Omnigeneity

Omnigeneity (sometimes also called omnigenity) is a property of a magnetic field inside a magnetic confinement fusion reactor. Such a magnetic field is called omnigenous if the path a single particle takes does not drift radially inwards or outwards on average.^[1] A particle is then confined to stay on a flux surface. All tokamaks are exactly omnigenous by virtue of their axisymmetry,^[2] and conversely an unoptimized stellarator is generally not omnigenous.

A flux surface of Wendelstein 7-X (yellow), a magnetic field line on that flux surface (green), and the coils needed to generate the magnetic field (blue). Wendelstein 7-X is designed to be nearly omnigenous.

Because an exactly omnigenous reactor has no neoclassical transport (in the collisionless limit),^[3] stellarators are usually optimized in a way such that this criterion is met. One way to achieve this is by making the magnetic field quasi-symmetric,^[4] and the Helically Symmetric eXperiment takes this approach. One can also achieve this property without quasi-symmetry, and Wendelstein 7-X is an example of a device which is close to omnigeneity without being quasi-symmetric.^[5]

Theory

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).