# On dynamical systems and climatology

That all of these are modeled on HPC gives me the tie-in to HPC that we need. I think we need more theoretical development, modeling, and model refinement. HPC systems accord us virtual laboratories that allow us to create and probe state spaces that may be impossible to consider in an experimental sense otherwise.

And this is, IMO, where we need to spend more time/effort/cycles.

Sadly we have political views impinging into scientific research, and this is problematic. Before I discuss this, let me talk briefly about dynamical systems, Poincare’ surfaces, and energy.

A pendulum is a very simple dynamical system. A double pendulum (a pendulum suspended from a pendulum) exhibits all of the features of more complex dynamical systems. You can add sources (additional energy/driving motion), and sinks (braking/retarding motion, removal of energy). You can model these.

Dynamical systems, well closed systems in equilibrium (e.g. the gahzintas = the gahzoutas … that is, the energy going into the system is balanced by the energy going out of the system … gahzintas is a play on Goes Into, and gahzoutas is a play on Goes Out Of) tend to follow particular paths, Poincare’ sections/surfaces. It doesn’t mean that you will always have regular repeating motion as in the simple pendulum. Specifically, the double pendulum exhibits motions that can be considered chaotic when sufficient energy is provided. In the non-chaotic regions, the motion may be ergodic, e.g. it effectively visits every point in the available parameter space, called phase space.

As in the above java applet, you can change parameters. Watch the phase space portrait (the paths taken by the variables) when you do. It is interesting. Change the lengths. Grab one of the masses and start it from a different location. Watch the velocity versus angle or velocity versus velocity and compare it to the simple default case.

In the default case, you see the phase space portrait as being ergodic. It explores a volume of phase space, fairly regular in shape.

Now add energy (e.g. move one of the masses to a nearly vertical position and let it go). What happens to the phase space portrait?

There is a point I am trying to back into here. Let me just state it, and then discuss it in the context of modeling other dynamical systems.

When you add (or remove) energy from a dynamical system (such as the double pendulum), all you do is alter the phase space portrait of motion, you don’t necessarily push the dynamical system to one “side” or the other (look at the pendulum when you do this … adding energy doesn’t bias it to remaining on one side or the other, or force the average height of the pendulum higher or lower).

Weird … huh?

Adding energy doesn’t shift the parameters. It shifts trajectories.

Of course we can add in sources of energy (say magnetic pushers/pullers), and sinks of energy (say an eddy current based brake with a copper sheet mass). When we do this, all we do is shift the trajectory.

If we are adding energy in net, gradually, our system will enter into chaotic parameter regimes. This means that nearby phase space trajectories will diverge from each other at an exponential rate in time.

If we are removing energy in net, gradually our system may settle down to metastable parameter regimes … it may get stuck in valleys … or ruts if you prefer. Think of a ball rolling back and forth between two hills, lacking sufficient energy to overcome the energy barrier and move out of the hilly area.

Ok, so now you are saying … sure, fine, … but what has this to do with climatology?

Ignoring the political aspect of the phase space portrait, the climate is a dynamical system with sinks and sources of energy. Climatological history suggests we have had many many trajectory regimes, and that there is coupling between multiple sinks/sources of energy. The processes are complex, not as well understood as we would like.

Again, ignoring the strong emotion behind the political projection of these phase space portraits, I humbly suggest that rather than talking about (dis)belief in AGW that we instead focus upon getting our Climatological models correct, able to be used as predictive tools that correctly model observed data from ice-cores and related data sets. Once we have the model(s) correct, we can use them for (hopefully) more accurate prediction. The HPC tie-in to this is that these models are tremendously computationally intensive. Possibly beyond our current ability to model them with any degree of confidence. I think we should be investing in designing HPC systems that can model this, and investing in the science underlying construction of these models.

I am not arguing for or against AGW in this piece, I am taking a decidedly neutral stance. I am pointing out that dynamical systems may behave differently than political types, who may not be aware of dynamical system behavior in the presence of sources and sinks of energy, may believe. This analysis is best left up to the scientific community. But you need to make sure that they have the funds and equipment to do the work. And hopefully, freedom from political biases while pursuing the research as well as getting funds for the research.

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