NUMERICAL MODELLING OF TURBULENCE AND TRANSPORT IN THE EDGE PLASMA OF TOKAMAKS: A MANDATORY BUT CHALLENGING TOOL FOR ITER

NUMERICAL MODELLING OF TURBULENCE AND TRANSPORT IN THE EDGE PLASMA OF TOKAMAKS: A MANDATORY BUT CHALLENGING TOOL FOR ITER Patrick Tamain on behalf of the SOLEDGE2D and TOKAM3X teams Séminaire Maison de la Simulation April 11 th 2016 CEA 10 AVRIL 2012 PAGE 1

Producing energy with nuclear fusion Energetically favourable to fuse light nuclei (mass < Fe) The most attractive reaction: 17.6 MeV Deuterium abundant Tritium produced from Lithium Li + n He + T Deuterium: 115ppm=33mg/L

What is necessary to make fusion? Condition 1: heat up hot plasma (>10keV) Condition 2: confine due to energy balance n e τ E = 12T E α L T P ext P fus P losses ~E plasma /τ E Plasma ~30keV

Making a star on Earth? Not exactly Every single star = (very) successful fusion reactor gravitational confinement => not reproducible on Earth sun: 6.10 11 kg/s vs 250 kg/year for power plant On Earth, 2 possible ways: large n e (~10 31 m -3 ), small τ E (~10-11 s): inertial confinement (Mégajoule) small n e (~10 20 m-3), large τ E (~1s): magnetic confinement

Magnetic confinement in tokamaks Cyclotron motion in magnetic field: helicoidal trajectory, radius ρ L = mv = Larmor radius qb for D + in 30keV and B = 8T : ρ L ~2mm ρ L centre guide Tokamak configuration = toroidal field (coils) + plasma current (transfo) helicoidal field around ~toroidal surfaces = magnetic (flux) surfaces Poloidal direction θ Field lines Toroidal direction φ Magnetic flux surfaces Radial direction

Confinement is determined by turbulence Tokamak plasma = out of thermodynamic equilibrium transverse transport => confinement degradation Measured transport >> collisional transport transverse transport dominated by turbulence ~1m n~10 29 m 3 T~0.03eV n~10 20 m 3 T~10keV n~10 18 m 3 T~10 100eV

ITER in sight ITER = next generation machine under construction in Cadarache Objectives: achieve for the first time Q>1 (target Q=10) assess technologies for reactors (e.g. Li breeding blankets) Dramatic change of scale with respect to current machines extrapolation based on scaling laws very uncertain need theoretical understanding to back-up

The edge plasma of tokamaks open field lines = Scrape-Off Layer (SOL) transition = Last Closed Flux Surface (LCFS) Two types of configurations: LCFS / separatrix JET Tore Supra limiter divertor

The wall, key actor of edge plasma physics Solid material = perfect sink for the plasma sink for particles, energy boundary conditions on fluxes v i = c s = T e + T i m i Stationnary shock wave Recycling of neutral particles complex atomic and molecular reactions when re-entering the plasma important particle source for the plasma (>95%) energy and momentum sink (up to 100%) sputtering of impurities limiter

The specific properties of the edge Multiple consequences to the presence of the wall: 2D equilibrium (r, θ) centre exponential decay of n e et T e/i => (relatively) cold and thin plasma with strong gradients different dynamics of turbulence Poloidal equilibrium of parallel velocity in limiter geometry δn n bord ~10 100% δn n centre < 1%

The edge plasma determines the life expectancy of the reactor Power flux per unit of surface determined by decay length λ (a.k.a. SOL width) λ q wall = P edge 2πR 0 λf geo

The edge plasma determines the life expectancy of the reactor Power flux per unit of surface determined by decay length λ (a.k.a. SOL width) q wall = 6m P edge 2πR 0 λf geo 1mm (?) 100MW ~200MW. m 2 ~10 Surface of the sun q sun ~60MW. m 2 Technological limits: q max ~10 20MW. m 2 Two axes of study: understanding of λ: equilibrium between and transport => turbulence? dissipate P edge before it reaches the wall by harnessing atomic and molecular processes

The edge plasma sets performances H-mode = improved confinement mode (High-confinement) bifurcation beyond some heating power threshold Large additional gradients in the edge plasma => n e and T e larger in the whole core regime expected for reactor Very limited understanding related to drop of turbulence + sheared rotation velocity in the edge = transport barrier? recover mode H = challenge for turbulence models

How to model the edge plasma? Main difficulties: multi-scale physics τ turb ~μs τ E ~s l turb ~ρ l ~mm r ~m multi-physics system: balance between plasma transport atomic and molecular processes complex geometry (plasma shape + anisotropy) Transport models Closure assumptions real time hours weeks years Scaling laws Operational codes Transport codes Full fluid codes Kinetic codes Luck : edge plasma cold => collisional (ω col T 3 2) fluid approach possible

How to model the edge plasma? Main difficulties: multi-scale physics τ turb ~μs τ E ~s l turb ~ρ l ~mm r ~m multi-physics system: balance between plasma transport atomic and molecular processes complex geometry (plasma shape + anisotropy) Transport models Closure assumptions real time hours weeks years Scaling laws Operational codes Transport codes Full fluid codes Kinetic codes SolEdge2D TOKAM3X Luck : edge plasma cold => collisional (ω col T 3 2) fluid approach possible

SOLEDGE2D-EIRENE: a coarse description of transverse transport SOLEDGE2D = edge plasma mean-field transport solver Solves only the fluctuation averaged fluid equations n t +. nu = S n, u, n = n + n u = u + u n t +. nu + nu t,r = S n, u, + S n, u, t,r + n +. nu + nu t = S

SOLEDGE2D-EIRENE: a coarse description of transverse transport SOLEDGE2D = edge plasma mean-field transport solver Solves only the fluctuation averaged fluid equations n t +. nu = S n, u, 3D, mm m μs s n = n + n u = u + u n t +. nu + nu t,r = S n, u, + S n, u, t,r + n +. nu + nu t = S 2D, cm m steady-state gradient-diffusion closure to model turbulent transport nu t,r = D n

but very detailed plasma-wall/neutral interaction physics 4 coupled tools for a precise treatment of the plasma-wall interaction extremely rich description of plasma solid interaction and neutral particles physics Direct coupling via STYX interface

Multi-fluid equations Multi-fluid mean-field model for arbitrary # of species follows each degree of ionization for each element described (ex: for carbon, 6 ionized species) 3 fluid equations per ion species: particle balance, momentum balance, energy balance + 3 equations for electrons n α t + n αu α b + n α v α = S nα t m αn α u α + m α n α u α u α b + v α = p α q α n α φ + ν α u α b + m α n α ν α u α + R α + S uα t 3 2 n αt α + 1 2 m αn α u α 2 + with 5 2 n αt α + 1 2 m αn α u α 2 u α b + 3 2 n αt α + 1 2 m αn α u α 2 v α = κ α T α b + n α χ α T α q α u b + v α = D α n α n α n T v q n u φ + R u + Q + S Parallel advection Perpendicular advection Collisional closure Gradient-diffusion closure 1 2 ν α u α 2 b + 1 2 m αn α ν α u α

SOLEDGE2D-EIRENE numerics Time discretization: 1 st order (steady-state!) semi-implicit operator splitting to separate fast (implicit) and slow terms (explicit) Spatial discretization: conservative finite differences structured mesh of quadrangles Parallelization: OpenMP (sub-domains) for SOLEDGE2D plasma solver MPI for EIRENE Monte-Carlo kinetic neutrals code typical run = 2 nodes of 12 cores for 10 to 30 days (mésocentre AMU) room for improvements although EIRENE = very old code!

Arbitrary 2D geometry with fluxsurface aligned grid Geometry: arbitrary 2D plasma equilibrium complex geometries made possible by domain decomposition COMPASS geometry DEMO geometry (zoom on divertor)

A penalization method for plasma-wall boundary conditions Problem: geometry of the wall (BC) often complex and different from the plasma magnetic geometry Solution: adapt to plasma equations the penalization method developed for neutral fluids [L. Isoardi, JCP 2010] [A. Paredes, JCP 2014] Allows treatment of very complex wall geometries with little impact on the mesh of the plasma solver

SOLEDGE applied to main present machines and involved in the European road-map

Example of application of the SOLEDGE2D-EIRENE package Confrontation between ASDEX-U discharge and SOLEDGE2D simulation

Example of application of the SOLEDGE2D-EIRENE package Confrontation between ASDEX-U discharge and SOLEDGE2D simulation Reproduces very well the divertor heat and particle loads captures accurately most of the main processes at play

The main limit of the mean-field approach No theoretical drive for transverse transport coefficients should be considered as free parameters depend on the machine, the plasma and even the position in the plasma => huge number of degrees of freedom

Poloidal asymmetries as a textbook example A marking example: the ballooning of transverse transport poloidal asymmetries of plasma flows observed in all machines observation of turbulence only on the outboard side of the poloidal section interpreted as enhanced turbulent transport on the outboard side b

Mean-field codes unable to capture poloidal asymmetries self-consistently Mean-field codes fail at reproducing the basic feature of poloidal flow asymmetries must enhance artificially transport coefficients by x100 on the outboard side, and trends not reproduced Parallel Mach number Radial direction [Erents et al., PPCF (2000)] [Erents et al., PPCF (2004)]

The promises of 3D global turbulence codes Turbulent transport is fundamentally not a diffusive process 2D turbulence codes data cannot be fitted with a Fick law Necessity to solve for turbulent transport selfconsistently Direct Numerical Simulations [Garcia et al., JNM (2007)] But tackling issues such as poloidal asymmetries or divertor heat loads requires to go 3D => TOKAM3X code

What does it take to treat electrostatic turbulence? Adjunctions to the SOLEDGE2D model to treat turbulent transport self-consistently: 1) 2 perpendicular directions (lift the toroidal symmetry) must be 3D if geometrical features are to be adressed B 2) drift velocities for perpendicular advection i/e i/e B Φ i/e 2T u i/e = ue + u B + up i u E = B 2 u B = ± B 3) a current balance equation to solve for the electrostatic potential B B B 2 t W +. W Γ N b +. Wu i i E + Wu B =. N u B e u B +. J b + D W 2 W with W =. 1 B 2 Φ + ln N

TOKAM3X vs SOLEDGE2D in a nutshell: geometry and physics SOLEDGE2D 2D, realistic magnetic geometry, realistic wall geometry Transport model: mean-field diffusion, no drift Multi-fluid (impurities) (new) A&M physics by coupling with EIRENE TOKAM3X 3D (full torus), realistic magnetic geometry (axisymmetric) Transport model: fluid-drift turbulence Two-fluid (single ion species) A&M physics by coupling with EIRENE (new)

TOKAM3X vs SOLEDGE2D in a nutshell: numerics SOLEDGE2D Explicit / implicit operator splitting Domain decomposition TOKAM3X Explicit / implicit operator splitting Multi-level domain decomposition

TOKAM3X vs SOLEDGE2D in a nutshell: numerics SOLEDGE2D Explicit / implicit operator splitting Domain decomposition fdsfdsfdsfds OpenMP parallelization dsqdsqdsqdqs Shock-capturing (Roe-Marquina) schemes for advection terms Coupled to EIRENE kinetic neutrals Monte-Carlo code (MPI) via unified interface STYX PASTIX sparse LU solver for implicit operators Penalization method for boundary conditions typical run: 24 CPUs, 2 weeks TOKAM3X Explicit / implicit operator splitting Multi-level domain decomposition Hybrid MPI+openMP parallelization Shock-capturing (Roe-Marquina) schemes for advection terms Coupled to EIRENE kinetic neutrals Monte-Carlo code (MPI) via unified interface STYX MUMPS or PASTIX sparse LU solver for implicit operators Standard boundary conditions typical run: 256 CPUs, 1 month

TOKAM3X applied to poloidal asymmetries (1) Parameters: ρ = 1/256, D = 2. 10 2, η = 1. 10 5, q parabolic 3-6 φ N A priori observations from simulations output: change of turbulence characteristics across separatrix filamentary (field-aligned) structures poloidal asymmetry (ballooning)

Density (au) TOKAM3X applied to poloidal asymmetries (2) Poloidal inhomogeneities of turbulent transport recovered consistently (no free parameter) Turbulence properties outboard inboard and core SOL Turbulence amplitude outboard >> inboard 8 0.25 6 4 2 N N t,φ 0 0 2.5 5 Time (10 5 /ω ci ) 0.05

TOKAM3X applied to poloidal asymmetries (2) Poloidal inhomogeneities of turbulent transport recovered consistently (no free parameter) Turbulence properties outboard inboard and core SOL Turbulence amplitude outboard >> inboard Transverse particle flux strongly ballooned (70% in ±60 sector) Γ r E B t,φ 0.025 70% 0. -0.005

TOKAM3X catching up with SOLEDGE2D When they have reached full maturity: SOLEDGE2D TOKAM3X until recently, TOKAM3X lagging behind SOLEDG2D on several aspect but progressively catching up Recent TOKAM3X advances: inclusion of neutrals physics via coupling with EIRENE Electron density (au) Microturbulence filaments Ionization source (log 10 scale, au) Wall recycling of filaments Limiter recycling

TOKAM3X catching up with SOLEDGE2D When they have reached full maturity: SOLEDGE2D TOKAM3X until recently, TOKAM3X lagging behind SOLEDG2D on several aspect but progressively catching up Recent TOKAM3X advances: inclusion of neutrals physics via coupling with EIRENE turbulent simulations in realistic X-point geometry φ (au)

TOKAM3X catching up with SOLEDGE2D on neutrals physics When they have reached full maturity: SOLEDGE2D TOKAM3X until recently, TOKAM3X lagging behind SOLEDG2D on several aspect but progressively catching up Recent TOKAM3X advances: inclusion of neutrals physics via coupling with EIRENE turbulent simulations in realistic X-point geometry non-isothermal model implemented and currently being tested but very costly (x5!) T e (au)

TOKAM3X s numerical knot Current balance equation contains extremely fast anisotropic operator (due to parallel dynamics) implicit scheme chosen to avoid crippling too much the time step t. 1 B 2 φ +. 1 η φ = RHS η ~10 9 10 6 1 sparse linear system (~10-5 ) with very large size (1 to 250 millions DF) and very ill-conditionned (κ>10 6 ) numerical error critical for physics Current method = direct solver (MUMPS / PASTIX) main limitation to parallel scalability (>1000 CPU) and to up-scaling of cases need to explore other solutions but no luck for now (EoCoE project)

Bridging the gap between mean-field and turbulence approaches Are 3D global turbulence codes going to replace 2D mean-field codes? Solve the main limitation of 2D codes Still lag behind in term of description of the plasma-wall and plasma-neutral interaction but should catch up within 2-3 years BUT: computing costs 3D turbulence codes >> 2D mean-field codes For the near/mid term, meanfield codes are the only ones able to tackle ITER/DEMO size machines for routine usage and/or design purposes 2D mean-field range 3D turbulence range

Can the solution come from other branches of physics? CFD community facing same type of issue for aeronautic and automotive industries 2D mean-field codes RANS models Plethora of solutions developed and widely used for 50 years to avoid the cost of direct numerical simulation (DNS)

The k-epsilon model philosophy Example: the k-epsilon turbulence model 2 additional equations: for turbulence intensity k = 1 2 u2 and damping rate ε = u + T u 2 T t nk + nku = μ t u + u T t nε + nεu = c 1ε μ t u + u 2 + 2 + μ t k σ k μ t ε σ ε nε c 2ε n ε2 k n u u = μ t u + u T + 2 3 nki μ t = c μ n k2 ε New model parameters introduced and tuned from experiments Remain valid for most applications => universal parameters (vs free parameters)

First application to the ballooning issue Attempt to integrate similar turbulence model in SOLEDGE2D Instability and damping mechanisms adapted to the specific case of magnetized plasmas and tuned from outputs of TOKAM3X Very promising first results Spontaneous ballooning of transverse transport recovered Amplitude and trends of parallel flow asymmetries recovered

Summary (1) The edge plasma of tokamaks is at the centre of key issues for the success of ITER and fusion in general: Life-expectancy of the walls of the reactor Plasma performances (i.e., energetic and economical yield of the machine) Numerical simulations with predictive capabilities mandatory for design and operation purposes in sight of ITER and future IRFM and its partners have strong position in the field with the current development of 2 complementary numerical tools: SOLEDGE2D-EIRENE, a 2D mean-field code TOKAM3X(-EIRENE), a 3D global turbulence code 3D turbulence codes solve the major physical limitation of 2D mean-field codes e.g. no need for arbitrary twicking of transport coefficients to match experiments

Summary (2) Tremendous progress of 3D global turbulence codes means that 2D mean-field model might soon just be a subset of 3D tools However, 3D turbulent simulations are not THE solution Computational cost hinders their use for design or routine usage Unlikely to change in the next few years Currently exploring a third way inspired from the history of CFD: RANS turbulence models, e.g. k-epsilon Very promising first application to the case of poloidal asymmetries Need to be pushed further For all these approaches (2D / 3D / advanced turbulence closure), very strong need for algorithmic and numerical optimization

Thank you for your attention! The SOLEDGE2D and TOKAM3X teams: C. Baudoin, H. Bufferand, C. Colin, G. Ciraolo, R. Futtersack, D. Galassi, Ph. Ghendrih, G. Latu, N. Nace, F. Schwander, E. Serre, P. Tamain PAGE 52 CEA 10 AVRIL 2012 Commissariat à l énergie atomique et aux énergies alternatives Centre de Cadarache 13108 Saint Paul Lez Durance Cedex T. +33 (0)4 42 25 46 59 F. +33 (0)4 42 25 64 21 DSM IRFM SIPP Etablissement public à caractère industriel et commercial RCS Paris B 775 685 019

Back-up slides PAGE 53 CEA 10 AVRIL 2012 Commissariat à l énergie atomique et aux énergies alternatives Centre de Cadarache 13108 Saint Paul Lez Durance Cedex T. +33 (0)4 42 25 46 59 F. +33 (0)4 42 25 64 21 DSM IRFM SIPP Etablissement public à caractère industriel et commercial RCS Paris B 775 685 019

Fundamentally anisotropic physics // direction free directions constrained by B Characteristic length Characteristic time Dir. L = 2πqR~100m τ = L v th ~10 5 s Dir. L = r~1m τ E ~100ms 1s Main consequence: flux surfaces = iso-n e /T e / plasma equilibrium determined by 1D radial profiles

Inputs / Outputs of a mesh generation tool Minimal input: a ψ R, Z map wall line coordinates and material Output: HDF5 format For each subdomain: position of nodes of the grid in (R,Z) plane ψ for each mesh flux surf. B R, B Z, B φ at each node the number of neighbouring subdomains in each direction EIRENE mesh in EIRENE specific format

A new mesh generation tool for SOLEDGE A grid tool has been developed in Matlab (user-friendly GUI) to generate grids for SOLEDGE2D-EIRENE able to mesh arbitrary number of X-points and generate domain decomposition automatically (inputs: psi map & wall geometry) can be fully automated or partially hand driven Limits: generated mesh can be quite irregular (problem for TOKAM3X)

Careful verification process using MMS Both codes verified using Method of Manufactured Solutions (MMS) Ex: TOKAM3X

Computing ressources: growing needs not simple to satisfy Typical production run in 2015, SOLEDGE 24 CPUs, 1 to 2 weeks => 4 to 10 kh.cpu TOKAM3X 200 CPUs, 1 to 2 weeks => 30 to 50 kh.cpu likely to triple for SOLEDGE and quintuple for TOKAM3X in 2016 Growing allocation and consumption but 2016 will be difficult 1000000 800000 600000 400000 200000 0 Alloc SOLEDGE Conso SOLEDGE Alloc TOKAM3X Conso TOKAM3X Key contribution of mesocentre AMU Difficulty of juggling with too many different machines (porting,reporting )

A strongly collaborative work Both codes developed in strong collaboration with Aix-Marseille University A fast-growing team: IRFM: H. Bufferand (SOLEDGE RO), C. Baudoin*, G. Ciraolo, J. Denis*, R. Futtersack*, Ph. Ghendrih, R. Mao*, N. Nace*, P. Tamain (TOKAM3X RO), M. Valentinnuzzi* M2P2: D. Galassi*, G. Giorgiani*, F. Schwander, E. Serre, J.-A. Soler* PIIM: Y. Marandet (STYX-EIRENE RO) + 7 PhDs & 1 post-doc already finished

3D edge turbulence codes: an unprecedented effort BOUT++ Drift-fluid / MHD SOL+core Flexible geometry Gradient or fluxdriven ATTEMPT Drift-fluid EM, isothermal Limiter SOL+core Gradient-driven GEMR Drift-gyrofluid EM, 8 fields Limiter SOL+core TOKAM3X Drift-fluid ES, SOL+core Flexible geometry Flux-driven GBS Drift-fluid EM, T i =0 Limiter SOL Flux-driven PAGE 60

Kinetic codes on the global scale: the BIT1 code Kinetic codes required to investigate assumptions made in MHD & fluid models for kinetic effects: Sheath boundary conditions Parallel transport and fluid closures Impact of non-maxwellian distribution functions on atomic physics Most advanced code: BIT1 [Tskhakaya, Contrib. Plasma Phys. 2012] 1D plasma, 2D neutrals and quasi-2d impurity models (3V) full size SOL with finest resolution (up to 200 000 cells in poloidal direction) Massively parallel runs up to 8096 processors Rich atomic physics and PSI database PAGE 61

Example of application: heat flux limiters must be revisited for consistent parallel transport Heat-flux limiters required in fluid codes to avoid non-realistic divergence of Spitzer-Harm heat fluxes at low collisionality: Characterized by free parameter α q 1 q SH 1 nv T T 1 q SH T s 0.1?? Application of BIT1 to heat flux limiters: Limiters are strongly nonuniform along the field lines and sensitive to plasma conditions in the SOL Need for a more precise model 10-1 α e α i Perspectives of development: Enrich atomic data base 2D3V BIT2 code under test 10-2 [Tskhakaya, JNM 2009] 10 0 10 1 10 2 PAGE 62

Edge gyrokinetic codes already available but cost will remain prohibitive in the near future 2 main motivations: GK = only approach able to combine selfconsistently the fore-mentioned physics Suspected interdependence of core and edge physics [Holland C., PoP 2009] [XGC1] On-going effort towards extending GK codes to the edge and SOL: Main existing projects: TEMPEST, XGC But edge GK comes at a huge cost: Non-adiabatic electrons necessary for sheath and interchange Extremely greedy in numerical resources (N proc ~100.000 for ~months) Prohibitive for parameter scans [Chang, PoP 16 (2009)] PAGE 63

Summary: where are we standing w/r to the heat exhaust issue? The multiple facets of heat exhaust issues commend the use of a broad range of numerical tools Scaling laws Fluid diffusive transport MHD routine week(s) / 1-2 years Fluid turbulence kinetic gyrokinetic 0D 2D 3D 3D 2D 4D 5D 1 st ppleness PAGE 64

Summary: where are we standing w/r to the heat exhaust issue? The multiple facets of heat exhaust issues commend the use of a broad range of numerical tools 3D fluid turbulence codes: cross-field transport, L-H transition Limiter configuration: readily available (GEMR) or within reach Divertor configuration: within reach for 2 codes (BOUT++, TOKAM3X) 1D / 2D (3V) PIC codes: parallel transport and kinetic effects Full SOL simulations available => feedback for fluid modelling 2D under test but costly Scaling laws Fluid diffusive transport MHD routine week(s) / 1-2 years Fluid turbulence week(s) / 1-5 years kinetic week(s) / 1-5 years gyrokinetic 0D 2D 3D 3D 2D 4D 5D 1 st ppleness PAGE 66

Summary: where are we standing w/r to the heat exhaust issue? The multiple facets of heat exhaust issues commend the use of a broad range of numerical tools 3D fluid turbulence codes: cross-field transport, L-H transition Limiter configuration: readily available (GEMR) or within reach Divertor configuration: within reach for 2 codes (BOUT++, TOKAM3X) 1D / 2D (3V) PIC codes: parallel transport and kinetic effects Full SOL simulations available => feedback for fluid modelling 2D under test but costly 3D (2V) gyrokinetic codes: fully consistent transport On-going but extremely challenging for edge issues Scaling laws Fluid diffusive transport MHD routine week(s) / 1-2 years Fluid turbulence week(s) / 1-5 years kinetic week(s) / 1-5 years gyrokinetic 0D 2D 3D 3D 2D 4D 5D month(s) / >5 years 1 st ppleness PAGE 67

A multi-facets issue The unavoidable EIRENE 3D fluid turbulence codes 3D (2V) gyrokinetic codes Pedestal Separatrix 2D + EIRENE 1D/2D kinetic 3D fluid 5D gyrokinetic Perpendicular transport 1D (3V) kinetic codes Parallel transport Atomic physics ~ ~ PAGE 68

A multi-facets issue The unavoidable EIRENE 3D fluid turbulence codes 3D (2V) gyrokinetic codes Pedestal Separatrix Empirical codes + EIRENE 2D + EIRENE 1D/2D kinetic 3D fluid 5D gyrokinetic Perpendicular transport 1D (3V) kinetic codes Parallel transport Atomic physics ~ ~ PAGE 69

The limits of the diffusive description Local Fick s law does not fit with 2D turbulence results TOKAM2D SOL interchange turbulence code Perpendicular transport better represented as convective process but what is the thermodynamic force? => non local operator?? need for new model or self-consistent simulations 4 3 2 1 0 0 50 100 150 200 250 PAGE 71

Pre-study: How can we improve transport description in empirical codes? What influence has a sampled diffusion coefficient on the large-scale system? Solving

La turbulence de bord : un phénomène 2D? Expérience : structures turbulentes alignées sur B ~ 2D Hypothèse : transport plus rapide que => découplage / & k = 0 équations moyennées le long de B instabilité de courbure B Conservation matière + charge t D N + φ, N = σne Λ φ + S t ν φ + φ, φ = σ 1 e Λ φ g y lnn advection puits (paroi) courbure ( B) T 1 ~ Rayleigh Bénard (g = courbure de B) T 2 > T 1

y θ Simulations numériques avec le code TOKAM2D Simulation d une boîte 2D dans les lignes de champ ouvertes au-delà d un seuil en gradient de pression, transport turbulent 250 B 200 150 100 50 0 0 50 100 x r 150 200 250

Les modèles 2D permettent de retrouver l essentiel des propriétés expérimentales Comparaison qualitative avec l expérience : forte intermittence (45% flux par évènements > 1rms) Flux (a.u.) 0.6 0.4 0.2 0.0 0.0 PDF 4 0.2 0.4 0.6 0.8 1.0 time (ms) 2 0.01 4 2 0.001 4 0.0 1.0 2.0 flux (u.a.)

Les modèles 2D permettent de retrouver l essentiel des propriétés expérimentales Comparaison qualitative avec l expérience : forte intermittence (45% flux par évènements > 1rms) profils exponentiels => détermination largeur de décroissance λ [Sarazin, JNM (2003)]

Propagation balistique de structures cohérentes 250 200 150 100 50 0 0 50 100 150 B 200 250 a r s 100 4 3 80 Phénomène d avalanches : structures perturbent l équilibre => conditions de maintien et de propagation fondamentalement multiéchelles 2 1 0 160 180 200 (r-a) r s n (a.u.) 2.0 1.5 1.0 0.5 front @ t 1 front @ t 2 = t1 + 600 / W i <n> ~ 40.2 r s Dr~20r s Transport = balistique diffusif M ~0.04 0.0 140 160 180 200 (r - a) / r s n / n ~ 4.6 r s