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 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.