Presentation on theme: "Recent progress in MHD simulations and open questions"— Presentation transcript:
1Recent progress in MHD simulations and open questions Valentin IgochineMax-Planck Institut für PlasmaphysikEURATOM-AssociationD Garching bei MünchenGermany
2Outline Introduction Linear and Non-linear simulations Recent results and open questionsSawtooth crashMagnetic reconnectionNeoclassical Tearing Modes (NTMs)Resistive Wall Modes (RWMs)Fast particle modes (TAEs, BAEs, EPMs,…)Edge Localized Modes (ELMs)DisruptionSummary
3Why we need computer simulations? Analytical derivation of the plasma behavior is possible only inin simplified geometrywith simplified profileswith simplified boundary conditionswith simplified plasma descriptionThe analytical approaches do not represent experimental situation and can not be used for prediction…Solution: We can do numerical simulations which takes into account realistic parameters and use analytical results to benchmark the codes.
4Different plasma descriptions Kinetic descriptionFluid descriptionVlasov equations,Fockker-Planck codesParticledescriptionHybriddescriptionMHDFluid parametersParticleparametersParticle and fluid parametersDistribution functionmore comp. powerless comp. power
5Different plasma descriptions Kinetic descriptionFluid descriptionVlasov equations,Fockker-Planck codesParticledescriptionHybriddescriptionMHDFluid parametersParticleparametersParticle and fluid parametersDistribution functionThis is typically sufficient for MHD instabilitiesmore comp. powerless comp. power
6Single fluid MHD equations Resistive MHDIdeal MHDIt is also possible to formulate two fluid MHD which will decouple electrons and ions dynamics (and this could be very important as we will see later!)
7Linear and non-linear evolution Mode amplitudeNon-linearLinear evolutionExponential growth of the instabilityLinearized MHD (eigenvalue problem, stable&unstable)timeNon-linear evolutionEquilibrium profile changes in time!Perturbations are not any more small!
8Our instabilities are mainly non-linear Non-linear instabilitiesSawtooth crashNTMsELMsFast particle modesDisruptionLinear instabilitiesRWMis very slow because of the wall(RWM is shown to be linear in RFPs. Is this true for tokamaks as well?)
10Sawtooth Kadomtsev model linearKadomtsev modelASDEXUpgradenonlinearq>1 after the reconnectionO-point becomes the new plasma centerThe model is in contradiction with experimental observationsPosition of (1,1) mode is the same before and after the crash![Igochine et.al. Phys. Plasmas 17 (2010)]
11Sawtooth modellingNonlinear MHD simulations (M3D code) show stochastisity.but .. „multiple time and space scales associated with the reconnection layer and growth time make this an extremely challenging computational problem. … and there still remain some resolution issues.”Small tokamak → small Lundquist number: S = 104 (big tokamaks 108)Lundquist number = (resistive diffusion time)/(Alfven transit time)[Breslau et.al. Phys. Plasmas 14, , 2007]Non-linear simulations of the sawtooth is very challenging task(even in a small tokamak).11
12Sawtooth modellingOhm‘s law, 2 fluid MHD…at least two fluid MHD with correct electron pressure description are necessary for reconnection region (fast crash time, smaller stochastic region)![Breslau et.al. Phys. Plasmas 14, , 2007]Stochastic region is too large,… much more then visible in the experiments (heat outflow is rather global instead of local as in the experiments). Magnetic reconnection is one of the key issues!12
13Magnetic reconnection changes topology Reconnection plays important role insawtooth crashseed island formation of NTMspenetration of the magnetic perturbations into the plasma (ELMs physics!)Reconnection allows to change magnetic topology and required for all resistive instabilities!Magnetic reconnection changes topology
14Magnetic reconnection redistributes energy Magnetic field linesplasmasling as a model Energy conversion from magnetic field into heating and acceleration of the plasma
15Structure of the reconnection region (MHD approx.) Ohm‘s lawAmper‘s law
16Structure of the reconnection region (MHD approx.) Ohm‘s lawAmper‘s lawConservationof mass
17Structure of the reconnection region (MHD approx.) Equation of motionThis is the maximal outflow velocity!
18One of the main questions: How one could explain fast reconnection? Reconnection rate for Sweet-Parker modelLundquistnumberIn our plasmas Lundquist numbers are very high:Fusion plasmasSpace plasmasExpected reconnection time for solar flaresMeasured reconnection timeSawtooth crash in JET:One of the main questions: How one could explain fast reconnection?
19Sawtooth crash time in Kadomtsev model Single fluid MHD calculations often show Kadomtsev reconnection processReconnectionregionq=1TCV:ASDEX:JET:O-point becomes the new plasma centerKadomtsev model = Sweet-Parker regime = single fluid MHD = SLOW!
20Plasma parameters for our experiments (MRX, tokamaks) The layer width is magnified by several orders of magnitude to make it visible!MHD is not enough. Single fluid picture is wrong for most plasmas of interest!
21Magnetic reconnection and different regions Electrons are not magnetizedIdeal MHDIons are not magnetizedSingle fluid MHD does not valid any more here!
22Ohm’s law (two fluid formulation). Compare different components with gradient of convective electric fieldSingle fluid MHDPriest and Forbes «Magnetic reconnection», 2000
23Magnetic reconnection and different regions Ions are not magnetized(ion diffusion region)Ideal MHDElectrons are not magnetized(electron diffusion region)
24Flows in reconnection region (computer simulations) [Pritchett Journal of Geophysical Reseach, 2001]
25Flows in reconnection region (computer simulations) electronIon diffusion regionElectron diffusion region[Pritchett Journal of Geophysical Reseach, 2001]
26Sweet-Parker (single fluid MHD) Is Sweet-Parker model always wrong?MRXYamada, PoP, 20062 fluid MHD simulationNormalized plasma resistivity(reconnection rate)Ion diffusion regionSweet-Parker layerSweet-Parker (single fluid MHD)High collisionalityLow collisionalitySweet-Parker is correct for collisional plasmas….Unfortunately, our plasmas are collisionless.
27Nonlinear simulations of (1,1) mode crashprecursorpostcursorTwo fluid MHDXTOR code
29Nonlinear simulations of (1,1) mode XTOR codeCompression of e-fluid parallel to the magnetic field↓Charge separationVariation of electric fieldIon polarization drift should be included to make fast crash!There are still missing parts regarding description of the reconnection region.
30Particle effects on (1,1) mode Motivation for DIII-D:or (1,1)Linear stability of the n=1 mode with and without energetic particle effectsusing the extended-MHD (XMHD) approach. (DIII-D case with NBI particles)Energetic particle densityplasma densityfast particles…but
31Particle effects on (1,1) mode Linear stability of the n=1 mode with and without energetic particle effectsusing the extended-MHD (XMHD) approach.(16%)MHD onlyMHD + particlesExperimentally we see n=1 mode here!MHD stable region becomes unstable if fast particles are considered
33Island width is a good measure of the reconnected flux Tearing ModeIsland width is a good measure of the reconnected fluxReconnection zonesZohm, MHDTearing mode: current driven, resistive instability.Neoclassical tearing mode: drive because of current deficiency in the island
34Triggers are the main drive for seed island formation! Reconnection in ASDEX Upgrade. Tearing mode.SXRFrom ECE in ASDEX Upgrade (#27257, I.Classen MATLAB script)Fast(2,1)Very fast(2,1)Slow(3,1)Island widthcoreMirnovSame βNtimesawteethTearing mode has different growth rates in different cases. Not only plasma profiles (Rutherford equation) determine the reconnection!Triggers are the main drive for seed island formation!
35Simulation of the triggered NTMs A slowly growing trigger drives a tearing modeA fast growing trigger drives a kink-like mode, which becomes a tearing mode later when the trigger’s growth slows down.the equilibrium plasma rotation frequency at q = 2 surfacelocal electron diamagnetic drift frequencyThe island widthobtained from the reduced MHD equations is muchsmaller than that obtained from two-fluid equations!Two fluid effects are important for prediction of the seed island width!
36Influence of static external perturbations Small static perturbations from the coils spin up the plasma(electron fluid at rest for penetration, there is a differential rotation between ions and electrons)Cylindrical (for current driven modes is sufficiently good aprox.)Two fluid, non-linear MHD code.Realistic Lundquist numbers are possible (very important! Not yet possible for toroidal cases)36
37Simulation of the (2,1) NTMs in JET XTOR, two fluid, non-linear, JET caseXTOR results and other approximationsRutherfod equation is not enough!Amplitude of n=1 magnetic perturbation from Mirnov coils localized on the HFS and LFSMissing bootstrap current in the island
42Linear simulations of fast particle modes Accurate description of ion drift orbits and the mode structure isused for calculating the wave-to-particle power transfer(results from CASTOR-K code)
43Results from linear calculations Eigenmode frequencies. These are robust for perturbative (TAE, EAE, Alfvén cascades etc.) and well measured in experiments. Usually, a good agreement is found between theory and experiment → Alfvén spectroscopy• Mode structure. Robust for perturbative modes, used not only in linear(MISHKA, CASTOR) but also in non-linear (e.g. HAGIS) modelling. Measured in experiment occasionally, a good agreement is found• Growth rates. Linear drive can be computed reliably but it may changequickly due to nonlinear effects• Damping rates. Except for electron collisional damping, the damping rates are exponentially sensitive to plasma parameters (ion Landau damping, radiative damping, continuum damping).
44Simulation of fast particle modes Linear Stability: basic mechanisms well understood, but lack of a comprehensive code which treats damping and drive non-perturbativelyNonlinear Physics: single mode saturation well understood, but lack of study for multiple mode dynamicsEffects of energetic particles on thermal plasmas: needs a lot of work
46Linear analysis of ELMs Stability boundaries can be identified with linear MHD codesImportant: Result is very sensitive to plasma boundary and number of the harmonicsTypical solution: reduced MHD approach (increased number of the mode) and accurate cut of the last close flux surface (99,99% of the total flux)JETType IType IIIL-modeSaarelma, PPCF, 2009
47Non-linear simulations of ELMs Non-linear MHD code JOREK solves the time evolution of the reduced MHD equations in general toroidal geometryDensitytimeHyusmans PPCF (2009)
48Non-linear simulations of ELMs Formation of density filaments expelled across the separatrix.Hyusmans PPCF (2009)123
49Non-linear simulations of ELMs Formation of density filaments expelled across the separatrix.Hyusmans PPCF (2009)All these results are in qualitative agreement with experiments, …but exact comparison for a particular case is necessary. One need a synthetic diagnostic comparison (the same approach as in MHD interpretation code but for edge region)
50Non-linear MHD simulations of pellets injected in the H-mode pedestal A strong pressure develops in the high density plasmoid, in this case the maximum pressure is aprox. 5 times the pressure on axis.There is a strong initial growth of the low-n modes followed by a growth phase of the higher-n modes ballooning like modes.The coupled toroidal harmonics lead to one single helical perturbation centred on the field line of the original pellet position.JOREKG T A Huysmans, PPCF 51 (2009)Simulations of pellets injected in the H-mode pedestal show that pellet perturbation can drive the plasma unstable to ballooning modes.50
51Simulation of ELMsQualitative agreement between non-linear simulations and experiments is foundQuantitative comparison should be doneInvestigation of pellets and resonant magnetic perturbations effects on the ELMs (the second is particular important, because of different results from different experiments)Penetration of the magnetic field into the plasma requires at least two fluid description (as discussed in the reconnection part)
53Resistive Wall Mode (RWM) Resistive wall mode is an external kink mode which interacts with the resistive wall.The mode will be stable in case of an perfectly conducting wall. Finite resistivity of the wall leads to mode growth.[T. Luce, PoP, 2011]DIII-DJET[I.T.Chapman, PPCF, 2009][M.Okabayashi, NF, 2009]RWM has global structure. This is important for “RWM ↔ plasma” interaction.
54Interaction of RWM with external perturbations Real vessel wallcurrents in the wall[F.Vilone, NF, 2010; E.Strumberger, PoP, 2008]Linear MHD code + finite element calculations for real wall.Coupling is done via boundary conditions.
55Application of both models for ITER [Liu, NF,2009, IAEA, 2010]black dots are stable RWMself-consistentperturbativeidealwallidealwallrotationrotationnowallnowallStable at low rotationRWM is stable at low plasma rotation up to without feedback due to mode resonance with the precession drifts of trapped particles.… but some important factors are missing (for example alpha particles are not taken into account).
56Possible variants of modeling Self-consistent modeling(MARS,…)Linear MHD + approximation for damping term(+) rotation influence on the mode eigenfunction(-) damping model is an approximationPerturbative approach(Hagis,…)Fixed linear MHD eigenfunctions as an input for a kinetic code(-) rotation does not influence on the mode eigenfunctions(+) damping is correctly described in kinetic code
57Possible variants of modeling Self-consistent modeling(MARS,…)Linear MHD + approximation for damping term(+) rotation influence on the mode eigenfunction(-) damping model is an approximationPerturbative approach(Hagis,…)Fixed linear MHD eigenfunctions as an input for a kinetic code(-) rotation does not influence on the mode eigenfunctions(+) damping is correctly described in kinetic codeWe need self-consistent kinetic modeling (probably very consuming in CPU power)Use this to check approximation!
59Simulation of the disruption Perturbed poloidal fluxTemperature
60Non-linear codesRSAE (D3D)TAE (NSTX)ELM (ITER)Sawteeth (NSTX)Non-linear MHD code is a powerful tool which could be applied to different problems (+ disruption + penetration of external field + particle effects + …)
61ITER priority Sawteeth NTMs RWMs Understanding of control ELMs Robust control,Good understanding, crash phase is not clearRobust control,Good understanding, seeding is not clearSawteethNTMsRWMsPlaned for the later operation phase.Influence of the particles is not clear.Robust control,poor understanding(especially for external perturb.)Understanding of controlELMsPhysical predictions are required.Preemptive ECCD control is possibleDisruptionmore urgentless urgent
62ConclusionsThere is a big progress during the last years in computer simulations of the MHD instabilitiesDepending on the situation and type of instabilitynon-linear evolutionparticle effectstwo-fluid effectscould be importantSelf-consistent non-linear simulation with particle effects will be the next step
64Non-linear calculations Up to now only hybrid simulations are possible (for example M3D code).Nonlinear evolution ofsingle n=2 mode in NSTXExperimentsimulationst=0.0t=336
65New model for rotation influence on the MHD modes (MARS-K) [Liu, PoP, 2008, Liu, IAEA, 2010]Kinetic effects are inside the pressurefull toroidal geometry in which the kinetic integrals are evaluated
66New model for rotation influence on the MHD modes (MARS-K) [Liu, PoP, 2008, Liu, IAEA, 2010]Kinetic effects are inside the pressurefull toroidal geometry in which the kinetic integrals are evaluated…but still some strong assumptions are made: neglects the perturbed electrostatic potential, zero banana width for trapped particles, no FLR corrections to the particle orbits. There is no guaranty that all important effects are inside.
67Linear simulations of fast particle modes Most often used are the CASTOR-K code (JET) and LIGKA code(IPP);• Equilibrium or equilibrium reconstruction codes for generating straight fieldline coordinate system: e.g. EFIT + HELENA in the case of CASTOR-K;• AE eigenfunctions are assumed to remain unchanged during nonlinearwave-particle interactionand are computed in MHD-type spectral approach;• Linear stability codes CASTOR-K or NOVA-K used fora) identifying the mode-particle resonances;b) computing energetic ion drive for AE;c) computing thermal plasma damping for AE;c) assessing stabilising effect of fast ions on sawtooth
68What to do in linear analysis? Comprehensive sensitivity study of instability boundaries to plasma parameters.• Combined effects of AE excitation by several energetic ion populations (alphas, NBI, ICRH-accelerated ions)• Mode suppression over a sufficiently broad radial interval to create a transport barrier for energetic ions. Either equilibrium effects (e.g. transport barrier at qmin found by Zonca et al.) or radial shift between different fast ion pressure gradients may be employed.…