Orbit physics in discontinuous elds: open questions · Orbit physics in discontinuous elds: open...

Orbit physics in discontinuous fields: openquestions

David Pfefferle1

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

Mini-course/workshop on the application of computational mathematics toplasma physics

June 24-27, 2019 - Canberra, Australia

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

∂tfhot + v · ∂xfhot + (E + v ×B) · ∂vfhot = C[fhot, fM ] + S(x,v, t)D.Pfefferle (UWA) Orbit physics ANU/MSI mini-course 2 / 14

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.Pfefferle (UWA) Orbit physics ANU/MSI mini-course 2 / 14

Magnetic confinementCharged 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(xy

)= R(−ωt)ρ⊥ +X

where R(θ) is the rotation matrixaround ez of angle θ

ω = qB/m the Larmor frequency

ρ⊥ = mqBb×v⊥ is the Larmor radius


Drifts due to non-uniform field“Grad-B” drift

upward drift due to non-uniform magnetic field

“grad-B” drift fromspatially-varying field-strength |B|

VB =µ

qb× ∇B

B

where µ =mv2

⊥2B

is the “mag-

netic moment”


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||

qBb× κ

where κ = b · ∇b is the field-

line curvature


Mirror trapping in “magnetic bottles”

consequence of magnetic momentand energy conservation

m

2v2|| + µB = E

where µ =mv2

⊥2B

is the magnetic moment

Mirror devices

• historically first magneticconfinement devices

• suffer from huge losses atboth ends


Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ passing orbits


Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ banana orbits


3D makes particle motion complexLack of symmetry results in chaotic dynamics

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


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

0

1ρ

ϑ

ϕ

rθ

ϕ

ϕ R Z

x

y

z

Φ


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) = 12mv · v + qA · v − qΦE

= 12gij u

iuj + qAiui − qΦE

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

• Euler-Lagrange equations yield ∇uu = qm(−dΦE + iuB)]

ui =q

m

(Ei + gijεjklu

k√gBl)

︸︷︷︸E+v×B

− umunΓimn︸︷︷︸inertiel forces

where the Christoffel symbol

Γimn = gijΓl,mn Γl,mn = ∂lx · ∂2

mnx


Advantages/drawbacksof solving orbits in toroidal coordinates

Advantages:

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

(ρ, ϑ, ϕ) 7→ x = Φ(ρ, ϑ, ϕ), x 7→ (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?)


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 ?


Particle motion in discontinuous fieldsconstant modulus, sheared field (current sheet)

B = cosαex +

{sinαey z > 0

− sinαey z < 0⇐⇒ j = ∇×B = −2δ(z) sinαex



B = cosαex +

{sinαey z > 0




B = cosαex +

{sinαey z > 0



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 particleconfinement in MRxMHD configurations

• guiding-centre approximation is invalid in discontinuous fields ⇒alternative kinetic models

• numerically integrating through ideal interface


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 6= 0 is the symplectic form for aHamiltonian system

dϑ

dϕ=∂Ψp

∂Ψt=

Ψ′pΨ′t

= ι(ρ)dΨt

dϕ= −∂Ψp

∂ϑ= 0


• 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


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