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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

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SLIDE 1

Magnetic confinement fusion: a perfect sand box for applied mathematicians

David Pfefferl´ e1

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

ANZIAM meeting, May 14 2019, UWA

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SLIDE 2

Outline

1 What is magnetic confinement fusion? 2 Stellarator vs Tokamak

D.Pfefferl´ e (UWA) Fusion ANZIAM 2 / 23

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SLIDE 3

Nuclear reactions and atomic energy

Fission Fusion 3.5 MeV 14.1 MeV

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SLIDE 4

Nuclear reactions and atomic energy

Fission Fusion

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SLIDE 5

Nuclear binding energy

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SLIDE 6

Fusion requires high temperature plasmas

separation of electrons from nucleus

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SLIDE 7

Plasma is most common state of matter

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SLIDE 8

Fusion power and cross-sections

Most probable fusion reaction

2 1H + 3 1H → 4 2He 3.5MeV

+ n

14.1MeV

fusion power volume = n1n2 < σv > Er ∼ 0.1 − 10[MW/m3]

where Er is energy/reaction, ni ∼ 1019[m−3] density of reactant i and < σv > reaction cross-section

D.Pfefferl´ e (UWA) Fusion ANZIAM 7 / 23

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SLIDE 9

Triple product (Lawson criterion)

Figure of merit 0D analysis

sustained fusion Pfusion ≥ Ploss

  • < σv >= aT 2, a = 1.1 · 10−24[m3/s]
  • 50/50 mix of D-T, nD = nT = n/2
  • quasi-neutrality ne = n and thermal

equilibrium Te = T

  • Ploss

V

= Wplasma

τE

=

3 2 neTe+ 3 2 nT

τE

= 3nT

τE

n2 4 aT 2 ≥ 3nT τE ⇐ ⇒ nTτE

triple product

≥ 3 · 1021[keVs/m3]

D.Pfefferl´ e (UWA) Fusion ANZIAM 8 / 23

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SLIDE 10

How to increase the triple product ?

n T τE particle source heating power complicated plasma instabilities, transport,. . . empirical scaling

τISS04 = 0.134a2.28R0.64P −0.61n0.54B0.84ι0.41

bigger plasma bigger B-field

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SLIDE 11

How to increase the triple product ?

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SLIDE 12

Magnetic confinement

Charged plasma particles wrap around magnetic field-lines

helical motion along uniform magnetic field

In uniform magnetic field B = Bez, particle motion is 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) Fusion ANZIAM 10 / 23

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SLIDE 13

Drifts due to non-uniform field

“Grad-B” drift

upward drift due to non-uniform magnetic field

when field-strength |B| is spatially varying VB = µ q b × ∇B B

where µ =

mv2

⊥

2B

is the “mag- netic moment”

D.Pfefferl´ e (UWA) Fusion ANZIAM 11 / 23

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SLIDE 14

Drifts due to non-uniform field

“Curvature” drift

upward drift due to curved magnetic field

when field-lines are bend- ing (curved) Vκ = mv2

||

qB b × κ

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

D.Pfefferl´ e (UWA) Fusion ANZIAM 12 / 23

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SLIDE 15

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) Fusion ANZIAM 13 / 23

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SLIDE 16

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

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SLIDE 17

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

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SLIDE 18

Figure-8 stellarator [Spitzer 1958]

Rearranging the coils so that “Grad-B” drift averages to zero

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SLIDE 19

Confinement optimised stellarators

see video of W7X assembly : https://youtu.be/u-fbBRAxJNk

D.Pfefferl´ e (UWA) Fusion ANZIAM 16 / 23

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SLIDE 20

3D makes particle motion complex

Lack of symmetry results in chaotic dynamics

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

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SLIDE 21

Tokamak (Toroidal magnetic chamber)

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SLIDE 22

Tokamak (Toroidal magnetic chamber)

poloidal fields induced by toroidal current

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SLIDE 23

Tokamak (Toroidal magnetic chamber)

toroidal + poloidal ≡ twisted magnetic fields ⇒ good confinement strong plasma current ⇒ instabilities (control issues)

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SLIDE 24

Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ passing orbits

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SLIDE 25

Particle motion in tokamaks

tokamak fields (toroidal + poloidal) ⇒ banana orbits

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SLIDE 26

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

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SLIDE 27

The ITER

International Thermonuclear Experimental Reactor

  • $20bn collaboration: EU, China, India, Japan, Russia, South Korea,

CH, US

  • major radius R = 6.2m, superconducting coils B = 11.8T
  • first plasma scheduled for 2025, burning plasma 2035

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SLIDE 28

ITER construction progress

Cadarache, France

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SLIDE 29

ITER construction progress

Cadarache, France

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SLIDE 30

ITER construction progress

Cadarache, France

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SLIDE 31

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

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SLIDE 32

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

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SLIDE 33

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) S[γ] = λ1

  • γ

ds

length

+ λ2

  • γ

τds

  • torsion

+ λ3

  • γ

1 2κ2ds

  • bending energy

D.Pfefferl´ e (UWA) Fusion ANZIAM 25 / 23

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SLIDE 34

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

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SLIDE 35

Less planar plasma ⇒ less complex coils

[Hudson et al., 2018]

ellipticity ǫ = 3 winding ι = 0.859 coil complexity C = 4.87. ellipticity ǫ = 1.73 winding ι = 1.6 coil complexity C = 0.674.

D.Pfefferl´ e (UWA) Fusion ANZIAM 27 / 23