  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.
