Turbulence & Transport in magnetised plasmas

AssociationEuratom-Cea Turbulence & Transportin magnetised plasmas Y. SarazinInstitut de Recherche sur la Fusion par confinement MagnétiqueCEA Cadarache, France Acknowledgements: P. Beyer, G. Dif-Pradalier,X. Garbet, Ph. Ghendrih, V. Grandgirard

Confinement governs tokamak performances • Economic viability of Fusion governed by tE Amplification Factor Q • Self-heating (ignition)  Upper bound for ni: b=nT/(B2/2m0) < 1tE ~ few sec.

Z q Non-circular poloidal cross-section Axi-symmetric X-point r Safetyfactor ri = miv/eB  10-3 m v// 3 q v┴ 2.5 j B 2 1.5 r R 1 0 0.2 0.4 0.6 0.8 1 Confinement ensured by large B field(~105 BEarth) • Helicoidal field lines generate toroidal flux surfaces • MHD equilibrium: Laplace force (jpB) = Expansion (P)  n,T are flux functions • Particle trajectories ~ magnetic field lines(// Transp. >> Transp.) Poloïdal angle q current jp Toroidal angle j

Transport is turbulent • Collisional transport negligible: Fusion plasmas weakly collisional • Heat losses are mainly convective: • Turbulent diffusivity cturb governs confinement properties ~102-103 s-1 ~105 s-1

Outline • Basicsof turbulent transport • Drift- Wave instabilities in tokamaks • Wave-particle resonance  k//~0 • Transport models: fluid vs. Gyrokinetic and numerical tools • Dimensionless scaling laws:similarity principle, experiments vs theory • Large scale structures: Zonal Flows & Avalanche-like events • Improved confinement, physics of Transport Barriers

Contour lines of iso-potential Test particle trajectory Electrostatic turbulence EB drift: . Turbulent field df Random walk Diffusion cES Correlation time of Turbulent convection cells Challenge: df, tcorrel?

Magnetic turbulence dBr fluctuations Radial component of v//: vr~ (dB/B) v// Random walk  Diffusion: . vr (dBr/B) v// dB Magnetic field line v// Beq Fast particles more sensitive to magnetic turbulence

Electrostatic vs Magnetic Transport m << es except at high b

Fluctuations and transport are correlated • Fluctuation magnitude:  when Padd  whenconfinement is improved • Cross-phase between pressure (density) and velocity is important e.g.  No transport of matter

20 15 10 5 1 Tore Supra reflectometer Fluctuation level % L mode ohmic r/a 0.5 0.7 1 Experimental characteristics of fluctuations Large scales are dominant Fluctuation level increases at the edge Tore Supra P. Hennequin ITG TEMETG

tE (s) Observed tE (s) ITER-FEAT 1 JET ? • Proposing routes towards high confinement regimes  Transport barriers Observed tE (s) 0.1 Autres machines 0.01 Fit tE(s) 0.01 0.1 1 First principle simulations Loi d'échelle Main challenges for transport simulations • Predicting transport/performances in next step devices: Gap  uncertainty  Requires understanding of the physics to validate the extrapolation

Time scale separation between cyclotron motion & Turbulence Adiabatic theory Broad range of space & time scales Sace  (m) lD~re~5.10-5 ri~10-3 a~1 ℓpm//~103 Frequency (s-1) wce~5.1011 wci~108 wturb~105 nii~102 1/tE~1

jc rc B Field line B Guiding Center Particle Particle drifts within adiabatic limit Adiabatic limit: wturb  105 s <wc= 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…

Particle drifts within adiabatic limit (cont.) • Transverse drifts: (limit b<<1) governs turbulent transport Vertical charge separation (Balanced by // current) • Parallel dynamics: Parallel trapping

Fluid drifts within adiabatic limit 1st order 2nd order ~ r* vT ~ r*2 vT diamagnetic drift current ensures MHD equilibrium: j*B=p polarisation (ions)

Main primary instabilities in tokamaks Strongly magnetised plasma  Drift Wave instabilities (adiabatic limit) • Homogeneous B field  DW instability, also "slab-ITG" • Inhomogeneous B field (curvature, grad-B)  Interchange Various species and classes of particles (passing, trapped) • Negative sheath resistivity (governed by plasma-wall interaction) • Kelvin-Hemoltz if plasma flow is large enough (?) Ion Temperature Gradient Core Edge • All of these have magnetic counterparts at large bêta (Drift Alfvén Waves, etc.)

Drift Wave instability • Unstable if Causes: viscosity, resonances... vEx < 0 • Isothermal // force balance:  adiabatic response

Interchange instability • B inhomogeneous centrifugal force ~ effective gravity Rayleigh-Bénard convection geff T1 Dense, heavy fluid Tokamak Top view gravity n2 > n1 n1 toroidal direction Hot, light fluid T2 (>T1) • Interchange is unstable on the low field side • Both regions are connected by // current  stabilising

B B  1/R ions B vE n électrons n Interchange instability (cont.) Field line curvature Vertical drift vgs  (BB)/es Polarisation provided j// small enough Electric drift vE = (Bf)/B2 Parametric instability Stable if on the (high field side)

EB advection: [F,f] = uE.f xFyf - yFxf • // conductivity: • Curvature: Competing instabilities: DW vs. Interchange • Extended Hasegawa-Wakatani model accounting for B curvature Continuity eq. Charge balance .j=0 2D, fluid

// conductivity scan curvature scan Density gradient scan Interchange Interchange Interchange DW DW DW DW Linear analysis: scanning control parameters • DW instability dominant at small resistivity (large C) & weak curvature g • Phase shift Djn,f: DW:small Djn,f Low transport Interchange: Djn,f~p/2 Maximum transport

Several 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  Linear growth rate

Stability diagram (Weiland model) -RT/T -Rn/n Electron and/or ion modes are unstable above a threshold • Instabilities  turbulent transport • Appear above a threshold c • Underlie particle, electron and ion heat transport: interplay between all channels

Wave-particle resonant interactions • Instability due to resonant energy exchange btw. waves & particles for wave q(r) • Tokamaks: resonances are localised in space Resonance: Resonant surface: rmn Landau damping Landau damping Supra-thermal particles give energy to the wave within (r-rmn) < few ri

q(r) m-2 m-1 m m+1 m+2 r D d Mode width & Chirikov parameter • Typical mode width d ~ region where Wpart.wave>0: • Distance D between adjacent modes (m,n) & (m+1,n): • Chirikov parameter (stochasticity threshold)  Mode overlap, Stochasticity Shear length: Poloidal wave vector: ~0.3 ~10-30

Z Z R R e.g. gyrokinetic code GYSELA Linear eigenmodes are global modes • Approximate form of an eigenmode: • But: not-periodic, assumes constant gradient • Exact solutions calculated numerically

Tokamaks:  ~ 1 ~ 1 / eddy turn-over time Transition towards strong turbulence • Decorrelation time between waves & particles: Turbulence  diffusion in velocity Dv phase diffusion  d[(w-kv) t] 2 k2 dv2t2  k2 Dvt3  tD (k2 Dv)-1/3(Dupree / Kolmogorov time) • Wave correlation time: tcDw-1 • Transition: • tc < tD Weak turbulence  quasi-linear • tc > tD Strong turbulence  non linear

degrees of freedom • Particlexj(t), vj(t) 6N mj dvj/dt = ej { E(xj) +vjB(xj) } • Kinetic fs(x,v,t) 6Ns Vlasov: dfs/dt= 0 Hamiltonian system C(f) collisions • Fluid ns(x,t), us(x,t), etc… 3Ns Moments of fs (or fGCs): M(k) vk fs d3v Infinite hierarchy a priori Closure ? Decreasing complexity First principle models Quasi-neutrality: Ampère: Maxwell Fluctuations of electro-magnetic field Fluctuations of charge & current densities dr, dj df, dA Plasma response? (Gyro-)Kinetic or Fluid

Gyro-Landau fluid models • Guiding-center approximation: Field "seen" by fluid particles is gyro-averaged (over cyclotron motion) • Adjunction of damping terms to mimic Landau resonances KineticFluid for imposes

[Dimits 2000] Turb. transport coefficient • Non linear threshold in kinetics Temperature gradient Linear threshold non-linear threshold Non-linear mismatches • Fluid models hardly account for Landau resonances (likely important in collisionless regime) Trapped & fast particles • Present fluid closures not sufficient • Large dispersion • Fluid over estimates transport level

q • Field aligned coordinates q, a=j-qq, h=q • Not periodic in h • High spatial resolution, appropriate for low r*= rc/a j Flux Tube Simulations initial • Drift waves: k//  0 Electric potential Fourier spectrum (Log) (GYSELA) Toroidal mode # n Poloidal mode # m final -2.5 -6 Slope 1/q Toroidal mode # n Poloidal mode # m

Several numerical techniques to solve a kinetic 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

Temperature radius Fixed Gradient vs Fixed Flux • Boundary condition at r=a: fixed fields, free gradients • 3 choices for the core: • prescribed central value • central value + self-adaptive source fixed gradient everywhere • prescribed flux Fixed boundaries (thermal baths) Prescribed flux (open system) Statistical equilibrium • No control of incoming flux • Profile relaxation If NO self-adaptive source • Close to experimental conditions

Challenges for simulations • 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

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 Connor-Taylor ‘77 Scale invariance (similarity principle for fluids) • 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 I. II. III. Kadomtsev ‘75

r* / r*Iter n* / n*Iter Main experimental trends Normalised gyroradius Electromagnetic effects Collisionality • JET & DIII-D: • Strong impact of r* • Consistent with electrostatic: lci and cR/cs • Iter: r* & n* will be smaller

GyroBohm scaling when r*0 a=0: Bohm; a=1: gyroBohm most favourable case for Iter  r* scaling in simulations (r*Iter~2.10-3) • Challenging in terms of numerical resources: r* / 2  N23, Dt / 2 E.g. r* = 1/256 5D grid: N ~ 1010 points CPU time ~ 50 000h (512 procs.~ 4 days) Plasma duration ~ 300 ms

No a definite scaling with * • cEis a decreasing function of * • Not a definite scaling cE  [*]-0.3 at low * cE  [*]-0.8 at high * • May reflect competing effects… McDonald '06

* scaling: trapped electrons vs. Zonal Flows Ryter '05 • Collisionality stabilizes TEM  cEshould be an increasing function of * • Should affect e more than i  might be invisible on E Lin '98 • Collisions damp zonal flows  cE should be a decreasing function of * • Found in numerical simulations Lin ‘98 , Falchetto ‘05 *

Collision operators • Full collision operator much too complex for numerical studies • Development of reduced models e.g. Constraints: Ensure momentum & energy conservation, ambipolarity Recover neoclassical theoretical results [Garbet '08] [Belli '08] [Dif-Pradalier '08] Rotation Transport vqNC / (T/eB)

Log E(k) k-5/3 k-3 Energie Enstrophie 1/L0 Log k Mode Condensation • Inverse cascade: formation of large scale structures • Exact for a 2D turbulence in a magnetised plasma • Persistent featureof most simulations

Regulate the transport Simple understanding: If ky=0 modes, other modes • Linearly undamped in collisionless regime  requires kinetic calculation  [Lin '98, Beyer '00] Electric potential Zonal Flows • Fluctuations of the poloidal velocity ky0 [Grandgirard '05]

Excitation of Zonal Flows GYSELA Several mechanisms : • Modulational instability + back-reaction on fluctuations • Kelvin-Helmholtz instability • Geodesic curvature: wGAM~cs/R Reynolds stress without ZF ZF included

>5 lcorr 24.103 wc t 5.103 V  1.5 r* vT 50 r / ri 170 Large 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 flux GYSELA (r* = 1/256)

Turbulence & Transport in magnetised plasmas