Fusion: Intro Fusion: Numerics and Asymptotics Fusion: Summary Superconductor: Problem Superconductor: Results

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1 Investigation into the Feasibility and Operation of a Magnetized Target Fusion Reactor, and Qualitative Predictions of Magnetic Field Profile Perturbations Induced by Surface Roughness in Type II Superconductors, PhD Candidate Brian Wetton, Supervisor Michael Ward, Committee Member Rob Kiefl, Committee Member May 7, 2015

2 Fusion energy context General Fusion (2002-): attempting to produce clean, sustainable fusion energy on earth.

3 Fusion Fusing atomic nuclei yield new nuclei plus energy Lawson criterion for energy yield: density temperature time cm 3 KeV s

4 General Fusion design Magnetized target fusion: magnetically confine plasma with magnetic field, implode in metal cavity

5 Lead-Lithium With density ρ, velocity v, and pressure P: ρ t + (ρv) = 0 (mass conservation) (ρv) t + (ρv v) + P = 0 (momentum conservation) Empirical fit to lead experiments P = P(ρ)

6 Pistons, plasma, and general simplifications Spherical symmetry Pressure: Piston (Gaussian), plasma (gas and magnetic) Reversible conditions; equilibrium initialization No mixing of plasma and lead-lithium: d dt r boundary(t) = v(r boundary (t), t)

7 Overall model In r L (t) < r < r R (t), t > 0, dimensionless system has form: ρ t + 1 r 2 (r 2 ρv) r = 0, (ρv) t + p r + 1 r 2 (r 2 ρv 2 ) r = 0 (1) dr L,R p = p(ρ), = v(r L,R (t), t) (2) dt p(r L (t), t) = p L (r L (t)), p(r R (t), t) = f (t) (3) v(r, 0) = 0, p(r, 0) constant (4) r L (0) given, r R (0) = 1 (5)

8 Finite volume methodology Conservation u t + (f (u)) x = 0: = ū j i k F j i+1/2 F j i 1/2 h ū j+1 i F combination of low/high resolution via limiters L 1 convergence: u num (x, t) u ex (x, t) dx = O(h p ) Fixed space domain via coordinate change Local linearized systems, approximate Riemann solvers Split stepping for geometric sources

9 Pulse profiles

10 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,

11 Qualitative story and techniques Matched asymptotics r min b4 χ 3 µ ɛ: reduced pulse time π ɛ = , sound speed b, radius χ, pressure µ - I formation: Riemann invariants - II focusing: velocity potential - III reflection: boundary conditions imply long-term velocity - IV/V compression: velocity radially dependent

12 Minimum radius 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

13 Key insights Almost all input energy reflected: E input 8π 3 b ɛ3/2, E compression 4π2 b 4 χ 3 ɛ5/2

14 Results and future work Results: - energy yield may be within reach - larger outer sphere radius and impact pressure noteworthy Future directions: - more physics - effects of imperfect spherical symmetry - more precise assessment of design

15 Fusion: Intro Fusion: Numerics and Asymptotics Fusion: Summary Superconductor: Problem Superconductor: Results Superconductor roughness context Superconductors expel magnetic fields, some unresolved questions that arise in comparing theory to experiment.

16 Overview Superconductors: - cold enough - no resistance, expel magnetic fields - YBCO studied experimentally, unexpected field profiles London Model: - field decays from applied value exponentially with length scale λ with flat surface - experiments find dead layer: could roughness cause this?

17 Methodology Use real AFM surface data to study how fitting parameters affected

18 Results λ may be underestimated, almost no dead layer: best fitting (λ, δ) are (0.956λ true, 0.016λ true ) Minute change in field orientation

19 Future work Extend simulations to spatially varying order-parameters Consider anisotropic superconductors

20 Thanks to... Thanks to the following people for helpful discussions, collaboration, and support: My committee: Brian Wetton, Michael Ward, Rob Kiefl (UBC) Alex Fang (UBC) Sandra Barsky and Aaron Froese (General Fusion) George Bluman (UBC) Randy LeVeque (University of Washington) My family, friends, colleagues, and fellow graduate students

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,