Guest Post by Willis Eschenbach
“Climate sensitivity” is the name for the measure of how much the earth’s surface is supposed to warm for a given change in what is called “forcing”. A change in forcing means a change in the net downwelling radiation at the top of the atmosphere, which includes both shortwave (solar) and longwave (“greenhouse”) radiation.
There is an interesting study of the earth’s radiation budget called “Long-term global distribution of Earth’s shortwave radiation budget at the top of atmosphere“, by N. Hatzianastassiou et al. Among other things it contains a look at the albedo by hemisphere for the period 1984-1998. I realized today that I could use that data, along with the NASA solar data, to calculate an observational estimate of equilibrium climate sensitivity.
Now, you can’t just look at the direct change in solar forcing versus the change in temperature to get the long-term sensitivity. All that will give you is the “instantaneous” climate sensitivity. The reason is that it takes a while for the earth to warm up or cool down, so the immediate change from an increase in forcing will be smaller than the eventual equilibrium change if that same forcing change is sustained over a long time period.
However, all is not lost. Figure 1 shows the annual cycle of solar forcing changes and temperature changes.
Figure 1. Lissajous figure of the change in solar forcing (horizontal axis) versus the change in temperature (vertical axis) on an annual average basis.
So … what are we looking at in Figure 1?
I began by combining the NASA solar data, which shows month-by-month changes in the solar energy hitting the earth, with the albedo data. The solar forcing in watts per square metre (W/m2) times (1 minus albedo) gives us the amount of incoming solar energy that actually makes it into the system. This is the actual net solar forcing, month by month.
Then I plotted the changes in that net solar forcing (after albedo reflections) against the corresponding changes in temperature, by hemisphere. First, a couple of comments about that plot.
The Northern Hemisphere (NH) has larger temperature swings (vertical axis) than does the Southern Hemisphere (SH). This is because more of the NH is land and more of the SH is ocean … and the ocean has a much larger specific heat. This means that the ocean takes more energy to heat it than does the land.
We can also see the same thing reflected in the slope of the ovals. The slope of the ovals is a measure of the “lag” in the system. The harder it is to warm or cool the hemisphere, the larger the lag, and the flatter the slope.
So that explains the red and the blue lines, which are the actual data for the NH and the SH respectively.
For the “lagged model”, I used the simplest of models. This uses an exponential function to approximate the lag, along with a variable “lambda_0” which is the instantaneous climate sensitivity. It models the process in which an object is warmed by incoming radiation. At first the warming is fairly fast, but then as time goes on the warming is slower and slower, until it finally reaches equilibrium. The length of time it takes to warm up is governed by a “time constant” called “tau”. I used the following formula:
ΔT(n+1) = λ∆F(n+1)/τ + ΔT(n) exp(-1/ τ)
where ∆T is change in temperature, ∆F is change in forcing, lambda (λ) is the instantaneous climate sensitivity, “n” and “n + 1” are the times of the observations,and tau (τ) is the time constant. I used Excel to calculate the values that give the best fit for both the NH and the SH, using the “Solver” tool. The fit is actually quite good, with an RMS error of only 0.2°C and 0.1°C for the NH and the SH respectively.
Now, as you might expect, we get different numbers for both lambda_0 and tau for the NH and the SH, as follows:
Hemisphere lambda_0 Tau (months)
NH 0.08 1.9
SH 0.04 2.4
Note that (as expected) it takes longer for the SH to warm or cool than for the NH (tau is larger for the SH). In addition, as expected, the SH changes less with a given amount of heating.
Now, bear in mind that lambda_0 is the instantaneous climate sensitivity. However, since we also know the time constant, we can use that to calculate the equilibrium sensitivity. I’m sure there is some easy way to do that, but I just used the same spreadsheet. To simulate a doubling of CO2, I gave it a one-time jump of 3.7 W/m2 of forcing.
The results were that the equilibrium climate sensitivity to a change in forcing from a doubling of CO2 (3.7 W/m2) are 0.4°C in the Northern Hemisphere, and 0.2°C in the Southern Hemisphere. This gives us an overall average global equilibrium climate sensitivity of 0.3°C for a doubling of CO2.
Comments and criticisms gladly accepted, this is how science works. I put my ideas out there, and y’all try to find holes in them.
w.
NOTE: The spreadsheet used to do the calculations and generate the graph is here.
NOTE: I also looked at modeling the change using the entire dataset which covers from 1984 to 1998, rather than just using the annual averages (not shown). The answers for lambda_0 and tau for the NH and the SH came out the same (to the accuracy reported above), despite the general warming over the time period. I am aware that the time constant “tau”, at only a few months, is shorter than other studies have shown. However … I’m just reporting what I found. When I try modeling it with a larger time constant, the angle comes out all wrong, much flatter.
While it is certainly possible that there are much longer-term periods for the warming, they are not evident in either of my analyses on this data. If such longer-term time lags exist, it appears that they are not significant enough to lengthen the lags shown in my analysis above. The details of the long-term analysis (as opposed to using the average as above) are shown in the spreadsheet.
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joeldshore:
At May 30, 2012 at 5:36 am you say to Willis:
Of dear! Even by you standards that is very wrong.
Firstly, climate sensitivity is NOT “known”.
If climate sensitivity were “known” then each model would use the same value of it.
But the climate models each use a different value of climate sensitivity that vary by a factor of 2.5.
(ref. Kiehl JT,Twentieth century climate model response and climate sensitivity. GRL vol.. 34, L22710, doi:10.1029/2007GL031383, 2007).
Secondly, if a method works then it works whether or not “the answer is known”.
Thirdly, … and etc. …
Richard
[SNIP: Off Topic. We have a Tips and Notes page here for things like this. -REP]
Billy Liar – “So where is the model (in control system terms) documented. Any idea?”
A reasonable resource is at http://stommel.tamu.edu/~baum/climate_modeling.html – links to major climate models. If you are interested in simpler models, I would suggest Googling 2-box to 6-box models; increasing model complexity does improve fidelity to observations, but even a zero dimensional flux model can provide some insight.
Willis Eschenbach – The major issue I have with your work is that it’s a “one-box” model, which fails entirely to capture the behavior of ocean heat transfer.
I might suggest you take a look at http://arthur.shumwaysmith.com/life/category/tags/two_box_model for a discussion/development of a somewhat more capable model – the behavior of which with reasonable heat transfer rates. Single box models will simply give unrealistic (as in your 0.3C/doubling) results.
richardscourtney says:
You are clearly not following the discussion. Let me explain it again:
(1) The climate sensitivity in the real world is not known and Willis claims to have found a method to determine it by looking at the seasonal cycle.
(2) The climate sensitivity in various climate models is known. For the purposes of this discussion, it is irrelevant whether that climate sensitivity corresponds to the climate sensitivity in the real world.
(3) Since the climate sensitivity in the various models is known, one can try to use Willis’s method and see if it correctly diagnoses the known climate sensitivity of a particular model when one looks at the seasonal cycle in the model.
(4) If Willis’ method does correctly diagnose the climate sensitivity in the model, then there is some reasonable hope it might also diagnose the climate sensitivity in the real world; if it doesn’t, there is little hope that it will magically do a better job in the more complicated real world.
What about this is difficult to understand?
joeldshore May 30, 2012 at 8:33 am – That is a major reason I pointed to the Knutti and Meehl paper. While all climate models (including, I’ll note, Willis Eschenbach’s single-box model as discussed in this thread) are inexact, comparing seasonal responses to forcings with a realistic climate model (one that at least considers several time constants) and it’s sensitivities would be an excellent test case for a seasonal method. That’s just what Knutti and Meehl 2006 did.
Annual cycles are just too short for significant heat penetration into the oceans, and hence cannot directly provide transient (20 year) climate sensitivity estimates. The thermal mass of the oceans acts as a flywheel or inductor – limiting responses to fast forcing changes.
Willis:
One should not imply the existence in nature of the equilibrium climate sensitivity (TECS) for TECS is defined in terms of the equilibrium temperature but this temperature is not an observable. As the equilibrium temperature is not an observable, claims made about the magnitude of TECS, including the claims that you make in this article, are not falsifiable, thus lying outside science.
Through the error of implying the existence of TECS, climatologists have neglected the necessity for providing a basis for falsifying the claims that are made by their models if these models are to be properly labelled as “scientific.” Were it to exist, this basis would be the underlying statistical population. Currently, there is no such population.
Magic Turtle says:
May 29, 2012 at 4:17 pm
To my saying (May 29, 2012 at 10:51 am):
“1. The amount of radiative forcing produced by a doubling of the CO2 concentration is not a regular fixed amount. The IPCC’s logarithmic formula from which the fixed amount results is false. The correct formula can be derived from the Beer-Lambert law of optics and it follows an inverse exponential law, not a logarithmic one.”
Phil says (May 29, 2012 at 12:56 pm):
‘ Not true, the logarithmic dependence does not come from the Beer-Lambert Law which applies in optically thin situations, but from the optically thick situation which applies to CO2 absorption in the IR in our atmosphere. Look up ‘Curve of Growth’ to see a derivation.
Basically, weak lines have a linear dependence on concentration, moderately strong lines have a logarithmic dependence and very strong lines a square root dependence.’
What I said is true. I never said that the logarithmic relationship is derived from the Beer-Lambert law, which I agree does apply to optically-thin lines.
No you said that the “correct formula can be derived from the Beer-Lambert law” which it can not because the absorption is not optically thin.
But it also applies to groups of optically-thin lines and for that reason it can be applied to entire absorption/emission wavebands. The thicknesses of the individual lines become irrelevant in that case. Such an application provides a straightforward method of deriving a general formula for the absorption of terrestrial surface radiance by atmospheric CO2.
However that absorption band is optically thick so your use of the Beer-Lambert Law to derive a relationship is flawed. The absorption by CO2 in the 15μm corresponds to the moderately strong region and hence has a logarithmic dependence which is what is found experimentally.
http://i302.photobucket.com/albums/nn107/Sprintstar400/CO2spectra-1.gif
A good derivation can be found here (note that it starts from the B-L Law):
http://www.physics.sfsu.edu/~lea/courses/grad/cog.PDF
@ur momisugly George E. Smith
Re. Your comments to me of May 30, 2012 at 3:15 am and 3:33 am.
“”””” the Beer-Lambert law is a theoretically-ideal expression for radiative absorption “””””
So what means “theoretically ideal expression”. Would it also be a practically ideal expression?
That depends on what you want to use it for. If you want to estimate the amount of radiant power that the CO2 in the earth’s atmosphere is absorbing it will only give you an approximation to the true value. But I contend that it would give you a better approximation than the IPCC’s logarithmic expression which does not do what is claimed for it and makes no sense to me.
You also state (or does Wikipedia, also stand in for you here) it is an expression for “radiative absorption ” ; OF WHAT ? The incident photons, or the energy they carry.
It’s both! The molecule absorbs both the photon and the energy it carries, perhaps like a flying duck ‘absorbs’ a shotgun pellet along with the energy that it carries.
If it is the latter, then B-L could also be used to compute THE ENERGY TRANSMISSION of solids, liquids and gases.
I’m not sure what you mean by ‘energy transmission’ here, but if you mean the amount of power that is transmitted by a beam of radiation through a medium and out the other side then, yes, that is how the B-L law can often be used (depending on the precise conditions of the case of course).
I think I understand your point about the inapplicability of Beer’s law to optical filters. However I don’t see it as a problem in relation the application of the Beer-Lambert law to the atmosphere because the Beer-Lambert law is not the same as Beer’s law. Under Beer’s law ‘the absorption follow(s) an exponential with THICKNESS’ relationship as you rightly say, but the B-L law is more fundamental and is not constrained in this way. Instead the B-L law allows that the total amount of power that is absorbed from a beam will depend (exponentially) on the total number of absorbent molecules in the beam’s path regardless of their spatial distribution along the beam. Therefore the B-L law is applicable to CO2 molecules in the earth’s atmosphere where the density of the gas varies whereas Beer’s law is not applicable to it as you say.
Sure the CO2 absorbs surface emitted LWIR photons; but they don’t stay absorbed; the energy is re-emitted at some other wavelength so it is not stopped by the CO2.
Quite possibly, but I am not arguing that CO2’s emission of radiation conforms to the Beer-Lambert law; only that its absorption of radiation does.
Just because something is in wikileaks does not make it reliable information.
Indeed so! But I didn’t refer you to wikileaks; I referred you to Wikipedia. Ha, ha!
However I acknowledge the validity of your point in relation to Wikipedia too and I don’t trust it blindly myself. I referred to it because it gives the most lucid and comprehensive explanation of the B-L law (derived from first principles) that I’ve seen to date and because it includes a section specifically on the application of the B-L law to the atmosphere (here: http://en.wikipedia.org/wiki/Beer-Lambert_law#Beer.E2.80.93Lambert_law_in_the_atmosphere). If you find a discussion somewhere else that disagrees with it please let me know.
ferd berple says:
May 29, 2012 at 6:53 pm
George E. Smith; says:
May 29, 2012 at 6:27 pm
That’s why the “CO2 saturation” notion is a non-starter. CO2 absorbs LWIR; re-emits some other wavelength, and then is ready to grab another LWIR photon from the surface; so it never really “saturates”.
=========
Why assume that the absorbed energy will come in the form of a photon from the surface? It is more likely the CO2 molecule will absorb kinetic energy from N2 and O2 and convert this into radiation, and about 1/2 of which will then be radiated into space that would otherwise remain in the atmosphere and be conducted back to the surface.
Absorbed LWIR excites the vibrational and rotational states which are therefore able to emit radiation if they are not collisionally deactivated first, transfer of translational kinetic energy from N2 and O2 does not necessarily excite the ro-vib levels in which case it will not emit, so no it is not more likely that the acquisition of kinetic energy will be converted into radiation.
Magic Turtle says:
May 30, 2012 at 10:09 am
That depends on what you want to use it for. If you want to estimate the amount of radiant power that the CO2 in the earth’s atmosphere is absorbing it will only give you an approximation to the true value. But I contend that it would give you a better approximation than the IPCC’s logarithmic expression which does not do what is claimed for it and makes no sense to me.
That’s because you don’t understand absorption in an optically thick situation which is the case for CO2 absorption in the atmosphere. You insist on applying B-L outside its range of applicability,
it might come close for the Martian atmosphere but not Earth’s.
Magic Turtle says:
May 30, 2012 at 10:09 am
Instead the B-L law allows that the total amount of power that is absorbed from a beam will depend (exponentially) on the total number of absorbent molecules in the beam’s path regardless of their spatial distribution along the beam. Therefore the B-L law is applicable to CO2 molecules in the earth’s atmosphere where the density of the gas varies whereas Beer’s law is not applicable to it as you say.
This is true only if that beam path is optically thin (read the page you cited), you have to take account of optical thickness, do yourself a favor and read the article I referenced.
Willis
please could you explain your formula
ΔT(n+1) = λ∆F(n+1)/τ + ΔT(n) exp(-1/ τ) ?
Do you have any sources for that?
When you look at the units in your formula, there must be something wrong. τ must be without unit, it can’t be a time constant. And if τ is just a factor, wouldn’t lambda be “your lambda”/ τ ? But even then, this lambda is not the usual climate sensitivity.
But even without this, your formula looks strange, I need some sources to understand.
PS:
You haven’t answered my question before, maybe you overlooked or was it a stupid question? My question was, if you considered heat exchange between the two hemispheres e.g. by winds or ocean currents.
KR:
In your post at May 30, 2012 at 8:27 am you reply to my having written
But, as I explained, they don’t.
And you rightly say I also wrote:
You reply with the incorrect assertion
And you then provide a series of numbered points. I address each of them in turn.
You say
I agree.
But I add the caveat that Willis is the third person I know who has attempted to determine climate sensitivity from the variation of solar forcing which results from Earth/Sun distance provided by the Earth’s orbit.
You say
No!
We are discussing the method Willis has used to the magnitude of “the climate sensitivity in the real world”
You say
Yes. But so what? We are discussing the method Willis has used to determine the magnitude of “the climate sensitivity in the real world” (n.b. NOT the climate sensitivity used in any “particular model”).
You say
Absolutely not!
Models are tested against observations and analyses of reality.
Analyses of reality are NOT tested against models.
This is because the analysis is of reality and a model is merely a representation of an idea about reality. Therefore, your assertion is the opposite of what is required. Your assertion would be correct if it said
If there is no flaw in Willis’ method then it does correctly diagnose the climate sensitivity in the real world. So, if a model uses the climate sensitivity indicated by Willis’ method then there is some reasonable hope that the model might emulate the real world; if it doesn’t, there is little hope that it will magically do a better job of emulating the more complicated real world.
You ask me
I have no difficulty understanding this, but clearly you do.
Richard
Willis Eschenbach
May 30, 2012 at 12:35 am
“Finally, as you likely know, the blackbody response from a doubling of CO2 is 0.7°C.”
And as you likely know, the blackbody sensitivity, expressed in units of watts/M^2 per degree change in temperature, depends on the assumed emission temperature, as given by:
dJ/dT = (2.268 X 10^-7) * T^3
For 255K emission (what approximately balances absorbed solar energy with net radiative loss to space) the sensitivity to a doubling is 3.76 watts/M^2/K, or about 3.71/3.76 = 0.987 C per doubling (which is why I say “about 1C”). It is only by assuming a much higher average emission temperature (eg 288K) that the blackbody sensitivity is close to 0.7C per doubling (actually 0.685C per doubling for 288K). But 288K is not the emission temperature for Earth.
“”””” Magic Turtle says:
May 30, 2012 at 10:09 am
@ur momisugly George E. Smith
Re. Your comments to me of May 30, 2012 at 3:15 am and 3:33 am.
“”””” the Beer-Lambert law is a theoretically-ideal expression for radiative absorption “””””
So what means “theoretically ideal expression”. Would it also be a practically ideal expression?
That depends on what you want to use it for. “””””
Well Magic, you are missing the whole point. Beer’s Law, and Lambert’s law (they are different laws) both assume that photons (and their energy) are absorbed proportionally to the number of absorbing species molecules or atoms they encounter; and that the ENERGY STAYS ABSORBED, which lowers the amount of propagating radiant energythat becomes available to the next bunch of molecules to take a shot at absorbing. So for example, in a stack of thin sheets, the EM radiant energy exiting each sheet in turn, would be reduced by the ABSORPTION in the sheet, so less energy enters the next sheet. But the photons are required to stay dead for the exponential decay relationship to apply for either law. But often they simply resurrect as a longer wavelength photon, which will alsmost surely have a different absorption probability to apply in the next sheet. So individual photons may be extinguished, but the energy continues to propagate, at a different wavelength,. The ORIGINAL SPECIES are absorbed in the usual exponentially decaying manner, so the laws apply to the absorption of the original species, but they don’t apply to the ENERGY TRANSMISSION of many materials.
For starters,there are deviations from the laws simply due to the Stokes shift when the subsequent emission of a longer wavelength photon occurrs. Eventually it all will be transmitted, just at different wavelengths, including Planckian thermal wavelengths from sample heating.
Photons don’t stay dead; not until the temperature reaches zero K.. But if you only account for the original impinging photon species, (absorption), measurements show quite good agreement with those laws. Try googling “blue pumped white LEDs” They depend on re-emission of longer wavelength photons, as well as transmission of the original blue (often 460 nm) photons, and Stokes shift losses are key to final efficiency.
“”””” re stevefitzpatrick………..For 255K emission (what approximately balances absorbed solar energy with net radiative loss to space) the sensitivity to a doubling is 3.76 watts/M^2/K, or about 3.71/3.76 = 0.987 C per doubling (which is why I say “about 1C”). It is only by assuming a much higher average emission temperature (eg 288K) that the blackbody sensitivity is close to 0.7C per doubling (actually 0.685C per doubling for 288K). But 288K is not the emission temperature for Earth. “””””
Well 288 K is certainly closer to the usually cited mean Temperature of earth, than is 255 K.. “””””
You say 255K (BB radiation) matches solar input, but that assumes that earth is indeed a single Temperature BB emitter which it isn’t. You have to believe that ALL of the roughly 5 to 80 micron spectrum emission from a 288 K Temperature source is absorbed in the atmosphere, and none escapes, to believe that earth looks like a 255 K BB. In fact plenty of energy escapes from earth at much higher effective source Temperatures from the hottest desert areas, where CO2 absorption is less of a factor.
The tropical hot areas are responsible for cooling the earth, not the cold polar areas, which often emit less than 10% of what the hot zones radiate.
richardscourtney May 30, 2012 at 10:49 am – I’m afraid that the comment you were replying to was from joeldshore, not me.
richardscourtney says:
How is one to know if there is a flaw in the method? The method, like a model, makes a zillion approximations, as we have been discussing. In fact, his particular method is based on fitting an extremely simple model to the data and then using that model to determine the climate sensitivity. There is no evidence whatsoever that his model can do a good job telling you the climate sensitivity when tuned to data on the seasonal cycle.
It is strange how you somehow believe that Willis’s model can tell you the correct climate sensitivity but a full-scale global climate model can’t…And, that in fact, you can’t even use a full-scale climate model as a testbed to see how well Willis’s simple technique works on a much more careful representation of reality. It seems the only reason that you believe this is that Willis’s model gives you your desired answer.
Willis did not do an “analysis of reality” any more than any data-fitting exercise with a simple model is an “analysis of reality”. Willis used a model that is completely untested to fit some data. (Actually, I think such a model has been tested to enough to know that it is way too simplistic for the task.) And now he wants to claim that the completely untested aspect of his model is correct…i.e., that his simple model tuned to the most basic aspects of the seasonal cycle can diagnose the correct climate sensitivity. There is no reason whatsoever to believe that it can and every reason to believe that it can’t.
The use of “synthetic data” produced by models as a way of evaluating a diagnostic technique has a rich and successful history in science. You want to overturn that because you want to elevate a simple model that gives the answer that you happen to like above all other models. It is really bizarre. You guys have an extreme allergic reaction to climate models unless said models are simplistic enough that they can be tuned to data to give answers that you like and then you apparently ready to believe them without even demanding any testing whatsoever!
Curiously enough, looking at some discussions of multiple box models, I ran across a reference to several earlier Eschenbach postings on forcings and modeling, and a fairly detailed reply (http://tinyurl.com/6tvk3wt) indicating that they all suffer from the same problem – using a single time constant, a single-box model. And just that doesn’t fit the data.
The model discussed in this thread also has a single time constant, and will, inevitably, also fail to fit the data accordingly.
A two-box model (as discussed in the link above), two time constants (still a fairly crude model, mind you), fits the data quite well with time constants of ~2 and ~45 years. And demonstrates a sensitivity of around 2.4C/doubling of CO2. That model is also a much better match to the actual physics – where the fast climate response is seen in portions of the climate (atmosphere) with low heat capacity, and the slow response time is seen in portions (oceans) with high heat capacity.
“Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.” – George Box.
A single box model is not terribly useful.
[UPDATE—ERROR] I erroneously stated above that the climate sensitivity found in my analysis of the climate models was 0.3 for a doubling of CO2. In fact, that was the sensitivity in degrees per W/m2, which means that the sensitivity for a doubling of CO2 is 1.1°C. -w.
@Phil, Re. Your comment to me May 30, 2012 at 9:54 am
MT: “What I said is true. I never said that the logarithmic relationship is derived from the Beer-Lambert law, which I agree does apply to optically-thin lines.”
Phil: ‘No you said that the “correct formula can be derived from the Beer-Lambert law” which it can not because the absorption is not optically thin.’
Sorry, but I am not following you. How does the optical thickness of CO2 absorption prevent one from deriving a valid formula for radiation-absorption by CO2 from the Beer-Lambert law? I cannot understand an argument that you are not putting, Phil!
MT: “But it also applies to groups of optically-thin lines and for that reason it can be applied to entire absorption/emission wavebands. The thicknesses of the individual lines become irrelevant in that case. Such an application provides a straightforward method of deriving a general formula for the absorption of terrestrial surface radiance by atmospheric CO2.”
Phil: ‘ However that absorption band is optically thick so your use of the Beer-Lambert Law to derive a relationship is flawed. The absorption by CO2 in the 15μm corresponds to the moderately strong region and hence has a logarithmic dependence which is what is found experimentally.’
My approach does not focus on one particular frequency of absorption as yours does, but focuses on the sizes of the absorption wavebands instead. I can imagine that the optical thickening effect would broaden that waveband to a degree and render it more shallow to a degree as well, but I doubt that such modifications would be significant. If they were the whole concept of absorption wavebands would be meaningless. In any case we can take them into account for practical purposes by treating them as effectively constant and incorporating them into the constant ‘g’ in the formula that I presented in my earlier post:
A = S.k.[1 – exp(-g.C)].
Phil: ‘http://i302.photobucket.com/albums/nn107/Sprintstar400/CO2spectra-1.gif’
Why have you given me this link to a couple of unexplained graphs that have no assigned authorship and give no explanation of what they are supposed to represent and how they have been generated? They are meaningless to me.
Phil: ‘A good derivation can be found here (note that it starts from the B-L Law):
http://www.physics.sfsu.edu/~lea/courses/grad/cog.PDF’
This link purports to give a derivation of the Curve of Growth. It does not give a derivation of the IPCC’s logarithmic formula for radiative forcing from CO2. And again, it is unattributed, has no explanatory introduction, does not state its assumptions and provides no references that might give some key to the context of thought in which its complex mathematical arguments might have been conceived. To me it is meaningless technobabble, I’m afraid.
I do not deny that the Curve of Growth for radiation absorption by CO2 at 15µm is logarithmic with respect to the relative intensities of the incident and absorbed radiant energies by the individual CO2 molecules. But that is not what the IPCC’s formula is referring to. It is portraying the combined effect of all the CO2-molecules in the atmosphere taken together as logarithmic! That is a completely different matter.
Re. Your comment to me May 30, 2012 at 10:25 am
MT: “But I contend that (the B-L formula) would give you a better approximation than the IPCC’s logarithmic expression which does not do what is claimed for it and makes no sense to me.”
Phil: ‘That’s because you don’t understand absorption in an optically thick situation which is the case for CO2 absorption in the atmosphere.’
Then pray enlighten me. How does absorption in an optically thick situation make the IPCC’s logarithmic formula sensible (instead of insensible) and more applicable to atmospheric CO2 than my formula derived from B-L? You still have not explained this and neither have any of the references that you have given me.
Re. Your comment to me May 30, 2012 at 10:34 am
MT: “Instead the B-L law allows that the total amount of power that is absorbed from a beam will depend (exponentially) on the total number of absorbent molecules in the beam’s path regardless of their spatial distribution along the beam. Therefore the B-L law is applicable to CO2 molecules in the earth’s atmosphere where the density of the gas varies whereas Beer’s law is not applicable to it as you say.”
Phil: ‘This is true only if that beam path is optically thin (read the page you cited), you have to take account of optical thickness, do yourself a favor and read the article I referenced.’
I have read the page that I cited! I have also read the article that you referenced and it doesn’t show how optical thickness makes the B-L law inapplicable to atmospheric CO2. Neither does it show how the IPCC’s formula has been derived! Surely you are the one who needs to do himself a favour by reading the article that he has referenced to me!
joeldshore says:
May 30, 2012 at 5:36 am
Much appreciated, Joel. What I reported is the sensitivity that fits the climate model results, with a correlation over 0.97 for each of the two models I studied … so you can claim it is “not the correct climate sensitivity”, but it is assuredly the sensitivity that works, and works exceedingly well.
So … why does it almost perfectly emulate the model if it is “not the correct climate sensitivity”??? Can you emulate the model results using what you say is the “correct climate sensitivity”? If not, why not?
Thanks,
w.
Willis Eschenbach says:
Wills: It did not perfectly emulate the model. It systematically misrepresented the magnitude of the model response to volcanic forcings, requiring you to put a “fudge” into your model to make it work better. In fact, that problem with the volcanic forcings was a signal that something was wrong…and that something, as KR is pointing out, is assuming a model with only one timescale when there are at least two timescales (associated with the heat capacities of the atmosphere and ocean mixed layer) that need to be considered.
I am puzzled why you would claim that you found the correct climate sensitivity for the GISS model. Do you think they are not telling you the truth when they run the model to determine the true climate sensitivity of the model? Clearly, the climate sensitivity that you find with your fit of a simpler model to their model, while it may do a reasonable job at fitting (if you ignore the one forcing that operates over a different timescale, the volcanic forcing), does not diagnose the correct equilibrium climate sensitivity of the model. I.e., they measured the equilibrium climate sensitivity and got a different result than you got from your fitting procedure. If you think they did it wrong, you have to demonstrate this. You haven’t because you haven’t directly determined an equilibrium climate sensitivity. You have determined it only indirectly through fitting of their model results to a simpler model. Your simpler model includes only one timescale and what it mainly shows is that such a model is too simple to accurately determine the actual equilibrium climate sensitivity of the model.
And, I should add that you now have corrected the record http://wattsupwiththat.com/2012/05/29/an-observational-estimate-of-climate-sensitivity/#comment-997301 (thanks) and noted that your two estimates of climate sensitivity, one by fitting your model to the GISS model emulation of the instrumental temperature record and the other by fitting your model to the seasonal cycle, produce estimates that differ by about a factor of four. This is further evidence of the deficiencies of your model.
And, they differ in just the way that we expect: I.e., the higher the frequency of response that you look at, the smaller the estimate of climate sensitivity that you come up with. That is because equilibrium climate sensitivity is the zero-frequency limit of the response…and will be systematically underestimated when you try to do it using higher frequency responses using a model that only allows for one relaxation time.