08/09/2008 E.Lazzaro EDF/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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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)

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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 J boot loss?

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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 (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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 5 Non ideal effects Helical Perturbations

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 6 isobaric surfaces of toroidal genus 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 The B field lines “pitch” is constant on each nested surface (Isobaric) magnetic surfaces where have a different topology: there are alternate O and X singular points that : axisymmetry is broken and divB=0 allows a B r component (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 B r 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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  dependent !

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 11 Current driven tearing modes physics Outer region - marginal ideal MHD - kink mode. The torque balance requires: Solved with proper boundary conditions to determine the discontinuity of the derivative 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  s: reconnected helicalflux

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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<<r s ) gives: Inner region Inner region - includes effects of inertia, resistivity, drifts, viscosity, etc  l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 x=r-r s slab coordinate from rational surface q(r s )=m/n helical flux reconnected on the rational surface integrals and averages on island  x

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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 

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 15 Summary of RMHD equations Resistive-neoclassical MHD fluid model

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 16 Bootstrap Current Generalised parallel Ohm’s law with electron viscosity effects Constant on magnetic surfaces Electron viscous stress damps the poloidal electron flow - new free energy source. Mechanism of bootstrap current

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 17 The NTM drive mechanism An initial perturbation( W seed ) 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  Poloidal angle

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 18 Construction of the nonlinear island equation 1-A nonlinear averaging operator over the helical angle  =m  -n  t  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 gives 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(  B r r s / B  nq / ) 1/2 Grad-Shafranov equation “neoclassical” currents R.F.Current drive

19 Modified Rutherford Equation NTM evolution (Integrating Faraday-Ampere on island) geom. factor (de)stabilising factor, <0 in NTM (# TM) J bootstrap Term >0 pressure gradient & curvature Term <0 Polarisation Term >0, 0, <0 Electron Cyclotron CD Term resistive wall Term G.Ramponi, E. Lazzaro, S. Nowak, PoP 1999

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 20 Threshold Physics Makes an NTM Linearly Stable and Non-linearly Unstable =  ’ r s + 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) w pol  (L q /L p ) 1/2  1/2   I ~ 2 cm

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 21 The Modified Rutherford Equation: discussion w 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? W sat W thres

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 22 Threshold Physics Makes an NTM Linearly Stable and Non-linearly Unstable =  ’r s + m/n=2/1  ’r s =-2  p =0.6  1/2  L q /L p  =0.56 c(  ) =1 r s =1.54 m a=2 m unstable stable

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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. r abs ≈ r O-point - 3 cm r abs ≈ r O-point r abs ≈ r O-point + 1 cm r abs ≈ r O-point + 2 cm

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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: H m,n = efficiency by which a helical component is created by island flux surface averaging H 0,0 =modification of equilibrium current profile

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 25 CD efficiency to replace the missing bootstrap current H m,n depends on: w/w cd 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 26 Larger CD efficiency with narrow J CD profiles Note: within the used model, in case of perfect alignment, the (2,1) mode is fully suppressed with 50% modulated EC power, I cd = 3% I p (r s ) (P EC ~ 7 MW by FS UL), when w cd =2.5 cm larger w cd would reduce the saturated island width (partial stabilization) narrow, well localized Jcd profiles are a major request for the ITER UL! --- stable

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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 r m/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 r dep, 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 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…

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 29 Linearized equation near threshold Dimensionless state variables and linearization near threshold W=W t  =  T Linear state system Control vector Mode amplitude x 1 and frequency x 2 are coupled through a 12 EC driven current External momentum input

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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 rank (Q)=2 if both b1 and b2 are non zero In our case the condition, mode rotation control is necessary amplitude control b 1 frequency control b 2 EC driven current External momentum input

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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)=x 0, 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, (1959); R.Bellman,I. Glicksberg O.Gross, “On the bang-bang control problem” Q. Appl. Math (1956)

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 33 Concept of experimental set-up for ECCD control of Tearing modes “Just align” strategy:Find optimal angles a,b to minimize when r dep (R M, Z M ) 

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 34 Estimate of “a priori” r dep (  ) Best poloidal angle  for three toroidal angles  (0,  /18.  /9) example of minimization of | rdep(a,b) – rm/n|2

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 35 Experiments of automatic TM stabilization by ECRH/CD on FTU

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 36 Island recognition with T e diagnostics T e flattening  loss of bootstrap current  rotating NTM  antisymmetric T e oscillations ECH (associated with ECCD) may mask strict antisymmetry Multiple zeros possible  Te/T 0 (r-r m,n )/W c

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 37 Correlation of the ECE fluctuations measured between nearby channels, both for natural and “heated” islands (e.g. r 1 =r s -x, r 2 =r s+ 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 r m/n of q=m/n Position r m/n,mea measurement

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 38 Principle of r island tracking algorithm P ij ≈ 1 if both i and j are on the same side with respect to the island O-point. P ij ≈ -1 if on opposite sides. A positive concavity in the P ij sequence locates the island. channels 1 0 P i j

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 39 Position r island 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 ECE n space Gain J. Berrino,E. Lazzaro,S. Cirant et al., Nucl. Fusion 45 (2005) 1350

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 40 Tracking of rational surfaces r m/n (2,1) axis (1,1) FTUAUG 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 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 r NTM, r dep Actuator basic control variables : beam steering angle , and Power modulation

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 42 Assimilation (Bayesian filtering) a-posteriori pdf likelihood 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 43 Algorithms for real time NTM control Cross-correlation Estimate for ECW power r dep in shot 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 44 Algorithms for real time NTM control Real time estimate ECW power r dep (t) for shot 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)

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 45 FTU: B tor = 5.6 T plasma axis Resonance 140GHz mirror ECE channels EC beam Gyrotron 1 Gyrotron ECRH power deposition at different R by changes of the angle of the mirror

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 46 P i,B P i,A FTU Shot 27714:real-time recognition r dep f mod,Gy1 = 100 Hz f mod,Gy3 = 110 Hz Plasma axis Correlation functions of the two gyrotrons The deposition radius of each beam is detected by the maximum in  T e,ECE -ECH correlation. Different beams are recognized by different ECH timing. Gy1 Gy3 Gyrotron 1

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 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

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 48 [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 References

08/09/2008 E.Lazzaro EDF/CEA/INRIA Summer School 49 FINE