Guest Post by Willis Eschenbach
“Climate sensitivity” is the name for the measure of how much the earth’s surface is supposed to warm for a given change in what is called “forcing”. A change in forcing means a change in the net downwelling radiation at the top of the atmosphere, which includes both shortwave (solar) and longwave (“greenhouse”) radiation.
There is an interesting study of the earth’s radiation budget called “Long-term global distribution of Earth’s shortwave radiation budget at the top of atmosphere“, by N. Hatzianastassiou et al. Among other things it contains a look at the albedo by hemisphere for the period 1984-1998. I realized today that I could use that data, along with the NASA solar data, to calculate an observational estimate of equilibrium climate sensitivity.
Now, you can’t just look at the direct change in solar forcing versus the change in temperature to get the long-term sensitivity. All that will give you is the “instantaneous” climate sensitivity. The reason is that it takes a while for the earth to warm up or cool down, so the immediate change from an increase in forcing will be smaller than the eventual equilibrium change if that same forcing change is sustained over a long time period.
However, all is not lost. Figure 1 shows the annual cycle of solar forcing changes and temperature changes.
Figure 1. Lissajous figure of the change in solar forcing (horizontal axis) versus the change in temperature (vertical axis) on an annual average basis.
So … what are we looking at in Figure 1?
I began by combining the NASA solar data, which shows month-by-month changes in the solar energy hitting the earth, with the albedo data. The solar forcing in watts per square metre (W/m2) times (1 minus albedo) gives us the amount of incoming solar energy that actually makes it into the system. This is the actual net solar forcing, month by month.
Then I plotted the changes in that net solar forcing (after albedo reflections) against the corresponding changes in temperature, by hemisphere. First, a couple of comments about that plot.
The Northern Hemisphere (NH) has larger temperature swings (vertical axis) than does the Southern Hemisphere (SH). This is because more of the NH is land and more of the SH is ocean … and the ocean has a much larger specific heat. This means that the ocean takes more energy to heat it than does the land.
We can also see the same thing reflected in the slope of the ovals. The slope of the ovals is a measure of the “lag” in the system. The harder it is to warm or cool the hemisphere, the larger the lag, and the flatter the slope.
So that explains the red and the blue lines, which are the actual data for the NH and the SH respectively.
For the “lagged model”, I used the simplest of models. This uses an exponential function to approximate the lag, along with a variable “lambda_0” which is the instantaneous climate sensitivity. It models the process in which an object is warmed by incoming radiation. At first the warming is fairly fast, but then as time goes on the warming is slower and slower, until it finally reaches equilibrium. The length of time it takes to warm up is governed by a “time constant” called “tau”. I used the following formula:
ΔT(n+1) = λ∆F(n+1)/τ + ΔT(n) exp(-1/ τ)
where ∆T is change in temperature, ∆F is change in forcing, lambda (λ) is the instantaneous climate sensitivity, “n” and “n + 1” are the times of the observations,and tau (τ) is the time constant. I used Excel to calculate the values that give the best fit for both the NH and the SH, using the “Solver” tool. The fit is actually quite good, with an RMS error of only 0.2°C and 0.1°C for the NH and the SH respectively.
Now, as you might expect, we get different numbers for both lambda_0 and tau for the NH and the SH, as follows:
Hemisphere lambda_0 Tau (months)
NH 0.08 1.9
SH 0.04 2.4
Note that (as expected) it takes longer for the SH to warm or cool than for the NH (tau is larger for the SH). In addition, as expected, the SH changes less with a given amount of heating.
Now, bear in mind that lambda_0 is the instantaneous climate sensitivity. However, since we also know the time constant, we can use that to calculate the equilibrium sensitivity. I’m sure there is some easy way to do that, but I just used the same spreadsheet. To simulate a doubling of CO2, I gave it a one-time jump of 3.7 W/m2 of forcing.
The results were that the equilibrium climate sensitivity to a change in forcing from a doubling of CO2 (3.7 W/m2) are 0.4°C in the Northern Hemisphere, and 0.2°C in the Southern Hemisphere. This gives us an overall average global equilibrium climate sensitivity of 0.3°C for a doubling of CO2.
Comments and criticisms gladly accepted, this is how science works. I put my ideas out there, and y’all try to find holes in them.
w.
NOTE: The spreadsheet used to do the calculations and generate the graph is here.
NOTE: I also looked at modeling the change using the entire dataset which covers from 1984 to 1998, rather than just using the annual averages (not shown). The answers for lambda_0 and tau for the NH and the SH came out the same (to the accuracy reported above), despite the general warming over the time period. I am aware that the time constant “tau”, at only a few months, is shorter than other studies have shown. However … I’m just reporting what I found. When I try modeling it with a larger time constant, the angle comes out all wrong, much flatter.
While it is certainly possible that there are much longer-term periods for the warming, they are not evident in either of my analyses on this data. If such longer-term time lags exist, it appears that they are not significant enough to lengthen the lags shown in my analysis above. The details of the long-term analysis (as opposed to using the average as above) are shown in the spreadsheet.
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Willis, I assumed that you want to do it yourself. If you want, you can send me the data in Excel or ascii (comma delimited) and I’ll fit the model in SAS.
Matthew R Marler says:
May 31, 2012 at 4:28 pm (Edit)
Matthew, many thanks for the offer. All of the data is in the spreadsheet that I referenced above, here it is again.
w.
CONTINUATION—I have just published a new post which answers many of the comments made on my first analysis. The post does the same analysis, but this time using the actual data rather than the averaged data. The post is “A Longer Look at Climate Sensitivity“.
w.
richardscourtney says:
May 31, 2012 at 3:29 pm
“KR:
I appreciate your taking the trouble to provide your answer to me that you have posted at May 31, 2012 at 1:04 pm.
You say a two-box model is “better” than a one-box model. Perhaps.
The point I (and I think also Terry Oldberg) am making is that there is no way to define what is an adequate model.
It is a fact that – as Willis says – Willis “one box” model DOES work. It DOES emulate physical reality. For sake of argument, I will accept that Tamino’s “two box” model also works (but I have not checked that). And, as I explained, other models are also possible by adding more “boxes”.
Which of the many models (each constrained by the available data) is preferable and for what?
The range of published values of climate sensitivity shows that the data allows interpretation such that the determinations of climate sensitivity vary by an order of magnitude. In my terms, that says the data permits almost any interpretation one wants to make.
At present, Willis’ determination of climate sensitivity is as valid as any other. Arm-waving about “add another box” or boxes does not change that.
Richard”
You are wrong about Willis’ calculation being as valid as any other. The Physics of the situation says that there are many different “boxes” that hold differing amounts of energy in the climate system. The speed of transmission of energy, between different boxes is dependent on their identity. You have the atmosphere, different land surfaces, the ocean surface layer which absorbs solar energy, the deep ocean etc. Each has a different heat capacity and coefficient of energy transfer between it and the adjacent layer. The time scale for setting up a steady state solution between each adjacent box is quite different. It is pretty clear from this physics that a single box model cannot accurately model the behavior of the earth’s climate system.
This is a pretty convincing argument to anyone with an understanding of the physics, and an open mind. It appears that you are lacking one or the other or both.
Eric Adler (May 31, 2012 at 6:12 pm):
It seems to me that there is not a conflict between Richard Courtney’s claim and yours. You claim that the discretization error declines with the number of boxes; that’s true. Richard claims that the numerical value of the equilibrium climate sensitivity (TECS) is indeterminant; that’s also true.
Willis, you asked for a citation for how I used the magnitude of the diurnal cycle of solar forcing or the daily variation of sea-surface temperature? Do you not believe the numbers I gave you, which are fairly straightforward? Given these two numbers you can derive a sensitivity either by your method or my simpler one. You will come out with a very small number to help your cause, but it doesn’t mean much less than using the annual cycle. The point is that you are not going to get climate sensitivity from the annual cycle, any more than from the diurnal cycle, because limiting the analysis to the annual frequency also limits the lag time to something shorter, which ends up making it irrelevant to the years of lag time that actually exist in the climate system, or are you advocating that the climate system lag time is really only a couple of months?
richardscourtney, Eric Adler
There are many ways to judge the relative merits of various models.
– Does one model track the data better than another (all the data, as opposed to the annual data used in testing this one-box model)? That’s a plus.
– Does one model work better than another on data that it hasn’t been refined on; refine it on part of the data set, feed it the forcings for another, and see how it works? For example, different subsets of temporal or frequency coverage? That’s a plus.
– Is one model based on more physics, and/or more constraints, than another? A plus.
– Does one model require ad hoc adjustments, such as arbitrarily rescaling certain forcings to improve fit? That’s a serious minus.
– Does the model have unconstrained free parameters? Not a problem here, but some of Dr. Spencer’s recent work includes models with >20 free parameters, unconstrained by observation – and hence leading to unphysical silliness such as 700 meter oceanic mixed layers. As John von Neumann said, “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk …”. Serious minus; it’s entirely too easy to over-fit the data.
– Does one model fit the observations with better statistical significance? This may require Monte Carlo testing or fairly significant work (judging the Akaike information criterion, for example) to establish, but a plus.
It’s actually fairly straightforward to compare models and judge which is better, which is more likely to be useful.
—
If, as you appear to, reject all modeling, I would despair of your ever being able to cross the street – since your expectations of the world around are, indeed, models.
Eric Adler – My apologies, I misread your last comment, and which parts were yours and Richard Courtney’s. I would agree with your post WRT Richard’s position.
KR:
As you point out, rhere are many ways in which to judge the relative merits of various models. As you also point out, prior to forming expectations about the world around us, we must select one of them. There is a logical approach to making this selection. Like researchers in many other fields of inquiry, climatological researchers make this selection illogically and to our disadvantage.
Joe Born
You helped a lot to clarify my thoughts. Thanks.
KR and Eric Adler:
Insults and childish snark are not an acceptable response to logical argument and criticism.
I have shown you are wrong. Live with it.
Richard
Jim D says:
May 31, 2012 at 6:14 pm
I’m sorry, Jim, but I don’t believe your numbers, for two reasons. First, you give no indication of where you got them. That’s why I asked for a citation, so I wouldn’t have to just take your word for something.
Second, I don’t believe you because I know something about Pacific temperatures. You say:
However, here’s a typical record of a couple of months of surface air temperatures from one of the TAO moored buoy array in the mid Pacific:
Note that swings of up to 4°C are not uncommon. As you can see, your claim of a 1° variation is simply not true.
Regarding the larger point, will I get a different climate sensitivity and time constant when I look at daily data? Quite possibly so, but unfortunately I don’t have the data to calculate it. Which is why I asked for a citation in the first place, because I thought you might have data that would settle the question.
Thanks,
w.
KR says:
May 31, 2012 at 8:11 pm
KR, I have used all the data that I have available. What other data are you referring to? What am I missing here?
Thanks,
w.
Willis: All of the data is in the spreadsheet that I referenced above, here it is again.
You had the hysteresis plots. Nice. Sorry I didn’t notice before.
Matt
Willis Eschenbach – “I have used all the data that I have available. What other data are you referring to?”
Data at different time scales, such as the GISS forcings you used in http://wattsupwiththat.com/2011/01/17/zero-point-three-times-the-forcing/. You’re going to have a very different calculated climate sensitivity for the GISS data with your single-box model (with or without your ad hoc volcanic adjustment) – and varying sensitivities for varying time scales indicate problems with the model. Let alone the ad hoc adjustments. And if transient sensitivities are as low as you have computed, recent warming could not have occurred with the forcings over the last 100 years – another data point worth considering. .
“”While it is certainly possible that there are much longer-term periods for the warming, they are not evident in either of my analyses on this data.
That’s because in this particular analysis you didn’t look at longer periods. If you got the same sensitivity (without ad hoc adjustments) for multiple time periods, that would be interesting. But you’re not likely to with a single-box model.
Again, as discussed before, 6-12 month forcing swings are measuring instantaneous sensitivity far more than the 20-year transient sensitivity you appear to be comparing your results to – not enough time for the ocean mixed layer to respond, or for mid-term feedbacks to occur. Your lags are not (IMO) going to be sufficient to extend your results to 20 years. About the only conclusion that can be drawn from such a short data span with a simple model is that transient sensitivity will be higher than instantaneous sensitivity (which is really what is being measured here), and equilibrium sensitivity will be higher still. You have, at best, computed a lower bound. Very interesting, mind you! But not a calculation of the transient sensitivity.
richardscourtney May 31, 2012 at 11:51 pm – “”Insults and childish snark..”
My apologies, I did not intend to be snarky. However, I am quite appalled that you seem to feel that there is no objective criteria for comparing models – if that was the case, you could hardly support Eschenbach’s model. Your posts supporting Eschenbach’s model despite (as I linked to) issues with different time scales seem to suggest confirmation bias.
I sincerely hope that is not the case – but you have not presented any reasons or criteria to prefer Eschenbach’s single-box model over two-box models – models with better performance in terms of the data at different time scales, without any ad hoc adjustments.
Models should be simple – but no so simple (http://tinyurl.com/6tvk3wt) that they miss major elements of the data.
Willis, you can estimate a diurnal forcing over the ocean without looking up references. The upward longwave part is constant and the downward one changes little, because the main variation is solar that changes from zero to maybe 1000 W/m2 depending what latitude you are at, perhaps 500 W/m2 further from the equator. The diurnal cycle is then easily several hundred W/m2. The ocean (water) temperature variation may be higher at some shallower points, but averaged over the ocean it would not be more than about a degree, and your buoy shows air temperature that has other things, probably related to local weather also going on because it is clearly not all diurnally forced. The variations, any way you compute them, give you a low sensitivity (even if you believe your four degree ocean diurnal cycle.) And, as a bonus, you also get a short lag time of a few hours. Why would this even shorter lag and lower sensitivity not apply to climate, you should ask? Then the same question could be applied to annual data.
KR says:
June 1, 2012 at 4:36 pm
I must be missing something here. What am I supposed to do with the GISS forcings? They are not of the slightest use to me or my method.
The only GISS data I know of are their global temperature gridded data. In addition to that, they have assumptions and approximation about forcing, and model results.
So what in all of that are you calling “data”, and what are you suggesting I do with it?
Regarding my study of the GISSE model, which found a very different sensitivity, that is a study of a model’s response to carefully selected forcings … am I supposed to be surprised that the result doesn’t resemble the real world?
“Recent warming”? The recent warming period, 1984-1997, is exactly the period I analyzed. So exactly why could it not have occurred with the forcings? I just showed that you can hindcast the recent warming almost exactly using only albedo forcings …
Please note that over the 14 years I analyzed, the temperature rose by about 0.4°C … which is about 2/3 of the total global warming for the century.
I looked at a fourteen year period, because that is all of the data I have. That is a reasonably long period, and my method calculated the temperature over that period almost exactly, for both the NH and the SH. So how long a period are you wanting me to look at?
If you have hemispheric albedo data for earlier or later periods, please let me know where it is … but until then, you’re asking for something impossible. I cannot do anything further without further data.
w.
Jim D says:
June 1, 2012 at 5:34 pm
Jim, as I demonstrated above, you’ve tried to estimate diurnal forcing and failed. So don’t try to lecture me about what you can estimate.
w.
Willis, you can probably get diurnal forcing from a high-school textbook. If you don’t believe even that the order of magnitude is a few hundred W/m2, you need to say so, because I can’t believe you are actually disputing that. The one degree average diurnal cycle over the oceans, shown by your buoy data backs up the point I made earlier.
I attempted to use the above approach on synthetic data, and it seemed to underestimate the lambda. The stimulus I applied was 500 steps of sin(2*pi/120) applied to a system whose transfer function is 0.2/(1+5s) + 0.4/(1+50s), i.e., that had a total lambda of 0.2 + 0.4 = 0.6 and time constants of 5 and 50. I seem to come up with a lambda estimate of 0.37 instead of 0.6. (The rms error was 0.0024 and the estimated time constant was 15.)
I need to check my work (and use values that make the comparison easier), but I probably won’t be able to return to this for another day or two. So perhaps someone else, who’s faster at this stuff than I, can try a similar test in the interim?
Actually, as I was heading out I realized that I’d made an error in simulating the system, so the results are doubtlessly flawed. I’ll return to this when I can, but, again, maybe someone else can do it?
I (think I) fixed my calculations, and it appears that, with synthetic data from a system whose lambda (long-term sensitivity) is actually 0.6, Mr. Eschenbach’s approach (if I’ve applied it correctly) infers a lambda less than 0.35. In two tests I applied sinusoids of periods 120 and 12, respectively, to a dual-lag system whose transfer function is 0.2/(1+2s) + 0.4/(1+50s), as the table below indicates. The rms errors were small in both cases, but the lambda inferred was wide of the mark.
Test No. 1 Test No. 2
sine period 120 12
lambda1 0.2 0.2
lambda2 0.4 0.4
tau1 2 2
tau2 50 50
lambda 0.6 0.6
estimated lambda 0.34 0.31
estimated tau 12.5 4.0
e_rms 1.07E-08 3.08E-07
Of course, I’d be more confident if someone else performed this exercise independently, but at this point it appears that Mr. Eschenbach’s demonstration, although certainly enlightening, proves rather less than I had at first thought.