Higher order Mori-Zwanzig models for the Euler equations

In a recent paper \cite{CHSS06}, an infinitely long memory model (the t-model) for the Euler equations was presented and analyzed. The model can be derived by keeping the zeroth order term in a Taylor expansion of the memory integrand in the Mori-Zwanzig formalism. We present here a collection of models for the Euler equations which are based also on the Mori-Zwanzig formalism. The models arise from a Taylor expansion of a different operator, the orthogonal dynamics evolution operator, which appears in the memory integrand. The zero, first and second order models are constructed and simulated numerically. The form of the nonlinearity in the Euler equations, the special properties of the projection operator used and the general properties of any projection operator can be exploited to facilitate the recursive calculation of even higher order models. We use our models to compute the rate of energy decay for the Taylor-Green vortex problem. The results are in good agreement with the theoretical estimates. The energy decay appears to be organized in "waves" of activity, i.e. alternating periods of fast and slow decay. Our results corroborate the assumption in \cite{CHSS06}, that the modeling of the 3D Euler equations by a few low wavenumber modes should include a long memory.