On topological fluid mechanics of non-ideal systems and virtual frozen-in dynamics
Abstract
Euler and Navier-Stokes have variant systems with dynamical invariance of helicity and thus (weak) topological equivalence, allowing a strong `frozen-in' (to, or, dually, `Lie-carried' by the \textit{virtual} velocity ) formulation of the vorticity with a flavor of `inverse Helmholtz theorem'. We remark on the non-ideal (statistical) topological fluid mechanics (TFM) for (1) the Constantin-Iyer formulation of Navier-Stokes, (2) our own extension of the Gallavotti-Cohen type dynamical ensembles of modified Navier-Stokes with energy-helicity constraints and (3) the Galerkin truncated Euler, as the typical case variants with dynamical time reversibility and helicity invariance. Ideal TFM is thus bridged with non-ideal flows. An example virtual (Lie-)carrier of the vorticity in a Galerkin-truncated Euler system is calculated to demonstrate the issue of determining .
Keywords
Cite
@article{arxiv.1702.04447,
title = {On topological fluid mechanics of non-ideal systems and virtual frozen-in dynamics},
author = {Jian-Zhou Zhu},
journal= {arXiv preprint arXiv:1702.04447},
year = {2018}
}
Comments
this and previous versions contain some technical error which will be fixed a bit later; closely relevant remarks concerning the virtual velocity are given in the appendix (A.2) of jfm14hydrochirality paper arXiv:1303.3823 and in the pof18Lie paper arXiv:1709.03356 Sec. III