Omnigeneity

The drifting of particles across flux surfaces is generally only a problem for trapped particles, which are trapped in a magnetic mirror. Untrapped (or passing) particles, which can circulate freely around the flux surface, are automatically confined to stay on a flux surface.[6] For trapped particles, omnigeneity relates closely to the second adiabatic invariant ${\displaystyle {\cal {J}}}$ (often called the parallel or longitudinal invariant).

One can show that the radial drift a particle experiences after one full bounce motion is simply related to a derivative of ${\displaystyle {\cal {J}}}$,[7]

$${\displaystyle {\frac {\partial {\cal {J}}}{\partial \alpha }}=q\Delta \psi }$$

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.[8] With this relation, omnigeneity can be expressed as the criterion that the second adiabatic invariant should be the same for all the magnetic field lines on a flux surface,

$${\displaystyle {\frac {\partial {\cal {J}}}{\partial \alpha }}=0}$$

This criterion is exactly met in axisymmetric systems, as the derivative with respect to ${\displaystyle \alpha }$ can be expressed as a derivative with respect to the toroidal angle (under which the system is invariant).