title: Number of degrees of freedom of two-dimensional turbulence.
abstract: We derive upper bounds for the number of degrees of freedom of two-dimensional Navier-Stokes turbulence freely decaying from a smooth initial vorticity field omega(x,y,0)=omega0. This number denoted by N is defined as the minimum dimension such that for n>or=N, arbitrary n-dimensional balls in phase space centered on the solution trajectory omega(x,y,t); for t>0, contract under the dynamics of the system linearized about omega(x,y,t). In other words, N is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that N<or=C1Re when the phase space is endowed with the energy norm and N<or=C2Re(1+ln Re)1/3 when the phase space is endowed with the enstrophy norm. Here C1 and C2 are constant and Re is the Reynolds number defined in terms of omega0, the system length scale, and the viscosity nu. The linear (or nearly linear) dependence of N on Re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number. This result is in a sharp contrast to the forced case, for which well-known estimates for the Hausdorff dimension DH of the global attractor scale highly superlinearly with nu(-1). We argue that the "extra" dependence of DH on nu(-1) is not an intrinsic property of the turbulent dynamics. Rather, it is a "removable artifact" brought about by the use of a time-independent forcing as a model for energy and enstrophy injection that drives the turbulence.
authors.affiliation: School of Mathematics and Statistics, University of St. Andrews, St. Andrews KY16 9SS, United Kingdom.
authors.forename: Chuong V
authors.initials: CV
authors.lastname: Tran
authors.forename: Luke
authors.initials: L
authors.lastname: Blackbourn