F. Imbeaux et al – Numerical Models for Controlled Fusion 1 Association Euratom-CEA 20-24 April 2008 Integrated modelling of tokamak plasmas : the CRONOS

20-24 April 2008F. Imbeaux et al – Numerical Models for Controlled Fusion 1

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AssociationEuratom-CEA

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-13108 Saint Paul Lez Durance, France


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OutlineOutline

• 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


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Introduction : what is integrated modelling ?Introduction : what is

integrated modelling ?


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AssociationEuratom-CEAPhysics problems are strongly

coupled integrationPhysics problems are strongly

coupled integration

Edge plasmaRadiation, recycling,

…

Plasma facingcomponents

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

TransportParticles, heat,

current,momentum


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AssociationEuratom-CEAIntegrated Modelling : a

realistic numerical experiment 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


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AssociationEuratom-CEAApplications of Integrated

Modelling 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, …)


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Various levels of descriptionVarious 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 spaceDetailed Integrated Modellingexample : CRONOS, ASTRA, JETTO, TRANSP, CORSICA, TOPICS, …

mix 0D/1DMix of scalings + 1-D description of profiles, simplified modules, fast calculations (< 1mn CPU)Flight simulator : preliminar scenario studies, pre-shot discharge assessment, data consistency testsexample : METIS (included in the CRONOS platform)

F. Imbeaux et al – Numerical Models for Controlled Fusion 8

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(,,p||,p)(x,y,z,px,py ,py) (nT)

Local description,Not measurable

gyro-averageToroidal axisymmetry

Fluid moments +flux surfaceaveraging

Macroscopic quantities,In most cases mesurable

Semi-macroscopic quantities,In some cases measurable

Kinetic description Fluid description

Reducing the dimensionality of the problem

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

rRB

B

Ip


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AssociationEuratom-CEA1D½ simulators usually organised simulators usually organised

around core transport equationsaround core transport equations1D½ simulators usually organised simulators usually organised around core transport equationsaround 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 ofFlux-surface geometry

→ equilibrium solverSource terms

→ source modulesTransported flux

→ neoclassical module + turbulent transport (the most simplified module)

<A> = flux surface average


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The CRONOS Integrated Modelling suite

The CRONOS Integrated Modelling suite


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The CRONOS suiteThe 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


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CRONOS : modular structure around core transport

equations

CRONOS : modular structure around core transport

equations

Transport coefficients: NCLASS...

Fusion power,particle dynamics,

SPOT

NBI deposition & distrib. function,

SINBAD

ICRF wave propagation, resonating ion distrib.

function, PION

LH wave propag. & absorp., el. distrib. func.

DELPHINE+DKE

ECRF wave propagation, REMA

Equilibrium solverHELENA

MHD stability, MISHKA

Sawteeth, ELMs, recon-

nections

Simulationoutput

Model of plasmafor edge+SOL,(SOL-ONE)

Pellet injection,

GLAQUELC

Impurities, radiations, ITC

Inputparameters

Transport solver t t+t

(1.5D)Linear stability,gyrokinetics, KINEZERO

Self-consistentcoupling

Transport coefficients: NCLASS...

Fusion power,particle dynamics,

SPOT

NBI deposition & distrib. function,

SINBAD

ICRF wave propagation, resonating ion distrib.

function, PION

LH wave propag. & absorp., el. distrib. func.

DELPHINE+DKE

ECRF wave propagation, REMA

Equilibrium solverHELENA

MHD stability, MISHKA

Sawteeth, ELMs, recon-

nections

Simulationoutput

Model of plasmafor edge+SOL,(SOL-ONE)

Pellet injection,

GLAQUELC

Impurities, radiations, ITC

Inputparameters

Transport solver t t+t

(1.5D)Linear stability,gyrokinetics, KINEZERO

Self-consistentcoupling


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General CRONOS workflowGeneral CRONOS workflow

Input file

Main loop on time

Post-processing

Result file

Initialisation

Equilibriumconvergence

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


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AssociationEuratom-CEATransport solver is the

simplest part !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


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AssociationEuratom-CEACoupling between source

modulesCoupling 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

ZZ

1.51.5

11

0.50.5

00

-0.5-0.5

-1-1

2 3 4 R2 3 4 R

Element ofpower Pp

Ray-tracing elements at places of strong interaction with alpha particles

ZZ

RR

Passing orbitPassing orbit

Trapped orbitTrapped orbit

Initial positionInitial position


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Application : modelling of current profile in tokamaksApplication : modelling of

current profile in tokamaks


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Current profile in tokamaksCurrent 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

rRB

B

Ip


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AssociationEuratom-CEADetermination of current

profileDetermination 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


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AssociationEuratom-CEACurrent profile shaping

experimentCurrent 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 start @ t = 1.8 s

ECCD

Sawteeth start @ t = 2.3 s


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AssociationEuratom-CEADetermination of current

profileDetermination 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 s t = 2 s t = 3 s


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CRONOS includes synthetised diagnostics + experimental data visualisation tools

-0.10

-0.05

0

0.05

0.10

MSE angles [rad.]

3.43.23.02.8

Major radius [m]

MSE polarisation angles#53521, t=5.5s

measured simulated

R (m)

.. ⃝.. Measured-- Simulated

JET shot with ITB #53521, t = 5.5 s

0.1

0.05

0

-0.05

-0.1

[X. Litaudon et al., Nucl. Fusion 44 (2002)]

MS

E a

ng

les

(rad

)

Current diffusion simulation validated by synthetised diagnostic comparison



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Coupling transport equations and free-boundary equilibriumCoupling transport equations and free-boundary equilibrium


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Free-boundary equilibriumFree-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


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AssociationEuratom-CEAFree-boundary equilibrium :

application to ITER ramp-upFree-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]


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TransportTransport


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Transport solverTransport 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

qTnqTn ,,,,,n,T, (t) n,T, (t+dt)


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First guess : adiabatic p1(t)

1= (p1(t),…)

n=α (pn(t),…) +(1-α) n-1

Convergence ?

p(t), (t) S(t), …

p(t-dt), (t-dt) , S(t-dt)

Solve transport equation pn(t)finite difference scheme, implicit or Crank-Nicholson

YES

NO

neo, sources

can be adapted inside convergence loop (optional)

p pressure diffusivityS source

Convergence loop in transport solver (e.g. heat)

Convergence loop in transport solver (e.g. heat)

Lo

op

on

ti

me

Co

nve

rgen

ce

0 < < 1 : damping non-linearities for faster convergence


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Duration of a simulationDuration 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 tmodel should be ~ 1-10 s (in order to have the result within ~ 1-12 days)

iterationssolver

plasmasimulation n

t

ttt

model

10-4 s

1-103 s

1-5


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AssociationEuratom-CEASimplified transport models

are usedSimplified 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 tmodel ~ 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


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AssociationEuratom-CEATransport is the Achille’s heel

of Integrated ModellingTransport 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)


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Conclusions and perspectivesConclusions and perspectives


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AssociationEuratom-CEAIntegrated Modelling : what it

is and what it is notIntegrated 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 !!


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AssociationEuratom-CEAPerspectives for Integrated

ModellingPerspectives 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


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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 !


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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


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Final wordsFinal 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


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Comparison to experiment : synthetised diagnostic

Comparison to experiment : synthetised diagnostic


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AssociationEuratom-CEADirect determination of profiles

from measurement is ambiguousDirect 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


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AssociationEuratom-CEARelevant comparison to experiment

requires synthetised diagnostics 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)

Relevantcomparison

• 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

Ambiguouscomparison


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CRONOS includes synthetised diagnostics + experimental data visualisation tools

-0.10

-0.05

0

0.05

0.10

MSE angles [rad.]

3.43.23.02.8

Major radius [m]

MSE polarisation angles#53521, t=5.5s

measured simulated

R (m)

.. ⃝.. Measured-- Simulated

JET shot with ITB #53521, t = 5.5 s

0.1

0.05

0

-0.05

-0.1

[X. Litaudon et al., Nucl. Fusion 44 (2002)]

MS

E a

ng

les

(rad

)




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Quasi-static assumptionQuasi-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


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∂ Ψ/ ∂ t + DΨ = S(t), metric @ t-dt

Ψdiff(t)

Use converged metric for next time step of the transport equations

Ψeq(t) & {new_metric}

Yes

No

2D equilibrium calculates {new_metric}=Feq.[Ψdiff, Ptot, J(Ψdiff,{prev_metric})]

J(Ψeq,{new_metric}) =? J(Ψdiff,{prev_metric})

Convergence loop on metric to conserve j(Ψ)

Convergence loop on metric to conserve j(Ψ)

Current diffusion t-dt t

Deduce J(Ψdiff,{metric@t-dt})]

Deduce J(Ψeq,{new_metric)]


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Free-boundary equilibriumFree-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


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Non-inductive current driveNon-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

BT = 3.4 TIp = 0.5 MAPLHCD = 3 MWVloop = 0RTC


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Test of current drive modelsTest 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]


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Test of current drive modelsTest 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 qmin

MHD markers and internal inductance in excellent agreement for the simulation using FEB

[Imbeaux, Peysson PPCF 2005]

Documents

F. Imbeaux et al – Numerical Models for Controlled Fusion 1 Association Euratom-CEA 20-24 April 2008 Integrated modelling of tokamak plasmas : the CRONOS