二维不可压缩流体动力学中的典型尺度分离
Canonical scale separation in two-dimensional incompressible hydrodynamics
AI总结：
本文证明二维无粘不可压缩 Euler 方程存在由 Poisson 括号与 Hamiltonian 定义的涡量典型分裂，可自然产生尺度分离并解释 Kraichnan 双重级串与能谱折线。
摘要：
二维无粘不可压缩流体由简单的规则支配。然而，刻画其长时间行为却是一个棘手问题。流体按照 Euler 方程演化：这是一个具有无穷多个守恒律的非线性 Hamiltonian 系统。在实验和数值模拟中，初始阶段之后都会出现相干涡结构（或称涡块）。这些结构主导大尺度动力学，但小尺度也持续存在。Kraichnan 在其经典工作中描述了拟涡能向更小尺度的正级串，以及能量向更大尺度的反级串。以往对 Kraichnan 双重级串进行建模的尝试采用滤波技术，从一开始就强制实现分离。本文表明，Euler 方程具有涡量函数的内在、典型分裂。该分裂在四个方面引人注目：(i) 它仅通过 Poisson 括号和 Hamiltonian 定义；(ii) 它刻画定常流；(iii) 它无需强加条件即可产生尺度分离，使 Kraichnan 定性描述背后的动力学得以实现；(iv) 它解释了实验和数值模拟中均观察到的能谱幂律中的“折线”。这种分裂源于 Zeitlin 的 Euler 方程截断模型与一种标准量子工具——Hermitian 矩阵的谱分解——的结合。除理论洞见外，这种尺度分离动力学还可用于随机模型约简，其中小尺度由乘性噪声建模。
英文摘要：
A two-dimensional inviscid incompressible fluid is governed by simple rules. Yet, to characterise its long-time behaviour is a knotty problem. The fluid evolves according to Euler's equations: a non-linear Hamiltonian system with infinitely many conservation laws. In both experiments and numerical simulations, coherent vortex structures, or blobs, emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also persist. Kraichnan describes in his classical work a forward cascade of enstrophy into smaller scales, and a backward cascade of energy into larger scales. Previous attempts to model Kraichnan's double cascade use filtering techniques that enforce separation from the outset. Here we show that Euler's equations posses an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely via the Poisson bracket and the Hamiltonian, (ii) it characterises steady flows, (iii) without imposition it yields a separation of scales, enabling the dynamics behind Kraichnan's qualitative description, and (iv) it accounts for the "broken line" in the power law for the energy spectrum, observed in both experiments and numerical simulations. The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum-tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics could be used for stochastic model reduction, where small scales are modelled by multiplicative noise.
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