二维turbulence的自由度
Number of degrees of freedom of two-dimensional turbulence
摘要：
本文从光滑的初始涡量场$ω(x,y,0)=ω_0$出发，导出了二维Navier-Stokes湍流自由衰减的自由度的上界。这个数字，由$N$表示，被定义为最小尺寸，使得对于$n\ge N$，以解轨迹$ω(x,y,t)$为中心的相空间中的任意$n$维球，对于$t>0$，在关于$ω(x,y,t)$线性化的系统动力学下收缩。换句话说，$N$是最大李雅普诺夫指数的最小数量，其总和变为负值。当相空间赋以能量范数时，发现$N\le C_1R_e$，而当相空间赋以拟能范数时，发现$N\le C_2R_e(1+\ln R_e)^{1/3}$。这里，$C_1$和$C_2$是常数，$R_e$是根据$ω_0$、系统长度尺度和粘度$ν$定义的雷诺数。$N$对$R_e$的线性（或近线性）依赖性与从最近的粘性耗散波数的数学界推导出的活跃模式的数量的估计是一致的。这一结果是在一个鲜明的对比，强迫的情况下，众所周知的估计Hausdorff维数$D_H$的全球吸引子规模高度超线性与$ν^{-1}$。我们认为$D_H$对$ν^{-1}$的“额外”依赖性不是湍流动力学的内在性质。相反，它是一个“可移动的人工制品”，通过使用与时间无关的强迫作为驱动湍流的能量和拟能注入的模型而产生。
英文摘要：
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.
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