2-D Turbulence is a Myth
Although there is no universal definition of the Turbulent state, the one property that is agreed upon is that a turbulent process is thermodynamically irreversible. Recall that chaos does not imply turbulence for there exist chaotic processes that are reversible.
It is known that chaos does not occur for a system that can be described in terms of two functions or less. It takes three (irreducible) functions as a necessary condition to produce chaos. The Pfaff dimension of a chaotic system is at least 3. An argument based upon the evolution of Cartan topology can be used to demonstrate that the Pfaff dimension of a chaotic system must be 3, and the Pfaff dimension of an irreversible process must be 4!

Hence 2-D turbulence is a myth.

This does not imply that experiments on thin films are not useful in developing an understanding of fluid flows, but the claim that the thin film is truely a time dependent, 2-D manifold, and yet admits turbulence cannot be strictly true. The thin film can admit chaos, as it has 2 spatial and 1 timelike dimension = Pfaff dimension 3. Variations in the thickness of the film imply the system is of Pfaff dimension 4, which therefore can exhibit turbulence. It is just that the turbulence - when observed - is not a 2 dimensional time dependent flow.

The criteria of what makes up an irreversible process can be determined from classical thermodynamics, without the use of statistics. An irreversible process is a process for which the heat 1-form does not admit an integrating factor. This statement can be translated into the language of differential forms, for it is the equivalent to the Frobenius integrability theorem. When Q^dQ <> 0, Q does not admit an intgerating factor.

It is easy to show that all symplectic processes ( which include extremal processes on contact structures, Bernoulli-Casimir processes, and -in general- all processes for which the open integrals of flux are invariant) are thermodynamically reversible. The argument applies to physical systems which can be described in terms of a 1-form of Action, A. The integrals of the derived 2-form, F=dA, define the flux.

For all C2 continuous processes, the closed integrals of flux are evolutionary deformation invariants. This statement is the cohomological equivalent of Helmholtz theorem.

However, the open integrals of flux are not necessarily evolutionary invariants, and it is this set of processes which are irreversible. It is extraordinary that evolution in the direction of the Torsion vector , A^F, on a 4-D domain where F^F <>0 will define a process that is thermodynamically irreversible, for then Q^dQ<>0.

When Q^dQ <> 0 there does not exist an integrating factor for the heat 1-form.


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Last update 01/23/2009
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