Models Methods Software

Dan Hughes

Chaos and Butterflies yet again

The NWP and GCM communities cannot think that a Butterfly will have any influence whatsoever on any physical phenomena or processes of interest. Instead the phenomenology of The Butterfly Effect as exhibited by the numerical calculations of some systems of ordinary differential equations is invoked by hypothesis into NWP and GCM models/methods/codes. I think we need to limit discussions to the Lorenz-like systems of ODEs, as these seem to be the basis for invoking the phenomenology into the NWP and GCM communities. Otherwise we will get side-tracked into discussions of the “chaotic response of complex dynamical systems” in general.

The phenomenology, however, is a strictly numerical artifact observed in some simplified and specialized systems of, generally, simple systems of non-linear ODEs. None of these equation systems are known to describe any known fluid flows. As noted by Jim Clarke above, the Butterfly Effect has not been observed in the natural systems of interest. The effect cannot be, and will not ever be, observed in the natural systems of interest.

Importantly, the phenomenology cannot be determined to be present in the continuous equations and its presence is based solely on observations of numbers produced by calculations by numerical solution methods. That is, there are no mathematical ‘necessary and sufficient’ tests that can be applied to the small simple systems of ODEs to know a priori that chaotic response is an expected and correct outcome. While complexity, non-linearity, and sensitivity to initial conditions are frequently used interchangeably with chaotic response, none of these alone, or even all together provide necessary and sufficient conditions.

In view of the necessity of determination of the phenomenon of chaotic response by calculated numbers it is imperative that the numbers be actual solutions to the continuous equations. It should also be necessary that the continuous equations be complete and accurate descriptions of the physical phenomena and processes of interest. In the absence of meeting these conditions, it seems that the phenomenology cannot be assigned to the physical systems of interest.

The uncertainty of this situation is compounded by the fact that NWP and AOLGCM continuous equations are large systems of complex PDEs plus ODEs plus numerous algebraic equations for the parameterizations. That is, these are not small systems of simple ODEs. And additionally compounded by the fact that it is known that the numerical methods in both the NWP and AOLGCM fields have yet to produce numbers that are not functions of the discrete increments in time and space. Thus the calculated numbers are known to not be solutions of even the discrete approximations and the convergence to solutions of the continuous equations is not even yet addressed. Under these conditions, the observed chaotic response is at the very best only some kind of response for the specific calculation. The observed chaotic response cannot in any way be assigned to the continuous equations and most certainly cannot be assigned to the physical systems.

And while the communities seem very eager to adapt the notion of chaotic response of dynamical systems, they do not seem to accept all the consequences that come with the total concept. The known fact that long-range predictability might not be possible for the specific applications of interest is an example. Dissipative systems have multiple attractors, if the concept of an attractor(s) can even be shown to apply to systems having an infinite number of dimensions, and an associated basin. There is no way to ensure to which basin assigned initial conditions belong and so even statistical properties based on the calculated numbers are not predictable.

The Global Climate Model community seems to be counting on the over-riding effects of the forced boundary conditions to control the course of a calculation as a new state of radiative equilibrium is approached. In a sense, the radiative equilibrium state represents the stable fixed point to which dynamic systems return in the absence of chaotic response. The numbers calculated from ODEs that exhibit chaotic response do not show that a fixed point is attained. Only some regions of the calculated response are approached, and then under only certain conditions, but a stationary fixed point is never attained.

The end states of thermodynamic systems are functions of the path taken (processes encountered) to those states. As each (never-ending) path in systems that exhibit chaotic response is different for each set of initial conditions, the thermodynamic states all along the paths are different. The internal adjustments to the calculated states of the materials that make up the physical systems, as determined by the mathematical representations of the physical phenomena and processes, will all be different for different initial conditions and the contents of the mathematical models. It is no wonder then that all the models/codes are calculating different numbers for the states of the systems. If the calculations were allowed to continue to the point that the applied boundary conditions begin to assert controlling guidance as radiative equilibrium is approached, the internal adjustments are very likely to be inconsistent with the actual state of the physical systems.

And we haven’t yet even started to investigate and discuss the over-riding importance of numerical solution methods on the calculated numbers. These methods are the ultimate source of the displayed results. To put it in another light, the thermodynamic states of the system as now calculated by the NWP and AOLGCM models/codes are functions of the discrete representations of the continuous equations. Thermodynamic principles cannot be allowed to be functions of approximate solutions to approximate representations of physical systems. The hypothesis that the physical systems of interest in NWP and GCMs exhibit chaotic response has yet to be tested and has yet to be proven correct.

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June 3, 2007 - Posted by | Chaos and Lorenz, Verification | , , , ,

2 Comments »

  1. Dan,

    Beautiful stuff.

    Reminds me of Krybczkinski: “The map is not the territory.”

    Being transcribed into your world: “A mathematical curiosity has no necessary analogue in Nature.”

    Why do natural scientists, apparently, receive no education in scientific method, the history of science, or in philosophical rigour (necessary and sufficient conditions)?

    Why so many plumbers and so few scientists?

    We are in the middle of a shift in the scientific paradigm, from simple, linear systems to complex, non-linear ones. Why don’t the mass of scientists apprehend this? And its implications?

    Mike

    Comment by Mike Hollinshead | September 2, 2007 | Reply

  2. Thanks Mike.

    Comment by Dan Hughes | September 4, 2007 | Reply


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