Particle motion in 3D MHD equilibria versus relaxed states e 1 - - PowerPoint PPT Presentation

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Particle motion in 3D MHD equilibria versus relaxed states e 1 - - PowerPoint PPT Presentation

Aug 04, 2023 115 likes •353 views

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

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SLIDE 1

Particle motion in 3D MHD equilibria versus relaxed states

David Pfefferl´ e1

1The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia

23rd Australian Institute of Physics Congress, 9-13 December, Perth, Australia

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SLIDE 2

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 winding via strong plasma current stellarator: 3D, no plasma current, helical winding via coils

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 2 / 18

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SLIDE 3

Charged particle dynamics

helical motion along uniform magnetic field

For uniform magnetic field B = Bez z = v||t + z0 x y

  • = R(−ωt)ρ⊥ + X

where R(θ) is the rotation matrix around ez of angle θ ω = qB/m the Larmor frequency ρ⊥ =

m qB b × v⊥ is the Larmor

radius

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 3 / 18

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SLIDE 4

Charged particle dynamics

upward drift due to non-uniform magnetic field

“Grad-B” drift due to non- uniform amplitude |B| VB = µ q b × ∇B B where µ =

mv2

⊥

2B

is the “magnetic moment”

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 4 / 18

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SLIDE 5

Charged particle dynamics

upward drift due to curved magnetic field

“Curvature” drift due to bending field-lines Vκ = mv2

||

qB b × κ where κ = b · ∇b is the field-line curvature

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 5 / 18

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SLIDE 6

Charged particle dynamics

“Mirror trapping” in magnetic bottles m 2 v2

|| + µB = const

where µ = mv2

⊥

2B

is the magnetic moment

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 6 / 18

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SLIDE 7

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

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SLIDE 8

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

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SLIDE 9

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

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SLIDE 10

Guiding-centre motion

Fast gyration can be “averaged out” via clever coordinate change L(x, v, t) = 1

2 m v2 + q A · v ⇒ (qA + mv||b) · ˙

X − ( 1

2mv2 || + µB)

10−8

  • expansion in m/q justifies adiabatic conservation of µ = mv2

⊥/2B

  • 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

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SLIDE 11

Drift-line topology and drift-surfaces

passing particles as “modified field-lines”

dX dl = B‡ B‡ , B‡ = ∇ × ( A +

m q v|| B B )

field-line perturbation axisymmetric flux-surfaces axisymmetric drift-surfaces

2 2.5 3 3.5 −1 −0.5 0.5 1 R [m] Z [m] flux−surface counter−passing co−passing

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 11 / 18

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SLIDE 12

Drift-line topology and drift-surfaces

passing particles as “modified field-lines”

dX dl = B‡ B‡ , B‡ = ∇ × ( A +

m q v|| B B )

field-line perturbation internally kinked flux-surfaces drift-islands

0.4 0.6 0.8 1 1.2 1.4 −0.6 −0.4 −0.2 0.2 0.4 0.6 R [m] Z [m] φ = 0o 0.4 0.6 0.8 1 1.2 1.4 R [m] φ = 120o 0.4 0.6 0.8 1 1.2 1.4 R [m] φ = 240o D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 11 / 18

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SLIDE 13

Drift-line topology and drift-surfaces

passing particles as “modified field-lines”

dX dl = B‡ B‡ , B‡ = ∇ × ( A +

m q v|| B B )

field-line perturbation internally kinked flux-surfaces drift-islands

0.4 0.6 0.8 1 1.2 1.4 −0.6 −0.4 −0.2 0.2 0.4 0.6 R [m] Z [m] φ = 0o 0.4 0.6 0.8 1 1.2 1.4 R [m] φ = 120o 0.4 0.6 0.8 1 1.2 1.4 R [m] φ = 240o

  • 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

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SLIDE 14

Symmetry is synonymous to the existence

  • f flux-surfaces

Symmetry reduces the 3D MHD equilibrium equations j × B = ∇p → ∆∗Ψ = F(Ψ) 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

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SLIDE 15

3D MHD equilibria are substantially different

courtesy of [Dennis et al., 2013]

Even the “simpler” force-free states (Taylor relaxed), where ∇ × B = αB, can display complex field-lines (chaotic/stochastic)

  • perfectly nested flux-surfaces

require extreme fine-tuning

  • 3D configurations are fragile

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 13 / 18

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SLIDE 16

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

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SLIDE 17

Particle motion in discontinuous fields

constant modulus, sheared field (current sheet)

B = cos αex +

  • sin αey

z > 0 − sin αey z < 0 ⇐ ⇒ j = ∇ × B = −2δ(z) sin αex

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18

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SLIDE 18

Particle motion in discontinuous fields

constant modulus, sheared field (current sheet)

B = cos αex +

  • sin αey

z > 0 − sin αey z < 0 ⇐ ⇒ j = ∇ × B = −2δ(z) sin αex

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18

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SLIDE 19

Particle motion in discontinuous fields

constant modulus, sheared field (current sheet)

B = cos αex +

  • sin αey

z > 0 − sin αey z < 0 ⇐ ⇒ j = ∇ × B = −2δ(z) sin αex

D.Pfefferl´ e (UWA) Particles in 3D MHD vs relaxed AIP Congress 2018 15 / 18

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SLIDE 20

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

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SLIDE 21

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

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SLIDE 22

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