AI 中文总结
该研究针对大尺度绝对平衡的二维湍流，证明了速度环量的面积规则严格成立，推导出仅依赖环周长的新周长规则，为二维环量方程提供了新稳态解，有助于探索湍流几何不变量。
AI 中文摘要
我们证明，传统上与湍流惯性区相关但被证明并非严格成立的速度环量面积规则，在具有拟能均分的二维均匀各向同性湍流的大尺度绝对平衡中严格成立。我们还从二维无粘性环量方程推导出一种新的周长规则，该规则指出速度环量的概率分布函数（PDF）仅取决于环的周长而非其面积。该周长规则在具有能量均分的大尺度绝对平衡中严格成立。在由拟能和能量共同决定的中间状态中，这两种状态被特征平衡尺度$l_\text{eq}$分隔：当$l\neq l_\text{eq}$时，面积规则支配环的统计特性；当$l\neq l_\text{eq}$时，周长规则显现。即使对于纵横比低至0.03的环，这些统计规律仍然稳健，而惯性区研究从未达到过该值。我们的发现为二维环量方程提供了一种新的稳态解，并暗示存在其他解，为探索湍流此前未发现的几何不变量铺平了道路。
英文摘要
We demonstrate that the area rule of velocity circulation -- traditionally associated with the turbulent inertial range but shown not to be exact -- is strictly satisfied in the large-scale absolute equilibrium of two-dimensional (2D) homogeneous isotropic turbulence under enstrophy equipartition. We also derive a novel perimeter rule from the 2D inviscid loop equation, which posits that the probability distribution function (PDF) of velocity circulation depends solely on the loop perimeter rather than its area. This perimeter rule holds strictly in the large-scale absolute equilibrium characterized by energy equipartition. At the intermediate states determined by both enstrophy and energy, these two regimes are separated by a characteristic equilibrium scale $l_\text{eq}$: the area rule governs loop statistics when $l\ll l_\text{eq}$, while the perimeter rule emerges for $l\gg l_\text{eq}$. These statistical laws remain robust even for loops with extreme aspect ratios as low as $0.03$, a value that inertial-range studies never achieved. Our findings provide a new steady-state solution to the 2D loop equation and suggest additional solutions, paving the way for exploring previously undiscovered geometric invariants of turbulence.
Comments7 pages, 4 figures
