  In ideal fluids, Clebsch potentials occur as paired canonical variables
associated with the Hamiltonian description of the Euler equations. This paper
explores the properties of the incompressible Navier-Stokes equations when the
velocity field is expressed through a complete set of paired Clebsch
potentials. First, it is shown that the incompressible Navier-Stokes equations
can be cast as a system of transport (convection-diffusion) equations where
each Clebsch potential plays the role of a generalized distribution function.
The diffusion operator associated with each Clebsch potential departs from the
standard Laplacian due to a term depending on the Lie-bracket of the
corresponding Clebsch pair. It is further shown that the Clebsch potentials can
be used to define a Shannon-type entropy measure, i.e. a functional, different
from energy and enstrophy, whose growth rate is non-negative. As a consequence,
the flow must vanish at equilibrium. This functional can be interpreted as a
measure of the topological complexity of the velocity field. In addition, the
Clebsch parametrization enables the identification of a class of flows, larger
than the class of two dimensional flows, possessing the property that the
vortex stretching term identically vanishes and the growth rate of entrophy is
non-positive.
