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CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008)Y. Sarazin 3 Confinement ensured by large B field Confinement ensured by large B field (~10 5 B Earth ) Helicoidal field lines generate toroidal flux surfaces MHD equilibrium: Laplace force (j p B) Expansion ( P) n,T are flux functions Particle trajectories ~ magnetic field lines ( Transp. Transp.) Poloïdal angle Toroidal angle v // v┴v┴ B i = m i v /eB 10 3 m current j p r Non-circular poloidal cross-section Axi-symmetric X-point Z R Safety factor q r
Citation preview
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 2
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 3
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 4
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 5
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 6
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 7
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 8
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 9
Electrostatic vs Electrostatic vs Magnetic Magnetic TransportTransport
m << es except at high
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 10
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 11
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 12
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 13
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 14
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 15
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 16
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 17
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)
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 18
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 19
Drift Wave Drift Wave instabilityinstability
Unstable if
Causes: viscosity, resonances...
< 0 Isothermal // force balance:
adiabatic response
vEx
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 20
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 21
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 22
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:
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 23
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 24
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 25
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 26
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 27
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 28
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 29
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 30
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 31
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 32
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 33
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 34
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 35
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 36
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 37
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 38
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 39
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 40
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 41
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 42
** 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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 43
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 44
* 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
*
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 45
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 46
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 47
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 48
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 49
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 50
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 51
AvalanchesAvalanches
Profile relaxations at all scales
Domino effect
Propagate at a fraction of the sound speed
steep gradients
Avalanche
Average profile
Radial direction
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 52
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 53
Interplay Between Avalanches and Interplay Between Avalanches and ZFZF
r/a
Thermal flux
time
Shear of ZF
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 54
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 55
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 56
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 57
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 58
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 59
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 60
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 61
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 62
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 63
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 64
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 65
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 66
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 ?
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 67
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]
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 68
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 69
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 70
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 71
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 72
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
)
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 73
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)
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 74
Critical Gradient Model Critical Gradient Model (cont.)
T sT
eBsR
Stiffness GyroBohm Threshold
RdTTdr
c
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 75
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 76
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 77
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 78
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 79
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 80
Particules piégées Particules piégées (II)(II)
Fréquence de rebond:
b v// /L// vth/qR
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 81
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
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 82
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 83
Experimental Experimental evidence is sparseevidence is sparse
Streamers not observed yet Zonal Flows measured with dual Heavy Ion Beam Probes
CHS
CEA-EDF-INRIA school "Numerical Models for Controlled Fusion", Nice (8-12 Sept. 2008) Y. Sarazin 84
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)
+ +