Ertel's vorticity theorems, the particle relabelling symmetry and the energy-vorticity theory of fluid mechanics
Ertel's vorticity theorems, the particle relabelling symmetry and the energy-vorticity theory of fluid mechanics
复制标题
Ertel 涡度定理、粒子重标记对称性和流体力学能量涡度理论
DOI：
10.1127/0941-2948/2004/0013-0485
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发表时间：
2004
影响因子：
1.2
通讯作者：
P. Névir
中科院分区：
文献类型：
--
作者：
P. Névir
ERTEL的各种涡度定理与流体力学的实质或拉格朗日描述中的独特对称性有关。根据诺特定理，这是粒子重新标记对称（PRS），也称为交换对称。缺少的是对正压流体和斜压流体的这种对称的分类，以及对欧拉框架的转换，揭示这种对称的普遍和实际意义。本文证明了重标对称是一种无限对称，类似于基本粒子物理中的规范对称。斜压流可以用一个自由任意函数来参数化，正压流可以用两个函数来参数化。在斜压情况下，我们得到ERTEL势熵在正压情况下，螺旋度作为全局守恒量。在这两种情况下，守恒定律都明确地依赖于物理相空间的任意函数。这种对称的产生器也可以转换为欧拉参考空间。作为一种特殊情况，我们可以推导出旋转和平移的产生器以及相关角动量和线动量守恒定律的约束条件。在这种情况下，任意函数只依赖于空间坐标。这个令人惊讶的结果表明，重新标记生成器具有伟大的普遍性。此外，这种重新标记的对称性负责流体力学的封闭欧拉描述，并给出了流体力学的能量涡量理论的理论解释。作为该理论的实际应用，可以定义动态状态指数（DSI）和化学状态指数（CSI）来诊断天气和气候系统。
The various vorticity theorems of ERTEL are related to a unique symmetry in the substantial or Lagrangian description of fluid mechanics. According to NOETHER'S theorem, this is the particle relabelling symmetry (PRS), also called exchange symmetry. What is missing is a classification of this symmetry for barotropic and baroclinic fluids and a transformation to the Eulerian frame, revealing the universal and practical implications of this symmetry. The paper shows that the relabelling symmetry is an infinite symmetry like the gauge symmetries in elementary particle physics. It can be parameterised by one free arbitrary function for baroclinic flows and two functions for barotropic flows. In the baroclinic case we get ERTEL'S potential enstrophy and in the barotropic case the helicity as global conserved quantities. In both cases the conservation laws are explicitly dependent upon arbitrary functions of the physical phase space. The generator of this symmetry can be also transformed to the Eulerian reference space. As a special case we can derive the generators of rotation and translation and the constraints for the related conservation laws of angular and linear momentum. In this case the arbitrary functions depend solely on the space coordinates. This surprising result is an indication that the relabelling generator is of great universal nature. Moreover, this relabelling symmetry is responsible for the closed Eulerian description of fluid mechanics and gives a theoretical explanation of the energy-vorticity theory of fluid mechanics. A dynamic state index (DSI) and a chemical state index (CSI) can be defined for the diagnosis of the weather and climate system as a practical application of this theory.