CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008)Y. Sarazin 1 Turbulence & Transport in magnetised plasmas Y. Sarazin

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 1

Turbulence & Turbulence & TransportTransportin magnetised plasmasin magnetised plasmas

Y. SarazinInstitut de Recherche sur la Fusion par confinement Magnétique

CEA Cadarache, France

AssociationEuratom-Cea

Acknowledgements: P. Beyer, G. Dif-Pradalier,X. Garbet, Ph. Ghendrih, V. Grandgirard


Confinement governs tokamak Confinement governs tokamak performancesperformances Economic viability of Fusion governed by E

Self-heating (ignition)

Upper bound for ni: nTB220 E ~ few sec.

Amplification Factor Q


Confinement ensured by large B fieldConfinement ensured by large B field (~105 BEarth)

Helicoidal field lines generate toroidal flux surfaces

MHD equilibrium:Laplace force (jpB) Expansion (P) n,T are flux functions

Particle trajectories ~ magnetic field lines ( Transp. Transp.)

Poloïdalangle

Toroidal angle

v//

v┴

B

i = miv/eB 103 m

current jp

r

Non-circular poloidal cross-section

Axi-symmetric X-point

Z

R

Safety factor

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

q

r


Transport is Transport is turbulentturbulent

Collisional transport negligible:Fusion plasmas weakly collisional

Heat losses are mainly convective:

Turbulent diffusivity turb governs confinement properties

~102-103 s1 ~105 s1


OutlinOutlinee

1. Basics of turbulent transport

2. Drift- Wave instabilities in tokamaks

3. Wave-particle resonance k//~0

4. Transport models: fluid vs. Gyrokinetic and numerical tools

5. Dimensionless scaling laws: similarity principle, experiments

vs theory

6. Large scale structures: Zonal Flows & Avalanche-like events

7. Improved confinement, physics of Transport Barriers


OutlinOutlinee





5. Dimensionless scaling laws: similarity principle, experiments vs

theory




Electrostatic Electrostatic turbulenceturbulence

EB drift: .

Turbulent field

Random walk Diffusion ES

Correlation time ofTurbulent convection cells

Challenge: correl?

Contour lines of iso-potential

Test particletrajectory


Br fluctuations

Radial component of v//:

vr ~ (B/B) v//

Random walk Diffusion: .

Magnetic Magnetic turbulenceturbulence

B

Beq

vr (Br/B) v//

v//

Magnetic field line

Fast particles more sensitiveto magnetic turbulence


Electrostatic vs Electrostatic vs Magnetic Magnetic TransportTransport

m << es except at high


Fluctuations and transport are Fluctuations and transport are correlatedcorrelated

Fluctuation magnitude: when Padd

when confinement is improved

Cross-phase between pressure (density) and velocity is important

e.g. No transportof matter


Tore Suprareflectometer

0.5 0.7 1 r/a

20

15

10

5

1

Flu c

t ua t

ion

leve

l %

L mode

ohmic

Experimental characteristics of fluctuationsExperimental characteristics of fluctuations

ITG TEM ETG

Tore Supra

P. Hennequin

Large scales are dominant Fluctuation level increasesat the edge


Loi d'échelle

E (s)

1

0.1

0.01

ITER-FEAT

0.01 0.1 1

?

Main challenges for transport Main challenges for transport simulationssimulations

Predicting transport/performances in next step devices:Gap uncertainty Requires understanding

of the physics tovalidate the extrapolation

JET

Autresmachines

Obs

erve

d E

(s)

Fit E (s)

Obs

erve

d E

(s)

First principle simulations

Proposing routes towards high confinement regimes Transport barriers


OutlinOutlinee






theory




Broad range of space & time Broad range of space & time scalesscales

ci~108 turb~105 1/E~1ii~102

i~103D~e~5.105 a~1 ℓpm//~103

ce~5.1011

Frequency (s1)

Sace (m)

Time scale separation betweencyclotron motion & Turbulence

Adiabatic theory


Particle drifts within adiabatic Particle drifts within adiabatic limitlimit

Adiabatic limit:turb 105 s c eB/mi 108 s

Phase space reduction

Additional invariant: .

( magn. flux enclosed by cyclotron motion)

3 invariants motion is integrable:Energy

Toroidal kin. Momentum

(axi-symmetry)

Velocity drifts of guiding center…

cccc

B

Particle

GuidingCenter

Field line B


Particle drifts within adiabatic limit Particle drifts within adiabatic limit (cont.)

(limit 1)

Transverse drifts:

governs turbulent transport

Vertical charge separation (Balanced by // current)

Parallel dynamics:

Parallel trapping


Fluid drifts within adiabatic Fluid drifts within adiabatic limitlimit 1st order 2nd order

* vT *2 vT

diamagnetic drift current ensures MHD

equilibrium: j*B=p

polarisation (ions)


Strongly magnetised plasma Drift Wave instabilities (adiabatic limit)

Homogeneous B field DW instability, also "slab-ITG"

Inhomogeneous B field (curvature, grad-B) InterchangeVarious species and classes of particles (passing, trapped)

Negative sheath resistivity (governed by plasma-wall interaction)

Kelvin-Hemoltz if plasma flow is large enough (?)

Main primary instabilities in Main primary instabilities in tokamakstokamaks

All of these have magnetic counterparts at large bêta(Drift Alfvén Waves, etc.)

Core

Edge

Ion Temperature Gradient


Drift Wave Drift Wave instabilityinstability

Unstable if

Causes: viscosity, resonances...

< 0 Isothermal // force balance:

adiabatic response

vEx


Interchange Interchange instabilityinstability

B inhomogeneous centrifugal force ~ effective gravity

TokamakTop view gravity

Dense, heavy fluid

Hot, light fluidT2 T1

T1

Interchange is unstable on the low field side

Both regions are connected by // current stabilising

geff

n2 > n1n1

toroidaldirection

Rayleigh-Bénard convection


B 1/R

n

Interchange instability Interchange instability (cont.)

Field line curvature

Vertical drift vgs (BB)/es

Polarisationprovided j// small enough

Electric drift vE (B)/B2

Parametric instability

Stable if on the (high field side)

nn

BB

BBions

électronsélectrons

vvEE


Competing instabilities: DW Competing instabilities: DW vs. InterchangeInterchange

Extended Hasegawa-Wakatani model accounting for B curvature

Continuity eq.

Charge balance .j=0

2D, fluid

EB advection:[,f] = uE.f xyf yxf

// conductivity:

Curvature:


Linear analysis: scanning control Linear analysis: scanning control parametersparameters

DW instability dominant at small resistivity (large C) & weak curvature g Phase shift n: DW: small n Low transport

Interchange: n Maximum transport

DW

Inte

rcha

nge

DW Inte

rcha

nge

DW DW

Inte

rcha

nge

curvature scan Density gradient scan// conductivity scan


Several branches are potentially unstableSeveral branches are potentially unstable

Ion Temperature Gradient modes: driven by passing ions, interchange + “ slab ”

Trapped Electron Modes: driven by trapped electrons, interchange

Electron Temperature Gradient modes: driven by passing electrons

Ballooning modes at high

ki1

driven modes (ITG)

i

driven modes (ETG)

e

Trapped Electron Modes (TEM)

Linear growth rate


Electron and/or ion modes are unstable Electron and/or ion modes are unstable above a thresholdabove a threshold

Instabilities turbulent transport

Appear above a threshold c

Underlie particle, electron and ion heat transport: interplay between all channels

15

10

5

06420

-Rd rL

og(T

)

-RdrLog(ne)

Ion Mode (ITG) Electron

Mode (TEM)

Ion + Electron Mode (ITG+TEM)

Stable

Stability diagram (Weiland model)

Rn/n

R

T/

T


OutlinOutlinee






theory




q(r)

nm

Wave-particle resonant Wave-particle resonant interactionsinteractions Instability due to resonant energy exchange btw. waves & particles

Resonance:Resonant surface:

for wave

Tokamaks: resonances are localised in space

Supra-thermal particles giveenergy to the wavewithin (rrmn) few i

rmn

Landaudamping

Landaudamping


Typical mode width region where Wpart.wave>0:

Distance between adjacent modes (m,n) & (m+1,n):

Chirikov parameter (stochasticity threshold)

Mode overlap,

Stochasticity

Poloidal wave vector:

Mode width & Chirikov Mode width & Chirikov parameterparameter

Shear length:

~0.3 ~10-30

q(r)

nm

m2 m1 m m+1 m+2

r


Linear eigenmodes are global Linear eigenmodes are global modesmodes

Approximate form of an eigenmode:

But: not -periodic, assumes constant

gradient

Exact solutions calculated numerically

R

Z

R

Z

e.g. gyrokinetic code GYSELA


Transition towards strong Transition towards strong turbulenceturbulence

Decorrelation time between waves & particles:

Turbulence diffusion in velocity Dv phase diffusion

kvt2 k2 v2 t2

k2 Dv t3 D

k2 Dv1/3 (Dupree / Kolmogorov time)

Wave correlation time: c 1 Transition:c < D Weak turbulence quasi-linear

c > D Strong turbulence non linear

Tokamaks:

~ 1 / eddy turn-over time

~ 1


OutlinOutlinee






theory




Plasmaresponse?

First principle First principle modelsmodels

, j , A

Maxwell

Fluctuations ofcharge & current

densities

Fluctuations ofelectro-magnetic

field

Quasi-neutrality:

Ampère:

(Gyro-)Kinetic or Fluid

degrees of freedom

Particle xj(t), vj(t)6N

mj dvj/dt ej { E(xj) vjB(xj) }

Kinetic fs(x,v,t)6Ns

Vlasov: dfs/dt = 0 Hamiltonian system

C(f) collisions

Fluid ns(x,t), us(x,t), etc…3Ns

Moments of fs (or fGCs): M(k)vk fs d3v

Infinite hierarchy a priori Closure ?

Dec

reas

ing

com

plex

ity


Gyro-Landau fluid Gyro-Landau fluid modelsmodels

Guiding-center approximation:Field "seen" by fluid particles is gyro-averaged (over cyclotron motion)

Adjunction of damping terms to mimic Landau resonances

Kinetic Fluid

for imposes


Non-linear Non-linear mismatchesmismatches Fluid models hardly account for

Landau resonances (likely important in collisionless regime)

Trapped & fast particles

Present fluid closures not sufficient[Dimits 2000]

Temperature gradient

Turb

. tra

nspo

rt co

effic

ient

Large dispersion

Fluid over estimates transport level

Non linear threshold in kinetics

Linear threshold non-linear threshold


Flux Tube SimulationsFlux Tube Simulations

Drift waves: k// 0

Poloidal mode # m

Poloidal mode # m

Slope1/q

Toro

idal

mod

e #

n

Toro

idal

mod

e #

n

Electric potentialFourier spectrum (Log)

(GYSELA)

initial

final-2.5

-6

Field aligned coordinatesq, q,

Not periodic in

High spatial resolution, appropriate for low

c/a


Several numerical techniques to solve a Several numerical techniques to solve a kinetickinetic equation equation

Noise reduction: computes f for guiding centers

Particle In Cell: pushes particles f e.m. field

Eulerian-Vlasov: solves Vlasov as a (complicated) differential equation

Semi-lagrangian: fixed grid, calculates trajectories backward


Fixed Gradient vs Fixed FluxFixed Gradient vs Fixed Flux

prescribed central value

central value + self-adaptive source fixed gradient everywhere

prescribed flux

Fixed boundaries(thermal baths)

No control of incoming flux Profile relaxation

If NO self-adaptive source

Prescribed flux(open system)

Close to experimentalconditions

Statisticalequilibrium

Boundary condition at r=a: fixed fields, free gradients

3 choices for the core:

radius

Tem

pera

ture


ChallenChallenges for ges for simulationssimulations Characterisation of turbulence features & transport dynamics:

Scaling laws extrapolation to Iter, etc…

Tendency for producing large scale structures: inverse cascade

Fluctuations of the poloidal flow: Zonal Flows. Reduce anomalous transport. Introduce non locality in k space

Large scale transport events: avalanches and streamers. Breaks locality and scaling of the correlation length

Transport barriers: Velocity shear Magnetic shear & low order rational surfaces


OutlinOutlinee






theory




Numbering of dimensionless parameters for a given set of plasma parameters

8 numbers for a pure e-i plasma

Implication on confinement time:

II & III given

Scale invariance Scale invariance (similarity principle for fluids)

I.

II.

III.

Kad

omts

ev ‘7

5

Analysis of scale invariance of Fokker-Planck equation coupled to Maxwell equations local relations

If geometry, profiles, & boundary conditions are fixed, plasma is

neutral, then Con

nor-

Tayl

or ‘7

7


JET & DIII-D:

Strong impact of

Consistent with electrostatic:ci and cR/cs

Main experimental Main experimental trendstrends

Normalised gyroradius

Electromagnetic effects

Collisionality

Iter: & will be smaller

Iter

Iter


** scaling in simulations scaling in simulations (*Iter~2.103)

Challenging in terms of numerical resources:* / 2 N23, t / 2E.g. * = 1/256

5D grid: N ~ 1010 pointsCPU time ~ 50 000h(512 procs. ~ 4 days)Plasma duration ~ 300 s

GyroBohm scaling when *0

=0: Bohm; =1: gyroBohmmost favourable case for Iter


No a definite scaling with No a definite scaling with * *

cE is a decreasing function of *

Not a definite scalingcE [*]-0.3 at low *

cE [*]-0.8 at high *

May reflect competing effects…

McDonald '06


* scaling: trapped electrons vs. Zonal * scaling: trapped electrons vs. Zonal FlowsFlows

Collisionality stabilizes TEM cE should be an increasing

function of *

Should affect e more than i

might be invisible on E

Ryter '05

Collisions damp zonal flows cE should be a decreasing

function of *

Found in numerical simulationsLin ‘98 , Falchetto ‘05

Lin '98

*


Collision Collision operatorsoperators

Full collision operator much too complex for numerical studies Development of reduced models

e.g.

Constraints:Ensure momentum & energy conservation, ambipolarityRecover neoclassical theoretical results

v

NC /

(T/

eB)

[Dif-Pradalier '08] [Belli '08]

[Garbet '08]

Transport Rotation


OutlinOutlinee






theory




Mode CondensationMode Condensation

Inverse cascade: formation of large scale structures

Exact for a 2D turbulence in a magnetised plasma

Persistent feature of most simulations

k5/3

k3

Log E(k)

Log k

Energie

Enstrophie

1/L0


Zonal Zonal FlowsFlows Electric potential

[Grandgirard '05]

Regulate the transport Simple understanding:

If ky=0 modes , other modes

Linearly undamped in collisionless regime requires kinetic calculation

30

20

10

010000

5

0

Eturb V

time

[Lin '98, Beyer '00]

Fluctuations of the poloidal velocity ky0


Excitation of Zonal Excitation of Zonal FlowsFlows

Several mechanisms :

Modulational instability

+ back-reaction on fluctuations

Kelvin-Helmholtz instability

Geodesic curvature: GAM~cs/R

Reynolds stress without ZF

ZF included

GYSELA


Large scale transport eventsLarge scale transport events

Events that take place over distances larger than a correlation length

Identified as avalanches streamers

May lead to enhanced transport and/or non local effects

Turbulent radial heat fluxGYSELA *

50 r / i 170

24.103

c t

5.103

corr

V * vT


AvalanchesAvalanches

Profile relaxations at all scales

Domino effect

Propagate at a fraction of the sound speed

steep gradients

Avalanche

Average profile

Radial direction


Streamers Streamers

Convective cells elongated in the radial direction kx0, aligned along the magnetic fieldReminiscence of linear eigenmodes ?

Boost the radial transport if the ExB velocity is large enough controversial

Radius

Pol

oida

l ang

le

RBM simulations


Interplay Between Avalanches and Interplay Between Avalanches and ZFZF

r/a

Thermal flux

time

Shear of ZF


OutlinOutlinee






theory




Several “regimes” in a tokamak Several “regimes” in a tokamak plasmaplasma

L-mode: basic plasma, turbulence everywhere

H-mode: low turbulent transport in the edge, formation of a pedestal

Internal Transport Barrier: low turbulent transport in the core, steep profiles

Normalised radiusNormalised radius r/a

Pla

sma

p res

s ure


Several mechanisms Several mechanisms may lead to improved may lead to improved confinementconfinement

Flow shear

Magnetic shear

Te/Ti, Zeff, density gradient, fast particles… : not generic

R

Z

VE


Flow Shear Flow Shear StabilisationStabilisation Shearing rate

Approximate criterion for stabilisation:

Biglari-Diamond-Terry '90 Waltz '94

Large convection cells are teared apart

Turbulent transport is reduced

Electric potential

[Figarella '03] vE=0 vE=0.9


Transport barrier Transport barrier relaxationsrelaxationsTransport barriers can exhibit quasi-periodic relaxations

vE=1.23

vE=1.52

vE=1.83

Turbulent flux at q=2.5

Time

Turbulent flux

Basic understanding: Predator (ZF) – prey (turbulence) model Time delay for EB stabilisation

Beyer '01


Force balance equation:

Power threshold

Controlling the FlowControlling the Flow

Fuelling Heating ToroidalMomentum

turbulence collisions

Flow generation

*=0.01; R/LT7Dif-Pradalier '08

0

0.02

r / a


Negative magnetic shear is stabilisingNegative magnetic shear is stabilising

Magnetic shear :

s<0 : favourable average of interchange drive (vEB)(vEp) along field lines

Enhanced by geometry effect. B.B.Kadomtsev, J.Connor, M.Beer,J.Drake, R.Waltz, A.Dimits, C.Bourdelle…

drdq

qrs

s=0 s>0unstable

s<0stable

Vortex distorsion


Dynamics of transport barriers is more Dynamics of transport barriers is more complex than s<0 and shear flowcomplex than s<0 and shear flow

JET- E. Joffrin

JET #51573

0.014

0.016

0.018

0.020

0.022

4.5 5 5.5 6 6.5 7 7.5

3.2

3.3

3.4

3.5

3.6

3.7

Time (s)

s>0

s<0

q=2

5MW ICRH + 11.5MW NBI

qmin

‘narrow’ ITB s<0 region

q=2 location from the MHD analysis

R(m

)

ee

c TT

Map of -cT/T : profile steepening


Density of rational Density of rational surfacessurfaces

Resonant surfaces far from each other:Close to low order rational surfacesIn vanishing shear region s=0

q(r)

m:0128 n:064

r / a

Can lead to transport barriers: Observed in fluid models Some support from experiments (JET) Not observed in gyrokinetic simulations

so far

still a matter of debate

Garbet '01

r / a

q(r)

Tem

pera

ture


Magnetic shear lowers critical shear flowMagnetic shear lowers critical shear flowat transitionat transition

Force balance equation

in a reactor plasma

adjustement of magnetic shear s to lower lin.

Shear flow rate vs. magnetic shear (JET)

1linE

0p)(en iii BVE0.5

0.4

0.3

0.2

0.1

0.00.80.60.40.20.0

s

R/cs


ConclusionConclusionss Turbulence simulations are efficient ways of testing

theories and building reduced transport models

Still the accuracy of transport models is not better than 20%

Generic mechanisms to control turbulence improved confinement

Turbulence simulations have tested the validity of various theoretical ideas

Still many issues remain unresolved



core

Edgepedestal

pcore = 3nT

ppedestal ~ 50% pcore

H-mode: edge transport barrierH-mode: edge transport barrier

“H-mode” = High confinementscenario, reference for ITER

Bifurcation (super / sub critical ?)

Spontaneaous* triggering of high confinement regimeWith low turbulence level

(* experimental control parameter: heating power )

What mecanism(s) ?

Stability of these regimes ?


Physics of L- to H-mode Physics of L- to H-mode transition?transition?

No self-consistent model so far Difficulty: Transition takes place at interface between open

(plasma-wall interaction) & closed (core) magn. surfaces

Players: Magnetic configuration (X point)

Role of electrique fieldTokam 3D [Tamain '07]

JOREK [Huysmans '06]


Quasi-periodic relaxations of H-mode barrier Huge energy losses (~1 MJ in JET) – short time (~200 s) Main concern for ITER (deterioration of plasma facing components)

Strong relaxations at the edge ("ELMs")Strong relaxations at the edge ("ELMs")

Tokamak JET During an ELM

[Ghendrih '03]

0

4

80

10

20

12 14 16 18 20 22time [s]

D

Energy [MJ]

heating powerNBI [MW]

SolarFlare


Constraints on plasma facing Constraints on plasma facing componentscomponents

Steady state:power to be extracted = alpha power Heat Flux ~ 10 MW.m2

> thermic protectionof space shuttle

Transient: ELMadapted from [Federici et al., 2003]

Energy loss per ELM (MJ)

Nb

of IT

ER

pul

ses

110

102

103

104



Critical Gradient Critical Gradient ModelModel

Rules for correlation length and time :

Mixing length estimate :

Can be extended to more complex models: Weiland, GLF23, …

Lc s csR

RdTTdr

c

T sT

eBsR

Stiffness GyroBohm Threshold

RdTTdr

c


A useful, but controversial, concept : A useful, but controversial, concept : mmarginal stabilityarginal stability

• Marginally stable profile

• Stiffness: tendency of profiles to stay close to marginal stability.

• Central temperature is improved if- threshold c is larger

- edge pedestal Ta is higher.

Rra

ac

eTT

68

1

2

4

68

10

2

1.00.80.60.40.20.0

c=5c=7

c=5

r/a

T(ke

V)

Ta=2keV

Ta

Normalised radius

Tem

pera

tur e

(keV

)


0.1

1.0

0 0.2 0.4 0.6 0.8 1

Ohmic0.8 MW1.6 MW

T e [

keV

]

tor

5.0

AUG 13556, 13558Ip = 1 MA, q95 = 3.5

mixASDEX Upgrade

Edge plasma gets closer to the threshold for high Tedge

Core plasma is subcritical.

Profiles are not marginally stable Profiles are not marginally stable everywhereeverywhere

Normalised radius

T

e (ke

V)


Critical Gradient Model Critical Gradient Model (cont.)

T sT

eBsR

Stiffness GyroBohm Threshold

RdTTdr

c


Modulation experiments provide a stringent Modulation experiments provide a stringent test of transport modelstest of transport models

• Localised electron heat modulation.

• Slope ~1/[hp]1/2

hp= +T/T

Assessment of transport models. stiffness s and threshold c.

Temperature vs time at several radius

Time (s)

Phase and amplitude vs radius

Phase

Amplitude

Normalised radius Normalised radius

Phase and amplitude vs radius

JET


SStiffness is found to be highly variabletiffness is found to be highly variable

• Critical gradient model: - threshold as expected.- large variation of stiffness.

• Reproduced by transport modeling and stability analysis

• Transition from electron to ion turbulence is key issue.

0

0.05

0.1

0.15

0.2

0.25

0.3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

Theory no collisions Theory with collisionsExperiment

Heat

flux

theo

ry [a

.u.]

Heat flux experiment [M

W/m

2]

R/LTe

ne =

± 0

.1

ASDEX Upgrade

-RTe/Te

Ele

ctro

n he

at fl

ux

With ion heating

Dominant electron heating

JET



Trois invariants du Trois invariants du mouvementmouvement

1.1. Moment magnétiqueMoment magnétique = 1/2 mv = 1/2 mv22 / B / B

Flux magnétique englobé par le mouvement cyclotronique

Invariant dit "adiabatique": limite (tB)/B<<c et (B)/B<<c

2.2. Energie Energie (ssi chauffage, rayonnement) E = 1/2 mvE = 1/2 mv////22 + + B + B +

eeEnergie cinétique & potentielle (si t=0)

2.2. Moment cinétique toroïdalMoment cinétique toroïdal M = mRvM = mRv + e + e

Axisymétrie du tokamak L/ d/dtL/

3 variables angulaires du mvt: c, et

Trajectoires intégrables inscrites sur des toresTrajectoires intégrables inscrites sur des tores


Particules piégées Particules piégées (I)(I)

Invariants du Invariants du mouvement:mouvement:

1.1. E = E = 1/21/2 mv mv////22 + + BB

= = 1/21/2 mv mv22 / B / B

3.3. M = mRvM = mRv + e + e

1-2) E < BM v// s'annule

Piégeage v///v

1/2

3) Largeur banane b q c 1/2

B

Z

R

R

BM

Bm

r

v//.t

E

B

bb

= r/R


Particules piégées Particules piégées (II)(II)

Fréquence de rebond:

b v// /L// vth/qR


Dépiégeage Dépiégeage collisionnelcollisionnel

Collisions v//2 = Dv.t vth

2 c.t

Dépiégeage pour v//2 v

2

Fréquence de dépiégeageeffeff cc / / Fraction de

particules piégées:

ffpp 1/21/2

= r/R B/B

Collisions marche au hasard dans l'espace des vitesses

v//

v

1/2Cône depiégeage



Experimental Experimental evidence is sparseevidence is sparse

Streamers not observed yet Zonal Flows measured with dual Heavy Ion Beam Probes

CHS


Modèle gyrocinétique Modèle gyrocinétique électrostatiqueélectrostatique"Ion Temperature Gradient (ITG) driven turbulence" Mouvement de dérive du centre-guide (théorie adiabatique tB /B c)

Equation gyrocinétique

Electroneutralité (Poisson dans la limite kD 1)

(limite 1)

+ +

Documents

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008)Y. Sarazin 1 Turbulence & Transport in magnetised plasmas Y. Sarazin