1. Recent progress in MHD simulations and open questions
Valentin Igochine
Max-Planck Institut für Plasmaphysik
EURATOM-Association
D 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
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 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)]

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, , 2007]
Non-linear simulations of the sawtooth is very challenging task
(even in a small tokamak). 11

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, , 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

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. 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?

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

26. 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.

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. 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

34. 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!

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) 36

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)
JET
Type I
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
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

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)

52. Resistive Wall Mode (RWM)

53. 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.

54. 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.

55. 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).

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

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!

58. Disruption

59. Simulation of the disruption
Perturbed poloidal flux
Temperature

60. 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 + …)

61. 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

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

64. 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

65. 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

66. 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.

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

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. …