Free and Controlled Dynamics of
Magnetic Islands in Tokamaks
E. Lazzaro
IFP “P.Caldirola”, Euratom-ENEA-CNR Association, Milano, Italy
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
1
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)
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
2
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 noninductive LOCAL current to balance the Jboot loss?
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
3
Tokamak magnetic confinement configuration
The most promising plasma (ideal) confinement
is obtained by magnetic field configurations
that permit a magnetoidrostatic balance of
fluid pressure gradient and magnetic force
J  B  grad p
rotB  0 J
div B  0
Since B  p  0 the isobaric surfaces (p=nT)
are “covered ergodically” by the lines of force
of B and since   B  0 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
Superficie poloidale
Superficie toroidale
B  BT  B p  F (  )    
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
4
Non ideal effects
Helical Perturbations
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
5
Overview of basic concepts
• Tokamaks have good confinement because the magnetic field lies on
isobaric
surfaces of toroidal genus
•The B field lines “pitch” dd  BB    q(0 )
is constant on each nested surface
•(Isobaric) magnetic surfaces where q()=m/n have a different topology: there are
alternate O and X singular points that do not exist on irrational surfaces: axisymmetry is
broken and divB=0 allows a Br 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
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
6
Tokamak equilibrium and helical perturbations
•
•
Tokamak Equilibrium Magnetic field in terms of axisymmetric flux function
B0  B0 e  e  0 ( r )
1° Force equilibrium
J 0  B0  cp
•
•
Field line pitch :

d B  

 q( 0 )
d B  
Helically perturbed field
B  B0 ( r,  )  B1 r,  
B1  e  ~( r, t ) cos  ,   m  n
•
2° Equilibrium condition (local torque balance)
  J  B  0  B  J  J  B
  J 0  B1    J1  B 0   B 0  J1  B1  J 0  J 0  B1  J1  B 0   0
•
To order(r/R)
08/09/2008
J 0 
E.Lazzaro  O( 2 )
B 0  J

B


1
1r
EDF/CEA/INRIA Summer School
r


7
Vanishing in axisymmetry

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

1 
E
  

1 A

c t
E   c t  
B  F    
 B    A

4




2



1,
A

4
 
J
  2A  

J
c
c


J // 
   J  0
B      J   0
B 

O (1) : B ,  m , 
• Ordering
Filters physics
08/09/2008
O ( ) : r R , B ,V , J  ,  t ,  //
O ( 2 ) : B1 ,Vr , J  , Acomp  R
S   R  A  1E.Lazzaro
EDF/CEA/INRIA Summer School
8
•
In Ideal MHD Plasma Magnetic topology is conserved
B
   V  B
t
Ideal and Resistive MHD
•
B is convected with V
•
 Magnetic field diffuses relative to plasma topology
In Resistive MHD
B
c 2 2
   V  B 
B
t
4
•
The evolution of linear magnetic perturbations B˜ is

B˜
c 2 2 ˜
˜
 B 0 b  V 
B
t
4

ik (x )0

//
•
Topology can change through reconnection of field lines in a “resistive” layer where

•
•
•
k // (x)  0
Resistive MHD E  V  B/c  J removes Ideal MHD constraint of preserved
magnetic topology allowing possible instabilities with small growth rates


Key parameter
08/09/2008
S   R  A  cr 2VA 4R ~ 10 5 10 8
E.Lazzaro
EDF/CEA/INRIA Summer School
9
Essential physics of tearing perturbations
•
•
Quasineutrality constraint
 J  0
J // 
B      J
B 
First order perturbation

J // 
J // 


J

B



B


 
  
 1
1


B 0
B 1

line bending
•
•
•
  J1
0
ion inertia
kink
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

c
c
d
c
J  2 B  p 2 B  (VE  Vp i )  2 B    
B
B
dt
B
diamagnetic
Inertia,ion polarization
neoclassicalviscosity
GGJeffect
current
enhanchced polarization
wdependent!
Tearing layer width is determined by balancing inertial and parallel current
contributions to quasineutrality R ~ r (mS) 2 / 5
The time
 evolution of the perturbations is governed by Faraday law and generalised
forms of Ohm’s law, including external non inductive contributions
08/09/2008
b  A
E.Lazzarob  E  b  J  J  J
 cb  E b  c
b
CD 
EDF/CEA/INRIA
Summer School
t
10
Current driven tearing modes physics
Boundary Layer problem
Outer region - marginal ideal MHD - kink mode. The torque balance requires:
B  J // B  *  ( 4c 2 * )  0
q(r)=2
2 *  F ( * )
1.5

A linear perturbation is governed by an equation that is
singular on the mode rational surface where k·B = 0
0
h =0
1
s
˜  F *
˜   
˜ dF drd* dr 
2

1

˜
2
4
c
0.5
˜ /B [1 nq /m]
J0z
0
0
0.2
0.4
0.6
1 r/a
0.8
Singularity at q=m/n !!
Solved with proper boundary conditions to determine the discontinuity of the derivative
'
 out 
08/09/2008
1 d 
|
 dx
s: reconnected helicalflux
E.Lazzaro
EDF/CEA/INRIA Summer School
11
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
d 2
dr 2

l
d
dr
•Ampere’s law relates the B perturbation to the current perturbation.
For long, thin
islands, it can be written:
l
1 d 2 B  (  B) 4


J||
R dr 2
B
c
•Integrating this over a period in x and out to a large distance, l, from the rational surface
(w<<l<<rs) gives:

4
 (w,S, ) 
˜ s

dx  dJ // cosm

in
c 
x  r  rs
•Inner region - includes effects of inertia, resistivity, drifts, viscosity, etc
Linear Dispersion relation:

Linear Growth rate:
08/09/2008
out ( A, J0// (rs ))  in (, R,)
2/5
 
m2 / 5
4 / 5 rq
  [rsout ] E.Lazzaro
 
3/5 2/5
 qSummer
  School
R A
EDF/CEA/INRIA
12
Geometry & Terminology
•
•
•
•
•
•
 B 
contours of constant helical flux    x  2L  ~ cos 
magnetic shear length L  qq' rR cylindrical safety factor q(r)  RrBB
island instantaneous phase  (t)  m  n   w(t')dt'

x=r-rs slab coordinate from rational surface q(rs)=m/n
w B

 
helical flux reconnected on the rational surface 16 L
integrals L   dx  L(x, )d and averages  L   L(x, )d on island
*
2
s
s
2

s
s
0 
t
0
2

s
s
 d


x

13
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 
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
14
Summary of RMHD equations
Resistive-neoclassical MHD fluid model

n
   nV  b   V// i  J // e  0

t

dV//iB

n

 B  p   B    s

0
   nV  0

dt

t
s

dV
1

 J  B  p     s
V

B   
  0

2

dt
B

s

B  F    
B A



J // 





B


A  
    J   0
,1
B 


J0
4


2




J //
1

A


E
 
c
c t


 1 

1
1
1
1
1
E  V  B  J 
J //   //
 b   
b  p e 
b    e 
p e 
  
cR

t
en
en




e
e
en e
en e
 c

 dV

1
i
J

B




p




 0

 s 

2
B
dt



s
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
15
Mechanism of bootstrap current
Bootstrap Current
c dp
Jb  
B dr
Constant on magnetic surfaces

Generalised parallel Ohm’s law with electron viscosity effects

1   1
1
0   
Electron viscous stress damps the poloidal
 b  p e  b    // e  J //

c t  en
en
electron flow - new free energy source.
electron viscosity
 bootstrapcurrent

c dp
b    //e  meneV e 
 (1  )J //
B
dr
1  
trapped particle
bootstrapcurrent
08/09/2008
V e  
c 
B
pe  pi    V e  V i 
enB r
B
effect
E.Lazzaro
EDF/CEA/INRIA Summer School

16
The NTM drive mechanism
Consider an initial small “seed” island:
Perturbed flux surfaces;
lines of constant W
Poloidal
angle
•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
Pressure
• Thus the bootstrap current is
removed inside the island
Pressure flattens across island
• This current perturbation amplifies
the magneticfield perturbation,i.e. the
island
Minor radius
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
17
Construction of the nonlinear island equation
•
•
•
1-A nonlinear averaging operator over the helical angle =m-nwt makes
 //  0
2-The parallel current is obtained solving the current closure (quasineutrality)
equation ,averaging and and inserting it in Ampere’s law
1

J
B

B
  J I  J p  J     // J nc B


//
//

J //  J //  J nc  J nc  J CD  


c
R  R 2
4
R.F.Current drive
•
“neoclassical” currents
Grad-Shafranov equation
3-Averaging Faraday law and eliminating <J//> gives

~
1  s
c
cos  
R R 2    J nc  J nc    J CD
c t
4
•
•
4-An integration weighted with cos,over 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
 dW
4 
R
rs dt
08/09/2008
˜ s(W)
 rs(W);




E.Lazzaro
EDF/CEA/INRIA Summer School
 dx  dJ // cos 
c 
18
Modified Rutherford Equation
NTM evolution (Integrating Faraday-Ampere on island)
4g1 dw rs2
 ['0 'bs 'GGJ ' pol 'EC Re 'wall ] g   dW  cos   d /W /   0.82
 dt  r
geom. factor

2
1
1
 w 
 0  m 1  (de)stabilising
 w s 
'
rs Lq
w
Rax L p w 2  ws2
' BS  aBS  p
 GGJ
'

factor, <0 in NTM (# TM)
 Lq  q /q', Lp  p / p', ws    /  //
 rs 2 Lq Lq
1
 aGGJ  p  
11/q 2 

w
Rax  rs L p
pressure gradient & curvature Term <0
2
 r 
' pol  apol  p  s 
Rax 
L  w  w
 1
m Te,rs
q
2
r
1
,
w






L
T
3
L  w w
rs e B Ln e
 w
 p 
T  T
' EC  'CD  ' H  aCD
Lq
ICD Lq
1
PˆEC

(w)

a
r
J //,r  H w
H s
I p,rs cd 2
w2
I p,rs n e,rs   Te,rs  s
3/2
2m
2m rs 
w 
 
rs d 
'
w  w   iw  w 
2
1 w  w 
2
Jbootstrap Term >0
Polarisation Term >0, <0
Electron Cyclotron CD Term
resistive wall Term
19
G.Ramponi, E. Lazzaro, S. Nowak, PoP 1999
Threshold Physics Makes an NTM Linearly
Stable and Non-linearly Unstable

 R dw
1/ 2 L q
= ’ rs + 
Lp
rs dt
2 

w
w
pol 
 p rs 
 c(w ,  i )
3 
w 2  w 2
w

d

transport threshold
polarization threshold
(R.Fitzpatrick ,1995)
•related to transverse
plasma heat

conductivity that partially removes
the pressure flattening
•ion polarization currents
(A.Smolyakov, E.Lazzaro et al, 1995)
for ions E X B drifts are stronger than for
electrons J is generated. J is not
divergence free J// varies such that
=0
Qu co
i ck Ti m ess™edan) dd eco
a
TI F
ar F
e (nUn
ee demdprt e
o s ee t h is pi m
ct p
ur re
e ss
. or
L2 r 2  1/ 4
s
  ~ 1 cm
w d  
 2  
// 
 m
08/09/2008
wpol (Lq/Lp) 1/2  1/2 I ~ 2
cm
c(w, ) : polarization term also depends on
i
frequency of rotating mode, stabilizing only
if 0>w>wi (J, Connor,H.R. Wilson et.al,1996)
E.Lazzaro
EDF/CEA/INRIA Summer School
20
The Modified Rutherford Equation: discussion
dw
dt
Need to generate “seed” island
 additional MHD event
 poorly understood?
Stable solution
 saturated island width
 well understood?
Wthres
Wsat
w
Unstable solution
 Threshold
 poorly understood
 needs improved transport model
 need improved polarisation current
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
21
Threshold Physics Makes an NTM Linearly
Stable and Non-linearly Unstable
2 

w
L
w
pol 
= ’rs +  1/ 2 q  p rs 
 c(w )
rs dt
3 
w 2  w 2
Lp
w

d

 R dw
10
w =0, w
8

 dw
R
r
s
dt
d
6

pol
=0
m/n=2/1
’rs=-2
unstable
w =1.5cm
4
d
rs=1.54 m
a=2 m
w =2cm
pol
2
p=0.6
0
stable
-2
1/2 Lq/Lp=0.56
-4
0
0.05
0.1
0.15
0.2
c(w) =1
w/a
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
22
rabs≈ rO-point- 3 cm
rabs≈ rO-point
The islands can be reduced in width
or completely suppressed by a
current driven by Electron
Cyclotron waves (ECCD)
accurately located within the island.
rabs≈ rO-point+ 1 cm
rabs≈ rO-point+ 2 cm
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
08/09/2008
-wave beam direction.
E.Lazzaro
EDF/CEA/INRIA Summer School
23
Co-CD can replace the missing bootstrap
current
Localized Co-CD at mode rational surface may both increase the linear stabili
and replace the missing bootstrap current
 R dw
rs dt

rs [ ' 0 ' bs ' pol ' CD ]
I CD L q
w cd 2
'
 CD  aCD
[  m,n (w / w cd )
  0,0 ]
2
2
I p (rs ) w cd
w
32
 wCD /rs  2 

Hm,n = efficiency by which a helical component is created by island flux sur
averaging
H0,0 =modification of equilibrium current profile

where:
08/09/2008
a CD  8


E.Lazzaro
EDF/CEA/INRIA Summer School
24
CD efficiency to replace the missing
bootstrap current
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
1.5
0.5
RF modulation sketch
No-misalignment
1
50/50 mod
0.4
0.5
H
0.3
CW
x/ 
m,n
RF
cd
0
0.2
-0.5

0.1
-1
0

W
W

0
1
2
w/w
08/09/2008
3
4
cd
-1.5
-200
-150
-100

mod
-50
0

E.Lazzaro
EDF/CEA/INRIA Summer School
50
mod
100
150
200
(deg)
25
Larger CD efficiency with narrow JCD profiles
7
NO_ECCD
cw_wcd=7.5cm
6
cw_wcd=5cm
mod_wcd=5cm
5
 dw
R
r
s
cw_wcd=2.5cm
mod_wcd=2.5cm
4
dt 3
I /Ip(r ) = 0.03
cd
2
s
1
0
-1
--- stable
0
0.05
0.1
0.15
0.2
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!
w/a
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
26
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 w.
•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
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
27
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
w 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…
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
28
Linearized equation near threshold
Dimensionless state variables and linearization near threshold W=Wt w=wT
X 1  W / rs ,
X 2  w  wT  / w*e ,
X 1  X t  x1,
X *  wT / w*e  w*i / w*e ,
X c  Wc / rs ,
X t  X c 0 b
X 2  X *  x2
Linear state system
dx a11 a12 

 x  b1 b2  u

d  0 a22 
Control vector
b2 1
b1u1   2 4 rs EC
0 X c
b2u2
08/09/2008
Mode amplitude x1
and frequency x2
are coupled through a12
EC driven current
External momentum input
rs b  302 2 
b3 X *
G  
a11  2 1  2 X c E.Lazzaro
, a12  rs b 3 6 , a22  6 v 2 R2
EDF/CEA/INRIA
Xc 
b
G1Gw rs X t
 Summer School 0 X c
29
Controllability and observability of the system
The dynamic system is controllable if its state variables respond to the control variables
According to Kalman controllability matrix Q= [b,Ab] must be of full rank
 b1
Q
b2
a11b1  a12b2 
a22b2 
amplitude control b1
frequency control b2
rank (Q)=2 if both b1 and b2 are non zero
In our case the condition, mode rotation control is necessary
b2 1
b1u1   2 4 rs EC
0 X c
b2u2
08/09/2008
EC driven current
External momentum input
E.Lazzaro
EDF/CEA/INRIA Summer School
30
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
dx
 A(t)x(t)  B(t)u(t)
dt
•
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 [*]
E.Lazzaro
•08/09/2008
[*] J.P. LaSalle, Proc. Nat. Acad. Of
Sciences
45, 573-577 (1959); R.Bellman ,I. Glicksberg 31
EDF/CEA/INRIA Summer School
O.Gross, “On the bang-bang control problem” Q. Appl. Math.14 11-18 (1956)
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
•
•
•
•
u(t)
t
•
Steering times can be chosen testing ||x(t|| < 
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
32
Concept of experimental set-up for ECCD control
of Tearing modes
(RM, ZM)

rdep
• “Just align” strategy:Find optimal angles a,b to minimize
J  rdep ( ,  )  rm / n
• when
08/09/2008
˜ Mirnov (t)  B
˜ t arg et  0
B
2
1     2 , 1     2
E.Lazzaro
EDF/CEA/INRIA Summer School
33
Estimate of “a priori” rdep(,)
2

w

w

rdep ( ,  )   ce 0  1 R02   Z M  tg  ce 0     RM 
 wce

 wce



2
 w


wce 
2
2 
ce 0

    R0 cos   
2 RM sin   2
R0   RM sin 2  cos2 

  wce

2wce 0 



2

7.652
example of minimization
of | rdep(a,b) – rm/n|2
8
6
dist2  , 0
(rdep-rs)
Best poloidal
angle  for three
toroidal angles
(0, /18. /9)


 

18 



9
dist2  ,
dist2  ,



4
2
0
 0.887
2
0.4
0.5
08/09/2008
2

 wce 0 

4
 
R0  cos  

 wce


E.Lazzaro
EDF/CEA/INRIA Summer School
0.6
0.8
1

Poloidal angle
1.2
1.4
34

2
Experiments of automatic TM stabilization
by ECRH/CD on FTU
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
35
Island recognition with Te diagnostics
T
T
e,=0
|dTe(KeV)|
0.6
e,=¹/m
25
ITER
0.5
Scenario 2
0.4
15
e
0.3
10
0.2
5
|Te(KeV)|
T (keV)
20
00
Te/T0
(r-r m,n)/Wc
0.1
0.5
1
1.5 2
 (m)
2.5
3
Multiple zeros possible
0
tor
Te flattening  loss of bootstrap current

rotating NTM

antisymmetric Te oscillations
08/09/2008
ECH (associated with ECCD) may
36
mask strict antisymmetry
E.Lazzaro
EDF/CEA/INRIA Summer School
Position rm/n,mea measurement
Correlation of the ECE fluctuations measured between nearby
channels , both for natural and “heated” islands (e.g. r1=rs-x, r2=rs+x)
P1,2  Te (r1. )Te (r2. )cos (t)cos(t)  12  T  Acos12 
|Te|
•The phase jump is effective on
detecting the q=m/n radius, but
not “unconditionally robust”
0.06
0.5
0.04
0
ij
P
•The concavity of the sequence
of Pij is a robust observable
that gives the radial position rm/n
of q=m/n
Pij
1
|Te(KeV)|

|Te,ECE|
0.02
0
-0.5
-1
0
0.5
1 1.5

2
2.5
3
tor
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
37
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.
Pi j
1
0
channels
-1
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
38
Position risland measurement from three ECE channels
Gain
Example of real-time data
processing for O-point
location in the ECEn space
High-pass filter
Correlation
Second derivative  maxima (minima)
J. Berrino,E. Lazzaro,S. Cirant et al., Nucl.
Fusion 45 (2005) 1350
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
39
Tracking of rational surfaces rm/n
FTU
AUG
(2,1)
(1,1)
axis
• Finite ECE resolution (channel width and separation)
• false positives (mode multiplicity, axis, sawteeth...)
• intermittancy of the measurement (small island or short
integration time...)
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
40
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
E.Lazzaro
08/09/2008
41
Actuator
basic control variables EDF/CEA/INRIA
: beam
steering
angle , and Power modulation
Summer School
Assimilation (Bayesian filtering)

likelihood function , measured data
a-posteriori pdf
Ld |    
p(  | d) 
 Ld |     
p(d)
p(d) 
evidence
08/09/2008
•
•
•
•

IR
Ld |    
a-priori pdf, estimated data
uncertainty reduction
continuity of the observation (even if there is no mode)
“regularize” the observation
evidence is available for confidence in the decisions
E.Lazzaro
EDF/CEA/INRIA Summer School
42
Algorithms for real time NTM control
a priori PDF
Likelihood
a posteriori PDF
•
•
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
08/09/2008
Bayesian Filter : p(r|d)=L(d|r)*p(r)/p(d)
L(d|r)*p(r)
E.Lazzaro
EDF/CEA/INRIA Summer School
43
Algorithms for real time NTM control
Bayesian Filter : p(r|d)=L(d|r)*p(r)/p(d)
•
•
Real time estimate ECW power rdep (t) for shot 17107 in ASDEX-U (G. D’Antona et al, Proc.,
Varenna 2007
Evidence p(d)
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
44
ECRH power deposition at different R
by changes of the angle of the mirror
FTU: Btor = 5.6 T
EC beam
ECE channels
1 2 3 4 5 6 7 8 9 10 11 12
Gyrotron
1
mirror
Resonance 140GHz
plasma axis
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
45
FTU Shot 27714:real-time recognition rdep
fmod,Gy1 = 100 Hz
fmod,Gy3 = 110 Hz
Gy1
Gy3
Shot 27712
The deposition radius of
each beam is detected
by the maximum in
Te,ECE -ECH correlation.
ECE&ECPower1 correlation
0.3
Correlation functions
of the two gyrotrons
Gyrotron 1
0.2
Gyrotron 1
0.1
Pi,A
0
Plasma axis
-0.1
1
2
3
4
5
6
7
ECE Channels
8
9
10
11
12
Different beams are
recognized by different
ECH timing.
ECE&ECPower3 correlation
0.15
Gyrotron 3
0.1
0
-0.05
08/09/2008
Pi,B
0.05
1
2
3
4
5
6
7
ECE Channels
E.Lazzaro
EDF/CEA/INRIA Summer School
8
9
10
11
12
46
MHD control in FTU (2 ECW beams)
ch.3 (gy.1 deposition)
ch.2
Mode hit and suppressed !
ch.1 (gy.3 deposition)
gy.3
gy 3 on
gy.1
Mode Trigger (sawtooth?)
08/09/2008
t feedback ON=0.4 s
E.Lazzaro
EDF/CEA/INRIA Summer School
action: low  high duty cycle
47
References
[1] Z.Chang and J.D.Callen, Nucl.Fusion 30,219, (1990)
[2] C.C.Hegna and J.D Callen, Phys. Plasmas 1, 2308 (1994)
[3] R. Fitzpatrick, Phys. Plasmas, 2, 825 (1995)
[4] A.I. Smolyakov, A. Hirose, E. Lazzaro, et al., Phys. Plasmas 2, 1581 (1995)
[5] H.R. Wilson et al., Plasma Phys. Control. Fusion 38, A149 (1996)
[6] G.Giruzzi et al., Nucl.Fusion 39, 107, (1999)
[7] G.Ramponi, E. Lazzaro, S.Nowak, Phys. Plasmas, 6, 3561 (1999)
[8] Smolyakov, E.Lazzaro et al., Plasma Phys. Contr. Fus. 43, 1669 (2001)
[9] H.Zohm et al., Nucl.Fusion 41, 197, (2001)
[10] A.I. Smolyakov, E. Lazzaro, Phys. Plasmas 11, 4353 (2004)
[11] O. Sauter, Phys. Plasmas, 11, 4808 (2004)
[12] R.J.Buttery et al., Nucl.Fusion 44, 678 (2004)
[13] H.R. Wilson, Transac. of Fusion Science and Tech. 49, 155 (2006)
[14] R.J. La Haye et al., Nucl. Fusion 46, 451 (2006)
[15] R.J. La Haye, Physics of Plasmas 13 (2006)
[16] J. Berrino, S. Cirant, F.Gandini, G. Granucci, E.Lazzaro ,F. Jannone,
P. Smeulders and G.D’Antona IEEE Trans 2005
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
48
FINE
08/09/2008
E.Lazzaro
EDF/CEA/INRIA Summer School
49