In Part Nine – Data I – Ts vs OLR we looked at the monthly surface temperature (“skin temperature”) from NCAR vs OLR measured by CERES. The slope of the data was about 2 W/m² per 1K surface temperature change. Commentators pointed out that this was really the seasonal relationship – it probably didn’t indicate anything further.

In Part Ten we looked at anomaly data: first where monthly means were removed; and then where daily means were removed. Mostly the data appeared to be a big noisy scatter plot with no slope. The main reason that I could see for this lack of relationship was that anomaly data didn’t “keep going” in one direction for more than a few days. So it’s perhaps unreasonable to expect that we would find any relationship, given that most circulation changes take time.

We haven’t yet looked at regional versions of Ts vs OLR, the main reason is I can’t yet see what we can usefully plot. A large amount of heat is exported from the tropics to the poles and so without being able to itemize the amount of heat lost from a tropical region or the amount of heat gained by a mid-latitude or polar region, what could we deduce? One solution is to look at the whole globe in totality – which is what we have done.

In this article we’ll look at the mean global annual data. We only have CERES data for complete years from 2001 to 2013 (data wasn’t available to end of the 2014 when I downloaded it).

Here are the time-series plots for surface temperature and OLR:

*Figure 1*

Here is the scatter plot of the above data, along with the best-fit linear interpolation:

*Figure 2*

The calculated slope is similar to the results we obtained from the monthly data (which probably showed the seasonal relationship). This is definitely the year to year data, but also gives us a slope that indicates positive feedback. The correlation is not strong, as indicated by the R² value of 0.37, but it exists.

As explained in previous posts, a change of 3.6 W/m² per 1K is a “no feedback” relationship, where a uniform 1K change in surface & atmospheric temperature causes an OLR increase of 3.6 W/m² due to increased surface and atmospheric radiation – a greater increase in OLR would be negative feedback and a smaller increase would be positive feedback (e.g. see Part Eight with the plot of OLR changes vs latitude and height, which integrated globally gives 3.6 W/m²).

The problem of the”no feedback” calculation is perhaps a bit more complicated and I want to dig into this calculation at some stage.

I haven’t looked at whether the result is sensitive to the date of the start of year. Next, I want to look at the changes in humidity, especially upper tropospheric water vapor, which is a key area for radiative changes. This will be a bit of work, because AIRS data comes in big files (there is a lot of data).

## Natural Variability and Chaos – Two – Lorenz 1963

Posted in Atmospheric Physics, Climate Models, Commentary on July 27, 2014 | 22 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:

## Articles in the Series

Natural Variability and Chaos – One – Introduction

Natural Variability and Chaos – Two – Lorenz 1963

Natural Variability and Chaos – Three – Attribution & Fingerprints

Natural Variability and Chaos – Four – The Thirty Year Myth

Natural Variability and Chaos – Five – Why Should Observations match Models?

Natural Variability and Chaos – Six – El Nino

Natural Variability and Chaos – Seven – Attribution & Fingerprints Or Shadows?

Natural Variability and Chaos – Eight – Abrupt Change

## 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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