## A Short Summary of Future Discussions

The chaotic phenomenology of small systems of non-linear ODEs is entirely numerical ODE chaos. And, the original Lorenz system of 1963 contains no physical phenomena or processes of interest in NWP and AOLGCM applications.

The only very rough physics contained in the original Lorenz 3D equations is the onset of motion for a differentially-heated fluid mass. The equations are a severely truncated series approximation for an already overly-simplified model of fluid flows. The published numbers obtained from the equations are generally for conditions that are not obtained for fluids of interest in the atmosphere or oceans; the Prandtl number is too large and the Rayleigh number is far outside the range of applicability of the ‘model’.

For me, lack of demonstrated convergence of numerical solutions for even the simple original Lorenz equations means that the equations have not been solved. The calculated numbers do not represent solutions to the continuous equations. If convergence is not demonstrated how can even the ‘phenomenology’ of the Lorenz equations be adapted for any applications whatsoever. How can past states of the global climate, for example, be labeled ‘an attractor’? I find the present state of this situation to be unparalleled with respect to the complete lack of even the faintest hint of acknowledgement that fundamental aspects of analysis have been ignored.

It is very significant that the first peer-reviewed paper to investigate convergence of numerical approximations to the original Lorenz system appeared in 2007; over 40 years after the original publication of the equations and some calculated numbers.

The lack of representation of any physical phenomena or processes of interest, and the lack of any actual solutions to the equations, seem to not have been a deterrent relative to adapting the ‘chaotic phenomenology’ that the calculated numbers exhibit. An over-riding aspect of these numbers is the apparent lack of convergence of the numerical ‘solutions’ to the continuous equations. They are simply numbers, do not represent solutions to the equations, are numerical noise, and represent ‘numerical chaos’ and nothing more.

Frequently statements along the line that all complex physical phenomena and processes, modeled with non-linear differential equations, exhibit chaotic responses. While I just took some liberty in stating the phrase, I’m sure I can find statements that are almost exactly that. However, non-linearity might be a necessary condition for chaotic responses, but it is not sufficient. Neither is sensitivity to initial conditions a sufficient condition. And it is also a true fact that not all complex non-linear model equations exhibit chaotic responses. If the archetypical original Lorenz equations are taken as an example, solutions to the equations have yet to appear. It is very likely that none of the low-order non-linear ODEs that are said to exhibit chaotic response have ever been correctly solved.

The mathematical models used in NWP and AOLGCMs consist of systems PDEs plus ODEs plus a very large number of algebraic equations. The dimensions of the discrete equations numbers in the millions. Nothing of a theoretical nature is known about the chaotic response of such equation systems; either the continuous or the discrete. How the phrase ‘an attractor’ or ‘the trajectory’ can even be applied to these systems is not at all clear. Generally, numbers calculated by use of complex computer models/codes, for which the initial conditions are unknown, including both those measured in the physical system and those that satisfy the models equations, seem to be declared ‘chaotic’ by osmosis. Certainly no theoretical basis has ever been presented. The invoked analogies to turbulent fluid flows and molecular dynamics are additionally totally false. There is not a peer-reviewed paper trail from the simple ODE systems to the very complex systems used in NWP and AOLGCMs. NWP and AOLGCM chaos is based entirely on the numbers calculated by these complex models/codes.

NWP and AOLGCM calculations are known to be under-resolved in both time and space. Convergence of the numerical methods in these codes has not yet been demonstrated. The ‘chaos’ is based entirely on all the potential artifacts of algebraic approximations to the continuous equations and the approximate numerical ‘solutions’ to enormous systems of algebraic equations. Again the calculated numbers are known and acknowledged to not be solutions of the discrete equations. Under-resolution has been demonstrated time and time again to have a great potential to introduce purely numerical chaos into numerical solutions in which chaos should not be present. Lorenz himself has demonstrated this phenomena.

The invocation of the original 1963 Lorenz equations as the phenomenology observed in the numbers from NWP and AOLGCM models/codes cannot be justified on any theoretical grounds; ‘it looks like chaotic response, so Lorenz is correct again’ does not qualify.

NWP and AOLGCM chaos is entirely and purely numerical chaos. Unfortunately, I think this statement cannot be shown to be false. But neither can the statement ‘the weather is chaotic’ be shown to be false.

Dan my take on this issue would be as follows .

First I do not think that there can be a mathematical definition of deterministic chaos .

It is true that everybody would agree on a number of necessary conditions – non linearity , sensitivity to initial conditions , exponential growth of errors …

Yet what everybody has in mind as sufficient condition is unpredictabilty .

The trouble with that is that it is circular – you don’t prove unpredictability , you observe it .

Also when you look at the whole zoo of chaos examples , you see that there is all sort of very different behaviors and all of them are experimentally observed .

So now the daring jump to the underlying mathematics .

What strikes me is that in all cases without exception , and I’ll limit myself to only continuous cases , the equations deemed to describe the observed chaotic system cannot be analytically solved and mostly you have not even basic proofs about existence , unicity and continuity .

One solution would be to say that the ODE and/or PED systems actually do not completely describe the system and that we miss something but nobody is going this way .

So you are only left with numerical methods .

Now those must necessarily fail because precisely the necessary condition for chaos to exist is the sensibility to initial conditions .

Whatever small numerical step you take , you will still be too far from the dt and dx that would be necessary and that will ensure by itself that the calculation will NEVER converge .

That’s why it is only fitting that when you run a calculation with different steps and however small they are , you obtain different “solutions” .

So as it is impossible to find an exact solution , you can begin to restrict the problem to finite times (that is the whole trick with Lyapounov coefficients) or go over to stochastics .

For me that transition is an important and relevant one because it completely changes the paradigm like in that very puzzling yet frequent statement “Weather is chaotic but climate is not .”

You no longer look for THE solution of the equations but you look for A function that has the same properties like what the unknown solution should have (f.ex has the right invariants) and its dynamics make it stay in the same subset of the phase space as the exact solution .

So assuming that you are able to describe the topology of the sub space where the solution(s) dwell , you can make some statistical statements about the relevance to the problem of functions who dwell in the same space .

That’s typically what the shadowing lemma does .

Now where I completely join you is that there is preciously little work being done on this field .

On a particular note – there are 2 views of the problem .

One is the relationship between physical chaotic systems and the equations supposed to completely describe them .

Second is the ability of numerical methods to approach with a suitable precision the continuous solutions (if they exist) of certain ODE/PDE systems like the Lorenz’s one regardless of the fact if they describe a real physical system or not .

Comment by Tom Vonk | March 30, 2007 |