Stellarator tools for neoclassical transport and flow interpretation in helical RFP plasmas M. Gobbin RFX-mod Programme Workshop 2011, February 7-9, Padova,

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


Transport and flow Transport and flow in 3D systemsin 3D systems

Neoclassical transport in 3D systemsNeoclassical transport in 3D systems


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


)()(1)()( mnffrBxB ht (r)ε(r)ε ht

TOROIDAL RIPPLETOROIDAL RIPPLE HELICAL RIPPLEHELICAL RIPPLE

Role of magnetic field configurationRole of magnetic field configuration


Tokamaks described by ~ 5 parameters

Tokamaks described by ~ 5 parameters(aspect ratio, ellipticity,

triangularity…)(aspect ratio, ellipticity,

triangularity…)





Role of magnetic field configurationRole of magnetic field configuration


+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


And

in

RFX-mod ?RFX-mod ?

RFX-modh

t

HSX

t

h

r/ar/a r/ar/a

higher h in the core


Helical ripple and |B| modulationHelical ripple and |B| modulation


|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



3D tools from stellarators : DKES/PENTA

DKESDKES


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)


Monoenergetic coefficientsat each magnetic surface


DKESDKES


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



- 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



PENTAPENTA


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


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




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


particle fluxes

)()( rire EE

improved confinementimproved confinement



Solution for ambipolar Er in RFX-modSolution for ambipolar Er in RFX-mod


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



- 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



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


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


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 Ti=0.7TeRFX-mod Ti=0.7Te

Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod


Experimental estimates ≈ 3km/s (poloidal)Experimental estimates ≈ 3km/s (poloidal)


- <v>- <v>

<v·b><v·b>

- <v>- <v>q’~0q’~0




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.


- <v>- <v>

- <v>- <v>




q’~0q’~0

<v·b><v·b>

Effect of Ti profile on flow componentsEffect of Ti profile on flow components


ei TT 7.0

lower ion temperature gradient decreasing v valueslower ion temperature gradient decreasing v values


- <v>- <v>

<v·b><v·b>

- <v>- <v>


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)



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?










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



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


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


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!



HSXHSX

From J.Lore talk From J.Lore talk

T

TnDDn

T

EqDnD r

11121111

ˆ2


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


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


|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


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)




~100km/s~100km/s

~5km/s~5km/s

~-30km/s~-30km/s

~-10km/s~-10km/s


from D.A.Spong PoP 12 (2005)from D.A.Spong PoP 12 (2005)

pol.pol.

tor.tor.

pol.pol.

tor.tor.


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


Radial transport coefficients Radial transport coefficients


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


Er/v=0Er/v=0

Er/v=10-3Er/v=10-3


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


DKES fails: at low collisionality locality


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




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


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)


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


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


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


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


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


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


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:


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


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


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


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


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)


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


<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


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.


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


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


/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)


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 )


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


#27730@64ms#27730@64ms



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


#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


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


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



15.4

mT

T

e

e

15.3

mT

T

e

e


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



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?


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


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.


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.


Documents

Stellarator tools for neoclassical transport and flow interpretation in helical RFP plasmas M. Gobbin RFX-mod Programme Workshop 2011, February 7-9, Padova,