hefei, china/ august 2012 / 7th lecturevalentin igochine 1 recent progress in mhd simulations and...
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Hefei, China/ August 2012 / 7th Lecture Valentin Igochine1
Recent progress in MHD simulations and open questions
Valentin Igochine
Max-Planck Institut für PlasmaphysikEURATOM-Association
D-85748 Garching bei München Germany
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine2
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine3
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.
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine4
Different plasma descriptions
Kinetic description Fluid description
Vlasov equations,
Fockker-Planck codes
Distribution function
MHDParticle description
Hybriddescription
Particle parameters
Particle and fluid parameters
Fluid parameters
less comp. powermore comp. power
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine5
Different plasma descriptions
Kinetic description Fluid description
Vlasov equations,
Fockker-Planck codes
Distribution function
MHDParticle description
Hybriddescription
Particle parameters
Particle and fluid parameters
Fluid parameters
less comp. powermore comp. power
This is typically sufficient for MHD instabilities
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine6
Single fluid MHD equations
Resistive MHD 0
Ideal MHD 0
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!)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine7
Linear and non-linear evolution
Mode amplitude
time
Non-linear
linear
Linear evolutionExponential growth of the instability
Linearized MHD (eigenvalue problem, stable&unstable)
0 1
0 1
tp p p e
p p
Non-linear evolution
Equilibrium profile changes in time!
Perturbations are not any more small!
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine8
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine9
Sawtooth crash
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine10
Sawtoothlinear
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine11
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, 056105, 2007]
Small tokamak → small Lundquist number: S = 104 (big tokamaks 108)Lundquist number = (resistive diffusion time)/(Alfven transit time)
Non-linear simulations of the sawtooth is very challenging task (even in a small tokamak).
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine12
Sawtooth modelling
[Breslau et.al. Phys. Plasmas 14, 056105, 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!
1 1
ee eidealresistive
MHDMHDHall electronterm pressure
term
j E v B j B pen en
Ohm‘s law, 2 fluid MHD
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine13
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine14
Magnetic field lines
plasma
Energy conversion from magnetic field into heating and acceleration of the plasma
sling as a model
Magnetic reconnection redistributes energy
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine15
Structure of the reconnection region (MHD approx.)
E v B j
in inE v B
E j
Ohm‘s law Amper‘s law inBj B
inv
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine16
Structure of the reconnection region (MHD approx.)
E v B j
in inE v B
E j
Ohm‘s law Amper‘s law inBj B
inv
Conservation of mass
in outLv v
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine17
Structure of the reconnection region (MHD approx.)
Equation of motion
outneglected jB
v v p j B 2
out in outv B B
L
inout
Bv
This is the maximal outflow velocity!
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine18
Reconnection rate for Sweet-Parker model
inout Alfven
Bv v
in outv v
L
inv
2 2 2 1Alfvenin out out out out
Alfven Alvfen
vv v v v v
L L v Lv S
1in
out
v
v S
2diff Alfven
Alfven Alfven
LvLS
L v
In our plasmas Lundquist numbers are very high:Fusion plasmasSpace plasmas
8 910 10S 11 1210 10S
Lundquist number
One of the main questions: How one could explain fast reconnection?
Expected reconnection time for solar flaresMeasured reconnection time 310 15mint s
710 0.3t s year
exp100predicted Sawtooth crash in JET:
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine19
Single fluid MHD calculations often show Kadomtsev reconnection process
*K R A
20 1
R
r
* 1
*0
A
r
B
*01 0.3B B q B
expK
exp100K
exp10 50K
TCV:
ASDEX:
JET:
q=1
O-point becomes the new plasma center
Sawtooth crash time in Kadomtsev model
1r
Kadomtsev model = Sweet-Parker regime = single fluid MHD = SLOW!
Reconnectionregion
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine20
Plasma parameters for our experiments (MRX, tokamaks)
9 610 10sp m
3 210 10L m
410e m 3 210 10i m
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!
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine21
Magnetic reconnection and different regions
Ideal MHD
0E v B Ions are not magnetized
Electrons are not magnetized
Single fluid MHD does not valid any more here!
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine22
Ohm’s law (two fluid formulation).
2
c
e e
Ohm Hall e vise inertia
m pj j j BE v B vj jv
ne t ne ne
�
2e
e inertia epe
mcL d
ne
2
pHall i
pi
mcL d
ne
sterss iLM
e icollision
mfp
LM
Priest and Forbes «Magnetic reconnection», 2000
Compare different components with gradient of convective electric field
Single fluid MHD
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine23
Magnetic reconnection and different regions
Ideal MHD 0E v B Ions are not magnetized
(ion diffusion region)
Electrons are not magnetized(electron diffusion region)
2
cos
e e
Ohm Hall e vis itye inertia
m pj j j BE v B vj jv
ne t ne ne
�
ipi
cd
epe
cd
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine24
Flows in reconnection region (computer simulations)
[Pritchett Journal of Geophysical Reseach, 2001]
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine25
Flows in reconnection region (computer simulations)
ion electron
[Pritchett Journal of Geophysical Reseach, 2001]
Ion diffusion region
Electron diffusion region
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine26
Sweet-Parker (single fluid MHD)
High collisionality Low collisionality
2 fluid MHD simulation
Is Sweet-Parker model always wrong?
MRXN
orm
aliz
ed
pla
sma
res
istiv
ity(r
eco
nn
ectio
n r
ate
)
Sweet-Parker is correct for collisional plasmas….Unfortunately, our plasmas are collisionless.
i spd i spd
Ion diffusion region
Sweet-Parker layer
Yamada, PoP, 2006
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine27
Nonlinear simulations of (1,1) mode
precursorcrash
postcursor
Two fluid MHDXTOR code
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine28
Nonlinear simulations of (1,1) mode
precursor crash postcursor
Two fluid MHDXTOR code
Large stochastic region
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine29
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine30
Particle effects on (1,1) mode
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 density plasma density
…but
fast particles
Motivation for DIII-D: or (1,1)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine31
Particle effects on (1,1) mode
Linear stability of the n=1 mode with and without energetic particle effectsusing 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%)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine32
Neoclassical Tearing Mode (NTM)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine33
Reconnection zones
Tearing Mode
Zohm, MHDTearing 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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine34
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!
Isla
nd w
idth
time sawteeth
Mirnov
SXR
core
Reconnection in ASDEX Upgrade. Tearing mode.
From ECE in ASDEX Upgrade (#27257, I.Classen MATLAB script)
Same βN
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine35
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 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!
local electron diamagnetic drift frequency
the equilibrium plasma rotation frequency at q = 2 surface
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine36
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine37
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 approximationsRutherfod equation is not enough!
XTOR, two fluid, non-linear, JET case
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine38
Interaction of several modes (FIR-NTM)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine39
Simulation of FIR-NTMs
Experimental reconstructionFor ASDEX Upgrade Predictions for ITER,
Non-linear MHD code XTOR
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine40
Fast particle modes (TAEs, …)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine41
Linear simulations of fast particle modes
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine42
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine43
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).
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine44
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine45
Edge Localized Modes
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine46
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine47
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine48
Hyusmans PPCF (2009)
1
2
3
Formation of density filaments expelled across the separatrix.
Non-linear simulations of ELMs
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine49
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine50
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine51
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)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine52
Resistive Wall Mode (RWM)
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• 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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine54
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine55
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).
0.4C
[Liu, NF,2009, IAEA, 2010]
perturbative self-consistent
idealwall
idealwall
nowall
nowall
rotationrotation
Stable at low rotation
black dots are stable
RWM
N N
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine56
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine57
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!
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine58
Disruption
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine59
Simulation of the disruption
Perturbed poloidal flux
Temperature
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine60
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 + …)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine61
ITER priority
more urgentless urgent
Disruption
ELMs
NTMsRWMs
Sawteeth
Un
der
stan
din
g o
f co
ntr
ol
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine62
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine63
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Non-linear calculations
Up to now only hybrid simulations are possible (for example M3D code).
2
0
-2
t = 0.267
108530
200 s
Experiment
simulations
t=0.0 t=336
Nonlinear evolution of single n=2 mode in NSTX
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine65
New model for rotation influence on the MHD modes (MARS-K)
• full toroidal geometry in which the kinetic integrals are evaluated • * 0, 0D
Kinetic effects are inside the pressure
[Liu, PoP, 2008, Liu, IAEA, 2010]
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine66
• 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.
* 0, 0D
Kinetic effects are inside the pressure
[Liu, PoP, 2008, Liu, IAEA, 2010]
New model for rotation influence on the MHD modes (MARS-K)
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine67
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
Hefei, China/ August 2012 / 7th Lecture Valentin Igochine68
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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