Single Helicity and QuasiSingle Helicity Computations for Reversed Field Pinches

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Single Helicity and Quasi-Single Helicity
Computations for Reversed Field Pinches

• Gian Luca Delzanno
• Luis Chacon
• John M. Finn
• Rick Nebel

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Outline

• Scope
• Paramagnetic pinch
• Linear theory
• PIXIE3D
• Mean-field linear theory
• Conclusions

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Scope
P. Martin et al, Nucl. Fus. 43, 1855 (2003)

• Quasi-Single Helicity (QSH) state
• Experimental results indicate QSH is more
  probable for higher current
• Numerical results show QSH when there is enough
  dissipation

Our goal gain understanding of the transitions,
expand the range of SH and QSH
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Mathematical model

• Dimensionless MHD
• Zero ß limit aspect ratio R/a5
• Hartmann number and Prandtl
  number

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Paramagnetic pinch magnetic field
Cylindrical ohmic state with no reversal

• Cylindrical pinch
• Electric field
• Force free

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Paramagnetic pinch fields

• Magnetic fields

• Density and flow

E0/?4
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Paramagnetic pinch q profile

• Dependence on E0/?

• Linear stability investigation of several
  resonances in the positive q region

ideal
E0/?4
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Linear stability
Paramagnetic pinch
No flow
Pr1
E0/?4
No flow constant ?
Benchmarked with PIXIE3D
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Linear stability E0/?
E0/?3.75
E0/?4
Pr1
E0/?3.5
No room for SH
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The tool PIXIE3D

• Parallel, implicit, extended 3D MHD code
• L. Chacon, Comp. Phys. Comm. 163 (3), 143-171
  (2004)
• Finite volume discretization
• Curvilinear geometry (fully general helical,
  cylindrical, toroidal)
• Conservative
• Linearly and nonlinearly stable without numerical
  dissipation
• Solenoidal in B and J
• Second-order implicit temporal discretization
• Newton-Krylov scheme
• Physics-based preconditioning (in progress)
• Parallelization with PETSc toolkit

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Mean-field q and ?

• Mean field computed from the saturated state of
  PIXIE3D in single helicity
• Paramagnetic pinch green
• Three helicities
• n9 blue
• n10 black
• n11 red
• Exploring the region (Pr1)
• H400
• H1300
• H2200
• H3100
• H4000

E0/?4
q(0) 0.113 n11 q(0) 0.110 n10 q(0) 0.104
n9 q(0)0.100 equil.
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Mean-field linear theory n9
E0/?4
Multiple unstable helicities
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Mean-field linear theory n10
E0/?4
Multiple (Quasi-single) unstable helicities
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Mean-field linear theory n11
E0/?4
Single unstable helicity
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Conclusions

• Benchmark with PIXIE3D
• Single helicity state not explained by the linear
  theory of the original equilibrium profile
• Conjecture transition MH?SH can be explained by
  the mean field profile unstable to a second mode
  (?)
• Ongoing fully 3D runs
• Physics based preconditioning

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Mean-field q and ?
q(0) 0.115 n10 q(0) 0.110 n9 q(0)0.107
equil.
E0/?3.75
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Mean-field linear theory n9
E0/?3.75
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Mean-field linear theory n10
E0/?3.75
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Introduction

• Comparable magnetic field components, mainly
  generated by internal currents
• Toroidal field at the edge opposite to that in
  the core
• High ß, low q plasma with respect to tokamak
• MHD instabilities resonances (m1,n) in positive
  (accumulating at the reversal point) and negative
  q regions
• Multiple Helicity (MH) state turbulence, wide
  spectrum of m0, m1 modes bad confinement
  properties

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Paramagnetic pinch flow

• Radial flow to vanish at the wall

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Linear stability Pr
Pr10
Pr1
E0/?4
Pr100
Marginal stability dependent on H-Pr
No room for SH