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.<ref>Cary, John R.; Shasharina, Svetlana G. (September 1997). "Omnigenity and quasihelicity in helical plasma confinement systems". Physics of Plasmas. 4 (9): 3323–3333. Bibcode:1997PhPl....4.3323C. doi:10.1063/1.872473. ISSN 1070-664X.</ref> A particle is then confined to stay on a flux surface. All tokamaks are exactly omnigenous by virtue of their axisymmetry,<ref>Landreman, Matt (2019). "Quasisymmetry: A hidden symmetry of magnetic fields" (PDF).</ref> and conversely an unoptimized stellarator is generally not omnigenous.
Because an exactly omnigenous reactor has no neoclassical transport (in the collisionless limit),<ref>Beidler, C.D.; Allmaier, K.; Isaev, M.Yu.; Kasilov, S.V.; Kernbichler, W.; Leitold, G.O.; Maaßberg, H.; Mikkelsen, D.R.; Murakami, S.; Schmidt, M.; Spong, D.A. (2011-07-01). "Benchmarking of the mono-energetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS)". Nuclear Fusion. 51 (7): 076001. Bibcode:2011NucFu..51g6001B. doi:10.1088/0029-5515/51/7/076001. hdl:11858/00-001M-0000-0026-E9C1-C. ISSN 0029-5515. S2CID 18084812.</ref> 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,<ref>Rodríguez, E.; Helander, P.; Bhattacharjee, A. (June 2020). "Necessary and sufficient conditions for quasisymmetry". Physics of Plasmas. 27 (6): 062501. arXiv:2004.11431. Bibcode:2020PhPl...27f2501R. doi:10.1063/5.0008551. ISSN 1070-664X. S2CID 216144539.</ref> 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.<ref>Nührenberg, Jürgen (2010-12-01). "Development of quasi-isodynamic stellarators". Plasma Physics and Controlled Fusion. 52 (12): 124003. Bibcode:2010PPCF...52l4003N. doi:10.1088/0741-3335/52/12/124003. ISSN 0741-3335. S2CID 54572939.</ref>
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.<ref>Helander, Per (2014-07-21). "Theory of plasma confinement in non-axisymmetric magnetic fields". Reports on Progress in Physics. 77 (8): 087001. Bibcode:2014RPPh...77h7001H. doi:10.1088/0034-4885/77/8/087001. hdl:11858/00-001M-0000-0023-C75B-7. ISSN 0034-4885. PMID 25047050. S2CID 33909405.</ref> For trapped particles, omnigeneity relates closely to the second adiabatic invariant <math>\cal{J}</math> (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 <math>\cal{J}</math>,<ref>Hall, Laurence S.; McNamara, Brendan (1975). "Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: Theory of the magnetic plasma". Physics of Fluids. 18 (5): 552. Bibcode:1975PhFl...18..552H. doi:10.1063/1.861189.</ref><math display="block">\frac{\partial \cal{J}}{\partial \alpha} = q \Delta \psi</math>where <math>q</math> is the charge of the particle, <math>\alpha</math> is the magnetic field line label, and <math>\Delta \psi</math> is the total radial drift expressed as a difference in toroidal flux.<ref>D'haeseleer, William Denis. (6 December 2012). Flux Coordinates and Magnetic Field Structure : A Guide to a Fundamental Tool of Plasma Theory. Springer. ISBN 978-3-642-75595-8. OCLC 1159739471.</ref> 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,<math display="block">\frac{\partial \cal{J}}{\partial \alpha} = 0</math>This criterion is exactly met in axisymmetric systems, as the derivative with respect to <math>\alpha</math> can be expressed as a derivative with respect to the toroidal angle (under which the system is invariant).
