Notes

Chapter 8: Implications for Everyday Systems

Section 4: Fluid Flow


Chaos theory and turbulence

The full Navier-Stokes equations for fluid flow are far from being amenable to traditional mathematical analysis. But some simplified ordinary differential equations which potentially approximate various situations in fluid flow can be more amenable to analysis—and can exhibit the chaos phenomenon. Work in the 1950s by Lev Landau, Andrei Kolmogorov and others focused on equations with periodic and quasiperiodic behavior. But in 1962 Edward Lorenz discovered more complicated behavior in computer experiments on equations related to fluid flow (see page 971). Analysis of this behavior was closely linked to the chaos phenomenon of sensitive dependence on initial conditions. And by the late 1970s it had become popular to believe that the randomness in fluid turbulence was somehow associated with this phenomenon.

Experiments in very restricted situations showed correspondence with iterated maps in which the chaos phenomenon is seen. But the details of the connection with true turbulence remained unclear. And as I argue in the main text, the chaos phenomenon in the end seems quite unlikely to explain most of the randomness we see in turbulence. The basic problem is that a complex pattern of flow in effect involves a huge amount of information—and to extract this information purely from initial conditions would require for example going to a submolecular level, far below where traditional models of fluids could possibly apply.

Even within the context of the Lorenz equations there are already indications of difficulties with the chaos explanation. The Lorenz equations represent a first-order approximation to certain Navier-Stokes-like equations, in which viscosity is ignored. And when one goes to higher orders progressively more account is taken of viscosity, but the chaos phenomenon becomes progressively weaker. I suspect that in the limit where viscosity is fully included most details of initial conditions will simply be damped out, as physical intuition suggests. Even within the Lorenz equations, however, one can see evidence of intrinsic randomness generation, in which randomness is produced without any need for randomness in initial conditions. And as it turns out I suspect that despite subsequent developments the original ideas of Andrei Kolmogorov about complicated behavior in ordinary differential equations were probably more in line with my notion of intrinsic randomness generation than with the chaos phenomenon.


From Stephen Wolfram: A New Kind of Science [citation]