A model for sawtooth oscillations in tokamak experiments is outlined. A threshold criterion for the onset of internal kink modes and a prescription for the relaxed profiles immediately after the sawtooth crash have been implemented in a transport code that evolves the relevant plasma parameters. In this paper, applications of this model to the prediction of the sawtooth period and amplitude in projected ITER discharges are discussed. It is found that sawteeth can be stabilized transiently by the fusion alpha particles in ITER for periods that are long on the energy confinement timescale (). The sawtooth period depends on the amount of reconnected flux at the preceding sawtooth crash. When Kadomtsev's full reconnection model is used, the period can exceed 100 s. The sawtooth mixing radius following long duration sawtooth ramps can easily exceed half the plasma minor radius, raising questions about the desirability of transient sawtooth suppression.

Model for the sawtooth period and amplitude

A principal pressure limit in tokamaks is set by the onset of neoclassical tearing modes (NTMs), which are destabilized and maintained by helical perturbations to the pressure-gradient driven “bootstrap” current. The resulting magnetic islands break up the magnetic surfaces that confine the plasma. The NTM is linearly stable but nonlinearly unstable, and generally requires a “seed” to destabilize a metastable state. In the past decade, NTM physics has been studied and its effects identified as performance degrading in many tokamaks. The validation of NTM physics, suppressing the NTMs, and/or avoiding them altogether are areas of active study and considerable progress. Recent joint experiments give new insight into the underlying physics, seeding, and threshold scaling of NTMs. The physics scales toward increased NTM susceptibility in ITER, underlying the importance of both further study and development of control strategies. These strategies include regulation of “sawteeth” to reduce seeding, using static...

http://absimage.aps.org/image/DPP05/MWS_DPP05-2005-000592.pdf

Neoclassical Tearing Modes and Their Control

http://www.dct.tue.nl/New/Wouw/Automatica2013_Haring.pdf

Extremum-seeking control for nonlinear systems with periodic steady-state outputs

The characteristics of core electron-root confinement (CERC) in helical devices are illustrated using results from four different experiments: the Compact Helical System, Large Helical Device, TJ-II, and Wendelstein 7-AS. Common features include strongly peaked electron temperature profiles and large positive radial electric fields E r in the core region for discharges with sufficient central electron cyclotron heating (ECH). Such observations are consistent with a transition to the electron-root solution of the ambipolarity condition for E r , a feature of neoclassical theory that is unique to nonaxisymmetric configurations. The magnetic topology of the configuration plays a role in this transition, and thresholds are found for the particle density and ECH power, in accordance with neoclassical expectations. Neoclassical theory alone cannot explain all observations, however, as CERC formation can also be influenced by ECH-driven convective fluxes of localized electrons and by the presence of magnetic islands in the core region. This is the first report describing collaborative activities within the framework of the International Stellarator Profile Data Base.

/pdf/common-features-of-core-electron-root-confinement-in-helical-416ygkw3ph.pdf

Common Features of Core Electron-Root Confinement in Helical Devices

The physics model of electron cyclotron heating (ECH) and current drive (ECCD) is becoming well validated through systematic comparisons of theory and experiment. This work has shown that ECH and ECCD can be highly localized and robustly controlled in toroidal plasma confinement systems, leading to applications including stabilization of magnetohydrodynamic instabilities like neoclassical tearing modes, control and sustainment of desired profiles of current density and plasma pressure, and studies of localized transport in laboratory plasmas. The experimental work was supported by a broad base of theory based on first principles which is now well encapsulated in linear ray tracing codes describing wave propagation, absorption, and current drive and in fully relativistic quasilinear Fokker–Planck codes describing in detail the response of the electrons to the energy transferred from the wave. The subtle balance between wave-induced diffusion and Coulomb relaxation in velocity space provides an understandin...

Heating and Current Drive by Electron Cyclotron Waves

This lecture gives an overview of heating and current drive with electron cyclotron waves. We present the main theoretical aspects of wave propagation, wave absorption, and non-inductive current drive, as well as important technical aspects for the application of high power electron cyclotron waves, and the major achievements in their experimental application.

/pdf/electron-cyclotron-waves-2xmwy2kq0f.pdf

Electron Cyclotron Waves

The systematic design of a robust adaptive control strategy for the sawtooth period using electron cyclotron current drive (ECCD) is presented. Recent developments in extremum seeking control (ESC) are employed to derive an optimized controller structure and offer practical tuning guidelines for its parameters. In this technique a cost function in terms of the desired sawtooth period is optimized online by changing the ECCD deposition location based on online estimations of the gradient of the cost function. The controller design does not require a detailed model of the sawtooth instability. Therefore, the proposed ESC is widely applicable to any sawtoothing plasma or plasma simulation and is inherently robust against uncertainties or plasma variations. Moreover, it can handle a broad class of disturbances. This is demonstrated by time-domain simulations, which show successful tracking of time-varying sawtooth period references throughout the whole operating space, even in the presence of variations in plasma parameters, disturbances and slow launcher mirror dynamics. Due to its simplicity and robustness the proposed ESC is a valuable sawtooth control candidate for any experimental tokamak plasma, and may even be applicable to other fusion-related control problems. © 2012 IAEA, Vienna.

http://www.dct.tue.nl/New/Wouw/NuclearFusion2012_Bolder.pdf

Robust sawtooth period control based on adaptive online optimization

Plasma heating and non-inductive current drive by waves in the electron cyclotron range of frequencies are reviewed. Both theoretical aspects concerning wave properties, heating and current drive mechanisms, as well as the major experimental results are summarized.

Electron cyclotron heating and current drive

Resistive magneto-hydrodynamic instabilities are expected to limit the performance of nuclear fusion reactors. Prime examples are the sawtooth instability and the neoclassical tearing modes. The sawtooth instability will affect the refueling of the plasma core and the fast particle concentration. In reactor relevant conditions, the sawtooth can also trigger secondary instabilities. These are the neo-classical tearing modes, which can deteriorate the plasma performance or even disrupt the discharge. These modes (sawteeth and tearing modes) appear at specific locations in the tokamak plasma, associated with the distribution of the current density and the toroidal magnetic field. Localized current drive from electron cyclotron waves is foreseen as a possible actuator for the width of the tearing modes and the period of the sawteeth. Magnetic pick-up coils, electron cyclotron emission (ECE) and soft X-ray emission are the most likely sensors. A system approach for real-time detection, localisation and control of resistive magneto-hydrodynamic modes in tokamaks is presented. The system combines an ECE diagnostic for sensing of the instabilities in the same sight-line with a steerable Electron Cyclotron Resonant Heating and Current Drive (ECRH/CD) antenna. A model for the sawtooth is used to derive the linearized input-output relations (transfer functions) from the varying deposition location of high power Electron Cyclotron waves to the sawtooth period. The transfer functions are derived around a large collection of operating points. Proportional-integral-integral (PII) action can be applied to achieve fast and perfect tracking, while satisfying robustness constraints. The launcher dynamics seriously affects the closed loop performance in present-day experiments. Special emphasis is put on the issue of real-time sawtooth period detection. An algorithm based on time-scale wavelet theory and edge detection for accurate real-time sawtooth period estimation has been developed. The period is estimated by detecting subsequent crashes. The realized accuracy of the detection algorithm is well below the uncertainty of the crash period for most crashes. Multi-resolution analysis enables distinction between different sizes of sawtooth crashes due to the different sizes of wavelets (scales), resulting in an algorithm, which is robust and accurate. A methodology for fast detection of q = m/n = 2/1 tearing modes and retrieval of their location, rotation frequency and phase is presented. Set-points to establish alignment of the ECRH/CD deposition location with the centre of the tearing mode are generated in real-time and forwarded in closed-loop to the steerable launcher and as a modulation pulse train to the gyrotron.

Control of sawteeth and neo-classical tearing modes in tokamaks using electron cyclotron waves

The hot plasma dielectric tensor is discussed in its various approximations. Collisionless cyclotron resonant damping and ion/ electron Bernstein waves are discussed to exemplify the significance of a kinetic description of plasma waves. 10 refs., 3 figs.

E. Westerhof

Papers

Electron Cyclotron Waves

Robust sawtooth period control based on adaptive online optimization

Electron cyclotron heating and current drive

Control of sawteeth and neo-classical tearing modes in tokamaks using electron cyclotron waves

Hot plasma dielectric tensor