X. Garbet22nd IAEA FEC, Geneva 2008 1

Theory of turbulence and gyrokinetic simulations

X. GarbetCEA-IRFM

AssociationEuratom-Cea

Thanks to: P. Angelino, C. Angioni, C. Bourdelle, A. Casati, J. Candy, P.H. Diamond, G. Dif-Pradalier, G. Falchetto, P. Ghendrih, O. Gürcan, Y. Idomura, F. Jenko, C.Nguyen, M. Ottaviani, Y. Sarazin, L. Villard, R.Waltz and T.-H. Watanabe.

X. Garbet22nd IAEA FEC, Geneva 2008 2

Confinement is a crucial issue for fusion• Triple product ITER nDTDτE≈3 1021m-3.keV.s• Confinement • Transport is turbulent: understanding, prediction, control.

ITERin 3.7salossespower contentenergy 2

E →χ

≈=τ

Outline1) Basics of turbulent transport.2) Turbulence self-organisation. 3) Outcomes of theory and simulations.4) Improved confinement : read the paper! 5) Prospects and conclusion

X. Garbet22nd IAEA FEC, Geneva 2008 3

γ

kρi1

∇Τ driven modes (ITG)

i

∇Τ driven modes (ETG)

e

Trapped Electron Modes (TEM)

6

4

2

0420

-R∇ n/n

-R∇ T/T

Stable

ITG

ITG+TEM

TEM

KinezeroSeveral modes can be

unstable above a threshold

• Instabilities → turbulent transport

•Appear above a threshold κ

c

.• Underlie particle, electron and

ion heat transport : interplay between all channels.

• Coherence: plasma response + Maxwell equations.

X. Garbet22nd IAEA FEC, Geneva 2008 4

Turbulent transport

• Eddies : ExB velocity

→ Assuming a random walk

δφ<0

vE

B

δφ>0

X. Garbet22nd IAEA FEC, Geneva 2008 5

Gyrokinetic framework Brizard and Hahm 08

( )t,,v,F //GG µx

Grandgirard 07

• Plasma is nearly collisionless → kinetic calculation.

• Calculation of 6D distribution function F(x,v,t) is costly.

• Compute instead the distribution of gyrocenters

• Gyrocenter and particle densities are different. → numerical challenge

X. Garbet22nd IAEA FEC, Geneva 2008 6

Several codes, but not all doing the same thing …

trapped electrons

Full F, neoclassical

GTC, ORB5

ELMFIRE , GYSELA, ORB5, TEMPEST, XGC

Local

electrostatic ITG case

Global

GENE, GKW, GS2, GYROelectromagnetic

GYRO

GEM, GT5D, UCAN GKV, PG3EQ

FEFI

Lagrangian

Semi-Lagrangian

Eulerian

From Falchetto 08

trapped electrons

GEM, GT3D

X. Garbet22nd IAEA FEC, Geneva 2008 7

II. Turbulence self-organization

• Why is it difficult? Disparity of spatial and time scales.

• Generation of structures, which back-react on turbulence background:

- Large scale transport events: feedback via profile relaxation.

- Zonal flows: feedback via flow shear

X. Garbet22nd IAEA FEC, Geneva 2008 8

Large scale transport events • Streamers: ExB eddies elongated in the radial direction

Beyer 00, Champeaux 00.

• Boost radial transport if ExB velocity is large enough. Jenko 00, Labit 03, Idomura 05, Lin 05, Candy 08.

• Radially propagating fronts (avalanches). Related to turbulence spreading. Garbet 94, Diamond 95, Hahm 04, Gurcan 06.

Radial direction

Pol

oida

l dire

ctio

n

Idomura 05 ETG turbulence

X. Garbet22nd IAEA FEC, Geneva 2008 9

0 0 . 2 0 . 4 0 . 6 0 . 8 1

0

0 . 5

1O R B 5 d a t a A

R f r o m R H t h e o r y

( 1 - AR

) e x p ( - γ G* t ) + A

R

t⋅ Ω ci⋅ 10-4V E

/VE(0

)

Zonal flowsDiamond, Itoh, Itoh & Hahm 05

• Low frequency fluctuations of the ExB poloidal velocity.

• Generated via Reynolds and Maxwell stresses.

Turbulent amplification ~ |φ|2Vθ’

• Damping is weak Rosenbluth & Hinton 98.

Jolliet 07

Geodesic Acoustic Modes

Residual flow

X. Garbet22nd IAEA FEC, Geneva 2008 10

Zonal Flows (cont.)

• Strong feedback on turbulence: ExB vortex shearing Biglari 90, Waltz

94, Hahm 95 Clearly seen in all turbulence simulations Hammett 93,

Lin 98, Kinsey 05, Villard 02 .

• Leads to a self-organized state.

Lin 98 - GTC

X. Garbet22nd IAEA FEC, Geneva 2008 11

III. Outcome of theory and simulations

- Dimensionless analysis: scaling with normalised gyroradius at fixed β and collisionality.

- Transport channels: ion and electron heat channels, particle transport, momentum transport.

X. Garbet22nd IAEA FEC, Geneva 2008 12

ρ*=2.10−2 ρ*=10−2 ρ*=5.10−3

1/ρ*

Sarazin 08 - GYSELA

Lin 02 - GTC

ITER

Candy 04

GS2

GYRO

• GyroBohm scaling law

• Gyrokinetic and fluid simulations find that the scaling is gyroBohm when ρ*→ 0

Garbet 96, Ottaviani 99, Lin 02, Candy 04.

•Scaling with ν

*

and β still under investigation.

Scaling laws Kadomtsev 75, Connor 77, Waltz 90

X. Garbet22nd IAEA FEC, Geneva 2008 13

Ion heat transport is rather well understood

• ITG dominated: quite well assessed. Now a test for gyrokinetic codes.

• Profiles are stiff.• Actual threshold ≥linear

value.

Dimits 00 – Cyclone project

Temperature gradient

Tur

b. tr

ansp

ort c

oeffi

cien

t

Linear threshold non-linear threshold

Fluid

Gyrokinetic

X. Garbet22nd IAEA FEC, Geneva 2008 14

Electron heat transport

Candy 06 - GYRO• Contribution from TEMs

when ∇n or ∇T

e

large enough Ernst 04, Merz 08, Lang

08

• Contribution from ETGs depends on parameters:

- small for ITG dominated turbulence Candy 06

- significant for TEM/ETG dominant modes Goerler 08

Goerler 08 - GENE

X. Garbet22nd IAEA FEC, Geneva 2008 15

• Particle flux :

• Complex physics of turbulent pinch velocity :

- can reverse direction.

- effect of collisions is important.Coppi 78, Tang 86, Garbet 03, Isichenko 96, Hallatschek 05, Dubuit 05, Estrada-Mila 05

Particle transport

ee

e Vndr

dnD +−=Γ

ν

eff

-V/D

Angioni 03 – GS2 Quasilinear

ITER

X. Garbet22nd IAEA FEC, Geneva 2008 16

Momentum transport and spontaneous spin-up

Peeters 08 - GKW

• Still an open issue Hahm 06, Gurcan 06, Peeters 07, Waltz 07, Peeters 08.

residual stress (ExB shear)

• A puzzling observation: toroidal rotation without external torque

• Structure of // momentum radial flux Dominguez 93, Diamond 94, Garbet 02

Momentum pinch

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IV. Where are we going to?

• Reduced transport models.

• Statistical models of turbulence

• Ab-initio simulations

X. Garbet22nd IAEA FEC, Geneva 2008 18

Dannert 05

Wav

e nu

mbe

r

a / LT i

= 3 . 0 : C r o s s - p h a s e δ Ei - δ φ

a n g le [ r a d ]

k θρ s- 3 - 2 - 1 0 1 2 3

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

1

2

3

4

5

6

7

8

9

x 1 0- 3

Casati 08 - GYRO

Cross phase δvE-δT

Phase

k θρ i

Reduced transport models• Flux • Quasi-linear theory: cross

phase given by linear theory Vedenov 61, Drummond 63, Horton 83 – some support from simulations Dannert 04, Lin 07, Casati 08.

• Fluctuation spectra: mixing-length rule Prandtl 25

• Basis of most transport models: GLF23, Weiland, CDBM, Qualikiz…Still in progress.

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A new generation of codes

• Aim:- full distribution function, full

torus.- steady-state, driven by sources- coupling to the SOL- neoclassical transport• Challenging - first results in the

literature. Ku 06, Heikkinen 08, Dif-Pradalier 08, Scott 08, McMillan 08, Xu 08.

Chang 08 - XGC

X. Garbet22nd IAEA FEC, Geneva 2008 20

Progress is tied to high performance computers

Simulation (numbers are indicative)

Time needed for 1 simulation with a 100TFlops HPC

Memory

Turbulence steady state electrostatic ITG/TEM, global 1/ρ*=512 (~ITER)

1 day 100 GB

Confinement time 1/ρ*=512 electrostatic ITG/TEM.

1 year 50 TB

Turbulence steady state, electromagnetic, ITG/TEM/ETG

k

max

ρ

i

=20

10 years 1 PB

→ Need for multi-petaflops computers

X. Garbet22nd IAEA FEC, Geneva 2008 21

Conclusions

• Dimensionless scaling law: gyroBohm at small ρ*. Scaling

with ν

*

and β still to be clarified.• Some consensus on ion heat transport , not far from

agreement for electron heat and particle transport, momentum transport still debated.

• Understanding of transport barriers: far from satisfactory –

e.g. LH transition and effect of rational q

min

for ITBs. • 2 approaches for predicting transport in fusion devices: - reduced transport models : fast but approximate.- full ab-initio simulations : accurate but costly.