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Stellarator tools for neoclassical transport and flow interpretation in helical RFP
plasmas
M. Gobbin
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
presented by presented by
Consorzio RFX, Associazione Euratom-ENEA sulla fusioneConsorzio RFX, Associazione Euratom-ENEA sulla fusione
Outline
Transport and flow in 3D systems
3D tools from stellarators: DKES/PENTA
Er and flow computation in RFX-mod
Open issues
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Transport and flow Transport and flow in 3D systemsin 3D systems
Neoclassical transport in 3D systemsNeoclassical transport in 3D systems
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Helical RFP and STELLARATORS share common
topics on
Helical RFP and STELLARATORS share common
topics on
neoclassical transport dominant in the core and near internal transport barriers (ITB);
neoclassical transport dominant in the core and near internal transport barriers (ITB);
neoclassical fluxes determine Er and averaged flows in non axisymmetry systems.
neoclassical fluxes determine Er and averaged flows in non axisymmetry systems.
ANOMALOUS TRANSPORT = EXPERIMENTAL – NEOCLASSICALANOMALOUS TRANSPORT = EXPERIMENTAL – NEOCLASSICAL
neoclassical
transport
neoclassical
transport
Tokamaks described by ~ 5 parameters
Tokamaks described by ~ 5 parameters(aspect ratio, ellipticity,
triangularity…)(aspect ratio, ellipticity,
triangularity…)
Stellarators described by ~ tens of parameters
Stellarators described by ~ tens of parameters
)()(1)()( mnffrBxB ht (r)ε(r)ε ht
TOROIDAL RIPPLETOROIDAL RIPPLE HELICAL RIPPLEHELICAL RIPPLE
Role of magnetic field configurationRole of magnetic field configuration
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Tokamaks described by ~ 5 parameters
Tokamaks described by ~ 5 parameters(aspect ratio, ellipticity,
triangularity…)(aspect ratio, ellipticity,
triangularity…)
Stellarators described by ~ tens of parameters
Stellarators described by ~ tens of parameters
)()(1)()( mnffrBxB ht (r)ε(r)ε ht
TOROIDAL RIPPLETOROIDAL RIPPLE HELICAL RIPPLEHELICAL RIPPLE
Role of magnetic field configurationRole of magnetic field configuration
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
+h
NCSX
0h
Quasi Axi. Quasi Helical
LHD
th
HSX 0 th |B|
parallel flow <v· b> strongly depends on the particular configurationparallel flow <v· b> strongly depends on the particular configuration
Helical ripple and |B| modulationHelical ripple and |B| modulation
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
And
in
RFX-mod ?RFX-mod ?
RFX-modh
t
HSX
t
h
r/ar/a r/ar/a
higher h in the core
higher h in the core
Helical ripple and |B| modulationHelical ripple and |B| modulation
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
|B||B|
q~1q~1 q < 0.13q < 0.13
in helical rfp plasmas: weak |B| modulation in the edge.
in helical rfp plasmas: weak |B| modulation in the edge.
No 1/ regimeNo 1/ regime
And
in
RFX-mod ?RFX-mod ?
Gobbin,Spizzo PRL 106 125001 (2011)Gobbin,Spizzo PRL 106 125001 (2011)
RFX-modh
t
HSX
t
h
r/ar/a r/ar/a
higher h in the core
higher h in the core
3D tools from stellarators : DKES/PENTA
DKESDKES
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Hirshman, Phys. Fluids 29 (1986)Hirshman, Phys. Fluids 29 (1986)
Transport codes adapted from stellarators community Transport codes adapted from stellarators community
DKESDKESDKESDKESHELICAL EQUILIBRI
A By VMEC
HELICAL EQUILIBRI
A By VMEC
- a linearized drift kinetic equation is solved with pitch angle scattering collision operator (NO MOMENTUM CONSERVATION)
- a linearized drift kinetic equation is solved with pitch angle scattering collision operator (NO MOMENTUM CONSERVATION)
Monoenergetic coefficientsat each magnetic surface
Monoenergetic coefficientsat each magnetic surface
DKESDKES
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Hirshman, Phys. Fluids 29 (1986)Hirshman, Phys. Fluids 29 (1986)
Transport codes adapted from stellarators community Transport codes adapted from stellarators community
DKESDKESDKESDKESHELICAL EQUILIBRI
A By VMEC
HELICAL EQUILIBRI
A By VMEC
Monoenergetic coefficientsat each magnetic surface
Monoenergetic coefficientsat each magnetic surface
- from the resulting distribution function:- from the resulting distribution function:
D11,12,21,22D11,12,21,22
D13, 23, 31, 32D13, 23, 31, 32
D33D33
radial transportradial transport
bootstrap currentbootstrap current
parallel transportparallel transport
…used for viscous and friction-flow relations …used for viscous and friction-flow relations
- a linearized drift kinetic equation is solved with pitch angle scattering collision operator (NO MOMENTUM CONSERVATION)
- a linearized drift kinetic equation is solved with pitch angle scattering collision operator (NO MOMENTUM CONSERVATION)
PENTAPENTA
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
D.A.Spong PoP 12 (2005)D.A.Spong PoP 12 (2005)
#27730@64ms#27730@64ms
Experimental Thomson scattering profiles are mapped on helical flux coordinates
Experimental Thomson scattering profiles are mapped on helical flux coordinates
No measures of Ti radial profile: guess and sensivity studies
No measures of Ti radial profile: guess and sensivity studies
PENTA computes the ambipolar radial field and neoclassical flow, including correction for the momentum conservation
PENTA computes the ambipolar radial field and neoclassical flow, including correction for the momentum conservation
Absolutely required for Quasi Symmetric systems!!
Absolutely required for Quasi Symmetric systems!!
INTRODUCTION INTRODUCTION to HELICAL RFP to HELICAL RFP
REGIMESREGIMESEr and flowand flow computation
T
TnDDn
T
EqDnD r
11121111
ˆ2
3ˆˆˆparticle fluxes
particle fluxes
)()( rire EE no impurity, ne=3·1019m-3, Ti=0.7Te
no impurity, ne=3·1019m-3, Ti=0.7Te
Solution for ambipolar ErSolution for ambipolar Er
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Solution for ambipolar ErSolution for ambipolar Er
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
StellaratorStellarator
Er (kV/m)
Ion root: Ion root:
Electron root: Electron root:
small negative solutionsmall negative solution
reduces the ion fluxreduces the ion flux
large positive solutionlarge positive solution
both fluxes are reduced both fluxes are reduced
T
TnDDn
T
EqDnD r
11121111
ˆ2
3ˆˆˆparticle fluxes
particle fluxes
)()( rire EE
improved confinementimproved confinement
no impurity, ne=3·1019m-3, Ti=0.7Te
no impurity, ne=3·1019m-3, Ti=0.7Te
Solution for ambipolar Er in RFX-modSolution for ambipolar Er in RFX-mod
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Er (
kV/m
)
r/a
r/a
Er (
kV/m
)
i>ei>e
i<ei<e
i=ei=e
i=
e
i=
e
contour of i-e as function of Er
and r/a
contour of i-e as function of Er
and r/a
zoomzoom
point of minimum Er ≈-2kV/m
around the ITB
point of minimum Er ≈-2kV/m
around the ITB
i-e
ITB ITB
Solution for ambipolar ErSolution for ambipolar Er
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
- in the helical core region (red/black lines) ion root solution at Er~-2kV/m
- in the helical core region (red/black lines) ion root solution at Er~-2kV/m
- |Er| decreases moving towards the quasi-axisymmetric edge
- |Er| decreases moving towards the quasi-axisymmetric edge
h>>th>>t
t>>ht>>hat r/a>0.8|Er |≤ 0.1kV/mat r/a>0.8|Er |≤ 0.1kV/m
RFX-modRFX-mod
ii
ee
r/a=0.35r/a=0.35
r/a
Er (
kV/m
)
i=ei=e
)()( rire EE
no impurity, ne=3·1019m-3, Ti=0.7Te
no impurity, ne=3·1019m-3, Ti=0.7Te
ITBeITBi TT ,,
-ions-ions -electrons-electrons
Er ≈ -1.6kV/mEr ≈ -1.6kV/m
i=e
Qi/Ti
Qe/Te
(ITB region)(ITB region)
ei TT 7.0
0 iT
ei TT 7.0
e,eff ≈ 1.5-3m2/s<10m2/s (experiment)e,eff ≈ 1.5-3m2/s<10m2/s (experiment)
Er depends on Ti profile Er depends on Ti profile
Ti profile Er(kV/m)
-1.75
-1.6
+0.2
Tn
Qeff
Effect of assumptions on the Ti profile Effect of assumptions on the Ti profile
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
TnDDnEqDnTDQ r
21222112
ˆ2
3ˆˆˆheat
fluxesheat
fluxes
INTRODUCTION INTRODUCTION to HELICAL RFP to HELICAL RFP
REGIMESREGIMESEErr and and flow computation(in progress)
Flow computationFlow computation
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Flow velocity given by: Flow velocity given by:
diamagneticpart
diamagneticpart flow velocity // field
(parallel viscous stress tensor)
flow velocity // field(parallel viscous stress tensor)
Pfirsch-Schluter flow velocity
Pfirsch-Schluter flow velocity
BB // B// B
for RFP also dynamo could play a role: not included nowfor RFP also dynamo could play a role: not included now
Flow components in RFX-modFlow components in RFX-mod
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
RFX-mod Ti=0.7TeRFX-mod Ti=0.7Te
Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod
Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod
Experimental estimates ≈ 3km/s (poloidal)Experimental estimates ≈ 3km/s (poloidal)
(ITB region)(ITB region)
- <v>- <v>
<v·b><v·b>
- <v>- <v>q’~0q’~0
RFX-mod Ti=0.7TeRFX-mod Ti=0.7Te
Flow components in RFX-modFlow components in RFX-mod
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
from D.A.Spong PoP 12 (2005)from D.A.Spong PoP 12 (2005)
HSX (ECH)HSX
(ECH)
HSX (ICH)HSX (ICH)
~30km/s~30km/s
~-5km/s~-5km/s
~0.5km/s~0.5km/s
~-6km/s~-6km/s
pol.pol.
tor.tor.
pol.pol.
tor.tor.
(ITB region)(ITB region)
- <v>- <v>
- <v>- <v>
Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod
Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod
Experimental estimates ≈ 3km/s (poloidal)Experimental estimates ≈ 3km/s (poloidal)
q’~0q’~0
<v·b><v·b>
Effect of Ti profile on flow componentsEffect of Ti profile on flow components
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
ei TT 7.0
lower ion temperature gradient decreasing v valueslower ion temperature gradient decreasing v values
(ITB region)(ITB region)
- <v>- <v>
<v·b><v·b>
- <v>- <v>
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
ei TT 7.0 0 iT
lower ion temperature gradient decreasing v valueslower ion temperature gradient decreasing v values
flow has opposite sign for zero ion temperature gradient (as Er )flow has opposite sign for zero ion temperature gradient (as Er )
(ITB region)(ITB region) (ITB region)(ITB region)
comparison with exp. data in progress: (Boozer coordinates in PENTA) comparison with exp. data in progress: (Boozer coordinates in PENTA)
- <v>- <v>
<v·b><v·b>
- <v>- <v>
+ <v>+ <v>
- <v·b>- <v·b>
+ <v>+ <v>
Effect of Ti profile on flow componentsEffect of Ti profile on flow components
Effect of residual chaos at the eITB (ORBIT)
Di,e computed locally near ITB by ORBIT with secondary
modes too ( at Er=0)
Di,e computed locally near ITB by ORBIT with secondary
modes too ( at Er=0)
with secondary modes
with secondary modes
ELECTRONSELECTRONS
pure helicalpure
helicalpure helical + secondary
modes
Di~0.5–1.5m2/s Di~0.5–1.5m2/s
De~0.04m2/s De~2-3m2/s
experimental transport exceeds the neoclassical one: effect of secondary modes?
experimental transport exceeds the neoclassical one: effect of secondary modes?
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Effect of residual chaos at the eITB (ORBIT)
Di,e computed locally near ITB by ORBIT with secondary
modes too ( at Er=0)
Di,e computed locally near ITB by ORBIT with secondary
modes too ( at Er=0)
with secondary modes
with secondary modes
ELECTRONSELECTRONS
pure helicalpure
helicalpure helical + secondary
modes
Di~0.5–1.5m2/s Di~0.5–1.5m2/s
De~0.04m2/s De~2-3m2/s
experimental transport exceeds the neoclassical one: effect of secondary modes?
experimental transport exceeds the neoclassical one: effect of secondary modes?
low Er required for ambipolarity for small level of secondary modes? What about the predicted flows?
low Er required for ambipolarity for small level of secondary modes? What about the predicted flows?
runs with helical Er≠0 in progressruns with helical Er≠0 in progress
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
New version of PENTA released by J.Lore: benchmark on going
New version of PENTA released by J.Lore: benchmark on going
simpler inclusion of impurity profiles in the new PENTA versionsimpler inclusion of impurity profiles in the new PENTA version
0)( rEq
, all species, all species
comparison with experiment in the right coordinate system.comparison with experiment in the right coordinate system.
evaluation of the single terms to the total flows, bootstrap current
evaluation of the single terms to the total flows, bootstrap current
Next stepsNext steps
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
application to more experimental scenariosapplication to more experimental scenarios(pellet, higher density, higher helical deformation …)(pellet, higher density, higher helical deformation …)
Thanks for your attention
Thanks for your attention
Ceterum censeo Chartaginem esse delendam!
Solution for ambipolar ErSolution for ambipolar Er
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
HSXHSX
From J.Lore talk From J.Lore talk
T
TnDDn
T
EqDnD r
11121111
ˆ2
3ˆˆˆparticle fluxes
particle fluxes
)()( rire EE no impurity, ne=3·1019m-3
no impurity, ne=3·1019m-3
RFX-modRFX-mod
ii
ee
r/a=0.35r/a=0.35
particle trajectoriesparticle trajectories
momentum transportmomentum transport
when a “straight” helical system is bent into a torus,
ripple trapped particles acquire non zero bounce averaged radial drift
when a “straight” helical system is bent into a torus,
ripple trapped particles acquire non zero bounce averaged radial drift
0
r
superbananas losses superbananas losses
asymmetry damping of plasma rotation both in poloidal and
toroidal directions;
asymmetry damping of plasma rotation both in poloidal and
toroidal directions;
Subjects of active research for TRANSPORT OPTIMIZATION in
Stellarators
Subjects of active research for TRANSPORT OPTIMIZATION in
Stellarators
Transport optimization in StellaratorsTransport optimization in Stellarators
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
flow shear and rotation allow :flow shear and rotation allow :
studying impurity transportstudying impurity transport
reducing micro-turbolencereducing micro-turbolence
preventing island formationpreventing island formation
….….
RFX-modh
t
HSX
t
h
And RFX-mod ?
on the contrary of Stellarators, in helical RFX-mod plasmas h is much higher in
the core and very low at the edge
on the contrary of Stellarators, in helical RFX-mod plasmas h is much higher in
the core and very low at the edge
r/ar/a r/ar/a
Ripples in RFX-modRipples in RFX-mod
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
|B||B|
HSX
HSX
RFX-modRFX-mod
in helical rfp plasmas: weak |B| modulation at the edge, stronger in the core. in helical rfp plasmas: weak |B| modulation at the edge, stronger in the core.
no 1/ regime by ORBIT simulations (Er=0): only for higher deformation of the helical surfaces losses due to superbana particles become important.
no 1/ regime by ORBIT simulations (Er=0): only for higher deformation of the helical surfaces losses due to superbana particles become important.
q~1q~1 q < 0.13q < 0.13
RFX-mod: effect of configuration on particle orbitsRFX-mod: effect of configuration on particle orbits
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Gobbin,Spizzo PRL 106 125001 (2011)Gobbin,Spizzo PRL 106 125001 (2011)
θeu
bu
eu
poloidalpoloidal
toroidaltoroidal
parallelparallel
NCSX (ECH)NCSX (ECH)
NCSX (ICH)NCSX (ICH)
RFX-mod Ti=0.7TeRFX-mod Ti=0.7Te
Flow components in RFX-modFlow components in RFX-mod
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
~100km/s~100km/s
~5km/s~5km/s
~-30km/s~-30km/s
~-10km/s~-10km/s
Experimental estimates ≈ 3km/s (poloidal)Experimental estimates ≈ 3km/s (poloidal)
from D.A.Spong PoP 12 (2005)from D.A.Spong PoP 12 (2005)
pol.pol.
tor.tor.
pol.pol.
tor.tor.
(ITB region)(ITB region)
Contravariant flow components computed at the ITB for RFX-modContravariant flow components computed at the ITB for RFX-mod
Er/v=0Er/v=0
Er/v=10-3Er/v=10-3
Er/v=0.1Er/v=0.1 Er/v=1Er/v=1
ITB surfaceITB surfaceD11≈ 0.5-1m2/s: good
agreement with ORBIT estimates (Er=0) in
experimental condition of density and temperature
D11≈ 0.5-1m2/s: good agreement with ORBIT
estimates (Er=0) in experimental condition of density and temperature
DKES fails at low collisionality: locality
assumption not valid in RFX-mod
DKES fails at low collisionality: locality
assumption not valid in RFX-mod
Radial transport coefficients Radial transport coefficients
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
evaluation of bootstrap current:evaluation of bootstrap current:
IONSIONS
ELECTRONSELECTRONS
TOTALTOTAL
~10-4-10-3Johmic~10-4-10-3Johmic
Bootstrap current computationBootstrap current computation
very small contribute with
ordinary temperature and
density experimental
profiles
very small contribute with
ordinary temperature and
density experimental
profiles
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
Er/v=0Er/v=0
Er/v=10-3Er/v=10-3
Er/v=0.1Er/v=0.1 Er/v=1Er/v=1
ITB surfaceITB surfaceD11≈ 0.5-1m2/s: good
agreement with ORBIT estimates (Er=0) in
experimental condition of density and temperature
D11≈ 0.5-1m2/s: good agreement with ORBIT
estimates (Er=0) in experimental condition of density and temperature
DKES fails: at low collisionality locality
assumption not valid in RFX-mod
DKES fails: at low collisionality locality
assumption not valid in RFX-mod
Radial transport coefficients Radial transport coefficients
00 dt
dFollowing a trapped particle with its helical flux
coordinate:
Following a trapped particle with its helical flux
coordinate:
TOP VIEWTOP VIEW Toroidal precessionToroidal precession Helical flux coordinateHelical flux coordinate
time (a.u.)time (a.u.)time (a.u.)time (a.u.)X(cm)X(cm)
Y(c
m)
Y(c
m)
the banana orbit is only slightly modified by the presenceof the helix, since |B| is essentially still axisymmetric
the banana orbit is only slightly modified by the presenceof the helix, since |B| is essentially still axisymmetric
only for Bh/B > 60 % superbananas can reach the wallonly for Bh/B > 60 % superbananas can reach the wall
SHAx =SHAx =
helical equilibrium with 1,7 periodicityhelical equilibrium with 1,7 periodicity
onset of internal electron transport barriersonset of internal electron transport barriers
helical magnetic surfaceshelical magnetic surfaces
low level of residual chaoslow level of residual chaos
At high plasma current a SINGLE saturated resistive kink mode drives most of the self organization process and gives the plasma a global helical symmetry.
Helical equilibrium
Helical equilibrium
Low residual magnetic chaos
Low residual magnetic chaos
• quasi-axisymmetric edge• quasi-axisymmetric edge
n
n=7
• helical core• helical core
RFX-modmode
spectrum
RFX-modmode
spectrumNeoclassical effects may
become relevant
Neoclassical effects may
become relevant
Helical rfp plasmas
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
RFP HELICAL STATES contribute to 3D3D physics studies in unexplored regions of the plasma parameter spaceRFP HELICAL STATES contribute to 3D3D physics studies in unexplored regions of the plasma parameter space
RFP safety factor is lower than in stellarators and tokamaks
RFP safety factor is lower than in stellarators and tokamaks
adapted from Fujisawa, PPCF,2001
RFPRFP
TOKAMAKTOKAMAK
STELLARATORSTELLARATOR
B≈ BB≈ B
RFX-MOD < 0.12 >8
CHS ~3.3
HSX ~1
LHD ~2
TJ-II ~0.7
W7-AS ~1.4-4
DEVICEDEVICE qq
RFP helical states features
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
)()(1)()( mnffrBxB ht (r)ε(r)ε ht
TOROIDAL RIPPLETOROIDAL RIPPLE HELICAL RIPPLEHELICAL RIPPLE
RFX-modh
t
HSX
t
h
radial coordinateradial coordinate radial coordinateradial coordinate
R0 = 2 m a = 0.459 m
RFX-mod device: main features
(still not optimized)(still not optimized)
Max Ip = 2 MAMax Ip = 2 MA
Now achieved!
Now achieved!
E up to 5 msE up to 5 ms
Largest RFPLargest RFPLargest RFPLargest RFP
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
plasma current up to 2MA
plasma current up to 2MA
mode (1,-7) is dominant for most of the discharge:
Quasi Single Helicity (QSH)
mode (1,-7) is dominant for most of the discharge:
Quasi Single Helicity (QSH)
low secondary modes
low secondary modes
electron density : 3-6·1019m-3
electron density : 3-6·1019m-3Back transitions from QSH to MH,
related to reconnection events, under investigation
Back transitions from QSH to MH, related to reconnection events,
under investigation
RFX-mod device: main features
(1,-7)(1,-7) (1,n <-7)(1,n <-7)
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
192 independently feedback controlled coils covering the whole torus. Digital Controller with Cycle frequency of 2.5 kHz.
ACTIVE COILSACTIVE COILS
control of the internally resonant
tearing modes
control of the internally resonant
tearing modes
RWM control both in RFP and Tokamak
configuration
RWM control both in RFP and Tokamak
configuration
control of helical magnetic field
reinforces persistency of 3D shaping
control of helical magnetic field
reinforces persistency of 3D shaping
RFX-mod device: active control
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
time (s)
IP
(MA)
br(a)/B(%)
1/-7phase(rad)
n/nGW
neTe
(kPa)
#28218
1/-7
1/-8 to -15
br1/-7(a)≠0
rotatingbr
1/-7(a)≠0static br
1/-7(a)=0
Experiments with br(a) ≠0 on the 1/-7
mode:
Experiments with br(a) ≠0 on the 1/-7
mode:
- high record valuesfor br
1,-7(a) - high record values
for br1,-7(a)
- secondary m=1 modes amplitude does not vary
significantly
- secondary m=1 modes amplitude does not vary
significantly
- long, rotating but also static, QSH obtained
- long, rotating but also static, QSH obtained
Active control with finite references
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Temperature and SXR emissivity are helical flux
function
Temperature and SXR emissivity are helical flux
function
SXRSXR
TeTe
steep gradientssteep gradients
high Te in the helical core
high Te in the helical core
Thermal evidences of 3D topology
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Electron Internal Transport Barriers
remapping on square root of helical flux
remapping on square root of helical flux
electron Internal Transport Barrier (eITBS) region :
link with magnetic topology?
electron Internal Transport Barrier (eITBS) region :
link with magnetic topology?
In Tokamak or Stellarators ITBs are associated to
effect of low shear on microinstability growth rates or in
reducing their radial extent.
suppression of microinstability induced
transport by sheared E×B flowsConnor et al. 1994
weak/negative s= (r/q) dq/drweak/negative s= (r/q) dq/dr shear flowsshear flows
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
br1,-7(a)/B(a)>4%br1,-7(a)/B(a)>4%
Depending on the amplitude of the (1,-7)
mode, two QSH configurations :
Depending on the amplitude of the (1,-7)
mode, two QSH configurations :
DAxDAx Double Axis Double Axis
Single Helical Axis
Single Helical Axis
Separatrix expulsion
Separatrix expulsion
br1,-7(a)/B(a)<4%br1,-7(a)/B(a)<4%
DAx and SHAx states in RFX-mod
SHAxSHAx
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
q in SHAX states determined from averaged winding of field lines in toroidal direction for poloidal turn
*
*kki rotation of poloidal
angle after the k-th toroidal transit
niq k
nk
n
)(lim
2
1/1
*1
n=7
AROUND THE HELICAL AXISAROUND THE HELICAL AXIS
Rotational transform in SHAx states
SHEAR REVERSALq<7
SHEAR REVERSALq<7
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
1/71/7
separatrixseparatrix
Around helical axis
Around geometrical (shifted) axis
In DAx states field lines wind around two axis and a single valued helical flux function cannot be defined.
q1/7 near the separatrix
shear reversal near the island
domain
….and in DAx cases:
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
SHAX
max(q) slightly exceeds ITB foot position in SHAx
max(q) preceeds the ITB foot position
in DAx
Link between eITBs and magnetic topology
DAx
Point of zero magnetic shear correlated with electron
transport barrier position
Similiarity with TOKAMAK ITBs
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Transport barriers are correlated to maximum flow shearTransport barriers are correlated to maximum flow shear
From 3D nonlinear visco-resistive MHD code (Specyl): EB flow with
maximum at the ITB
From 3D nonlinear visco-resistive MHD code (Specyl): EB flow with
maximum at the ITB
Flow shear at the barrier: like in Stellarators?
From experiment: m=1 flow reconstruction
shows a poloidal flow inversion close to the ITB
From experiment: m=1 flow reconstruction
shows a poloidal flow inversion close to the ITB
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Perturbative Approach by SHEq code
The helical state is well described in terms of a helical flux mn with m=1,n=7:
7=constant7=constant
definition allows a faster reconstruction of q profile in
SHAx states
Field lineSHEq
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
F0; 0 TOROIDAL, POLOIDAL fluxes
,,, rnm nminmnm ernfrm )()( ,,
Dominant mode
)(F)( 00 rnrm Axi-symmetric
++
INPUT constraints:
1. q(s) 1/(s)
2. Pressure profile
3. Total Toroidal flux
4. Plasma boundary shapein terms of harmonic components(LCFS)
INPUT guess:Magnetic axis shape
Configuration periodicity:Dominant mode helicity (Nfp=7)
Configuration periodicity:Dominant mode helicity (Nfp=7)
VMEC has been ported to RFP equilibrium, using
the poloidal flux coordinate instead than
the toroidal one
The VMEC code for RFP helical states
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Benchmark
Comparison of magnetic fields between VMEC and expmeriments
The EXTENDER code is used for computing fields at sensors for VMEC
The EXTENDER code is used for computing fields at sensors for VMEC
A better matching requires taking into account the effect of passives structures on the fields produced by the active control coils.
A better matching requires taking into account the effect of passives structures on the fields produced by the active control coils.
Eigenfunctions (1,-7) and (0,7)Eigenfunctions (1,-7) and (0,7)
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
At present VMEC uses the q profile obtained with the NCT/SHEq code.At present VMEC uses the q profile obtained with the NCT/SHEq code.
cubic-spline fit with 5 parameters
cubic-spline fit with 5 parameters
A point of zero magnetic shear appears to be
essential in order to obtain a helical equilibrium.
A point of zero magnetic shear appears to be
essential in order to obtain a helical equilibrium.
The helical displacement decreases as long as the q0
increases and eventually an axi-simmetric equilibrium is
recovered.
The helical displacement decreases as long as the q0
increases and eventually an axi-simmetric equilibrium is
recovered.
Work is in progress in order to determine a parametric
representation of the q profile that can be matched with
experimental measurements.
Work is in progress in order to determine a parametric
representation of the q profile that can be matched with
experimental measurements.
VMEC-experimental measures matching
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
<D
i>h
elORBIT code adapted to 3D geometry
Guiding center code ORBIT modified to deal with helical surfaces
Montecarlo simulations to evaluate an averaged ion diffusion coefficient over the helical core <Di>hel
Montecarlo simulations to evaluate an averaged ion diffusion coefficient over the helical core <Di>hel
collisions included
mono-energetic ions
no radial electric field
collisions included
mono-energetic ions
no radial electric field
Experimental range
Chaotic MH
Helical
no 1/ regime at low collisions, which is a concern for stellarators optimization
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Lack of fast superbanana losses
superbananas%superbananas%HELICAL PERT x1HELICAL PERT x1 HELICAL PERT x 7HELICAL PERT x 7
superbananas%superbananas%
HELICALHELICAL
TOROIDAL (n=0)
TOROIDAL (n=0)
(1,-7)(1,-7)
RIP
PLE
RIP
PLE
r/a
Positive effect of axisymmetric outer region with decreasing helical ripple.
Positive effect of axisymmetric outer region with decreasing helical ripple.
Helical trapped particles with significant radial drift and lost in few bounces are a small fraction (~0.3%) till higher perturbation are considered.
Helical trapped particles with significant radial drift and lost in few bounces are a small fraction (~0.3%) till higher perturbation are considered.
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
NEOCLASSICAL TRANSPORT STUDIES
How far is RFX-mod transport from NEOCLASSICAL estimates in the
barrier region?
Use of DKES code to evaluate local mono-energetic coefficients (pure SH case)
Use of PENTA code for ambipolar constraint and fluxes determination (pure SH case)
Local estimates by the ORBIT code (with also secondary modes inclusion)
Use of DKES code to evaluate local mono-energetic coefficients (pure SH case)
Use of PENTA code for ambipolar constraint and fluxes determination (pure SH case)
Local estimates by the ORBIT code (with also secondary modes inclusion)
D11,D31,D33 (s,/v,Er/v)
D11,D31,D33 (s,/v,Er/v)
i,e,Qi,Qei,e,Qi,Qe
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Radial transport coefficients at the ITB
HELICAL EQUILIBRIA
VMEC
HELICAL EQUILIBRIA
VMECDKESDKESDKESDKES
Monoenergetic coefficients at
each magnetic surfacesD11(/v,Er/v)
Monoenergetic coefficients at
each magnetic surfacesD11(/v,Er/v)
Er/v=0Er/v=0
Er/v=10-3Er/v=10-3
Er/v=0.1Er/v=0.1 Er/v=1Er/v=1
/v=10-4/v=10-4
/v=0.1/v=0.1
/v=40/v=40EXPEXP
EXPEXP
D11≈0.5-1m2/s good agreement with ORBIT estimates (Er=0)
D11≈0.5-1m2/s good agreement with ORBIT estimates (Er=0)
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Plot of D11 at the experimental
collisionality in helical RFX-mod plasmas
Plot of D11 at the experimental
collisionality in helical RFX-mod plasmas
D11 spatial dependence
Strong reduction of D11 at s > 0.5 where the field
becomes more axisymmetric even at Er=0
Strong reduction of D11 at s > 0.5 where the field
becomes more axisymmetric even at Er=0
( normalized poloidal flux )
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Neoclassical transport studies by PENTA
TnDDnEqDnTDQ r
21222112
ˆ2
3ˆˆˆ
D11 by DKESVMEC EQUILIBRIA
D11 by DKESVMEC EQUILIBRIA
EXPERIMENTAL Te , n
EXPERIMENTAL Te , n
PENTAPENTAPENTAPENTA
dEEDKgeED ijjiKTE
ij )()(ˆ /
0
T
TnDDn
T
EqDnD r
11121111
ˆ2
3ˆˆˆ
ri
ii EZ 0 ambipolar radial electric fieldambipolar radial electric field
particle fluxesparticle fluxes
heat fluxesheat fluxes
Dij covolution with local maxwellianDij covolution with local maxwellian
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
#27730@64ms#27730@64ms
Experimental Thomson scattering profiles are mapped on helical flux coordinates
Experimental Thomson scattering profiles are mapped on helical flux coordinates
Linear fit at the ITB
Linear fit at the ITB
A linear fit of the gradient is performed at the ITB, the region considered for PENTA application
A linear fit of the gradient is performed at the ITB, the region considered for PENTA application
No measures of Ti radial profile: guess and sensivity studiesNo measures of Ti radial profile: guess and sensivity studies
Neoclassical transport studies by PENTA/2
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
#27730@64ms#27730@64ms
Ti(r)=0.7 Te(r)Ti(r)=0.7 Te(r) -ions-ions -electrons-electrons Estimated ambipolar Er:
Estimated ambipolar Er:
≈ -1.75kV/m≈ -1.75kV/m
i=e
Qi/Ti
Qe/Te
ne=3·1019m-3ne=3·1019m-3
Thermal diffusivities
estimated as:
Thermal diffusivities
estimated as:
e,eff ≈ 3.1±0.8m2/se,eff ≈ 3.1±0.8m2/s
i,eff ≈ 5±1m2/si,eff ≈ 5±1m2/s
Fluxes and electric field at the ITB
18.2
mT
T
e
e
Tn
Qeff
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Sensivity on Ti guess
Radial electric field required for
ambipolarity depend on Ti profile both for sign and amplitude
Radial electric field required for
ambipolarity depend on Ti profile both for sign and amplitude
Effective electron thermal diffusivity does not show a
significative dependence on Ti
Effective electron thermal diffusivity does not show a
significative dependence on Ti
0, ITBiT ITBeITBi TT ,,
≈ +0.2kV/m≈ +0.2kV/m ≈-1.6kV/m≈-1.6kV/m
e,eff ≈ 2.6±0.6m2/se,eff ≈ 2.6±0.6m2/s
i,eff ≈ 10-15m2/si,eff ≈ 10-15m2/s
e,eff ≈ 2.7±0.5m2/se,eff ≈ 2.7±0.5m2/s
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Test on more cases (assuming Ti=0.7 Te)
#28676@235ms#28676@235ms
#27838@135ms#27838@135ms
Er≈-2.5kV/mEr≈-2.5kV/m
Er≈-1.8kV/mEr≈-1.8kV/m
e,eff ≈ 2.5±0.7m2/se,eff ≈ 2.5±0.7m2/s
e,eff ≈ 3.1±0.8m2/se,eff ≈ 3.1±0.8m2/s
i,eff ≈ 4±1m2/si,eff ≈ 4±1m2/s
i,eff ≈ 5±1m2/si,eff ≈ 5±1m2/s
15.4
mT
T
e
e
15.3
mT
T
e
e
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Power balance from experimental data and VMEC equilibria agree with transport interpretations provided by the ASTRA code: e,ITB ~ 10 m2/s.
)('
')'()'()'('
)( 0
2
sVs
Tgn
dssJssV
sss
e
s
ASTRAASTRA VMECVMEC
>10m2/s
e from power balance exceeds PENTA about a factor 5!
Power balance estimates
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Effect of residual chaos at the eITB (ORBIT)
1122334455
Di,e computed locally near ITB with secondary modes too (Er=0)
Di,e computed locally near ITB with secondary modes too (Er=0)
No effect of secondary modes on ions diffusion
No effect of secondary modes on ions diffusion
In SHAx states De ~ Di:
In SHAx states De ~ Di:
SHSHSHAxSHAx
SHSH
SHAxSHAx
IONSIONS
ELEC.ELEC.
low Er required for ambipolarity?low Er required for ambipolarity?
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Small scale instabilities
The configuration is subcritical for ITG, but low/null magnetic shear as well as impurities are destabilizing.
Steeper density gradients are required to trigger the Trapped Electron Mode instability.
GS2 code applied to axisymmetric RFX-mod plasmas show that micro-tearing modes are unstable for a significant range of wavenumbers on the barrier if LTe ≤ 0.2 m.
The electron thermal conductivity across the barrier can be quasi-linearly estimated e ~ 10 m2/s.
[S. C. Guo, PoP 15,122510 (2008), I. Predebon et al., PoP 17, 012304 (2010), F. Sattin et al., submitted]
MICROTEARING MICROTEARING
ITG and TEM ITG and TEM
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
cccc
cccc
cccc
cccc
cccc
Conclusions
Electron internal transport barriers are a robust evidence in self-organized helical equilibria of high current RFP.
RFPs have started to share tools and knowledge with tokamak and stellarator community on 3D physics
Role of magnetic and flow shear shares important analogies with Tokamak and Stellarator physics.
Axisymmetric edge with low helical ripple at the edge: reduction of number and losses of superbanana orbits.
Adaptation of VMEC to RFP allow access to many codes used in the stellarator community: neoclassical estimates of thermal diffusivity at the barrier performed with DKES and PENTA have been performed.
Residual chaos and microteraing modes could explain the gap between neoclassical estimates and power balance calculations.
52° APS Conference, 8-12 November 2010, Chicago, USA52° APS Conference, 8-12 November 2010, Chicago, USA
Conclusions
Neoclassical transport is being investigated in the helical plasmas of RFX-mod adapting stellarator tools like DKES and PENTA. From the experimental data: electron temperature profiles (Thomson scattering) and density. Guess on Ti.
The corresponding radial electric ambipolar field near the ITB without impurities is of the order of ~-2kV/m depending on the assumed Ti profile.
The same holds for toroidal and poloidal flow components are of ~ 10-20km/s and 2-8km/s respectively, decreasing in magnitude with lower ion temperature gradients.
Presence of residual chaos could significatly affect the ambipolar radial field (reducing it) and flow components. Work in progress with ORBIT.
RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy
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