Bifurcated Helical Core Equilibrium States in Tokamaks W. A. Cooper - - PowerPoint PPT Presentation

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24th IAEA Fusion Energy Conference, October 8-13, 2012

Bifurcated Helical Core Equilibrium States in Tokamaks

• W. A. Cooper1, J. P. Graves1, H. Reimerdes1, O. Sauter1, M. Albergante1,
• D. Brunetti1, F. Halpern1, D. Pfefferl´

e1, T. M. Tran1, J. Rossel 1, S. Coda1,

• B. P. Duval1, A. Pochelon 1, B. Labit1, O. Schmitz2, I. T. Chapman3,
• A. D. Turnbull4, T. E. Evans4, L. Lao4, R. Buttery4, J. R. Ferron4, E. Hollman4,
• C. Petty4, M. van Zeeland4, E. A. Lazarus5, F. Turco6, J. Hanson6
• M. E. Fenstermacher7, M. J. Lanctot7, A. J. Cole8, S. C. Jardin9, B. J. Tobias9

1Ecole Polytechnique F´

ed´ erale de Lausanne, Association EURATOM-Conf´ ed´ eration Suisse, Centre de Recherches en Physique des Plasmas, CH1015 Lausanne, Switzerland

2Forschungzentrum J¨

ulich, J¨ ulich, Germany

3CCFE, Abingdon, UK 4General Atomics, San Diego, USA 5Oak Ridge National Laboratory, Oak Ridge, USA 6Columbia University, New York, USA 7Lawrence Livermore National Laboratory, Livermore, USA 8University of Wisconsin, Madison, USA 9Princeton Plasma Physics Laboratory, Princeton, USA

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Principal 3D Effects in Tokamaks

• Toroidal Magnetic Field Ripple

Periodicity ∝ Number of toroidal coils

• Test Blanket Modules, Ferritic Inserts, Toroidal Coil Quench

Periodicity typically n = 1

• ELM Control – RMP Coils

Periodicity typically n = 3 − 4

• Spontaneous Internal Helical Structure Formation — typically

’Snakes’ Periodicity n = 1

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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0.275s 0.335s 0.365s

• MAST Long Lived Mode resembles n=m=1 saturated helical mode
• As qmin approaches unity, LLM appears and fast ions are expelled from

the plasma core (fast ions distribution represented by neutron emissivity)

Magnetic axis

IT Chapman et al, Nucl Fusion, 2010

• MAST neutron emissivity
• MAST frequency spectrum

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MOTIVATION

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Introduction

• Investigate 3D helical distortions.
• Long-Lived Modes in hybrid scenarios.
• Use MHD equilibrium approach. Compare with standard initial value

nonlinear stability.

• Free boundary calculations to include RMP and ripple effects.
• Fast particle confinement in static 3D equilibrium fields.
• 3D distortion in tokamak similar to SHAx in RFP.
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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(1) Assume standard tokamak coils (almost axisymmetric boundary) (2) Solve for internal flux surfaces in equilibrium:

• Relax axisymmetry constraint in the vacuum and plasma
• Two solutions possible. One axisymmetric, the other is helical, with

displacement amplitude δH

P B J dt dV ∇ − × = ρ

TCV

δH =

• R2

01(s = 0) + Z2 01(s = 0)/a

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Equilibrium Description

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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⊲ Use Fourier decomposition in the periodic angular variables u and v and a spe-

cial finite difference scheme for the radial discretisation. Implemented in the VMEC/ANIMEC codes.

⊲ Variation of the energy

dW dt = − dsdudv

• FR

∂R ∂t + FZ ∂Z ∂t + Fλ ∂λ ∂t

• −

s=1

dudv

• R
• p⊥ + B2

2µ0 ∂R ∂u ∂Z ∂t − ∂Z ∂u ∂R ∂t

• ⊲ Solve inverse equilibrium problem : R = R(s, u, v) , Z = Z(s, u, v).

⊲ Minimise energy of the system

W = d3x B2 2µ0 + p(s, B) Γ − 1

• ⊲ Impose nested magnetic surfaces and single magnetic axis

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Magnetohydrodynamic Equilibria — 3D

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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δW/W vs Ip δH vs βN δH vs qmin Z R R R R φ = π φ = 2π/3 φ = π/3 φ = 0

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Fixed Boundary DIII-D Computations

• β ≃ 0.89%

; Ip = 1.43MA ; qmin ∼ 1 near half radius

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Free Boundary TCV Computations

• TCV coil system
• Toroidal coils modelled with 4 filaments carrying a total of 358kA
• There are 16 poloidal field coils that typically allow up to 238kA
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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δH vs qmin q vs √s Ip vs √s

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TCV Profiles and Axis Excursion versus qmin

• Pressure profile prescribed as p(s) = p(0)(1 − s)(1 − s4)
• Large helical core for 0.96 < qmin < 1.01 for β > 0.6%
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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+ n = 3 RMP coils +PF coils TF coils

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Free Boundary MAST Computations

• MAST coil system — 4 filaments per coil
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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MAST profiles

• MAST profiles: pressure (p), toroidal current (j · ∇v) and q versus √s.
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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MAST pressure distribution at the midplane

• “Axisymmetric” Branch

Helical Branch Helical Branch

• ripple

no RMP + RMP

• Boundary modulation due to core snake structure is weak.
• External perturbation does not disturb helical core.
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Displacement δH qmin ANIMEC 3D equilibrium XTOR nonlinear MHD

• D. Brunetti et al., 2012 Varenna-Lausanne Theory of Fusion Plasmas Workshop

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ITER 3D Helical Core Simulations

• Comparison of the magnetic axis displacement between the ANIMEC 3D heli-

cal branch equilibrium solution and the nonlinear saturated state evolved with the XTOR initial value stability code (H. L¨ utjens and J. F. Luciani, J. Com-

• put. Phys. 227 (2008) 6944) of the axisymmetric branch equilibrium solution in a

fixed boundary ITER simulation.

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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0.275s 0.335s 0.365s

Magnetic axis MAST neutron emissivity

• MAST neutron emissivity
• fast particle density
• D. Pfefferl´

e et al., Varenna-Lausanne International Workshop on Theory of Fusion Plasmas, 2012.

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MAST Fast Particle Guiding Centre Orbits

• Coordinate independent noncanonical phase space Lagrangian formulation of guid-

ing centre orbit theory (Littlejohn, J. Plasma Phys. 29 (1983) 111) implemented in the VENUS-LEVIS code.

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 Boozer R [m] Z [m] 0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 ANIMEC R [m] Z [m]

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TCV Boozer mesh grid

• Boozer coordinate mesh grid shows distortions at the interface of the helical core

and the axisymmetric mantle

• Boozer coordinate spectrum may not be optimal for energetic particle guiding

centre orbit confinement analysis.

• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Summary

• Axisymmetric tokamak systems: 2 solutions
• 2D axisymmetric branch
• 3D helical core branch
• Helical core predicted for
• hybrid scenario.
• standard scenario before first sawtooth crash (MAST).
• Reversed magnetic shear with qmin ∼ 1 off-axis can trigger a core

helical structure solution similar to a snake.

• The predictions are relevant for the ITER hybrid scenario operation.
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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Conclusions

• Internal helical structures weakly modulate plasma-vacuum interface.
• External perturbations do not alter 3D helical core.
• Standard nonlinear stability calculation consistent with 3D helical

core equilibrium states.

• Helical core degrades fast ion confinement.
• W. Anthony Cooper, CRPP/EPFL; 24th IAEA Fusion Energy Conference, October 8-13 2012

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