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Rotation of the plasma and MHD modes in tokamaks has been shown to stabilize resistive wall and tearing modes as well as improve confinement through suppression of edge turbulence. In this work, we control mode rotation with a biased electrode inserted into the plasma of the High Beta Tokamak-Extended Pulse's facility in conjunction with its active GPU (Graphical Processing Unit) feedback system. We first characterize a negative linear relationship between the electrode voltage and mode rotation. Using this relationship, we design, simulate, and implement a proof-of-concept, GPU-based active-control system, which shows consistent success in controlling mode rotation in both feedforward and feedback operation. Controllability is limited by operating conditions, the electrode's voltage range, and by the electrode's proximity to the vessel's walls. The final control system has a 15 μs cycle time, but the addition of various signal filters results in a full cycle latency of 200 μs.

Investigations of biasing induced H-mode discharges on HBT-EP show that the edge turbulence is consistent with the ion temperature gradient) mode and have allowed for the controlled observation of the effect of applied flow shear on the turbulence. Measurements of the radial wavenumber spectrum of floating potentials at the edge show that the turbulence intensity decreases with increasing shift in the spectrum average ⟨kr⟩ when increasing amounts of bias probe voltage (and increasing amounts of flow shear) is applied. This is in agreement with the spectral shift model [Staebler et al., Phys. Rev. Lett. 110, 055003 2013] for turbulence suppression via sheared flow. A shift in the wavenumber spectrum occurs at applied electrode voltages and currents below the threshold needed for an L–H transition, and a dithering transition is obtained when biasing near the threshold. Suppression of blob-filament turbulence in the scrape-off layer (SOL) precedes the L–H transition, with the SOL turbulence remaining low throughout the dithering phase, despite the modulation of turbulence levels in the nearby edge. This demonstrates that the SOL turbulence “decouples” from the edge turbulence. The spectral shift in the measured radial wavenumber is corroborated by the direct measurement of eddy tilt angle using a novel time delay analysis technique first developed for Doppler reflectometry [Pinzon et al., Plasma Phys. Controlled Fusion 61, 105009 (2019)] but adapted here for floating potential measurements.

Measurements of the plasma response to applied n = 1 perturbations from DIII-D ITER baseline scenario (IBS) demonstration discharges exhibit increasing amplitude prior to disruption inducing mode locking events. Simulations reveal that changes in ideal and resistive stability impact the response in the DIII-D IBS regime, well below the pressure limit of the external kink mode. The dependencies of the response measurements on the plasma normalized internal inductance l i and beta β N are qualitatively consistent with ideal MHD, although in most cases the amplitude of the measurements exceeds predictions, indicating that the experimental discharges are less stable than expected. Resistive MHD simulations of the response show improved compatibility with the measurements, and the closest agreement is obtained by including the experimental plasma rotation in the simulations. Although the input neutral beam torque is near zero, the simulations show that the corresponding level of rotation can lead to nearly complete screening of the pitch-resonant field component at the q = 2 surface. However, the simulation results exhibit a significant weakening of this screening effect at the lowest rotation levels in the dataset, concurrent with predictions of elevated response amplitudes that are consistent with measurements. These comparisons with DIII-D data provide a qualitative validation of a linear, resistive MHD response model.

DIII-D experiments show that the resistive wall mode (RWM) can remain stable in high β scenarios despite a low net torque from nearly balanced neutral beam injection heating. The minimization of magnetic field asymmetries is essential for operation at the resulting low plasma rotation of less than 20 krad s−1 (measured with charge exchange recombination spectroscopy using C VI emission) corresponding to less than 1% of the Alfven velocity or less than 10% of the ion thermal velocity. In the presence of n = 1 field asymmetries the rotation required for stability is significantly higher and depends on the torque input and momentum confinement, which suggests that a loss of torque-balance can lead to an effective rotation threshold above the linear RWM stability threshold. Without an externally applied field the measured rotation can be too low to neglect the diamagnetic rotation. A comparison of the instability onset in plasmas rotating with and against the direction of the plasma current indicates the importance of the toroidal flow driven by the radial electric field in the stabilization process. Observed rotation thresholds are compared with predictions for the semi-kinetic damping model, which generally underestimates the rotation required for stability. A more detailed modeling of kinetic damping including diamagnetic and precession drift frequencies can lead to stability without plasma rotation. However, even with corrected error fields and fast plasma rotation, plasma generated perturbations, such as edge localized modes, can nonlinearly destabilize the RWM. In these cases feedback control can increase the damping of the magnetic perturbation and is effective in extending the duration of high β discharges.

Recent high-β DIII-D (Luxon J.L.2002Nucl. Fusion4264) experiments with the new capability of balanced neutral beam injection show that the resistive wall mode (RWM) remains stable when the plasma rotation is lowered to a fraction of a per cent of the Alfvén frequency by reducing the injection of angular momentum in discharges with minimized magnetic field errors. Previous DIII-D experiments yielded a high plasma rotation threshold (of order a few per cent of the Alfvén frequency) for RWM stabilization when resonant magnetic braking was applied to lower the plasma rotation. We propose that the previously observed rotation threshold can be explained as the entrance into a forbidden band of rotation that results from torque balance including the resonant field amplification by the stable RWM. Resonant braking can also occur naturally in a plasma subject to magnetic instabilities with a zero frequency component, such as edge localized modes. In DIII-D, robust RWM stabilization can be achieved using simultaneous feedback control of the two sets of non-axisymmetric coils. Slow feedback control of the external coils is used for dynamic error field correction; fast feedback control of the internal non-axisymmetric coils provides RWM stabilization during transient periods of low rotation. This method of active control of then= 1 RWM has opened access to new regimes of high performance in DIII-D. Very high plasma pressure combined with elevatedqminfor high bootstrap current fraction, and internal transport barriers for high energy confinement, are sustained for almost 2 s, or 10 energy confinement times, suggesting a possible path to high fusion performance, steady-state tokamak scenarios.

The critical beta, which is limited by an external kink mode with Alfven wave time scale, is shown to be decreased by ferromagnetic effect by about 8% for μ/μ0 ~ 2 where μ and μ0 denote the permeability of a ferromagnetic wall and vacuum, respectively, for a tokamak of aspect ratio 3. The existence of the stability window in the wall distance for resistive wall mode opened by the effects of both the toroidal plasma rotation and the plasma dissipation, which was not observed for the high aspect ratio tokamak, is found for the tokamak of aspect ratio 3. The effect of ferromagnetism on them is also investigated. The critical beta analyses of the National Centralized Tokamak (NCT) plasma using the VALEN code are started by stabilizing plate and vacuum vessel geometry with finite resistivity, and the results of the passive effect of the stabilizing plate are obtained. The calculations including the stabilizing effect of the vacuum vessel and also active feedback control are performed for a design of the NCT plasma.

Experiments in the DIII-D tokamak show that the n = 1 ideal kink can be stabilized by a resistive wall if the plasma is rotating fast enough. A database of the onset of the n = 1 resistive wall mode as a function of the equilibrium toroidal magnetic field, the plasma density and the toroidal rotation has been assembled for plasmas with beta between the theoretically predicted no wall and ideal wall stability limits. The critical rotation frequency is found to scale as the inverse of the Alfven time with ? ?A 0.02 (evaluated at the q = 2 surface at ? 0.6) or ? ?S 0.7, where ?S is the sound time. The dependence of ? ? A or ? ?S on ?N/?N,no wall from 1?2 is weak and suggests the plasmas are in the 'intermediate dissipation' regime.

Steady-state operation of the advanced tokamak reactor relies on maintaining plasma stability with respect to the resistive wall mode ~RWM!. Active magnetic feedback and plasma rotation are the two methods proposed and demonstrated for this purpose. A comprehensive modeling effort including both magnetic feedback and plasma rotation is needed for understanding the physical mechanisms of the stabilization and to project to future devices. For plasma with low rotation, a complete solution for the feedback issue is obtained by assuming the plasma obeys ideal magnetohydrodynamics ~MHDs! and utilizing a normal mode approach ~NMA! @M. S. Chu et al., Nucl. Fusion 43, 441 ~2003!#. It is found that poloidal sensors are more effective than radial sensors and coils inside of the vacuum vessel more effective than outside. For plasmas with non-negligible rotation, a comprehensive linear nonideal MHD code, the MARS-F has been found to be suitable. MARS-F @Y. Q. Liu et al., Phys. Plasmas 7, 3681 ~2000!# has been benchmarked in the ideal MHD limit against the NMA. The effect of rotation stabilization of the plasma depends on the plasma dissipation model. Broad qualitative features of the experiment are reproduced. Rotation reduces the feedback gain required for RWM stabilization. Reduction is significant when rotation is near the critical rotation speed needed for stabilization. The International Thermonuclear Experimental Reactor ~ITER! @R. Aymar et al., Plasma Phys. Controlled Fusion 44, 519 ~2002!# ~scenario IV for advanced tokamak operation! may be feedback stabilized with babove the no wall limit and up to an increment of ;50% towards the ideal limit. Rotation further improves the stability.

This paper describes the modelling of the feedback control and rotational stabilization of the resistive wall mode (RWM) in tokamaks. A normal mode theory for the feedback stabilization of the RWM has been developed for an ideal plasma with no toroidal rotation. This theory has been numerically implemented for general tokamak geometry and applied to the DIII-D tokamak. A general formulation is further developed for the feedback stabilization of a tokamak with toroidal rotation and plasma dissipation. It has been used to understand the role of the external resonant field in affecting the plasma stability and compared with the resonant field amplification phenomenon observed in DIII-D. The effectiveness of a differentially rotating resistive wall in stabilizing the RWM has also been studied numerically. It is found that for a non-circular tokamak, a wide range of flow patterns are all effective. The structure of the RWM predicted from ideal MHD theory has been compared with signals from various diagnostics. It is also projected that based on DIII-D results scaled up to the ITER-FEAT, 33 MW of 1 MeV negative neutral beam injection will be able to sustain plasma rotation sufficient to stabilize the RWM without relying on feedback.

This study explores the effects of plasma shaping on magnetohydrodynamic mode stability and rotational stabilization in a tokamak, including both plasma and wall resistivity. Depending upon the plasma shape, safety factor, and distance from the wall, the β-limit for rotational stabilization is given by either the resistive-plasma ideal-wall (tearing mode) limit or the ideal-plasma resistive-wall (resistive wall mode) limit. In order to explore this broad parameter space, a sharp-boundary model is developed with a realistic geometry, resonant tearing surfaces, and a resistive wall. The β-limit achievable in the presence of stabilization by rigid plasma rotation, or by an equivalent feedback control with imaginary normal-field gain, is shown to peak at specific values of elongation and triangularity. It is shown that the optimal shaping with rotation typically coincides with transitions between tearing-dominated and wall-dominated mode behavior.

Results from an array of theoretical and computational tools developed to treat the instabilities of most interest for high performance tokamak discharges are described. The theory and experimental diagnostic capabilities have now been developed to the point where detailed predictions can be productively tested so that competing effects can be isolated and either eliminated or confirmed. The linear MHD stability predictions using high quality discharge equilibrium reconstructions are tested against the observations for the principal limiting phenomena in DIII-D: L mode negative central shear (NCS) disruptions, H mode NCS edge instabilities, and tearing and resistive wall modes (RWMs) in long pulse discharges. In the case of predominantly ideal plasma MHD instabilities, agreement between the code predictions and experimentally observed stability limits and thresholds can now be obtained to within several per cent, and the predicted fluctuations and growth rates to within the estimated experimental errors. Edge instabilities can be explained by a new model for edge localized modes as predominantly ideal instabilities with low to intermediate toroidal mode number. Accurate ideal calculations are critical to demonstrating RWM stabilization by plasma rotation, and the ideal eigenfunctions provide a good representation of the RWM structure when the plasma rotation slows. Ideal eigenfunctions can then be used to predict stabilization using active feedback. For non-ideal modes, the agreement in some cases is promising. Δ' calculations, for example, indicate that some discharges are linearly unstable to classical tearing modes, consistent with the observed growth of islands in those discharges. Nevertheless, there is still a great deal of improvement required before the non-ideal predictive capability can routinely approach levels similar to those for the ideal comparisons.

Research on the stability of spherical torus plasmas at and above the no-wall beta limit is being addressed on the National Spherical Torus Experiment [M. Ono et al., Nucl. Fusion 40, 557 (2000)], that has produced low aspect ratio plasmas, R/a∼1.27 at plasma current exceeding 1.4 MA with high energy confinement (TauE/TauE_ITER89P>2). Toroidal and normalized beta have exceeded 25% and 4.3, respectively, in q∼7 plasmas. The beta limit is observed to increase and then saturate with increasing li. The stability factor βN/li has reached 6, limited by sudden beta collapses. Increased pressure peaking leads to a decrease in βN. Ideal stability analysis of equilibria reconstructed with EFIT [L. L. Lao et al., Nucl. Fusion 25, 1611 (1985)] shows that the plasmas are at the no-wall beta limit for the n=1 kink/ballooning mode. Low aspect ratio and high edge q theoretically alter the plasma stability and mode structure compared to standard tokamak configurations. Below the no-wall limit, stability calculations show the perturbed radial field is maximized near the center column and mode stability is not highly effected by a nearby conducting wall due to the short poloidal wavelength in this region. In contrast, as beta reaches and exceeds the no-wall limit, the mode becomes strongly ballooning with long poloidal wavelength at large major radius and is highly wall stabilized. In this way, wall stabilization is more effective at higher beta in low aspect ratio geometry. The resistive wall mode has been observed in plasmas exceeding the ideal no-wall beta limit and leads to rapid toroidal rotation damping across the plasma core.

A proof of principle magnetic feedback stabilization experiment has been carried out to suppress the resistive wall mode (RWM), a branch of the ideal magnetohydrodynamic (MHD) kink mode under the influence of a stabilizing resistive wall, on the DIII-D tokamak device [Plasma Phys. Controlled Fusion Research (International Atomic Energy Agency, Vienna, 1986), p. 159; Phys. Plasmas 1, 1415 (1994)]. The RWM was successfully suppressed and the high beta duration above the no-wall limit was extended to more than 50 times the resistive wall flux diffusion time. It was observed that the mode structure was well preserved during the time of the feedback application. Several lumped parameter formulations were used to study the feedback process. The observed feedback characteristics are in good qualitative agreement with the analysis. These results provide encouragement to future efforts towards optimizing the RWM feedback methodology in parallel to what has been successfully developed for the n=0 vertical positional control. Newly developed MHD codes have been extremely useful in guiding the experiments and in providing possible paths for the next step.

A system of coils, sensors and amplifiers has been installed on the DIII-D tokamak to study the physics of feedback stabilization of low-frequency MHD modes such as the resistive wall mode (RWM). Experiments are being performed to assess the effectiveness of this minimal system and benchmark the predictions of theoretical models and codes. In the last campaign the experiments had been extended to a regime where the RWM threshold is lowered by a fast ramp of the plasma current. In these experiments the onset time of the RWM is very reproducible. With this system, the onset of the RWM had been delayed by up to 100 ms without degrading the plasma performance. The growth rate of the mode increases proportional to the length of the delay, suggesting that the plasma is evolving towards a more unstable configuration. The present results have suggested directions for improving the feedback system, including better sensors and improved feedback algorithms.

Recent DIII-D and RFX-mod experiments have demonstrated stable tokamak operation at very low values of the edge safety factor q(a) near and below 2. The onset of n = 1 resistive wall mode (RWM) kink instabilities leads to a disruptive stability limit, encountered at q(a) = 2 (limiter plasmas) and q 95 = 2 (divertor plasmas). However, passively stable operation can be attained for q(a) and q 95 values as low as 2.2. RWM damping in the q(a) = 2 regime was measured using active MHD spectroscopy. Although consistent with theoretical predictions, the amplitude of the damped response does not increase significantly as the q(a) = 2 limit is approached, in contrast with damping measurements made approaching the pressure-driven RWM limit. Applying proportional gain magnetic feedback control of the n = 1 modes has resulted in stabilized operation with q 95 values reaching as low as 1.9 in DIII-D and q(a) reaching 1.55 in RFX-mod. In addition to being consistent with the q(a) = 2 external kink mode stability limit, the unstable modes have growth rates on the order of the characteristic wall eddy-current decay timescale in both devices, and a dominant m = 2 poloidal structure that is consistent with ideal MHD predictions. The experiments contribute to validating MHD stability theory and demonstrate that a key tokamak stability limit can be overcome with feedback.

The confinement and the stability properties of the DIII-D tokamak high performance discharges are evaluated in terms of rotational and magnetic shear with emphasis on the recent experimental results obtained from the negative central magnetic shear (NCS) experiments. In NCS discharges, a core transport barrier is often observed to form inside the NCS region accompanied by a reduction in core fluctuation amplitudes. Increasing negative magnetic shear contributes to the formation of this core transport barrier, but by itself is not sufficient to fully stabilize the toroidal drift mode (trapped- electron-{eta}{sub i}mode) to explain this formation. Comparison of the Doppler shift shear rate to the growth rate of the {eta}{sub i} mode suggests that the large core {bold E x B} flow shear can stabilize this mode and broaden the region of reduced core transport . Ideal and resistive stability analysis indicates the performance of NCS discharges with strongly peaked pressure profiles is limited by the resistive interchange mode to low {Beta}{sub N} {lt} 2.3. This mode is insensitive to the details of the rotational and the magnetic shear profiles. A new class of discharges which has a broad region of weak or slightly negative magnetic shear (WNS) is described. The WNS discharges have broader pressure profiles and higher values than the NCS discharges together with high confinement and high fusion reactivity.

The experimental program of the Princeton Beta Experiment‐Modified (PBX‐M) tokamak [Phys. Fluids B2, 1271 (1990)] is directed toward tailoring plasma profiles to achieve greater stability and confinement and to gain access to the second stability region. Modification of the current density profile has been achieved with lower‐hybrid current drive (LHCD), leading to a regime free of global magnetohydrodynamic modes, while raising the value of q(0) above unity. The diffusion of the fast electrons produced by LHCD has been examined using two‐dimensional hard x‐ray imaging. Ion Bernstein waves (IBW) have been used for ion heating: a preliminary analysis shows that ion heating was spatially localized and in agreement with theoretical calculations. Divertor biasing has modified the electric field inside the last closed surface, resulting in the formation of a transport barrier, which in turn has reduced the threshold power of neutral beam injection (NBI) for H‐mode transition by 25%.

Medium‐n (toroidal mode number) instabilities with ballooning characteristics were investigated using x‐ray and electron‐cyclotron emission (ECE) imaging techniques in the Tokamak Fusion Test Reactor (TFTR) [R. J. Hawryluk et al., Plasma Phys. Controlled Fusion 33, 1509 (1991)]. The poloidal mode numbers were determined by fitting the data to soft x‐ray signals simulated with a wave‐packet model, and the toroidal mode numbers were determined from the phase difference between the soft x‐ray and ECE signals. The modes are identified as ideal ballooning modes from the observation that the identified mode numbers are much higher than that of the usual kink or tearing modes, the mode has a strong ballooning characteristic, the growth rate is consistent with the theoretical prediction, and the equilibrium is theoretically predicted to be marginal to the infinite n ballooning mode. The ballooning mode accompanies a β collapse which happens near the Troyon limit in TFTR. The ballooning mode has also been observed just before a major disruption in TFTR.

Recent DIII-D experiments with reduced neutral beam torque and minimum nonaxisymmetric perturbations of the magnetic field show a significant reduction of the toroidal plasma rotation required for the stabilization of the resistive-wall mode (RWM) below the threshold values observed in experiments that apply nonaxisymmetric magnetic fields to slow the plasma rotation. A toroidal rotation frequency of less than 10 krad/s at the q=2 surface (measured with charge exchange recombination spectroscopy using C VI) corresponding to 0.3% of the inverse of the toroidal Alfven time is sufficient to sustain the plasma pressure above the ideal MHD no-wall stability limit. The low-rotation threshold is found to be consistent with predictions by a kinetic model of RWM damping.

The stability of the resistive-wall mode (RWM) in DIII-D plasmas above the conventional pressure limit, where toroidal plasma rotation in the order of a few percent of the Alfve n velocity is sufficient to stabilize the n=1 RWM, has been probed using the technique of active MHD spectroscopy at frequencies of a few Hertz. The measured frequency spectrum of the plasma response to externally applied rotating resonant magnetic fields is well described by a single-mode approach and provides an absolute measurement of the damping rate and the natural mode rotation frequency of the stable RWM.

A beta degradation event in a high {beta}p, TFTR tokamak plasma is analyzed with X-ray and electron cyclotron emission (ECE) imaging techniques. Medium-n (toroidal mode number) instabilities with ballooning characteristics are observed near and within the q =1.5 surface during a slow degradation in the plasma {beta} and precede a sudden partial collapse in the central plasma pressure. This is the first reported observation of a ballooning instability in the interior of a large, collisionless tokamak plasma.

A review of TFTR plasma transport studies is presented. Parallel transport and the confinement of suprathermal ions are found to be relatively well described by theory. Cross-field transport of the thermal plasma, however, is anomalous with the momentum diffusivity being comparable to the ion thermal diffusivity and larger than the electron thermal diffusivity in neutral beam heated discharges. Perturbative experiments have studied nonlinear dependencies in the transport coefficients and examined the role of possible nonlocal phenomena. The underlying turbulence has been studied using microwave scattering, beam emission spectroscopy and microwave reflectometry over a much broader range in k perpendicular to than previously possible. Results indicate the existence of large-wavelength fluctuations correlated with enhanced transport.

Recent experiments in the DIII-D tokamak [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] show that the resistive wall mode (RWM) can be stabilized by smaller values of plasma rotation than previously reported. Stable discharges have been observed with beta up to 1.4 times the no-wall kink stability limit and ion rotation velocity (measured from CVI emission) less than 0.3% of the Alfven speed at all integer rational surfaces, in contrast with previous DIII-D experiments that indicated critical values of 0.7%–2.5% of the local Alfven speed. Preliminary stability calculations for these discharges, using ideal magnetohydrodynamics with a drift-kinetic dissipation model, are consistent with the new experimental results. A key feature of these experiments is that slow plasma rotation was achieved by reducing the neutral beam torque. Earlier experiments with strong neutral beam torque used “magnetic braking” by applied magnetic perturbations to slow the rotation, and resonant effects of these perturbations may have ...

Density fluctuations, δn n ≲ 20%, were observed to have no effect on the classical diffusion of a collisional helium plasma across a poloidal octupole field with an additional weak toroidal field. The implication of these results on tokamak scattering experiments is discussed.

External n = 1 magnetic fields are applied in RFX-mod and DIII-D low safety factor Tokamak plasmas to investigate their interaction with the internal MHD dynamics and in particular with the sawtooth instability. In these experiments the applied magnetic fields cause a reduction of both the sawtooth amplitude and period, leading to an overall stabilizing effect on the oscillations. In RFX-mod sawteeth eventually disappear and are replaced by a stationary m = 1, n = 1 helical equilibrium without an increase in disruptivity. However toroidal rotation is significantly reduced in these plasmas, thus it is likely that the sawtooth mitigation in these experiments is due to the combination of the helically deformed core and the reduced rotation. The former effect is qualitatively well reproduced by nonlinear MHD simulations performed with the PIXIE3D code. The results obtained in these RFX-mod experiments motivated similar ones in DIII-D L-mode diverted Tokamak plasmas at low q 95. These experiments succeeded in reproducing the sawtooth mitigation with the approach developed in RFX-mod. In DIII-D this effect is correlated with a clear increase of the n = 1 plasma response, that indicates an enhancement of the coupling to the marginally stable n = 1 external kink, as simulations with the linear MHD code IPEC suggest. A significant rotation braking in the plasma core is also observed in DIII-D. Numerical calculations of the neoclassical toroidal viscosity (NTV) carried out with PENT identify this torque as a possible contributor for this effect.