Number of degrees of freedom of two-dimensional turbulence

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)=omega(0). This number denoted by N is defined as the minimum dimension such that for n >= 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 <= C1Re when the phase space is endowed with the energy norm and N <= C2Re(1+ln Re)(1/3) when the phase space is endowed with the enstrophy norm. Here C-1 and C-2 are constant and Re is the Reynolds number defined in terms of omega(0), 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 D-H of the global attractor scale highly superlinearly with nu(-1). We argue that the "extra" dependence of D-H 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.