Omnigeneity Explained

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.

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

\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

\cal{J} ,^[7] $$\backslash frac\; =\; q\; \backslash Delta\; \backslash psi$$where q is the charge of the particle, \alpha is the magnetic field line label, and \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,$$\backslash frac\; =\; 0$$This criterion is exactly met in axisymmetric systems, as the derivative with respect to \alpha can be expressed as a derivative with respect to the toroidal angle (under which the system is invariant).

Notes and References

1. Cary . John R. . Shasharina . Svetlana G. . September 1997 . Omnigenity and quasihelicity in helical plasma confinement systems . Physics of Plasmas . en . 4 . 9 . 3323–3333 . 10.1063/1.872473 . 1997PhPl....4.3323C . 1070-664X.
2. Web site: Landreman . Matt . 2019 . Quasisymmetry: A hidden symmetry of magnetic fields .
3. 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 . 10.1088/0029-5515/51/7/076001 . 2011NucFu..51g6001B . 11858/00-001M-0000-0026-E9C1-C . 18084812 . 0029-5515. free .
4. Rodríguez . E. . Helander . P. . Bhattacharjee . A. . June 2020 . Necessary and sufficient conditions for quasisymmetry . Physics of Plasmas . en . 27 . 6 . 062501 . 10.1063/5.0008551 . 2004.11431 . 2020PhPl...27f2501R . 216144539 . 1070-664X.
5. Nührenberg . Jürgen . 2010-12-01 . Development of quasi-isodynamic stellarators . Plasma Physics and Controlled Fusion . 52 . 12 . 124003 . 10.1088/0741-3335/52/12/124003 . 2010PPCF...52l4003N . 54572939 . 0741-3335.
6. Helander . Per . 2014-07-21 . Theory of plasma confinement in non-axisymmetric magnetic fields . Reports on Progress in Physics . en . 77 . 8 . 087001 . 10.1088/0034-4885/77/8/087001 . 25047050 . 2014RPPh...77h7001H . 11858/00-001M-0000-0023-C75B-7 . 33909405 . 0034-4885. free .
7. 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 . en . 18 . 5 . 552 . 10.1063/1.861189. 1975PhFl...18..552H .
8. Book: D'haeseleer, William Denis. . Flux Coordinates and Magnetic Field Structure : A Guide to a Fundamental Tool of Plasma Theory . 6 December 2012 . Springer . 978-3-642-75595-8 . 1159739471.