E.Lazzaro

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 effectsformation 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 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
• Since the isobaric surfaces
  (pnT) 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

5
Non ideal effects Helical Perturbations
6
Overview of basic concepts

• Tokamaks have good confinement because the
  magnetic field lies on isobaric surfaces of
  toroidal genus
• The B field lines pitch
  is constant on each nested
  surface
• (Isobaric) magnetic surfaces where q(Y)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 divB0
  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

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)

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

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

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
  Ohms law, including external non inductive
  contributions

w dependent!
11
Current driven tearing modes physics
Boundary Layer problem
Outer region - marginal ideal MHD - kink mode.
The torque balance requires
A linear perturbation is governed by an
equation that is singular on the mode rational
surface where kB 0
ys
Singularity at qm/n !!
Solved with proper boundary conditions to
determine the discontinuity of the derivative
Ys reconnected helicalflux
12
Current driven tearing modes physics

• The discontinuous derivative equivalent to
  currents, localised in a layer across the qm/n
  surface, where ideal MHD breaks down
• Amperes law relates the dB 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
  (wltltlltltrs) gives
• Inner region - includes effects of inertia,
  resistivity, drifts, viscosity, etc

y
-l
l
Linear Dispersion relation
Linear
Growth rate
13
Geometry Terminology

• contours of constant helical flux
• magnetic shear length cylindrical safety
  factor
• island instantaneous phase
• xr-rs slab coordinate from rational surface
  q(rs)m/n
• helical flux reconnected on the rational surface
• integrals and averages
  on island

x
?
14
Neoclassical Tearing Modes (NTM)

• In a tearing-stable plasma (?0lt0)
• Initial island large enough to flatten the local
  pressure
• gt 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 model

16
Bootstrap Current
Mechanism of bootstrap current
Constant on magnetic surfaces
Generalised parallel Ohms law with electron
viscosity effects
Electron viscous stress damps the poloidal
electron flow - new free energy source.
17
The NTM drive mechanism
Consider an initial small seed island
Perturbed flux surfaces lines of constant W

• 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

18
Construction of the nonlinear island equation

• 1-A nonlinear averaging operator over the helical
  angle xmq-nf-wt makes
• 2-The parallel current is obtained solving the
  current closure (quasineutrality) equation
  ,averaging and and inserting it in Amperes law
• 3-Averaging Faraday law and eliminating ltJ//gt
  gives
• 4-An integration weighted with cosx, 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

neoclassical currents
R.F.Current drive
Grad-Shafranov equation
19
Modified Rutherford Equation
NTM evolution (Integrating Faraday-Ampere on
island)
geom. factor
(de)stabilising factor, lt0 in NTM ( TM)
Jbootstrap Term gt0
pressure gradient curvature Term lt0
Polarisation Term gt0, lt0
Electron Cyclotron CD Term
resistive wall Term
G.Ramponi, E. Lazzaro, S. Nowak, PoP 1999
20
Threshold Physics Makes an NTM Linearly Stable
and Non-linearly Unstable
? rs

• 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

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

wpol? (Lq/Lp) 1/2 ? 1/2 ??I 2 cm

• c(?,?i) polarization term also depends on
  frequency of rotating mode, stabilizing only if
  0gt?gt??i (J, Connor,H.R. Wilson et.al,1996)

21
The Modified Rutherford Equation discussion

• Need to generate seed island
• additional MHD event
• poorly understood?

• Stable solution
• saturated island width
• well understood?

w
Wsat
Wthres

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

22
Threshold Physics Makes an NTM Linearly Stable
and Non-linearly Unstable
?rs
m/n2/1
unstable

?rs-2
rs1.54 m a2 m
?p0.6
stable
?1/2 ?Lq/Lp?0.56 c(?) 1
23
rabs rO-point- 3 cm
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
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 -wave beam
direction.
24
Co-CD can replace the missing bootstrap current
Localized Co-CD at mode rational surface may both
increase the linear stability and replace the
missing bootstrap current

where Hm,n efficiency by which a helical
component is created by island flux surface
averaging H0,0 modification of equilibrium
current profile
25
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 qm/n

50 on - 50 off
No-misalignment

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
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 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

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 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

29
Linearized equation near threshold
Dimensionless state variables and linearization
near threshold WWt wwT
Linear state system
Mode amplitude x1 and frequency x2 are coupled
through a12
Control vector
EC driven current
External momentum input
30
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
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
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 qm/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)

32
Formal aspects of the control problem

• Definition An admissible (piecewise
  measurable in a set ? ) steering function u is
  optimal if for some tgt0 x(t,u) 0 and if
  x(t,u)?0 for 0lt tlt t for all u(t) ? ?
• 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) ? ? such that x(t,u)0,
  for tgt0, then there is an optimal steering
  function u in ?.
• 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 lt e

u(t)
t
33
Concept of experimental set-up for ECCD control
of Tearing modes
(RM, ZM)
a
rdep

• Just align strategyFind optimal angles a,b to
  minimize
• when

34
Estimate of a priori rdep(a,b)
example of minimization of rdep(a,b) rm/n2
Best poloidal angle a for three toroidal angles
b(0, p/18. p/9)
35
Experiments of automatic TM stabilization by
ECRH/CD on FTU
36
Island recognition with Te diagnostics
dTe/T0
(r-r m,n)/Wc
Multiple zeros possible
Te flattening ? loss of bootstrap
current ? rotating NTM ? antisymmetric Te
oscillations
ECH (associated with ECCD) may mask strict
antisymmetry
37
Correlation of the ECE fluctuations measured
between nearby channels , both for natural and
heated islands (e.g. r1rs-x, r2rsx)
Position rm/n,mea measurement

• The phase jump is effective on detecting the
  qm/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 qm/n

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.
39
Position risland measurement from three ECE
channels
Example of real-time data processing for O-point
location in the ECEn space
Gain
High-pass filter
Correlation
Second derivative ? maxima (minima)
J. Berrino,E. Lazzaro,S. Cirant et al., Nucl.
Fusion 45 (2005) 1350
40
Tracking of rational surfaces rm/n
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 a, and Power modulation

42
Assimilation (Bayesian filtering)
likelihood function , measured data
a-posteriori pdf
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
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

Bayesian Filter p(rd)L(dr)p(r)/p(d)?
L(dr)p(r)
44
Algorithms for real time NTM control
Bayesian Filter p(rd)L(dr)p(r)/p(d)

• Real time estimate ECW power rdep (t) for shot
  17107 in ASDEX-U (G. DAntona et al, Proc.,
  Varenna 2007
• Evidence p(d)

45
ECRH power deposition at different R by changes
of the angle of the mirror
FTU Btor 5.6 T
EC beam
Gyrotron 3
ECE channels
Gyrotron 1
mirror
1 2 3 4 5 6 7 8 9 10 11
12
Resonance 140GHz
plasma axis
46
FTU Shot 27714real-time recognition rdep
fmod,Gy1 100 Hz fmod,Gy3 110 Hz
Correlation functions of the two gyrotrons
Gyrotron 1
The deposition radius of each beam is detected
by the maximum in ?Te,ECE -ECH correlation.
Different beams are recognized by different
ECH timing.
Pi,A
Plasma axis
Pi,B
47
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?)
action low ? high duty cycle
t feedback ON0.4 s
48
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.DAntona IEEE Trans 2005
49
FINE