A logarithmic relationship between CO2, radiative forcing and direct impact is reported by IPCC and widely relied on. While this may well be a plausible relationship (Lubo, for one, endorses it), it is not easy finding a proof of the relationship. In a recent post, I noted this in connection with IPCC AR1 (1990), where I reported their references. Today, I’m going to discuss the handling of the logarithm formula in AR4, AR3 and then report on today’s search for the source of the Nile.

**AR4**

AR4 stated that the logarithmic formula of TAR remained valid as follows:

(2.3.1) The simple formulae for radiative forcing of the LLGHG [long-lived greenhouse gases] quoted in Ramaswamy et al. (2001) [IPCC TAR] are still valid. These formulae are based on global radiative forcing calculations where clouds, stratospheric adjustment and solar absorption are included, and give an RF of +3.7 W m2 for a doubling in the CO2 mixing ratio. (The formula used for the CO2 RF calculation in this chapter is the IPCC (1990) expression as revised in the TAR. Note that for CO2, radiative forcing increases

logarithmicallywith mixing ratio.)

TAR

Section 1.3.1 of TAR stated:

If the amount of carbon dioxide were doubled instantaneously, with everything else remaining the same, the outgoing infrared radiation would be reduced by about 4 Wm-2. In other words, the radiative forcing corresponding to a doubling of the CO2 concentration would be 4 Wm-2. To counteract this imbalance, the temperature of the surface-troposphere system would have to increase by 1.2°C (with an accuracy of ±10%), in the absence of other changes. In reality, due to feedbacks, the response of the climate system is much more complex. It is believed that the overall effect of the feedbacks amplifies the temperature increase to 1.5 to 4.5°C. A significant part of this uncertainty range arises from our limited knowledge of clouds and their interactions with radiation. …

It has been suggested that the absorption by CO2 is already saturated so that an increase would have no effect. This, however, is not the case. Carbon dioxide absorbs infrared radiation in the middle of its 15 mm band to the extent that radiation in the middle of this band cannot escape unimpeded: this absorption is saturated. This, however, is not the case for the bands wings. It is because of these effects of partial saturation that the radiative forcing is not proportional to the increase in the carbon dioxide concentration

but shows a logarithmic dependence.Every further doubling adds an additional 4 Wm-2 to the radiative forcing.

**SAR**

I have not been able to locate any mention of the logarithmic relationship in SAR (or for that matter any use of the quantity 4.37 wm-2 for doubled CO2). Nor in IPCC (1994).

**AR1**

I discussed this aspect IPCC AR1 (1990) here giving an extended quotation. I observed that IPCC (1990) cited two references for the logarithmic relationship:

1) Wigley Climate Monitor 1987. Climate Monitor is a house publication of CRU, where Wigley was then employed. It is not carried at U of Toronto and Wigley said that he no longer had a copy.

2) Hansen et al 1988. As noted in previous post, Appendix B of Hansen et al 1988 stated the logarithmic relationship referring to Lacis et al 1981 as authority:

Radiative forcing of the climate system can be specified by the global surface air temperature change ΔT0 that would be required to maintain energy balance with space if no climate feedbacks occurred (paper 2). Radiative forcings for a variety of changes of climate boundary conditions are compared in Figure B1, based on calculations with a one-dimensional radiative-convective model (Lacis et al, 1981). The following formulas approximate the ΔT0 from the 1D RC model within about 1% for the indicated range of composition. The absolute accuracy of these forcings is of the order of 10% because of uncertainties in the absorption coefficients and approximations in the 1D calculations:

CO2:

where x_0=315 ppmv; X<1000 ppmv.

**Lacis et al GRL 1981**

I obtained Lacis et al 1981 today. It did not **mention** the logarithmic relationship anywhere. It contained a very sketchy description of their 1-D radiative-convective model.

**TAR 6.3.5 Simplified Expressions **

As noted in comment #3 below, TAR 6.3.5 also contains the following statement:

IPCC (1990) used simplified analytical expressions for the well-mixed greenhouse gases based in part on Hansen et al. (1988). With updates of the radiative forcing, the simplified expressions need to be reconsidered, especially for CO2 and N2O. Shi (1992) investigated simplified expressions for the well-mixed greenhouse gases and Hansen et al. (1988, 1998) presented a simplified expression for CO2. Myhre et al. (1998b) used the previous IPCC expressions with new constants, finding good agreement (within 5%) with high spectral resolution radiative transfer calculations. The already well established and simple functional forms of the expressions used in IPCC (1990), and their excellent agreement with explicit radiative transfer calculations, are strong bases for their continued usage, albeit with revised values of the constants, as listed in Table 6.2. Shi (1992) has suggested more physically based and accurate expressions which account for (i) additional absorption bands that could yield a separate functional form besides the one in IPCC (1990), and (ii) a better treatment of the overlap between gases. WMO (1999) used a simplified expression for CO2 based on Hansen et al. (1988) and this simplified expression is used in the calculations of GWP in Section 6.12. For CO2 the simplified expressions from Shi (1992) and Hansen et al. (1988) are also listed alongside the IPCC (1990)-like expression for CO2 in Table 6.2. Compared to IPCC (1990) and the SAR and for similar changes in the concentrations of well-mixed greenhouse gases, the improved simplified expressions result in a 15% decrease in the estimate of the radiative forcing by CO2 (first row in Table 6.2), a 15% decrease in the case of N2O, an increase of 10 to 15% in the case of CFC-11 and CFC-12, and no change in the case of CH4.

Their Table 6.2 entitled “Simplified expressions for calculation of radiative forcing due to CO2, CH4, N2O, and halocarbons” contains the following expressions described as follows”

The first row for CO2 lists an expression with a form similar to IPCC (1990) but with newer values of the constants. The second row for CO2 is a more complete and updated expression similar in form to that of Shi (1992). The third row expression for CO2 is from WMO (1999), based in turn on Hansen et al. (1988).

F = 5.35 ln(C/C0)

F= 4.841 ln(C/C0) + 0.0906 (C – C0)

F= 3.35 *( ln(1+1.2C+0.005C2 +1.4 x 10-6C3) – ln(1+1.2C0)+0.005C0)2 +1.4 x 10-6C0)3))

The third row expression appeared previously in Appendix B of Hansen et al 1988 (See above). So that doesn’t help.

**Current Provenance**

The IPCC 1990 form and the WMO (1999 – Ozone Depletion) both derive from Hansen et al 1988, which cites Lacis et al 1981, which turns out to be a dead end with no mention of logarithms. I haven’t located Shi (1992) yet and again will update this when I do so. It’s quite possible, perhaps even plausible, that there should be a logarithmic relationship between CO2 and direct temperature impact, but it’s certainly not easy to locate a clear derivation.

**New Reference:**

Shi, G., 1992: Radiative forcing and greenhouse effect due to the atmospheric trace gases. Science in China (Series B), 35, 217-229.

WMO, 1999: Scientific Assessment of Ozone Depletion: 1998, Global Ozone Research and Monitoring Project, World Meteorological Organization, Report No. 44, Geneva, Switzerland.

## 84 Comments

Steve -

I guess I’m confused. Doesn’t section 6.3.5 of IPCC TAR discuss

the origin of this?

http://www.grida.no/climate/ipcc_tar/wg1/222.htm#635

Erik

Um, no. It simply references the same things that Steve noted above. The paper trail seems to stop where Steve has indicated.

Mark

Mark, there are many particular issues in the climate situation that you’re not addressing. “impedance of power” in the climate situation seems like an analogy to me – in any event, while the example may be helpful to you, it’s not helpful to me as I’m more familiar with the climate situation than with the impedance of power situation.

But regardless, the question is not really whether you are able to derive the relationship – and I’d expect more than the back-of-the envelope calculation in your post above – but whether IPCC has derived the relationship.

#1. 6.3.5 does not “discuss the origin” of this. However it states two references that I hadn’t canvassed. I’ve updated the text to reflect this section.

By the way, Venkatachalam Ramaswamy is one of the two coordinating lead authors of chapter 2 of the latest IPCC report (AR4-WG1).

The TAR formulas for radiative forcing are given in this table (section 6.3.5):

http://www.grida.no/climate/ipcc_tar/wg1/222.htm#635

- note that 3 formulas are shown for CO2, only one of which is purely logarithmic. The citations for the three are:

alpha ln(C/C0) – IPCC (1990) with updated proportionality constant

alpha ln(C/C0) + beta (sqrt(C) – sqrt(C0)) – Shi (1992)

alpha (g(C) – g(C0)) where g(C) = ln(1 + 1.2C + 0.005 C^2 +1.4 x 10-6 C^3) – WMO (1999) based on Hansen et al (1988)

References are:

IPCC, 1990: Climate Change 1990: The Intergovernmental Panel on Climate Change Scientific Assessment [Houghton, J.T., B.A. Callander, and S.K. Varney (eds)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.

Shi, G., 1992: Radiative forcing and greenhouse effect due to the atmospheric trace gases. Science in China (Series B), 35, 217-229.

WMO, 1999: Scientific Assessment of Ozone Depletion: 1998, Global Ozone Research and Monitoring Project, World Meteorological Organization, Report No. 44, Geneva, Switzerland.

Hansen, J., I. Fung, A. Lacis, D. Rind, S. Lebedeff, R. Ruedy, G. Russell, and P. Stone, 1988: Global climate changes as forecast by Goddard Institute for Space Studies 3-dimensional model. J. Geophys. Res., 93, 9341-9364.

—–

But these are simplified formulas and apparently are *not* what is actually used in the climate models – rather they are pulled out of the climate models based on more complex radiation models (MODTRAN/HITRAN data, etc.). The purpose of pulling out the radiative numbers to get a “forcing” is to provide some explanatory power beyond the greenhouse gas concentrations themselves. In TAR, section 6.3.1 goes into this in some more detail – http://www.grida.no/climate/ipcc_tar/wg1/219.htm -

The section after the AR4 section you quote also indicates this – though I absolutely agree this could be said much more clearly:

ie. it appears the radiative forcing numbers are *approximated* by the logarithmic rule for the range of CO2 concentrations of interest (280 to 600 or so), but that is verified by comparison to GCM’s, which disagree among one another on the matter by 10-20%.

At least that’s my interpretation reading all these different reviews. Now, perhaps we should invite Ramaswamy or somebody else who’s worked on this to explain?

The earliest mention of a logarithmic relationship was over 100 years ago by Arrhenius, but it appears that he didn’t derive it, but simply observed it.

http://en.wikipedia.org/wiki/Svante_Arrhenius#Greenhouse_effect_as_cause_for_ice_ages

He was proposing at the time that CO2 caused ice ages through some sort of reverse feedback mechanism. Thus, his coefficient was rather high. It’s now understood that that wasn’t the cause of ice ages, and the coefficient is wrong.

At the risk of submitting a “primary school” or “baby food” explanation, isn’t it a case of a constant proportion of radiation being absorbed by each ppm of CO2? Imagine a column of CO2 that absorbed 50% of IR radiation. If you place a second column on top, it does not absorb the other 50%, but rather 50% of the remainder. Hence the logarithmic relationship.

I’m sure I must be missing a point somewhere. Back to lurking.

Thats interesting that in the updated version of the IPCC tart that they now include a square root function because as I previously mentioned the width of Cauchy distributed spectral peaks varies with the square root of the CO2 concentration.

#10. I’m not asking for people to provide their own explanations in 3 sentences. I’m looking for the provenance as IPCC used it.

#10 the relationship you describe is negative exponential. That is it is given by:

Io*(1-exp(-lambda*(Co2 concentration))

The peaks first increase in hight via the negative exponential relationship then once they become saturated the width of the peaks, grows by a square(CO2 concentration) for a Cauchy distributed absorption peak, and by a sqrt(log(CO2 concentration)) for a Gaussian distributed absorption peek.

I’d also add that I wouldn’t expect the forcing to follow exactly the shape of the absorption peaks for the whole atmosphere as that neglects the process of re-emission and convection .

#8. Much of your post is simply repeating what I’d already posted. You say:

That’s not what they say and is inconsistent with the history. The formula seems to be derived somehow from the Lacis et al 1981 1-D radiative-convective model and not from a GCM.

Secondly, to my knowledge, GCMs don’t use HITRAN; as far as I knoiw, they use narrow-band parameterizations (and formerly even broadband parameterizations), which are surveyed from time to time for consistency with LBL models.

Thirdly, since a number of IPCC results are derived from the Wigley and Raper simple emulations of climate models, one would need to examine this model to see whether this form has been used there as well.

John Creighton has sent me some notes on the functional form which he’s asked me to post and I’m getting to after some more review.

Steve, the “1 D” models sound interesting. Do you have any good links for me to look at?

The 1-D literature is quite interesting – I have many articles in paper format since they tend to be from the 1970s and 1980s. You can locate some Ramanathan articles from the 1970s online if you google “journal climate online” and do a search there. Start with the 1970s.

Btw, if you assume a stable system, it is possible to prove that a) the relationship is

at mosta logarithmic function and b) if it is asymptotic, the slope of the asymptote is 0.Mark

#19 sounds interesting. Where can I find the proof?

#21 I think I know what you’re getting at. To me forcing is the magnitude of the feedback at a given temperature. Although to other people it may mean the sensitivity (i.e. the gain) or it could be the magnitude of the feedback at equilibrium.

If there is a limiting factor (like a window or re-emission) such that the magnitude of the positive feedback can only approach some fixed distance from one call it w then the gain will asymptotically approach:

1/w

Steve #15 – my #8 was a repeat mainly because you updated the main post while I was writing it (to refer to the TAR section 6.3.5 formulas) – anyway, my fault for not refreshing before posting the comment.

While the original source of the fitting to a logarithm clearly dates back to early papers that had nothing to do with GCM’s, my comment was based on the pretty clear statement in the TAR:

In the TAR formulas they *changed* the value of the paramter ‘alpha’ from the number in the 1990 IPCC report, so clearly the precise formula, including parameter, was new at that point, even though the basic logarithmic form was old. And the reason they changed it was because “several studies, including some using GCMs” found a lower radiative forcing. That seems pretty clear to me that the formula is a *fit* to model results, not derived from some fundamental theory.

In any case, given that statement, it’s also clear that these simple formulas are not what’s used in the GCM’s, as you confirm. So actually I’m not quite sure what all the fuss is about questioning the formula – it’s apparently a useful thing to have at hand to estimate the effects of additional CO2, but it’s not anything fundamental…

Nope, it’s got nothing to do with feedback, though we do assume a stable system which means the sum of all the feedbacks results in a gain of less than 1. If we didn’t have that assumption, then our climate would exponentially diverge which we are not seeing. Anyway, simply assuming stability is sufficient for what I’m getting at, though the “gain” could be either really small or really large or anywhere in between (it does not matter). Also, we assume the maximum amount of power in the spectrum that CO2 can affect is bounded, i.e. P_max. If it is not bounded, then our system is not bounded input and the whole idea of stability likewise goes out the window (hence the system/control theory phrase “bounded input-bounded output stable”). The power trapped, P_t, is a function of CO2 by design, or P_t = f(CO2).

Anyway, first assume that the relationship between CO2 and the total power that CO2 traps is a line with slope a, i.e. P_t = a*CO2. We could also assume some higher order function, but that only speeds the decay and what I’m getting to will necessarily be less than a line anyway.

Now, for any a, we can find some value of CO2 at which P_t = P_max. Rather than point out that further increases in CO2 content would by definition trap more heat, which is a contradiction since P_max is bounded, which necessarily bounds P_t, we can simply note that this situation results in an asymptotic function with asymptote of 0. As a -> 0, CO2 -> inf, but there is always some point at which a*CO2 = P_max. Since an asymptote is a line, then it is proved that if the function is asymptotic, it must have a slope of zero since any non-zero slope will ultimately result in P_t > P_max OR no further power being trapped which is the same as an asymptote of 0. Since the slope of the line cannot be asymptotic, then it must be continually (monotonically) decreasing. The largest such function with a monotonically decreasing slope is a logarithm. Therefore, an upper bound of a logarithmic function is established. The actual function must be somewhere in between a flat line and a logarithm (with arbitrary scale).

Now, the part I have not convinced myself of is whether or not a logarithm decays sufficiently fast enough. In particular, is it enough that the slope of the function approaches zero, or should the function be asymptotic? The problem with the logarithm is that it has infinite range, as does a line, which is physically impossible when we know the input power is bounded… OR, does the logarithm simply serve as an approximation for now? I need to dwell on this final thought some more either way.

Mark

Oh, by this:

I meant feedback gains, not closed loop gain of less than 1. This simply guarantees no poles in the RHP, which likewise guarantees stability, which was a condition of my proof (justified, IMO).

Mark

If the TAR really says

“It has been suggested that the absorption by CO2 is already saturated so that an increase would have no effect. This, however, is not the case. Carbon dioxide absorbs infrared radiation in the middle of its 15 mm band to the extent that radiation in the middle of this band cannot escape unimpeded: this absorption is saturated. This, however, is not the case for the bands wings. It is because of these effects of partial saturation that the radiative forcing is not proportional to the increase in the carbon dioxide concentration but shows a logarithmic dependence. Every further doubling adds an additional 4 Wm-2 to the radiative forcing.”then the TAR must clearly be wrong? When the absorption is far from saturated it must be a (fairly) linear response to more added CO2 (since the odds of a photon with the particular wave length encountering a CO2 molecule depend on the likelihood of a molecule in the outgoing radiative path). I.e. the contribution from the outer band wings will be linear, though miniscule and probably not so linear when other greenhouse gases, water vapour and clouds are taken in to the picture. When the mean free path is equal to the vertical length of the atmosphere, the linear response model have broken down; the odds of a photon encountering a CO2 molecule is 0,5 and the response is logaritmic – if other effects are neglected, which again is unphysical. In the much larger saturated bands however, the response should be exponential. The odds of a photon smashing into a CO2 is represented by the mean free path and – with other effects neglected – should be of the form 0,5^(atmospheric depth/mean free path).

So, the logaritmic response is valid – if other effects are neglected which they shouldn’t be – only to the middle of the side slopes of the absorbtion bands. But the side slopes of the 4 um band matters very little as the spectral intensity is very low, at least for temperatures above -20°C. And the side slopes of the 15 um band are overlaped by water vapour, partly by oxygene and active oxygene, and obviously by clouds. And the response should be considered with convection and latent heat as well, making an overall logaritmic response impossible. So how is the logaritmic response calculated?

I would like to add that I’m humble about this, I’m not a climate scientist though I work with climate models of sorts (they all suck), and this is my coffee break… I’m greatful if anyone can explain why I’m being stupid ;-)

#28,

It’s always logarithmic, it’s just that when you are far from saturation, the slope of the function is nearly linear.

Another thought, the primary respons from CO2 is dependent on both temperature (the lower the higher the response) and the presence of water vapour and clouds (the more of those the lower the response). Antarctis being a vast dry desert and significantly colder than Arctis, one would expect a more significant response in Antarctis than Arctis. Does anybody know why most climate models show less warming in Antarctis than Arctis? Are they tuned or are there physical reasons as to why such a behavior would be expected?

MarkW, presumably you mean the slope of the tangent line. Well, a line is

alwayslinear.It seems to me that the relationship was derived not from observation or experimentation, but by logic alonw.

The seem to be using the reasoning (good, but not necessarily correct) that it shouldn’t increase linearly becuase it would absorb all the radiation available way before it reaches high concentrations. While I can see where one would think so, I don’t think this is properly demonstrated. But I tentatively eqcept the theory, becuase I think it seems pluasible and makes logic sense to me.

But what other way to describe it could there be? I can’t think of anything. (You might suggest a function like 1/q, which would tell you the effect of each new molecule, but this isn’t really different, becuase if you integrate it, it’s a log again ;) )

krypgrund avfuktare vind, maybe becuase that’s eqactly what’s happening and they want to match reality?

Actually, I think they’ve got this part a bit messed up presently. I’m pretty sure the models predict the same amount of warming at both poles but, problematically, that doesn’t apper to be happening.

Feel free to correct me if I’m wrong, anyone. But have proof!

the logarithmic relationship between IR absorption and CO2 concentration was already experimentally confirmed by John Koch in Sweden in 1901.

ref

John Koch, Beiträge zur Kenntnis der Wärmeabsorption in Kohlensäure., Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlinger, 1901. N:o 6 p 475-488

Hans Erren, what’s the sensitivity on that graph?

#34 Hans Erren:

“Having the look” of a logarithmic relationship is not quite the same as actually having said relationship.

If you look to the Box-Cox family of transformations (of which the “natural” log is a member), with the right data set, a confidence interval can be calculated to determine whether or not the log is a plausible function.

of course is isn’t the exact relationship it’s an emperical approximation of the integral of the quantummechanical absorptionfunction. That’s the difference between pure and applied physics.

Be my guest to derive the exact formula. (integral over wavenumbers of (the planck function * lambert-beer(quantum specific absorptivity))

#28. Please stop creating straw men.

I specifically did not say “then the TAR must clearly be wrong?”. Indeed, I’ve only said that they do not derive the formula themselves and their authorities state but do not derive the formula. I’;ve allowed for the possibility of a derivation of the formula elsewhere in the literature/

re 34:

What do you mean by sensitivity? It’s infrared absorption of a tube filled with co2 for various tube lengths.

This is used as input for an atmospheric model, from which forcing can be calculated per CO2 doubling.

The logarithmic relationship was probably used because the Beer-Lambert law is used in radiation shielding. It is the same problem but the photon energies are much higher when you’re trying to stop gamma-rays.

The main difference is that radiation shielding is primarily concerned with solids, not gases. Gases have absorption characteristics that are functions of temperature (Doppler broadening) and pressure (pressure broadening). I looked at using a couple of MathCad files that I had developed for radiation effects but the characteristics of gases (particularly mixed gases that have very different absorption spectra) make this a very complicated problem.

Apparently, the high energy gamma ray spectra can be calculated for almost any substance, but I have not found anything useful for the photon energies encountered for the sun’s spectrum. I suspect that it is something that needs to be measured.

The NIST website that covers radiation shielding is

hereIt may, or may not be useful for the physicists online here. Maybe someone really skilled at navigating sites can find what I couldn’t.

Steve: I think vind was asking in #28 if that IPCC paragraph was wrong, not that you’d

said it. Anyway, good luck with finding Shi 92 and maybe getting this out of the “derivational dead end”.

Hans: Isn’t that more so “Forcing can be calculated per doubling of CO2 as regards to IR absorption in a tube of various lengths filled with CO2, for use in models.” :)

Mark T: Given all the other variables at play here, getting a sensibly physical good approximation is probably the golden standard at this time. If only all the variables had such.

I was doing some work, on the side, in academia, early in my career. This was during the early to mid 1980s. I was specifically doing work in the realm of geophysical math research. The inklings which I got of “mainstream thought” regarding climate change at the time, was “runaway greenhouse” or at least, “very energetic / hot greenhouse.” Clearly, the view was one of strong positive feedback playing a role. The clear candidate for the main causal was H20 in all forms increasing in the atmosphere, and being viewed in all forms as a positive feedback on multidecadal and greater time frames. This inspired fiction such as Dakota James “Greenhouse” as well as science such as that promoted by Hansen, at the time.

RE: #6 – As those who deal with geophysical prospecting methods know, we also use the concept of acoustic impedance to understand reflecting seismological outputs (and thereby ascertain subsurface acoustic impedance contrasts and hence, an inkling of structures and tectonic features). Anything having to do with energy flows can rightfully allude to impedance.

Sorry, Hans, I was confused by it. I meant what the amount of increase was from CO2 doubling according to the graphic, but I think it’s somewhat irrelevant, now.

Oh, one other thing. It looks like, on your second graphic, some spectral lines have more than 100% absorption. That can’t be right, can it?

I don’t disagree. I was merely putting a limit on the function, i.e. it cannot be a line because that would result in the physical impossibility of trapping more power than is available, and it can’t be asymptotic unless the asymptote is 0.

Exactly.

Mark

Mark T is giving the logical argument I speculated that the IPCC must have used earlier. Well, okay, good working model. But how do you know that it is in fact a simple function? Rather than being asymptotic, might it just hit an upper limit and afterwards do nothing? I know that a lot of people dispute that it’s physically possible, and suspect they are right. So is it derivable that it is asymptotic, or logarithmic? Perhaps it is in fact linear up to a certain point, after which it suddenly begins to slow its growth. How do we know that it follows a slowing, smooth curve? How do we know these things? More than likely the log/asymptotic relationship would emerge from the actual calculations, but they are to complicated, arent they, to be solved?

re: log realation. The following link gives an opening derivation from light absorbance. There are more complete lectures out there from astronomy profs.

http://plus.maths.org/issue13/features/garbett/index.html

Climate science begins with signal attenuation and uses Kirchoff’s black body equations and proceeds to slab integrations through the depth of the atmosphere which is attempted by various and sundry means.

Mark T: I was agreeing with you (paraphrasing sort of) and making the observation that good enough might be the best we can hope for. :) Just saying I wish we could get a physically sensibled approximation that’s “good enough” for every variable, maybe we could do better than guess if in the system CO2 (or any other substance) gives us .5C or 5C per doubling, or if we can get a better range than 2.75 +/- 2.25 (or whatever).

Long live limits of a function at infinity!

I understand and agree. I think I mention somewhere that the log is really an approximation of what we see. It may well decay to “doing nothing” (which really means “to an asymptote”) at some point when CO2 concentrations are high enough.

^Andrew: “Rather than being asymptotic, might it just hit an upper limit and afterwards do nothing?”

Well, that’s really what an asymptote is, no? :) You’re probably right with what you say as well as the implications of your questions, IMO, and the “smoothness” of the curve would certainly be impacted by other influences (perhaps an oscillation about the underlying function), though there should still exist some general relationship which is on average pretty smooth. Nature and discontinuities don’t get along well until you get to the quantum level.

The problem is that even if everyone on the planet agrees there is some logarithmic or logarithmic-like function, that doesn’t necessarily give you a real clue what the actual scaling factors are.

Nice link, Gary. Some simple differential equation work in there as well.

Mark

Maybe I should clarify, Mark, that when I think of asymptotic I mean the mathematically precise meaning, that there is an upper limit which is approached, but not reached. What I was describing was similar, but I literally meant it actually

reachesthe limit, then flattens out. But your probably right about the smoothness. Nature is pretty continuous.Mark T, please do not post any more right now on your proposed derivation of a log formula. You’re hijacking the thread. What we’re doing right now is seeing what climate scientists say about the log formula. Maybe they should have done it your way, maybe not. Maybe you can do something that they can’t, but hold your horses.

#33. Hans, what were the assumptions and atmospheric model used in this article?

Could be, Andrew… I get your distinction.

Sorry Steve. I was just working from simple physical principals. I would venture that Gary’s link is where the original idea came from (at least, the math/physics of the explanation), and my contribution was simply to put some logic on why it is likely logarithmic or similar. That link is very informative, btw, and an easy read. There’s nothing really unique in what I proposed, at least not anything that climate science cannot do (or has not done). I suppose the goal is to find out which rabbit hole they actually went down… :)

Btw, it is an inherent problem in nearly all signal processing problems to establish what is known as the Cramer-Rao bound, which puts a limit on your expected performance for a given processing method. I’ll be needing to delve into that shortly for the algorithm I’m proposing for my school work. It is often an understandably nasty proposition which I am dreading – coffee and insomnia will be the norm for a few months. :)

Mark

Steve, Vincent Gray wrote an interesting analysis of the forcing equations for John Daly’s “Still Waiting for Greenhouse”, posted here: http://www.john-daly.com/bull-121.htm. He cites Wigley’s 1987 Climate Monitor paper as though he has a copy of it. Maybe he does, and could send you a copy. His email address is at the bottom of the essay.

#8, Ramaswamy, LLGHG

I’m supposing this is Lower-level GHG’s??

Not a single hit for the expanded version on the Web…

Cheers — Pete the Simple

The strange thing about

6..3.5 Simplified Equationsthat gets me is why should CO2, CH4 and N2O have different equational forms? And what would be the physical basis for raising something to the 0.75 or 1.52 power? The whole thing looksad hocas if someone was insistently forcing a linear regression fit.28,37,41: CO2 absorption “wings”

Yes, that’s what I thought, too. Don’t bite the newbies!

Vind, welcome to CA! The boss gets a little grouchy with us slowpokes sometimes, but he’s worth putting up with, for all the interesting stuff he churns out ;-)

I assume this is your employer? http://www.climacure.com/content/view/23/37/lang,en/

Cheers — Pete Tillman

Gunnar: “average temperature has meaning from an information point of view, but not as a physical attribute of reality that drives physical processes.” My point exactly about the “global mean temperature anomaly” not being “meaningful”.

As far as the wing discussion, some of it is here I believe:

Frontiers of Climate ModelingThe radiatve forcing due to clouds and water vapor V. Ramanathan and Anand Inamdar

snip

I moved Mark’s discussion to Unthreaded since people have continued discussing this after my request not to. You can discuss dividing by 4 on Unthreaded.

I understand that, and the actual per unit area irradiance will vary significantly over the whole surface (which is half of a sphere anyway).

Mark

I tried to get out Steve, but they kept dragging me back in! :)

Mark

You may need to go farther back. After all Manabe was the first to derive a CO2 temperature signal in a GCM in the 1970s. I believe Sir JohnHoughton in 1997 cited:

Manabe, S., and Wetherald, R. T.: 1975 The effects of doubling the CO2 concentration on the climate of a general circulation model, J. Atm. Sci. 32, 315.

as the most complete treatment. There may be more useful background in:

Manabe, S., and Wetherald, R. T.: 1967, Thermal equilibrium of the atmosphere with a given distribution of relative humidity, J. Atm. Sci. 24, 241259.

I apologize in advance if these aren’t what you are looking for.

Peter – #49 – in LLGHG the “LL” refers to “long-lived” – gases that stay in the atmosphere on average for years at a time. I.e. not water.

On the discussion of the meaning of “average temperature” (Urbinto #52, and other discussions here) a perhaps more meaningful number would be the total internal energy of the Earth; obviously that’s dominated by the hot core, but the only way energy escapes is through the surface, so the *changes* in total energy should be measurable. The only way change happens is through radiative imbalance (since Earth has no other substantial connections to the rest of the universe); increasing mean surface temperature is a sign that the energy is going up, not down. But so would melting glaciers with no mean surface temp increase. I think it would be useful to promote this as a much more meaningful alternative measure…

Arthur: Energy balance as an indictator is far better (or probably both, to varify/validate each other for a “sanity check”. However, I’ll also say that if you send warmer wind over, and more rain and particulates onto ice, it will melt more regardless of the temperature. If you have ice where there’s no wind, no rain or snow, and no substances covering it, even an increase in sunlight won’ have much of an effect upon it.

Just sayin’

Isn’t the logarithmic relationship between absorption and concentration just Beer’s Law?

That’s correct, that’s also why the section is called “6.3.5 Simplified expressions[sic]” The equations are different because the spectral lines are different.

Lubos just posted a thread on this at his blog:

http://motls.blogspot.com/

Keith, I mentioned Beer’s law in a comment on Unthreaded #27, and showed (in perhaps excessive detail) that it implies a exponential, not a logarithmic, relationship. Others argued that Beer’s law has nothing to do with the CO2 response curve, but I tend to think the analogy is close enough that it probably gives a good first-order approximation.

Phil asserted with considerable certainty, but no evidence, that spectral broadening resulted in a logarithmic relationship:

I asked for some support for the claim, but none was offered.

Since Steve hasn’t posted the derivation yet, I’m going to give it out as an exercise. It is on clear if the windowing effect is the reason behind the logarithmic relationship but it is easy to see how the area of the absorption peeks grows when the optical density increases.

Beers law states that the amount of light absorbed is given by Io*(1-exp(-tau x)) where

tau is the absorptivity

x is the amount of greenhouse gas that the light travels through.

tau*x is the optical desnity

The width of the absorption peek can be the distance between the two frequency points where the optical density equal to some predefined value. For instance someone might choose the two points where tau*x=0.01.

If tau has a Gaussian or Cauchy distribution with respect to frequency it is easy to derive a functional form for the width of the absorption peak. I leave it as an exercise to show that for a peak that if lambda is Gaussian distributed then for large optical deaths the peak varries with sqrt(log(CO2 concentration) and if lambda is Cauchy distributed then the absorption peak then the width varies with sqrt(CO2 concentration).

As a side note, for small optical depths we can find an analytic expression for the area under the give by:

-Plugging the value -lambda*x into the Tayler series expansion of and exponential

-Then over frequency from 0 to infinity.

Also note that, the expression for the width of the absorption peaks is quite simple. If you approximate the area as K*width then it should be easy to get the total power emitted from the earth by integrating k1*T^4*(1-k(width)) backwards toward earth from the tropapause.

#62, I agree that is a perfectly plausible explanation of the logarithmic relationship. I’m curious though that the exact function by which the tropopause rise. Will it rise in such a way to preserver the optical depth between the tropopause and space?

If it rises in that fashion then the explanation seems more plausible but if it doesn’t then more UV will radiation will penetrate into the tropopause and more CO2 will escape into space above the tropopause. Assuming the tropopause occurs at a constant optical depth then we should be able to calculate the hight of the tropopause based on the temperature, lapse rate and desnity of air on the ground.

My thought is that raising the CO2 concentration will mean that a greater percentage of infra red light will be radiated from the stratasphere wheree the temperature is actually warmer then the tropopause. This could counteract to some extent the raising of the tropopause.

This makes one wonder, is the hight of the tropopause is determined more by the optical depth to the tropopause as seen from space with respect to UV radiation, or the optical depth as seen from space with respect to IR radiation?

I may be missing something, but to the extent I follow Lubos’s argument, I don’t buy it. Maybe it makes sense for an atmosphere of pure CO2, but for our atmosphere, where CO2 is a minor constituent, doubling the concentration of CO2 has no meaningful effect on the height of the tropopause or the lapse rate. He seems (for whatever reason) to be using the term

tropopausein a somewhat quirky manner, but the lapse rate still depends on all the gases in the atmosphere, so any argument that relies on the lapse rate changing significantly if the concentration of CO2 doubles seems dubious.While the enthusiasm is high, a lot of these posts are not relevant to the header and a lot display misunderstanding of the physics. I have forgetten more than I learned, but I did do 2 years of CO2 laser research and a dozen years of lab spectroscopy. I’m pulling out of the discussion. Shining a near parallel beam of monochromatic light down a short tube of pure gas under controlled temperature and pressure does not yield the same results as atmospheric conditions impose. It is a way to start the study, but believe me, it is a small first step and there are plenty of people out here who can describe the whole atmospheric story better than many of the above posts. Including mine. Whether they are right remains to be seen.

Re #66, #65 & #64

Indeed and the lapse rate will in particular depend on humidity. I don’t buy Motl’s explanation since you can get the log relationship to occur in a cuvette so clearly it doesn’t depend on the vertical structure of the atmosphere. As well as John’s derivation you can get similar results by numerically integrating over a single line (either Gaussian or Lorentzian).

Steve

I think I know what your problem is here. You are trying to resolve all of this from the top down. A much more logical, and in the end time efficient method, would be to do it from the bottom up. For every answer you get at present, you only get another question as you drill down. Do it from the bottom up, as you do in a University course, you get a much more logical learning path, and a much more efficient one.

Re: #69

I agree. Textbooks first, literature later. Unless you stick with freely available papers, a couple of false leads on papers and you’ve paid for a decent textbook.

The logarithmic relationship between CO2 and temperature is probably roughly correct for the present atmospheric levels. However it cannot be correct for very low levels. In the limit when CO2 levels approach zero the logarithmic formula gives the nonsensical value of ‘-infinity’. So, it needs to be questioned on that basis alone.

The simple answer is that there is a linear region at low concentrations. Then, as concentrations increase, the relationship looks more logarithmic. But what defines the two regions? The form of Lubos Motl’s reducing exponential equation is good one too. However the constants “1.5″ and “200′ are open to some debate. If these are changed to “5″ and “300″ the graph looks indistinguishable, over the 280 to 560 ppmv CO2 range, from one the IPCC wouldn’t disaprove of. So it doesn’t prove anything either way, unfortunately.

http://farm4.static.flickr.com/3380/3582363543_ee6a6872d4_o.png

Just following on from the previous post, I recently heard an analogy of paint on a window used to justify the logarithmic relationship between CO2 levels and temperature.

The argument was that the first coat of paint absorbed half the light. Then the window was painted again and the second coat absorbed half of what was transmitted through the first coat. And so on with each increasing layer of paint.

OK is this a correct explanation? Let’s put in some numbers:

Incident light = 32 units

Light

transmittedthrough each successive layer = 32,16,8,4,2……Light

absorbedas layers are increased = 0,16,24,28…..It is worth noting that Arrhenius talked, at least in the references I have seen, about geometric and arithmetic progressions rather logarithmic relationships as such.

Sure, a geometric progression can be regarded as a logarithmic series, but let’s not be too loose in our thinking. It should be noted that the formulae to calculate light transmitted and absorbed are based on exponentials rather than logarithms:

Light out = Lightin * exp (-N/k) where N is the number of layers and K is a constant

Light absorbed = Lightin – Lightout = Lightin * (1-exp(-N/k))

Steve:

Don’t know if you are still following this thread. I have developed a graphical relation between CO2 and “forcing”, based on the work done by Leckner and Hottell. The graphs, etc., developed, initially by Hottell and later refined by Leckner, are used in engineering (for instance, blast furnace design) to determine the impact of CO2 on radiant heat transfer. My graph closely approximates the IPCC graphs. In particular, the pure logarithmic graph (delta F = ln (C / C nought). I would like some serious critical review of this work. I can find many things wrong with my work, but none that make it invalid. For instance, my method uses a large number of calculations that would probably be better done either by direct calculus or limits.

The critical thing is: If you use my method, the impact of CO2 levels off much more radically than the IPCC trends. Their graphs are very close, and hence, if mine is correct (a huge arrogance on my part), then it would fit within the error of the IPCC graphs at current and historic levels, but would seriously diverge at higher levels.

I’d be pleased to e-mail you my report and supporting spreadsheet, including appropriate references. I’m still working on this, so it is a tad rough (I have a real job too!).

Best Regards

John Eggert P.Eng. (mining, minerals processing option, 1990 Queen’s University, not that that is relevant)

Dear Sir;

I think the key point of your work is that

‘the IPCC and everyone else plots CO2 on a linear scale’

despite their repeated assertions that temperature will change by ‘x’ degrees for every doubling of CO2.

I even got Gavin at RC to agree, that CO2 should be plotted on a log scale when on the same graph as observed temperatures. (before he started deleting my questions).

At this point, I think the question is about how to regraph the ‘data’ they have, in a more realistic way.

Talk is cheep.

I appreciate the science you are offering to bring to the table.

But the debate seems to be all about what the graphs look like, and how did they ‘fool themselves’ with their own graphs.

This, from a guy who heard Feynman’s Cargo Cult Science lecture in person in 1974.

TL

The question is not “what is the correct relationship?”, or. worse, “where do you think it comes from?”. The question is “from where does IPCC get its assumptions”? Follow the trail until it runs dry. This is library work, not bench science. Where does the trail run dry? At that point you need to write to the chapter author.

I wonder if anyone is still reading the comments here.. because the answer is simple.

I’ve been doing my own research of old papers for http://scienceofdoom.com and the “source” is clear.

IPCC TAR cites “New estimates of radiative forcing due to well mixes greenhouse gases” Myhre et al 1998.

Their paper, published in GRL, explains clearly what they do. At least, they follow on from a couple of decades of work with “radiative-convective models”, so a few things might be implied rather than explicit.

They are running the RTE (radiative transfer equations) using both LBL (line by line) and well-proven band models. They use the results from a few standardized profiles because the results from one “average” profile have been demonstrated to not match the average of results from many profiles.

And the “simplified expression” – the holy grail – comes from running these averaged 1-d simulations for different CO2 values and plotting the results.

Consequently, they find the new coefficient for the ln(C/Co) formula which best fits.

Delta F = 5.35 ln(C/Co)

Hi SoD.

Just ran into this post tonight. I was wondering if Steve had ever come to a ‘conclusion’ regarding the plausibility of logarithmic forcing.

So, yeah, I appreciate the update. Even by a stranger, passing in the night.

Ron Broberg – you can see the post at http://scienceofdoom.com/2010/02/19/co2-an-insignificant-trace-gas-part-seven-the-boring-numbers/

- I don’t go into the IPCC expression a lot more but there is an extract from the paper Myhre and some background. Feel free to ask a question over at that post..

F = 5.35 ln(C/C0)

What units is this in? K, F, or C?

Radiative forcing, F, is in W/m^2.

The factor: ln(C/C0) is obviously dimensionless, but the constant (5.35) is in units of W/m^2.

It is the net change in rate of energy transfer per unit area due to an instantaneous change in CO2 concentrations.

This equation is not an analytical solution to the radiative transfer equations. This equation is an empirical fit to change in CO2 concentrations over a certain range – with the results for this empirical fit derived from the radiative transfer equations – see, for example, Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.

Steve: I know that it is an “empirical fit”. That was one of the purposes of this post.

Thanks for noticing and caring, SoD.

Thanks!

http://www.sciencemag.org/content/173/3992/138.abstract

Science 9 July 1971:

Vol. 173 no. 3992 pp. 138-141

DOI: 10.1126/science.173.3992.138

Atmospheric Carbon Dioxide and Aerosols: Effects of Large Increases on Global Climate

1. S. I. Rasool1,

2. S. H. Schneider1

+ Author Affiliations

1. 1Institute for Space Studies, Goddard Space Flight Center, National Aeronautics and Space Administration, New York 10025

Abstract

Effects on the global temperature of large increases in carbon dioxide and aerosol densities in the atmosphere of Earth have been computed. It is found that, although the addition of carbon dioxide in the atmosphere does increase the surface temperature, the rate of temperature increase diminishes with increasing carbon dioxide in the atmosphere. For aerosols, however, the net effect of increase in density is to reduce the surface temperature of Earth. Because of the exponential dependence of the backscattering, the rate of temperature decrease is augmented with increasing aerosol content. An increase by only a factor of 4 in global aerosol background concentration may be sufficient to reduce the surface temperature by as much as 3.5 ° K. If sustained over a period of several years, such a temperature decrease over the whole globe is believed to be sufficient to trigger an ice age.

James Hansen, as well as Steven Schneider, was involved in the above research,

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