  We consider the question of fundamental limitations on the performance of
eddy-viscosity closure models for turbulent flows, focusing on the Leith model
for 2D {Large-Eddy Simulation}. Optimal eddy viscosities depending on the
magnitude of the vorticity gradient are determined subject to minimum
assumptions by solving PDE-constrained optimization problems defined such that
the corresponding optimal Large-Eddy Simulation best matches the filtered
Direct Numerical Simulation. First, we consider pointwise match in the physical
space and the main finding is that with a fixed cutoff wavenumber $k_c$, the
performance of the Large-Eddy Simulation systematically improves as the
regularization in the solution of the optimization problem is reduced and this
is achieved with the optimal eddy viscosities exhibiting increasingly irregular
behavior with rapid oscillations. Since the optimal eddy viscosities do not
converge to a well-defined limit as the regularization vanishes, we conclude
that in this case the problem of finding an optimal eddy viscosity does not in
fact have a solution and is thus ill-posed. We argue that this observation is
consistent with the physical intuition concerning closure problems. The second
problem we consider involves matching time-averaged vorticity spectra over
small wavenumbers. It is shown to be better behaved and to produce physically
reasonable optimal eddy viscosities. We conclude that while better behaved and
hence practically more useful eddy viscosities can be obtained with stronger
regularization or by matching quantities defined in a statistical sense, the
corresponding Large-Eddy Simulations will not achieve their theoretical
performance limits.
