In Part Three we looked at attribution in the early work on this topic by Hegerl et al 1996. I started to write Part Four as the follow up on Attribution as explained in the 5th IPCC report (AR5), but got caught up in the many volumes of AR5.

And instead for this article I decided to focus on what might seem like an obscure point. I hope readers stay with me because it is important.

Here is a graphic from chapter 11 of IPCC AR5:

*Figure 1*

And in the introduction, chapter 1:

Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The relevant quantities are most often surface variables such as temperature, precipitation and wind.

Classically the period for averaging these variables is 30 years, as defined by the World Meteorological Organization.Climate in a wider sense also includes not just the mean conditions, but also the associated statistics (frequency, magnitude, persistence, trends, etc.), often combining parameters to describe phenomena such as droughts. Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer.

[Emphasis added].

### Weather is an Initial Value Problem, Climate is a Boundary Value Problem

The idea is fundamental, the implementation is problematic.

As explained in Natural Variability and Chaos – Two – Lorenz 1963, there are two key points about a chaotic system:

- With even a minute uncertainty in the initial starting condition, the predictability of future states is very limited
- Over a long time period the statistics of the system are well-defined

(Being technical, the statistics are well-defined in a *transitive* system).

So in essence, we can’t predict the exact state of the future – from the current conditions – beyond a certain timescale which might be quite small. In fact, in current weather prediction this time period is about one week.

After a week we might as well say either “the weather on that day will be the same as now” or “the weather on that day will be the climatological average” – and either of these will be better than trying to predict the weather based on the initial state.

No one disagrees on this first point.

In current climate science and meteorology the term used is the **skill** of the forecast. Skill means, not how good is the forecast, but how much **better** is it than a **naive** approach like, “it’s July in New York City so the maximum air temperature today will be 28ºC”.

What happens in practice, as can be seen in the simple Lorenz system shown in Part Two, is a tiny uncertainty about the starting condition gets amplified. Two almost identical starting conditions will diverge rapidly – the “butterfly effect”. Eventually these two conditions are no more alike than one of the conditions and a time chosen at random from the future.

The wide divergence doesn’t mean that the future state can be anything. Here’s an example from the simple Lorenz system for three slightly different initial conditions:

*Figure 2*

We can see that the three conditions that looked identical for the first 20 seconds (see figure 2 in Part Two) have diverged. The values are bounded but at any given time we can’t predict what the value will be.

On the second point – the statistics of the system, there is a tiny hiccup.

But first let’s review what is agreed upon. Climate is the statistics of weather. Weather is unpredictable more than a week ahead. Climate, as the **statistics of weather**, might be predictable. That is, just because weather is unpredictable, it doesn’t mean (or prove) that climate is also unpredictable.

This is what we find with simple chaotic systems.

So in the endeavor of climate modeling the best we can hope for is a probabilistic forecast. We have to run “a lot” of simulations and review the statistics of the parameter we are trying to measure.

To give a concrete example, we might determine from model simulations that the mean sea surface temperature in the western Pacific (between a certain latitude and longitude) in July has a mean of 29ºC with a standard deviation of 0.5ºC, while for a certain part of the north Atlantic it is 6ºC with a standard deviation of 3ºC. In the first case the spread of results tells us – if we are confident in our predictions – that we know the western Pacific SST quite accurately, but the north Atlantic SST has a lot of uncertainty. We can’t do anything about the model spread. In the end, the statistics are knowable (in theory), but the actual value on a given day or month or year are not.

Now onto the hiccup.

With “simple” chaotic systems that we can perfectly model (note 1) we don’t know in advance the timescale of “predictable statistics”. We have to run lots of simulations over long time periods until the statistics converge on the same result. If we have parameter uncertainty (see Ensemble Forecasting) this means we also have to run simulations over the spread of parameters.

Here’s my suggested alternative of the initial value vs boundary value problem:

*Figure 3*

So one body made an ad hoc definition of climate as the 30-year average of weather.

If this definition is correct and accepted then “climate” is not a “boundary value problem” at all. Climate is an initial value problem and therefore a massive problem given our ability to forecast only one week ahead.

Suppose, equally reasonably, that the statistics of weather (=climate), given constant forcing (note 2), are **predictable over a 10,000 year period**.

In that case we can be confident that, with near perfect models, we have the ability to be confident about the averages, standard deviations, skews, etc of the temperature at various locations on the globe over a 10,000 year period.

### Conclusion

The fact that chaotic systems exhibit certain behavior doesn’t mean that 30-year statistics of weather can be reliably predicted.

30-year statistics might be just as dependent on the initial state as the weather three weeks from today.

### Notes

**Note 1**: The climate system is obviously imperfectly modeled by GCMs, and this will always be the case. The advantage of a simple model is we can state that the model is a perfect representation of the system – it is just a definition for convenience. It allows us to evaluate how slight changes in initial conditions or parameters affect our ability to predict the future.

The IPCC report also has continual reminders that the model is not reality, for example, chapter 11, p. 982:

For the remaining projections in this chapter the spread among the CMIP5 models is used as a simple, but crude, measure of uncertainty. The extent of agreement between the CMIP5 projections provides rough guidance about the likelihood of a particular outcome. But — as partly illustrated by the discussion above —

it must be kept firmly in mindthat the real world could fall outside of the range spanned by these particular models.

[Emphasis added].

Chapter 1, p.138:

Model spread is often used as a measure of climate response uncertainty, but such a measure is crude as it takes no account of factors such as model quality (Chapter 9) or model independence (e.g., Masson and Knutti, 2011; Pennell and Reichler, 2011), and not all variables of interest are adequately simulated by global climate models..

..Climate varies naturally on nearly all time and space scales, and quantifying precisely the nature of this variability is challenging, and is characterized by considerable uncertainty.

I haven’t yet been able to determine how these firmly noted and challenging uncertainties have been factored into the quantification of 95-100%, 99-100%, etc, in the various chapters of the IPCC report.

**Note 2**: There are some complications with defining exactly what system is under review. For example, do we take the current solar output, current obliquity,precession and eccentricity as fixed? If so, then any statistics will be calculated for a condition that will anyway be changing. Alternatively, we can take these values as changing inputs in so far as we know the changes – which is true for obliquity, precession and eccentricity but not for solar output.

The details don’t really alter the main point of this article.

## Natural Variability and Chaos – Two – Lorenz 1963

Posted in Atmospheric Physics, Climate Models, Commentary on July 27, 2014 | 21 Comments »

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:

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

Figure 1Steven 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, 0We 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 3Going 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 expandTwo 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 expandThe 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:

Repeat for the data from the other initial condition.

Here is the result:

Figure 6To make it easier to see, here is the difference between the two sets of histograms, normalized by the maximum value in each set:

Figure 7This 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 2Without 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 expandHere’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:

From Strogatz 1994

Figure 10A dynamic visualization on Youtube with 500,000 initial conditions:

Figure 11There’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:

## 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.

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