Effect of helical magnetic perturbations on the 3D MHD self ......2015/05/03 · 7th IAEA Technical...

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 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]

3D effects in axisymmetric fusion devices: external MPs

3D effects can result from plasma self-organization (long-lived helical states)…

… and from external magnetic perturbations (MPs) with non-axisymmetric coils

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

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

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

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

Outline of the talk

3. Tokamak modelling. Effect of MPs:

Mitigation of the sawtooth activity

Plasma shaping with X-points

Outline of the talk

4. Summary and perspectives

Outline of the talk

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

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

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

Outline of the talk

SPECYL, PIXIE3D and their benchmark

The RFX-mod device in Padova (Italy)

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

The RFX-mod device in Padova (Italy)

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)

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

Low current discharge: MH regime (low confinement due to stochastic transport)

MHD modes with poloidal

periodicity m=1

toroidal

periodicity

plasma current waveform

High-current discharge: QSH (improved confinement) with back-transitions

MHD modes with poloidal

periodicity m=1

toroidal

periodicity

Systematic QSH states

with preferred helicity

plasma current waveform

m=1, m=1,

High-current discharge: QSH (improved confinement) with back-transitions

Hot helical core

enclosed by ITBs

SHAx state

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]

Low dissipation: sawtoothing MH with quasi-periodic reconnection events

edge safety factor: sawteeth

m=1, reconnection events

Low dissipation: sawtoothing MH with quasi-periodic reconnection events

High dissipation: spontaneous SH Ohmic equilibrium (related to RFX-mod!?)

SH equilibrium

edge safety factor: sawteeth

m=1, reconnection events

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

Helical MPs with m=1, n=7 periodicity: sawtoothing QSH states

Systematic helical state

Sawtooth amplitude & period reduced

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]

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

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]

MHD simulation: helical MPs with n=7 first, then n=6 periodicity

n=7 MPs n=6 MPs

MHD simulation: helical MPs with n=7 first, then n=6 periodicity

RFX-mod discharge: n=6 RWM feedback-controlled at finite amplitude

n=7 n=7

n=7 MPs n=6 MPs

n=6 MPs

Outline of the talk

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]

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

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

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

core safety factor

n=1, sawtooth crashes

Magnetic topology before sawtooth crash: circular magnetic surfaces

Poincaré plot by field-line tracing code NEMATO [J. Finn and L. Chacón, PoP 2005]

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 with 1,1 MPs: sawtooth mitigation

MPs with 1,1 periodicity and 0.1% amplitude: sawtooth mitigation like in RFX-mod

Sawtooth amplitude & period reduced

sawtooth crashes n=1,

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

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

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

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

core safety factor

n=1, sawtooth crashes

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]

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]

Outline of the talk

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

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:

Classical stellarator

ongoing PIXIE3D applications

Helical RFP Diverted tokamak

Spare slides: stellarator

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:

Only dominant MP Secondary MP=0.02 % Secondary MP=0.2 %

Stellarator studies: PIXIE3D simulations in toroidal geometry

Spare slides: tokamak

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)

2,1 Br (%)

time (s)

with mode control

Courtesy of P. Piovesan

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

Courtesy of P. Piovesan

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%:

qmin=1.08 qmin=1.06 qmin=1.03

Red dots: stable; green dots: unstable internal kink simulations.

Stability diagram

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

sawtooth crashes

core safety factor

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

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.

Shaped PIXIE3D simulations: axisymmetric equilibria

See proceeding paper, this meeting. Two cases:

Purely elongated

D-Shaped

The elongated equilibrium is vertically unstable

D-shaped PIXIE3D simulations: low dissipation w/o MPs

See proceeding paper, this meeting.

D-shaped PIXIE3D simulations: high dissipation w/o MPs

D-shaped PIXIE3D simulations: low dissipation, 1/1 MPs

Spare slides: RFP

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

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

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.

0 r/a 1

Tokamak

B >> B B

0 0 r/a 1

B ≈ B

Safety factor profile in RFX-mod and main resonances

The RFP configuration is prone to the onset of several MHD resistive kink / tearing

modes with m=1 and m=0

Safety factor:

Experimental scaling of dominant and secondary modes

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

Separatrix expulsion and ITBs in the experiment

Poincaré plots made with ORBIT code as shown in [P. Piovesan et al., NF 2009]

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

with separatrix: magnetic island

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

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

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

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]

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]

= v v =

H is the important parameter

when inertia is negligible! )vH(

1 1-2= BJt

1, p 0

)H()v( 1- JBtB =

( , ) (H , P)

Rescaling of the model equations

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:

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

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]

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]

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]

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

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

Effect of helical magnetic perturbations on the 3D MHD self ......2015/05/03 · 7th IAEA Technical...

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