hefei, china/ august 2012 / 7th lecturevalentin igochine 1 recent progress in mhd simulations and...

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


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


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.


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


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


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!)


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!


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


Sawtooth crash


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


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).


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


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


Magnetic field lines

plasma

Energy conversion from magnetic field into heating and acceleration of the plasma

sling as a model

Magnetic reconnection redistributes energy


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



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



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!


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:


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


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!


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!


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


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


Flows in reconnection region (computer simulations)

[Pritchett Journal of Geophysical Reseach, 2001]


Flows in reconnection region (computer simulations)

ion electron

[Pritchett Journal of Geophysical Reseach, 2001]

Ion diffusion region

Electron diffusion region


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


Nonlinear simulations of (1,1) mode

precursorcrash

postcursor

Two fluid MHDXTOR code



precursor crash postcursor

Two fluid MHDXTOR code

Large stochastic region



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


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)


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%)


Neoclassical Tearing Mode (NTM)


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


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


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


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)


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


Interaction of several modes (FIR-NTM)


Simulation of FIR-NTMs

Experimental reconstructionFor ASDEX Upgrade Predictions for ITER,

Non-linear MHD code XTOR


Fast particle modes (TAEs, …)


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)


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).


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


Edge Localized Modes


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)


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


Hyusmans PPCF (2009)

1

2

3

Formation of density filaments expelled across the separatrix.



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


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)


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)


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)


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.

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


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


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


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


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!


Disruption


Simulation of the disruption

Perturbed poloidal flux

Temperature


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 + …)


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


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


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


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]


• 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)



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


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.

• …

hefei, china/ august 2012 / 7th lecturevalentin igochine 1 recent progress in mhd simulations and...

Documents

th lecturevalentin igochine

mhd simulations

mhd instabilities

power slide

sawtooth crash slide

simplified plasma description

outline introduction

account realistic parameters