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Hefei, China/ August 2012 / 7th LectureValentin Igochine 1 Recent progress in MHD simulations and open questions Valentin Igochine Max-Planck Institut für Plasmaphysik EURATOM-Association D Garching bei München Germany

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 2 Outline Introduction Linear and Non-linear simulations Recent results and open questions Sawtooth crash Magnetic reconnection Neoclassical Tearing Modes (NTMs) Resistive Wall Modes (RWMs) Fast particle modes (TAEs, BAEs, EPMs,…) Edge Localized Modes (ELMs) Disruption Summary

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 3 Why we need computer simulations? Analytical derivation of the plasma behavior is possible only in in simplified geometry with simplified profiles with simplified boundary conditions with simplified plasma description The 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.

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 4 Different plasma descriptions Kinetic descriptionFluid description Vlasov equations, Fockker-Planck codes Distribution function MHD Particle description Hybrid description Particle parameters Particle and fluid parameters Fluid parameters less comp. powermore comp. power

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 5 Different plasma descriptions Kinetic descriptionFluid description Vlasov equations, Fockker-Planck codes Distribution function MHD Particle description Hybrid description Particle parameters Particle and fluid parameters Fluid parameters less comp. powermore comp. power This is typically sufficient for MHD instabilities

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 6 Single fluid MHD equations Resistive MHD Ideal MHD It 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!)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 7 Linear and non-linear evolution Mode amplitude time Non-linear linear Linear evolution Exponential growth of the instability Linearized MHD (eigenvalue problem, stable&unstable) Non-linear evolution Equilibrium profile changes in time! Perturbations are not any more small!

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 8 Our instabilities are mainly non-linear Linear instabilities RWM is very slow because of the wall (RWM is shown to be linear in RFPs. Is this true for tokamaks as well?) Non-linear instabilities Sawtooth crash NTMs ELMs Fast particle modes Disruption

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 9 Sawtooth crash

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 10 Sawtooth linear nonlinear ASDEX Upgrade O-point becomes the new plasma center q>1 after the reconnection Kadomtsev model [Igochine et.al. Phys. Plasmas 17 (2010)] Position of (1,1) mode is the same before and after the crash! The model is in contradiction with experimental observations

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 11 Sawtooth modelling Nonlinear 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.” [Breslau et.al. Phys. Plasmas 14, , 2007] Small tokamak → small Lundquist number: S = 10 4 (big tokamaks 10 8 ) Lundquist number = (resistive diffusion time)/(Alfven transit time) Non-linear simulations of the sawtooth is very challenging task (even in a small tokamak).

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 12 Sawtooth modelling [Breslau et.al. Phys. Plasmas 14, , 2007] …at least two fluid MHD with correct electron pressure description are necessary for reconnection region (fast crash time, smaller stochastic region)! 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! Ohm‘s law, 2 fluid MHD

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 13 Magnetic reconnection changes topology Reconnection plays important role in sawtooth crash seed island formation of NTMs penetration 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

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 14 Magnetic field lines plasma Energy conversion from magnetic field into heating and acceleration of the plasma sling as a model Magnetic reconnection redistributes energy

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 15 Structure of the reconnection region (MHD approx.) Ohm‘s law Amper‘s law

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 16 Structure of the reconnection region (MHD approx.) Ohm‘s law Amper‘s law Conservation of mass

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 17 Structure of the reconnection region (MHD approx.) Equation of motion This is the maximal outflow velocity!

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 18 Reconnection rate for Sweet-Parker model In our plasmas Lundquist numbers are very high: Fusion plasmas Space plasmas Lundquist number One of the main questions: How one could explain fast reconnection? Expected reconnection time for solar flares Measured reconnection time Sawtooth crash in JET:

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 19 Single fluid MHD calculations often show Kadomtsev reconnection process TCV: ASDEX: JET: q=1 O-point becomes the new plasma center Sawtooth crash time in Kadomtsev model Kadomtsev model = Sweet-Parker regime = single fluid MHD = SLOW! Reconnection region

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 20 Plasma 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!

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 21 Magnetic reconnection and different regions Ideal MHD Ions are not magnetized Electrons are not magnetized Single fluid MHD does not valid any more here!

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 22 Ohm’s law (two fluid formulation). Priest and Forbes «Magnetic reconnection», 2000 Compare different components with gradient of convective electric field Single fluid MHD

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 23 Magnetic reconnection and different regions Ideal MHD Ions are not magnetized (ion diffusion region) Electrons are not magnetized (electron diffusion region)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 24 Flows in reconnection region (computer simulations) [Pritchett Journal of Geophysical Reseach, 2001]

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 25 Flows in reconnection region (computer simulations) ion electron [Pritchett Journal of Geophysical Reseach, 2001] Ion diffusion region Electron diffusion region

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 26 Sweet-Parker (single fluid MHD) High collisionalityLow collisionality 2 fluid MHD simulation Is Sweet-Parker model always wrong? MRX Normalized plasma resistivity (reconnection rate) Sweet-Parker is correct for collisional plasmas….Unfortunately, our plasmas are collisionless. Ion diffusion region Sweet-Parker layer Yamada, PoP, 2006

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 27 Nonlinear simulations of (1,1) mode precursor crash postcursor Two fluid MHD XTOR code

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 28 Nonlinear simulations of (1,1) mode precursor crash postcursor Two fluid MHD XTOR code Large stochastic region

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 29 Nonlinear simulations of (1,1) mode Compression of e-fluid parallel to the magnetic field ↓ Charge separation ↓ Variation of electric field ↓ Ion polarization drift should be included to make fast crash! There are still missing parts regarding description of the reconnection region. XTOR code

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 30 Particle effects on (1,1) mode Linear stability of the n=1 mode with and without energetic particle effects using the extended-MHD (XMHD) approach. (DIII-D case with NBI particles) Energetic particle densityplasma density …but fast particles Motivation for DIII-D: or (1,1)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 31 Particle effects on (1,1) mode Linear stability of the n=1 mode with and without energetic particle effects using the extended-MHD (XMHD) approach. MHD stable region becomes unstable if fast particles are considered MHD only MHD + particles Experimentally we see n=1 mode here! (16%)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 32 Neoclassical Tearing Mode (NTM)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 33 Reconnection zones Tearing Mode Zohm, MHD Tearing mode: current driven, resistive instability. Neoclassical tearing mode: drive because of current deficiency in the island Island width is a good measure of the reconnected flux

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 34 Fast (2,1) Slow (3,1) Very fast (2,1) Tearing 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! Island width timesawteeth Mirnov SXR core Reconnection in ASDEX Upgrade. Tearing mode. From ECE in ASDEX Upgrade (#27257, I.Classen MATLAB script) Same β N

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 35 Simulation of the triggered NTMs A slowly growing trigger drives a tearing mode A fast growing trigger drives a kink-like mode, which becomes a tearing mode later when the trigger’s growth slows down. The island width obtained from the reduced MHD equations is much smaller than that obtained from two-fluid equations! Two fluid effects are important for prediction of the seed island width! local electron diamagnetic drift frequency the equilibrium plasma rotation frequency at q = 2 surface

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 36 Influence 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)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 37 Simulation of the (2,1) NTMs in JET Missing bootstrap current in the island Amplitude of n=1 magnetic perturbation from Mirnov coils localized on the HFS and LFS XTOR results and other approximations Rutherfod equation is not enough! XTOR, two fluid, non-linear, JET case

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 38 Interaction of several modes (FIR-NTM)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 39 Simulation of FIR-NTMs Experimental reconstruction For ASDEX Upgrade Predictions for ITER, Non-linear MHD code XTOR

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 40 Fast particle modes (TAEs, …)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 41 Linear simulations of fast particle modes

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 42 Linear simulations of fast particle modes Accurate description of ion drift orbits and the mode structure is used for calculating the wave-to-particle power transfer (results from CASTOR-K code)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 43 Results 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 change quickly 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).

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 44 Simulation of fast particle modes Linear Stability: basic mechanisms well understood, but lack of a comprehensive code which treats damping and drive non-perturbatively Nonlinear Physics: single mode saturation well understood, but lack of study for multiple mode dynamics Effects of energetic particles on thermal plasmas: needs a lot of work

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 45 Edge Localized Modes

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 46 Linear analysis of ELMs JET Type I Type III L-mode Saarelma, PPCF, 2009 Stability boundaries can be identified with linear MHD codes Important: Result is very sensitive to plasma boundary and number of the harmonics Typical solution: reduced MHD approach (increased number of the mode) and accurate cut of the last close flux surface (99,99% of the total flux)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 47 Hyusmans PPCF (2009) time Non-linear MHD code JOREK solves the time evolution of the reduced MHD equations in general toroidal geometry Density Non-linear simulations of ELMs

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 48 Hyusmans PPCF (2009) Formation of density filaments expelled across the separatrix. Non-linear simulations of ELMs

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 49 Hyusmans PPCF (2009) Formation of density filaments expelled across the separatrix. Non-linear simulations of ELMs 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)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 50 Non-linear MHD simulations of pellets injected in the H-mode pedestal Simulations of pellets injected in the H-mode pedestal show that pellet perturbation can drive the plasma unstable to ballooning modes. JOREK 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. G T A Huysmans, PPCF 51 (2009)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 51 Simulation of ELMs Qualitative agreement between non-linear simulations and experiments is found Quantitative comparison should be done Investigation 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)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 52 Resistive Wall Mode (RWM)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 53 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. Resistive Wall Mode (RWM) [M.Okabayashi, NF, 2009] [T. Luce, PoP, 2011] RWM has global structure. This is important for “RWM ↔ plasma” interaction. DIII-D [I.T.Chapman, PPCF, 2009] JET

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 54 Interaction of RWM with external perturbations Linear MHD code + finite element calculations for real wall. Coupling is done via boundary conditions. Real vessel wall [F.Vilone, NF, 2010; E.Strumberger, PoP, 2008] currents in the wall

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 55 Application of both models for ITER RWM 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). [Liu, NF,2009, IAEA, 2010] perturbative self-consistent ideal wall ideal wall no wall no wall rotation Stable at low rotation black dots are stable RWM

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 56 Possible variants of modeling Self-consistent modeling (MARS,…) Linear MHD + approximation for damping term (+) rotation influence on the mode eigenfunction (-) damping model is an approximation Perturbative 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

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 57 Possible variants of modeling Self-consistent modeling (MARS,…) Linear MHD + approximation for damping term (+) rotation influence on the mode eigenfunction (-) damping model is an approximation Perturbative 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 We need self-consistent kinetic modeling (probably very consuming in CPU power) Use this to check approximation!

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 58 Disruption

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 59 Simulation of the disruption Perturbed poloidal flux Temperature

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 60 Non-linear codes Sawteeth (NSTX) RSAE (D3D) TAE (NSTX) ELM (ITER) Non-linear MHD code is a powerful tool which could be applied to different problems (+ disruption + penetration of external field + particle effects + …)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 61 ITER priority more urgent less urgent Disruption ELMs NTMs RWMs Sawteeth Understanding of control Planed for the later operation phase. Influence of the particles is not clear. Robust control, Good understanding, crash phase is not clear Robust control, Good understanding, seeding is not clear Robust control, poor understanding (especially for external perturb.) Physical predictions are required. Preemptive ECCD control is possible

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 62 Conclusions There is a big progress during the last years in computer simulations of the MHD instabilities Depending on the situation and type of instability non-linear evolution particle effects two-fluid effects could be important Self-consistent non-linear simulation with particle effects will be the next step

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 63

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 64 Non-linear calculations Up to now only hybrid simulations are possible (for example M3D code). Experiment simulations t=0.0t=336 Nonlinear evolution of single n=2 mode in NSTX

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 65 New model for rotation influence on the MHD modes (MARS-K) full toroidal geometry in which the kinetic integrals are evaluated Kinetic effects are inside the pressure [Liu, PoP, 2008, Liu, IAEA, 2010]

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 66 full 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. Kinetic effects are inside the pressure [Liu, PoP, 2008, Liu, IAEA, 2010] New model for rotation influence on the MHD modes (MARS-K)

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 67 Linear 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 field line coordinate system: e.g. EFIT + HELENA in the case of CASTOR-K; AE eigenfunctions are assumed to remain unchanged during nonlinear wave-particle interactionand are computed in MHD-type spectral approach; Linear stability codes CASTOR-K or NOVA-K used for a) 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

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Hefei, China/ August 2012 / 7th LectureValentin Igochine 68 What 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. …

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