Frobenius Integrability and the Transition to Turbulence.

A streamline flow is related to a 1-form of Action such that A^dA = 0. The velocity field and the vorticity reside on a streamline surface. When A^dA = 0, it is said that the 1-form satisfies the Frobenious requirements of unique integrability. That is, in 3D for example, there is a single function whose zero set defines a 2 dimensional surface, and whose normal field (gradient of the function) is orthogonal to the surface. The velocity field and the vorticity reside in this surface in streamline flow.

Now the antithesis of a streamline flow is a turbulent flow. it follows that the turbulent flow must be assoviated with situations where the Frobenius conditions are not satisfied. The decay of turbulence is a transition from a state where the Frobenius theorem is not satisfied to a state where the Frobeniou theorem is satisfied (and the flow is no longer turbulent ).

The transition to Turbulence implies that the original integrable process no longer satisfies the Frobenius . It is possible to demonstrate that the stremline flow generates a connected topology, while the non-streamline flow generates a non-connected topology.

It is possible to study the decay of turbulence by a continuous process, but it is impossible to study the creation of turbulence by a continuous process.

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Last update 07/02/2001
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