Component-wise dimensionally reduced flows and helicity conservation

It is proved, with a no-go theorem of transforming all one type of real Schur matrices into the other type by the same (orthogonal) transformation, that the so-called real Schur flows (RSFs) corresponding to the two types of uniformly real Schur form velocity gradient matrices are different; on the other hand, the further component-wise dimensionally reduced lone Schur flow (LSF)'' is unique in the sense that simple uniform transformations such as the switch of the coordinate axes are sufficient to unify them. One type of RSFs can have closed streamlines only on the equilibrium planes of the velocity component dimensionally reduced to be one-dimensonal. The theorem of no closed streamlines in LSFs leads to a simple (re)definition of vortex’’ and swirl''. Not all component-wise dimensionally reduced flows (CWDRFs) associated to the Euler equation correspond to an invariant manifold of the latter, but the intersection" of the two types of RSFs do. Previous proofs, by Moffatt and by Khesin & Chekanov, of the helicity invariance in barotropic ideal flows were overkilling in the sense of using the unnecessary condition of local mass conservation, while our new ``sharper’’ proof without invoking the latter carries over to CWDRFs.