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Quasisymmetry Matt Landreman University of Maryland 2 3 Guiding-center trajectories & neoclassical transport* in quasisymmetry are isomorphic to those in axisymmetry. • The equations of motion for a guiding center, when expressed in Boozer coordinates, depend on position only through s and B. • The gyro-averaged Lagrangian, i.e. the Lagrangian for guiding center motion, when expressed in Boozer coordinates, depends on position only through s and B. • In quasisymmetric magnetic fields, there is a conserved quantity analogous to canonical angular momentum that can guarantee confinement of charged particles. • When B is expressed as a function of Boozer coordinates, neoclassical radial fluxes and parallel flows depend on the magnetic field geometry only through B and functions of s. * “Neoclassical”: Fluxes due to guiding center orbits and collisions, without turbulence. 4 Outline • Axisymmetry can give confinement. • Guiding-center motion in Boozer coordinates depends only on B=|B|. • Lagrangian perspective. • Equivalent conditions for quasisymmetry. • How accurately has quasisymmetry been achieved? 5 Particle orbits in nonuniform B fields are complicated Magnetic field line Charged particle Given this complexity, how can we hope to confine particles? 6 Idea: To confine particles, we can constrain their position with a conservation law. Noether’s theorem: For each continuous symmetry of a system, there is a corresponding conserved quantity. Emmy Noether (1882-1935) 7 Lagrangian Mechanics & Noether’s Theorem 8 “True axisymmetry” Not axisymmetric Discrete rotational symmetry doesn’t count 9 Axisymmetry + Noether’s Theorem is one way to achieve confinement. Continuous rotational symmetry ⇒ Canonical angular momentum is conserved: If AϕR surfaces are bounded like this, then particles will be confined: 10 In axisymmetry, particles are confined (close) to AφR surfaces, despite complicated orbits. AϕR surface Magnetic field line Particle orbits 11 Particles are actually confined this way, in nature and in the laboratory. Tokamak (Fusion) Van Allen (1959) 12 Problem: Axisymmetric confinement requires an internal current. AϕR surface I Integration contour I 13 “Quasisymmetric” experiments to date actually have significant departures from symmetry. 14 Quasisymmetry can be achieved to any given accuracy, at least at high enough aspect ratio A. 15.

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