MHD and plasma control
Plasma control
Magnetically confined plasmas are prone to magneto-hydrodynamic (MHD) instability at the high temperatures and densities required for a fusion reactor. These instabilities reduce or destroy the performance of the reactor. In a Tokamak, the pressure of the hot plasma core displays, beyond a certain threshold value, a saw-tooth like relaxation oscillation. During the crash of the saw-tooth the core of the plasma is mixed with colder parts, lowering the temperature of the plasma core. More outward in the plasma, magnetic flux surfaces made up of helical field lines, may break up by the growth of neoclassical tearing modes (NTMs). These modes become unstable when beta, the ratio of kinetic over magnetic pressure, exceeds a threshold value. Nonlinear growth of the NTMs, often triggered by a saw-tooth crash, drops the pressure and eventually destroys the plasma.
In addition to limiting the central plasma pressure, saw-tooth oscillations can cause loss of energetic alpha particles generated by D-T fusion reactions. However, saw-teeth also may have a beneficial side; they remove the cooled down alpha particles from the plasma core and thus prevent the fusion reaction to choke in its own ashes. Beside saw-teeth and NTMs, many more MHD instabilities are predicted on ITER, for example, edge localized modes (ELMs), fishbone modes and various types of Alfvén modes. Many of these modes are destabilized by the energetic ion population produced by fusion.
The ITER machine will reach an unexplored regime of burning plasma in which the complex interplay between all these instabilities and their effect on plasma confinement and on the energetic particles population plays a key role. Operating such burning plasma requires a control strategy that manages these effects. In particular, an active control of saw-teeth and NTMs is essential for the development of a Tokamak fusion reactor.
In this wide field of research, the Tokamak Physics Group focuses on the control of NTM and sawteeth instabilities by modelling and experiment employing ECRH and ECCD. Complementary to this effort, the Computational Plasma Physics High Temperature Group concentrates on the theory and modelling of NTMs and saw-teeth. Experiments are carried out on external Tokamak facilities including TEXTOR, ASDEX Upgrade, MAST, ToreSupra and, in future, ITER.
MHD Modelling and Control
The FOM programme includes control of MHD instabilities by electron cyclotron resonance heating (ECRH) and current drive (ECCD). This particular heating and current drive method is favoured over other schemes for its much localised power deposition deep inside the plasma, for its absence of physical contact and impedance matching problems between antenna and plasma and for its modular hardware architecture. Theory of heating and current drive efficiency requires modelling of the electron cyclotron wave propagation and absorption by the plasma to obtain the power deposition profile coupled with Fokker-Planck calculations of the current density profile generated, for details see Electron Cyclotron Waves. Here, the focus is on the control of MHD instabilities by local current drive from ECCD. The modelling activities are naturally divided into two categories: 1) physics based MHD modelling, and 2) modelling of the complete feedback loop. Where the first aims at physics understanding of the processes involved, the second aims at the development and test of a robust control system.
Neoclassical tearing modes
The modified Rutherford equation provides the canonical model for the nonlinear evolution of NTMs. Symbolically, it is written
where each term on the right hand side describes the individual effect of mode growth caused by a particular physical process: 
is the classical tearing mode stability parameter; 
provides the nonlinear destabilization by the neoclassical bootstrap current, whilst the last two terms describe the stabilizing effects from local non-inductive current drive or heating inside the magnetic island, respectively. The NTM modelling work is focused on these last two effects. Although the current drive and heating terms enter separately into the modified Rutherford equation, in the experiment the non-inductive current drive will always be accompanied by the effect of the heating. Detailed analyses of the relative importance of these two effects indicate that the effect of the heating is wrongly neglected in much of the current literature.
The studies of mode stabilization as described by the modified Rutherford equation will be supplemented by full 3D MHD modelling of NTMs. This will serve to benchmark the relevant terms in the modified Rutherford equation against a more complete physics model. Finally all modelling results are to be compared to experiments.
A detailed study of the effect of localized ECCD on saw-teeth was performed on the TEXTOR Tokamak sited at the Forschungszentrum Jülich GmbH, Germany. The results allow interpretation in terms of the critical shear model for the onset of the saw-tooth crash. Also the effect of localized ECCD and ECRH on the growth of tearing modes (NTM) was investigated in detail. The work was validated by demonstrating effective NTM feed back control.
Experimental work on the feedback control of MHD instabilities by mm-wave beams will be extended to ASDEX Upgrade (AUG) at the Max-Planck Institute for Plasma Physics, Garching, Germany. An in-line ECE diagnostic suitable for integration into the AUG ECRH system is being developed. This poses additional requirements on continuous high power operation of the frequency selective coupler in the transmission line that separates the low power ECE radiation from the high power gyrotron radiation. The design pursued will be compatible with the ITER ECRH system.
Feedback control
When studying feedback control strategies, simplified models capturing the essential physics of all elements involved is required. Such control inspired models of the complete feedback loop modelling each individual element is employed for NTM stabilization and sawtooth period control. Figure 1 schematically shows the main elements involved in an NTM feedback control loop.
Figure 1: Feedback control loop
The feedback control loop of Figure 1 includes the plasma dynamics (the growth of the NTM as given by the modified Rutherford equation), the diagnostics acting as sensor in the control loop (for example, ECE and Mirnov coils), the data acquisition and analysis (to extract mode location and phase), the actuators (ECRH launcher and gyrotron power) and their control to achieve the required radial localization (through launcher steering) and deposition in the proper phase of the mode (through timing of the gyrotron power). Finally, the entire feedback loop is closed with the response of the plasma dynamics to these actions. A similar diagram can be made for saw-tooth control. Models for each of these components are being developed and implemented in a SIMULINK environment. These models serve to test various control strategies in order to develop a robust controller.
Neo-classical Tearing Mode suppression
A typical NTM control system consists of an ECCD actuator and a sensor to detect and localize the mode, often electron cyclotron emission (ECE) diagnostics. The position of the (rotating) mode is inferred from telltale perturbations on one of the ECE channels. Real-time equilibrium reconstruction and ray tracing are then used to translate mode location into an actuator setting, in this case the orientation of the mirror that launches the mm-wave beam.
Errors creep into this loop at many points. For example, the position of the ECRH layer is a function of the local magnetic field strength, which can be inferred real time from the current flowing through the toroidal magnetic field coils. Corrections, however, for the time varying poloidal magnet currents, the diamagnetic and paramagnetic plasma currents are necessary. Adding to these are relativistic and Doppler shift corrections on the electron cyclotron frequency observed.
On the actuator side, beam steering corrections are needed for mechanical errors caused by inertia, friction, hysteresis and thermal deformation. More importantly, however, is the refraction of the beam by the plasma density profile, which bends the beam away from a straight-line trajectory. Therefore, measurement of the plasma density profile accompanied by beam trajectory calculation needs to be carried out in real time to correct beam steering.
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In-transmision-line ECE
Inherent accurate beam steering is obtained by using the same line of sight for the ECE sensor signal as is used for the high power beam actuator. The principle is shown in Figure 2. The actuator beam is steered such that the NTM located at a specific ECE frequency matches with the actuator frequency. Here, it is assumed that electron cyclotron emission and absorption profiles are identical (true in case of constant temperature over the region). The island phase is derived from the phase of the ECE signal measured.
Figure 2: Sketch of a real-time control system for tearing modes using the same sight line for the ECRH actuator and the ECE sensor. A high power mm-wave beam (red) generated by a gyrotron is injected into the plasma by a focusing- and a steering-mirror. Where the heating beam crosses the locus of electron cyclotron resonance, power is deposited in the plasma and drives a non-inductive current. The ECE radiation (yellow beam), observed along the same optical path at slightly different frequency, is separated from the heating beam by a resonant dielectric plate (Fabry-Perot interference filter) and, after further filtering, detected by the ECE radiometer. The spectrum obtained is processed to generate feedback signals to the steering mirror and the gyrotron power supply.
An in-transmission-line ECE diagnostic has been developed and commissioned on TEXTOR. Employing a resonant dielectric quartz plate of the Fabry-Perot interference type, the high power beam is separated from ECE frequencies. The plate is highly transparent (> 95%) at the beam frequency of 140 GHz, but has maxima in reflection (~35%) at 132.5, 135.5, 138.5, 141.5, 144.5, and 147.5 GHz at the emitted ECE frequencies, selected such as to yield required spatial resolution. A second resonant dielectric plate in tandem with the first, followed by an 80dB notch filter protects the sensitive ECE radiometer from any remaining 140 GHz stray radiation. Figure 3 show the set-up of this quasi-optical system and a picture of the hardware as installed on TEXTOR.
Figure 3: Left: Schematic of the ECE diagnostics beam path (green) separated and travelling in reverse direction of the high power ECRH input beam (red). Right: actual hardware on TEXTOR of the in-line ECE diagnostics.
Extensive tests have been performed, which have shown that the system performs according to expectation. In particular, the gyrotron stray radiation at the radiometer horn is successfully reduced to levels below 1 W, which guarantees the safe operation of the ECE radiometer. A further 80 dB notch-filter ensures that the gyrotron stray radiation entering the radiometer is well below the level of the ECE radiation to be measured. Measurements of the ECE spectrum have been obtained during high power ECRH pulses demonstrating the feasibility of in-line ECE measurements.
Launcher control
Dedicated controllers have been developed for the TEXTOR ECRH launcher (J.W. Oosterbeek, et al., Rev. Sci. Instr., 2008). Dynamical properties are obtained from Frequency Response Function measurements. Physical system limitations introduced by friction, nonlinear dynamics, resonances, time delays etc. are thereby identified. Frequency domain tuning procedures are used to derive controllers which account for these limitations and result in optimal performance in terms of operational stability, maximum bandwidth and positioning accuracy. Simulation models, representing the TEXTOR ECRH launcher, serve in the controller design process. The TEXTOR mm-wave beam launcher and steering mirror are shown in Figure 4.
The set points for steering of the TEXTOR ECRH launcher are generated from the inline ECE data. The controller executes the following steps. First, the signals from the six ECE channels are fed into a tearing mode recognition and localization algorithm. When an NTM is detected, the algorithm provides the ECE frequency at which the mode is located. This frequency corresponds to a unique position in the plasma determined by the position of electron cyclotron resonance in the magnetic field topology. Subsequently, this frequency is compared with the reference 140 GHz actuator frequency. The difference signal provides the correction for the elevation angle of the launching mirror. When alignment is achieved, i.e. when the ECE frequency at which the mode is detected matches with the ECRH frequency, the ECRH power is switched on. During control, the feed-back loop remains active, keeping the ECRH beam aligned with the mode. These control steps are hard-wired into a field programmable gate array (FPGA).
Figure 4: Left: sketch of the TEXTOR ECRH launcher. Right: photograph of the steering mirror unit.
Suppression of m=2, n=1 NTMs by localized ECRH on TEXTOR
Although TEXTOR plasmas generally do not reach the beta values required to trigger NTMs, reproducible tearing modes can be generated with the help of Resonant Magnetic Perturbations (RMP) produced by the Dynamic Ergodic Divertor (DED). The latter consists of a set of helical coils that are powered to excite a m/n = 3/1, 6/2 or 12/4 RMP at the edge of the plasma. In its 3/1 configuration, the 2/1 side band provides the dominant RMP inside the plasma and can routinely trigger a m=2, n=1 tearing mode. These modes have formed the basis for an extensive study on the effects of ECRH and ECCD on their stability (Reference: E. Westerhof, et al. Nucl. Fusion 47 (2007) 85).
It was shown that under TEXTOR conditions the mode suppression can almost entirely be attributed to the effect of localized heating inside the island, whereas the effect of the non-inductively driven current is negligible. 2D ECE-Imaging measurements have been used to diagnose the local temperature perturbation inside the island. The measured temperature perturbations and the observed changes in the growth rate of the magnetic island are consistent with modelling of these results on the basis of modified Rutherford equation (Reference: I.G.J. Classen, et al., Phys. Rev. Letters 98 (2007) 035001).
Figure 5: The figure shows the width of the m=2, n=1 magnetic island island as a function of the location of the ECRH power deposition. Efficient suppression of the mode is only obtained when the power deposition coincides almost exactly with the position of the island. This indicates that the suppression of the island is mainly caused by the power deposited inside the island.
Figure 6: Results are shown of in-line ECE measurements on TEXTOR discharge #106913, applying a 2 s, 400 kW ECRH pulse at t=2s. Prior to application of ECRH, a clear sawtooth instability is observed and sawtooth inversion is seen to occur between the ECE channels 132.5 and 135.5 GHz, see expanded view on the right. The 140 GHz resonance is on the high field side (inner part) of the tokamak, this indicates that the 140 GHz ECRH power deposition will occur well outside the sawtooth inversion area. After switch-on of ECRH at t=2s the sawteeth disappear as predicted for heating outside the inversion radius.
Collaborations
- Institute for Energy Research 4 – Plasma Physics, Forschungszentrum Jülich GmbH, Jülich, Germany: TEXTOR-team
- Systems and Control Engineering, TU/e, Eindhoven, Netherlands: P. Nuij, M. Steinbuch, E.M.M. Demarteau
- TNO, Delft, Netherlands: N. Doelman, G. Witvoet
- Max-Planck Institute for Plasma Physics, Garching, Germany: AUG-, and ECRH-teams
- Istituto di Fisica del Plasma, CNR Milano, Milano, Italy: S. Cirant, S. Nowak
- DRFC, CEA-Cadarche, St. Paul-lez-Durance, France: G.T.A. Huysmans
- EFDA JET Taskforce on MHD
- EFDA Topical Group on MHD
- EFDA Taskforce on Integrated Tokamak Modelling