Model-based design of multi-mode feedback control in the RFX-mod experiment

Active feedback stabilization of multiple resistive wall modes (RWMs) has been successfully proven in the EXTRAP T2R reversed field pinch. One of the features of plasma discharges operated with active feedback stabilization, in addition to the prolongation of the plasma discharge, is the sustainment of the plasma rotation. Sustained rotation is observed both for the internally resonant tearing modes (TMs) and the intrinsic impurity oxygen ions. Good quantitative agreement between the toroidal rotation velocities of both is found: the toroidal rotation is characterized by an acceleration phase followed, after one wall time, by a deceleration phase that is slower than in standard discharges. The TMs and the impurity ions rotate in the same poloidal direction with also similar velocities. Poloidal and toroidal velocities have comparable amplitudes and a simple model of their radial profile reproduces the main features of the helical angular phase velocity. RWMs feedback does not qualitatively change the TMs behaviour and typical phenomena such as the dynamo and the ‘slinky’ are still observed. The improved sustainment of the plasma and TMs rotation occurs also when feedback only acts on internally non-resonant RWMs. This may be due to an indirect positive effect, through non-linear coupling between TMs and RWMs, of feedback on the TMs or to a reduced plasma-wall interaction affecting the plasma flow rotation. Electromagnetic torque calculations show that with active feedback stabilization the TMs amplitude remains well below the locking threshold condition for a thick shell. Finally, it is suggested that active feedback stabilization of RWMs and current profile control techniques can be employed simultaneously thus improving both the plasma duration and its confinement properties.

The reversed-field pinch (RFP) can spontaneously transition from an axisymmetric magnetic topology to a 3D-helical geometry. Investigations on fast ion transport associated with energetic particle driven Alfvén instabilities, tearing mode induced stochasticity, and neoclassical effects have been performed on the Madison Symmetric Torus. STELLGAP produced shear-Alfvén continua seeded with V3FIT 3D-equilibrium reconstructions describe the response of Alfvénic bursting activity as a direct consequence of the equilibrium change on the fast ion resonance. Far infrared interferometry resolved electron density perturbations associated with the bursts provide a spatial measurement of the mode structure and support the reconstructions. The bursts produce no global resonant fast ion transport; however, their disappearance at a high core-resonant amplitude implies other transport mechanisms at play. Neutral particle analysis and neutron signals suggest fast ion losses at sufficient core tearing mode strength, supporting the lack of Alfvénic activity. The guiding-center code ORBIT corroborates rapid fast ion loss times in the helical state largely as a consequence of remnant tearing modes. Additionally, ORBIT simulations demonstrate little neoclassical enhancement of particle transport. While superbanana orbits may exist, the growth in the core-resonant fast ion island and the associated secondary mode overlap govern the largest transport process, leading to robust fast ion losses in the 3D-RFP.

A non-linear, two-dimensional, resistive MHD computer code to study the non-linear behaviour of the m = 0, 1, and 2 tearing modes in a reversed field pinch is used. For the cases m ≠ 0, a co-ordinate transformation which assumes helical symmetry is employed to reduce the dimensionality of the problem from three to two. The force-free Bessel function model equilibrium is used. It is found that the most dangerous resistive instability in this case is the m = 1 tearing mode, because of its larger growth rate and extended period of exponential growth.

Tokamak plasma experiments on the DIII-D device (Luxon et al 2005 Fusion Sci. Tech. 48 807) demonstrate high-performance, negative central shear (NCS) equilibria with enhanced stability when the minimum safety factor exceeds 2, qualitatively confirming theoretical predictions of favorable stability in the NCS regime. The discharges exhibit good confinement with an L-mode enhancement factor H89 = 2.5, and are ultimately limited by the ideal-wall external kink stability boundary as predicted by ideal MHD theory, as long as tearing mode (TM) locking events, resistive wall modes (RWMs), and internal kink modes are properly avoided or controlled. Although the discharges exhibit rotating TMs, locking events are avoided as long as a threshold minimum safety factor value is maintained. Fast timescale magnetic feedback control ameliorates RWM activity, expanding the stable operating space and allowing access to values approaching the ideal-wall limit. Quickly growing and rotating instabilities consistent with internal kink mode dynamics are encountered when the ideal-wall limit is reached. The RWM events largely occur between the no- and ideal-wall pressure limits predicted by ideal MHD. However, evaluating kinetic contributions to the RWM dispersion relation results in a prediction of passive stability in this regime due to high plasma rotation. In addition, the ideal MHD stability analysis predicts that the ideal-wall limit can be further increased to by broadening the current profile. This path toward improved stability has the potential advantage of being compatible with the bootstrap-dominated equilibria envisioned for advanced tokamak (AT) fusion reactors.

An investigation of the effect of resistivity on the linear stability of line-tied magnetohydrodynamic (MHD) modes is presented in cylindrical geometry, based on the method recently developed in the paper by Evstatiev et al. [Phys. Plasmas 13, 072902 (2006)]. The method uses an expansion of the full solution of the problem in one-dimensional radial eigenfunctions. This method is applied to study sausage modes (m=0, m being the poloidal wavenumber), kink modes (m=1), and m=2 modes. All these modes can be resistively unstable. It is found that m≠0 modes can be unstable below the ideal MHD threshold due to resistive diffusion of the field lines, with growth rates proportional to resistivity. For these resistive modes, there is no indication of tearing, i.e., current sheets or boundary layers due to ideal MHD singularities. That is, resistivity acts globally on the whole plasma column and not in layers. Modes with m=0, on the other hand, can exist as tearing modes if the equilibrium axial magnetic field reverses sign within the plasma.

Controlled magnetohydrodynamic mode sustainment in the reversed-field pinch: Theory, design and experiments

The interaction of a static resonant magnetic perturbation (RMP) with a tearing mode (TM) is becoming a relevant topic in fusion plasma physics. RMPs can be generated by active coils and then used to affect the properties of TMs and of the corresponding magnetic islands. This paper shows how the feedback system of the EXTRAP T2R reversed field pinch (RFP) can produce a RMP that affects a rotating TM and stimulate the transition to the so-called quasi-single helicity (QSH) regime, a RFP plasma state characterized by a magnetic island surrounded by low magnetic chaos. The application of the RMP can increase the QSH probability up to 10% and enlarge the size of the corresponding island. Part of the experimental results are supported by a theoretical study that models the effect of the active coils on the magnetic island.

Reversed field pinches (RFPs) share with tokamaks and stellarators the experimental evidence of an upper limit for the maximum value of the electron density at which they can operate. Above a certain density level, well described by the Greenwald law for tokamaks and RFPs, a radiative collapse with strong plasma cooling is observed, predominantly due to processes occurring at the plasma boundary. In the RFX-mod RFP close to the density limit a radiating belt, poloidally symmetric and toroidally localized, develops in the region where the plasma is shrunk as an effect of the m = 0 tearing modes. The phenomenology recalls that of MARFES or plasma detachment, though, unlike tokamaks, the appearance of the radiating belt is associated with a soft landing of the plasma discharge.The paper reports the experimental pattern of the RFX-mod plasmas close to the density limit, including density and radiation profiles, plasma flow and turbulence. Particles are toroidally conveyed towards the region of maximum shrinking of the plasma column where they accumulate. The interpretation is related to the topology of MHD m = 0 and m = 1 modes: the reconstruction of the magnetic topology shows that the highly radiating region corresponds to the presence of peripheral m = 0 magnetic islands well detached from the wall. The emerging indication is that in RFPs a reduction of the m = 0 activity could be a way to overcome the density limit.

It is well known that stabilization of the resistive wall mode (RWM) may allow fusion power to be significantly increased for a given magnetic field in advanced tokamak operation. The principle of stabilization of the RWM by rotation has been established both experimentally and theoretically. Recent experimental results have indicated stabilization of the RWM has been achieved with very small levels of rotation using balanced neutral beam injection. The framework of Connor et al (Connor et al 1988 Phys. Fluids 31 577) is used to develop two ideal plasma analytic toroidal models where stepped pressure profiles and careful ordering of terms are used to simplify the analysis. The first model has one resonant layer in the plasma and two resistive walls and the second has two resonant layers and one resistive wall. The RWM can be stabilized with slow rotation (∼0.5%ΩA, where ΩA is the Alfvénic rotation frequency) of a secondary resistive wall. A secordary rotating resistive wall can only stabilize the plasma if a perfectly conducting wall at that location would stabilize the plasma. Differential rotation in the plasma is investigated by rotating two resonant layers in the plasma at different rates. It is found in this model that differential rotation of the outer resonant surface can stabilize the RWM, with no rotation required at the inner surface. However, stabilization is not possible with the outer surface static and the inner surface rotating. Experimental rotation profiles for balanced neutral beam shots have higher levels of rotation near the plasma edge showing differential rotation between the outer surface and the wall may be part of the RWM stabilization mechanism.

Advanced tokamaks (AT) are distinguished from conventional tokamaks by their high degree of shaping, achievement of profiles optimized for high confinement and stability characteristics, and active stabilization of MHD instabilities to attain high values of normalized beta and confinement. These high performance fusion devices thus require accurate regulation of the plasma boundary, internal profiles, pumping, fueling, and heating, as well as simultaneous and well-coordinated MHD control action to stabilize such instabilities as tearing modes and resistive wall modes. Satisfying the simultaneous demands on control accuracy, reliability, and performance for all of these subsystems requires a high degree of integration in both design and operation of the plasma control system in an advanced tokamak. The present work describes the approach, benefits, and progress made in integrated plasma control with application examples drawn from the DIII-D tokamak. The approach includes construction of plasma and system response models, validation of models against operating experiments, design of integrated controllers which operate in concert with one another as well as with supervisory modules, simulation of control action against off-line and actual machine control platforms, and iteration of the design-test loop to optimize performance.

Numerical results are presented on control of magnetohydrodynamic (MHD) modes in reversed field pinches (RFPs) for a geometry with two resistive walls. We use measurements of the normal component of the magnetic field and introduce the use of both tangential components. In Richardson et al (2010 Phys. Plasmas 17 112511), RFP control studies were performed sensing the radial (normal) component of the magnetic field and a single tangential component just inside the wall, showing that it is possible to stabilize the MHD modes in an RFP for current up to the ideal plasma–ideal wall limit in that configuration. Here, we extend our modeling by including two resistive walls, in a configuration relevant to experiments such as RFX-mod, and measuring all three magnetic field components, i.e. including a second tangential component, as an exploratory effort. We present our study incrementally, starting with a single resistive wall, and conclude that with the first tangential sensor located inside the wall, the plasma can be stabilized up to the ideal plasma–ideal wall limit, as in Richardson et al. With the first tangential sensor outside the wall, stabilization is possible only up to the ideal wall–resistive plasma (tearing) limit. We then show that for experimentally relevant parameters the thin-wall approximation is indeed valid for the MHD modes of interest but invalid for the high-frequency magnetosonic mode (Richardson et al) driven by the (first) tangential component feedback. In fact, when a thick-wall formulation with realistic parameters is considered, the high-frequency magnetosonic mode is found to be destabilized only for a very high gain parameter, and we conclude that this mode can be ignored for an experimentally relevant analysis. Consequently, the plasma can be stabilized in a much larger region of feedback gain parameter space than found in Richardson et al. In the presence of two walls, with the first tangential component measured just outside the inner wall and with RFX-mod relevant time constants, we show that feedback control can stabilize the plasma at currents much larger than the ideal wall–resistive plasma limit. The current limit is still less, however, than the ideal plasma–ideal wall limit. Use of the second tangential component appears in all cases to lead to significantly different but not necessarily improved feedback stabilization. These results may lead to better understanding and improved stability properties in current-day RFP experiments through robust access to quasi-single-helicity states.

Strong self-heating of ions is observed in the reversed field pinch (RFP). During a sawtooth crash in the Madison Symmetric Torus RFP, the ion temperature can spontaneously double in ∼100μs. It is also observed that high Z impurities are heated more strongly than bulk ions. The possibility of ion heating due to tearing instabilities at sawtooth crash is examined. Heating scenarios due to viscous damping of strongly localized perpendicular and parallel flows driven in the vicinity of resonant surface in tearing mode are considered. Flow amplitudes and spatial scales are estimated from linear and nonlinear resistive magnetohydrodynamic modeling. The heating rates are found from kinetic models with different levels of approximation, up to solving kinetic equation with a Landau collision operator. Results show reasonable agreement of the modeled impurity heating rate with the experiment, while the estimated bulk ions heating is somewhat weaker than in the experiment. Further theoretical and experimental study are required for a more definite conclusion as to whether it is the main ion heating mechanism or if there is some other important ion heating scenario.

We calculate momentum transport from tearing fluctuations in a reversed field pinch with sheared flow, considering both the effect of a single tearing mode (through quasilinear theory and MHD computation) and multiple tearing modes (through nonlinear MHD computation). A single tearing mode transports momentum, via Maxwell and Reynolds stresses, more rapidly than classical viscous forces. Moreover, the transport is enhanced by nonlinear coupling of multiple modes.

Magnetohydrodynamic computation demonstrates that feedback can sustain reversal and reduce the loop voltage in resistive shell reversed field pinch (RFP) plasmas. Edge feedback on approximately 2R/a tearing modes resonant near the axis is found to restore plasma parameters to nearly their levels obtained with a close fitting conducting shell. When original dynamo modes are stabilized, neighbouring tearing modes grow to maintain the RFP dynamo more efficiently. This suggests that the experimentally observed limits on RFP pulse lengths to the order of the shell time can be overcome by applying feedback to a few helical modes

Confinement in the reversed field pinch (RFP) has been shown to increase strongly with current profile control. The MST RFP can operate in two regimes: the standard regime with a naturally occurring current density profile, robust reconnection and dynamo activity; and the improved confinement regime with strong reduction in reconnection, dynamo and transport. New results in standard plasmas include the observation of a strong two-fluid Hall effect in reconnection and dynamo, the determination that the m = 0 edge resonant mode is nonlinearly driven, and the determination that tearing modes can lock to the wall via eddy currents in the shell. New results in improved confinement plasmas include observations that such plasmas are essentially dynamo-free, contain several isolated magnetic islands (as opposed to a stochastic field) and contain reduced high frequency turbulence. Auxiliary current drive and heating is now critical to RFP research. In MST, a programme to apply auxiliary systems to the RFP is underway and progress has accrued in several techniques, including lower hybrid and electron Bernstein wave injection, ac helicity injection current drive, pellet injection and neutral beam injection.