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 $ω(x,y,0)=ω_0$. This number, denoted by $N$, is defined as the minimum dimension such that for $n\ge N$, arbitrary $n$-dimensional balls in phase space centred on the solution trajectory $ω(x,y,t)$, for $t>0$, contract under the dynamics of the system linearized about $ω(x,y,t)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that $N\le C_1R_e$ when the phase space is endowed with the energy norm, and $N\le C_2R_e(1+\ln R_e)^{1/3}$ when the phase space is endowed with the enstrophy norm. Here $C_1$ and $C_2$ are constant and $R_e$ is the Reynolds number defined in terms of $ω_0$, the system length scale, and the viscosity $ν$. The linear (or nearly linear) dependence of $N$ on $R_e$ 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 $ν^{-1}$. We argue that the "extra" dependence of $D_H$ on $ν^{-1}$ 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.