  We discuss a theoretical framework to define an optimal sub-grid closure for
shell models of turbulence. The closure is based on the ansatz that consecutive
shell multipliers are short-range correlated, following the third hypothesis of
Kolmogorov formulated for similar quantities for the original three-dimensional
Navier-Stokes turbulence. We also propose a series of systematic approximations
to the optimal model by assuming different degrees of correlations across
scales among amplitudes and phases of consecutive multipliers. We show
numerically that such low-order closures work well, reproducing all known
properties of the large-scale dynamics including anomalous scaling. We found
small but systematic discrepancies only for a range of scales close to the
sub-grid threshold, which do not tend to disappear by increasing the order of
the approximation. We speculate that the lack of convergence might be due to a
structural instability, at least for the evolution of very fast degrees of
freedom at small scales. Connections with similar problems for Large Eddy
Simulations of the three-dimensional Navier-Stokes equations are also
discussed. Postprint version of the article published on Phys. Rev. E 95,
043108 (2017) DOI: 10.1103/PhysRevE.00.003100
