Mathematical Modelling of a Magnetized Target Fusion Reactor

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1 Mathematical Modelling of a Magnetized Target Fusion Reactor Michael PIC Assistant Adjunct Professor, Mathematics UCLA April 20, 2016

2 Context A Canadian research company General Fusion (2002-Present, Michel Laberge) is designing a reactor to yield sustainable fusion energy, which has never been achieved before. Success with this could hold immense promise for clean and sustainable energy sources in the future. In this talk, we will study the reactor design by: Deriving a reactor model Studying the model numerically with finite volumes Qualitatively describing the operation with formal asymptotic approach Assessing the effects of instability via linearization

3 Thanks to... Thanks to the following people for helpful discussions, collaboration, and support: Sandra Barsky and Aaron Froese, General Fusion George Bluman, Michael Ward, and Brian Wetton, UBC Math Department Randy LeVeque, University of Washington Applied Mathematics Department Russ Caflisch, John Garnett, James Ralston, Marcus Roper, UCLA Math Department

4 Fusion Fusion occurs naturally in sun: pressure 300 G atm, temperature 10 7 K Potential clean energy source Tritium-deuterium plasma reaction: 3 1 H +2 1 H 4 2 He +n MeV of energy Complex: thermal distribution, radiative losses, etc. Lawson criterion 1 for energy yield: density temperature time cm 3 KeV s

5 General Fusion design How feasible is such a design?

6 Lead-Lithium Density ρ, velocity v, and pressure P: ρ t + (ρv) = 0 (ρv) t + (ρv v) + P = 0 Quadratic fit to equation of state 2 P = P(ρ) (mass) (momentum)

7 Assumptions and simplifications Spherical symmetry Piston pressure: P(t) = P atmospheric + (P impact P atmospheric )e t2 /t 2 0 θ(t) Adiabatic: plasma gas pressure V 5/3 Plasma magnetic pressure 3 R 4 plasma Initially P gas 0.1P magnetic Initial plasma temperature 100 ev System starts in equilibrium No mixing/escape yields free boundary problems: d dt r boundary(t) = v(r boundary (t), t) Neglect thermal and radiative energy losses, and slow rotation

8 Overall model

9 Coordinate transformation Transform from moving to fixed computational domain Let τ = t, (τ) = r R (τ) r L (τ), Γ(τ) = v R (τ) v L (τ) Set y = r r L(τ) (τ), y [0, 1] p(r R (t), t) = p(1, τ), etc. New conservation laws e.g. mass: ρ τ + { 1 [ (v L + Γy)ρ + ρv]} y = 2 r L + y Γ ρ

10 Finite volume approach Split stepping: update via ρ τ + { 1 [ (v L + Γy)ρ + ρv]} y = 0 then via ρ τ = ρ - similarly for momentum 2 r L + y Γ Code summary 4 : - constant extrapolation to ghost points - upwind homogeneous system with second-order correction: use local eigenvector bases - use approximate Riemann solver - update with source term - update time and boundary data

11 Pulse profiles

12 Abridged sensitivity analysis Table: Min radius R min, Lawson triple product Π L, impact pressure P impact, initial plasma radius R plasma,0, initial sphere radius R lead,0. System R min (cm) Π L (10 15 kev s cm 3 ) Baseline R plasma, P impact R lead, P impact R lead,

13 Big picture Modifications: - Gaussian impulse pressure without heaviside - linearized P(ρ) equation of state - only magnetic pressure Relevant scales: - asymptotic parameter ɛ big sound speed: bɛ 1/2 - impulse pressure: O(1) - small initial plasma radius: χɛ 1/2 - very small impulse time: O(ɛ) - very, very small initial pressure: µɛ 3/2 Matched asymptotics 5

14 Asymptotic regimes

15 Pulse formation and focusing Formation: - ρ(y, τ) 1 + ɛρ 1 + ɛ 3/2 ρ 2, v(y, τ) ɛ 1/2 v 0 + ɛv 1 - Riemann Invariants 6 - plane wave solutions in (, 0] [0, ): ρ 1 (y, τ) = 1 b 2 e (τ+y/b)2, - need divergent ρ 2, v 1 for matching Focusing: - amplitude growth: - linear acoustic limit 7 : v 0 (y, τ) = 1 2 b e (τ+y/b) ρ 1 + ɛρ 1 + ɛ 3/2 ρ 2, v ɛ 1/2 v 0 + ɛv 1 (outer) ρ 1 + ɛ 1/2 ρ 1 + ɛρ 2, v v 0 + ɛ 1/2 v 1 (inner) ρ 1,ˆt + v 0,σ + 2 σ v 0 = 0, v 0,ˆt + b2 ρ 1,σ = 0

16 Pulse reflection and plasma compression Complete reflection of ρ 1 and v 0 but v 1,ˆt + b2 ρ 2,σ = (ρ 1 v 0 )ˆt (v 2 0 ) σ 2 σ v 2 0 Find v 1 (σ, ) = 2π b 2 χσ 2 so v = O(ɛ 1/2 ) Rayleigh argument 8 or more dynamics: take v L = v L (r L ). - ρ 1 + ɛ 2 ˆρ, v ɛ 1/2ˆv: plasma radius σ L - find ˆv(σ L ) = ˆv L = 2π b 2 χ 3/2 σ 3/2 L Finally - ρ 1 + ɛ 1/2 ˆρ, v ɛ 1/4ˆv: plasma radius z L - find ˆv(z L ) = ˆv L = 2Az L 2µ z 2 L Matching gives A: turnaround point when ˆv L = 0

17 Minimum radius Minimum radius: r min b4 χ 3 µ ɛ π Agreement with numerics as ɛ 0 with b = 1.05, χ = 0.937, µ = π: ɛ r min,num r min,asy

18 Key insights Dimensional minimum radius R min C4 s P plasma,0 R 7 plasma,0 ϱ3 0 πp 4 impact R4 lead,0 t2 0 = 1.6 cm Symbol Meaning Symbol Meaning C s lead sound speed P plasma,0 initial pressure R plasma,0 initial plasma radius ϱ 0 lead density P impact piston pressure R lead,0 initial lead radius t 0 impulse time scale Qualitative consistency with numerics for all parameters

19 Key insights consistent compression profile wave-like behaviour: sound speed dominates almost all input energy reflected: E input 8π 3 b ɛ 3/2, E compression 4π2 ɛ 5/2 b 4 χ 3

$Pistons cover fraction F of angle 2π/N c : P(r R, θ, t) = P atm + (P impact P atm ) e t2 /t0 2 (F + k=1 2 πk sin(kπf)re(eikncθ )), N c 15 Dimensionless: study boundary pressure$

20 Asymmetric context Finite ( 100 pistons on sphere), time differences, etc. Perturbations to axially symmetric collapse: linearize! Pistons cover fraction F of angle 2π/N c : P(r R, θ, t) = P atm + (P impact P atm ) e t2 /t0 2 (F + k=1 2 πk sin(kπf)re(eikncθ )), N c 15 Dimensionless: study boundary pressure p(1, θ, t) = (1 + ηe imθ )e t2 /τ 2 η 1 controllable

21 Perturbed inviscid Burgers equation Consider u t + ( 1 2 u2 ) x = u 2 with a perturbation parameter η with u η (x, 0) = (1 η)θ( x) + ηθ(x) { Gateaux derivative (u η u 0 )/η as η 0: 1/(1 t) 2, x < xs 0 (t) 1, x > xs 0 t2 δ(x x 0 2(1 t) (t) 2 s (t))

22 Perturbed inviscid Burgers equation If u = u 0 + ηu 1 then ut 1 + (u 1 u 0 ) x = 2u 0 u 1 Let u 1 = ũ 1 + Mδ(x xs 0 ), a < xs 0 < b then d b dt a u1 dx = dm dt + xs 0 a ũt 1dx + b xs 0 ũt 1dx +ũ1 (xs 0 )x 0 s (t) ũ 1 (xs 0 + )x 0 s (t) d b dt a u1 dx = (u 0 u 1 ) (u 0 u 1 ) + + b a 2u0 u 1 dx. Obtain dm dt = xs 0 (t)[u 1 ] [u 0 u 1 ] (2u0 + 2u 0+ )M. Burger: M(t) = η η=0 [u 0 ] = gap ηm(t)/[u 0 ]

23 Perturbed equations Density ρ = ρ 0 (r, t) + η ρ(r, t)e imθ, momentum density in ˆr- and ˆθ-directions µ = µ 0 (r, t) + ηs(r, t)e imθ, ηψ(r, t)e imθ. Obtain 3 3 perturbed system

24 Abridged sensitivities Dimensionless time m ( ρ) η (s) η η : sensitivity of deviation amplitude with respect to perturbation size Suponitsky et al. 9 suggest low mode numbers interacting with plasma pose few problems for operation Work here suggests high mode numbers damped out

25 Pulse formation as m Near outer boundary with m acoustic model: p TT = p xx p, p(x, 0) = p t (x, 0) = 0, p(0, t) = 1 for x, T 0 p e x 2 + π ( 1 ˆσ/T 4 2 sin( 2ˆσ + ˆσ 2 /T 2 + π/4) 2ˆσ+ˆσ 2 /T sin( 2ˆσ + π/4)) 1 2ˆσ π F (ˆσ), x < T (T ) ρ = 1/2, x = T ρ = 0, x > T F(z) = Ω (sin(zω + 1 2Ω ) + sin( z Ω + Ω 2 ))dω.

26 Results and future work Results: - numerics and asymptotics consistent - sensitivity to parameters, energy yield may be within reach - larger outer sphere radius and impact pressure appear important - much energy is reflected - asymmetries may be dampened Future directions: - incorporate more plasma physics - more precise assessment of design

27 References [1]Lawson, Some Criteria for a Power..., Proc. Phys Society B, [2]Rothman et al., Measurement of principal isentropes..., J. Phys D, [3]Woodruff et al., Adiabatic compression of a doublet field..., JOFE, [4]LeVeque, Finite Volume Methods for Hyperbolic Problems, [5]Hinch, Perturbation Methods, [6]Whitham, Linear and Nonlinear Waves, [7]Landau and Lifschitz, Fluid Mechanics, [8]Lord Rayleigh, On the Pressure Developed..., Phil. Mag. Series 6, [9] Suponitsky et al., Richtmer-Meshkov instability..., Computers & Fluids, 2014.

Magnetized Target Fusion: Insights from Mathematical Modelling

Magnetized Target Fusion: Insights from Mathematical Modelling : Insights from Mathematical Modelling, UBC, PhD Candidate - Applied Math Brian Wetton, UBC, Supervisor Michael Ward, UBC, Committee Member Sandra Barsky, General Fusion, Committee Member November 28,