Particle motion in 3D MHD equilibria versus relaxed states e 1 David Pfefferl´ 1 The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia 23 rd Australian Institute of Physics Congress, 9-13 December, Perth, Australia
Magnetic fields to foster fusion reactions • magnetic confinement is most promising path to viable fusion energy • on topological grounds, the doughnut (torus) is only shape that can support a vector field without singularities tokamak: axisymmetric, helical stellarator: 3D, no plasma current, winding via strong plasma current helical winding via coils D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 2 / 18
Charged particle dynamics For uniform magnetic field B = B e z z = v || t + z 0 � x � = R ( − ωt ) ρ ⊥ + X y where R ( θ ) is the rotation matrix around e z of angle θ ω = qB/m the Larmor frequency m ρ ⊥ = qB b × v ⊥ is the Larmor radius helical motion along uniform magnetic field D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 3 / 18
Charged particle dynamics “Grad-B” drift due to non- uniform amplitude | B | V B = µ q b × ∇ B B mv 2 where µ = is the ⊥ 2 B “magnetic moment” upward drift due to non-uniform magnetic field D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 4 / 18
Charged particle dynamics “Curvature” drift due to bending field-lines mv 2 || V κ = qB b × κ where κ = b · ∇ b is the field-line curvature upward drift due to curved magnetic field D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 5 / 18
Charged particle dynamics “Mirror trapping” in magnetic bottles m 2 v 2 || + µB = const where µ = mv 2 is the magnetic ⊥ 2 B moment D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 6 / 18
Particle motion in tokamaks “passing” particles closely follow field-lines D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 7 / 18
Particle motion in tokamaks “trapped” particles bounce back and forth in B ∼ 1 /R (banana orbit) D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 8 / 18
Particle motion in stellarator stellarator 3D fields ⇒ complex motion, detrapping, magnetic wells,. . . D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 9 / 18
Guiding-centre motion Fast gyration can be “averaged out” via clever coordinate change 2 m v 2 + q A · v ⇒ ( q A + mv || b ) · ˙ L ( x , v , t ) = 1 X − ( 1 2 mv 2 || + µB ) 10 − 8 • expansion in m/q justifies adiabatic conservation of µ = mv 2 ⊥ / 2 B • requires sufficiently smooth magnetic field | ρ · ∇ B | /B ≪ 1 m q − → 0 D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 10 / 18
Drift-line topology and drift-surfaces passing particles as “modified field-lines” perturbation field-line dl = B ‡ d X B ‡ = ∇ × ( A + m B , q v || B ) B ‡ axisymmetric flux-surfaces axisymmetric drift-surfaces flux−surface counter−passing 1 co−passing 0.5 Z [m] 0 −0.5 −1 2 2.5 3 3.5 R [m] D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 11 / 18
Drift-line topology and drift-surfaces passing particles as “modified field-lines” perturbation field-line dl = B ‡ d X B ‡ = ∇ × ( A + m B , q v || B ) B ‡ internally kinked flux-surfaces drift-islands φ = 0 o φ = 120 o φ = 240 o 0.6 0.4 0.2 Z [m] 0 −0.2 −0.4 −0.6 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4 R [m] R [m] R [m] D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 11 / 18
Drift-line topology and drift-surfaces passing particles as “modified field-lines” perturbation field-line dl = B ‡ d X B ‡ = ∇ × ( A + m B , q v || B ) B ‡ internally kinked flux-surfaces drift-islands φ = 0 o φ = 120 o φ = 240 o 0.6 0.4 0.2 Z [m] 0 −0.2 −0.4 −0.6 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4 R [m] R [m] R [m] • symmetry protects the topology of both field-lines and drift-lines • 3D enables changes in topology between field-lines and drift-lines D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 11 / 18
Symmetry is synonymous to the existence of flux-surfaces Symmetry reduces the 3D MHD equilibrium equations ∆ ∗ Ψ = F (Ψ) j × B = ∇ p → to a 2D scalar elliptic PDE: Grad-Shafranov equation • contours of Ψ in the 2D plane extrude out in the direction of the symmetry • ≡ flux-surfaces D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 12 / 18
3D MHD equilibria are substantially different Even the “simpler” force-free states (Taylor relaxed), where ∇ × B = α B , can display complex field-lines (chaotic/stochastic) • perfectly nested flux-surfaces require extreme fine-tuning • 3D configurations are fragile courtesy of [Dennis et al., 2013] D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 13 / 18
Ideal MHD motion leads to singularities Hahm-Kulsrud-Taylor problem • shaping of the boundary leads to break-up of topology • width of the ruptured contours scales as √ ǫ where ǫ is amplitude of boundary motion • preservation of topology requires singularities, current sheets, discontinuous magnetic fields (multi-region relaxed MHD) D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 14 / 18
Particle motion in discontinuous fields constant modulus, sheared field (current sheet) � sin α e y z > 0 B = cos α e x + z < 0 ⇐ ⇒ j = ∇ × B = − 2 δ ( z ) sin α e x − sin α e y D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18
Particle motion in discontinuous fields constant modulus, sheared field (current sheet) � sin α e y z > 0 B = cos α e x + z < 0 ⇐ ⇒ j = ∇ × B = − 2 δ ( z ) sin α e x − sin α e y D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18
Particle motion in discontinuous fields constant modulus, sheared field (current sheet) � sin α e y z > 0 B = cos α e x + z < 0 ⇐ ⇒ j = ∇ × B = − 2 δ ( z ) sin α e x − sin α e y D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18
Conclusion • charged particle motion is essentially governed by magnetic configuration • 3D MHD equilibria are unlikely to have a “symmetry” (flux-surfaces) • drift-line topology can differ from field-line topology in 3D • nested flux-surfaces will generically be accompanied by singularities, current sheets, discontinuous fields • guiding-centre approximation is invalid in discontinuous fields D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 16 / 18
Prospective work and opportunities • well-posedness of MHD equilibrium equations • investigation of particle motion in discontinous fields • interface between particle simulation codes ( LEVIS ) and relaxed MHD solvers ( SPEC ) D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 17 / 18
Bibliography I G. R. Dennis, S. R. Hudson, D. Terranova, P. Franz, R. L. Dewar, and M. J. Hole, Phys. Rev. Lett. 111 , 055003 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.111.055003 . D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 18 / 18
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