bifurcated helical core equilibrium states in tokamaks...internal helical structures weakly modulate...

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

Bifurcated Helical Core EquilibriumStates in Tokamaks

W. A. Cooper1, J. P. Graves1, H. Reimerdes1, O. Sauter1, M. Albergante1,D. Brunetti1, F. Halpern1, D. Pfefferle1, 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 Federale de Lausanne, Association EURATOM-Confederation Suisse, Centre de Recherches en

Physique des Plasmas, CH1015 Lausanne, Switzerland

2Forschungzentrum Julich, Julich, 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 TH/7-1 1

CRPP 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


0.275s0.335s0.365s

• 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

CRPP MOTIVATION


CRPP 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.


(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

PBJdtdV

∇−×=ρ0

TCVδH

=√R2

01(s = 0) + Z201(s = 0)/a

CRPP Equilibrium Description


. Use Fourier decomposition in the periodic angular variables u and v and a spe-cial finite difference scheme for the radial discretisation. Implemented in theVMEC/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

CRPP Magnetohydrodynamic Equilibria — 3D


δW/W vs IpδH vs βNδH vs qmin

Z

RRRR

φ = πφ = 2π/3φ = π/3φ = 0

CRPP Fixed Boundary DIII-D Computations

• 〈β〉 ' 0.89% ; Ip = 1.43MA ; qmin ∼ 1 near half radius


CRPP 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


δH vs qminq vs√sIp vs

√s

CRPP 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%


+ n = 3 RMP coils

+PF coils

TF coils

CRPP Free Boundary MAST Computations

• MAST coil system — 4 filaments per coil


CRPP MAST profiles

• MAST profiles: pressure (p), toroidal current (〈j · ∇v〉) and q versus√s.


CRPP 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.


Dis

plac

emen

t δH

qmin

ANIMEC 3D equilibrium

XTOR nonlinear MHD

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

CRPP 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 withthe XTOR initial value stability code (H. Lutjens and J. F. Luciani, J. Com-put. Phys. 227 (2008) 6944) of the axisymmetric branch equilibrium solution in afixed boundary ITER simulation.


0.275s0.335s0.365s

Magnetic axis

MAST neutronemissivity

•MAST neutron emissivity•fast particle density

D. Pfefferle et al., Varenna-Lausanne International Workshop on Theory of FusionPlasmas, 2012.

CRPP 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) implementedin the VENUS-LEVIS code.


0.7 0.8 0.9 1

−0.4

−0.3

−0.2

−0.1

0

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

0.1

0.2

0.3

0.4

ANIMEC

R [m]

Z [m

]

CRPP TCV Boozer mesh grid

• Boozer coordinate mesh grid shows distortions at the interface of the helical coreand the axisymmetric mantle

• Boozer coordinate spectrum may not be optimal for energetic particle guidingcentre orbit confinement analysis.


CRPP 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.


CRPP 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.


bifurcated helical core equilibrium states in tokamaks...internal helical structures weakly modulate...

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