Local invariants in non-ideal flows of neutral fluids and two-fluid plasmas

Local structures, beyond the well-known `frozen-in' to the barotropic flows of the generalized vorticities, of the two-fluid model of plasma flows are presented. More general non-barotropic situations are also considered. A modified Euler equation [T. Tao, Ann. PDE \textbf{2}, 9 (2016)] is also accordingly analyzed and remarked from the angle of view of two-fluid model, with emphasis on the local structures. And, the local constraints of high-order differential forms such as helicity, among others, find simple formulation for possible applications in modelling the dynamics. Thus, the Cauchy invariants equation [N. Besse and U. Frisch, J. Fluid Mech. \textbf{825}, 412 (2017)] may find practical application in non-ideal flows. Some formal examples are offered to outline the calculations, and particularly interestingly the two-dimensional-three-component (2D3C) or the 2D passive scalar problem presents that a locally invariant $\Theta = 2\theta \zeta$, with $\theta$ and $\zeta$ being respectively the scalar value of the `vertical velocity' (or the passive scalar) and the `vertical vorticity', may be used as if it were the spatial density of the globally invariant helicity, providing a Lagrangian prescription to control the latter in some situations of studying its physical effects in rapidly rotating flows (ubiquitous in atmosphere of astrophysical objects) with marked 2D3C vortical modes or in purely 2D passive scalars.