  Visual manifestations of intermittency in computations of three dimensional
Navier-Stokes fluid turbulence appear as the low-dimensional or `thin'
filamentary sets on which vorticity and strain accumulate as energy cascades
down to small scales. In order to study this phenomenon, the first task of this
paper is to investigate how weak solutions of the Navier-Stokes equations can
be associated with a cascade and, as a consequence, with an infinite sequence
of inverse length scales. It turns out that this sequence converges to a finite
limit. The second task is to show how these results scale with integer
dimension $D=1,\,2,\,3$ and, in the light of the occurrence of thin sets, to
discuss the mechanism of how the fluid might find the smoothest, most
dissipative class of solutions rather than the most singular.
