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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