Number of degrees of freedom of two-dimensional turbulence

TL;DR

This paper derives upper bounds for the number of degrees of freedom in two-dimensional turbulence, showing a linear or near-linear dependence on the Reynolds number, contrasting with the superlinear estimates in forced turbulence.

Contribution

It provides new bounds on the degrees of freedom in unforced 2D turbulence, highlighting differences from forced cases and challenging previous assumptions about attractor dimensions.

Findings

Number of degrees of freedom scales linearly with Reynolds number in unforced turbulence.

Contrasts with superlinear scaling of attractor dimension in forced turbulence.

Suggests the superlinear scaling is due to artificial forcing, not intrinsic turbulence properties.

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 $\omega(x,y,0)=\omega_0$. 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 $\omega(x,y,t)$, 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 $N\le C_1R_e$ 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 $\omega_0$, the system length scale, and the viscosity $\nu$. 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 $\nu^{-1}$. 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.