Title: Single Helicity and QuasiSingle Helicity Computations for Reversed Field Pinches
1Single Helicity and Quasi-Single Helicity
Computations for Reversed Field Pinches
- Gian Luca Delzanno
- Luis Chacon
- John M. Finn
- Rick Nebel
2Outline
- Scope
- Paramagnetic pinch
- Linear theory
- PIXIE3D
- Mean-field linear theory
- Conclusions
3Scope
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
4Mathematical model
- Dimensionless MHD
- Zero ß limit aspect ratio R/a5
-
- Hartmann number and Prandtl
number
5Paramagnetic pinch magnetic field
Cylindrical ohmic state with no reversal
- Cylindrical pinch
- Electric field
- Force free
-
6Paramagnetic pinch fields
E0/?4
7Paramagnetic pinch q profile
- Linear stability investigation of several
resonances in the positive q region
ideal
E0/?4
8Linear stability
Paramagnetic pinch
No flow
Pr1
E0/?4
No flow constant ?
Benchmarked with PIXIE3D
9Linear stability E0/?
E0/?3.75
E0/?4
Pr1
E0/?3.5
No room for SH
10The 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
11Mean-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.
12Mean-field linear theory n9
E0/?4
Multiple unstable helicities
13Mean-field linear theory n10
E0/?4
Multiple (Quasi-single) unstable helicities
14Mean-field linear theory n11
E0/?4
Single unstable helicity
15Conclusions
- 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
16Mean-field q and ?
q(0) 0.115 n10 q(0) 0.110 n9 q(0)0.107
equil.
E0/?3.75
17Mean-field linear theory n9
E0/?3.75
18Mean-field linear theory n10
E0/?3.75
19Introduction
- 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
20Paramagnetic pinch flow
- Radial flow to vanish at the wall
21Linear stability Pr
Pr10
Pr1
E0/?4
Pr100
Marginal stability dependent on H-Pr
No room for SH