Piecewise omnigenity

The drifting of particles across flux surfaces is generally only a problem for trapped particles, which are trapped in a magnetic mirror. Passing particles, which can circulate freely around the flux surface, are automatically confined to stay on a flux surface.^[3] For trapped particles, omnigeneity and piecewise omnigenity relate closely to the second adiabatic invariant ${\displaystyle {\cal {J}}}$.

One can show that the radial drift a particle experiences after one full bounce motion is simply related to a derivative of ${\displaystyle {\cal {J}}}$,^[5]$${\displaystyle {\frac {\partial {\cal {J}}}{\partial \alpha }}=q\Delta \psi }$$where ${\displaystyle q}$ is the charge of the particle, ${\displaystyle \alpha }$ is the magnetic field line label, and ${\displaystyle \Delta \psi }$ is the total radial drift expressed as a difference in toroidal flux.^[6] In a piecewise omnigenous field, the flux surface of the stellarator is composed of several regions; within each of them, the second adiabatic invariant should be the same for all the magnetic field lines, and thus$${\displaystyle {\frac {\partial {\cal {J}}}{\partial \alpha }}=0}$$inside each region. The junctures between regions can be shown not to cause neoclassical transport.