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

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 physics June 24-27, 2019 - Canberra, Australia

VENUS-LEVIS : energetic particles 1 guiding-centre / full-orbit • non-canonical Hamiltonian formulation ( 2 nd order) • curvilinear coordinates • switching algorithm in high field-variations D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 2 / 14 ∂ t f hot + v · ∂ x f hot + ( E + v × B ) · ∂ v f hot = C [ f hot , f M ] + S ( x , v , t )

VENUS-LEVIS : energetic particles 1 guiding-centre / full-orbit • non-canonical Hamiltonian formulation ( 2 nd 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 ∂ t f hot + v · ∂ x f hot + ( E + v × B ) · ∂ v f hot = C [ f hot , f M ] + S ( x , v , t ) D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 2 / 14

Magnetic confinement Charged plasma particles wrap around magnetic field-lines In uniform magnetic field B = B e z , motion is helical 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) Orbit physics ANU/MSI mini-course 3 / 14

Drifts due to non-uniform field “Grad-B” drift “grad-B” drift from spatially-varying field- strength | B | V B = µ q b × ∇ B B mv 2 where µ = is the “mag- ⊥ 2 B netic moment” upward drift due to non-uniform magnetic field D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 4 / 14

Drifts due to non-uniform field “Curvature” drift curvature drift when field- lines are bending (curved) 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) Orbit physics ANU/MSI mini-course 5 / 14

Mirror trapping in “magnetic bottles” consequence of magnetic moment and energy conservation m 2 v 2 || + µB = E mv 2 where µ = 2 B 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

Particle motion in tokamaks tokamak fields (toroidal + poloidal) ⇒ passing orbits D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 7 / 14

Particle motion in tokamaks tokamak fields (toroidal + poloidal) ⇒ banana orbits D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 7 / 14

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

Toroidal coordinates in magnetic fusion 2 π • toroidal systems often use chart map ϕ 1 → M ⊂ R 3 Φ : (0 , 1) × (0 , 2 π ) × (0 , 2 π ) ρ � �� � � �� � � �� � 0 ϑ 2 π ρ ϕ ϑ x = R ( ρ, ϑ, ϕ ) cos( ϕ ) y = R ( ρ, ϑ, ϕ ) sin( ϕ ) Φ z = Z ( ρ, ϑ, ϕ ) r θ ϕ R ( ρ, ϑ, ϕ ) = R 0 ( ϕ ) + r ( ρ, ϑ, ϕ ) cos[ θ ( ρ, ϑ, ϕ )] Z ( ρ, ϑ, ϕ ) = Z 0 ( ϕ ) + r ( ρ, ϑ, ϕ ) sin[ θ ( ρ, ϑ, ϕ )] z • vector potential (1-form) A ( ρ, ϑ, ϕ ) = A ρ dρ + A ϑ dϑ + A ϕ dϕ y ϕ R Z • magnetic flux (2-form) B = dA ⇐ ⇒ B = ∇ × A x D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 9 / 14

Full-orbit curvilinear equations of motion • u = ( u 1 , u 2 , u 3 ) = ( ρ, ϑ, ϕ ) ∈ (0 , 1) × (0 , 2 π ) × (0 , 2 π ) • Lagrangian for single charged particle L ( u i , ˙ u i , t ) = 1 2 m v · v + q A · v − q Φ E u i ˙ u j + qA i ˙ u i − q Φ E = 1 2 g ij ˙ g ij = ∂ i x · ∂ j x is the metric tensor (pullback metric) u = q u B ) ♯ • Euler-Lagrange equations yield ∇ ˙ u ˙ m ( − d Φ E + i ˙ u i = q � u k √ gB l � E i + g ij ǫ jkl ˙ u m ˙ u n Γ i ¨ − ˙ mn m � �� � � �� � inertiel forces E + v × B where the Christoffel symbol Γ i mn = g ij Γ l,mn Γ l,mn = ∂ l x · ∂ 2 mn x D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 10 / 14

Advantages/drawbacks of solving orbits in toroidal coordinates Advantages: • mapping forward via Φ is easy, but computing the inverse is not x �→ ( ρ, ϑ, ϕ ) = Φ − 1 ( x ) ( ρ, ϑ, ϕ ) �→ x = Φ( ρ, ϑ, ϕ ) , 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

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

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

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

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

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

Nested toroidal flux surfaces (VMEC) • prescribing the vector potential to be the 1 -form A = Ψ t ( ρ ) dϑ − Ψ p ( ρ ) dϕ • magnetic flux is ( B = ∇ × A ) B = dA = Ψ ′ t dρ ∧ dϑ + Ψ ′ p dϕ ∧ 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 = Ψ ′ dϕ = ∂ Ψ p dϑ d Ψ t dϕ = − ∂ Ψ p p = ι ( ρ ) ∂ϑ = 0 Ψ ′ ∂ Ψ t t D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 15 / 14

� � • Ψ t is the toroidal flux, B = A = 2 π Ψ t P ( ρ ) ∂P ( ρ ) � � • Ψ p is the poloidal flux, B = A = 2 π Ψ p T ( ρ ) ∂T ( ρ ) � B T P D.Pfefferl´ e (UWA) Orbit physics ANU/MSI mini-course 16 / 14