Dissipation in Turbulent Solutions of 2-D Euler

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 an appropriate notion of ``dissipative Euler solution'' in 2D. However, we show that the DiPerna-Majda solutions with vorticity in $L^p$ are conservative and have zero defect. We formulate a notion of weak Euler solution for distributional vorticities with zero H\"{o}lder exponent in space for which we conjecture dissipation in the strict sense may occur.