F. Imbeaux et al – Numerical Models for Controlled Fusion 1 Association Euratom-CEA April 2008 Integrated modelling of tokamak plasmas : the CRONOS code, F. Imbeaux, J. F. Artaud, V. Basiuk, T. Aniel, J. Decker, G. Garcia, G. Giruzzi, P. Huynh, G. Huysmans, R. Masset, Y. Peysson, M. Schneider, G. Selig CEA, IRFM, F Saint Paul Lez Durance, France

F. Imbeaux et al – Numerical Models for Controlled Fusion 2 Association Euratom-CEA April 2008 Outline Integrated Modelling : definition and motivation An example : the CRONOS suite Organisation and workflow Examples of applications : current diffusion, comparison to experiment coupled transport + free-boundary equilibrium calculations Numerical aspects : convergence loops : transport solver Conclusions and perspectives Limitations Perspectives for the future

F. Imbeaux et al – Numerical Models for Controlled Fusion 3 Association Euratom-CEA April 2008 Introduction : what is integrated modelling ?

F. Imbeaux et al – Numerical Models for Controlled Fusion 4 Association Euratom-CEA April 2008 Physics problems are strongly coupled  integration Edge plasma Radiation, recycling, … Plasma facing components Heat load, erosion, … Sources Particles, heat, current, momentum Fusion reactions Equilibrium Core plasma Transport equations for particles, heat, current, momentum  -heating Numerical tokamak : must include all physics coupling … and also extend to technology : heating systems, magnetic field coils, diagnostics MHD limits‏ Transport Particles, heat, current, momentum

F. Imbeaux et al – Numerical Models for Controlled Fusion 5 Association Euratom-CEA April 2008 Integrated Modelling : a realistic numerical experiment Fully coupled physics : describe all interactions between various physical phenomena Fully self-consistent calculations Cost : reduced dimensionality (mix 1-D / 2-D), use often simplified models (e.g. turbulence, MHD, …) Realistic configuration of sub-systems (mainly : heating, magnetic field coils, diagnostics, wall, …) Time and space scales of a tokamak experiment : time : a few seconds to several minutes space : full plasma, including edge and subsystems

F. Imbeaux et al – Numerical Models for Controlled Fusion 6 Association Euratom-CEA April 2008 Applications of Integrated Modelling Analysis of existing experiments (interpretation) Data validation : multiple measurements put together to check their consistency (mainly through current diffusion simulations) Testing the accuracy of models (e.g. transport) Predicting future experiments  saving the cost of a real experiment !! Experiment preparation Extrapolation to future devices, scenario design Develop feedback control schemes Sub-system design (e.g. heating, diagnostic, …)

F. Imbeaux et al – Numerical Models for Controlled Fusion 7 Association Euratom-CEA April 2008 Various levels of description 0D No space dependence, purely based on 0D scalings Basic device design (major radius, plasma current, …) example : HELIOS 1D½ Core transport equations are fluid and 1-D in space, Sophisticated source / equilibrium modules can be kinetic, 2-D or 3- D in space Detailed Integrated Modelling example : CRONOS, ASTRA, JETTO, TRANSP, CORSICA, TOPICS, … mix 0D/1D Mix of scalings + 1-D description of profiles, simplified modules, fast calculations (< 1mn CPU)‏ Flight simulator : preliminar scenario studies, pre-shot discharge assessment, data consistency tests example : METIS (included in the CRONOS platform)

F. Imbeaux et al – Numerical Models for Controlled Fusion 8 Association Euratom-CEA ( , ,p ||,p  )‏ (x,y,z,p x,p y,p y )‏ (  n  T)‏ Local description, Not measurable gyro-average Toroidal axisymmetry Fluid moments + flux surface averaging Macroscopic quantities, In most cases mesurable Semi-macroscopic quantities, In some cases measurable Kinetic description Fluid description Reducing the dimensionality of the problem Some modules : heating, equilibrium Core transport equations Core plasma : 1-D fluid description for the transport equations Magnetic confinement : closed nested magnetic surfaces, labeled by a radial coordinate. Transport in the perpendicular direction. Thermal populations are Maxwellian, fluid quantities such as density and temperature are constant on a flux surface r R BB   BB IpIp

F. Imbeaux et al – Numerical Models for Controlled Fusion 9 Association Euratom-CEA April 2008 simulators usually organised around core transport equations 1D½ simulators usually organised around core transport equations Main loop : solve a set of radial continuity equations for poloidal flux(current), energy, particles, toroidal momentum Usual diffusive-convective form of the flux : Modular structure for self-consistent evaluation of Flux-surface geometry → equilibrium solver Source terms → source modules Transported flux → neoclassical module + turbulent transport (the most simplified module) = flux surface average

F. Imbeaux et al – Numerical Models for Controlled Fusion 10 Association Euratom-CEA April 2008 The CRONOS Integrated Modelling suite

F. Imbeaux et al – Numerical Models for Controlled Fusion 11 Association Euratom-CEA April 2008 The CRONOS suite Continuous development since 1999 at CEA-IRFM Essentially core transport Modern code design : high modularity, object-oriented data model, dynamic generation of graphical interfaces and of data management routines Strong link to experiment : versatile interpretative / predictive simulation platform Input : experimental database access, profile fitting Output : comparison to experimental signals, synthetised diagnostics Graphics : Matlab environment allows high flexibility in visualisation / data edition / interactive simulation / debugging Features sophisticated source and equilibrium modules, mostly developed at CEA-IRFM : on-site expertise, avoid « black-box » calculations

F. Imbeaux et al – Numerical Models for Controlled Fusion 12 Association Euratom-CEA April 2008 CRONOS : modular structure around core transport equations

F. Imbeaux et al – Numerical Models for Controlled Fusion 13 Association Euratom-CEA April 2008 General CRONOS workflow Input file Main loop on time Post- processing Result file Initialisation Equilibrium convergence Initial source module calls Determination of optimal time step (transport equation convergence) Plasma events Pellets MHD Within one time step Transport equation solver (finite differences, coupled equations, convergence on non-linearities of transport model) Equilibrium convergence Source module calls Transport coefficients Equilibrium Neoclassic Additional sources Edge Impurity content, radiation « External » modules

F. Imbeaux et al – Numerical Models for Controlled Fusion 14 Association Euratom-CEA April 2008 Transport solver is the simplest part ! Core transport equations are 1D, fluid (thermal plasma) Other modules can be much more sophisticated Equilibrium : 2D NBI, wave solvers : often 3D for the propagation Modelling of fast particles : Fokker-Planck solvers Monte-Carlo solvers  source terms to the thermal plasma (transport equations) 3D configuration of JET beam lines

F. Imbeaux et al – Numerical Models for Controlled Fusion 15 Association Euratom-CEA April 2008 Coupling between source modules Parasitic absorption of LH waves by fusion born alpha particles Ray-tracing + Fokker-Planck for LH propagation and absorption on electrons (current drive) Orbit following code for fusion-born alpha particles (orbit effects are important), Monte Carlo operators for i) collisions and ii) quasilinear interaction with RF waves Z R Element of power  P p Ray-tracing elements at places of strong interaction with alpha particles Z R Passing orbit Trapped orbit Initial position

F. Imbeaux et al – Numerical Models for Controlled Fusion 16 Association Euratom-CEA April 2008 Application : modelling of current profile in tokamaks

F. Imbeaux et al – Numerical Models for Controlled Fusion 17 Association Euratom-CEA April 2008 Current profile in tokamaks Current profile is an important quantity for confinement properties. Turbulence sensitive on current profile Safety factor : q-profile : magnetic field line topology, closely related to the current profile : number of toroidal turns / number of poloidal turns q = 2 surface/field line High order rational values of q : closed field lines; MHD activity can develop  characterisation of q-profile r R BB   BB IpIp

F. Imbeaux et al – Numerical Models for Controlled Fusion 18 Association Euratom-CEA April 2008 Determination of current profile Current profile not routinely measured  useful to calculate it from Integrated Simulations MHD markers are extremely useful to validate the simulation Neoclassical resistivity describes well current diffusion in tokamaks – plays the role of a diffusion coefficient Current diffusion simulations are used : To check data consistency (e.g. Zeff, Te in ohmic discharges, total energy content versus magnetics) To determine the current profile of an experiment To determine the « experimental » transport coefficients (all source terms calculated) To test current drive models against experiment

F. Imbeaux et al – Numerical Models for Controlled Fusion 19 Association Euratom-CEA April 2008 Current profile shaping experiment Current ramp at the beginning of the discharge, modified by the injection of a small amount of co-ECCD (  = 0.3) Appearance time of sawtooth (MHD linked to q = 1) delayed by 0.5 s Ip ramp, stops at t = 1 s (0.9 MA) Sawteeth t = 1.8 s ECCD Sawteeth t = 2.3 s

F. Imbeaux et al – Numerical Models for Controlled Fusion 20 Association Euratom-CEA April 2008 Determination of current profile Current diffusion + EC current source term calculated by CRONOS Calculated dynamics of current profile are in very good agreement with MHD markers (time of sawtooth onset + position of the q = 1 surface) t = 1 st = 2 st = 3 s

F. Imbeaux et al – Numerical Models for Controlled Fusion 21 Association Euratom-CEA April 2008 CRONOS includes synthetised diagnostics + experimental data visualisation tools R (m) ‏.. ⃝.. Measured -  - Simulated JET shot with ITB #53521, t = 5.5 s [X. Litaudon et al., Nucl. Fusion 44 (2002)] ‏ MSE angles (rad) ‏ Current diffusion simulation validated by synthetised diagnostic comparison

F. Imbeaux et al – Numerical Models for Controlled Fusion 22 Association Euratom-CEA April 2008 Coupling transport equations and free-boundary equilibrium

F. Imbeaux et al – Numerical Models for Controlled Fusion 23 Association Euratom-CEA April 2008 Free-boundary equilibrium Equilibrium and pressure / current transport are tightly coupled  specific convergence loop Fixed-boundary : plasma separatrix is prescribed, equilibrium solved only inside separatrix Free-boundary : plasma separatrix is calculated by the equilibrium which uses : Boundary conditions : poloidal field coil currents Constraint : j,p profiles in the plasma Application : realistic simulation of the whole plasma including its boundary (depends significantly on plasma profiles), integrated simulation of poloidal field coils circuits CS1 CS2U CS2L CS3L CS3U PF1 PF6 PF5 PF4 PF3 PF2 ITER

F. Imbeaux et al – Numerical Models for Controlled Fusion 24 Association Euratom-CEA April 2008 Free-boundary equilibrium : application to ITER ramp-up CRONOS-DINA simulator : scenario optimisation of the current ramp-up in ITER (scenario 2) Application of LHCD decreases internal inductance (reduces vertical instabilities) and saves flux Plasma boundary controlled by prescribing « gaps ». Feedback control on the poloidal coil voltages. Coil currents calculated, remain within operational limits. X-point formation and shape evolution strongly depends on the plasma profiles [S.H. Kim et al, accepted PPCF 2009]

F. Imbeaux et al – Numerical Models for Controlled Fusion 25 Association Euratom-CEA April 2008 Transport

F. Imbeaux et al – Numerical Models for Controlled Fusion 26 Association Euratom-CEA April 2008 Transport solver Though written as diffusive-convective, transport is in fact much more complex : transport coefficients feature parametric dependencies on the transported quantities and their gradients Coupling between transport equations Anomalous transport models (turbulence) usually quite sensitive to the transported quantities and their gradients (non-linear dependencies, including threshold effects) Transport equation using n,T,  (t) n,T,  (t+dt)

F. Imbeaux et al – Numerical Models for Controlled Fusion 27 Association Euratom-CEA April 2008 First guess : adiabatic p 1 (t)‏  1 =  (p 1 (t),…)  n =α  (p n (t),…) +(1-α)  n-1 Convergence ? p(t),  (t)  S(t), … p(t-dt),  (t-dt), S(t-dt)‏ Solve transport equation  p n (t)‏ finite difference scheme, implicit or Crank-Nicholson‏ YES NO  neo, sources can be adapted inside convergence loop (optional) p  pressure   diffusivity S  source Convergence loop in transport solver (e.g. heat) Loop on time Convergence 0 <  < 1 : damping non-linearities for faster convergence

F. Imbeaux et al – Numerical Models for Controlled Fusion 28 Association Euratom-CEA April 2008 Duration of a simulation Sophisticated source modules can be time-consuming. However, source evolve usually much slower than transported quantities  source modules not called at each time step (user’s choice) ! For reasonable calling frequency of source models, the main contributor to the computation time is the anomalous transport model (turbulence), since it is called in the innermost loop : main time loop + convergence on non-linearities loop So for a 10 s plasma : Acceptable t model should be ~ 1-10 s (in order to have the result within ~ 1-12 days) s s 1-5

F. Imbeaux et al – Numerical Models for Controlled Fusion 29 Association Euratom-CEA April 2008 Simplified transport models are used Theoretical description of plasma turbulence : Vlasov + Maxwell equations Full non-linear treatment of these equations (either fluid or gyrokinetic formulation) is orders of magnitude beyond the requested computation time t model ~ 1-10 s Need for an intermediate degree of complexity : quasi- linear approximation : 1990’s models : Weiland model, GLF23 New generation : TGLF, Qualikiz  more sophisticated, require some parallelisation

F. Imbeaux et al – Numerical Models for Controlled Fusion 30 Association Euratom-CEA April 2008 Transport is the Achille’s heel of Integrated Modelling Simplified turbulence models are too simple : lack of reliability for the prediction of core transport H-mode pedestal still poorly understood from first principles All phenomena coupled in a simulation  fully predictive transport modelling is highly uncertain [Imbeaux et al, PPCF 2005] (GLF23 : prediction inside  < 0.8 only)

F. Imbeaux et al – Numerical Models for Controlled Fusion 31 Association Euratom-CEA April 2008 Conclusions and perspectives

F. Imbeaux et al – Numerical Models for Controlled Fusion 32 Association Euratom-CEA April 2008 Integrated Modelling : what it is and what it is not Integrated Modelling is a sophisticated way of coupling many physics modules, mandatory since physics phenomena are coupled Ideal framework for working : Closest to realistic experimental conditions With a guarantee of consistency of input/output between physics modules Sophisticated coupling gives an impression of global predictive capability Several individual models are far from 100% reliability (in particular transport models, but not only)  be aware of the limitations !!

F. Imbeaux et al – Numerical Models for Controlled Fusion 33 Association Euratom-CEA April 2008 Perspectives for Integrated Modelling Progress in computer performances  more and more sophisticated modules Main time loop cannot be parallelised, but sophisticated individual modules can (already the case in CRONOS) : Increase link with High Performance Computing Present weaknesses in individual models must be overcome By closer interaction with First Principles calculations and Theory Extensive model testing against existing experiments Present Integrated Modelling codes are built around core transport equations Build a fully flexible Integrated Modelling platform

F. Imbeaux et al – Numerical Models for Controlled Fusion 34 Association Euratom-CEA April 2008 Since 2004, the ITM-TF aims at defining and setting up the ideal Integrated Modelling platform Unique data format, object-oriented, logically structured to represent physics elementary problems Fully flexible and modular workflow, connected to HPC  could be used even as a framework for First Principles calculations Transparent use of multiple programming languages Transparent data access and unique representation of any Tokamak Synthetised diagnostics, technological modelling  ideal tool for model testing and improving our understanding !

F. Imbeaux et al – Numerical Models for Controlled Fusion 35 Association Euratom-CEA April 2008 This is just the beginning of this endeavour Many obstacles, large ressources needed in computer science for the development of the platform Graphical workflow design : prototype of the European Transport Solver

F. Imbeaux et al – Numerical Models for Controlled Fusion 36 Association Euratom-CEA April 2008 Final words CRONOS : a sophisticated and mature plasma core Integrated Solver Strong link to experiment Versatile interpretative / predictive simulations Ongoing developments : free-boundary equilibria, quasi- linear transport models, impurity transport, basic plasma edge modules … Used for interpretation of existing experiments and ITER scenario design CRONOS development team strongly involved in the preparation of the Next Step : support the ITM-TF with 10 years of CRONOS developments and experience

F. Imbeaux et al – Numerical Models for Controlled Fusion 37 Association Euratom-CEA April 2008 Comparison to experiment : synthetised diagnostic

F. Imbeaux et al – Numerical Models for Controlled Fusion 38 Association Euratom-CEA April 2008 Direct determination of profiles from measurement is ambiguous Physicists like to think in terms of radial profiles of fluid quantities n, T, j, … « Profile measurement » in tokamaks is Utopia Non-local measurements : line-integrated diagnostics (interferometry, polarimetry, radiation measurements), global measurements with no spatial resolution (neutron diagnostics)  conversion to profiles not unique (Abel inversion, dependence on multiple quantities …) Local measurements : all require mapping on an equilibrium. Some are localised by magnetic field (ECE, reflectometry, …)  even more dependent on equilibrium assumptions

F. Imbeaux et al – Numerical Models for Controlled Fusion 39 Association Euratom-CEA April 2008 Relevant comparison to experiment requires synthetised diagnostics Diagnostic #1 Diagnostic #3‏ Diagnostic model (CRONOS post-processing) Set of consistent profiles + equilibrium (result of a simulation) Relevant comparison Simulation codes provide profiles. All quantities are known and self- consistent  Much more valid comparison to experiment is obtained by recalculating the diagnostic measurements from a set of consistent profiles and equilibrium  Instead of trying to obtain directly those profiles from the measurement. Application : data consistency, model testing, diagnostic design, feedback control Diagnostic #2 Ambiguous comparison

F. Imbeaux et al – Numerical Models for Controlled Fusion 40 Association Euratom-CEA April 2008 CRONOS includes synthetised diagnostics + experimental data visualisation tools R (m) ‏.. ⃝.. Measured -  - Simulated JET shot with ITB #53521, t = 5.5 s [X. Litaudon et al., Nucl. Fusion 44 (2002)] ‏ MSE angles (rad) ‏ Current diffusion simulation validated by synthetised diagnostic comparison

F. Imbeaux et al – Numerical Models for Controlled Fusion 41 Association Euratom-CEA April 2008 Quasi-static assumption Equilibrium : force balance between kinetic and magnetic pressure  magnetic field topology (magnetic surfaces) Plasma equilibrium is established on much faster time scales (Alfven, 10-6 s) than transport time scales (> 0.1 s) Quasi-static assumption : Transport equations are evolved at constant magnetic surface topology Equilibrium is recalculated when a significant change of the plasma profiles (pressure and current density) has occurred  new topology for the subsequent evolution of plasma profiles Difficulty : p(  ),j(  )  topology. j(  ) must be conserved during the equilibrium recalculation, but depends on the topology  not guaranteed by a single pass in the equilibrium module  convergence loop on the topology

F. Imbeaux et al – Numerical Models for Controlled Fusion 42 Association Euratom-CEA April 2008 ∂ Ψ/ ∂ t + DΨ = S(t), t-dt Ψ diff (t)‏ Use converged metric for next time step of the transport equations Ψ eq (t) & {new_metric} Yes No 2D equilibrium calculates {new_metric}=F eq. [Ψ diff, P tot, J(Ψ diff,{prev_metric})] J(Ψ eq,{new_metric}) =? J(Ψ diff,{prev_metric})‏ Convergence loop on metric to conserve j(Ψ) Current diffusion t-dt  t Deduce J(Ψ Deduce J(Ψ eq,{new_metric)]

F. Imbeaux et al – Numerical Models for Controlled Fusion 43 Association Euratom-CEA April 2008 Free-boundary equilibrium Key issue for coupling to equilibrium : current diffusion and topology must remain consistent No guarantee that the poloidal flux is the same at separatrix between i) the transport equation and ii) the free- boundary solver  specific convergence loop needed Current diffusion itself (and the other transport equations) are recalculated with the new topology until convergence on  (  ) between two iterations on the topology On-going work in collaboration with Université de Nice

F. Imbeaux et al – Numerical Models for Controlled Fusion 44 Association Euratom-CEA April 2008 Non-inductive current drive Tokamaks rely on toroidal current for confinement Driven by inductive means  current diffusion Steady-state operation requires to drive current by non- inductive means Tore Supra : 6min 20 s of plasmas sustained fully non- inductively, 85 % LHCD and 15 % bootstrap TS#32299 B T = 3.4 T I p = 0.5 MA P LHCD = 3 MW V loop = 0 RTC

F. Imbeaux et al – Numerical Models for Controlled Fusion 45 Association Euratom-CEA April 2008 Test of current drive models LHCD : Delphine RT/FP solver Calculated every 0.1 s Coupled (indirectly) to antenna solver SWAN to use realistic injected wave spectrum Modelling of fully non-inductive discharge is challenging Self-consistent and sensitive loop : Non-inductive source  current profile A typical integrated modelling problem [Imbeaux, Peysson PPCF 2005]

F. Imbeaux et al – Numerical Models for Controlled Fusion 46 Association Euratom-CEA April 2008 Test of current drive models Integrated current diffusion simulation with comparison to measurements show the limitations of the models RT/FP simulation LH driven current density assumed homothetic to Fast Electron Bremsstrahlung measurements MHD marker for q min MHD markers and internal inductance in excellent agreement for the simulation using FEB [Imbeaux, Peysson PPCF 2005]