effect of helical magnetic perturbations on the 3d mhd self ......2015/05/03 · 7th iaea technical...
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7th IAEA Technical Meeting on “Theory of Plasma Instabilities”, Frascati – March 4-6, 2015
Effect of helical magnetic perturbations on the
3D MHD self-organization of fusion plasmas
Daniele Bonfiglio1,
S. Cappello1, M. Veranda1, L. Chacón2 and D. F. Escande3,1
1Consorzio RFX, Euratom-ENEA Association, Padova, Italy
2Los Alamos National Laboratory, Los Alamos, USA
3Aix-Marseille Université, CNRS, PIIM, Marseille, France
Helical RFP Diverted tokamak
3D effects in axisymmetric fusion devices
3D effects play an important role in nominally axisymmetric toroidal configurations
for magnetic confinement, such as the tokamak and the reversed-field pinch (RFP)
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 1
3D effects in axisymmetric fusion devices: self-organization
3D effects play an important role in nominally axisymmetric toroidal configurations
for magnetic confinement, such as the tokamak and the reversed-field pinch (RFP)
3D effects can result from plasma self-organization (long-lived helical states)…
Tokamak:
Density snake [A. Weller et al., PRL 1987;
L. Delgado-Aparicio et al., NF 2013]
Saturated ideal modes in
advanced (high-) regimes [I. T. Chapman et al., NF 2010;
W. A. Cooper et al., NF 2013]
Reversed-field pinch (RFP):
Single-helical axis (SHAx)
state with ITBs in RFX-mod [R. Lorenzini et al., Nature Phys.
2009; J. S. Sarff et al., NF 2013]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 1
3D effects in axisymmetric fusion devices: external MPs
3D effects play an important role in nominally axisymmetric toroidal configurations
for magnetic confinement, such as the tokamak and the reversed-field pinch (RFP)
3D effects can result from plasma self-organization (long-lived helical states)…
… and from external magnetic perturbations (MPs) with non-axisymmetric coils
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 1
3D effects in axisymmetric fusion devices: external MPs
3D effects in nominally axisymmetric toroidal configurations for magnetic
confinement, such as the reversed-field pinch (RFP) and the tokamak
3D effects can result from plasma self-organization…
… and from external magnetic perturbations (MPs) with non-axisymmetric coils
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 1
Non-axisymmetric coils are used for:
Error field correction and feedback
control of MHD instabilities [M. S. Chu and M. Okabayashi, PPCF 2010]
Edge-localized modes (ELMs)
suppression [T. E. Evans et al., PRL 2004]
Drive of neoclassical toroidal
rotation [A. M. Garofalo et al., PRL 2008]
Alfvén modes mitigation [A. Bortolon et al., PRL 2013]
…
RMPs in MHD simulations with JOREK [F. Orain et al.,, PoP 2013; M. Bécoulet, this morning]
Coupling of internal and external 3D effects
External magnetic perturbations can stimulate and control helical self-organization
RFP. RFX-mod:
MPs with the same helicity as spontaneous helical mode:
persistence of spontaneous SHAx states increased
MPs with different helicities:
helical states with the chosen helicity induced
MPs mitigate the spontaneous sawtooth activity of RFP plasmas
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 2
Coupling of internal and external 3D effects
External magnetic perturbations can stimulate and control helical self-organization
RFP. RFX-mod:
MPs with the same helicity as spontaneous mode:
persistence of spontaneous SHAx states increased
MPs with different helicity:
helical states with the chosen helicity induced
MPs mitigate the spontaneous sawtooth activity of RFP plasmas
Tokamak. RFX-mod and DIII-D:
n = 1 MPs mitigate the sawtooth activity associated with the internal kink mode
Experimental findings qualitatively reproduced by nonlinear 3D MHD
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 2
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their verification benchmark
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 3
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their verification benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
Outline of the talk
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 3
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their verification benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 3
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their verification benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
4. Summary and perspectives
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 3
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their verification benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
4. Summary and perspectives
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
SPECYL code [S. Cappello and D. Biskamp, NF 1996]:
Solves the equations of the nonlinear visco-resistive MHD model
tv + (v) v = JB + 2v momentum balance
tB = E = (vB J) Faraday-Ohm eq.
B = 0, J = B
Resistivity: = A/R S-1 (inverse Lundquist number)
Viscosity: = A/V M-1 (inverse viscous Lundquist number)
Hartmann number (inverse “-dissipation”): H ()-½ = (SM)½
Approximations: cylindrical geometry, const, p 0
Magnetic BCs: ideal wall or helical MPs brm,n(a)=c [D. Bonfiglio et al., NF 2011]
Nonlinear 3D MHD modelling: the SPECYL code
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 4
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
PIXIE3D code [L. Chacón, PoP 2008 and refs. therein]:
Takes into account additional MHD terms
t + (v) = 0 continuity equation
t(v) + (vv) = JB – p + (v) momentum balance
tT + vT + (1)[Tv–(T+Q)/2n]=0 temperature eq.
tB = E = (vB J) Faraday-Ohm eq.
B = 0, J = B
Finite volume, fully implicit, general curvilinear formulation:
Both cylindrical and toroidal geometries allowed
Same magnetic BCs as SPECYL: ideal wall or helical MPs
First PIXIE3D finite- with isotropic transport [D. Bonfiglio et al., PPCF 2015]
In this talk: constant density, =0, toroidal geometry
Nonlinear 3D MHD modelling: the PIXIE3D code
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 5
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Nonlinear verification benchmark: SPECYL – PIXIE3D
Nonlinear verification benchmark performed in the common limit of application of
the two codes [D. Bonfiglio, L. Chacón and S. Cappello, PoP 2010]
Examples: helical (2D) simulations in cylindrical geometry
Temporal evolution of the magnetic energy associated with helical harmonics
SPECYL (black) and PIXIE3D (red curves) superimposed
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
6
loga
rithm
ic s
cale
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
4. Summary and perspectives
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
The RFX-mod device in Padova (Italy)
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 7
The largest RFP in operation:
R0 = 2 m, a = 0.46 m
Max IP = 2 MA
Max B = 0.7 T
ne 1÷51019 m-3
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
The RFX-mod device in Padova (Italy)
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 7
The largest RFP in operation:
R0 = 2 m, a = 0.46 m
Max IP = 2 MA
Max B = 0.7 T
ne 1÷51019 m-3
Fully covered by saddle coils for MHD
control and magnetic perturbations:
Also operated as Ohmic tokamak:
(ideal test bed for code validation)
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Helical RFP self-organization in the experiment
In RFX-mod and other RFP devices, the plasma current rules the transition from
multiple helicity (MH) to quasi-single helicity (QSH) states
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 8
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
In RFX-mod and other RFP devices, the plasma current rules the transition from
multiple helicity (MH) to quasi-single helicity (QSH) states
Low current discharge: MH regime (low confinement due to stochastic transport)
Helical RFP self-organization in the experiment
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 8
MHD modes with poloidal
periodicity m=1
toroidal
periodicity
plasma current waveform
edge
toro
idal
fie
ld (
%)
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1,
In RFX-mod and other RFP devices, the plasma current rules the transition from
multiple helicity (MH) to quasi-single helicity (QSH) states
Low current discharge: MH regime (low confinement due to stochastic transport)
High-current discharge: QSH (improved confinement) with back-transitions
Helical RFP self-organization in the experiment
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 8
MHD modes with poloidal
periodicity m=1
toroidal
periodicity
Systematic QSH states
with preferred helicity
plasma current waveform
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
edge
toro
idal
fie
ld (
%)
m=1, m=1,
In RFX-mod and other RFP devices, the plasma current rules the transition from
multiple helicity (MH) to quasi-single helicity (QSH) states
Low current discharge: MH regime (low confinement due to stochastic transport)
High-current discharge: QSH (improved confinement) with back-transitions
Helical RFP self-organization in the experiment
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 8
Hot helical core
enclosed by ITBs
SHAx state
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1,
SPECYL w/o MPs: helical RFP self-organization
In this talk: SPECYL simulations with 225 Fourier harmonics with 0≤m≤4
In 3D MHD with ideal wall, the visco-resistive dissipation rules the transition from
multiple helicity (MH) to single helicity (SH) states [S. Cappello, D. Escande, PRL 2000]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 9
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
In 3D MHD with ideal wall, the visco-resistive dissipation rules the transition from
multiple helicity (MH) to single helicity (SH) states [S. Cappello, D. Escande, PRL 2000]
Low dissipation: sawtoothing MH with quasi-periodic reconnection events
SPECYL w/o MPs: helical RFP self-organization
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 9
edge safety factor: sawteeth
edge
toro
idal
fie
ld (
%)
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1, reconnection events
SPECYL w/o MPs: helical RFP self-organization
In 3D MHD with ideal wall, the visco-resistive dissipation rules the transition from
multiple helicity (MH) to single helicity (SH) states [S. Cappello, D. Escande, PRL 2000]
Low dissipation: sawtoothing MH with quasi-periodic reconnection events
High dissipation: spontaneous SH Ohmic equilibrium (related to RFX-mod!?)
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 9
SH equilibrium
edge safety factor: sawteeth
edge
toro
idal
fie
ld (
%)
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1, reconnection events
m=1,
SPECYL with 1,7 MPs: systematic QSH states
Low dissipation regime: systematic repetition of QSH states with chosen helicity in
between reconnection events [M. Veranda et al., PPCF 2013; D. Bonfiglio et al., PRL 2013]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 10
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
SPECYL with 1,7 MPs: systematic QSH states
Low dissipation regime: systematic repetition of QSH states with chosen helicity in
between reconnection events [M. Veranda et al., PPCF 2013; D. Bonfiglio et al., PRL 2013]
Ideal wall: sawtoothing MH with quasi-periodic reconnection events
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 10
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1,
Low dissipation regime: systematic repetition of QSH states with chosen helicity in
between reconnection events [M. Veranda et al., PPCF 2013; D. Bonfiglio et al., PRL 2013]
Ideal wall: sawtoothing MH with quasi-periodic reconnection events
Helical MPs with m=1, n=7 periodicity: sawtoothing QSH states
SPECYL with 1,7 MPs: systematic QSH states
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 10
Systematic helical state
Sawtooth amplitude & period reduced
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1, m=1,
Comparison with spontaneous helical states in RFX-mod
Qualitative agreement with the experiment is obtained [D. Bonfiglio et al., PRL 2013]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 11
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Comparison with spontaneous helical states in RFX-mod
Qualitative agreement with the experiment is obtained [D. Bonfiglio et al., PRL 2013]
MHD simulation with helical MPs and realistic S=107
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 11
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1,
Qualitative agreement with the experiment is obtained [D. Bonfiglio et al., PRL 2013]
MHD simulation with helical MPs and realistic S=107
Reference RFX-mod discharge at high current (S=1.5107)
Main quantitative differences: range of q(a), secondary modes amplitude
Comparison with spontaneous helical states in RFX-mod
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 11
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
m=1, m=1,
SPECYL with 1,6 MPs: stimulated helical states
By using helical MPs, QSH states with the chosen helicity can be stimulated, as
confirmed on RFX-mod [S. Cappello et al., IAEA 2012; M. Veranda et al., PPCF 2013]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 12
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
SPECYL with 1,6 MPs: stimulated helical states
By using helical MPs, QSH states with the chosen helicity can be stimulated, as
confirmed on RFX-mod [S. Cappello et al., IAEA 2012; M. Veranda et al., PPCF 2013]
MHD simulation: helical MPs with n=7 first, then n=6 periodicity
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 12
n=7 MPs n=6 MPs
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
By using helical MPs, QSH states with the chosen helicity can be stimulated, as
confirmed on RFX-mod [S. Cappello et al., IAEA 2012; M. Veranda et al., PPCF 2013]
MHD simulation: helical MPs with n=7 first, then n=6 periodicity
RFX-mod discharge: n=6 RWM feedback-controlled at finite amplitude
SPECYL with 1,6 MPs: stimulated helical states
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 12
n=7 n=7
n=7 MPs n=6 MPs
n=6 MPs
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
4. Summary and perspectives
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
RFX-mod: effect of n=1 MPs in Ohmic tokamak experiments
In RFX-mod and DIII-D, sawtooth oscillations are mitigated by n=1 MPs [P. Martin et al., IAEA 2014]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
13
Sawtooth amplitude and
period significantly reduced
Sawtoothing 1,1 internal
kink replaced by a more
continuous 1,1 helical core
Amplitude of the
applied 1,1 MP
Phase of the
applied 1,1 MP
SX
R p
rofil
e
Similar phenomenology observed in MHD modelling [D. Bonfiglio et al., EPS 2013]
Tokamak part: PIXIE3D in toroidal geometry (mesh resolution nr×n×n=128×64×64)
Axisymmetric circular equilibrium with q profile consistent with RFX-mod expts:
qa1.8 and "vacuum" region with high between plasma and the wall
q=2 resonance located in the vacuum region
PIXIE3D w/o MPs: RFX-mod tokamak equilibrium
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 14
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
PIXIE3D w/o MPs: spontaneous sawtooth cycles
Like in the RFP, with ideal wall MHD ruled by dissipation [D. Bonfiglio et al., PoP 2010]
Low dissipation: periodic sawtooth cycles of internal kink mode like in RFX-mod
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
15
core safety factor
core
n=
1 ra
dial
fiel
d (%
)
n=1, sawtooth crashes
Like in the RFP, with ideal wall MHD ruled by dissipation [D. Bonfiglio et al., PoP 2010]
Low dissipation: periodic sawtooth cycles of internal kink mode like in RFX-mod
Magnetic topology before sawtooth crash: circular magnetic surfaces
Poincaré plot by field-line tracing code NEMATO [J. Finn and L. Chacón, PoP 2005]
PIXIE3D w/o MPs: spontaneous sawtooth cycles
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
15
n=1,
Like in the RFP, with ideal wall MHD ruled by dissipation [D. Bonfiglio et al., PoP 2010]
Low dissipation: periodic sawtooth cycles of internal kink mode like in RFX-mod
Magnetic topology before sawtooth crash: circular magnetic surfaces
At sawtooth crash: large 1,1 island, small 3,2 and 2,1 islands by toroidal coupling
PIXIE3D w/o MPs: spontaneous sawtooth cycles
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
15
n=1,
PIXIE3D with 1,1 MPs: sawtooth mitigation
MPs with 1,1 periodicity and 0.1% amplitude: sawtooth mitigation like in RFX-mod
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
16
Sawtooth amplitude & period reduced
sawtooth crashes n=1,
PIXIE3D with 1,1 MPs: sawtooth mitigation
MPs with 1,1 periodicity and 0.1% amplitude: sawtooth mitigation like in RFX-mod
Magnetic topology at sawtooth crash: larger 2,1 island by toroidal coupling
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
16
n=1,
Helical MP with m=1, n=1 periodicity and 0.1 % amplitude: sawtooth mitigation
Magnetic topology at sawtooth crash: larger 2,1 island by toroidal coupling
Magnetic topology after sawtooth crash: large 1,1 island still there
Like in RFX-mod, helical core deformation always present
PIXIE3D with 1,1 MPs: sawtooth mitigation
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
16
n=1,
n=0 MPs shaped tokamak plasmas within a circular domain [this meeting proceedings]
In this example, n=0 MPs with 4% m=2 (elongation) and 8% m=3 (triangularity):
D-shaped, double-null diverted tokamak (plasma aspect ratio 2.5)
PIXIE3D with n=0 MPs: diverted tokamak equilibrium
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 17
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
PIXIE3D with n=0 MPs: diverted tokamak equilibrium
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 17
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
n=0 MPs shaped tokamak plasmas within a circular domain [this meeting proceedings]
In this example, n=0 MPs with 4% m=2 (elongation) and 8% m=3 (triangularity):
D-shaped, double-null diverted tokamak (plasma aspect ratio 2.5)
Fixed resistivity profile such that initial q0<1 and q95=46
Nonlinear 3D MHD dynamics similar to the circular case. Without helical MPs:
low visco-resistive dissipation: sawtooth cycles
PIXIE3D with n=0 MPs: magnetic topology
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
18
core safety factor
core
rad
ial f
ield
(%
)
n=1, sawtooth crashes
Nonlinear 3D MHD dynamics similar to the circular case. Without helical MPs:
low visco-resistive dissipation: sawtooth cycles
high dissipation: stationary equilibrium with helical core
similar findings in ANIMEC [W. Cooper et al., NF 2013] and XTOR [D. Brunetti et al., NF 2014]
PIXIE3D with n=0 MPs: magnetic topology
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
18
n=1,
PIXIE3D with n=0 MPs: magnetic topology
Nonlinear 3D MHD dynamics similar to the circular case. Without helical MPs:
low visco-resistive dissipation: sawtooth cycles
high dissipation: stationary equilibrium with helical core
With n=1 helical MPs, at low dissipation: sawtooth mitigation, edge stochastization
2,1 MP: homoclinic lobes in connection length as in [O. Schmitz et al., PPCF 2008]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
18
Outline of the talk
1. Modelling tools. Nonlinear 3D MHD codes:
SPECYL, PIXIE3D and their benchmark
2. RFP modelling. Effect of MPs:
Spontaneous helical states
Stimulated helical states with chosen helicity
3. Tokamak modelling. Effect of MPs:
Mitigation of the sawtooth activity
Plasma shaping with X-points
4. Summary and perspectives
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Summary
The effect of magnetic perturbations on helical self-organization of fusion plasmas
has been discussed in the framework of nonlinear 3D MHD
Magnetic perturbations:
enlarge the parameter space of long-lived helical states towards low dissipation
RFP configuration:
systematic repetition of helical states consistent with experimental ones
Tokamak:
sawtooth mitigation experiments reproduced
shaped plasmas with X-points (diverted tokamak)
Synergy between RFP and tokamak research
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 19
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Future perspectives
Approximations (in present study) to be removed:
finite for MHD equilibrium and stability
plasma rotation for MPs screening
two-fluid and kinetic effects for realistic low-collisionality conditions
coupling with kinetic codes for fast particles?
RFP, tokamak and stellarator studies:
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas 20
1) Modelling tools 2) RFP modelling with MPs 3) Tokamak modelling with MPs 4) Summary
Classical stellarator
ongoing PIXIE3D applications
Helical RFP Diverted tokamak
Spare slides: stellarator
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Stellarator studies: effect of MPs on magnetic topology
Stellarator-like simulations with SpeCyl [D. Brunetti et al., NF 2014]:
Dominant MP → straight stellarator field with helical symmetry in vacuum
Secondary MP → open up magnetic islands at resonant surfaces
SpeCyl simulations with fixed m=2, n=-2 dominant MP and increasing amplitude of
m=2, n=-1 secondary MP:
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Only dominant MP Secondary MP=0.02 % Secondary MP=0.2 %
Stellarator studies: PIXIE3D simulations in toroidal geometry
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Spare slides: tokamak
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
RFX-mod: q(a)<2 tokamak operation by 2/1 RWM control
Successful even with partial coil coverage down to 3%
Potential impact for high-current, high-fusion gain scenarios, not steady-state
#29774
#29797
IP (MA)
q(a)
2,1 Br (%)
time (s)
with mode control
Courtesy of P. Piovesan
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
RFX-mod: effect of n=1 MPs in Ohmic tokamak experiments
#30398, #30403 2,1 Br (%)
2,1 phase (rad)
core SXR
time (s)
2,1 RWM maintained at finite
amplitude by feedback control
Sawtooth amplitude significantly
reduced
Sawtoothing 1,1 internal kink
replaced by a more continuous
1,1 helical core
SXR
Courtesy of P. Piovesan
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
In RFX-mod, stable Ohmic tokamak operation with q(a)<2 is demonstrated with
feedback control of the 2,1 resistive-wall mode (RWM). Reproduced in DIII-D.
If the 2,1 resistive-wall mode (RWM) is controlled at finite amplitude sawtooth
oscillations of the 1,1 internal kink are made more frequent and less intense.
Finite and coupling with heat transport: helical Tokamak
Saturated pressure driven internal kink in tokamak with hollow q profile and qmin1
(helical hybrid scenario [A. C. C. Sips et al., PPCF 2002])
Main PIXIE3D results (in agreement with XTOR simulations [D. Brunetti et al., NF 2014]):
internal kink linearly unstable provided qmin is close to 1 and is large
saturated state: helical core with largest displacement when qmin is close to 1
Saturated helical states for PIXIE3D simulations with total =5.5%:
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
qmin=1.08 qmin=1.06 qmin=1.03
Red dots: stable; green dots: unstable internal kink simulations.
Stability diagram
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Similar physics in Tokamak. With ideal wall, the visco-resistive dissipation rules the
transition from sawtoothing dynamics to helical equilibrium [D. Bonfiglio et al., PoP 2010]
Low dissipation: periodic sawtooth cycles of internal kink mode
High dissipation: stationary helical state. Also in XTOR-2F [F. Halpern et al., PPCF 2011]
Tokamak simulations of internal kink: helical self-organization
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
sawtooth crashes
core safety factor
core
rad
ial f
ield
(%
)
Snake-like
equilibrium
Low dissipation: mitigation of sawtooth dynamics and stimulation of the transition to
the helical state [M. Veranda et al., EPS 2012, D. Bonfiglio et al., EPS 2013]
Helical MP with m=1, n=1 periodicity and 0.1 % amplitude: sawtooth mitigation
Helical MP with 0.3 % amplitude: helical equilibrium, also in RFX-mod
Tokamak simulations of internal kink: effect of MPs
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
More frequent & less intense sawteeth
Snake-like
equilibrium
Effect of helical MPs on the sawtooth dynamics: PIXIE3D
Observed mitigation of the sawtooth dynamics in tokamak experiments on RFX-
mod reproduced in MHD [D. Bonfiglio, P. Martin and P. Piovesan, EPS 2013]
MHD simulations with PIXIE3D in toroidal geometry, S=105 and P=30
black: no external field applied blue: 2,1 external field with 0.1% ampl.
green: 1,1 external field with 0.1% red: 1,1 external field with 0.3% ampl.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Shaped PIXIE3D simulations: axisymmetric equilibria
See proceeding paper, this meeting. Two cases:
Purely elongated
D-Shaped
The elongated equilibrium is vertically unstable
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: low dissipation w/o MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: high dissipation w/o MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: low dissipation, 1/1 MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: low dissipation, 2/1 MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: low dissipation, 3/1 MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
D-shaped PIXIE3D simulations: low dissipation, 4/1 MPs
See proceeding paper, this meeting.
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Spare slides: RFP
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Dominant helical mode only:
magnetic surfaces with helical core enclosed by an almost axis-symmetric edge
q profile with core shear reversal also similar to experimental reconstructions
SPECYL with 1,7 MPs: magnetic topology of QSH states
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Cylindrical MHD configuration bent into a torus
Dominant helical mode only:
magnetic surfaces with helical core enclosed by an almost axis-symmetric edge
q profile with core shear reversal also similar to experimental reconstructions
All modes 0 ≤ m ≤ 4: field-line tracing code NEMATO [J. Finn and L. Chacón, PoP 2005]
Secondary modes 4 (to match RFX-mod)
SPECYL with 1,7 MPs: magnetic topology of QSH states
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
edge m=0 islands
chaotic region
core n=7 structures
transport barrier
The reversed-field pinch (RFP)
Is a toroidal device like the tokamak, but for a given core toroidal toroidal field:
the plasma current is larger in the RFP
the edge toroidal field is small and reversed.
In principle, the RFP might provide a cheap and safe reactor concept:
no need of superconductive coils and additional heating, no disruptions
but requires stabilizing shell / feedback system to control current-driven MHD
modes and a confinement level higher than in present devices.
Helical RFP states are associated with improved confinement properties.
B
B
0
0 r/a 1
Tokamak
B >> B B
B
0 0 r/a 1
RFP
B ≈ B
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Safety factor profile in RFX-mod and main resonances
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
The RFP configuration is prone to the onset of several MHD resistive kink / tearing
modes with m=1 and m=0
( )rB
q rRB
=
Safety factor:
Experimental scaling of dominant and secondary modes
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Separatrix expulsion and ITBs in the experiment
In RFX-mod, the ITB formation happens in concomitance with the expulsion of the
dominant mode’s separatrix [R. Lorenzini et al., PRL 2008]
b / B 4%
with separatrix
w/o separatrix
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Separatrix expulsion and ITBs in the experiment
Poincaré plots made with ORBIT code as shown in [P. Piovesan et al., NF 2009]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
b / B 3%
Larger region with
conserved helical
flux surfaces
+
broad region of sticky
magnetic field lines b / B 5%
Remnant helical flux
surfaces
Thomson scattering
An increased resilience to chaos is observed in MHD modeling after the dominant
mode’s separatrix expulsion [D. F. Escande et al., PRL 2000; D. Bonfiglio et al., JPCS 2010]
QSH states: chaos healing after separatrix expulsion
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
with separatrix: magnetic island
chaos
dominant mode amplitude:
small large
w/o separatrix: single helical axis
conserved helical structure
QSH at high IP: correlation between ITB and q profile
Like in the tokamak, a null of the magnetic
shear could help to reduce transport at the
ITB
In fact, the position of the ITB is correlated
with the maximum of the safety factor
profile [M. E. Puiatti et al., PPCF 2009; M. Gobbin et al.,
PRL 2011]
The q profile is computed as the number
of toroidal turns field lines perform for one
poloidal turn around the helical axis. =
effective radial coordinate starting from
the helical axis
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Helical magnetic boundary in RFX-mod
br1,-7 (a) up to 1.5% for RFX-mod standard operation [P. Zanca et al., NF 2007]
br1,-7 (a) up to 3% with 3D shaping [P. Piovesan et al., PPCF 2011]
Secondary modes have much smaller br1,-7 (a) amplitudes
Left: RFX-mod with standard operation; right: 3D shaping by external coils
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Helical RFP self-organization in 3D MHD modelling
Theoretical studies triggered the experimental discovery of the helical RFP:
Spontaneous SH equilibrium solutions of nonlinear 3D MHD with ideal wall
[S. Cappello and R. Paccagnella, Varenna 1990, PoF B 1992; J. Finn, R. Nebel and C. Bathke, PoF B 1992]
The MHD bifurcation is mainly ruled by the Hartmann number H ()-½, as
shown in [S. Cappello, D. Escande, PRL 2000; PPCF 2004; PoP 2006]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
high dissipation low high dissipation low
If time is rescaled as it follows that
Definitions: Harmann number H ()-1/2 ; Prandtl number P /
The model equations are rescaled as:
Shown in [S. Cappello and D. Escande, PRL 2000], highlighted before by [D. Montgomery et al. PPCF
92-93, Tebaldi and Ottaviani JPP 99]
ttt
= v v =
v
H is the important parameter
when inertia is negligible! )vH(
dvd
P
1 1-2= BJt
1, p 0
)H()v( 1- JBtB =
( , ) (H , P)
Rescaling of the model equations
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Helical boundary conditions in SpeCyl
Analytical studies suggest that finite edge radial magnetic field favours achieving
helical RFP equilibria [D. Bonfiglio, D. Escande, P. Zanca and S. Cappello, NF 2011]
Helical MPs are a schematic representation of the RFX-mod magnetic boundary:
br1,-7 (a) up to 1.5% for RFX-mod standard operation [P. Zanca et al., NF 2007]
br1,-7 (a) up to 3% with 3D shaping [P. Piovesan et al., PPCF 2011]
n=-10 SH equilibria with varying br1,-10(a)=0, 4, 8, 12, 16% from SpeCyl:
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
High dissipation regime
Stimulation of SH states with helicity different than the spontaneous one
[M. Veranda et al., EPS 2012; PPCF 2013; EPS 2013]
Below threshold helical BC amplitude: the spontaneous mode survives
Above threshold: a SH state with the chosen helicity is stimulated
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Magnetic reconnection: energy convertion and current sheets
Signature of quasi-periodic magnetic reconnection events in RFP simulations:
Conversion of magnetic energy into kinetic energy
Formation of current sheets, both m=0 as in figure and m=1 [S. Cappello, PPFC 2004]
also observed in RFX-mod [M. Zuin et al., PPCF 2009]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Low dissipation, n=-7 helical MPs: secondary modes scaling
The total secondary modes amplitude decreases with H [D. Bonfiglio et al., PRL 2013]
The resulting dependence with S at fixed M is rather weak, with S at fixed P is
closer to the experiment: hidden viscosity effect in RFX-mod scaling with S?
Quantitative comparison: need for a more complete model. Two-fluid effects reduce
amplitudes by a factor of two [J. King, C. Sovinec and V. Mirnov, PoP 2012]
Secondary modes vs H for simulations with 2% helical MPs on n=-7 mode
Secondary modes vs S, comparison with RFX-mod [P. Piovesan et al., NF 2011]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Sensitivity to perturbation amplitude and periodicity
Dominant and secondary modes with perturbation ampl [M. Veranda et al., PPCF 2013]
Dominant mode peak and average amplitude with n: n=-8 is the most reactive
n = -8 was one of the preferred modes in RFX [D. Escande, P. Martin et al., PRL 2000]
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
Magnetic topology of stimulated helical states
Stimulated helical states with non-resonant dominant mode seem to have a better
core magnetic topology: broader conserved helical structures
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas
n=-6 n=-5 n=-7
RFP simulations: effect of increasing helical MP amplitude
q(a) sawteeth are more frequent and less intense, QSH states more stationary
[M. Veranda et al., PPCF 2013]. Similar to experiment [P. Piovesan et al., PPCF 2011]
MHD simulation with helical MP amplitude br(a)=6%
Helical MP amplitude br(a)=10%: bifurcation to stationary helical state
D. Bonfiglio et al. Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas