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.