Orbit physics in discontinuous fields: open questions e 1 David - - PowerPoint PPT Presentation

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Orbit physics in discontinuous fields: open questions e 1 David Pfefferl 1 The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia Mini-course/workshop on the application of computational mathematics to plasma

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SLIDE 1

Orbit physics in discontinuous fields: open questions

David Pfefferl´ e1

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

Mini-course/workshop on the application of computational mathematics to plasma physics June 24-27, 2019 - Canberra, Australia

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SLIDE 2

VENUS-LEVIS: energetic particles

1 guiding-centre / full-orbit

  • non-canonical Hamiltonian

formulation (2nd order)

  • curvilinear coordinates
  • switching algorithm in high

field-variations

∂tfhot + v · ∂xfhot + (E + v × B) · ∂vfhot = C[fhot, fM] + S(x, v, t)

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 2 / 14

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SLIDE 3

VENUS-LEVIS: energetic particles

1 guiding-centre / full-orbit

  • non-canonical Hamiltonian

formulation (2nd order)

  • curvilinear coordinates
  • switching algorithm in high

field-variations

2 supra-thermal populations

  • NBI, ICRH, fusion alphas
  • full-f slowing-down / delta-f
  • ∼ ASCOT, ORBIT,SPIRAL

Vlasov-Boltzmann via PIC ∂tfhot + v · ∂xfhot + (E + v × B) · ∂vfhot = C[fhot, fM] + S(x, v, t)

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 2 / 14

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SLIDE 4

Magnetic confinement

Charged plasma particles wrap around magnetic field-lines

helical motion along uniform magnetic field

In uniform magnetic field B = Bez, motion is helical 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) Orbit physics ANU/MSI mini-course 3 / 14

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SLIDE 5

Drifts due to non-uniform field

“Grad-B” drift

upward drift due to non-uniform magnetic field

“grad-B” drift from spatially-varying field- strength |B| VB = µ q b × ∇B B

where µ =

mv2

⊥

2B

is the “mag- netic moment”

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 4 / 14

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SLIDE 6

Drifts due to non-uniform field

“Curvature” drift

upward drift due to curved magnetic field

curvature drift when field- lines are bending (curved) Vκ = mv2

||

qB b × κ

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

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 5 / 14

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SLIDE 7

Mirror trapping in “magnetic bottles”

consequence of magnetic moment and energy conservation m 2 v2

|| + µB = E

where µ =

mv2

⊥

2B is the magnetic moment

Mirror devices

  • historically first magnetic

confinement devices

  • suffer from huge losses at

both ends

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 6 / 14

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SLIDE 8

Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ passing orbits

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 7 / 14

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SLIDE 9

Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ banana orbits

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 7 / 14

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SLIDE 10

3D makes particle motion complex

Lack of symmetry results in chaotic dynamics

stellarator 3D fields ⇒ complex motion, detrapping, magnetic wells,. . .

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 8 / 14

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SLIDE 11

Toroidal coordinates in magnetic fusion

  • toroidal systems often use chart map

Φ : (0, 1)

ρ

× (0, 2π)

ϑ

× (0, 2π)

ϕ

→ M ⊂ R3

x = R(ρ, ϑ, ϕ) cos(ϕ) y = R(ρ, ϑ, ϕ) sin(ϕ) z = Z(ρ, ϑ, ϕ) R(ρ, ϑ, ϕ) = R0(ϕ) + r(ρ, ϑ, ϕ) cos[θ(ρ, ϑ, ϕ)] Z(ρ, ϑ, ϕ) = Z0(ϕ) + r(ρ, ϑ, ϕ) sin[θ(ρ, ϑ, ϕ)]

  • vector potential (1-form)

A(ρ, ϑ, ϕ) = Aρdρ + Aϑdϑ + Aϕdϕ

  • magnetic flux (2-form)

B = dA ⇐ ⇒ B = ∇ × A

2π 2π 1 ρ ϑ ϕ r θ ϕ ϕ R Z x y z Φ

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 9 / 14

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SLIDE 12

Full-orbit curvilinear equations of motion

  • u = (u1, u2, u3) = (ρ, ϑ, ϕ) ∈ (0, 1) × (0, 2π) × (0, 2π)
  • Lagrangian for single charged particle

L(ui, ˙ ui, t) = 1

2mv · v + qA · v − qΦE

= 1

2gij ˙

ui ˙ uj + qAi ˙ ui − qΦE

gij = ∂ix · ∂jx is the metric tensor (pullback metric)

  • Euler-Lagrange equations yield ∇ ˙

u ˙

u = q

m(−dΦE + i ˙ uB)♯

¨ ui = q m

  • Ei + gijǫjkl ˙

uk√gBl

  • E+v×B

− ˙ um ˙ unΓi

mn

  • inertiel forces

where the Christoffel symbol Γi

mn = gijΓl,mn

Γl,mn = ∂lx · ∂2

mnx D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 10 / 14

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SLIDE 13

Advantages/drawbacks

  • f solving orbits in toroidal coordinates

Advantages:

  • mapping forward via Φ is easy, but computing the inverse is not

(ρ, ϑ, ϕ) → x = Φ(ρ, ϑ, ϕ), x → (ρ, ϑ, ϕ) = Φ−1(x)

One could pre-evaluate the inverse on a grid and interpolate, but then it is hard to ensure ∇ · B = 0 (and other properties).

  • fields are semi-analytic functions (Fourier, splines, polynomials)

⇒ high accuracy of derivatives Drawbacks

  • evaluation of metric (Christoffel) prone to numerical error
  • integrators (Boris-Buneman?)

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 11 / 14

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SLIDE 14

Orbits in SPEC fields

  • interface between MRxMHD equilibrium SPEC and VENUS-LEVIS

(Zhisong, Dean)

  • energetic particle confinement in Taylor-relaxed states

Open questions:

  • SPEC nested toroidal annuli with ideal interfaces ⇒ discontinuous B
  • full-orbit OK, but gyrokinetics KO: reduced kinetic model ?
  • numerically integrating across interface ?

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 12 / 14

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SLIDE 15

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) Orbit physics ANU/MSI mini-course 13 / 14

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SLIDE 16

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) Orbit physics ANU/MSI mini-course 13 / 14

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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) Orbit physics ANU/MSI mini-course 13 / 14

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SLIDE 18

Conclusions and prospective work

  • charged particle motion is governed by magnetic field
  • full-orbit motion can be solved in curvilinear coordinates
  • SPEC/LEVIS interface underway to study energetic particle

confinement in MRxMHD configurations

  • guiding-centre approximation is invalid in discontinuous fields ⇒

alternative kinetic models

  • numerically integrating through ideal interface

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 14 / 14

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SLIDE 19

Nested toroidal flux surfaces (VMEC)

  • prescribing the vector potential to be the 1-form

A = Ψt(ρ)dϑ − Ψp(ρ)dϕ

  • magnetic flux is (B = ∇ × A)

B = dA = Ψ′

tdρ ∧ dϑ + Ψ′ pdϕ ∧ dρ

  • field-lines lie on surfaces of constant ρ(x, y, z) = C

dρ ∧ B = 0 ⇒ B · ∇ρ = 0

  • restricted to those surface, B = 0 is the symplectic form for a

Hamiltonian system dϑ dϕ = ∂Ψp ∂Ψt = Ψ′

p

Ψ′

t

= ι(ρ) dΨt dϕ = −∂Ψp ∂ϑ = 0

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 15 / 14

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SLIDE 20
  • Ψt is the toroidal flux,
  • P(ρ)

B =

  • ∂P(ρ)

A = 2πΨt

  • Ψp is the poloidal flux,
  • T(ρ)

B =

  • ∂T(ρ)

A = 2πΨp

P T

  • B

D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 16 / 14