Climate and the Lorenz attractor, 3D interactive model.

In my previous post, I described some recent blog discussion of chaos and climate models (GCMs), and gave my views on the relation to Navier-Stokes solution and CFD. People who write about chaos tend to focus on the trajectories, and their touchy relations to starting point, claiming this undermines GCMs. They should instead focus on the attractors, which are independent of start point sensitivity, and are the analogue of climate. And I contend that attractors have a manageable relation (see Appendix for math) to parameters that may vary - forcings for climate, or coefficients in a chaos differential equation.

In this post, I'll focus on the Lorenz DE's



These have, for various parameters values, chaotic solutions with interesting trajectory paths often shown:

A trajectory for the standard Lorenz parameters σ=10, β=8/3, ρ=28. Often displayed without mentioning that specific parameters are required.	An attractor (due to Anders Sandberg, Oxford) parameters not specified but seems close to standard.

This post provides below a Javascript interactive display of the Lorenz system. You can choose parameters and start points. It is built on the WebGL of my standard Earth view, so you can rotate with mouse as if it were in a trackball, and also magnify or reduce by right button dragged vertically. There is also provision for viewing separate trajectories, and for running an animation of their evolution.

The general idea is that you can compare the effect of changing start points, with comparison red/blue trajectories, and also see the very great range of different attractors that result when the parameters are changed, However, the changes are continuous. What I'd like to get to eventually (future post) is a possible relation between an average of trajectories and the attractor. That would help understand how GCM runs can be averaged to get a climate evolution.

So here is the interactive WebGL gadget, starting with the standard parameter values:

The plot works as a trackball; you can drag a virtual enclosing sphere with mouse left button down; there is a little set of unit length colored axes (x-red, y-green, z-blue) in the centre which show what is happening. By dragging vertically with the right button, you can change the scale.

Data and controls are on the right. You can enter parameter values or initial (x,y,z) values. Click Go to show a new plot with values you have entered. By default up to two trajectories will show, one red, one blue. If you enter more, the last two will show. You can remove either using the red/blue X buttons.

The solver works by fourth order Runge Kutta from supplied initial point; N is the number of steps that you can enter. Each trajectory starts out with faint color, then heavier toward the end.

You can click the Orient button at any time to restore the y-axis to vertical. The bottom button with a ↻ rotates between a x-y, y-z and z-x view.

You can start an animation by pressing the "Track" button. The trajectories on show will be traced out. The time interval per frame is in the text box dt in units of millisecs nf is the total number of frames for the trajectory.

So do experiment. I'd suggest start with varying the initial point to compare trajectories, then try varying parameters gradually to change the attractor shape. next in the series will, if I can work out a way, show in magnification how trajectories cluster around an attractor. I'd like to explore whether the attractor does emerge from averaging trajectories. That would help with the question of averaging GCM ensemble results to derive climate.

As always, the Javascript for this gadget, which does all the calculation, is available by displaying the HTML (Ctrl-U) and following links. But most of it is here.

UpdateThere is some fairly simple maths that picks out a few points on the shapes. The centres of the spirals are where the derivative is zero, and this comes out to a cubic in x:
y=x,z=x*x/β, x*(ρ-x*x/β-1)=0
The solutions are x=y=z=0 and z=ρ-1;x=±sqrt(βz),y=x.
For standard params, z=27, x=y=±sqrt(72) marks the centres. The other root, (0,0,0) is not an attractor point.

Update Large values of N (100000 say) make pretty pictures and don't take too long to compute. But they can drain resources. Usually they will show initially, but dragging the picture may cause it to vanish, and animating may do similar. The problems seem to accumulate. In Javascript, you have to rely on the system garbage collection, which can be slow to clean up. If this starts happening, best to just refresh the page and start again.

Appendix - attractors

There has been much study of the nature of Lorenz attractors and dependence on parameters. Guckenheimer and Williams, 1979 created a geometric analogue for which the dependence could be shown. Steve Smale included proving the equivalence of this analogue in his interesting problems of next century list - it was solved with due promptness by Warwick Tucker in 2002. There is a more complete description here on the attractors and their behaviour in the various parameter regions.