On topological fluid mechanics of non-ideal systems and virtual frozen-in dynamics

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 $V$) 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 $V$.