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**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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**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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**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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**Different plasma descriptions**

Kinetic description Fluid description Vlasov equations, Fockker-Planck codes Particle description Hybrid description MHD Fluid parameters Particle parameters Particle and fluid parameters Distribution function more comp. power less comp. power

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**Different plasma descriptions**

Kinetic description Fluid description Vlasov equations, Fockker-Planck codes Particle description Hybrid description MHD Fluid parameters Particle parameters Particle and fluid parameters Distribution function This is typically sufficient for MHD instabilities more comp. power less comp. power

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**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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**Linear and non-linear evolution**

Mode amplitude Non-linear Linear evolution Exponential growth of the instability Linearized MHD (eigenvalue problem, stable&unstable) time Non-linear evolution Equilibrium profile changes in time! Perturbations are not any more small!

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**Our instabilities are mainly non-linear**

Non-linear instabilities Sawtooth crash NTMs ELMs Fast particle modes Disruption 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?)

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Sawtooth crash

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**Sawtooth Kadomtsev model**

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

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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.” 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

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Sawtooth modelling Ohm‘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

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**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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**Magnetic reconnection redistributes energy**

Magnetic field lines plasma sling as a model Energy conversion from magnetic field into heating and acceleration of the plasma

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**Structure of the reconnection region (MHD approx.)**

Ohm‘s law Amper‘s law

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**Structure of the reconnection region (MHD approx.)**

Ohm‘s law Amper‘s law Conservation of mass

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**Structure of the reconnection region (MHD approx.)**

Equation of motion This is the maximal outflow velocity!

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**One of the main questions: How one could explain fast reconnection?**

Reconnection rate for Sweet-Parker model Lundquist number In our plasmas Lundquist numbers are very high: Fusion plasmas Space plasmas Expected reconnection time for solar flares Measured reconnection time Sawtooth crash in JET: One of the main questions: How one could explain fast reconnection?

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**Sawtooth crash time in Kadomtsev model**

Single fluid MHD calculations often show Kadomtsev reconnection process Reconnection region q=1 TCV: ASDEX: JET: O-point becomes the new plasma center Kadomtsev model = Sweet-Parker regime = single fluid MHD = SLOW!

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**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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**Magnetic reconnection and different regions**

Electrons are not magnetized Ideal MHD Ions are not magnetized Single fluid MHD does not valid any more here!

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**Ohm’s law (two fluid formulation).**

Compare different components with gradient of convective electric field Single fluid MHD Priest and Forbes «Magnetic reconnection», 2000

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**Magnetic reconnection and different regions**

Ions are not magnetized (ion diffusion region) Ideal MHD Electrons are not magnetized (electron diffusion region)

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**Flows in reconnection region (computer simulations)**

[Pritchett Journal of Geophysical Reseach, 2001]

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**Flows in reconnection region (computer simulations)**

electron Ion diffusion region Electron diffusion region [Pritchett Journal of Geophysical Reseach, 2001]

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**Sweet-Parker (single fluid MHD)**

Is Sweet-Parker model always wrong? MRX Yamada, PoP, 2006 2 fluid MHD simulation Normalized plasma resistivity (reconnection rate) Ion diffusion region Sweet-Parker layer Sweet-Parker (single fluid MHD) High collisionality Low collisionality Sweet-Parker is correct for collisional plasmas….Unfortunately, our plasmas are collisionless.

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**Nonlinear simulations of (1,1) mode**

crash precursor postcursor Two fluid MHD XTOR code

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**Nonlinear simulations of (1,1) mode**

crash precursor postcursor Large stochastic region Two fluid MHD XTOR code

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**Nonlinear simulations of (1,1) mode**

XTOR code 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.

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**Particle effects on (1,1) mode**

Motivation for DIII-D: or (1,1) 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 density plasma density fast particles …but

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**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. (16%) MHD only MHD + particles Experimentally we see n=1 mode here! MHD stable region becomes unstable if fast particles are considered

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**Neoclassical Tearing Mode (NTM)**

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**Island width is a good measure of the reconnected flux**

Tearing Mode Island width is a good measure of the reconnected flux Reconnection zones Zohm, MHD Tearing mode: current driven, resistive instability. Neoclassical tearing mode: drive because of current deficiency in the island

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**Triggers are the main drive for seed island formation!**

Reconnection in ASDEX Upgrade. Tearing mode. SXR From ECE in ASDEX Upgrade (#27257, I.Classen MATLAB script) Fast (2,1) Very fast (2,1) Slow (3,1) Island width core Mirnov Same βN time sawteeth 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!

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**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 equilibrium plasma rotation frequency at q = 2 surface local electron diamagnetic drift frequency 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!

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**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) 36

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**Simulation of the (2,1) NTMs in JET**

XTOR, two fluid, non-linear, JET case XTOR results and other approximations Rutherfod equation is not enough! Amplitude of n=1 magnetic perturbation from Mirnov coils localized on the HFS and LFS Missing bootstrap current in the island

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**Interaction of several modes (FIR-NTM)**

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**Simulation of FIR-NTMs**

Experimental reconstruction For ASDEX Upgrade Predictions for ITER, Non-linear MHD code XTOR

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**Fast particle modes (TAEs, …)**

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**Linear simulations of fast particle modes**

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**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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**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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**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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Edge Localized Modes

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**Linear analysis of ELMs**

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) JET Type I Type III L-mode Saarelma, PPCF, 2009

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**Non-linear simulations of ELMs**

Non-linear MHD code JOREK solves the time evolution of the reduced MHD equations in general toroidal geometry Density time Hyusmans PPCF (2009)

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**Non-linear simulations of ELMs**

Formation of density filaments expelled across the separatrix. Hyusmans PPCF (2009) 1 2 3

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**Non-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)

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**Non-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. JOREK G 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

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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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**Resistive Wall Mode (RWM)**

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**Resistive 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-D JET [I.T.Chapman, PPCF, 2009] [M.Okabayashi, NF, 2009] RWM has global structure. This is important for “RWM ↔ plasma” interaction.

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**Interaction of RWM with external perturbations**

Real vessel wall currents 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.

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**Application of both models for ITER**

[Liu, NF,2009, IAEA, 2010] black dots are stable RWM self-consistent perturbative ideal wall ideal wall rotation rotation no wall no wall Stable at low rotation 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).

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**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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**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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Disruption

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**Simulation of the disruption**

Perturbed poloidal flux Temperature

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Non-linear codes RSAE (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 + …)

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**ITER priority Sawteeth NTMs RWMs Understanding of control ELMs**

Robust control, Good understanding, crash phase is not clear Robust control, Good understanding, seeding is not clear Sawteeth NTMs RWMs Planed for the later operation phase. Influence of the particles is not clear. Robust control, poor understanding (especially for external perturb.) Understanding of control ELMs Physical predictions are required. Preemptive ECCD control is possible Disruption more urgent less urgent

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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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**Non-linear calculations**

Up to now only hybrid simulations are possible (for example M3D code). Nonlinear evolution of single n=2 mode in NSTX Experiment simulations t=0.0 t=336

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**New model for rotation influence on the MHD modes (MARS-K)**

[Liu, PoP, 2008, Liu, IAEA, 2010] Kinetic effects are inside the pressure full toroidal geometry in which the kinetic integrals are evaluated

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**New model for rotation influence on the MHD modes (MARS-K)**

[Liu, PoP, 2008, Liu, IAEA, 2010] Kinetic effects are inside the pressure 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.

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**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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**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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