  Enstrophy, half the integral of the square of vorticity, plays a role in 2D
turbulence theory analogous to that played by kinetic energy in the Kolmogorov
theory of 3D turbulence. It is therefore interesting to obtain a description of
the way enstrophy is dissipated at high Reynolds number. In this article we
explore the notions of viscous and transport enstrophy defect, which model the
spatial structure of the dissipation of enstrophy. These notions were
introduced by G. Eyink in an attempt to reconcile the Kraichnan-Batchelor
theory of 2D turbulence with current knowledge of the properties of weak
solutions of the equations of incompressible and ideal fluid motion. Three
natural questions arise from Eyink's theory: (1) Existence of the enstrophy
defects (2) Conditions for the equality of transport and viscous enstrophy
defects (3) Conditions for the vanishing of the enstrophy defects. In
[Nonlinearity, v 14 (2001) 787-802], Eyink proved a number of results related
to these questions and formulated a conjecture on how to answer these problems
in a physically meaningful context. In the present article we improve and
extend some of Eyink's results and present a counterexample to his conjecture.
