08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 1 Free and Controlled Dynamics of Magnetic Islands...

08/09/2008 E.LazzaroEDF/CEA/INRIA Summer School

1

Free and Controlled Dynamics of Magnetic Islands in Tokamaks

E. Lazzaro

IFP “P.Caldirola”, Euratom-ENEA-CNR Association, Milano, Italy


2

Outline• Brief reminder of tokamak ideal equilibrium• Nonideal effects:formation of magnetic island in

tokamaks through magnetic reconnection• Classical and neoclassical tearing modes• Useful mathematical models of mode dynamics• Problems and strategies of control by EC Current

Drive • Recent results from of experiments (FTU,ASDEX,DIII-D tokamaks)


3

Motivation and Objectives

• The reliability of Plasma Confinement in tokamaks is limited by the occurrence of

MHD instabilities that appear as growing and rotating MAGNETIC ISLANDS LOCALIZED on special isobaric surfaces and contribute to serious energy losses and can lead to DISRUPTION of the tokamak discharged.

• They are observed both as MIRNOV magnetic oscillations and as perturbations of Electron Cyclotron Emission and Soft X-ray signals

• They are associated with LOCALIZED perturbation of the current J ,e.g. J bootstrap

• Is it possible to stabilize or quench these instabilities by LOCALIZED injection of wave power (E.C.), heating locally or driving a non-inductive LOCAL current to balance the Jboot loss?


4

Tokamak magnetic confinement configurationThe most promising plasma (ideal) confinement

is obtained by magnetic field configurations that permit a magnetoidrostatic balance of fluid pressure gradient and magnetic force

Since the isobaric surfaces (p=nT) are “covered ergodically” by the lines of force of B and since the nested surfaces are of toroidal genus

The B field can be expressed through the the magnetic flux (R) through a poloidal section and (F(R,Z))through a toroidal section

00

B

JrotB

gradBJ

div

p

0 pB

Superficie poloidale

Superficie toroidale

0 B

)(FpT BBB


5

Non ideal effectsHelical Perturbations


6

• Tokamaks have good confinement because the magnetic field lies on isobaric isobaric surfacessurfaces of of toroidal genustoroidal genus

•The The BB field lines “pitch” is constant on each nested surface field lines “pitch” is constant on each nested surface

•(Isobaric) magnetic surfaces where (Isobaric) magnetic surfaces where q()=m/n have a different topologyhave a different topology: there are : there are alternate alternate OO and and XX singular pointssingular points that that do not exist on irrational surfaces: axisymmetry is : axisymmetry is broken and divbroken and divBB=0 allows a B=0 allows a Brr component component

•If current flows preferentially along certain field lines, magnetic islands form

•The contour of the island region is an isobar (and isotemperature)

• As a result, the plasma pressure tends to flatten across the island region, (thermal short-circuit) and energy confinement is degraded

Overview of basic concepts

)( 0

qd

d

B

B


7

Tokamak equilibrium and helical perturbations• Tokamak Equilibrium Magnetic field in terms of axisymmetric flux function

• 1° Force equilibrium

• Field line pitch :

• Helically perturbed field

• 2° Equilibrium condition (local torque balance)

• To order(r/R)

)(000 rB eeB

JB 0 BJ J B

J0 B1 J1 B0 B0 J1 B1 J0 J0 B1 J1 B0 0

nmtr

rr

,cos),(~,),(

1

10

eB

BBB

J0 B0 cp

)( 0

qd

d

B

B

B0 J1 B1r

J 0

r

O(2) Vanishing in axisymmetry


8

Basic Formalism of evolution equations

• Reduced Resistive MHD Equations: from vector to scalar system

• Compressional Alfven waves are removed• Closure of system with fluid equations

• Ordering Filters physics

E 1

c

A

t

B A

2A 4c

J

J 0

1, A

E 1

c

t

B F

2 4c

J

BJ //

B

J 0

1

,,,:)(

,,,,,:)(

,,:)1(

12

//

AR

RAcompr

m

S

JVBO

tJVBRrO

BO


9

Ideal and Resistive MHD• In Ideal MHD Plasma Magnetic topology is conserved

• B is convected with V

• In Resistive MHD Magnetic field diffuses relative to plasma topology

• The evolution of linear magnetic perturbations is

• Topology can change through reconnection of field lines in a “resistive” layer where

• Resistive MHD removes Ideal MHD constraint of preserved magnetic topology allowing possible instabilities with small growth rates

• • Key parameter

B

t V B

B

t V B c2

42B

˜ B

tB0 b ˜ V

ik // (x ) 0

c2

42 ˜ B

k //(x) 0

E V B /c J

S R A cr2VA 4R ~ 105 108

˜ B


10

Essential physics of tearing perturbations

• Quasineutrality constraint

• First order perturbation

• Competition of a stabilizing line bending term and a kink term feeding instabilities: perpendicular current may alter balance, through ion polarization current and neoclassical viscosity

• Tearing layer width is determined by balancing inertial and parallel current contributions to quasineutrality

• The time evolution of the perturbations is governed by Faraday law and generalised forms of Ohm’s law, including external non inductive contributions

BJ //

B

J

J 1 BJ //

B

1line bending

B1 J //

B

0kink

J1ion inertia

0

J c

B2 BpdiamagneticGGJ effect

c

B2 Bd

dt(VE Vp i

Inertia,ion polarizationcurrent

)c

B2 B

neoclassical viscos ityenhanchced polarization

R ~ r (mS)2 / 5

dependent

!

bAt

cbE bc bE bJ Jb JCD

J 0


11

Current driven tearing modes physics

Outer region - marginal ideal MHD - kink mode. The torque balance requires:

)(

0)(**2

*24

*//

F

BJ cB

Solved with proper boundary conditions to determine the discontinuity of the derivative

out' 1

ddx

|

0

0.5

1

1.5

0

h

0.2 0.4 0.6 0.8 1

0

r/a

=0

q(r)=2

2 ˜ F*

˜ ˜ dF dr d* dr 1

2 ˜ 4c

J 0z˜ /B [1 nq /m]

A linear perturbation is governed by an equation that is singular on the mode rational surface where k·B = 0

Singularity at q=m/n !!

Boundary Layer problem

s

s: reconnected helicalflux


12

Current driven tearing modes physics

• The discontinuous derivative equivalent to currents, localised in a layer across the q=m/n surface, where ideal MHD breaks down

•Ampere’s law relates the B perturbation to the current perturbation. For long, thin islands, it can be written:

•Integrating this over a period in x and out to a large distance, l, from the rational surface (w<<l<<rs) gives:

•Inner regionInner region - includes effects of inertia, resistivity, drifts, viscosity, etc

1

R

d2dr2

B(B)

B

4c

J||

˜ s in (,S,) 4

cdx

dJ // cosm x r rs

dr

d

2

2

dr

d

l l

Linear Dispersion relation:

Linear Growth rate:

out (A, J 0 //(rs)) in (,R,)

[rs out ]4 / 5 r q

q

2 / 5m2 / 5

R3 / 5A

2 / 5

13

Geometry & Terminology

• contours of constant helical flux• magnetic shear length cylindrical safety factor• island instantaneous phase

• x=r-rs slab coordinate from rational surface q(rs)=m/n

• helical flux reconnected on the rational surface• integrals and averages on island

cos~2

2*s

sL

Bx

ss rq

RqL

'

2

q(r) rB

R0B

(t) m n (t ')dt'0

t

s w2

16

B

Ls

xx

L L(x,)d

d

L dx L(x,)d


14

Neoclassical Tearing Modes (NTM)

• In a tearing-stable plasma (0’<0) • Initial island large enough to flatten the local

pressure• => loss of bootstrap current inside the island

sustains perturbation• Instability due to local flattening of bootstrap

current profile• Typically islands with m/n: 2/1 or 3/2 periodicity

Can prevent tokamaks from reaching high


15

Summary of RMHD equations

Resistive-neoclassical MHD fluid modelResistive-neoclassical MHD fluid model

nt

nV 0

0

dV

dtJB p

s

s

B A

A J 0

E 1cAt

E 1

cV BJ

1

ene

pe 1

ene

,1

nt

nV b V// i J // e 0

0

dV// iB

dt Bp B

s

s

V 1B2 B

B F

BJ //

B

J 0

24c

J //

J // //

1

cR

t

b1

ene

bpe 1

ene

b

e

J 1

B2 B 0

dVi

dtp

s

s


16

Bootstrap Current

J b c

B

dp

dr

Generalised parallel Ohm’s law with electron viscosity effects

Constant on magnetic surfaces

0 1

c

t

1

enbpe

1

enb

// e

electron viscos itybootstrap current

J //

b

// e meneV e

1 c

B

dp

drbootstrap current

(1 )J //trapped particleeffect

V e c

enB

r

pe pi B

B

V e V i

Electron viscous stress damps the poloidal electron flow - new free energy source.

Mechanism of bootstrap current


17

The NTM drive mechanism

•An initial perturbation( Wseed) leads to the formation of a magnetic island

• The pressure is flattened within the island at the O point, not at X point

• Thus the bootstrap current is removed inside the island

• This current perturbation amplifies the magneticfield perturbation,i.e. the island

Consider an initial small “seed” island:

Perturbed flux surfaces;lines of constant

Pressure flattens across island

Minor radius

Pre

ssur

e

Poloidalangle


18

Construction of the nonlinear island equation

• 1-A nonlinear averaging operator over the helical angle =m-nt makes

• 2-The parallel current is obtained solving the current closure (quasineutrality) equation ,averaging and and inserting it in Ampere’s law

•

• 3-Averaging Faraday law and eliminating <J//> gives

• 4-An integration weighted with cosover the radial extent of the nonlinear reconnection layer (island ), one obtains the basic Rutherford Equation for

• W(t) = 4(Br rs / B nq/)1/2

// J // B B 1J I Jp J // J nc B

J // J // J nc J nc J CD c

4RR 2

R

rs

dW

dtrs (W); ˜ s (W) 4

cdx

dJ // cos

// 0

CDncncs JJJRR

c

tc

2

4cos

~1

Grad-Shafranov equation

“neoclassical” currents

R.F.Current drive

19

Modified Rutherford Equation

4g1

dw

dt

rs2

r

['0 'bs 'GGJ 'pol 'EC Re 'wall ]

g1 d 1

cos 2 d / / 0.82

0' m 1

w

ws

NTM evolution (Integrating Faraday-Ampere on island)NTM evolution (Integrating Faraday-Ampere on island)

geom. factorgeom. factor

BS' aBS p

rs

Rax

Lq

Lp

w

w2 ws2 Lq q /q', Lp p / p', ws / //

(de)stabilising factor, <0 in NTM (# TM)(de)stabilising factor, <0 in NTM (# TM)

GGJ' aGGJ p

rs

Rax

2Lq

rs

Lq

Lp

1 1/q2 1

w

JJbootstrapbootstrap Term >0 Term >0

pol' apol p

rs

Rax

3 / 2Lq

Lp

2

rL2 T

T

1

1

w3 , T m

rs

Te,rs

e B Lne

pressure gradient & curvature pressure gradient & curvature Term <0Term <0

Polarisation Term >0, Polarisation Term >0, <0<0

EC' CD

' H' aCD

ICD

Ip,rs

Lq

cd2 (w)

1

w2 aH rs

Lq

Ip,rs

ˆ P EC

ne,rsTe,rs

J//,rsH w

Electron Cyclotron CD Electron Cyclotron CD TermTerm

w'

2m

rs

rs

d

2m w 2 i w 1 w 2 resistive wall Termresistive wall Term

G.Ramponi, E. Lazzaro, S. Nowak, PoP 1999

amc0

presentazioneanimazione personalizzatastruttura


20

Threshold Physics Makes an NTM Linearly Stable and Non-linearly Unstable

Rrs

dw

dt= ’ rs +

1/ 2 Lq

L p prs

w

w2 wd2 c(, i )

w pol2

w3

transport threshold(R.Fitzpatrick ,1995)•related to transverse plasma heat conductivity that partially removes the pressure flattening

~ 1 cm

polarization threshold(A.Smolyakov, E.Lazzaro et al, 1995)•ion polarization currents for ions E X B drifts are stronger than for electrons J is generated. J is not divergence free J// varies such that =0

c(,i) : polarization term also depends on

frequency of rotating mode, stabilizing only if 0>>i (J, Connor,H.R. Wilson et.al,1996)

wd Ls2r2

m2 //

1/ 4wpol (Lq/Lp)

1/2 1/2 I ~ 2

cm

Q ui ckTi me™ and aTI FF (Uncompressed) decompressorare needed to see thi s pi cture.


21

The Modified Rutherford Equation: discussion

w

dt

dw

Unstable solution Threshold poorly understood needs improved transport model need improved polarisation current

Stable solution saturated island width well understood?

Need to generate “seed” island additional MHD event poorly understood?

WsatWthres


22

Threshold Physics Makes an NTM Linearly Stable and Non-linearly Unstable

Rrs

dw

dt= ’rs +

1/ 2 Lq

L p prs

w

w2 wd2 c( )

w pol2

w3

-4

-2

0

2

4

6

8

10

0 0.05 0.1 0.15 0.2

Rrs dt

dw

w / a

wd=1.5cm

wpol

=2cm

wd=0, w

pol=0

m/n=2/1

’rs=-2

p=0.6

1/2 Lq/Lp=0.56c() =1

rs=1.54 ma=2 m

unstable

stable


23

The islands can be reduced in width or completely suppressed by a current driven by Electron Cyclotron waves (ECCD) accurately located within the island.

A requisite for an effective control action is the ability of identifying the relevant state variables in “real time”

-radial location -EC power absorption radius - frequency and phase

and vary accordingly the control variables -wave beam power modulation -wave beam direction.

rabs≈ rO-point- 3 cm

rabs≈ rO-point

rabs≈ rO-point+ 1 cm

rabs≈ rO-point+ 2 cm


24

Co-CD can replace the missing bootstrap

current

Localized Co-CD at mode rational surface may both increase the linear stabilityand replace the missing bootstrap current

CD' aCD

ICDI p (rs )

Lq

wcd2

[m,n (w /wcd )wcd

2

w20,0 ]

where:

Hm,n = efficiency by which a helical component is created by island flux surface averaging H0,0 =modification of equilibrium current profile

Rrs

dw

dtrs ['0 'bs ' pol 'CD ]

aCD 832

wCD / rs 2


25

CD efficiency to replace the missing bootstrap current

-1.5

-1

-0.5

0

0.5

1

1.5

-200 -150 -100 -50 0 50 100 150 200

RF modulation sketch

x/cd

(deg)

RF

mod

mod

Hm,n depends on:

• w/wcd

• whether the CD is continuous or modulated to turn it on in phase with

the rotating O-point

• on the radial misalignment of CD w.r.t. the rational surface q=m/n 50% on - 50% off

No-misalignment

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4w/w

cd

Hm,n

50/50 mod

CW


26

Larger CD efficiency with narrow JCD profiles

Note: •within the used model, in case of perfect alignment, the (2,1) mode is fully suppressed with 50% modulated EC power, Icd=

3% Ip(rs) (PEC~ 7 MW by FS

UL), when wcd =2.5 cm

•larger wcd would reduce the

saturated island width (partial stabilization)•narrow, well localized Jcd profiles are a major request for the ITER UL!

--- stable-1

0

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2

NO_ECCD

cw_wcd=7.5cm

cw_wcd=5cm

mod_wcd=5cm

cw_wcd=2.5cm

mod_wcd=2.5cm

Rrs dt

dw

w / a

Icd

/Ip(rs) = 0.03


27

Elements of the problem of control of NTM by Local absorption of EC waves

•The STATE variables of the process are the mode helicity numbers (m,n), the radial location rm/n, the width W (in cm!) of the island, and its

rotation frequency .

•The CONTROL variables of the system dedicated to island chase & suppression are: the radius rdep, of deposition the wave beam power depending

on the wave BEAM LAUNCHING ANGLES , the power pulse rate (CW or modulated)

•It is necessary to define and design real-time diagnostic and predictive methods for the dynamics of the process and of the controlling action, considering available alternatives and complementary possibilities


28

Approach to the problem

• One of the most important objectives of the control task is to prevent an island to grow to its nonlinear saturation level (that is too large)

• It is necessary to detect its size W, and its rotation frequency as early as possible after some trigger event has started the instability.

• Therefore the analysis of dynamics in the linear range near the threshold is important to be able to construct a useful real-time predictor algorithm.

• Key questions then are: observability and controllability• The work is in progress…


29

Linearized equation near threshold

221

226*

30

3

122

2

20

211 6,,3

1t

R

sw

v

c

bbsc

bc

bs

XrGG

Ga

X

XraX

X

ra

uxx

21

22

1211

0bb

a

aa

d

d

2*211

0*****21

,

,/,//,/,/

xXXxXX

XXrWXXXrWX

t

bctscceieTeTs

Dimensionless state variables and linearization near threshold W=Wt =T

Linear state system

Control vector

22

420

2

11

1

ub

rX

ub ECsc

b

Mode amplitude x1

and frequency x2

are coupled through a12

EC driven current

External momentum input


30

Controllability and observability of the system

2222

2121111

bab

bababQ

The dynamic system is controllable if its state variables respond to the control variablesAccording to Kalman controllability matrix Q= [b,Ab] must be of full rank

rank (Q)=2 if both b1 and b2 are non zero

In our case the condition, mode rotation control is necessary

22

420

2

11

1

ub

rX

ub ECsc

b

amplitude control b1

frequency control b2

EC driven current

External momentum input


31

Formal aspects of the control problem

• The physical objective is to reduce the ECE fluctuation to zero in minimal time using ECRH /ECCD on the position q=m/n identified by the phase jump method

• The TM control problem in the extended Rutherford form, belongs to a general class multistage decision processes [*] . In a linearized form the governing equation for the state vector x(t) is

• with the initial condition x(0)=x0, and a control variable (steering function) u(t).

• The formal problem consists in reducing the state x(t) to zero in minimal time by a suitable choice of the steering function u(t)

• Several interesting properties of this problem have been studied [*]

• [*] J.P. LaSalle, Proc. Nat. Acad. Of Sciences 45, 573-577 (1959); R.Bellman ,I. Glicksberg O.Gross, “On the bang-bang control problem” Q. Appl. Math.14 11-18 (1956)

dx

dtA(t)x(t)B(t)u(t)


32

Formal aspects of the control problem

• Definition [*]: An admissible (piecewise measurable in a set Ω ) steering function u* is optimal if for some t*>0 x(t*,u*) =0 and if x(t,u)≠0 for 0< t< t* for all u(t) Ω

• Theorem 1 [*]:• “ Anything that can be done by an admissible steering function can also

be done by a bang-bang function”• Theorem 2 [*]:• “If for the control problem there exists a steering function u(t) Ω such

that x(t,u)=0, for t>0, then there is an optimal steering function u* in Ω.

• “All optimal steering functions u* are of the bang-bang form”• Thus the only way of reaching the objective in minimum time is by using

properly all the power available

• Steering times can be chosen testing ||x(t|| <

u(t)

t


33

Concept of experimental set-up for ECCD control of Tearing modes

• “Just align” strategy:Find optimal angles a,b to minimize

• when 2121

2

/

,

),(

nmdep rrJ

˜ B Mirnov (t) ˜ B t arg et 0

rdep

(RM, ZM)


34

Estimate of “a priori” rdep()

4

2

00222

2

002

00

2

020

2

0

coscossin2sin22

cos

1),(

RRRRR

RtgZRr

ce

ceM

ce

ceM

ce

ce

Mce

ceM

ce

cedep

0.4 0.6 0.8 1 1.2 1.42

0

2

4

6

8

Poloidal angle

(rde

p-rs

)

7.652

0.887

dist2 0

dist2 18

dist2 9

2

0.5

Best poloidal angle for three toroidal angles (0, /18. /9)

example of minimization of | rdep(a,b) – rm/n|2


35

Experiments of automatic TM stabilizationby ECRH/CD on FTU


36

Island recognition with Te diagnostics

0

5

10

15

20

25

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3

Te,=0

Te,=¹/m

|dTe(KeV)|

Te(k

eV)

|Te(KeV

)|

tor

(m)

ITERScenario 2

Te flattening loss of bootstrap current

rotating NTM

antisymmetric Te oscillations

ECH (associated with ECCD) may mask strict antisymmetry

Multiple zeros possible

Te/T0

(r-r m,n)/Wc


37

0

0.02

0.04

0.06

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

|Te| |T

e,ECE| Pij

|Te

(KeV

)|

Pij

tor

P1,2 Te(r1.)Te(r2.)cos(t)cos (t) 1 2 TAcos 1 2

Correlation of the ECE fluctuations measured between nearby channels , both for natural and “heated” islands (e.g. r1=rs-x, r2=rs+x)

•The phase jump is effective on detecting the q=m/n radius, but not “unconditionally robust”

•The concavity of the sequence of Pij is a robust observable that gives the radial position rm/n of q=m/n

Position rm/n,mea measurement


38

Principle of risland tracking algorithm

Pij≈ 1 if both i and j are on the same side with respect to the island O-point.

Pij≈ -1 if on opposite sides.

A positive concavity in the Pij sequence locates the island.

channels

1

-1

0

Pi j


39

Position risland measurement from three ECE channels

High-pass filter

Correlation

Second derivative maxima (minima)

Example of real-time data processing for O-point location in the ECEn space

Gain

J. Berrino,E. Lazzaro,S. Cirant et al., Nucl. Fusion 45 (2005) 1350


40

Tracking of rational surfaces rm/n

(2,1)

axis(1,1)

FTU AUG

• Finite ECE resolution (channel width and separation)• false positives (mode multiplicity, axis, sawteeth...)• intermittancy of the measurement (small island or short

integration time...)


41

Algorithms for real time NTM control

• Information for control from : diagnostic & process model, assimilated in a Bayesian approach

• Control/Decision variables : mode amplitude W(t) , frequency and radial locations rNTM, rdep

• Actuator basic control variables : beam steering angle , and Power modulation


42

Assimilation (Bayesian filtering)

p( | d)L d |

p(d) L d |

p(d) L d | IR

a-posteriori pdflikelihood function , measured data

a-priori pdf, estimated data

evidence • uncertainty reduction• continuity of the observation (even if there is no mode)• “regularize” the observation• evidence is available for confidence in the decisions


43


• Cross-correlation Estimate for ECW power rdep in shot 17107 in ASDEX -U • From left: chann.-Xcorrelation, “a priori” PDF, chann. Likelihood, “a posteriori” PDF

Bayesian Filter : p(r|d)=L(d|r)*p(r)/p(d) L(d|r)*p(r)

a priori PDF

a posteriori PDF

Likelihood


44


• Real time estimate ECW power rdep (t) for shot 17107 in ASDEX-U (G. D’Antona et al, Proc., Varenna 2007

• Evidence p(d)

Bayesian Filter : p(r|d)=L(d|r)*p(r)/p(d)


45

FTU: Btor = 5.6 T

plasma axisResonance 140GHz

mirror

ECE channels

EC beam

Gyrotron 1

Gyrotron 3

1 2 3 4 5 6 7 8 9 10 11 12

ECRH power deposition at different R by changes of the angle of the mirror


46

1 2 3 4 5 6 7 8 9 10 11 12-0.05

0

0.05

0.1

0.15

ECE Channels

EC

E&

EC

Pow

er3

corr

ela

tion

1 2 3 4 5 6 7 8 9 10 11 12-0.1

0

0.1

0.2

0.3

Shot 27712

ECE Channels

EC

E&

EC

Pow

er1

corr

ela

tion Gyrotron 1

Gyrotron 3

Pi,B

Pi,A

FTU Shot 27714:real-time recognition rdep

fmod,Gy1 = 100 Hz

fmod,Gy3 = 110 Hz

Plasma axis

Correlation functions of the two gyrotrons

The deposition radius of each beam is detected by the maximum in Te,ECE -ECH correlation.

Different beams are recognized by different

ECH timing.

Gy1 Gy3

Gyrotron 1


47

MHD control in FTU (2 ECW beams)

t feedback ON=0.4 s action: low high duty cycle

ch.3 (gy.1 deposition)

ch.2

ch.1 (gy.3 deposition)

gy.3

gy.1

Mode Trigger (sawtooth?)

Mode hit and suppressed !

gy 3 on


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