A streamline flow is related to a 1form 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 1form 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 nonstreamline flow
generates a nonconnected 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.
