Optimal eddy viscosity in closure models for two-dimensional turbulent flows

We consider the question of fundamental limitations on the performance of eddy-viscosity closure models for turbulent flows, focusing on the Leith model for two-dimensional 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 wave number kc, 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 wave numbers. 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.