  Smooth solutions of the forced incompressible Euler equations satisfy an
energy balance, where the rate-of-change in time of the kinetic energy equals
the work done by the force per unit time. Interesting phenomena such as
turbulence are closely linked with rough solutions which may exhibit {\it
inviscid dissipation}, or, in other words, for which energy balance does not
hold.
  This article provides a characterization of energy balance for physically
realizable weak solutions of the forced incompressible Euler equations, i.e.
solutions which are obtained in the limit of vanishing viscosity. More
precisely, we show that, in the two-dimensional periodic setting, strong
convergence of the zero-viscosity limit is both necessary and sufficient for
energy balance of the limiting solution, under suitable conditions on the
external force. As a consequence, we prove energy balance for a general class
of solutions with initial vorticity belonging to rearrangement-invariant
spaces, and going beyond Onsager's critical regularity.
