A banana kinetic equation in tokamaks that includes effects of the finite banana width is derived for the electromagnetic waves with frequencies lower than the gyro-frequency and the bounce frequency of the trapped particles. The radial wavelengths are assumed to be either comparable to or shorter than the banana width, but much wider than the gyro-radius. One of the consequences of the banana kinetics is that the parallel component of the vector potential is not annihilated by the orbit averaging process and appears in the banana kinetic equation. The equation is solved to calculate the neoclassical quasilinear transport fluxes in the superbanana plateau regime caused by electromagnetic waves. The transport fluxes can be used to model electromagnetic wave and the chaotic magnetic field induced thermal particle or energetic alpha particle losses in tokamaks. It is shown that the parallel component of the vector potential enhances losses when it is the sole transport mechanism. In particular, the fact that the drift resonance can cause significant transport losses in the chaotic magnetic field in the hitherto unknown low collisionality regimes is emphasized.

A simple approach to efficiently solve a coupled set of 1-D diffusion-type transport equations with a stiff transport model for tokamak plasmas is presented based on the 4th order accurate Interpolated Differential Operator scheme along with a nonlinear iteration method derived from a root-finding algorithm. Numerical tests using the Trapped Gyro-Landau-Fluid model show that the presented high order method provides an accurate transport solution using a small number of grid points with robust nonlinear convergence.

Bounce-transit and drift resonance can be important to plasma confinement in tokamaks with a broken symmetry. The resonance usually is either treated by integrating along the unperturbed orbits or calculated using an action-angle approach. An Eulerian approach has been developed to take into account the momentum conservation property of the Coulomb collision operator. The difference between the Eulerian approach and other approaches is in the thermodynamic forces of the transport fluxes, and the corresponding toroidal plasma viscosity. The mass and heat flows that are parallel to the equilibrium magnetic field B appear in the thermodynamic forces in the Eulerian approach. However, in the existing Eulerian approach, only the E × B drift is kept in the theory; the magnetic drifts, i.e., ∇B, and curvature drifts are neglected by adopting the large aspect ratio assumption, where E is the electric field and B = |B|. Here, the Eulerian approach is extended to include the magnetic drifts, which is important for energetic alpha particles as well, to calculate neoclassical toroidal plasma viscosity in finite aspect ratio tokamaks. The relation to the nonlinear plasma viscosity in the plateau regime will also be discussed.

The nonlinear response of a low-beta tokamak plasma to non-axisymmetric fields offers an alternative to direct measurement of the non-axisymmetric part of the vacuum magnetic fields, often termed ‘error fields’. Possible approaches are discussed for determination of error fields and the required current in non-axisymmetric correction coils, with an emphasis on two relatively new methods: measurement of the torque balance on a saturated magnetic island, and measurement of the braking of plasma rotation in the absence of an island. The former is well suited to ohmically heated discharges, while the latter is more appropriate for discharges with a modest amount of neutral beam heating to drive rotation. Both can potentially provide continuous measurements during a discharge, subject to the limitation of a minimum averaging time. The applicability of these methods to ITER is discussed, and an estimate is made of their uncertainties in light of the specifications of ITER's diagnostic systems. The use of plasma response-based techniques in normal ITER operational scenarios may allow identification of the error field contributions by individual central solenoid coils, but identification of the individual contributions by the outer poloidal field coils or other sources is less likely to be feasible.

Error fields and resistive magnetohydrodynamic modes are ubiquitous in real tokamaks. They break the toroidal symmetry in | B | in tokamaks. Here, B is the magnetic field. There are two mechanisms that break the symmetry on the perturbed magnetic surface: one is the perturbed field itself and the other results from the distortion of the magnetic surface due to the perturbed field. The broken toroidal symmetry leads to enhanced neoclassical toroidal plasma viscosity and consequently the rate of the toroidal flow damping. The neoclassical toroidal plasma viscosity also results in a steady-state toroidal plasma flow. In addition, the neoclassical toroidal plasma viscosity in the vicinity of the magnetic islands provides a mechanism to determine the island rotation frequency, which is an important quantity for the island stability. Here, the theory for neoclassical toroidal plasma viscosity in the vicinity of the magnetic island is extended to include the effects of the collisional boundary layer that lead to scaling in the transport fluxes, where ν is the collision frequency.

The kinetic effects of thermal particles on the resistive wall mode (RWM) are studied in reversed field pinch (RFP) by adopting the toroidal hybrid stability code MARS-K (Liu et al 2008 Phys. Plasmas 15 112503) where the drift kinetic effects are included self-consistently. It is found that the transit ion resonance can provide ion acoustic Landau damping to stabilize the RWMs in high-beta plasmas. The trapped particles do not play a significant role in kinetic stabilization. The required critical flow rotation frequency is in the ion acoustic range which is much smaller than that predicted previously by fluid theory (Guo et al 1999 Phys. Plasmas 6 3868). The most unstable mode, having its rational surface closest to the plasma, can be stabilized for the wall near the plasma (e.g. n = 6 mode in RFX-mod) with the flow rotation in a few per cent of the Alfvén frequency. For other RWMs with different toroidal wave numbers n the stabilization conditions depend on the wall position and plasma βp. An analysis based on the potential energy components is carried out for physical understanding. The preliminary study on the effects of collisionality is also presented.

Motivated by the recent experimental observation that plasma stability can be improved by external magnetic perturbations, the general problem of plasma response to external magnetic perturbations is investigated. Different (vacuum, ideal and resistive) plasma response models are considered and compared. Plasma response, in experiments where stabilization was achieved, is obtained through computation using the MARS-F code, with a plasma model that includes both plasma resistivity and rotation. The resultant magnetic field line stochasticity is much reduced from that obtained formerly using the vacuum plasma model. This reduced stochasticity is more consistent with the favourable experimental observation of enhanced stability. Examples are given for the response of an ITER plasma to perturbations generated by the correction coils; and the response of a plasma to external coils (antenna) up to the Alfvén frequency.

A theory for the superbanana plateau resonance that occurs at the phase space boundary is presented to refine the comprehensive theory for neoclassical toroidal plasma viscosity in tokamaks. The results of the theory reproduce those of the standard superbanana plateau resonance theory when the resonance occurs away from the boundary. It shows that the strength of the superbanana plateau resonance weakens in the vicinity of the phase space boundary. It also indicates that it is important to know the resonance pitch angle parameter when comparing the results of the superbanana plateau resonance with those of the experiments. The theory is used to refine the kernel for the superbanana plateau resonance in the approximate expression for the neoclassical toroidal plasma viscosity.

The extension of the EFIT equilibrium reconstruction code to fine spatialgrid resolutions is discussed. The residue in the force-balance relation of the Grad-Shafranov (G-S) equation and the convergence property of these fine spatial-grid EFIT equilibria are studied in detail. The results suggest that fine spatial-grid equilibria generally better satisfy the force-balance constraint described by the G-S equation. Finer spatial-grid equilibria have typically smaller average error in satisfying the force-balance equation than coarse-grid equilibria and those extrapolated from coarse-grid results. Analysis of the equilibrium iteration algorithm employed in EFIT reveals that the iteration process is related to the spatial feedback stabilization of the plasma with flux control at various specified locations. Thus, for a converged equilibrium, axisymmetric stability is generally expected with feedback. The iteration error decreases self-similarly in the final stage of the iteration process and is related to the least stable axisymmetric mode in the feedback-stabilized equilibrium.

New processes have been discovered in the interaction of 3D fields with tearing mode stability at low torque and modest β on DIII-D and NSTX. These are thought to arise from the plasma response at the tearing resonant surface, which theoretically is expected to depend strongly on plasma rotation and underlying intrinsic tearing stability. This leads to sensitivities additional to those previously identified at low density where the plasma rotation is more readily stopped, or at high βN where ideal MHD responses amplify the fields (where βN is the plasma β divided by the ratio of plasma current to minor radius multiplied by toroidal field). It is found that the threshold size for 3D fields to induce modes tends to zero as the natural tearing βN limit is approached. 3D field sensitivity is further enhanced at low rotation, with magnetic probing detecting an increased response to applied fields in such regimes. Modelling with the MARS-F code confirms the interpretation with the usual plasma screening response breaking down in low rotation plasmas and a tearing response developing, opening the door to additional sensitivities to β and the current profile. Typical field thresholds to induce modes in torque-free βN ∼ 1.5 H-modes are well below those in ohmic plasmas or plasmas near the ideal βN limit. The strong interaction with the tearing mode βN limit is identified through rotation shear, which is decreased by the 3D field, leading to decreased tearing stability. Thus both locked and rotating mode field thresholds can be considered in terms of a torque balance, with sufficient braking leading to destabilization of a mode. On this basis new measurements of the principal parameter scalings for error field threshold have been obtained in torque-free H-modes leading to new predictions for error field sensitivity in ITER. The scalings have similar exponents to ohmic plasmas, but with seven times lower threshold at the ITER baseline βN value of 1.8, and a linear dependence on proximity to the tearing mode βN limit (∼2.2 at zero torque). This reinforces the need to optimize error field correction strategies in ITER, and implement sources to drive plasma rotation.

A detailed experiment-theory comparison reveals that linear ideal MHD theory is in quantitative agreement with external magnetic and internal soft x-ray measurements of the plasma response to externally applied non-axisymmetric fields over a broad range of beta and rotation. This result represents a significant step toward the goal of advancing the understanding of three-dimensional tokamak equilibria. Both the magnetic and soft x-ray measurements show the driven plasma perturbation increases linearly with the applied perturbation, suggesting the relevance of linear plasma response models. The magnetic and soft x-ray measurements are made at multiple toroidal and poloidal locations, allowing well resolved measurements of the global structure. The comparison also highlights the need to include kinetic effects in the MHD model once beta exceeds 80% of the kink mode limit without a conducting wall. Two distinct types of response fields are identified by the linear ideal MHD model: one that consists of localized currents at the rational surfaces that cancel the applied resonant field and another that is excited by the components of the external field that couple to the kink mode. Numerical simulations show these two fields have similar amplitudes in ITER-shaped DIII-D discharges where n = 3 fields are used to suppress edge localized modes.

The effects of finite gradient B drift are included in the collisional boundary layer analysis to improve the accuracy of the neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities present. Depending on the sign of the electric charge of the species and that of the radial electric field, the effects of finite gradient B drift can either reduce, if the E × B drift is in the same direction of the gradient B drift, or enhance, if these two drifts are in the opposite direction, the magnitude of the neoclassical toroidal plasma viscosity. Here, E is the electric field and B is the magnetic field. However, because the gradient B drift depends on the effective pitch angle, the net effects have to be properly weighted by integrating over the particle energy.

Two issues of the resistive wall mode (RWM) control code maturity are addressed: the inclusion of advanced mode damping physics beyond the ideal MHD description, and the possibility of taking into account the influence of 3D features of the conducting structures on the mode stability and control. Examples of formulations and computational results are given, using the MARS-F/K codes and the CarMa code. The MARS-K calculations for a DIII-D plasma shows that the fast ion contributions, which can give additional drift kinetic stabilization in the perturbative approach, also drive an extra unstable branch of mode in the self-consistent kinetic modelling. The CarMa modelling for the ITER steady state advanced plasmas shows about 20% reduction in the RWM growth rate by the volumetric blanket modules. The multi-mode analysis predicts a weak interaction between the n = 0 and the n = 1 RWMs, due to the 3D ITER walls. The CarMa code is also successfully applied to model the realistic feedback experiments in RFX.

Recent DIII-D experiments deliver strong evidence for the importance of kinetic modifications of ideal MHD three-dimensional (3D) equilibrium and resistive wall mode (RWM) stability models at high plasma pressure. Magnetic measurements of the plasma response to externally applied long-wavelength perturbations of the order of B BT 10 3 show that linear ideal MHD can only describe three-dimensional equilibria for plasma pressures up to 80% of the ideal MHD stability limit. At higher pressure, the observed stability of the RWM over a wide range of plasma rotation profiles and the measured dependence of the plasma response on plasma pressure and rotation are both explained by including wave-particle resonances due to the quasi-static perturbation of the equilibrium field in the model. The DIII-D results highlight the necessity to extend the original kinetic RWM stability model [A. Bondeson, M.S. Chu, Phys. Plasmas 3, 3013 (1996)] by including the precession motion of trapped particles [B. Hu, R. Betti, Phys. Rev. Lett. 93, 105002 (2004)] as well as the effect of trapped fast ions. Experimental validation of these kinetic effects is essential in developing confidence in predictions for reactor scenarios that rely on wall-stabilization, such as the steady-state scenario in ITER.

The simultaneous use of feedback control for error field correction (EFC) and stabilization of an unstable resistive wall mode (RWM) has been demonstrated in DIII-D. While the conventional EFC method addresses error fields in a pre-programmed manner, it is challenged when an unstable RWM becomes dominant, because a weakly stable or feedback-stabilized RWM becomes extremely sensitive to any small, uncorrected resonant error field. Since the DIII-D tokamak is uniquely equipped with internal coils for fast time response and external feedback coils for slower time response, independent magnetic feedback control in low and high frequency ranges allows us to explore the specific roles of EFC and direct feedback (DF) in active RWM control in stable, marginal and unstable RWM regimes. For an unstable RWM at the edge safety factor q95~3, the simultaneous operation of DF with the internal coils and dynamic (feedback-controlled) EFC with the external coils enabled us not only to stabilize the unstable RWM but also to determine the necessary EFC in the presence of a feedback-stabilized RWM. The gain dependence of the feedback-stabilized RWM differs from those of stable and marginal RWMs. Recent experiments and modeling show that the gain increase at stable RWMs (at q95~5 or 6) leads to the coil current increase, while a marginal RWM (at q95~4) is insensitive to the feedback gains. In contrast, according to an analytic cylindrical model, the EFC in the unstable RWM regime is predicted to require high gain to approach the desired correction current. This is also consistent with numerical RWM feedback modeling using the MARS-F code. It has been shown that a choice of an “under-relaxation” factor with high feedback gain could achieve “fast-track” EFC, requiring far fewer iterations than the conventional EFC method. Broadband magnetic feedback beyond a wall characteristic frequency enhanced the decay rates of resonant magnetic perturbations induced by various bursty MHD events, helping to create and sustain highperformance plasmas. The established methodology to determine the optimized EFC waveform with the simultaneous use of feedback control of EFC and DF is applicable for various operational scenarios with pressure beyond the no-wall ideal stability limit. In particular, it would be highly valuable when the onset of unstable MHD is sensitive to the quality of EFC.

The tokamak plasmas in most of the present experiments and those considered for the future reactors are up-down asymmetric in nature. This asymmetry arises due to external coils and conducting structures which surround the plasma. The analytical description of these equilibria using mathematically simpler methods is useful for the theoretical study of stability and transport. Such a tokamak equilibrium has been constructed analytically for arbitrary aspect ratios. The asymmetric nature arises through the homogeneous part of the exact solution of the Grad-Shafranov equation with a choice of pressure and toroidal function as linear in poloidal flux. These solutions can describe both single-and double-null divertor plasmas with an appropriate choice of plasma boundary. This has been used to construct DIII-D and ITER-like equilibrium and compared with those computed numerically.

The requirements for active resistive wall mode (RWM) feedback control have been systematically investigated and established using highly reproducible current-driven RWMs in ohmic discharges in DIII-D. The unambiguous evaluation of active RWM feedback control was not possible in previous RWM studies primarily due to the variability of the onset of the pressure-driven RWMs; the stability of the pressure-driven RWM is thought to be sensitive to various passive stabilization mechanisms. Both feedback control specifications and physics requirements for RWM stabilization have been clarified using the current-driven RWMs in ohmic discharges, when little or no passive stabilization effects are present. The use of derivative gain on top of proportional gain is found to be advantageous. An effective feedback control system should be equipped with a power supply with bandwidth greater than the RWM growth rate. It is beneficial to apply a feedback field that is toroidally phase-shifted from the measured RWM phase in the same direction as the plasma current. The efficacy of the RWM feedback control will ultimately be determined by the plasma fluctuations on internal diagnostics, as well as on external magnetics. The proximity of the feedback coils to the plasma appears to be an important factor in determining the effectiveness of the RWM feedback coils. It is desirable that an RWM feedback control system simultaneously handles error field correction at a low frequency, along with direct RWM feedback at a high frequency. There is an indication of the influence of a second least stable RWM, which had been theoretically predicted but never identified in experiments. A preliminary investigation based on active MHD spectroscopic measurement showed a strong plasma response around 400 Hz where the typical plasma response associated with the first least stable RWM was expected to be negligible. Present active feedback control requirements are based on a single mode assumption, so the investigation of the second least stable RWM is of high interest.

The objective of the Fusion Development Facility (FDF) under consideration is to carry forward advanced tokamak physics for optimization of fusion reactors and enable development of fusion's energy applications. A concept of FDF based on the tokamak approach with conservative expressions of advanced physics and nonsuper-conducting magnet technology is presented. It is envisioned to nominally provide 2 MW/m(2) of neutron wall loading and operate continuously for up to 2 weeks as required for fusion nuclear component research and development. FDF will have tritium breeding capability with a goal of addressing the tritium self-sufficiency issue for fusion energy. A zero-dimensional system study using extrapolations of current physics and technology is used to optimize FDF for reasonable power consumption and moderate size. It projects a device that is between the DIII-D tokamak (major radius 1.8 m) [J. L. Luxon, Nucl. Fusion, Vol. 42, p. 614 (2002)] and the Joint European Torus (major radius 3 m) [P. H. Rebut, R. J. Bickerton, and B. E. Keen, Nucl. Fusion, Vol. 25, p. 1011 (1985)] in size, with an aspect ratio A of 3.5 and a fusion gain Q of 2 to 5. Theory-based stability and transport modeling is used to complement the system study and to address physics issues related to specific design points. It is demonstrated that the FDF magnetohydrodynamic stability limits can be readily met with conservative stabilizing conducting wall placement. Transport analysis using a drift-wave-based model with an edge boundary condition consistent with the pedestal stability limit indicates that the FDF confinement requirement can also be readily satisfied. A surprising finding is that the toroidal Alfven eigenmodes are stabilized by strong ion Landau damping. Analysis of vertical stability control indicates that the basis configuration with an elongation kappa(X) similar to 2.35 can be controlled using a power supply technology similar to that used in DIII-D. Peak heat fluxes to the divertor are somewhat lower than those of ITER (R. Aymar, P. Barabaschi, and Y. Shimomura, Plasma Phys. Control. Fusion, Vol. 44, p. 519 (2002)), but FDF will operate with a higher duty factor.

An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities is presented. The expression smoothly joins transport fluxes or plasma viscosity in all the known collisionality regimes derived from the solution of the bounce averaged drift kinetic equation and should be useful in modelling results of existing and future tokamak experiments. It also incorporates some of the extensions of the known expressions to include the effects of finite ∇Bdrift in the non-resonant transport processes. Here,Bis the magnitude of the magnetic field. The toroidal momentum balance equation is a nonlinear function of the radial electric field when the neoclassical plasma viscosity is dominant. It can have bifurcated solutions for the radial electric field and may lead to better plasma confinement as a result.

Visible imaging has been used to provide the 2D spatial structure and temporal evolution of the profile of high-energy neutrals introduced by neutral beam injection, the fast ion profile and a variety of plasma instabilities in DIII-D plasmas; the combination of these techniques form a comprehensive fast ion physics diagnostic suite. The injected neutral profile is imaged in Doppler shifted D α light induced by collisional excitation. Fast ion profile information was obtained through imaging of Doppler shifted fast ion D α light (FIDA) emitted by re-neutralized energetic ions. Imaging of FIDA emission during sawtooth events shows a large central depletion following sawtooth crashes—indicative of a broad redistribution of fast ions. Two examples of instability structure measurements are given. Measurements of the detailed 2D poloidal structure of rotating tearing modes were obtained using spectrally filtered fast imaging of broadband visible bremsstrahlung emission, a method which is capable of imaging with high resolution the structure of coherent oscillations in the core of current and next-step fusion plasma experiments and can be applied to virtually any mode with a finite perturbed bremsstrahlung emissivity and frequency in the laboratory frame. Measurements are also presented of the n = 0 energetic particle geodesic acoustic mode which were made by observing fluctuations in active emission.