Title: 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 $V$)
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 $V$.