  We prove the conservation of energy for weak and statistical solutions of the
two-dimensional Euler equations, generated as strong (in an appropriate
topology) limits of the underlying Navier-Stokes equations and a Monte
Carlo-Spectral Viscosity numerical approximation, respectively. We characterize
this conservation of energy in terms of a uniform decay of the so-called
structure function, allowing us to extend existing results on energy
conservation. Moreover, we present numerical experiments with a wide variety of
initial data to validate our theory and to observe energy conservation in a
large class of two-dimensional incompressible flows.
