R. M. Kiehn, "Topology and Turbulence" APS 1992

**Abstract: **Over a given regular domain of independent variables {x,y,z,t},
every covariant vector field of flow can be constructed in terms of at most four
functions and their first derivatives. These functions may be used to construct
a differential 1-form of Action. The irreducible number of such functions
required to describe the topological properties of the 1-form of Action is
defined as the Pfaff dimension of the domain. These functions, their first
derivatives, and certain algebraic intersections of these sets, may be used to
construct a Cartan topology, and thereby define the four basic topological
equivalence classes of Pfaff dimension one, two, three and four on space time.
Those vector field solutions to the Navier-Stokes equations which are uniquely
integrable in the sense of Frobenious emulate potential flows or streamline
processes; they generate an Action 1-form of Pfaff dimension 1 and 2,
respectively. Chaotic solutions to the Navier-Stokes equations must be
associated with Action domains of Pfaff dimension 3; but such flows are
deterministic and reversible in space time. Only domains of Pfaff dimension 4
admit of evolutionary flow processes that can break time reversal and parity
symmetry, and therefore may be used to describe the irreversible processes
inherent in turbulent flows. When the Pfaff dimension is 2 or less, the Cartan
Topology induced by the vector field is connected. When the Pfaff dimension is
greater than 2, the induced Cartan topology is disconnected. The topological
implication is that the creation of turbulence (a state of Pfaff dimension 4 and
a disconnected Cartan topology) from a streamline flow (a state of Pfaff
dimension 2 and a connected topology) can take place only by discontinuous
processes which induce shocks and tangential discontinuities. On the otherhand,
the decay of turbulence can be described by continuous, but irreversible,
processes. Numerical procedures that force continuity of slope and value cannot
in principle describe the creation of turbulence, but such techniques of forced
continuity can be used to describe the decay of turbulence.