate4you.com/images/HadCRUT3%20and%20TropicalCloudCoverHIGH-MEDIUM-LOW%20ISCCP.gif
It seems there has been a decline of approximate 2% between 1985 and 1998.
If we assume the IPCC figure of 1.6 W/m² for the net anthropogenic forcing.
Would it be correct to assume there has been a tropical cloud effect at least 6 times the proposed anthropogenic forcing over the period? Can this account for the correlation in temperature rise without anthropogenic forcing?
T.I.A. much appreciated.
Steve Keohane says:
January 4, 2011 at 5:13 am
“If CO2 is as interactive with IR as the CAGW people profess, it impacts incoming as well as outgoing, ie. blocking IR coming in.”
That’s true and it’s even truer for water vapor but very little of the sun’s energy is thermal IR. The sun presents itself as a nearly ideal 5000K blackbody continuous spectrum with peak emission at 0.5 micrometer (yellow). Emitted energy falls off rapidly on either side of the peak. CO2 absorption begins at 5 micrometers and longer. There is essentially no energy in sunlight at wavelengths that long. Energy emitted by the earth however approximates a continuous 255K blackbody spectrum with peak emission at 12 micrometers which is very near CO2 peak absorption wavelength of 15 micrometers. Water vapor however overlaps about half of CO2’s absorption peak so when and where water vapor is present (which is most of the time in most places) it negates much of CO2’s potential as an insulating greenhouse gas.
I’m not sure it’s fair to treat the atmosphere as a single thin layer, or to say that the up-going radiation from the atmosphere is the same as the down-going:
http://barrettbellamyclimate.com/page5.htm
I asked Jack Barrett why, and he kindly explained to me that it was because the emissions to space are from the much colder top of the atmosphere.
I guess it’s all to do with optical density and lapse rate…
Willis,
I’m amazed science has yet to understand the importance of pressure fluctuations in a enclosed biosphere. The more gases building in the atmosphere, the more density to wind and the changes to the surface salt on the oceans.
Growth up mountainsides shows the expansion of the atmosphere has exerted against the ceiling of this biosphere.
John of Kent says:
January 4, 2011 at 2:28 am
“This whole article is negated by the fact that back radiation CANNOT cause warming of the earths surface.”
Technically correct. What back radiation does is slows down the rate of cooling. Just like a layer of clothing doesn’t actually warm your body but rather slows down how fast your body loses warmth. Greenhouse gases are insulators. Insulation doesn’t supply heat it just slows down the loss of heat.
So a climate sensitivity of 1 K (W m-2)-1 is exactly equivalent to a change of 3.7°C from a 3.7 W m-2 change from doubling of CO2. To convert from one to the other, you multiply or divide by 3.7.
I see what you mean. But on your drawing, S is the climate sensitivity induced by all greenhouse gas effect. Even with the exact same units (K.w-1.m2) I do not see how you can easily switch from S to Sco2 alone.
The equation works with S, but not sure why it should work with SCO2 alone, excepted on Venus maybe.
S (K/Wm-2) = How much temperature increases when we increase the forcing by 1 Wm-2.
1/S = How many Wm-2 of forcing are needed for a 1K increase in temperature.
If we call E to the total energy per unit of area that GHG reradiate, i.e. E=T/S, it results that:
E=T/S -> 1/S = E/T, therefore
Watts per 1K of Temperature change = Total energy / total temperature CHANGE.
Therefore Willis, in your T/S expression, T is not the absolute temperature, but the temperature change due to GHGs, i.e. about 33K. This results in a minimum posible sensitivity which is an order of magnitude smaller than you calculated.