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

Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials