2004/02/29 by Chuong V. Tran, Theodore G. Shepherd, Han-Ru Cho +1
Computer Science · Engineering · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #nlin.CD
paper · pdf · doi:10.1016/j.physd.2004.01.002
No figures, 14 pages
openalex publication_date 2004/02/29 · arxiv created 2004/03/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This study is concerned with how the attractor dimension of the two-dimensional Navier--Stokes equations depends on characteristic length scales, including the system integral length scale, the forcing length scale, and the dissipation length scale. Upper bounds on the attractor dimension derived by Constantin--Foias--Temam are analysed. It is shown that the optimal attractor-dimension estimate grows linearly with the domain area (suggestive of extensive chaos), for a sufficiently large domain, if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed. For sufficiently small domain area, a slightly ``super-extensive'' estimate becomes optimal. In the extensive regime, the attractor-dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined, on physical grounds, in terms of the average rate of shear. This dissipation length scale (which is not necessarily the scale at which the energy or enstrophy dissipation takes place) can be identified with the dimension correlation length scale, the square of which is interpreted, according to the concept of extensive chaos, as the area of a subsystem with one degree of freedom. Furthermore, these length scales can be identified with a ``minimum length scale'' of the flow, which is rigorously deduced from the concept of determining nodes.