In Part One we had a look at some introductory ideas. In this article we will look at one of the ground-breaking papers in chaos theory – Deterministic nonperiodic flow, Edward Lorenz (1963). It has been cited more than 13,500 times.

There might be some introductory books on non-linear dynamics and chaos that don’t include a discussion of this paper – or at least a mention – but they will be in a small minority.

Lorenz was thinking about convection in the atmosphere, or any fluid heated from below, and reduced the problem to just three simple equations. However, the equations were still non-linear and because of this they exhibit chaotic behavior.

Cencini et al describe Lorenz’s problem:

Consider a fluid, initially at rest, constrained by two infinite horizontal plates maintained at constant temperature and at a fixed distance from each other. Gravity acts on the system perpendicular to the plates. If the upper plate is maintained hotter than the lower one, the fluid remains at rest and in a state of

conduction, i.e., a linear temperature gradient establishes between the two plates.If the temperatures are inverted, gravity induced buoyancy forces tend to rise toward the top the hotter, and thus lighter fluid, that is at the bottom. This tendency is contrasted by viscous and dissipative forces of the fluid so that the conduction state may persist.

However, as the temperature differential exceeds a certain amount, the conduction state is replaced by a steady

convectionstate: the fluid motion consists of steady counter-rotating vortices (rolls) which transport upwards the hot/light fluid in contact with the bottom plate and downwards the cold heavy fluid in contact with the upper one.The steady convection state remains stable up to another critical temperature difference above which it becomes unsteady, very irregular and hardly predictable.

Willem Malkus and Lou Howard of MIT came up with an equivalent system – the simplest version is shown in this video:

*Figure 1*

Steven Strogatz (1994), an excellent introduction to dynamic and chaotic systems – explains and derives the equivalence between the classic Lorenz equations and this tilted waterwheel.

L63 (as I’ll call these equations) has three variables apart from time: intensity of convection (x), temperature difference between ascending and descending currents (y), deviation of temperature from a linear profile (z).

Here are some calculated results for L63 for the “classic” parameter values and three very slightly different initial conditions (blue, red, green in each plot) over 5,000 seconds, showing the start and end 50 seconds – click to expand:

*Figure 2 – click to expand – initial conditions x,y,z = 0, 1, 0; 0, 1.001, 0; 0, 1.002, 0*

We can see that quite early on the two conditions diverge, and 5000 seconds later the system still exhibits similar “non-periodic” characteristics.

For interest let’s zoom in on just over 10 seconds of ‘x’ near the start and end:

*Figure 3*

Going back to an important point from the first post, some chaotic systems will have predictable statistics even if the actual state at any future time is impossible to determine (due to uncertainty over the initial conditions).

So we’ll take a look at the statistics via a running average – click to expand:

*Figure 4 – click to expand*

Two things stand out – first of all the running average over more than 100 “oscillations” still shows a large amount of variability. So at any one time, if we were to calculate the average from our current and historical experience we could easily end up calculating a value that was far from the “long term average”. Second – the “short term” average, if we can call it that, shows large variation at any given time between our slightly divergent initial conditions.

So we might believe – and be correct – that the long term statistics of slightly different initial conditions are identical, yet be fooled in practice.

Of course, surely it sorts itself out over a longer time scale?

I ran the same simulation (with just the first two starting conditions) for 25,000 seconds and then used a filter window of 1,000 seconds – click to expand:

* Figure 5 – click to expand*

The total variability is less, but we have a similar problem – it’s just lower in magnitude. Again we see that the statistics of two slightly different initial conditions – if we were to view them by the running average at any one time – are likely to be different even over this much longer time frame.

From this 25,000 second simulation:

- take 10,000 random samples each of 25 second length and plot a histogram of the means of each sample (the sample means)
- same again for 100 seconds
- same again for 500 seconds
- same again for 3,000 seconds

Repeat for the data from the other initial condition.

Here is the result:

*Figure 6*

To make it easier to see, here is the difference between the two sets of histograms, normalized by the maximum value in each set:

*Figure 7*

This is a different way of viewing what we saw in figures 4 & 5.

The spread of sample means shrinks as we increase the time period but the difference between the two data sets doesn’t seem to disappear (note 2).

### Attractors and Phase Space

The above plots show how variables change with time. There’s another way to view the evolution of system dynamics and that is by “phase space”. It’s a name for a different kind of plot.

So instead of plotting x vs time, y vs time and z vs time – let’s plot x vs y vs z – click to expand:

*Figure 8 – Click to expand – the colors blue, red & green represent the same initial conditions as in figure 2*

Without some dynamic animation we can’t now tell how fast the system evolves. But we learn something else that turns out to be quite amazing. The system always end up on the same “phase space”. Perhaps that doesn’t seem amazing yet..

Figure 7 was with three initial conditions that are almost identical. Let’s look at three initial conditions that are very different: x,y,z = 0, 1, 0; 5, 5, 5; 20, 8, 1:

*Figure 9 - Click to expand*

Here’s an example (similar to figure 7) from Strogatz – a set of 10,000 closely separated initial conditions and how they separate at 3, 6, 9 and 15 seconds. The two key points:

- the fast separation of initial conditions
- the long term position of any of the initial conditions is still on the “attractor”

*Figure 10*

A dynamic visualization on Youtube with 500,000 initial conditions:

*Figure 11*

There’s lot of theory around all of this as you might expect. But in brief, in a “dissipative system” the “phase volume” contracts exponentially to zero. Yet for the Lorenz system somehow it doesn’t quite manage that. Instead, there are an infinite number of 2-d surfaces. Or something. For the sake of a not overly complex discussion a wide range of initial conditions ends up on something very close to a 2-d surface.

This is known as a **strange attractor**. And the Lorenz strange attractor looks like a butterfly.

### Conclusion

Lorenz 1963 reduced convective flow (e.g., heating an atmosphere from the bottom) to a simple set of equations. Obviously these equations are a massively over-simplified version of anything like the real atmosphere. Yet, even with this very simple set of equations we find chaotic behavior.

Chaotic behavior in this example means:

- very small differences get amplified extremely quickly so that no matter how much you increase your knowledge of your starting conditions it doesn’t help much (note 3)
- starting conditions within certain boundaries will always end up within “attractor” boundaries, even though there might be non-periodic oscillations around this attractor
- the long term (infinite) statistics can be deterministic but over any “smaller” time period the statistics can be highly variable

### References

Deterministic nonperiodic flow, EN Lorenz, *Journal of the Atmospheric Sciences* (1963)

*Chaos: From Simple Models to Complex Systems*, Cencini, Cecconi & Vulpiani, *Series on Advances in Statistical Mechanics – Vol. 17* (2010)

*Non Linear Dynamics and Chaos*, Steven H. Strogatz, *Perseus Books* (1994)

### Notes

**Note 1**: The Lorenz equations:

dx/dt = σ (y-x)

dy/dt = rx – y – xz

dz/dt = xy – bz

where

x = intensity of convection

y = temperature difference between ascending and descending currents

z = devision of temperature from a linear profile

σ = Prandtl number, ratio of momentum diffusivity to thermal diffusivity

r = Rayleigh number

b = “another parameter”

And the “classic parameters” are σ=10, b = 8/3, r = 28

**Note 2**: Lorenz 1963 has over 13,000 citations so I haven’t been able to find out if this system of equations is transitive or intransitive. Running Matlab on a home Mac reaches some limitations and I maxed out at 25,000 second simulations mapped onto a 0.01 second time step.

However, I’m not trying to prove anything specifically about the Lorenz 1963 equations, more illustrating some important characteristics of chaotic systems

**Note 3**: Small differences in initial conditions grow exponentially, until we reach the limits of the attractor. So it’s easy to show the “benefit” of more accurate data on initial conditions.

If we increase our precision on initial conditions by 1,000,000 times the increase in prediction time is a massive 2½ times longer.

## The Holocaust, Climate Science and Proof

Posted in Commentary on February 4, 2015 | 512 Comments »

I’ve been a student of history for a long time and have read quite a bit about Nazi Germany and WWII. In fact right now, having found audible.com I’m listening to an audio book

The Coming of the Third Reich, by Richard Evans, while I walk, drive and exercise.It’s heartbreaking to read about the war and to read about the Holocaust. Words fail me to describe the awfulness of that regime and what they did.

But it’s pretty easy for someone who is curious about evidence, or who has had someone question whether or not the Holocaust actually took place, to find and understand the proof.

The photos. The bodies. The survivors’ accounts. The thousands of eyewitness accounts. The army reports. The stated aims of Hitler and many of the leading Nazis in their own words.

We can all understand how to weigh up witness accounts and photos. It’s intrinsic to our nature.

People who don’t believe the Nazis murdered millions of Jews are denying simple and overwhelming evidence.

Let’s compare that with the evidence behind the science of anthropogenic global warming (AGW) and the inevitability of a 2-6ºC rise in temperature if we continue to add CO2 and other GHGs to the atmosphere.

## Step 1 – The ‘greenhouse’ effect

To accept AGW of course you need to accept the ‘greenhouse’ effect. It’s fundamental science and not in question but what if you don’t take my word for it? What if you want to check for yourself?

And by the way, the complexity of the subject for many people becomes clear even at this stage, with countless hordes not even clear that the ‘greenhouse’ effect is a just a building block for AGW. It is not itself AGW.

AGW relies on the ‘greenhouse’ effect but also on other considerations.

I wrote The “Greenhouse” Effect Explained in Simple Terms to make it simple, yet not too simple. But that article relies on (and references) many basics – radiation, absorption and emission of radiation through gases, heat transfer and convection. All of those are necessary to understand the greenhouse effect.

Many people have conceptual misunderstandings of “basic” physics. In reading comments on this blog and on other blogs I often see fundamental misunderstanding of how heat transfer works. No space here for that.

But the difficulty of communicating a physics idea is very real. Once someone has a conceptual block because they think some process works a subtly different way, the only way to resolve the question is with equations. It is further complicated because these misunderstandings are often unstated by the commenter – they don’t realize they see the world differently from physics basics.

So when we need to demonstrate that the greenhouse effect is real, and that it increases with more GHGs we need some equations. And by ‘increases’ I mean more GHGs mean a higher surface temperature, all other things being equal. (Which, of course, they never are).

The equations are crystal clear and no one over the age of 10 could possibly be confused. I show the equations for radiative transfer (and their derivation) in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations:

I

_{λ}(0) = I_{λ}(τ_{m})e^{-τm}+ ∫ B_{λ}(T)e^{-τ}dτ [16]The terms are explained in that article. In brief, the equation shows how the intensity of radiation at the top of the atmosphere at one wavelength is affected by the number of absorbing molecules in the atmosphere. And, obviously, you have to integrate it over all wavelengths. Why do I even bring that up, it’s so simple?

Voila.

And equally obviously, anyone questioning the validity of the equation, or the results from the equation, is doing so from evil motives.

I do need to add that we have to prescribe the temperature profile in the atmosphere (and the GHG concentration) to be able to solve this equation. The temperature profile is known as the lapse rate – temperature reduces as you go up in altitude. In the tropical regions where convection is stronger we can come up with a decent equation for the lapse rate.

All you have to know is the first law of thermodynamics, the ideal gas law and the equation for the change in pressure vs height due to the mass of the atmosphere. Everyone can do this in their heads of course. But here it is:

So with these two elementary principles we can prove that more GHGs means a higher surface temperature

beforeany feedbacks. That’s the ‘greenhouse’ effect.## Step 2 – AGW = ‘Greenhouse effect’ plus feedbacks

This is so simple. Feedbacks are things like – a hotter world probably has more water vapor in the atmosphere, and water vapor is the most important GHG, so this amplifies the ‘greenhouse’ effect of increasing CO2. Calculating the changes is only a little more difficult than the super simple equations I showed earlier.

You just need a GCM – a climate model run on a supercomputer. That’s all.

There are many misconceptions about climate models but only people who are determined to believe a lie can possibly believe them.

As an example, many people think that the amplifying effect, or positive feedback, of water vapor is programmed into the GCMs. All they have to do is have a quick read through the 200-page technical summary of a model like say CAM (community atmosphere model).

Here is an extract from Description of the NCAR Community Atmosphere Model (CAM 3.0), W.D. Collins (2004):

As soon as anyone reads this – and if they can’t be bothered to find the reference via Google Scholar and read it, well, what can you say about such people – as soon as they read it, of course, it’s crystal clear that positive feedback isn’t “programmed in” to climate models.

So GCMs all come to the conclusion that more GHGs results in a hotter world (2-6ºC). They solve basic physics equations in a “grid” fashion, stepping forward in time, and so the result is clear and indisputable.

## Step 3 – Attribution Studies

I recently spent some time reading AR4 and AR5 (the IPCC reports) on Attribution (Natural Variability and Chaos – Seven – Attribution & Fingerprints Or Shadows? and Natural Variability and Chaos – Three – Attribution & Fingerprints).

This is the work of attributing the last century’s rise in temperature to the increases in anthropogenic GHGs. I followed the trail of papers back and found one of the source papers by Hasselmann from 1993. In it we can clearly see the basis for attribution studies:

Now it’s very difficult to believe that anyone questioning attribution studies isn’t of evil intent. After all, there is the basic principle in black and white. Who could be confused?

As a side note, to excuse my own irredeemable article on the topic, the actual basis of attribution isn’t just in these equations, it is also in the assumption that climate models accurately calculate the statistics of natural variability. The IPCC chapter on attribution doesn’t really make this clear, yet in another chapter (11) different authors suggest completely restating the statistical certainty claimed in the attribution chapter because “..it is explicitly recognized that there are sources of uncertainty not simulated by the models”. Their ad hoc restatement, while more accurate than the executive summary, still needs to be justified.

However, none of this can offer me redemption.

## Step 4 – Unprecedented Temperature Rises

(This could probably be switched around with step 3. The order here is not important).

Once people have seen the unprecedented rise in temperature this century, how could they not align themselves with the forces of good?

Anthropogenic warming ‘writ large’ (AR5, chapter 2):

There’s the problem. The last 400,000 years were quite static by comparison:

From ‘800,000 Years of Abrupt Climate Variability’, Barker et al (2011)The red is a Greenland ice core proxy for temperature, the green is a mid-latitude SST estimate – and it’s important to understand that calculating global annual temperatures is quite difficult and not done here.

So no one who looks at climate history can possibly be excused for not agreeing with consensus climate science, whatever that is when we come to “consensus paleoclimate”.. It was helpful to read Chapter 5 of AR5:

I’ve only read about 350 papers on paleoclimate and I’m confused about the origin of the high confidence as I explained in Ghosts of Climate Past -Eighteen – “Probably Nonlinearity” of Unknown Origin.

Anyway, the key takeaway message is that the recent temperature history is another demonstration that anyone not in line with consensus climate science is clearly acting from evil motives.

## Conclusion

I thought about putting a photo of the Holocaust from a concentration camp next to a few pages of mathematical equations – to make a point. But that would be truly awful.

That would trivialize the memory of the terrible suffering of millions of people under one of the most evil regimes the world has seen.

And that, in fact, is my point.

I can’t find words to describe how I feel about the apologists for the Nazi regime, and those who deny that the holocaust took place. The evidence for the genocide is overwhelming and everyone can understand it.

On the other hand, those who ascribe the word ‘denier’ to people not in agreement with consensus climate science are trivializing the suffering and deaths of millions of people. Everyone knows what this word means. It means people who are apologists for those evil jackbooted thugs who carried the swastika and cheered as they sent six million people to their execution.

By comparison, understanding climate means understanding maths, physics and statistics. This is hard, very hard. It’s time consuming, requires some training (although people can be self-taught), actually requires academic access to be able to follow the thread of an argument through papers over a few decades – and lots and lots of dedication.

The worst you could say is people who don’t accept ‘consensus climate science’ are likely finding basic – or advanced – thermodynamics, fluid mechanics, heat transfer and statistics a little difficult and might have misunderstood, or missed, a step somewhere.

The best you could say is with such a complex subject straddling so many different disciplines, they might be entitled to have a point.

If you have no soul and no empathy for the suffering of millions under the Third Reich, keep calling people who don’t accept consensus climate science ‘deniers’.

Otherwise, just stop.

Important Note: The moderation filter on comments is setup to catch the ‘D..r’ word specifically because such name calling is not accepted on this blog. This article is an exception to the norm, but I can’t change the filter for one article.

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