  This document is on considerations and findings on modelling of spinning
beams. Spinning has been proposed for stabilizing beams against perturbations
notably risen by non-linear space charge forces, see [Y.-L. Cheon et al.,
Effects of beam spinning on the fourth-order particle resonance of 3D bunched
beams in high-intensity linear accelerators, Phys. Rev. Accel. & Beams 25,
064002 (2022)]. Although not further treated therein, spinning can be
quantified by angular momentumor by vorticity. Considering vorticity revealed
that the latter has remarkable similarity w.r.t. its modelling along solenoid
channels to modelling the beam envelope. Matrices of vorticity transport,
corresponding phase advances, and Twiss parameters look very similar and are
partially even identical to their counterparts concerning envelopes.
Corresponding to emittance, the quantity of vortissance, being a constant of
motion, is defined. Unlike emittance, for vorticity-dominated beams it may take
imaginary values, causing Twiss parameters, and negative or zero phase advances
along a finite beam line section. This imposes considerable consequences on
respective periodic solutions.
