Association EuratomCea Turbulence Transport in magnetised plasmas Y

Association Euratom-Cea Turbulence & Transport in magnetised plasmas Y. Sarazin Institut de Recherche sur la Fusion par confinement Magnétique CEA Cadarache, France Acknowledgements: P. Beyer, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, V. Grandgirard CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 1

Confinement governs tokamak performances q Economic viability of Fusion governed by E Amplification Factor Q q Self-heating (ignition) Upper bound for ni: =n. T/(B 2/2 0) 1 t. E ~ few sec. CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 2

Confinement ensured by large B field (~105 BEarth) q Helicoidal field lines generate toroidal flux surfaces q MHD equilibrium: Non-circular poloidal cross-section Z Laplace force (jp B) = Expansion ( P) n, T are flux functions Axi-symmetric X-point q Particle trajectories ~ magnetic field lines (// Transp. >> Transp. ) r Safety factor i = miv /e. B 10 -3 m Poloïdal angle 3 v// q v┴ 2. 5 Bj 2 1. 5 current j 0 0. 2 p 1 r 0. 4 0. 6 0. 8 R 1 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Toroidal angle j Y. Sarazin 3

Transport is turbulent q Collisional transport negligible: Fusion plasmas weakly collisional ~102 -103 s-1 ~105 s-1 q Heat losses are mainly convective: q Turbulent diffusivity turb governs confinement properties CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 4

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 5

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 6

Electrostatic turbulence E B drift: . Turbulent field f Contour lines of iso-potential Random walk Diffusion ES Correlation time of Turbulent convection cells Test particle trajectory Challenge: df, tcorrel? CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 7

Magnetic turbulence vr (d. Br/B) v// d. B Magnetic field line Beq v// Br fluctuations Radial component of v//: Fast particles more sensitive to magnetic turbulence vr ~ ( B/B) v// Random walk Diffusion: . CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 8

Electrostatic vs Magnetic Transport m << es except at high CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 9

Fluctuations and transport are correlated q q Fluctuation magnitude: when Padd when confinement is improved Cross-phase between pressure (density) and velocity is important e. g. No transport of matter CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 10

Experimental characteristics of fluctuations Fluctuation level increases at the edge Large scales are dominant Tore Supra Fluctuation level % 20 15 Tore Supra reflectometer 10 L mode 5 1 ohmic 0. 5 0. 7 1 ITG TEM ETG CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) r/a P. Hennequin Y. Sarazin 11

Main challenges for transport simulations Predicting transport/performances in next step devices: Gap uncertainty Requires understanding E (s) ITER-FEAT of the physics to 1 validate the extrapolation ? q Proposing routes towards high confinement regimes Transport barriers Observed E (s) q 0. 1 0. 01 First principle simulations CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) JET Autres machines 0. 1 1 Loi d'échelle Fit E (s) Y. Sarazin 12

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 13

Broad range of space & time scales Sace (m) l. D~ e~5. 10 -5 i~10 -3 a~1 ℓpm//~103 Frequency (s-1) ce~5. 1011 ci~108 turb~105 Time scale separation between cyclotron motion & Turbulence CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) ii~102 1/ E~1 Adiabatic theory Y. Sarazin 14

Particle drifts within adiabatic limit Adiabatic limit: turb 105 s c = e. B/mi 108 s jc c B Phase space reduction q Additional invariant: . Guiding Center ( magn. flux enclosed by cyclotron motion) Field line B Particle 3 invariants motion is integrable: Energy Toroidal kin. Momentum q (axi-symmetry) Velocity drifts of guiding center… CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 15

Particle drifts within adiabatic limit (cont. ) q Transverse drifts: (limit 1) governs turbulent transport q Vertical charge separation (Balanced by // current) Parallel dynamics: Parallel trapping CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 16

Fluid drifts within adiabatic limit 1 st order ~ * v. T ~ *2 v. T 2 nd order diamagnetic drift current ensures MHD equilibrium: j* B= p polarisation (ions) CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 17

Main primary instabilities in tokamaks Strongly magnetised plasma Drift Wave instabilities (adiabatic limit) q Core Homogeneous B field DW instability, also "slab-ITG" Ion Temperature Gradient q Inhomogeneous B field (curvature, grad-B) Interchange Various species and classes of particles (passing, trapped) Edge q Negative sheath resistivity (governed by plasma-wall interaction) q Kelvin-Hemoltz if plasma flow is large enough (? ) q All of these have magnetic counterparts at large bêta (Drift Alfvén Waves, etc. ) CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 18

Drift Wave instability v. Ex < 0 q Isothermal // force balance: adiabatic response CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) q Unstable if Causes: viscosity, resonances. . . Y. Sarazin 19

Interchange instability q B inhomogeneous centrifugal force ~ effective gravity geff Rayleigh-Bénard convection T 1 Dense, heavy fluid Tokamak Top view gravity n 1 n 2 > n 1 toroidal direction Hot, light fluid T 2 (> T 1) q Interchange is unstable on the low field side q Both regions are connected by // current stabilising CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 20

Interchange instability (cont. ) Field line curvature B B 1/R Vertical drift vgs (B B)/es Polarisation provided j// small enough ions B v. E n Electric drift v. E = (B f)/B 2 électrons Parametric instability n CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Stable if on the (high field side) Y. Sarazin 21

Competing instabilities: DW vs. Interchange q Extended Hasegawa-Wakatani model accounting for B curvature Continuity eq. 2 D, fluid Charge balance . j=0 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) q E B advection: [F, f] = u. E. f x. F yf - y. F xf q // conductivity: q Curvature: Y. Sarazin 22

Linear analysis: scanning control parameters DW DW Density gradient scan DW Interchange curvature scan Interchange // conductivity scan DW DW instability dominant at small resistivity (large C) & weak curvature g q Phase shift Djn, f: DW: small Djn, f Low transport Interchange: Djn, f~p/2 Maximum transport q CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 23

Several branches are potentially unstable q Ion Temperature Gradient modes: driven by passing ions, interchange + “ slab ” q Trapped Electron Modes: driven by trapped electrons, interchange q Electron Temperature Gradient modes: driven by passing electrons q Ballooning modes at high CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Linear growth rate Y. Sarazin 24

Electron and/or ion modes are unstable above a threshold q Instabilities turbulent transport q Appear above a threshold c q Underlie particle, electron and ion heat transport: interplay between all channels -R T/T Stability diagram (Weiland model) -R n/n CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 25

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 26

Wave-particle resonant interactions q Instability due to resonant energy exchange btw. waves & particles for wave q Tokamaks: resonances are localised in space q(r) Resonance: rmn Resonant surface: Supra-thermal particles give energy to the wave within (r-rmn) few i CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Landau damping Y. Sarazin Landau damping 27

Mode width & Chirikov parameter q Typical mode width ~ region where Wpart. wave>0: Shear length: q Distance D between adjacent modes (m, n) & (m+1, n): Poloidal wave vector: q(r) q Chirikov parameter (stochasticity threshold) D Mode overlap, d Stochasticity m-2 m-1 m m+1 m+2 ~0. 3 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) r ~10 -30 Y. Sarazin 28

Linear eigenmodes are global modes q Approximate form of an eigenmode: Z q q But: not -periodic, assumes constant gradient R Exact solutions calculated numerically e. g. gyrokinetic code GYSELA CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 29

Transition towards strong turbulence q Decorrelation time between waves & particles: Turbulence diffusion in velocity Dv phase diffusion [( -kv) t] 2 k 2 v 2 t 2 k 2 Dv t 3 D (k 2 Dv)-1/3 (Dupree / Kolmogorov time) q Wave correlation time: q Transition: c < D c > D c D -1 Weak turbulence Strong turbulence quasi-linear non linear Tokamaks: ~ 1 / eddy turn-over time CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 30

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 31

First principle models Decreasing complexity Quasi-neutrality: Ampère: q Particle degrees of freedom xj(t), vj(t) 6 N mj dvj/dt = ej { E(xj) + vj B(x j) } Maxwell Kinetic Fluctuations of Vlasov: dfs/dt = , j 0 charge & current C(f) densities q q Fluid fs(x, v, t) Hamiltonian system f, A collisions Plasma ns(x, t), us(x, t), etc… response? (k) k 6 Ns Fluctuations of electro-magnetic field 3 Ns Moments of fs (or f. GCs): M v fs d 3 v Infinite hierarchy a priori Closure ? (Gyro-)Kinetic or Fluid CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 32

Gyro-Landau fluid models q Guiding-center approximation: Field "seen" by fluid particles is gyro-averaged (over cyclotron motion) q Adjunction of damping terms to mimic Landau resonances Kinetic Fluid for CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) imposes Y. Sarazin 33

Non-linear mismatches q Fluid models hardly account for Landau resonances (likely important in collisionless regime) Trapped & fast particles q Present fluid closures not sufficient Turb. transport coefficient [Dimits 2000] q Large dispersion q Fluid over estimates transport level q Non linear threshold in kinetics Temperature gradient Linear threshold non-linear threshold CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 34

q Drift waves: k// 0 q Field aligned coordinates q, a=j-q , h= q initial Electric potential Fourier spectrum (Log) (GYSELA) Poloidal mode # m j final Not periodic in h High spatial resolution, appropriate for low *= c/a -2. 5 Toroidal mode # n q Toroidal mode # n Flux Tube Simulations Slope 1/q -6 Poloidal mode # m CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 35

Several numerical techniques to solve a kinetic equation q Noise reduction: computes f for guiding centers q Particle In Cell: pushes particles f e. m. field q Eulerian-Vlasov: solves Vlasov as a (complicated) differential equation q Semi-lagrangian: fixed grid, calculates trajectories backward CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 36

Fixed Gradient vs Fixed Flux q Boundary condition at r=a: fixed fields, free gradients q 3 choices for the core: Fixed boundaries (thermal baths) Prescribed flux (open system) Temperature Statistical equilibrium radius No control of incoming flux q Close to experimental q Profile relaxation conditions If NO self-adaptive source q CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin § prescribed central value § central value + selfadaptive source fixed gradient everywhere § prescribed flux 37

Challenges for simulations q 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 q Transport barriers: § Velocity shear § Magnetic shear & low order rational surfaces CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 38

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 39

Scale invariance (similarity principle for fluids) Numbering of dimensionless parameters for a given set of plasma parameters q 8 numbers for a pure e-i plasma Kadomtsev ‘ 75 q I. II. Connor-Taylor ‘ 77 III. q Implication on confinement time: II & III given q Analysis of scale invariance of Fokker-Planck equation coupled to Maxwell equations local relations q If geometry, profiles, & boundary conditions are fixed, plasma is neutral, then CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 40

Main experimental trends Normalised gyroradius Electromagnetic effects Collisionality JET & DIII-D: § Strong impact of r* § Consistent with electrostatic: lc i and c R/cs q Iter: r* / r*Iter q r* & * will be smaller n* / n*Iter CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 41

r* scaling in simulations q ( *Iter~2. 10 -3) Challenging in terms of numerical resources: * / 2 N 23, Dt / 2 E. g. * = 1/256 5 D grid: N ~ 1010 points CPU time ~ 50 000 h (512 procs. ~ 4 days) Plasma duration ~ 300 s q Gyro. Bohm scaling when * 0 a=0: Bohm; a=1: gyro. Bohm most favourable case for Iter CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 42

No a definite scaling with * q c E is a decreasing function of * q Not a definite scaling c E [ *]-0. 3 at low * Mc. Donald '06 c E [ *]-0. 8 at high * q May reflect competing effects… CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 43

* scaling: trapped electrons vs. Zonal Flows q Ryter '05 Collisionality stabilizes TEM c E should be an increasing function of * q Should affect e more than i might be invisible on E Lin '98 * CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) q Collisions damp zonal flows c E should be a decreasing function of * q Found in numerical simulations Lin ‘ 98 , Falchetto ‘ 05 Y. Sarazin 44

Collision operators Full collision operator much too complex for numerical studies q Development of reduced models q e. g. [Garbet '08] Constraints: Ensure momentum & energy conservation, ambipolarity Recover neoclassical theoretical results [Belli '08] [Dif-Pradalier '08] CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Rotation v NC / ( T/e. B) Transport Y. Sarazin 45

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 46

Mode Condensation q q q Inverse cascade: formation of large scale structures Exact for a 2 D turbulence in a magnetised plasma Log E(k) k-5/3 Energie Enstrophie Persistent feature of most simulations CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) k-3 1/L 0 Y. Sarazin Log k 47

Zonal Flows Electric potential q Fluctuations of the poloidal velocity ky 0 q Regulate the transport [Grandgirard '05] Simple understanding: [Lin '98, Beyer '00] If ky=0 modes , other modes q Linearly undamped in collisionless regime requires kinetic calculation CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 48

Excitation of Zonal Flows GYSELA Several mechanisms : q Modulational instability without ZF Reynolds stress + back-reaction on fluctuations q Kelvin-Helmholtz instability q Geodesic curvature: GAM~cs/R ZF included CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 49

Large scale transport events q Events that take place over distances larger than a correlation length q q Turbulent radial heat flux GYSELA ( * = 1/256) Identified as § avalanches § streamers May lead to enhanced transport and/or non local effects > 5 lcorr 24. 103 V 1. 5 r* v. T c t 5. 103 50 r / i CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 170 50

Avalanches q Profile relaxations at all scales q Domino effect q Propagate at a fraction of the sound speed Radial direction CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 51

Streamers q q Convective cells elongated in the radial direction kx 0, aligned along the magnetic field Reminiscence of linear eigenmodes ? Poloidal angle RBM simulations Radius Boost the radial transport if the Ex. B velocity is large enough controversial CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 52

Interplay Between Avalanches and ZF Shear of ZF time Thermal flux r/a CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 53

Outline 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 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 54

q L-mode: basic plasma, turbulence everywhere q H-mode: low turbulent transport in the edge, formation of a pedestal q Internal Transport Barrier: low turbulent transport in the core, steep profiles Plasma pressure Several “regimes” in a tokamak plasma Normalised radius r/a CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 55

Several mechanisms may lead to improved confinement q Flow shear q Magnetic shear q Te/Ti, Zeff, density gradient, fast particles… : not generic CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 56

Flow Shear Stabilisation q Shearing rate q Approximate criterion for stabilisation: Biglari-Diamond-Terry '90 [Figarella '03] v. E =0 Waltz '94 v. E =0. 9 § Large convection cells are teared apart § Turbulent transport is reduced Electric potential CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 57

Transport barrier relaxations Transport barriers can exhibit quasi-periodic relaxations q Turbulent flux at q=2. 5 v. E =1. 83 Basic understanding: § Predator (ZF) – prey (turbulence) model § Time delay for E B stabilisation Turbulent flux v. E =1. 52 Beyer '01 v. E =1. 23 Time CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 58

Controlling the Flow q Force balance equation: Fuelling Heating Toroidal Momentum Power threshold 0 q Dif-Pradalier '08 *=0. 01; R/LT 7 Flow generation -0. 02 turbulence collisions r / a CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 59

Negative magnetic shear is stabilising q q q Magnetic shear : s<0 : favourable average of interchange drive (v. E B)(v. E p) along field lines s<0 s=0 stable s>0 unstable j Enhanced by geometry effect. B. B. Kadomtsev, J. Connor, M. Beer, J. Drake, R. Waltz, A. Dimits, C. Bourdelle… Vortex distorsion CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 60

Dynamics of transport barriers is more complex than s<0 and shear flow Map of - c T/T : profile steepening JET #51573 q=2 3. 7 s>0 R(m) 3. 6 q=2 location from the MHD analysis s<0 3. 4 3. 2 0. 020 qmin 3. 5 3. 3 0. 022 ‘narrow’ ITB s<0 region 5 MW ICRH + 11. 5 MW NBI Time (s) 4. 5 5 5. 5 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) 0. 018 0. 016 0. 014 6 6. 5 7 Y. Sarazin 7. 5 JET- E. Joffrin 61

Density of rational surfaces q m: 0 128 n: 0 64 Resonant surfaces far from each other: Close to low order rational surfaces In vanishing shear region s=0 q(r) Garbet '01 q(r) r / a q Can lead to transport barriers: Temperature § Observed in fluid models § Some support from experiments (JET) § Not observed in gyrokinetic simulations so far r / a still a matter of debate CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 62

Magnetic shear lowers critical shear flow at transition Shear flow rate vs. magnetic shear (JET) q Force balance equation in a reactor plasma adjustement of magnetic shear s to lower lin. CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 63

Conclusions q Turbulence simulations are efficient ways of testing theories and building reduced transport models q Still the accuracy of transport models is not better than 20% q Generic mechanisms to control turbulence improved confinement q Turbulence simulations have tested the validity of various theoretical ideas q Still many issues remain unresolved CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 64

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 65

H-mode: edge transport barrier Spontaneaous* triggering of high confinement regime With low turbulence level (* experimental control parameter: heating power ) q q “H-mode” = High confinement scenario, reference for ITER Bifurcation (super / sub critical ? ) pcore = 3 n. T core ppedestal ~ 50% pcore Edge pedestal What mecanism(s) ? Stability of these regimes ? CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 66

Physics of L- to H-mode transition? q No self-consistent model so far q Difficulty: Transition takes place at interface between open (plasma-wall interaction) & closed (core) magn. surfaces q Players: Magnetic configuration (X point) Role of electrique field Tokam 3 D [Tamain '07] JOREK [Huysmans '06] CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 67

Strong relaxations at the edge ("ELMs") Quasi-periodic relaxations of H-mode barrier q Huge energy losses (~1 MJ in JET) – short time (~200 s) q Main concern for ITER (deterioration of plasma facing components) q Tokamak JET During an ELM Solar Flare [Ghendrih '03] CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 68

Constraints on plasma facing components q Steady state: power to be extracted = alpha power Heat Flux ~ 10 MW. m-2 > q thermic protection of space shuttle Transient: ELM 104 103 102 10 1 Nb of ITER pulses adapted from [Federici et al. , 2003] Energy loss per ELM (MJ) CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 69

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 70

Critical Gradient Model q Rules for correlation length and time : q Mixing length estimate : æ T s ö æ Rd. T ö s T = ç ÷ ç - c ÷ s è e. B R ø è Tdr ø Stiffness q Gyro. Bohm Threshold Can be extended to more complex models: Weiland, GLF 23, … CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 71

A useful, but controversial, concept : marginal stability • Stiffness: tendency of profiles to stay close to marginal stability. • Central temperature is improved if - threshold c is larger - edge pedestal Ta is higher. Temperature (ke. V) • Marginally stable profile Ta Normalised radius CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 72

q Edge plasma gets closer to the threshold for high Tedge q Core plasma is subcritical. Te (ke. V) Profiles are not marginally stable everywhere Normalised radius CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 73

Critical Gradient Model (cont. ) æ T s ö æ Rd. T ö s T = ç ÷ ç - c ÷ s è e. B R ø è Tdr ø Stiffness Gyro. Bohm CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Threshold Y. Sarazin 74

Modulation experiments provide a stringent test of transport models • Localised electron heat modulation. • Slope ~1/[ hp]1/2 Phase and amplitude radius Phase and amplitude vs vs radius Temperature vs time at several radius Phase JET hp= + T / T Assessment of transport models. stiffness s and threshold c. Amplitude Time (s) Normalised radius CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 75

Stiffness is found to be highly variable • Critical gradient model: - threshold as expected. - large variation of stiffness. • Transition from electron to ion turbulence is key issue. Electron heat flux • Reproduced by transport modeling and stability analysis With ion heating JET Dominant electron heating -R Te/Te CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 76

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 77

Trois invariants du mouvement 1. Moment magnétique = 1/2 mv 2 / B Flux magnétique englobé par le mouvement cyclotronique Invariant dit "adiabatique": limite ( t. B)/B<< c et ( B)/B<< c-1 2. Energie (ssi chauffage, rayonnement) E = 1/2 mv//2 + B + ef Energie cinétique & potentielle (si tf=0) 2. Moment cinétique toroïdal M = m. Rvj + e. Y . Axisymétrie du tokamak L/ j - d/dt{ L/ j} = 0 3 variables angulaires du mvt: jc, et j Trajectoires intégrables inscrites sur des tores CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 78

Particules piégées (I) B BM e = r/R Invariants du mouvement: Bm R Z db 1. E = 1/2 mv//2 + m. B 2. m = 1/2 mv 2 / B 3. M = m. Rvj + e. Y r R E 1 -2) E < m. BM Piégeage m. B 3) v// s'annule v///v |q=0 < e 1/2 Largeur banane db q rc e-1/2 v//. t CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 79

Particules piégées (II) Fréquence de rebond: b v// /L// e 1/2 vth/q. R CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 80

Dépiégeage collisionnel Collisions marche au hasard dans l'espace des vitesses v Cône de piégeage e-1/2 q Collisions q Dépiégeage pour v// Fraction de particules piégées: fp e 1/2 CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) v//2 = Dv. t vth 2 c. t v//2 e v 2 e = r/R B/B Fréquence de dépiégeage eff c / e Y. Sarazin 81

CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 82

Experimental evidence is sparse Streamers not observed yet q Zonal Flows measured with dual Heavy Ion Beam Probes q CHS CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin 83

Modèle gyrocinétique électrostatique "Ion Temperature Gradient (ITG) driven turbulence" q Mouvement de dérive du centre-guide (théorie adiabatique t. B /B c) (limite 1) q Equation gyrocinétique q Electroneutralité (Poisson dans la limite k l. D 1) CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8 -12 Sept. 2008) Y. Sarazin + + 84