  We establish local balance equations for smooth functions of the vorticity in
the DiPerna-Majda weak solutions of 2D incompressible Euler, analogous to the
balance proved by Duchon and Robert for kinetic energy in 3D. The anomalous
term or defect distribution therein corresponds to the ``enstrophy cascade'' of
2D turbulence. It is used to define a rather natural notion of ``dissipative
Euler solution'' in 2D. However, we show that the DiPerna-Majda solutions with
vorticity in $L^p$ for $p>2$ are conservative and have zero defect. Instead, we
must seek an alternative approach to dissipative solutions in 2D. If we assume
an upper bound on the energy spectrum of 2D incompressible Navier-Stokes
solutions by the Kraichnan-Batchelor $k^{-3}$ spectrum, uniformly for high
Reynolds number, then we show that the zero viscosity limits of the
Navier-Stokes solutions exist, with vorticities in the zero-index Besov space
$B^{0,\infty}_2$, and that these give a weak solution of the 2D incompressible
Euler equations. We conjecture that for this class of weak solutions enstrophy
dissipation may indeed occur, in a sense which is made precise.
