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.
Willis, I may be a bit late but: The circumference of the earth at the equator is 24,901.55 miles (40,075.16 kilometers). But, if you measure the earth through the poles the circumference is a bit shorter – 24,859.82 miles (40,008 km). Thus the earth is a tad wider than it is tall, giving it a slight bulge at the equator. This shape is known as an ellipsoid or more properly, geoid (earth-like).
“Well unfortunately, that odd density temperature function ONLY applies to fresh water, and low salinity salt water.”
Yes. True. I should have been a bit clearer – you’ll note of course that I didn’t actually specify whether it was salt or fresh water (I said “water” and was thinking in general physical terms), but I agree that it might have been assumed to be salt given that the oceans are the biggest part of the effect of all this on climate.
I am aware that I have an unfortunate tendency towards extremely long posts in which I am tempted to cram in all the necessary caveats and clarifications, distractingly diverting to discuss interesting but peripheral side-topics, which urges I try very hard to resist, so I normally try to keep caveats brief. They are only intended as a marker to the unwary that there is potentially more going on here than I am saying – and if you want to know more, to look it up, or to have a longer think about it. But it’s always more complicated. When sea ice melts, for example, the surface salinity can get quite low.
It doesn’t affect the main content of the comment, which was intended to be entirely about the simpler circumstances where such complications don’t apply. But I appreciate it that people are taking the time to read the comments and make improvements. My thanks.
hr,
my purpose wasn’t to provide a theory or accurate results beyond simply disproving the cartoon assumption Willis posted.
h2o vapor/water cycle is extremely important near the surface but gradually fades out towards the tropopause
George E. Smith says January 4, 2011 at 3:12 pm:
“You obviously didn’t read what I said. TSI averages around 1366 W/m^2, and varies over an eleven year cycle by about 0.1% and of course seasonally due to the small changing sun earth distance change.
Your figure of 342 W/m^2 has nothing to do with earth flatness or roundness; it has to do with a completely fictional and quite absurd assumptiuon that the sun shines on the entire earth surface 24/7/365; from pole to pole and even on Antarctica in the depths of Winter midnight.
It doesn’t do that, on either this planet or any other planet in the universe. the earth rotates, and the sun that reaches the ground (clear air) packs about 1000 W/m^2 of projected area (normal to the sun-earth line), everywhere on the sunlit half of the planet.”
Well said George 342 W/m² is the same for the moon and would also be the same for a tennis ball in earth’s orbit. It has nothing to do with what the earth’s surface gets or receives to absorb.
I do not believe any one of the W/m² values given in these energy flow plans has been actually measured. Apart from may be the Solar Constant (S) of 1366 W/m².
Ken Coffman says:
January 4, 2011 at 11:41 am
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).”
So let me get this straight. Using earth as a model, CO2 alone can cause the earth’s temperature to increase to disastrous levels. No feedbacks nor feedforwards nor interactions are required. But as a single factor in a greenhouse experiment, the same CO2 will not produce the temperature effect? I guess because there aren’t all the complicated real world interactions. So if you need all those complicated real work interactions in order for CO2 to manifest itself into higher temps, then the AGW theory and supposed knowledge is even more lacking than previously thought and the initial claim that CO2 alone can cause the runaway greenhouse effect is completely wrong. Once again, they can’t have it both ways. It’s either a main effect or it’s not. It can’t be a main effect in earths atmosphere but not a main effect in a greenhouse experiment.
Emitted power is proportional to temperature to the fourth power in the case of a cavity radiator only when the entire spectrum is present. In the case of such a radiator “looking through” a band limited window, as that in the atmosphere, the relationship is more like temperature to the 4.6 power, with a constant of proportionality quite a lot smaller than the stefan constant.
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?
What you are missing is that S (sensitivity) is not a constant it is a variable with both negative and positive bounds. Sensitivity is dependent on other variables that are not in the simplistic equation, such as humidity, associated atmospheric heat content and albedo. As you have yourself stated in previous posts as heat content increases atmospheric sensitivity (the positive feedback) initially can assist heating then becomes a negative feedback to the extent that inbound heating is locally massively reduced.
David L: you are not crazy. The “atmospheric greenhouse effect” fails EXACTLY for the same reasons that the “real” greenhouse effect doesn’t work when you open the windows. Convection, sir! If CO2 had an effect, it would have to change the lapse rate. It hasn’t. The models are wrong. There is no “hot spot” at 5 km in mid-latitudes. It doesn’t get hotter in Sterling, Colorado on a clear humid July day than on an extremely dry clear day. The “atmospheric greenhouse effect” cannot be demonstrated, and is probably nonsense. But I’m still open to some proof….
Hey, moderators: you are withholding a comment, no? It is about [Jeff’s] failure to consider thermalization, capiche??? This is not “normal” for this site and will render this site similar to Realclimate in my eyes–i.e., totally biased and irrelevantm unless it is fixed. Please post the comment! Jeff, you need to study thermalization!
REPLY: Fixed the name per your request, I don’t see any other comments in moderation from you. OTOH we’ve been hit with a barrage of spam comments today, so possible it got lost in that. Feel free to resubmit – Anthony
Sorry, Anthony, I meant Jeff. He has some “splaining” to do.
If there are 3 degrees per doubling of CO2, then if you reduced the CO2 to 3ppm, the temperature of the earth would be less than the black body radiation
Secretly I’m a little bit naive. So i’ll need some help through this. I had heard that E=mc2 (that’s squared). So does taht mean that some energy in the system is turned into matter? Could a 6% growth in world biomass in the last few years be an absorption of the extra energy?
Nullius in Verba says: “…I am aware that I have an unfortunate tendency towards extremely long posts in which I am tempted to cram in all the necessary caveats and clarifications…”
That’s okay, Nullius. I only read the first 16 lines of a comment. A lot less than that if there are a lot of words in all caps or quote marks.
ferd berple says: January 4, 2011 at 9:42 am
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.
…or fundamental-misunderstandings/incorrect-assumptions are present in the argument above. Please, for those who are truly skeptical, when someone presents a half-page/one-line proof that AGW is false it is wise to dig a little deeper. In this case, the assumption of linearity across the entire range of temperature response is in error.
Repeating charlie: January 4, 2011 at 9:16 am
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.
A thought experiment:
If we start with a cooler sea than current (for whatever reason), there is less cloud cover / rain / storms at the equator as there is less evaporation and latent heat, the trade winds from the equator are relaxed and moisture air reaches the current desert areas of the earth, they are green. The northern and southern extremes are cold in winter as there the current warm currents have slowed, but also clearer in summer due to the lower humidity. The sun is beaming energy at a constant rate, the clear sky at the equator allows much of the incoming energy to be absorbed into the ocean, this energy is conveyed via currents to the northern and southern extremes, as the sea warms more the currents start to struggle to keep up, the seas at the equator keep warmer, the air is moister, cloud cover more common, storms more frequent, h20 and co2 increase in the atmosphere, wind patterns accelerate, soil erosion increase, more sediment is washed into the equatorial seas changing its albedo and absorption characteristics, dust changes occur and effect the atmospheric albedo due to changed wind patterns, a monsoon belt forms shading the band closet to the sun, less energy starts to come in and the oceans locally receive less energy, due to increased rainfall drier air reaches the desert areas changing them from green to yellow and changing the overall albedo of earth, meanwhile warm water is still being transported to the extreme latitudes lagging behind the cooling at the equator, the currents slow as energy at the equator reduces and the system keeps going, undulating up and down trying to find that balance, only being thrown out by small changes in TSI, millancovitch cycles, changes in cloud cover due to GCR variation etc… the results in a constantly moving unstable system were many factors all effect each other in non-linear ways – then some one has tried to convert to a ridiculously simplistic equation using mostly assumed values and we ask whats wrong? We know very little about the climate full stop.
I think I know what puzzles me with this drawing. On your drawing, you consider S as a constant, but it is a function of T. Even for a black body, S is a function of T, as T does not move lineraly with incoming radiation.
For an accurate drawing, you should consider the system in equilibrium, with all flows constant, a given temperature, and add a little deltaForcing, for which you consider T will move of deltaT and S(t+deltaT)=S(T).
I suppose the equation will be far different.
Oops, I see many commenters write the same thing with better words 🙂
I think the whole debate and take on this has been somewhat “politicized/radicalized” made up and set up by “parties/elements” that that have a political agenda.
On average temperature decreases by 2 degrees C every 300 meter upwards. And visa versa.
If you added 40 hPa more air to our atmosphere so that the global standard air pressure at sea level increased to 1055 hPa, instead of today’s 1013 hPa, the global temperature at any given place would increase by 2 degrees C and the global temperature would increase from 14-15 degrees C to 16-17 degrees C.
This means on average that the more column of air you have above you the warmer it gets. The atmosphere also has an isolation effect. Just like in an air tight house in cold weather with a 2000w heating and the radical difference in temperature with no isolation in the walls and 40 cm isolation in the walls.
Temperature decreases by 2 degrees C every 300 meter upwards. Fig 1 claims “Most upwelling radiation absorbed by greenhouse gases and re-radiated equally upwards and downward.”
?
Radiation goes from warm to cold. Never from cold to warm.
The window area in a house during winter isn’t colder because cold radiation from outside is coming in. It’s cold because the house “warmness” in that area is radiating out.
A fridge does not get cold because you add coldness/ cold radiation. A fridge gets cold because you radiate out the warmness inside the fridge.
So when we know that normally the air above is colder it is technically difficult for me to comprehend that the surface is warmed by cold radiation from above?
?
Bryan says, Joel Shore;
What would happen if one of the models was left to run with the IR radiative effects of CO2 shut down?
H2O still operating with radiative properties in the IR and the usual phase change effects still intact.
Joel replied; “This recent paper addresses that issue:”….
http://www.sciencemag.org/content/330/6002/356.abstract
Unfortunately it is behind a pay wall and written by the “usual suspects”.
I cannot tell therefor if their prediction of the end of all human life is justified.
Common sense tells me that the consequences of such a minor change would be negligable.
Jon-Anders Grannes said: “So when we know that normally the air above is colder it is technically difficult for me to comprehend that the surface is warmed by cold radiation from above?”
That’s not too difficult to get, once you build the correct mental picture:
Figure 2 cases, and consider only radiative transfer:
a) a body B at T1 in a medium M1 at T2<T1
b) same body B at T1 in a medium M2 at T3<T2.
Do you concur that the body B will get colder more quick in case (a) than in case (b)? If yes, you have understood why atmospheric greenhouse effect does not violate thermodynamics, and why radiation must indeed occur from a cold body to a warm body (only less so than from warm to cold):
A photon coming out of the body at T1 does not know the temperature of the surrounding medium. For him, only molecular energy levels of body B count, he does not see the surrounding before it is emitted and afterward, too late to come back to body B 😉
But so does the photons emitted by the surrounding medium: they care only about the temperature of the medium that emit them.
So in case (a) and (b), body B emit the same amount of radiation toward the outside. But the outside medium M1, being at higher temp T2, emit more radiation in all direction (including body B), than medium M2 at temp T3. So body B gets colder in (b), not because it radiate more, but because it absorb less from the surrounding medium. when B is at the same T as tis surrounding, B temperature does not change anymore, even if it radiate the same, because at this point it absorb exactly the same energy as it radiates.
So in this case, the colder atmosphere really radiates back to the warmer ground, but of course not as much as the ground radiates up. Ground is thus not really heated, it just get colder more slow, and the less cold the surround medium is, the slower the ground will get colder.
“Radiation goes from warm to cold. Never from cold to warm.”
Radiation goes from anything above absolute zero, outwards – irrespective of the temperature of its surroundings. But warm radiates more than cold, so the net energy transfer between warm and cold is always from warm to cold.
“A fridge gets cold because you radiate out the warmness inside the fridge.”
A fridge gets cold because gases get hot when they’re compressed and they cool when they’re allowed to expand, just like air that rises or falls in the atmosphere.
“So when we know that normally the air above is colder it is technically difficult for me to comprehend that the surface is warmed by cold radiation from above?”
It isn’t. But if radiation was the only thing that mattered, it would be the case that that the surface would cool less quickly with a merely cold sky above it rather than one at close to absolute zero. When you wear clothes, the cloth is cooler than your skin, but is nevertheless warmer than the snowy-white landscape of knee-deep global warming that surrounds you. Clothes don’t generate any heat, and there is no net transfer of heat from cool cloth to warm you, but they still keep you warm.
But as I’ve said elsewhere, this argument about back-radiation is all an annoying distraction, because it isn’t how the greenhouse effect really works anyway. Climate scientists use a completely different model for calculations – they just use this pure radiative model in simple explanations. It’s like physicists saying that objects all fall at 9.8 m/s^2 under gravity, and applying this theory to the fall of feathers when explaining to the general public. The physicists already know perfectly well that feather fall doesn’t work that way. But huge amounts of sceptic time are wasted by people doing the equivalent of applying 9.8 m/s^2 to the fall of various objects and complaining that it doesn’t fit. The equations when extrapolated to extremes end up predicting ridiculous results.
The basic problem here is that they haven’t bothered to explain it properly.
Dave Springer:
since when? See Table 8a-1: Density of water molecules at various temperatures…
http://www.physicalgeography.net/fundamentals/8a.html
Nullius in Verba says:
January 4, 2011 at 4:49 pm
“Yes. True. I should have been a bit clearer – you’ll note of course that I didn’t actually specify whether it was salt or fresh water (I said “water” and was thinking in general physical terms), but I agree that it might have been assumed to be salt given that the oceans are the biggest part of the effect of all this on climate.”
No. False. What you should do is admit you made a mistake. Seawater density increases constantly until it freezes and the freezing temperature is about 2 degrees C below zero. Not knowing that led you to say that the deep ocean temperature is 3C because that’s the point of grestest density.
Man up and admit you made a mistake.
Rabe says:
January 5, 2011 at 4:01 am
“since when? See Table 8a-1: Density of water molecules at various temperatures…”
I was referring to seawater and its since always. Salinity changes the temperature/density relationship. At 3.5% salinity (that of the global ocean) density increases all the way to the freezing and the freezing temperature is -1.8C.
JAE says:
January 4, 2011 at 1:10 pm
“Well, ALL the gases in the atmosphere function as “insulation,” not just the GHGs (remember, the N2 and O2 are heated by collisions/thermalization).”
No they DO NOT act as insulators. They are buffers. They reduce the temperature differential between day and night and nothing more.
Graph of temperature vs. density of seawater
I’m gobsmacked at how many times this must be repeated. Seawater increases in density all the way to the freezing point. In this respect it is quite unlike fresh water.
Ninety percent of the global ocean lies below the thermocline at nearly a nearly constant temperature of 3C. It isn’t because that’s the maximum density of seawater. Maximum density is at -1.8C. It’s 3C because that’s the average temperature of the surface waters when the average is taken over the entire course of a glacial/interglacial period (100,000+ years). Over that length of time convection and conduction are sufficient to equalize the temperature of the deep waters with the surface waters. If there is any other possible cause for the measured temperature of the ocean below the thermocline I’ve yet to hear it.