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Climate Insensitivity

Filed under: — group @ 16 September 2007

Guest post by Tamino

In a paper, “Heat Capacity, Time Constant, and Sensitivity of Earth’s Climate System” soon to be published in the Journal of Geophysical Research (and discussed briefly at RealClimate a few weeks back), Stephen Schwartz of Brookhaven National Laboratory estimates climate sensitivity using observed 20th-century data on ocean heat content and global surface temperature. He arrives at the estimate 1.1±0.5 deg C for a doubling of CO2 concentration (0.3 deg C for every 1 W/m^2 of climate forcing), a figure far lower than most estimates, which fall generally in the range 2 to 4.5 deg C for doubling CO2. This paper has been heralded by global-warming denialists as the death-knell for global warming theory (as most such papers are).


Schwartz’s results would imply two important things. First, that the impact of adding greenhouse gases to the atmosphere will be much smaller than most estimates; second, that almost all of the warming due to the greenhouse gases we’ve put in the atmosphere so far has already been felt, so there’s almost no warming “in the pipeline” due to greenhouse gases already in the air. Both ideas contradict the consensus view of climate scientists, and both ideas give global-warming skeptics a warm fuzzy feeling (but not too warm).

Despite the celebratory reaction from the denialist blogosphere (and U.S. Senator James Inhofe), this is not a “denialist” paper. Schwartz is a highly respected researcher (deservedly so) in atmospheric physics, mainly working on aerosols. He doesn’t pretend to smite global-warming theories with a single blow, he simply explores one way to estimate climate sensitivity and reports his results. He seems quite aware of many of the caveats inherent in his method, and invites further study, saying in the “conclusions” section:

Finally, as the present analysis rests on a simple single-compartment energy balance model, the question must inevitably arise whether the rather obdurate climate system might be amenable to determination of its key properties through empirical analysis based on such a simple model. In response to that question it might have to be said that it remains to be seen. In this context it is hoped that the present study might stimulate further work along these lines with more complex models.

What is Schwartz’s method? First, assume that the climate system can be effectively modeled as a zero-dimensional energy balance model. This would mean that there would be a single effective heat capacity for the climate system, and a single effective time constant for the system as well. Climate sensitivity will then be

S=τ/C

where S is the climate sensitivity, τ is the time constant, and C is the heat capacity. Simple!

To estimate those parameters, Schwartz uses observed climate data. He assumes that the time series of global temperature can effectively be modeled as a linear trend, plus a one-dimensional, first-order “autoregressive” or “Markov” or simply “AR(1)” process [an AR(1) process is a random process with some 'memory' of its previous value; subsequent values y_t are statistically dependent on the immediately preceding value y_(t-1) through an equation of the form y_t = ρ y_(t-1) + ε, where ρ is typically required to be between 0 and 1, and ε is a series of random values conforming to a normal distribution. The AR(1) model is a special case of a more general class of linear time series models known as "Autoregressive moving average" models].

In such as case, the autocorrelation of the global temperature time series (its correlation with a time-delayed copy of itself) can be analyzed to determine the time constant τ. He further assumes that ocean heat content represents the bulk of the heat absorbed by the planet due to climate forces, and that its changes are roughly proportional to the observed surface temperature change; the constant of proportionality gives the heat capacity. The conclusion is that the time constant of the planet is 5±1 years and its heat capacity is 16.7±7 W • yr / (dec C • m^2), so climate sensitivity is 5/16.7 = 0.3 deg C/(W/m^2).

One of the biggest problems with this method is that it assumes that the climate system has only one “time scale,” and that time scale determines its long-term, equilibrium response to changes in climate forcing. But the global heat budget has many components, which respond faster or slower to heat input: the atmosphere, land, upper ocean, deep ocean, and cryosphere all act with their own time scales. The atmosphere responds quickly, the land not quite so fast, the deep ocean and cryosphere very slowly. In fact, it’s because it takes so long for heat to penetrate deep into the ocean that most climate scientists believe we have not yet experienced all the warming due from the greenhouse gases we’ve already emitted [Hansen et al. 2005].

Schwartz’s analysis depends on assuming that the global temperature time series has a single time scale, and modelling it as a linear trend plus an AR(1) process. There’s a straightforward way to test at least the possibility that it obeys the stated assumption. If the linearly detrended temperature data really do behave like an AR(1) process, then the autocorrelation at lag Δt which we can call rt), will be related to the time constant τ by the simple formula

rt)= exp{-Δt/τ}.

In that case,

τ = – Δt / ln(r),

for any and all lags Δt. This is the formula used to estimate the time constant τ.

And what, you wonder, are the estimated values of the time constant from the temperature time series? Using annual average temperature anomaly from NASA GISS (one of the data sets Schwartz uses), after detrending by removing a linear fit, Schwartz arrives at his Figure 5g:

Using the monthly rather than annual averages gives Schwartz’s Figure 7:

If the temperature follows the assumed model, then the estimated time constant should be the same for all lags, until the lag gets large enough that the probable error invalidates the result. But it’s clear from these figures that this is not the case. Rather, the estimated τ increases with increasing lag. Schwartz himself says:

As seen in Figure 5g, values of τ were found to increase with increasing lag time from about 2 years at lag time Δt = 1 yr, reaching an asymptotic value of about 5 years by about lag time Δt= 8 yr. As similar results were obtained with various subsets of the data (first and second halves of the time series; data for Northern and Southern Hemispheres, Figure 6) and for the de-seasonalized monthly data, Figure 7, this estimate of the time constant would appear to be robust.

If the time series of global temperature really did follow an AR(1) process, what would the graphs look like? We ran 5 simulations of an AR(1) process with a 5-year time scale, generating monthly data for 125 years, then estimated the time scale using Schwartz’s method. We also applied the method to GISTEMP monthly data (the results are slightly different from Schwartz’s because we used data through July 2007). Here’s how they compare:

This makes it abundantly clear that if temperature did follow the stated assumption, it would not give the results reported by Schwartz. The conclusion is inescapable, that global temperature cannot be adequately modeled as a linear trend plus AR(1) process.

You probably also noticed that for the simulated AR(1) process, the estimated time scale is consistently less than the true value (which for the simulations, is known to be exactly 5 years, or 60 months), and that the estimate decreases as lag increases. This is because the usual estimate of autocorrelation coefficients is a biased estimate. The word “bias” is used in its statistical sense, that the expected result of the calculation is not the true value. As the lag gets higher, the impact of the bias increases and the estimated time scale decreases. When the time series is long and the time scale is short, the bias is negligible, but when the time scale is any significant fraction of the length of the time series, the bias can be quite large. In fact, both simulations and theoretical calculations demonstrate that for 125 years of a genuine AR(1) process, if the time scale were 30 years (not an unrealistic value for global climate), we would expect the estimate from autocorrelation values to be less than half the true value.

Earlier in the paper, the AR(1) assumption is justified by regressing each year’s average temperature anomaly against the previous year’s and studying the residuals from that fit:

Satisfaction of the assumption of a first-order Markov process was assessed by examination of the residuals of the lag-1 regression, which were found to exhibit no further significant autocorrelation.

The result for this test is graphed in his Figure 5f:

Alas, it seems this test was applied only to the annual averages. For that data, there are only 125 data points, so the uncertainty in an autocorrelation estimate is as big as ±0.2, much too large to reveal whatever autocorrelation might remain. Applying the test to the monthly data, the larger number of data points would have given this more precise result:

The very first value, at lag 1 month, is way outside the limit of “no further significant autocorrelation,” and in fact most of the low-lag values are outside the 95% confidence limits (indicated by the dashed lines).

In short, the global temperature time series clearly does not follow the model adopted in Schwartz’s analysis. It’s further clear that even if it did, the method is unable to diagnose the right time scale. Add to that the fact that assuming a single time scale for the global climate system contradicts what we know about the response time of the different components of the earth, and it adds up to only one conclusion: Schwartz’s estimate of climate sensitivity is unreliable. We see no evidence from this analysis to indicate that climate sensitivity is any different from the best estimates of sensible research, somewhere within the range of 2 to 4.5 deg C for a doubling of CO2.

A response to the paper, raising these (and other) issues, has already been submitted to the Journal of Geophysical Research, and another response (by a team in Switzerland) is in the works. It’s important to note that this is the way science works. An idea is proposed and explored, the results are reported, the methodology is probed and critiqued by others, and their results are reported; in the process, we hope to learn more about how the world really works.

That Schwartz’s result is heralded as the death-knell of global warming by denialist blogs and Sen. Inhofe, even before it has been officially published (let alone before the scientific community has responded) says more about the denialist movement than about the sensitivity of earth’s climate system. But, that’s how politics works.


370 Responses to “Climate Insensitivity”

  1. 201
    Timothy Chase says:

    Vernon (#195) wrote:

    I saw this and had to ask, which Tamino is right?

    So which is it?

    Someone who is capable self-transcendance and rebirth: a living mind with the power of self-correction. Not much use in trying to explain this to you though, judging from your most recent remarks – and long well-established pattern of behavior.

  2. 202

    [[We see that in fact, all decades are in accord with the modern-era rate; every one of them gives an error range for the rate that includes the modern-era value. From this I conclude that there is no statistically significant evidence that temperature from 1975 to the present deviates from a linear trend plus red noise.]]

    Why don’t you try graphing all the points from 1880 onwards?

  3. 203
    Rod B says:

    Ray says, “Rod B., Let’s make it clear:
    5.35*ln(C/Co)=ln[(C/Co)^5.35]. That is it is the ratio of the concentrations that is raised to the power, not the log thereof.”

    Yes, I think that’s what I said; maybe my parens were not clear

  4. 204
    Timothy Chase says:

    Rod B (#193) wrote:

    Timothy (#177), this would imply (actually you state) a linear relationship between CO2 concentration and temperature. Are you sure this is correct??

    Rod, I believe you may have misread me.

    I stated that given the large distance from absolute zero (nearly 300 Kelvin), a linear increase in the forcing implies a near linear increase in the temperature, at least over the range of a few degrees. One sentence later I state that the linear and near linear functions are with respect to the log of the ratio of the concentrations of carbon dioxide.

    Now what does a curve look like if it is gradual with respect to the scale upon which you zoom in on it? A straight line. We aren’t speaking of fractals, at least not at this point. So while the linear function of the log of the ratios is an approximation in the case of temperature, it is a fairly good approximation over any range of temperatures we might be dealing with. (See #177 which you were responding to.)

    But even in the case of forcing, the relationship starts to breakdown at higher pressures where the bands begin to overlap. Not really something we have to worry about here, though, not for a few doublings at least. The opacity of CO2 concentration is nearly logarithmic since (as the center of any given broadening line of absorption becomes saturated towards a widening center) absorption shifts to the wings where emissivity falls nearly as an exponential of the negative of the distance from saturation.

    But as I have said, the real-wold consequences will be quite serious after the first couple of doublings of CO2, anthropogenic emissions – plus the feedback from the carbon cycle. At that point we would be speaking of around 1000 ppm. That would be a little over two degrees due to the direct effects of CO2, but about six degrees Celsius once one takes into account all the feedbacks and the now well-established ~3 degrees per doubling of CO2 for the past half-million years – for which the positions of the continents have been roughly constant. (See #174.)

    *

    Rod B (#191) wrote:

    The 235 watts leaving will always be the same whether CO2 doubles, halves, or stays the same. If it jumps to 239 watts, the earth’s climate system will cool, because the 235 watts incoming will stay the same I think this affects the rest of your calculations, though there may be some semblance of accuracy or insight in them. In any event, I haven’t got to the forcing going to temperature yet; I’m still pursuing concentration going to forcing.

    The 235 watts per square meter rate at which energy (thermal radiation) will be lost at the new equilibrium will be where it is escaping the atmosphere – at the top. The 235 watts per square meter rate at which this energy enters the climate system – at the bottom (plus what little absorption occurs in the atmosphere such as the absorption of solar radiation by ozone, and where the per square meter is measured relative to the average atmospheric column which is a square meter at the base). Given the increasing opacity of the atmosphere and consequently the increase in the downwelling radiation from the atmosphere, the surface must radiate thermal energy at a higher rate for the rate at which thermal energy leaving the climate system to balance the rate at which it leaves the system.

    Until this new balance is achieved, the climate system will appear dimmer as viewed outside it than it was prior to the doubling of CO2 concentration. Once this balance is achieved the climate system will cease to become warmer. Nevertheless, the surface will be warmer at this new equilibrium.

    For that surface to cool off back to the temperature which it had at the old equilibrium, it would have to radiate thermal energy at an even higher rate – as some the thermal energy which it radiation will be absorbed by the more opaque atmosphere, and some of that will be radiated back to the surface. (Half of the radiation – if one does not take into account other effects, such as moist air convection.)

    Once the rate at which thermal energy is leaving the climate system equals the rate at which thermal energy is entering the climate system, it will already be at the new higher equilibrium – where by “higher” we are refering to the higher temperature at the surface. As such the rate at which the surface emits thermal radiation won’t climb any higher. As such, the surface will never radiate thermal energy at a rate that would be required for it to cool off to the original temperature.

    *

    Anyway, with the problem you posed in 191 it might help to look at Gavin’s Learning from a Simple Model as well as my spreadsheet analysis of the greenhouse effect for the simple model which I believe complements Gavin’s analysis.

  5. 205
    Eli Rabett says:

    WRT 183. There is CO2 and there are all the greenhouse gases. When we count in methane, CFCs, nitrogen oxides and more, we are a considerable way to 2x. For just CO2 alone we have gone from about 280 ppm in the 19th century to about 380 ppm today. My forgettery says that with everything put together we are effectively at ~450 ppm (subject to correction on that one, but I don’t think a very large one).

  6. 206
    pete best says:

    Lots of other articles are springing up concerning the increased amounts of water vapour in the atmosphere which is a prediction of AGW is it not ?

  7. 207
    Rod B says:

    re 204 (Timothy): I see your point on the linearity, though it is almost linear for almost just two doublings, then falls off the linear chart noticeably. Though this is not my real (initial) point, which was that saying that forcing is a logarithmic function of CO2 concentration is misleading, though maybe for simplifying things rather than being nefarious. There is a major difference between being log related and being 5 to 6 times log related. The forcing is 5 to 6 times greater with the latter and I don’t think that should be sugar-coated.

    I have yet to follow up on some earlier (and maybe later) posts, so I’m still in the analytical stage. Nor have I gotten to the temperature as a function of forcing question yet, and don’t know if I have a big (or any) problem here. Though I do raise my eyebrows with the “well established” mantra.

    My 235 watts in = 235 watts out was referring to steady state. I understand the transient process (and thanks for your explanation), (though I might have disagreements with the numbers), which says 1) greenhouse gases go up; 2) more surface radiation is absorbed and partially radiated back to the earth (though how that happens is still an unanswered question of mine in this or maybe an earlier thread); 3) since the top of the atmosphere is temporarily fed less infrared radiation, its 235 watts outgoing is temporarily reduced (and dims a bit to the outside observer with infrared glasses, as you say); 4) outgoing radiation is temporarily less than the absorbed incoming, and the earth system must start to heat up; 5) as the system heats up the surface radiation increases, eventually feeds more radiation to the top of the atmosphere, which can now get back to a stable 235 watts output — though now with the earth system a bit warmer.

    Yell if the above is stated incorrectly.

  8. 208
    garhane says:

    I really do not think it helps that this fellow is polite or tosses out several caveats at the end of his paper. One wants to tell him to stop pissing about and start rowing.

    Of course science this and science that. Sure. But we do get to a point that it becomes necessary to say no private wars are permitted.

  9. 209
    David Warkentin says:

    re 207 (and elsewhere) -
    Rod B, I think you may have lost sight of the fact that relative forcing is expressed in physical units, so the (5.35) in the formula is really (5.35 W/m^2). As such, it doesn’t make physical sense to try to move it inside the logarithm (there it would be an exponent, but with units attached, which doesn’t work.)

    Another way of looking at it – if we chose the units to be kilowatts per square meter, we’d have RF = 0.00535 ln(C/C0), which makes it look like the CO2 ratio should have a tiny exponent, but really describes exactly the same forcing effect.

  10. 210
    Rod B says:

    David (209), you make a good point, the units should match up. None-the-less, mathematically, of course, N times the log of X is equivalent to the log of X to the Nth power. So as I stated to Timothy et al saying “forcing is a function of the log of concentration ratio” is, in part, misleading. Numerically, forcing is a function of the log of: concentration ratio raised to some power.

  11. 211
    Hank Roberts says:

    Rod, here’s a course dealing with your questions, just for an idea of the prerequisites required. It may be useful perspective:
    http://www.uio.no/studier/emner/matnat/fys/FYS9630/index.xml

  12. 212
    Timothy Chase says:

    Rod B (#207) wrote:

    re 204 (Timothy): I see your point on the linearity, though it is almost linear for almost just two doublings, then falls off the linear chart noticeably. Though this is not my real (initial) point, which was that saying that forcing is a logarithmic function of CO2 concentration is misleading, though maybe for simplifying things rather than being nefarious. There is a major difference between being log related and being 5 to 6 times log related. The forcing is 5 to 6 times greater with the latter and I don’t think that should be sugar-coated.

    No, I am afraid you are still missing the point.

    Forcing is a linear function of the log. This is what is stated and the statement is good approximation. And while the formula relating concentration to temperature is a rougher approximation, for the accuracy with which concentration is related to temperature, the formula works for at least five doublings (that is 2 X 2 X 2 X 2 X 2 or alternatively 32 X – plus), not 5 X or 6 X.

    *

    A little over 3 doublings (our CO2 emissions plus an increasing degree of CO2 feedback from the carbon cycle) would put us within the same range as the Permian-Triassic extinction (also known as the Great Dying) – 250 million years ago. This is when it appears that fungus ruled the earth.

    At that point the fact that the function relating CO2-concentration to temperature is only an approximation will probably be the least of our worries. As it is, a little more than one doubling would probably be more than enough to cause an economic crisis deeper than the Great Depression that would last for several decades – and no doubt this would have severe political ramifications.

    *

    However, it worth keeping in mind that ultimately the logarithmic function with which forcing is related to the concentration must ultimately break down due to bands in the spectra of carbon dioxide beginning to overlap. But this is not something we have to worry about under earth-like conditions. Also I suspect the non-LTE which becomes increasingly important at higher altitudes will cause some deviation. Then there is the distribution of atmospheric constituents, etc..

    But it would be silly to expect the climate system to act in as straightforward a fashion as dropping a rock. Then again even Newton’s gravitational law (or for that matter, classical mechanics) is just approximation which breaks down at high speeds, in strong gravitational fields and at quantum and cosmological scales. I wouldn’t consider the fact that these are approximations a “sugar coating” or recommend abandoning them we erecting a building.

    *

    Rod B (#207) wrote:

    My 235 watts in = 235 watts out was referring to steady state. I understand the transient process (and thanks for your explanation), (though I might have disagreements with the numbers), which says 1) greenhouse gases go up; 2) more surface radiation is absorbed and partially radiated back to the earth (though how that happens is still an unanswered question of mine in this or maybe an earlier thread); 3) since the top of the atmosphere is temporarily fed less infrared radiation, its 235 watts outgoing is temporarily reduced (and dims a bit to the outside observer with infrared glasses, as you say); 4) outgoing radiation is temporarily less than the absorbed incoming, and the earth system must start to heat up; 5) as the system heats up the surface radiation increases, eventually feeds more radiation to the top of the atmosphere, which can now get back to a stable 235 watts output — though now with the earth system a bit warmer.

    Seems accurate.

    Above you state:

    2) more surface radiation is absorbed and partially radiated back to the earth (though how that happens is still an unanswered question of mine in this or maybe an earlier thread); …

    As I said, at higher greenhouse gas concentrations the atmosphere becomes increasingly opaque to thermal radiation.

    The more opaque the atmosphere the more back radiation there will be where the atmosphere reradiates thermal radiation towards the surface. We went through the details involving spectra, absorption and reemission in the Part II: What Ångström didn’t know. As far as the thermal absorption and reradiation between the atmosphere raising the surface temperature is concerned, in 204 I recommended Gavin’s Learning from a Simple Model and my spreadsheet analysis of the greenhouse effect for the simple model that is intended to complement Gavin’s analysis.

  13. 213
    Timothy Chase says:

    Rod B (#210) wrote:

    David (209), you make a good point, the units should match up. None-the-less, mathematically, of course, N times the log of X is equivalent to the log of X to the Nth power. So as I stated to Timothy et al saying “forcing is a function of the log of concentration ratio” is, in part, misleading. Numerically, forcing is a function of the log of: concentration ratio raised to some power.

    If I state that five cars with four wheels each have 4 X 5 wheels total, would you consider this misleading and state that they have 5 X 4 wheels? In either case there is a total of 20 wheels.

    The same principle applies here:

    n X log(r) = log(rn)

  14. 214
    Ellis says:

    There seems to be alot of talk about the atmospheric system being out of equillibrium due to AGW. This seems nonsensical to me considering that since the Earth has had an atmosphere it has never been in equillibrium. Of course, I may be wrong, and if so I would appreciate someone explaining the state of equillibrium that exsisted prior to the industrial revolution.

  15. 215
    David B. Benson says:

    Ellis (214) — More precisely, the climate was nearly in equilibrium, most of the time. Nearly, not exactly. Exceptions occured due to large meteorite impacts, super-volcano eruptions, etc. The pulse of additional carbon anthropogenically added to the active carbon cycle is such an exception.

  16. 216
    Hank Roberts says:

    You may be puzzled because there are different meanings of the word “equilibrium” — for example “radiative equilibrium.” Try the AIP History (first link under Science at right) and the “Start Here” links (button at top of page) if you’re not clear on that sense.

  17. 217
    Ellis says:

    Yes, nearly in equillibrium, which, of course, is the long way of saying not in equillibrium. And, this has nothing to do with large catastrophic events, but has everything to do with having an atmosphere. The greenhouse effect exsisted long before man, and will continue to exsist long after our extinction. My only point is that people here and elsewhere continue to believe in a mythical equillibrium that exsisted before man screwed it up, and that is not science, but faith.

  18. 218
    Ellis says:

    Hank thank you for your response, however, if you could be a little more specific as to where to look. I appreciate that my question seems a bit simple, however, I really do not see anything about the different meanings of equillibrium in the links provided. I’ll admit that I am a little to lazy to go through at least six hours of reading on every tidbit of the theory of global warming, but am more than willing to read anything that can clear up my confusion on the subject of the different scientific meanings of equillibrium.

  19. 219
    Hank Roberts says:

    > My only point is

    Good.

    Bye.

  20. 220
    David B. Benson says:

    Ellis (217) — Nearly in equilibrium implies that changes are slow, moving toward the equilibrium, but then the forcings slowly change so the slimate system is always changing toward a new target.

    That is vastly different than being a long way from equilibrium. A long way from equilibrium means changes towards a new equilibrium are rapid. Large catastropic events suddenly cause the climate system to be far from equilibrium. The anthropogenic carbon slug added over the last 250 years (and mostly in the last 50 years or so) is such a large catastrophic event.

    And that is science, not faith.

    (I second Hank Roberts suggest that you read the AIP Discovery of Global Warming site.)

  21. 221
    Jim Eager says:

    Re 214 Ellis: “There seems to be alot of talk about the atmospheric system being out of equillibrium due to AGW. This seems nonsensical to me considering that since the Earth has had an atmosphere it has never been in equillibrium.”

    Ellis, as Hank wrote, the equilibrium referred to is “radiative equilibrium”, i.e. the amount of incoming solar energy falling on Earth must equal the amount of outgoing energy reflected and reradiated by Earth. If the outgoing is not as great as the incoming then the atmosphere will warm until it is equal to the incoming (the state we are current in). If the outgoing is greater than the incoming than the atmosphere will cool until the outgoing is equal to the incoming (the state during an ice age).

  22. 222
    Ellis says:

    Perhaps, to make my point I should let Gavins’ words speak for me.

    http://www.realclimate.org/index.php/archives/2007/08/the-co2-problem-in-6-easy-steps/#more-462

    “The fact that there is a natural greenhouse effect (that the atmosphere restricts the passage of long wave (LW) radiation from the Earth’s surface to space) is easily deducible from i) the mean temperature of the surface (around 15ºC) and ii) knowing that the planet is roughly in radiative equilibrium. This means that there is an upward surface flux of LW around (~390 W/m2), while the outward flux at the top of the atmosphere (TOA) is roughly equivalent to the net solar radiation coming in (1-a)S/4 (~240 W/m2). Thus there is a large amount of LW absorbed by the atmosphere (around 150 W/m2) – a number that would be zero in the absence of any greenhouse substances.”

    And, again my point, roughly equivalent means not equivalent, roughly in radiative equilibrium is not in equilibrium. But, if the scientist say it is close enough, then I ought to believe them.

  23. 223
    Rod B says:

    Timothy, No, your missing my point. My point, at its core has nothing directly to do with climate, greenhouse gases, geological periods, spectra bands, non-LTE, Newton, etc. It has to do with Algebra101.

    Technically, in a narrow sense, you might be correct saying that “forcing is a linear function of a log [of the concentration ratio]“, in that F(x) = a[ln(x)] might be classified as a linear log function, though some would say that’s a contradiction in terms. I’m just saying the phrase “forcing is a function of the log…..”, even with “linear” thrown in, common interpretation is that forcing would increase as the log of the concentration increases, or forcing increases much slower (by the log) than concentration, even though you might be technically correct. If I have F(x) = 1,000,000ln(x), blandly stating “F(x) is a linear function of the log of x” just doesn’t present the clear picture.

    Maybe you were getting ahead of me explaining where the particular exponent (5.35 as one) comes from. I’ll have to go back and re-read it.

    Later you say, “If I state that five cars with four wheels each have 4 X 5 wheels total, would you consider this misleading and state that they have 5 X 4 wheels? In either case there is a total of 20 wheels.” I say a[ln(x)] = ln[(x)^a]. Seems similar. I have no idea why this math rule upsets some people so… Other than when I say “forcing…. the the 5th power of concentration”, it sounds really bad and maybe they’d like it to sound better. Just musing.

  24. 224
    Hank Roberts says:

    Ellis, read it again and look at the temperatures, eight different reconstructions here:

    http://www.globalwarmingart.com/wiki/Image:Holocene_Temperature_Variations_Rev_png

    And look at the other time series.

    The planet’s in radiative equilibrium when the temperature doesn’t change.

    Read the caption:

    “…. eight records of local temperature variability on multi-centennial scales throughout the course of the Holocene, and an average of these (thick dark line). The records are plotted with respect to the mid 20th century average temperatures, and the global average temperature in 2004 is indicated. The inset plot compares the most recent two millennium of the average to other high resolution reconstructions of this period.

    At the far left of the main plot climate emerges from the last glacial period of the current ice age into the relative stability of the current interglacial. ….”

  25. 225
    Hank Roberts says:

    Oh, and Ellis:
    http://tamino.wordpress.com/2007/09/21/cheaper-by-the-decade/#more-376

    You remember those problems in math from high school, where there was a bucket with several different sized holes at different heights and a couple of different water sources adding water, and the task was to figure out if the level was rising, falling, or staying the same? Remember how you solved those?

  26. 226
    David Warkentin says:

    Rod -

    The reason I’d object to pulling the 5.35 inside the logarithm is that, as I noted, its value is entirely dependent on the units you choose – it’s really (5.35 W/m^2) – so you’re trying to get ln[(C/C0)^(5.35 W/m^2)]. This is problematic, because in physics (at least in any application I can think of) exponents should have no units, arguments of logarithms should have no units, and logarithms themselves have no units.

    Really, I don’t think there’s any slight-of-hand or deprecation of the strength of the effect going on here. If we wanted it to sound better, we could change units to kW/m^2, and say it’s the log of the concentration ratio raised to the power of (0.00535 kW/m^2), but this would be objectionable for the same reasons. Saying that forcing is proportional to the log of the concentration ratio and that the constant of proportionality is (5.35 W/m^2) seems perfectly standard to me.

  27. 227
    Timothy Chase says:

    Rod (#223) wrote:

    Maybe you were getting ahead of me explaining where the particular exponent (5.35 as one) comes from. I’ll have to go back and re-read it.

    Rod, if you want something to read, you might try:

    Climate Change 2001:
    Working Group I: The Scientific Basis
    6.3.5 Simplified Expressions
    http://www.grida.no/climate/ipcc_tar/wg1/222.htm

    Someone recommended it earlier in this thread, and according to this the f = 5.35 X ln(C/Co) doesn’t work that well after all. I don’t know the reasons yet, though. Looks like I will have to do some digging.

  28. 228

    [[My only point is that people here and elsewhere continue to believe in a mythical equillibrium that exsisted before man screwed it up, and that is not science, but faith.]]

    The climate system in the absence of anthropogenic effects would not be the same as the one we are now experiencing. The fact that climate changed before man was around doesn’t mean man can’t change the climate, or that climate change is always beneficial. Global warming is real, human technology is causing it, and it’s a serious problem. That’s the fact. Deal with it.

  29. 229
    Ray Ladbury says:

    Ellis, The concern is not that we are throwing the climate “out of equilibrium.” Rather, the reason to be concerned is because the past 10000 years have seen an especially stable climate by standards of geologic history. They also coincide with the period during which human civilization and all of its infrastructure developed. From the point of view of “Earth” or even of the biosphere, clomate change is nothing to worry about. It is only if you prefer a world with more biodiversity than rats, cockroaches, poison ivy and kudzu that you should be concerned. Since it is likely that only humans (and evidently only a few of them) have such a level of cognizance, this is a problem for humans, or at least, intelligent ones.

  30. 230
    Jim Eager says:

    Re 222 Ellis: “And, again my point, roughly equivalent means not equivalent, roughly in radiative equilibrium is not in equilibrium. But, if the scientist say it is close enough, then I ought to believe them.”

    Ellis, you foremost need to keep in mind that “equilibrium” is a dynamic concept, not a static one. When ever a system is disturbed it seeks a new equilibrium, and complex systems like our atmosphere are continually being disturbed.

    For example, when the Milankovetch orbital and axial cycles reduce the amount of solar insolation, the atmosphere then cools until a new equilibrium is reached. When those same cycles then increase the amount of solar insolation, the atmosphere then warms until a new equilibrium is reached. When the amount of greenhouse gasses in the atmosphere is increased, either by that warming, or, as at present, by our own direct actions, then the atmosphere will warm until it reaches a new equilibrium. As the present warming continues natural carbon sink and albedo feedbacks are being triggered, which will further disturb the system and lead to yet a different equilibrium. Thanks to diurnal and seasonal cycles alone there is and never has been a magic point where all is in static balance, although there have been relatively long time spans of stability. As Ray pointed out, human civilization developed in just such an era of stability. We disturb that stability at our own peril.

  31. 231
    Rod B says:

    Dave (226), you say, “…Saying that forcing is proportional to the log of the concentration ratio and that the constant of proportionality is (5.35 W/m^2) seems perfectly standard to me….”

    I simply disagree from a clarity viewpoint, though you are technically correct.

    You also said, “…The reason I’d object to pulling the 5.35 inside the logarithm is….”

    Oddly, I agree the equation ought to show the 5.35 outside of the log; it’s easier to read — you don’t need all of those confusing parens and carets. But I still think the words are more accurately descriptive (though not exact) with the “…to the 5.35th power…” Just my opinion.

  32. 232
    Ray Ladbury says:

    Rod B, David Warkentin is right. Exponents must be dimensionless. The 5.35 is a coefficient of the log term.

  33. 233
    Rod B says:

    > Ray (232)
    Yeh, but numerically the equation holds: …concentration ratio to the Nth power. I agree(d) the term should be shown as a coefficient — for clarity, and also for units I suppose, though, frankly, not absolutely necessary (though that too makes it clearer).

  34. 234
    David Warkentin says:

    Actually, Rod, I’m pretty sure it’s important for clarity not to treat the 5.35 as an exponent, precisely because its value depends on the units used. For instance: one Joule is 0.239 calories, so (5.35 W/m^2) is the same as (1.28 (cal/s)/m^2), and we could write the forcing as
    RF = (1.28 (cal/s)/m^2) ln(C/C0)
    just as well as the previous
    RF = (5.35 W/m^2) ln(C/C0)
    We’ve only changed the units of energy; we could have picked many others, each giving a different numerical value for the coefficient. But there’s no reason for a change in units to cause the functional relationship to change, as it would appear to if we treated the coefficient as an exponent inside the log. (Note the difference with the expression for kinetic energy, KE = (m v^2)/2; in this case, the exponent of 2 remains the same, whether we choose the units of velocity to be m/s or furlongs/fortnight.)

  35. 235
    Vernon says:

    I know your not posting anything that you cannot answer, so lets try again. Lets discuss the science.

    Now call me simple but if Hansen is right in his 2000 paper “Global warming in the twenty-first century: An alternative scenario” then how does anyone prove there is CO2 based warming by observing the climate?

    To quote from Hansen in the report:
    “Our estimates of global climate forcings indicate that it is the non-CO2 GHGs that have caused most observed global warming.”

    Further, Hansen went on to say:

    “Fossil fuel use is the main source of both CO2 and aerosols, with land conversion and biomass burning also contributing to both forcings. Although fossil fuels contribute to growth of some of the other GHGs, it follows that the net global climate forcing due to processes that produced CO2 in the past century probably is much less than 1.4 W/m2. ”

    Which reads as burning fossil fuels produces CO2 and aerosols which cancel either other out. That leaves only the other GHGs as the source of 20th century warming.

    The IPCC assumes a 4 W/m2 forcing but as the Hansen found “Most climate simulations, as summarized by the IPCC, do not include all of the negative forcings; indeed, if they did, and other forcings were unchanged, little global warming would be obtained.” In this study Hansen predicted that “Global warming at a rate 0.15 +/- 0.05 degrees C per decade will occur over the next several decades.” This works out to being 1.5 degrees +/- .5 C.

    Now as to AR1, the consensus from the IPCC is that AR1 plus the linear trend is good enough. This can be seen in IPCC AR4, where the caption to Table 3.2 says:

    Annual averages, with estimates of uncertainties for CRU and HadSST2, were used to estimate. Trends with 5 to 95% confidence intervals and levels of significance (bold: less than 1%; italic, 1 – 5 %) were estimated by Restricted Maximum Likelihood (REML; see Appendix 3.A), which allows for serial correlation (first order auto regression AR1) in the residuals of the data about the linear trend.

    The IPCC agrees that there are problems with the instrumented and proxy readings for the 20th century and it can be seen here

    In their large-scale reconstructions based on tree ring density data, Briffa et al. (2001) specifically excluded the post-1960 data in their calibration against instrumental records, to avoid biasing the estimation of the earlier reconstructions (hence they are not shown in Figure 6.10), implicitly assuming that the ‘divergence’ was a uniquely recent phenomenon, as has also been argued by Cook et al. (2004a). Others, however, argue for a breakdown in the assumed linear tree growth response to continued warming, invoking a possible threshold exceedance beyond which moisture stress now limits further growth (D’Arrigo et al., 2004). If true, this would imply a similar limit on the potential to reconstruct possible warm periods in earlier times at such sites. At this time there is no consensus on these issues (for further references see NRC, 2006) and the possibility of investigating them further is restricted by the lack of recent tree ring data at most of the sites from which tree ring data discussed in this chapter were acquired.

    So either the proxy readings are right and there is something wrong with the way we are doing the direct instrumented readings, or the proxy readings are wrong and we just lost the basis to say anything out of the ordinary is happening now.

    Warming in the Antarctic is mainly limited to the Antarctic Peninsula and even then manly in the portion that is outside the Antarctic Circle as can be clearly seen here. The interior of Antarctic is clearly cooling. This is not consistent with any of the GCMs. If the CO2 theory is correct with the sensitivity that is expected by the proponents, then warming should be happening at both poles.

    So, why is he wrong again? Do not seem to be able to get there from here.

  36. 236
    Ray Ladbury says:

    Vernon, The fact that the Arctic ice sheet is breaking up should be all the proof a rational person needs that something extraordinary is going on. If not, you also have the extinctions of amphibians in the cloud forests of Central America as they lose habitat, the shortening Winters and a raft of other evidence. It is beyond question that something unprecedented is occurring, and if the tools we have been using to estimate its severity are flawed, all that does is raise the level of risk.
    Re: Antarctic climate, see:
    http://www.realclimate.org/index.php/archives/2006/08/antarctica-snowfall/
    and several other posts on this site. Climate models, proxy reconstructions are not required to establish that the climate is warming. That is an empirical fact. The fact that CO2 is playing a role–well, physics works pretty well there. And your characterization of Hansen’s work demonstrates that you have not understood his point–the negative forcers have a much shorter lifetime than does CO2. That means there’s a lot more warming in the pipeline. And if we don’t have tools to accurately guage how much, we must prepare for the worst–a much more expensive proposition than would arise if we could do reliable risk assessment.

  37. 237
    John L. McCormick says:

    RE # 235

    Vernon, you said [So, why is he wrong again? Do not seem to be able to get there from here]

    Answer your own question rather than making comments that are solely your opinion.

    Do the research. Publish your findings. Be sure of your sources and stop throwing down pronouncements without coughing up some peer-reviewed conclusions that agree with your opinions.

    Start with the stratospheric cooling in the Antarctic and continued presence of wide ozone hole. Get constructive.

    Maybe you have something to tell us but your opinion has to be verifiable to more than yourself.

  38. 238
    Hank Roberts says:

    People get confused when they run back and forth between climateaudit and realclimate, asking for help understanding something, taking each answer back to the other place as the basis for their next question.
    Eric Berne used to call the game “let’s you and him fight.”

  39. 239
    Ray Ladbury says:

    An optimist sees the glass as half full. The pessimist sees the glass as half empty. The denialist sees the half empty glass and says, “Wow, look at all the space I have to fill up again!”–implicitly assuming that there will be wine to do the job.

  40. 240
    Timothy Chase says:

    Vernon (#235) wrote:

    To quote from Hansen in the report:
    “Our estimates of global climate forcings indicate that it is the non-CO2 GHGs that have caused most observed global warming.”

    Further, Hansen went on to say:…”

    Why don’t we look at what Hansen actually goes on to say – in the same paragraph. In fact, let’s look at the entire paragraph.

    Hansen, et al wrote:

    We suggest equal emphasis on an alternative, more optimistic, scenario. This scenario focuses on reducing non-CO2 GHGs and black carbon during the next 50 years. Our estimates of global climate forcings indicate that it is the non-CO2 GHGs that have caused most observed global warming. This interpretation does not alter the desirability of limiting CO2 emissions, because the future balance of forcings is likely to shift toward dominance of CO2 over aerosols. However, we suggest that it is more practical to slow global warming than is sometimes assumed.

    Global warming in the twenty-first century: An alternative scenario
    PNAS | August 29, 2000 | vol. 97 | no. 18 | 9875-9880
    http://www.pnas.org/cgi/content/full/97/18/9875

    Now you (#235) go on to say:

    Further, Hansen went on to say:

    “Fossil fuel use is the main source of both CO2 and aerosols, with land conversion and biomass burning also contributing to both forcings. Although fossil fuels contribute to growth of some of the other GHGs, it follows that the net global climate forcing due to processes that produced CO2 in the past century probably is much less than 1.4 W/m2.”

    [Veron concludes:] Which reads as burning fossil fuels produces CO2 and aerosols which cancel either other out. That leaves only the other GHGs as the source of 20th century warming.

    First, he is not stating that CO2 and aerosols cancel each other out, but that the processes by which both CO2 and aerosols have tended to cancel each other out during the twentieth century. But this does not imply that they will continue to cancel each other out. And as a matter of fact, Hansen says so quite explicitly when he states, “This interpretation does not alter the desirability of limiting CO2 emssions, because the future balance of forcings is likely to shift toward dominance of CO2 over aerosols.” Second, he is not stating that “only the other GHGs are the source of 20th century warming,” but that one should include the other GHGs if one intends to explain the warming which occured in the last century. Third, he is not stating that the reduction in CO2 emissions would in any way be undesirable but that other GHGs and black carbon should also be reduced. In fact he states quite the opposite in the abstract itself.

    Hansen et al wrote:

    The growth rate of non-CO2 GHGs has declined in the past decade. If sources of CH4 and O3 precursors were reduced in the future, the change in climate forcing by non-CO2 GHGs in the next 50 years could be near zero. Combined with a reduction of black carbon emissions and plausible success in slowing CO2 emissions, this reduction of non-CO2 GHGs could lead to a decline in the rate of global warming, reducing the danger of dramatic climate change.

    However, your claim that carbon dioxide and aerosols cancel each other out would suggest quite the opposite. It is also worth noting that there has been a reduction in the production of aerosols as of 1970, and as such, the effects of carbon dioxide have been masked to a smaller degree since then. Likewise he believes that other GHGs are playing less of a role now as is indicated by the sentence “The growth rate of non-CO2 GHGs has declined in the past decade.”

    Furthermore the authors later write:

    Climate forcing by CO2 is the largest forcing, but it does not dwarf the others (Fig. 1). Forcing by CH4 (0.7 W/m2) is half as large as that of CO2, and the total forcing by non-CO2 GHGs (1.4 W/m2) equals that of CO2.

    Now you (#235) go on to state,

    The IPCC assumes a 4 W/m2 forcing but as the Hansen found “Most climate simulations, as summarized by the IPCC, do not include all of the negative forcings; indeed, if they did, and other forcings were unchanged, little global warming would be obtained.” In this study Hansen predicted that “Global warming at a rate 0.15 +/- 0.05 degrees C per decade will occur over the next several decades.” This works out to being 1.5 degrees +/- .5 C.

    As stated above, if one did not include the other GHGs and looked at only the effects of CO2 and the aerosols which get produced during the same processes, the processes themselves, including the effects of both CO2 and the aerosols would leave much of the twentieth century warming unexplained. But this does not deny the role of CO2 itself. Instead he is arguing for the inclusion of both aerosols and other GHGs in the analysis of 20th century warming – something which is quite explicitly done in AR4. Moreover, the estimate of the forcing due to carbon dioxide and that of AR4 are essentially the same – roughly 4 w/m2.

    You (#235) continue:

    Now as to AR1, the consensus from the IPCC is that AR1 plus the linear trend is good enough. This can be seen in IPCC AR4, where the caption to Table 3.2 says:

    “Annual averages, with estimates of uncertainties for CRU and HadSST2, were used to estimate. Trends with 5 to 95% confidence intervals and levels of significance (bold: less than 1%; italic, 1 – 5 %) were estimated by Restricted Maximum Likelihood (REML; see Appendix 3.A), which allows for serial correlation (first order auto regression AR1) in the residuals of the data about the linear trend.”

    The trends are roughly the same. However, in AR4, this does not involve the omission of the effects of other greenhouse gases. Namely, gases like methane, CFCs, tropospheric ozone as well as the effects of black carbon. Likewise, the estimates of forcing due to carbon dioxide are roughly the same. As such your criticism of AR4 is null and void.

    *

    You (#235) write:

    The IPCC agrees that there are problems with the instrumented and proxy readings for the 20th century and it can be seen here:

    So either the proxy readings are right and there is something wrong with the way we are doing the direct instrumented readings, or the proxy readings are wrong and we just lost the basis to say anything out of the ordinary is happening now.

    In the long passage that you quote (which I am omitting for the sake of brevity – but which I link to so that people can go back and read it for themselves), they are admitting that there are uncertainties with respect to using tree rings as proxies, not that one must throw out either instrumental readings or all proxies as such. Moreover, the total acceptance of the absolute reliability of instrumental readings would in no way render even tree rings worthless – but would simply limit to some extent their reliability as proxies where uncertainties regarding moisture become a major factor. But there are other proxies – ratios of different isotopes of oxygen, the sizes of various microscopic organisms, and tree rings are usually fairly reliable and are by no means worthless.

    You (#235) write:

    Warming in the Antarctic is mainly limited to the Antarctic Peninsula and even then mainly in the portion that is outside the Antarctic Circle as can be clearly seen here.

    The link is to:

    Antarctic Heating and Cooling Trends
    http://svs.gsfc.nasa.gov/vis/a000000/a003100/a003188/index.html

    First Hansen and now Goddard – I am glad that you hold the views expressed by NASA in such high esteem! Actually judging from the images on that webpage, only a little more than half of the warming in the Southern ocean and Antarctica is happening outside of the Antartic Circle.

    You (#235) continue:

    The interior of Antarctic is clearly cooling. This is not consistent with any of the GCMs. If the CO2 theory is correct with the sensitivity that is expected by the proponents, then warming should be happening at both poles.

    Much of the interior of the Antarctic is cooling – but as the result of a lower stratosphere and the destruction of stratospheric ozone. This is well-understood – and it has been pointed out to you previously in this thread – and is incorporated into GCMs. Likewise, GCMs are not simply consistent with the Arctic and Antarctic showing differences (e.g., with some cooling occuring in the interior of Antarctica) but actually predict and explain it.

    You (#235) continue:

    The interior of Antarctic is clearly cooling. This is not consistent with any of the GCMs. If the CO2 theory is correct with the sensitivity that is expected by the proponents, then warming should be happening at both poles.

    Although I certainly don’t think that Hansen is infallible, I am not sure that I would disagree with any of the claims he makes in climatology. I personally think that he is one of the best the field has to offer.

    You (#235) began your post with the sentence:

    I know your not posting anything that you cannot answer, so lets try again. Lets discuss the science.

    After having analyzed your post in quite some detail, I will let the reader decide for himself just how much weight to assign to this – or for that matter any of the statements you make regarding climatology.

  41. 241
    David B. Benson says:

    Vernon — You really should

    (1) Go to the ‘Start Here’ link at the top of the page and start reading;

    (2) Go to the ‘AIP Discovery of Global Warming’ link in the Science section of the sidebar to read, at least, the page entitled “Carbon dioxide as a greenhouse gas”.

  42. 242
    Timothy Chase says:

    Gavin,

    The links on the right to comments have a percentage sign that prevents the webpages from coming up. The links look like this:

    http://www.realclimate.org/index.php/archives/2007/09/worth-a-look/langswitch_lang/%E#comment-55891

    It is the last bit:

    …/%E#comment-55891

    Anyway, welcome back!

  43. 243
    Raplh Smythe says:

    The glass at 50% of its capacity issue:

    If it started empty and got filled up halfway, it’s half full. If it started out full and was half taken out, it’s half empty.

    :D

  44. 244
    Rod B says:

    Ray (236), I’m neither agreeing or disagreeing with Vernon (235), but your claim that if there is a (partial) void of numerical analysis, using a few anecdotes with a wildly stretched inductive reasoning to fill the void, I submit, is faulty. Not terribly far off from, “Cherry blossoms arrived earlier this year! Batten down the hatches!” logic. Maybe, if they were truly unprecedented, but you have virtually no decent empirical rationale to claim, for example, the current altering of Arctic ice is unprecedented.

    Vernon’s assertion has far better empiricism (again, I am not verifying his claim) than your rebuttal.

    Just trying to keep things and the (pretty) straight and (kinda) narrow. No personal offense meant.

  45. 245
    Rod B says:

    Ray (239), then does the protagonist see a full glass and say, “Boy! We got to do something about that!” ??

  46. 246
    dhogaza says:

    I’m neither agreeing or disagreeing with Vernon (235), but your claim that if there is a (partial) void of numerical analysis, using a few anecdotes with a wildly stretched inductive reasoning to fill the void, I submit, is faulty.

    The problem is that there’s a lot more than a “few anecdotes” available. We have good, solid records on all sorts of biological phenomena. Migration timing of birds. First spring hatch of various insects. First spring mowing of the garden in the UK, first date on which roses bloom in various parts of the UK, etc (apparently there are a lot of fanatical record-keepers in the UK, people did a study …).

    The list goes on and on and on.

    They all point to a warming planet.

  47. 247
    David B. Benson says:

    T.J. Crowley
    Pliocene climate: the nature of the problem
    Mar. Micropaleontol. v. 27 (1996), 3–12

    seems to state that during the middle Pliocene the Arctic was ice free, when the temperature was about 2–3 degrees Celcius warmer than today and the sea stand was approximately 25 meters higher.

    (This is from the Hansen et al. 2007 paper ccited several times on the latest ‘Friday Roundup’ thread.)

  48. 248
    ray ladbury says:

    Rod B., If the cherry blossoms come early one year, it is not a trend. However, when Winters get shorter every year for decades, ome might think something is up. If one year a little more sea ice melts in the Arctic, it’s not a trend. When there is less ice pretty much every succeeding Summer, one might wonder what’s up. Or is it your contention that the globe is not warming?
    If the globe is warming, then it is natural to ask why. Energy is conserved, after all. Well, there’s one candidate explanation that deserves the name scientific.
    Rod, if the glass started out full of Arctic ice, it definitely ain’t full now.

  49. 249
    Philippe Chantreau says:

    David Benson: from what I read, there is reason to believe that the closing of the Isthmus of Panama during the Pliocene played a significant role in initiating the glaciation cycles. In other words, those cycles and the presence of an arctic ice cap is most likely a normal feature for a relative state of equilibrium with the current land distribution.
    http://www.agu.org/pubs/crossref/1997/96GL03950.shtml
    http://www.whoi.edu/oceanus/viewArticle.do?id=2508
    http://adsabs.harvard.edu/abs/1978Geo…..6..630K

  50. 250
    Philippe Chantreau says:

    I should have looked closer, it seems that this article is the most interesting that you can get in 10 min of basic googling:
    http://marine.rutgers.edu/faculty/rosentha/rosenthal_files/Lear_NADW_inpress.pdf


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