LENGTH SCALES IN SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

A set of ladder inequalities for the 2d and 3d forced Navier-Stokes eq uations on a periodic domain (0, L]d is developed, leading to a natura l definition of a set of length scales. We discuss what happens to the se scales if intermittent fluctuations in the vorticity field occur, a nd we consider how these scales compare to those derived from the attr actor dimension and the number of determining modes. Our methods are b ased on estimates of ratios of norms which appear to play a natural ro le and which make many of the calculations comparatively easy. In 3d w e cannot preclude length scales which are significantly shorter than t he Kolmogorov length. In 2d our estimate for a length scale l turns ou t to be (l/L)-2 less-than-or-equal-to cG(1 + log G)1/2 where G is the Grashof number. This estimate of l is shorter than that derived from t he attractor dimension. The reason for this is discussed in detail.