  The Gaussian-filtered Navier-Stokes equations are examined theoretically and
a generalized theory of their numerical stability is proposed. Using the exact
expansion series of subfilter-scale stresses or integration by parts, the terms
describing the interaction between the mean and fluctuation portions in a
statistically steady state are theoretically rewritten into a closed form in
terms of the known filtered quantities. This process involves high-order
derivatives with time-independent coefficients. Detailed stability analyses of
the closed formulas are presented for determining whether a filtered system is
numerically stable when finite difference schemes or others are used to solve
it. It is shown that by the Gaussian filtering operation, second and higher
even-order derivatives are derived that always exhibit numerical instability in
a fixed range of directions; hence, if the filter widths are unsuitably large,
the filtered Navier-Stokes equations can in certain cases be unconditionally
unstable even though there is no error in modeling the subfilter-scale stress
terms. As is proved by a simple example, the essence of the present discussion
can be applied to any other smooth filters; that is, such a numerical
instability problem can arise whenever the dependent variables are smoothed out
by a filter.
