omnigeneity

File:W7X-Spulen Plasma blau gelb.jpg

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.{{Cite journal |last1=Cary |first1=John R. |last2=Shasharina |first2=Svetlana G. |date=September 1997 |title=Omnigenity and quasihelicity in helical plasma confinement systems |url=http://aip.scitation.org/doi/10.1063/1.872473 |journal=Physics of Plasmas |language=en |volume=4 |issue=9 |pages=3323–3333 |doi=10.1063/1.872473 |bibcode=1997PhPl....4.3323C |issn=1070-664X|url-access=subscription }} A particle is then confined to stay on a flux surface. All tokamaks are exactly omnigenous by virtue of their axisymmetry,{{Cite web |last=Landreman |first=Matt |date=2019 |title=Quasisymmetry: A hidden symmetry of magnetic fields |url=https://hiddensymmetries.princeton.edu/sites/g/files/toruqf1546/files/landreman_-_introduction_to_quasisymmetry.pdf }} and conversely an unoptimized stellarator is generally not omnigenous.

Because an exactly omnigenous reactor has no neoclassical transport (in the collisionless limit),{{Cite journal |last1=Beidler |first1=C.D. |last2=Allmaier |first2=K. |last3=Isaev |first3=M.Yu. |last4=Kasilov |first4=S.V. |last5=Kernbichler |first5=W. |last6=Leitold |first6=G.O. |last7=Maaßberg |first7=H. |last8=Mikkelsen |first8=D.R. |last9=Murakami |first9=S. |last10=Schmidt |first10=M. |last11=Spong |first11=D.A. |date=2011-07-01 |title=Benchmarking of the mono-energetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS) |url=https://iopscience.iop.org/article/10.1088/0029-5515/51/7/076001 |journal=Nuclear Fusion |volume=51 |issue=7 |pages=076001 |doi=10.1088/0029-5515/51/7/076001 |bibcode=2011NucFu..51g6001B |hdl=11858/00-001M-0000-0026-E9C1-C |s2cid=18084812 |issn=0029-5515|hdl-access=free }} 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,{{Cite journal |last1=Rodríguez |first1=E. |last2=Helander |first2=P. |last3=Bhattacharjee |first3=A. |date=June 2020 |title=Necessary and sufficient conditions for quasisymmetry |url=http://aip.scitation.org/doi/10.1063/5.0008551 |journal=Physics of Plasmas |language=en |volume=27 |issue=6 |pages=062501 |doi=10.1063/5.0008551 |arxiv=2004.11431 |bibcode=2020PhPl...27f2501R |s2cid=216144539 |issn=1070-664X}} 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.{{Cite journal |last=Nührenberg |first=Jürgen |date=2010-12-01 |title=Development of quasi-isodynamic stellarators |url=https://iopscience.iop.org/article/10.1088/0741-3335/52/12/124003 |journal=Plasma Physics and Controlled Fusion |volume=52 |issue=12 |pages=124003 |doi=10.1088/0741-3335/52/12/124003 |bibcode=2010PPCF...52l4003N |s2cid=54572939 |issn=0741-3335|url-access=subscription }}

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.{{Cite journal |last=Helander |first=Per |date=2014-07-21 |title=Theory of plasma confinement in non-axisymmetric magnetic fields |url=https://doi.org/10.1088/0034-4885/77/8/087001 |journal=Reports on Progress in Physics |language=en |volume=77 |issue=8 |pages=087001 |doi=10.1088/0034-4885/77/8/087001 |pmid=25047050 |bibcode=2014RPPh...77h7001H |hdl=11858/00-001M-0000-0023-C75B-7 |s2cid=33909405 |issn=0034-4885|hdl-access=free }} For trapped particles, omnigeneity relates closely to the second adiabatic invariant $\cal{J}$ (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 $\cal{J}$ ,{{Cite journal |last1=Hall |first1=Laurence S. |last2=McNamara |first2=Brendan |date=1975 |title=Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: Theory of the magnetic plasma |url=https://aip.scitation.org/doi/10.1063/1.861189 |journal=Physics of Fluids |language=en |volume=18 |issue=5 |pages=552 |doi=10.1063/1.861189|bibcode=1975PhFl...18..552H |url-access=subscription }} $\frac{\partial \cal{J}}{\partial \alpha} = q \Delta \psi$ 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.{{Cite book |author=D'haeseleer, William Denis. |url=http://worldcat.org/oclc/1159739471 |title=Flux Coordinates and Magnetic Field Structure : A Guide to a Fundamental Tool of Plasma Theory |date=6 December 2012 |publisher=Springer |isbn=978-3-642-75595-8 |oclc=1159739471}} 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, $\frac{\partial \cal{J}}{\partial \alpha} = 0$ 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).

References

Category:Fusion reactors

Category:Electromagnetism