Recent progress in MHD simulations and open questions

1. Recent progress in MHD simulations and open questions Valentin Igochine Max-Planck Institut für Plasmaphysik EURATOM-Association D-85748 Garching bei München Germany

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

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.

4. 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

5. 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

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!)

7. Linear and non-linear evolution linear 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!

8. 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?)

9. Sawtooth crash

10. Sawtooth 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)]

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.” Small tokamak → small Lundquist number: S = 104 (big tokamaks 108) Lundquist number = (resistive diffusion time)/(Alfven transit time) [Breslau et.al. Phys. Plasmas 14, 056105, 2007] Non-linear simulations of the sawtooth is very challenging task (even in a small tokamak).

12. 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, 056105, 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!

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

14. Magnetic reconnection redistributes energy Magnetic field lines plasma sling as a model  Energy conversion from magnetic field into heating and acceleration of the plasma

15. Structure of the reconnection region (MHD approx.) Ohm‘s law Amper‘s law

16. Structure of the reconnection region (MHD approx.) Ohm‘s law Amper‘s law Conservation of mass

17. Structure of the reconnection region (MHD approx.) Equation of motion This is the maximal outflow velocity!

18. 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?

19. 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!

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!

21. Magnetic reconnection and different regions Electrons are not magnetized Ideal MHD Ions are not magnetized Single fluid MHD does not valid any more here!

22. Ohm’s law (two fluid formulation). Compare different components with gradient of convective electric field Single fluid MHD Priest and Forbes «Magnetic reconnection», 2000

23. Magnetic reconnection and different regions Ions are not magnetized (ion diffusion region) Ideal MHD Electrons are not magnetized (electron diffusion region)

24. Flows in reconnection region (computer simulations) [Pritchett Journal of Geophysical Reseach, 2001]

25. Flows in reconnection region (computer simulations) ion electron Ion diffusion region Electron diffusion region [Pritchett Journal of Geophysical Reseach, 2001]

26. 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.

27. Nonlinear simulations of (1,1) mode crash precursor postcursor Two fluid MHD XTOR code

28. Nonlinear simulations of (1,1) mode crash precursor postcursor Large stochastic region Two fluid MHD XTOR code

29. 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.

30. 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

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

32. Neoclassical Tearing Mode (NTM)

33. 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

34. 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!

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 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!

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)

37. 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

38. Interaction of several modes (FIR-NTM)

39. Simulation of FIR-NTMs Experimental reconstruction For ASDEX Upgrade Predictions for ITER, Non-linear MHD code XTOR

40. Fast particle modes (TAEs, …)

41. Linear simulations of fast particle modes

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)

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).

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

45. Edge Localized Modes

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

47. 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)

48. Non-linear simulations of ELMs Formation of density filaments expelled across the separatrix. Hyusmans PPCF (2009) 1 2 3

49. 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)

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. • 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)