二维截断欧拉流的几何微正则理论
Geometric microcanonical theory of two-dimensional Truncated Euler flows
摘要：
本文提出了一个几何微正则系综的角度来看二维截断欧拉流，其中包含有限数量的（傅立叶）模式和守恒能量和拟能。我们明确执行相空间体积积分壳的恒定能量和拟能。考虑了两个应用程序。在第一部分中，我们确定的平均能谱高度凝聚流配置，并表明，结果是一致的Kraichnan的正则系综描述，尽管没有热力学限制调用的事实。在第二部分中，我们计算的概率密度为最大规模的模式的自由滑流在一个广场上，它显示反转。我们测试的结果对数值模拟的极小模型，并发现极好的协议与微正则理论，不像正则理论，它不能描述双峰统计。本文是“物理流体动力学中的数学问题“专题的一部分。
英文摘要：
This paper presents a geometric microcanonical ensemble perspective on two-dimensional Truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In a first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In a second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue "Mathematical problems in physical fluid dynamics".
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