Magnetic confinement fusion: a perfect sand box for applied mathematicians e 1 David Pfefferl´ 1 The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia ANZIAM meeting, May 14 2019, UWA
Outline 1 What is magnetic confinement fusion? 2 Stellarator vs Tokamak D.Pfefferl´ e (UWA) Fusion ANZIAM 2 / 23
Nuclear reactions and atomic energy Fission Fusion 3 . 5 MeV 14 . 1 MeV D.Pfefferl´ e (UWA) Fusion ANZIAM 3 / 23
Nuclear reactions and atomic energy Fission Fusion D.Pfefferl´ e (UWA) Fusion ANZIAM 3 / 23
Nuclear binding energy D.Pfefferl´ e (UWA) Fusion ANZIAM 4 / 23
Fusion requires high temperature plasmas separation of electrons from nucleus D.Pfefferl´ e (UWA) Fusion ANZIAM 5 / 23
Plasma is most common state of matter D.Pfefferl´ e (UWA) Fusion ANZIAM 6 / 23
Fusion power and cross-sections Most probable fusion reaction 2 1 H + 3 4 1 H → 2 He + n 14 . 1 MeV 3 . 5 MeV fusion power = n 1 n 2 < σv > E r volume ∼ 0 . 1 − 10[ MW/m 3 ] where E r is energy/reaction, n i ∼ 10 19 [ m − 3 ] density of reactant i and < σv > reaction cross-section D.Pfefferl´ e (UWA) Fusion ANZIAM 7 / 23
Triple product (Lawson criterion) Figure of merit 0 D analysis sustained fusion P fusion ≥ P loss • < σv > = aT 2 , a = 1 . 1 · 10 − 24 [ m 3 /s ] • 50/50 mix of D-T, n D = n T = n/ 2 • quasi-neutrality n e = n and thermal equilibrium T e = T 3 2 n e T e + 3 = W plasma 2 nT • P loss = 3 nT = V τ E τ E τ E n 2 4 aT 2 ≥ 3 nT ⇐ ⇒ τ E ≥ 3 · 10 21 [ keV s/m 3 ] nTτ E � �� � triple product D.Pfefferl´ e (UWA) Fusion ANZIAM 8 / 23
How to increase the triple product ? n T τ E particle source heating power complicated plasma instabilities, transport,. . . empirical scaling τ ISS04 = 0 . 134 a 2 . 28 R 0 . 64 P − 0 . 61 n 0 . 54 B 0 . 84 ι 0 . 41 bigger plasma bigger B -field D.Pfefferl´ e (UWA) Fusion ANZIAM 9 / 23
How to increase the triple product ? D.Pfefferl´ e (UWA) Fusion ANZIAM 9 / 23
Magnetic confinement Charged plasma particles wrap around magnetic field-lines In uniform magnetic field B = B e z , particle motion is 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) Fusion ANZIAM 10 / 23
Drifts due to non-uniform field “Grad-B” drift when field-strength | B | is spatially varying 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) Fusion ANZIAM 11 / 23
Drifts due to non-uniform field “Curvature” drift when field-lines are bend- ing (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) Fusion ANZIAM 12 / 23
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) Fusion ANZIAM 13 / 23
Poincar´ e-Hopf theorem justifies torus toroidal fields alone do not provide plasma confinement | B | ∼ 1 /R ⇒ strong vertical “Grad-B” drift D.Pfefferl´ e (UWA) Fusion ANZIAM 14 / 23
Poincar´ e-Hopf theorem justifies torus toroidal fields alone do not provide plasma confinement | B | ∼ 1 /R ⇒ strong vertical “Grad-B” drift D.Pfefferl´ e (UWA) Fusion ANZIAM 14 / 23
Figure-8 stellarator [Spitzer 1958] Rearranging the coils so that “Grad-B” drift averages to zero D.Pfefferl´ e (UWA) Fusion ANZIAM 15 / 23
Confinement optimised stellarators see video of W7X assembly : https://youtu.be/u-fbBRAxJNk D.Pfefferl´ e (UWA) Fusion ANZIAM 16 / 23
3D makes particle motion complex Lack of symmetry results in chaotic dynamics stellarator 3D fields ⇒ complex motion, detrapping, magnetic wells,. . . D.Pfefferl´ e (UWA) Fusion ANZIAM 17 / 23
Tokamak (Toroidal magnetic chamber) D.Pfefferl´ e (UWA) Fusion ANZIAM 18 / 23
Tokamak (Toroidal magnetic chamber) poloidal fields induced by toroidal current D.Pfefferl´ e (UWA) Fusion ANZIAM 18 / 23
Tokamak (Toroidal magnetic chamber) toroidal + poloidal ≡ twisted magnetic fields ⇒ good confinement strong plasma current ⇒ instabilities (control issues) D.Pfefferl´ e (UWA) Fusion ANZIAM 18 / 23
Particle motion in tokamaks tokamak fields (toroidal + poloidal) ⇒ passing orbits D.Pfefferl´ e (UWA) Fusion ANZIAM 19 / 23
Particle motion in tokamaks tokamak fields (toroidal + poloidal) ⇒ banana orbits D.Pfefferl´ e (UWA) Fusion ANZIAM 19 / 23
Tokamak discharge is limited in time Longest steady-state in tokamaks by Chinese EAST 1 ramp-up phase 2 flat-top • quasi steady-state • heating, particle injection • fusion ignition 3 ramp-down D.Pfefferl´ e (UWA) Fusion ANZIAM 20 / 23
The ITER International Thermonuclear Experimental Reactor • $20bn collaboration: EU, China, India, Japan, Russia, South Korea, CH, US • major radius R = 6 . 2 m, superconducting coils B = 11 . 8 T • first plasma scheduled for 2025, burning plasma 2035 D.Pfefferl´ e (UWA) Fusion ANZIAM 21 / 23
ITER construction progress Cadarache, France D.Pfefferl´ e (UWA) Fusion ANZIAM 22 / 23
ITER construction progress Cadarache, France D.Pfefferl´ e (UWA) Fusion ANZIAM 22 / 23
ITER construction progress Cadarache, France D.Pfefferl´ e (UWA) Fusion ANZIAM 22 / 23
Bibliography I C. Mercier, Nuclear Fusion 4 , 213 (1964). D. Pfefferl´ e, L. Gunderson, S. R. Hudson, and L. Noakes, Physics of Plasmas 25 , 092508 (2018). J. Langer and D. A. Singer, Journal of the London Mathematical Society s2-30 , 512 (1984), ISSN 1469-7750. S. Hudson, C. Zhu, D. Pfefferl´ e, and L. Gunderson, Physics Letters A 382 , 2732 (2018), ISSN 0375-9601. D.Pfefferl´ e (UWA) Fusion ANZIAM 23 / 23
Stellarators achieve confinement through helical winding of magnetic field [Mercier, 1964] • rotating elliptic boundary, e.g. LHD • non-planar magnetic axis, e.g. W7X D.Pfefferl´ e (UWA) Fusion ANZIAM 24 / 23
Optimal “magnetic axis” as elasticae [Pfefferl´ e et al., 2018] Mathematical problem: Find all possible • closed curves • of fixed length • with minimum bending energy • while yielding a fixed amount of integrated torsion Solution via variational approach (least action principle) � � � 2 κ 2 ds 1 S [ γ ] = λ 1 ds + λ 2 τds + λ 3 γ γ γ � �� � � �� � � �� � length torsion bending energy D.Pfefferl´ e (UWA) Fusion ANZIAM 25 / 23
Variational problem has analytic solution Elasticae and Jacobi elliptic functions[Langer and Singer, 1984] One-parameter families of magnetic axis with increasing “winding number” D.Pfefferl´ e (UWA) Fusion ANZIAM 26 / 23
Less planar plasma ⇒ less complex coils [Hudson et al., 2018] ellipticity ǫ = 3 ellipticity ǫ = 1 . 73 winding ι = 0 . 859 winding ι = 1 . 6 coil complexity C = 4 . 87 . coil complexity C = 0 . 674 . D.Pfefferl´ e (UWA) Fusion ANZIAM 27 / 23
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