values to zero (a procedure called "clipping") with or without the averaging also resulted in stable calculations. Meneveau proposed an averaging over Lagrangian fluid trajectories with an exponentially decaying "memory". This can be applied to problems lacking homogeneous directions and can be stable if the effective time over which the averaging is done is long enough and yet not so long as to smooth out spatial inhomogenieties of interest. Lilly's modification of the Germano method followed by a statistical averaging or synthetic removal of negative viscosity regions seems ad hoc, even if it could be made to "work". An alternate formulation of the least square minimization procedure known as the "Dynamic Localization Model" (DLM) was suggested by Ghosal et al. In this approach one first defines a quantity with the tensors and replaced by the appropriate SGS model. This tensor then represents the amount by which the subgrid model fails to respect the Germano identity at each spatial location. In Lilly's approach, is then pulled out of the hat operator making an algebraic function of which is then determined by requiring that considered as a function of C have the least possible value. However, since the thus obtained turns out to be just as variable as any other fluctuating quantity in turbulence, the original assumption of the constancy of cannot be justified a posteriori. In the DLM approach one avoids this inconsistency by not invoking the step of removing C from the test filtering operation. Instead, one defines a global error over the entire flow domain by the quantity where the integral ranges over the whole fluid volume. This global error is then a functional of the spatially varying function (here the time instant, , is fixed and therefore appears just as a parameter) which is determined so as to minimize this functional. The solution to this variational problem is that must satisfy a Fredholm integral equation of the second kind where the functions and are defined in terms of the resolved fields and are therefore known at each time step and the integral ranges over the whole fluid domain. The integral equation is solved numerically by an iteration procedure and convergence was found to be generally rapid if used with a pre-conditioning scheme. Even though this variational approach removes an inherent inconsistency in Lilly's approach, the obtained from the integral equation still displayed the instability associated with negative viscosities.