  We estimate the maximal Lyapunov exponent at different resolutions and
Reynolds numbers in large eddy (LES) and direct numerical simulations (DNS) of
sinusoidally-driven Navier--Stokes equations in three dimensions. Independent
of the Reynolds number when nondimensionalized by Kolmogorov units, the LES
Lyapunov exponent diverges as an inverse power of the effective grid spacing
showing that the fine scale structures exhibit much faster error growth rates
than the larger ones. Effectively, i.e., ignoring the cut-off of this
phenomenon at the Kolmogorov scale, this behavior introduces an upper bound to
the prediction horizon that can be achieved by improving the precision of
initial conditions through refining of the measurement grid.
