Low dimensional attractors arise from forcing at small scales

Standard estimates of the dimension of the attractor of the 2D Navier-Stokes equations are given in terms of a dimensionless Grashof number that measures the total amount of energy injected into the flow. However, this result takes no I account of whether the forcing is concentrated at large or small scales. By a simple modification of the usual argument, this paper provides a bound that is linear in a modified Grashof number that reflects the spatial structure of the forcing. In particular, as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases. (C) 2003 Elsevier Science B.V. All rights reserved