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Internal coils, 'I-Coils', were installed inside the vacuum vessel of the DIII-D device to generate non-axisymmetric magnetic fields to act directly on the plasma. These fields are predicted to stabilize the resistive wall mode (RWM) branch of the long-wavelength external kink mode with plasma beta close to the ideal wall limit. Feedback using these I-Coils was found to be more effective as compared to using external coils located outside the vacuum vessel. Locating the coils inside the vessel allows for a faster response and the coil geometry also allows for better coupling to the helical mode structure. Initial results were reported previously (Strait E.J. et al 2004 Phys. Plasmas 11 2505). This paper reports on results from extended feedback stabilization operations, achieving plasma parameters up to the regime of Cβ ≈ 1.0 and open loop growth rates of γopenτw ≳ 25 where the RWM was predicted to be unstable with only the 'rotational viscous stabilization mechanism'. Here Cβ ≈ (β - βno-wall.limit)/(βideal.wall.limit - βno-wall.limit) is a measure of the beta relative to the stability limits without a wall and with a perfectly conducting wall, and τw is the resistive flux penetration time of the wall. These feedback experimental results clarified the processes of dynamic error field correction and direct RWM stabilization, both of which took place simultaneously during RWM feedback stabilization operation. MARS-F modelling provides a critical rotation velocity in reasonable agreement with the experiment and predicts that the growth rate increases rapidly as rotation decreases below the critical. The MARS-F code also predicted that for successful RWM magnetic feedback, the characteristic time of the power supply should be limited to a fraction of the growth time of the targeted RWM. The possibility of further improvements in the presently achievable range of operation of feedback gain values is also discussed.

Toroidal plasma rotation of the order of a few per cent of the Alfven velocity can stabilize the resistive wall mode (RWM) and extend the operating regime of tokamaks from the conventional, ideal magnetohydrodynamic (MHD) no-wall limit up to the ideal MHD ideal-wall limit. The stabilizing effect has been measured in DIII-D passively by measuring the critical plasma rotation required for stability and actively by probing the plasma with externally applied resonant magnetic fields. The comparison of these measurements to predictions of rotational stabilization of the sound wave damping and of the kinetic damping model using the MARS-F code results in qualitative agreement, but also indicates the need for further refinement of the measurements and models.

A set of twelve coils for stability control has recently been installed inside the DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] vacuum vessel, offering faster time response and a wider range of applied mode spectra than the previous external coils. Stabilization of the n=1 ideal kink mode is crucial to many high beta, steady-state tokamak scenarios. A resistive wall converts the kink to a slowly growing resistive wall mode (RWM). With feedback-controlled error field correction, rotational stabilization of the RWM has been sustained for more than 2.5 s. Using the internal coils, the required correction field is smaller than with the external coils, consistent with a better match to the mode spectrum of the error field. Initial experiments in direct feedback control have stabilized the RWMs at higher beta and lower rotation than could be achieved by the external coils in similar plasmas, in qualitative agreement with numerical modeling. The new coils have also allowed wall stabilization in plasmas with...

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.

Feedback stabilization of the resistive wall mode (RWM) of a plasma in a general feedback configuration is formulated in terms of the normal modes of the plasma-resistive wall system. The growth/damping rates and the eigenfunctions of the normal modes are determined by an extended energy principle for the plasma during its open (feedback) loop operation. A set of equations are derived for the time evolution of these normal modes with currents in the feedback coils. The dynamics of the feedback system is completed by the prescription of the feedback logic. The feasibility of the feedback is evaluated by using the Nyquist diagram method or by solving the characteristic equations. The elements of the characteristic equations are formed from the growth and damping rates of the normal modes, the sensor matrix of the perturbation fluxes detected by the sensor loops, the excitation matrix of the energy input to the normal modes by the external feedback coils, and the feedback logic. (The RWM is also predicted to be excited by an external error field to a large amplitude when it is close to marginal stability.) This formulation has been implemented numerically and applied to the DIII-D tokamak. It is found that feedback with poloidal sensors is much more effective than feedback with radial sensors. Using radial sensors, increasing the number of feedback coils from a central band on the outboard side to include an upper and a lower band can substantially increase the effectiveness of the feedback system. The strength of the RWM that can be stabilized is increased from γτw = 1 to 30 (γ is the growth rate of the RWM in the absence of feedback and τw is the resistive wall time constant) Using poloidal sensors, just one central band of feedback coils is sufficient for the stabilization of the RWM with γτw = 30.

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.

Suppression of the resistive wall mode (RWM) has been successfully demonstrated in the DIII–D tokamak by using rotational stabilization in conjunction with a close-fitting vacuum vessel wall. The duration of the high-pressure discharge was extended to hundreds of times the wall skin time. Frequently, the plasma pressure reached the ideal-wall magnetohydrodynamic (MHD) kink limit. The confined pressure is up to twice as high as the no-wall ideal MHD kink limit. Near its marginal stability point, the RWM is found to resonate with residual non-axisymmetric fields (e.g. components of the error field). A magnetic feedback system has been used to identify and compensate for the residual non-axisymmetric fields. This is to the best of our knowledge, the first demonstration of the sustainment of a stable plasma with pressure at levels well above the no-wall pressure limit. This technique is expected to be applicable to other toroidal devices.

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.

A theoretical framework for understanding the feedback mechanism for stabilization of external MHD modes has been formulated. Efficient computational tools - the GATO stability code coupled with a substantially modified VACUUM code - have been developed to effectively design viable feedback systems against these modes. The analysis assumed a thin resistive shell and a feedback coil structure accurately modelled in θ and , albeit with only a single harmonic variation in . Time constants and induced currents in the enclosing resistive shell are calculated. An optimized configuration based on an idealized model has been computed for the DIII-D device. Up to 90% of the effectiveness of an ideal wall can be achieved.

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.

The amplitude and frequency of modes driven in the edge region of tokamak high mode (H-mode) discharges [type I edge-localized modes (ELMs)] are shown to depend on the discharge shape. The measured pressure gradient threshold for instability and its scaling with discharge shape are compared with predictions from ideal magnetohydrodynamic theory for low toroidal mode number (n) instabilities driven by pressure gradient and current density and good agreement is found. Reductions in mode amplitude are observed in discharge shapes with either high squareness or low triangularity where the stability threshold in the edge pressure gradient is predicted to be reduced and the most unstable mode is expected to have higher values of n. The importance of access to the ballooning mode second stability regime is demonstrated through the changes in the ELM character that occur when second regime access is not available. An edge stability model is presented that predicts that there is a threshold value of n for second r...

A report is given of: (a) a substantial transport reduction by the turbulence driven E × B flows observed in three dimensional non-linear gyrokinetic simulations of microturbulence in magnetically confined toroidal plasmas; (b) an analytical derivation of the effective shearing rate for the time dependent E × B flow; (c) an interpretation of experimental data using linear gyrokinetic microinstability rotation models of E × B shear; (d) other developments in gyrokinetic theory and simulation.

The effect of the passive plate stabilizer on the ideal magnetohydrodynamic (MHD) stability is numerically studied in order to guide its design in the proposed Korea Superconducting Tokamak Advanced Research (KSTAR). The parametric study is systematically performed by taking into account the two major variables: the plate’s distance from the plasma surface, and the poloidal (toroidally continuous) vacuum gap required for the access of neutral beam ports and other diagnostics near the outboard mid-plane. Extensive calculations are carried out on the plasma beta limits for low-n (n being toroidal mode number) MHD modes in several major operating regimes. The results lead to a practical and optimistic design point: plasma-wall separation ≈10 cm, and the outboard gap having a vertical separation of about 80 cm between the upper and lower plates. This is one of the choices that will make it feasible to achieve the advanced operation with bootstrap current fraction of over 80% at βn=5 in the reversed shear mode in KSTAR.

Sheared rotation dynamics are widely believed to have signficant influence on experimentally observed confinement transitions in advanced operating modes in major tokamak experiments, such as the Tokamak Fusion Test Reactor (TFTR) [D.J. Grove and D.M. Meade, Nuclear Fusion 25, 1167 (1985)], with reversed magnetic shear regions in the plasma interior. The high-n toroidal drift modes destabilized by the combined effects of ion temperature gradients and trapped particles in toroidal geometry can be strongly affected by radially sheared toroidal and poloidal plasma rotation. In previous work with the FULL linear microinstability code, a simplified rotation model including only toroidal rotation was employed, and results were obtained. Here, a more complete rotation model, that includes contributions from toroidal and poloidal rotation and the ion pressure gradient to the total radial electric field, is used for a proper self-consistent treatment of this key problem. Relevant advanced operating mode cases for TFTR are presented. In addition, the complementary problem of the dynamics of fluctuation-driven E x B flow is investigated by an integrated program of gyrokinetic simulation in annulus geometry and gyrofluid simulation in flux tube geometry.

The results of two-dimensional calculations of eddy currents induced on external conducting walls surrounding a tokamak are reported. The computed eddy currents are generated by low-n (n=1,2,3) external ideal-magnetohydrodynamic (MHD) instabilities. For a given toroidal mode number n the eddy current patterns are found to be very similar in a variety of plasma configurations, e.g., different edge safety factors and different plasma–wall separation distances, in high beta plasmas. This result is promising for the design of active feedback coils for the stabilization of the resistive wall mode. Also, the effects of having a partial wall that has a poloidal gap on the outboard side are considered. Using the expected gap size in the proposed Korea Superconducting Tokamak Advanced Research (KSTAR) [“The KSTAR tokmak,” in Proceedings of the 17th Symposium on Fusion Engineering, San Diego, 1997 (Institute of Electrical and Electronics Engineers, New York, in press), Paper No. O3.1], the calculation shows that ac...

The resistive wall mode (RWM) and neoclassical tearing mode (NTM) have been simultaneously suppressed in the DIII-D for durations of over 2 s at beta values 20% above the no-wall limit with modest electron cyclotron current drive and very low plasma rotation. The achieved plasma rotation was significantly lower than reported previously. However, in this regime where stable operation is obtained, it is not unconditionally guaranteed. Various MHD activities, such as edge localized modes (ELMs) and fishbones, begin to couple to the RWM branch near the no-wall limit; feedback has been useful in improving the discharge stability to such perturbations. Simultaneous operation of slow dynamic error field correction and fast feedback suppressed the pile-up of ELM-induced RWM at a series of ELM events. This result implies that successful feedback operation requires not only direct feedback against unstable RWM but also careful control of MHD-induced RWM aftermath, which is the dynamical response to a small-uncorrected error field near the no-wall beta limit. These findings are extremely useful in defining the challenge of control of the RWM and NTM in the unexplored physics territory of burning plasmas in ITER.

The occurrence of giant edge localized modes (ELMs) in DIII-D has previously been correlated with the violation of the ballooning stability criterion at the plasma edge. These results are extended in the paper. It is demonstrated theoretically that flux surfaces near the separatrix of properly elongated and triangulated plasmas may be moved into the connection region between the first and second stability regions for ideal MHD ballooning modes, when q is high and the shear is low near the plasma surface. The edge flux surfaces are then predicted to have no limit to the sustainable pressure gradient. Experimentally, giant ELMs disappear in these highly shaped plasmas. However, the edge pressure gradient does not increase and 'grassy' ELM behaviour appears instead. These results lend further support to the hypothesis that giant ELMs in DIII-D are triggered by ideal ballooning mode instabilities, but they indicate that giant ELM and grassy ELM behaviour may arise from somewhat different mechanisms.

Motivated by experimental observations of apparently triggerless tearing modes, we have performed linear and nonlinear MHD analysis showing that a non-resonant mode with toroidal mode number n = 1 can develop in the National Spherical Torus eXperiment (NSTX) at moderate normalized βN when the shear is low and the central safety factor q 0 is close to but greater than one. This mode, which is related to previously identified ‘infernal’ modes, will saturate and persist, and can develop poloidal mode number m = 2 magnetic islands in agreement with experiments. We have also extended this analysis by performing a free-boundary transport simulation of an entire discharge and showing that, with reasonable assumptions, we can predict the time of mode onset.

The MHD spectrum associated with small perturbations about an analytic equilibrium is determined for a configuration with magnetic flux surfaces which are nested similar elliptic cylinders generated by a uniform axial current. Since the system is periodic, it models the essential features of a toroid. Both analytic and numerical techniques are used to investigate many properties of the modes, including the continuous shear Alfven and slow acoustic spectra as well as the discrete modes associated with fast magnetosonic waves, kinks, and instabilities where the interchange criterion is violated. In particular, the relationship between discrete and continuum modes is described both qualitatively and quantitatively, and it is shown how the unstable interchange modes emerge from the shear-Alfven branch of the spectrum.

A new type of axisymmetric magnetohydrodynamic equilibrium is presented. It is characterized by a region of pressure and safety factor variation with a short scale length imposed as a perturbation. The equilibrium consistent with these profile variations can be calculated by means of an asymptotic expansion. The flexibility obtained by generating such equilibria allows for a close examination of the mechanisms that are relevant to ballooning instabilities – ideal-MHD modes with large toroidal mode number. The so-called first and second regions of stability against these modes are seen well within the limits of validity of the asymptotic expansion. It appears that the modes must be localized in regions with small values of the local shear of the magnetic field. The second region of stability occurs where the local shear is large throughout the range where the magnetic-field-line curvature is destabilizing.

We studied ballooning instabilities in tokamaks of arbitrary cross sections and finite shear. These azimuthally localized, ideal magnetohydrodynamic modes have large toroidal-mode numbers, but finite variation along the field and across the flux surfaces. Stability is determined by solving a second-order ordinary differential equation on each flux surface, subject to the proper boundary conditions. Qualitative agreement is achieved with the Princeton PEST stability code.

Princeton Beta Experiment (PBX) discharges run at the lowest q and highest βt always terminated in a hard disruption. The discharges, with βt values of up to 5.5% and q-values down to 2.2, were obtained by employing large current ramps and large gas feed rates during neutral beam injection. Previous work has indicated that the achieved βt values were consistent with the limit imposed by the n = l ideal external kink with a conducting wall at b/a = 2. The authors of the paper investigate further the validity of ideal MHD theory in explaining the low qψ disruptions. In particular, the characteristics of the pre-disruption MHD activity in these low-q discharges, specifically the time-scale of growth and internal and external mode structures, are compared with those determined from theoretical calculations. The results of these comparisons indicate that non-ideal effects must be considered in order to obtain detailed agreement between theory and experiment.

The MHD equilibrium and stability of a vertically elongated tokamak configuration are analysed in the one-dimensional limit corresponding to infinite elongation. Stability against all ideal-MHD modes can be obtained for beta values arbitrarily close to unity. In the finite-resistivity stability analysis, axisymmetric (m = 0) tearing modes, centred on the null trace of the poloidal field, can be stabilized by a loosely fitting conducting shell. The presence of the toroidal field component, however, introduces the possibility of non-symmetric tearing modes (m ? 0), centred away from the null trace. These modes can be stabilized only by a more tightly fitting shell, plus reliance on finite-pressure effects on the small-major-radius side of the plasma profile. Under these conditions, stable configurations with peak beta values approaching unity are readily found.

A comprehensive analysis of the stability properties of the appropriate kinetically generalized form of MHD ballooning modes, together with the usual trapped-particle drift modes, is presented. The calculations are fully electromagnetic and include the complete dynamics associated with compressional ion acoustic waves. Trapped-particle effects, together with all forms of collisionless dissipation, are taken into account without approximations. The influence of collisions is estimated with a model Krook operator. Results from the application of this analysis to realistic tokamak operating conditions indicate that unstable short-wavelength modes with significant growth rates can extend from β = 0 to values above the upper ideal-MHD critical beta associated with the 'second stability regime'. Since the maximum growth rates of the relevant modes appear to vary gradually with beta, these results support a 'soft' beta-limit picture involving a continuous (rather than abrupt or 'hard') modification of anomalous transport already present in low-beta tokamaks. However, at higher beta the increasing dominance of the electromagnetic component of the perturbations indicated by these calculations could also imply significantly different transport scaling properties.

Extensive numerical studies of ideal-MHD instabilities have been carried out to gain insight into the parametric dependence of critical β's in tokamaks. The large number of interrelated equilibrium quantities involved in establishing a critical β has demanded a careful, systematic survey in order to isolate this dependence. The results of this survey establish the scaling with geometrical quantities including aspect ratio, elongation, and triangularity in the parameter regimes appropriate to both current and reactor-sized plasmas. A moderate dependence on the pressure profile and a strong variation with the current profile is found. The principal result is that, for aspect ratio R/a ≈ 3, critical β's are of the order of 2% for circular cross-sections and 5% for plasmas with elongation K ≈2; somewhat higher values could be achieved with more optimal shaping. Finally, sequences of equilibria have been analysed to compare critical β as a function of toroidal mode number n. It is concluded that the infinite-n analytic ballooning theory provides a sufficient condition for ideal-MHD internal-mode stability. Low-n free-boundary modes appear to set a lower limit.

Magnetohydrodynamic instabilities limit the material that can be contained in a tokamak. The nature of the dangerous kink and ballooning modes is investigated for a typical configuration. In high-beta systems the most dangerous instabilities are ballooning modes that form large convective cells concentrated in regions of unfavorable magnetic field-line curvature.

A study of the MHD stability properties of bean-shaped tokamak plasmas is presented. For ballooning modes, while increased indentation gives larger stable-beta configurations, the existence and accessibility of the second stable region are sensitive to the pressure and safety factor profiles. The second stable region appears at lower beta values for large aspect ratio and moderately high q-values. Finite-Larmor-radius kinetic effects can significantly improve the stability properties. For low-q ( 1) operation, long-wavelength (n ~ 2, 3) internal-pressure-driven modes occur at modest βp values and accessibility to higher beta operation is unlikely. Indentation modifies the nature of the usually vertical axisymmetric instability, but the mode can be passively stabilized by placing highly conducting plates near the tips of the plasma bean. At constant q, indentation has a stabilizing effect on tearing modes.

Stability criteria for localized instabilities for both ideal and resistive models are derived and evaluated for general axisymmetric configurations. A numerical scheme for determining the spectrum in these configurations is developed. Application is made to the special case of an elliptical cylindrical plasma column, with finite pressure and length, carrying a uniform axial current. (auth)

A rearrangement of the divertor coils in PDX will enable a test in 1984 of the MHD stability properties of deeply indented bean-shaped plasmas. The goal is a beta of 10%. Indentation is expected to counter the deterioration of MHD stability against pressure driven modes that is occasioned by the larger aspect ratios typical of anticipated reactor oriented devices. Indeed, as shown by M. Chance et al., indentation may offer direct access to the second region of stability for ballooning modes, and numerical analyses with PEST show the internal kink to be stabilized completely with even relatively modest indentation. The internal kink is implicated in the loss of beam ions in PDX. In this report the theoretical basis for the forthcoming experiment, and the design considerations underlying the modification from PDX to PBX, are described in detail. Additional theoretical material, including an analysis of particle orbits in an indented tokamak plasma, is appended.

A study of the MHD stability properties of bean-shaped tokamak plasmas is presented. For ballooning modes, while increased indentation gives larger ..beta.. stable configurations, the existence and accessibility of the second stable region is sensitive to the pressure and safety factor profiles. The second stable region appears at lower ..beta.. values for large aspect ratio and moderately high q-values. Finite-Larmor-radius (FLR) kinetic effects can significantly improve the stability properties. For low q (< 1) operation, long wavelength (n approx. 2,3) internal pressure driven modes occur at modest ..beta../sub p/ values and accessibility to higher ..beta.. operation is unlikely. Indentation modifies the nature of the usually vertical axisymmetric instability, but the mode can be passively stabilized by placing highly conducting plates near to the tips of the plasma bean. At constant q, indentation has a stabilizing effect on tearing modes.

Various toroidal reversed‐field pinch (RFP) equilibria stable to ideal Mercier and ballooning modes are subjected to resistive ballooning analysis. The stability properties of these configurations are thus investigated as a function of quantities that are experimentally measurable. Resistive ballooning is seen as a likely candidate for the interpretation of a certain class of density fluctuation phenomena observed in these devices.

Minimum‐energy equilibrium states of a pressureless cylindrical plasma are constructed subject to a recently proposed set of constraints appropriate to turbulent relaxation dominated by a tearing mode of single helicity. Stability to ideal and resistive modes is examined. It is found that the relaxed states of minimum energy are always stable to the dominant mode, as well as a class of other low m and n modes, but that there are typically high n residual instabilities in the case of the pitch branch, and a residual m=3, n=2 resistive instability is found for an otherwise stable narrow window in the tokamak branch. It is argued that the residual instabilities may be compatible with experimental observations.

A comprehensive analysis of low‐frequency, high‐toroidal‐mode‐number linear eigenmodes for tokamaks is presented. The most significant new features of this stability study are that the calculation is interfaced with a general numerical magnetohydrodynamic equilibrium, and that it is fully electromagnetic. The ballooning formalism is employed and all important kinetic effects, including those of trapped particles, are retained. In particular, the familiar trapped‐electron drift‐wave frequency regime is considered and results are presented for three sequences of artificial equilibria; one of increasing β(≡plasma pressure/magnetic pressure) values; one of varying equilibrium shape, from inverse‐D to circular to normal‐D; and one of increasing vertical ellipticity. The analysis is then applied to a realistic (self‐consistent) high‐β equilibrium generated with data obtained from the ISX‐B tokamak experiment. Here it is found that for the usual trapped‐electron branch, kinetic microinstabilities appear to be present over a wide range of toroidal mode numbers.