association euratom-cea electromagnetic self-organization and turbulent transport in tokamaks g....

AssociationEURATOM-CEA

Electromagnetic Self-Organization and Turbulent Transport in Tokamaks

G. Fuhr, S. Benkadda, P. BeyerG. Fuhr, S. Benkadda, P. BeyerFrance Japan Magnetic Fusion Laboratory,France Japan Magnetic Fusion Laboratory,

LPIIM, CNRS – Université de Provence,LPIIM, CNRS – Université de Provence,

Marseille, FranceMarseille, France

X. GarbetX. GarbetIRFM, Association Euratom-CEA, IRFM, Association Euratom-CEA,

CEA Cadarache, FranceCEA Cadarache, France

Numerical Models for Controlled Fusion, NMCF '09

Outline

• IntroductionIntroduction•Resistive Ballooning ModelResistive Ballooning Model•Characterization of TransportCharacterization of Transport•Transport Barrier (T.B.) DynamicsTransport Barrier (T.B.) Dynamics•ConclusionConclusion

Why are we interested in turbulence?

•Radial diffusion phenomena Radial diffusion phenomena evacuation of evacuation of contents in energy and particles from plasma contents in energy and particles from plasma core to the edgescore to the edges

•Diffusion coefficient : Diffusion coefficient : DDexpexp>>>>DDcol col abnormal abnormal transport, turbulence possible candidatetransport, turbulence possible candidate

•Presence of instabilitiesPresence of instabilities abnormal transport of heating and particlesabnormal transport of heating and particles

confinement timeconfinement time

Numerical Models for Controlled Fusion, 20-24 Avril 2009


Introduction

• Experimental measurements of fluctuation levelsExperimental measurements of fluctuation levels• Electrostatic fluctuations important (50% in TEXT), Electrostatic fluctuations important (50% in TEXT),

electrostatic modelselectrostatic models [Ritz, Phys. Fluids, 27, 2956 (1984)][Ritz, Phys. Fluids, 27, 2956 (1984)]

• Low level of magnetic fluctuationsLow level of magnetic fluctuations (~10 (~10-4-4 in Tore Supra) in Tore Supra) [Zou et al.,Phys. Rev. Let., 75, 1090 (1995)][Zou et al.,Phys. Rev. Let., 75, 1090 (1995)]

• Magnetic fluctuations => local modification of magnetic Magnetic fluctuations => local modification of magnetic topologytopology

• Important role with respect to the transport properties of the Important role with respect to the transport properties of the plasmaplasma

•Presented work :Presented work :1.1.Impact of magnetic fluctuations on turbulent self-Impact of magnetic fluctuations on turbulent self-

organizationorganization

2.2.Dynamics of transport barriersDynamics of transport barriers

Turbulence Modelization

• Fluid approximationFluid approximation=>description via density =>description via density nn, velocity , velocity uue,ie,i, pressure , pressure ppe,ie,i

• Drift approximation : characteristic frequencies << gyration Drift approximation : characteristic frequencies << gyration frequencyfrequency


dt

ud

eB

Bmu

neB

pBu

B

Bu

uuuu

Eii

ieie

E

iieEie

pol

dia

poldia

2

2,

,

2

,,

EExxBB drift drift

diamagnetic driftdiamagnetic drift

polarization driftpolarization drift

Instabilities


• Interchange Instability Interchange Instability (Resistive Ballooning)(Resistive Ballooning)

• Apparition of fluctuations Apparition of fluctuations at the interface between at the interface between two magnetic flux tubestwo magnetic flux tubes

• ““Main” mechanism of Main” mechanism of instability : curvature vs. instability : curvature vs. gradient ( of pressure, gradient ( of pressure, density…) density…)


Model for Resistive Ballooning

,.1

²

,.)1

(

sincos

||

y

OE

yz

x

Bu

qB

G

B

B

B

2||

22||||

42||

2

1: Law sOhm'

)(: BalanceEnergy

1)(

0j

: Balance Charge

Nt

cEt

NEt

SppGpu

sourcedissdt

dp

Gpu

•Reduced MHD equation for pressure p, electrostatic potential and electromagnetic potential

•Complete fields (profiles + fluctuations) for p and , only fluctuations for

gradient pressure normalized : ,2 Nq

: linked with the parameter : ratio of kinetic pressure to magnetic pressure-Future machines : ↗

Normalizations and Geometry

• Normalization used in the systemNormalization used in the system

• Time => resistive interchange time (Time => resistive interchange time (intint))

• Perpendicular length scales =>resistive ballooning length Perpendicular length scales =>resistive ballooning length (())

• Parallel length scales =>magnetic shear length (LParallel length scales =>magnetic shear length (Lss))

• GeometryGeometry• Magnetic flux surfaces Magnetic flux surfaces

=> set of concentric circular torii=> set of concentric circular torii


sL

Rz

ry

rrx

000 ,,


EMEDGE3D Characteristics

• Numerical code characteristics :Numerical code characteristics :• 3D, global3D, global• Flux driven (source term)Flux driven (source term)• ElectromagneticElectromagnetic

• Time advance scheme :Time advance scheme :• Staggered leap frogStaggered leap frog

• Space Discretization :Space Discretization :• Spectral representation in toroidal and poloidal directionSpectral representation in toroidal and poloidal direction• Finite Difference in radial directionFinite Difference in radial direction


Schemes

• Nonlinearities represented by the Poisson bracketsNonlinearities represented by the Poisson brackets

• Spatial schemeSpatial scheme• Centered differences for linear termsCentered differences for linear terms• Suppression of aliasing effects via Suppression of aliasing effects via “2/3 rule”“2/3 rule”• Arakawa scheme for Poisson brackets :Arakawa scheme for Poisson brackets :

•Conservation of Jacobian properties and Kinetic Conservation of Jacobian properties and Kinetic energyenergy

• Boundary Conditions :Boundary Conditions :

x

f

yy

f

xf

,

00~0~0

00~0~0

max

min

ppxxx

ppxx

Simulation Domain

• Radial zone : rRadial zone : rq=2q=2<r<r<r<rq=3 q=3

• « Corresponding » tokamak parameters« Corresponding » tokamak parametersNumerical Models for Controlled Fusion, NMCF '09

mLmms

TBmnTTeVTk

mrmrmR

s

ieeB

simu

101101101

110.550

06.035.05.2

int

0319

0000

00

TURBULENT SELF-ORGANIZATION


Self-Organization of Plasma

• Competition between Competition between different mechanisms for driving different mechanisms for driving flows :flows :

• Definition of « energy » and confinement timeDefinition of « energy » and confinement time


Vtot

conf dVtrp ),,,(1

R

drtrtE2

),()( V

dVtrtE2

~ ),,,(~)(

dVrSVtot )(

« electrostatic » turbulence


« electromagnetic » turbulence


Confinement Time

• Important parameter Important parameter N N , importance of , importance of E.M. effects, here :E.M. effects, here :

• Role of Role of NN on confinement time on confinement time

• Linear phase : no effect on growth rateLinear phase : no effect on growth rate

• Strong impact observed on quality of Strong impact observed on quality of confinementconfinement

• Statistitically stationnary state : Statistitically stationnary state : decrease of the order of 40%decrease of the order of 40%


Impact of N on poloidal rotation

• Mean Electrostatic PotentialMean Electrostatic Potential• Linked with poloidal Linked with poloidal

rotationrotation

• ↘ ↘ EE => ↘ poloidal => ↘ poloidal rotationrotation

• ↘ ↘ poloidal rotation => ↗ poloidal rotation => ↗ level of fluctuationslevel of fluctuations

• No parallel diffusivity in the No parallel diffusivity in the simulated systemsimulated system

• =>No direct coupling =>No direct coupling between pressure and between pressure and electromagnetic electromagnetic fluctuationsfluctuations

• =>cannot explain decrease =>cannot explain decrease of of confconf


Mean Flow EnergyMean Flow Energy Electromagnetic Electromagnetic FluctuationsFluctuations

N=0.01

N=0.1

N=0.2

N=0.3

Equations for Transport

• Average in the poloidal and toroidal directions (noted Average in the poloidal and toroidal directions (noted <><>) ) of the equations for pressure and electrostatic potentialof the equations for pressure and electrostatic potential


SGp

GpTTTv

cBCollConvrt

VMRrt

TTrr : Reynolds stress : Reynolds stressTTmm : Maxwell Tensor : Maxwell TensorTTvv : « viscosity : « viscosity Tensor »Tensor »

convconv : convective flux : convective fluxcollcoll : collisional damping : collisional dampingBB : flux due to parallel diffusivity : flux due to parallel diffusivity ((B B =0 here)=0 here)

• Relative importance of TRelative importance of Trr and T and Tmm in the transport dynamics in the transport dynamics

• Time averaged value for a tensor F :Time averaged value for a tensor F :

• Increase of Maxwell Tensor in competition with increasing Increase of Maxwell Tensor in competition with increasing Reynolds stressReynolds stressNumerical Models for Controlled Fusion, NMCF '09

tRt

drtrFF 22 ,

• Ratio between Maxwell Tensor and Reynolds StressRatio between Maxwell Tensor and Reynolds Stress

• Reynolds stress => drive poloidal flowReynolds stress => drive poloidal flow

• Low value of Low value of NN => Reynolds stress dominant => Reynolds stress dominant

• Decrease of poloidal flow energy with increasing Decrease of poloidal flow energy with increasing NN linked to a linked to a competition between Reynolds and Maxwell stresscompetition between Reynolds and Maxwell stress



Characterization of turbulent transport

• NN parameter : parameter :• Directly linked to the Directly linked to the parameter parameter• Importance of electromagnetic effectsImportance of electromagnetic effects• More and more important for future machinesMore and more important for future machines

• Increase of Increase of NN : :• Degradation of confinement Degradation of confinement • Increase of magnetic fluctuationsIncrease of magnetic fluctuations• Mainly due to decrease of sheared plasma rotation Mainly due to decrease of sheared plasma rotation • Due to increase of Maxwell tensor in competition with Due to increase of Maxwell tensor in competition with

increasing Reynolds stressincreasing Reynolds stress

DYNAMICS OF TRANSPORT BARRIERS

Numerical Models for Controlled Fusion, NMCF ’09


• Intermittent destruction Intermittent destruction of the barrier :of the barrier :

• Destructive effect : high Destructive effect : high increase of energy and increase of energy and particles flux on wallsparticles flux on walls

• Beneficial effect : Beneficial effect : evacuation of fusion evacuation of fusion products presents in the products presents in the plasmaplasma

• Operating mode for ITEROperating mode for ITER

Transport Barrier in fusion plasma

• CharacteristicsCharacteristics• Amelioration of Amelioration of

confinementconfinement• Steepness of profilesSteepness of profiles• Barrier associated with Barrier associated with

a localized velocity a localized velocity shearshear

• Local reduction of Local reduction of transporttransport

• H mode specific instability H mode specific instability : ELMs: ELMs

• Type relaxation Type relaxation oscillationsoscillations

• Accompanied by Accompanied by magnetic fluctuationsmagnetic fluctuations

Pressure profile0=plasma core, 1= edge

• Early 70’s, discharges Early 70’s, discharges with degraded with degraded confinementconfinement

ELMs


•Type IType I• Frequency increases with Frequency increases with

heating powerheating power

•Type IIIType III• Frequency decreases with Frequency decreases with

heating powerheating power

•Classification criteria• Frequency dependency with input power• Presence of a magnetic precursor• MHD stability

Dynamics of Transport Barrier

• Local reduction of turbulence associated with a growth of Local reduction of turbulence associated with a growth of pressure and density gradientpressure and density gradient

• From an experimental side,From an experimental side,• Spontaneous apparition during Spontaneous apparition during a transition from low a transition from low

confinement to high confinement (L-H transition)confinement to high confinement (L-H transition)• Triggered by externally Triggered by externally driving driving EE××BB shear flow shear flow

• In numerical simulation, self generation of transport barrier In numerical simulation, self generation of transport barrier has not yet been reproducedhas not yet been reproduced


Transport Barrier, an illustration



Transport Barrier Generation

)(added drivemean

)(ithequation w flowMean

'state stationary Statist.

)(equationTransport

min

Uu

TTTu

constSdr

Sp

VMRrt

r

r

totBcollconv

Bcollconvrt

• Ext. Imposed flow U, locally shearedExt. Imposed flow U, locally sheared• U profile in agreement with experimental U profile in agreement with experimental

observations, observations, [Ch. P. Ritz, [Ch. P. Ritz, Phys. Rev. Let.Phys. Rev. Let., 65, 2543 , 65, 2543 (1990)](1990)]

•

2/5qat localizedBarrier : Here

barriertransport

p

p

r

rcollcoll

conv

Imposed flow U

Transport Barrier Relaxations Phenomenon


• Evolution of confinement time Evolution of confinement time dynamic of pressure dynamic of pressure gradientgradient

• Observation of Quasi-periodic relaxationsObservation of Quasi-periodic relaxations

• Growth on a long time scale of Growth on a long time scale of conf conf followed by a fast followed by a fast degradationdegradation

• Relaxation frequency : , of the order of few Relaxation frequency : , of the order of few kHzkHz • Value in agreement with experimental frequencies of type III Value in agreement with experimental frequencies of type III

ELMsELMs [G.Fuhr [G.Fuhr et al.et al., , Phys. Rev. Let.Phys. Rev. Let., 101, 195001 (2008)], 101, 195001 (2008)]

1int006.0 relaxf

• Reduction of convective flux in the region where velocity Reduction of convective flux in the region where velocity shear is imposedshear is imposed

• Appearance of flux peaks Appearance of flux peaks increase of turbulent transport through increase of turbulent transport through the barrierthe barrier

• Propagation : radial propagation from barrier center to outer Propagation : radial propagation from barrier center to outer regionsregions

Spatial Evolution


Radial Structure of Fluxes Radial Structure of Pressure Fluctuations

Analysis of cross correlation (I)

• Cross correlation functions => establishment of connection Cross correlation functions => establishment of connection between velocity shear and components of systembetween velocity shear and components of system

• No correlation between No correlation between EExxBB and Reynolds stress / Maxwell and Reynolds stress / Maxwell tensortensor

• EExxB B and magnetic flux are correlatedand magnetic flux are correlated

• EExxBB and convective flux are anti-correlated and convective flux are anti-correlatedNumerical Models for Controlled Fusion, NMCF '09

Analysis of cross correlation (II)

• Strong correlation between magnetic flux and turbulent fluxStrong correlation between magnetic flux and turbulent flux

• =>These part of the dynamics is governed by the velocity =>These part of the dynamics is governed by the velocity shearshear

• Concerning Reynolds stress and Maxwell tensor :Concerning Reynolds stress and Maxwell tensor :• Weak correlation (~30%)Weak correlation (~30%)• Finite phase shift (max. around x=-0.2)Finite phase shift (max. around x=-0.2)


Evolution of Energy


• Same kind of dynamics for energy and fluxSame kind of dynamics for energy and flux

• =>Increase of fluctuations level during relaxation =>Increase of fluctuations level during relaxation eventsevents

• =>Enhancement of magnetic activity, as it is the case in =>Enhancement of magnetic activity, as it is the case in experimental observations of transport barrier relaxationsexperimental observations of transport barrier relaxations

Evolution of Energy


• ratio between level of electrostatic and electromagnetic ratio between level of electrostatic and electromagnetic fluctuationsfluctuations

• Relative importance of ES and EM fluctuations is similar to Relative importance of ES and EM fluctuations is similar to the one observed in a purely EM phenomenon (Alfven the one observed in a purely EM phenomenon (Alfven waves)waves)

~~ / EE

Mode Amplitude


• Power Spectra of EM fluctuations for 3 phases :Power Spectra of EM fluctuations for 3 phases :

• before, during and after a relaxation eventbefore, during and after a relaxation event

• Growth of a mode at barrier center during the relaxation,Growth of a mode at barrier center during the relaxation,

• Mode which is a resonant mode on the surface defined by Mode which is a resonant mode on the surface defined by q=m/n=10/4=2.5q=m/n=10/4=2.5

Mode Amplitude


• Mode growth at barrier center already observed in ES Mode growth at barrier center already observed in ES simulationssimulations

• =>Main mechanism similar in both cases : =>Main mechanism similar in both cases :

• perturbation at the barrier center exhibits a transitory perturbation at the barrier center exhibits a transitory growth due to a time delay in the shear flow stabilizationgrowth due to a time delay in the shear flow stabilization

• As the ES model is a limit case of the EM one, extrapolation As the ES model is a limit case of the EM one, extrapolation of the ES results is far from being straightforwardof the ES results is far from being straightforward

Conclusions

•Study of turbulence simulations of a tokamak edge Study of turbulence simulations of a tokamak edge plasma focusing on impact of electromagnetic plasma focusing on impact of electromagnetic fluctuationsfluctuations

•Role of Role of NN analyzed analyzed

• Modification of relative importance of Reynolds stress and Modification of relative importance of Reynolds stress and Maxwell TensorsMaxwell Tensors

• Decrease of confinement time with increasing Decrease of confinement time with increasing NN

•Transport Barrier dynamicsTransport Barrier dynamics• Enhancement of magnetic activity during relaxationsEnhancement of magnetic activity during relaxations• Frequency range in agreement with type III Edge Localized Frequency range in agreement with type III Edge Localized

modes (ELMs)modes (ELMs)• Physical mechanism underlying these relaxations remains Physical mechanism underlying these relaxations remains

the same for ES and EM simulationsthe same for ES and EM simulationsNumerical Models for Controlled Fusion, NMCF '09

association euratom-cea electromagnetic self-organization and turbulent transport in tokamaks g....

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