Falkovich Gregory, Kadish Yotam, Vladimirova Natalia
Weizmann Institute of Science, Rehovot 76100, Israel.
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia.
Phys Rev E. 2023 Jul;108(1-2):015103. doi: 10.1103/PhysRevE.108.015103.
We suggest a new focus for turbulence studies-multimode correlations-which reveal the hitherto hidden nature of turbulent state. We apply this approach to shell models describing basic properties of turbulence. The family of such models allows one to study turbulence close to thermal equilibrium, which happens when the interaction time weakly depends on the mode number. As the number of modes increases, the one-mode statistics approaches Gaussian (like in weak turbulence), the occupation numbers grow, while the three-mode cumulant describing the energy flux stays constant. Yet we find that higher multimode cumulants grow with the order. We derive analytically and confirm numerically the scaling law of such growth. The sum of all squared dimensionless cumulants is equal to the relative entropy between the full multimode distribution and the Gaussian approximation of independent modes; we argue that the relative entropy could grow as the logarithm of the number of modes, similar to the entanglement entropy in critical phenomena. Therefore, the multimode correlations give the new way to characterize turbulence states and possibly divide them into universality classes.
我们建议为湍流研究设定一个新的重点——多模关联，它揭示了湍流状态迄今隐藏的本质。我们将这种方法应用于描述湍流基本特性的壳模型。这类模型家族使人们能够研究接近热平衡的湍流，这种情况发生在相互作用时间对模式数的依赖较弱时。随着模式数的增加，单模统计趋近于高斯分布（如在弱湍流中），占据数增加，而描述能量通量的三模累积量保持不变。然而，我们发现更高阶的多模累积量随阶数增长。我们通过解析推导并通过数值验证了这种增长的标度律。所有无量纲累积量平方的总和等于全多模分布与独立模式的高斯近似之间的相对熵；我们认为相对熵可能会随着模式数的对数增长，类似于临界现象中的纠缠熵。因此，多模关联为表征湍流状态提供了新方法，并可能将它们划分为不同的普适类。
