Wednesday, August 22, 2012

Eli's Three Laws or Hansen Simply Explained

There is an interesting to and fro over at Real Climate about the recent Hansen, Ruedy and Sato paper on an increase in climate extremes.  Eli had a go at the preprint.  Tamino has entered a demurral, which Eli suspects has an answer in what is below.  Now some, not Eli to be sure, think the Rabett is on occasion cryptic.  So be it, but in this case he has not quite worked through all the implications and it is better to be thought mysterious than foolish.  Still the Bunny is somewhat skilled in boiling things down, and at RC pointed out that there are three simple points in the Hansen paper

First if there is an increasing/decreasing linear trend on a noisy series, the probability of reaching new maxima or minima increases with time

Second, if you put more energy into a system variability increases.

Third, if you put more energy into a system variability increases asymmetrically towards the direction favored by higher energy
 Hank added a codicil
Fourth, if you keep going in the direction you’re headed, you’ll get there.
 The first point has been a standard here at RR and also in various comments about the web in the recent discussions about why there are more temperature maxima than minima.  The second is related to the idea that if the parametric landscape affecting anything is not smooth, excursions from a stable state require additional energy to overcome "attractions".

For example, if you are in a low energy valley, as you add energy, the system moves higher up the sides and the range of motion becomes larger.  As examples consider potential energy surfaces for any process such as chemical reactions, protein unfolding, etc.  The third comes from the same general area, that if there are barriers to some excursions, you enhance the probability of exceeding them only with additional energy, e.g. you have to climb the mountain to get to the next valley, Hank's Law.

UPDATE:  Perhaps a hint of where the second and third law come from in answer to some questions

taken from protein folding paper.  The same arguments cover many situations including climate.  If we start at the bottom of the U(nfolded) well with only a small amount of energy available, the motion is roughly harmonic.  This is a logical consequence of the stability, because the point at the bottom of the well is where the system would come to rest if energy were zero and the first term in the Taylor series expansion of the potential would be quadratic  Small additional amounts of energy result in harmonic (symmetric) motion in first approximation (brown and red  line. 

As more energy is added you get pushed to the level of the green line, where there is more variation in the reaction coordinate as in the second law.  Finally if you reach the level of the red line, further areas of the parameter space become available and you can access the folded form.  The ability to access new  areas of the potential is what drives the asymmetry.


Anonymous said...

The Maxwell-Boltzmann distribution shows the general changes with increased system energy/temperature shown by Hansen (increased mean, increased variance, rightward skew)

MB statistics are obviously very relevant with regard to chemical reactions.


Anonymous said...

The first law is clear. Don't the others depend on circumstances? Over the past million years of glacial cycles, have not the cold periods been the most subject to dramatic variations?

Pete Dunkelberg

cRR Kampen said...

"Third, if you put more energy into a system variability increases asymmetrically towards the direction favored by higher energy"

Plausible but questionable. Because system variability was nicely symmetric coincidentally(?) when there was somewhat less energy in the system, say 30-60 years ago.

A clause may be needed to account for the system that is out of equilibrium and I dare say that is the reason for recent asymmetrising.

Tom Fiddaman said...

Another reason for increasing variance in systems generically is approach to a bifurcation. There's no real evidence that this is happening, except maybe for Arctic sea ice, but it's certainly possible.

Tom Fiddaman said...

The direction of the variance in the MB distribution is right, but isn't the change too small to account for any observed increase in variance? For example, if temp rises 1C, you'd expect the standard deviation to go up 288K/287K-1 = 0.3%.

a_ray_in_dilbert_space said...

As a general rule, isn't it true that it is only for the Normal distribution that the location parameter coincides with the first moment of the distribution (mean). Thus, of necessity, doesn't one affect the higher moments of the distribution if one changes the mean?

In effect, I think what Eli is saying is that if you increase (decrease) the mean of the system, you must also increase (decrease) the variance and skew the distribution to the right (left).

Tom Fiddaman said...

@ a_ray - I don't disagree that the variance changes with the mean. I'm just pointing out that the MB effect should be too small to see.

I think this is related to Tamino's point, that the observed change in variance could have two components, change in spatial means & change in local variances, with the former greater than the latter.

These Recaptchas are getting hard.

Anonymous said...

You are right, Tom.

Any change due to MB would not be noticeable in this case.


Jeffrey Davis said...

I wonder if, as a practical matter, people expect scant variance because they're expecting there to be some difficulty in changing state, i.e. our climate is basically stable and that it will take a lot of energy to push it into different states. I imagine the opposite is true. The climate is basically unstable and we've (civilization) advanced during an era of relatively unprecedented stability. When I look at those charts of temps and CO2 concentrations over the last 400,000 years I see huge swings in temps, and temps climb almost instantly. Up like an elevator and down like steps. The amount of energy that pushes the climate out of the glacial deep freeze is teensy. (Milankovich forcings are so small they're almost not there.)

badger badger badger said...

There was a paper, or maybe just a suggestion, that took this abstract energetic view of climate and IIRC suggested that oscillations in a system would slow as the energy approached the point where the system bifurcates, for similar reasons that the statistical skew is introduced (imagine a marble on the track in the update). Anybody have a lead on that?

Anonymous said...


Haven't we been through this before?

Air mass exchange is NOT driven by global heat content but by the GRADIENT in temperature.

In a world with no spatial temperature variation, no gradients would exist and no winds would blow and no exchange would take place and variability would be zero.

It doesn't matter if that world was hot or cold, if the gradients were zero, so would the variability.


Anonymous said...

Gradients being important, one could wonder about what a 2x CO2 world.

The line before has been that Arctic Amplification would reduce the N pole to equator gradient, which would reduce variability.

There is another possibility.

Increasing CO2 actually leads to COOLING over the high terrain of the winter pole (Antarctica, Greenland ).

I could imagine that would lead to increased variability, though mostly in winter, not summer. That also has some ramifications on the extent of CO2 warming.


EliRabett said...

Doing away with the lapse rate Eunie?

Anonymous said...

If Tamino is right, the increase in variance that Hansen found is largely (if not completely) an artifact of the way hansen did his analysis. which case, any speculation about "mechanisms" that might "explain" increased variance would not be particularly meaningful.

But let's assume the increased variance that hansen showed is real.

How would the analogy between that last diagram and temperature anomalies work?

How would increasing the energy of the climate increase the range of possible surface air temperatures (especially on the high side)?

What would be the physical mechnism?


EliRabett said...

Badger, the velocity of the passage along the reaction coordinate (in chemical dynamics language) is set by the difference between the total energy and the potential energy. Thus by necessity as you approach the high point (the transition state) the speed slows. It's a feature, not a bug

Anonymous said...

"Doing away with the lapse rate Eunie?"

If we're going to have an isothermal atmosphere, I guess we'd better.

badger badger badger said...

Right, I just wonder if anybody got anywhere applying the notion to weather/climate. Or even formulating what to look for.

Tom Fiddaman said...


That sounds like the phenomenon of "critical slowing down" - an increase of the time constant over which the system returns from a disturbance, as a bifurcation approaches. That's what I had in mind in my original comment. Don't know the paper unfortunately. There is evidence of a period-doubling bifurcation in Arctic ice - see .

Tom Fiddaman said...

@anonymous Tamino's argument doesn't render the search for mechanism irrelevant, it just shifts it to a different scale. If the means at different sites change, that still begs the question, why? The answer could be noise or tipping points or some other real mechanism.

Anonymous said...


Not sure what you mean by "shifts it to a different scale"

But, unless I have misunderstood what Tamino did and showed, his investigation (in two posts) indicated that the apparent change in variability that Hansen claimed was probably due to Hansen's failure to properly account for the effect of baseline.

Look at the graph of "de-trended Standardized anomaly" in Tamino's post (Increased variability)

Under the graph he states that

"There’s no visible sign of any change in the amount of temperature variability from one 11-year period to any other. Also, there’s no issue about baseline period, since de-trending the divisional anomalies before standardizing and combining removes the influence of the choice of baseline period (but in case you’re interested, the baseline period I used was the entire time span 1895-2012.5).

Tamino's second post is a further investigation of the baseline issue (basically supporting his earlier hunch).

These captcha's are nearly impossible to read (even if you're not a robot)

Windchasers said...

Increased variability's not necessarily a bad thing, in some contexts. I think it means the Rossby waves move faster, which means a greater variation in hot/cold, wet/dry.

I.e., a warming Arctic means a lower gradient of temperature from the Equator to the poles. This causes the peaks and troughs of the jet stream (Rossby waves) to have a greater N-S amplitude, and are more likely to move slowly or get stuck. Slow-moving Rossby waves could mean that one summer, hurricanes are all deflected away from the eastern US seaboard, or another summer, many more make landfall than usual. Or it means that a blocking pattern sets up over Greenland one winter, and the eastern US gets significantly more snow than normal.

That is to say, less variability in the jet stream may mean longer streaks of weather, which means more droughts, floods, etc., which is an increased variability of a different kind.

Examples of Chemical Changes said...

These topics are described here with a good examples and theory.

Tom Fiddaman said...

@ Anonymous

Here's how I'm thinking about it.

Consider Tamino's two-site thought experiment. Variance at each site stays the same, but total variance goes up because the mean of one site goes up.

But that still leaves the question, why did the mean of one site shift? That could be spurious (>decadal-scale noise), or it could be some real change in the structure of atmospheric circulation, which is what I meant by "different scale."

I think Tamino's baseline approach is the right way to look at local short term variability, but that's not the total perception of weather for a local person (which includes the trend), nor the entire story of variance.

Anonymous said...


While the reasons why variance "appears" to have changed may be interesting in their own right, the main issue in this case is whether variance has actually changed (and in particular, whether Hansen actually showed that it has).

In order to really be confident (one way or the other), one needs to perform detailed statistical analysis.

And in order to do that, one needs first to eliminate confounding factors like baseline issues (as Tamino did)

Tamino's detrended standardized anomaly may not be proof that variance has not changed over the period in question, but it sure suggests that -- and quite the opposite of what Hansen showed with his broadening distribution.

And while Tamino has not "proved" that the baseline issue is the source of the apparent increase in variability shown by Hansen, Tamino's argument makes logical sense.


badger badger badger said...

Thanks for the tip. searching on "nerc NE/F005474/1" (funding for that paper) turns up


Tom Fiddaman said...

@Anon The essence of T's baseline critique is that variance, narrowly defined as variability around the trend at individual sites, has not gone up, which certainly appears to be the case. But as far as I can tell, the thermodynamic arguments in this post are not restricted to this narrow interpretation of variance. Nor is an individual's perception of temperature neatly separable into trend and sd terms. Therefore questions about how aggregate variance, e.g. from divergence of means or trends, might go up is still interesting.

Jeffrey Davis said...

I think the issue is clouded because the year has a finite range. There's only 365 days for "variation" to occur in. And that's subdivided into seasons. A warm day in December would look quite differently than a warm day in May.

Tom Fiddaman said...


Fig 1 in your first link is a nice illustration of the reason for critical slowing down.

It would be interesting to see these methods applied to the global obs temp dataset.

Anonymous said...

Tom, We'll have to agree to disagree (and leave it at that) because the point Eli raised (...simple points in the Hansen paper..") and the point I was addressing and the point Tamino addressed IS the Hansen paper, specifically the claim that the variance has increased.

Not that "perceived variance" has increased (that certainly goes up if one does not remove the trends), but the actual variance.


badger badger badger said...


Yes. They mention skew as an indication, and as Eli has pointed out, there appears to be skew in Hansen et al.'s distributions. Which struck me as more unexpected than the variance which is or isn't there.

The second paper is the one I originally had in mind, so mission accomplished!

David B. Benson said...

In chapter 6 of Ray Pierrehumbert's "Principles of Planetary Climate" one discovers that total precipitation increases approximately as the square of dT, the temperature increase.

Hank Roberts said...


Relationship between hourly extreme precipitation and local air temperature in the United States

Key Points

Strong relationship between hourly extreme precipitation and temperature
Regression slopes are higher in summer than winter
Stations in the northern U.S. show higher slopes

Hank Roberts said...

PS, although I don't use a mobile phone, I had to give Google a made-up mobile phone number in order to be allowed to post above today. They gave three prompts urging me to give them a mobile number, then gave an error message when I continued to decline to do that. Eager, they are, to track everyone.

Anonymous said...

Hank, beware the hackers who take over your Gmail account. A text is a security feature which this person hadn't activated. I thought Google's response was so shoddy, though, I closed my account.