Ertel's vorticity theorems, the particle relabelling symmetry and the energy-vorticity theory of fluid mechanics

Abstract

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.