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High fidelity kinetic plasma equilibrium reconstructions are an essential requirement for accurate stability and disruption prediction analyses to support continuous operation of high beta tokamak plasmas. The present kinetic equilibrium reconstructions of plasmas in the KSTAR device include plasma density and temperature profiles from Thomson scattering and ion temperature from charge exchange spectroscopy diagnostics, and allowance for fast particle pressure. In addition, up to 25 channels of motional Stark effect (MSE) diagnostic data are used to constrain the magnetic field pitch angle profile in the plasma to produce a reliable computation of the safety factor, q, profile. H-mode plasmas exhibit clear pedestal characteristics in the reconstructed pressure profile compared to internal transport barrier or L-mode plasmas. The plasma configuration and vertical position of inner strike points are validated by CCD and infrared camera images. Ideal and resistive MHD stability analyses using the DCON and resistive DCON codes utilize these kinetic equilibrium reconstructions to compare to experimental plasma stability. Equilibria with sufficiently low convergence error can provide reliable computation of ideal and resistive magnetohydrodynamic (MHD) stability analysis.

The National Spherical Torus Experiment (NSTX) offers an operational space characterized by high-beta (βt= 39%, βN> 7,) and low aspect ratio (A> 1.27) to leverage the plasma parameter dependences of RWM stabilization and plasma rotation damping physics giving greater confidence for extrapolation to ITER. Significant new capability for RWM research has been added to the device with the commissioning of a set of six non-axisymmetric magnetic field coils, allowing generation of fields with dominant toroidal mode number,n, of 1–3. These coils have been used to study the dependence of resonant field amplification on applied field frequency and RWM stabilization physics by reducing the toroidal rotation profile below its steady-state value through non-resonant magnetic braking. Modification of plasma rotation profiles shows that rotation outsideq= 2.5 is not required for passive RWM stability and there is large variation in the RWM critical rotation at theq= 2 surface, both of which are consistent with distributed dissipation models.

The National Spherical Torus Experiment (NSTX) has demonstrated the advantages of low aspect ratio geometry in accessing high toroidal and normalized plasma beta, and βN ≡ 10 8〈βt〉 aB0/Ip. Experiments have reached βt = 39% and βN = 7.2 through boundary and profile optimization. High βN plasmas can exceed the ideal no-wall stability limit, βNno-wall, for periods much greater than the wall eddy current decay time. Resistive wall mode (RWM) physics is studied to understand mode stabilization in these plasmas. The toroidal mode spectrum of unstable RWMs has been measured with mode number n up to 3. The critical rotation frequency of Bondeson-Chu, Ωcrit = ωA/(4q2), describes well the RWM stability of NSTX plasmas when applied over the entire rotation profile and in conjunction with the ideal stability criterion. Rotation damping and global rotation collapse observed in plasmas exceeding βNno-wall differs from the damping observed during tearing mode activity and can be described qualitatively by drag due to neoclassical toroidal viscosity in the helically perturbed field of an ideal displacement. Resonant field amplification of an applied n = 1 field perturbation has been measured and increases with increasing βN. Equilibria are reconstructed including measured ion and electron pressure, toroidal rotation and flux isotherm constraint in plasmas with core rotation ω/ωA up to 0.48. Peak pressure shifts of 18% of the minor radius from the magnetic axis have been reconstructed.

The major objective of the National Spherical Torus Experiment (NSTX) is to understand basic toroidal confinement physics at low aspect ratio and high βT in order to advance the spherical torus (ST) concept. In order to do this, NSTX utilizes up to 7.5 MW of neutral beam injection, up to 6 MW of high harmonic fast waves (HHFWs), and it operates with plasma currents up to 1.5 MA and elongations of up to 2.6 at a toroidal field up to 0.45 T. New facility, and diagnostic and modelling capabilities developed over the past two years have enabled the NSTX research team to make significant progress towards establishing this physics basis for future ST devices. Improvements in plasma control have led to more routine operation at high elongation and high βT (up to ~40%) lasting for many energy confinement times. βT can be limited by either internal or external modes. The installation of an active error field (EF) correction coil pair has expanded the operating regime at low density and has allowed for initial resonant EF amplification experiments. The determination of the confinement and transport properties of NSTX plasmas has benefitted greatly from the implementation of higher spatial resolution kinetic diagnostics. The parametric variation of confinement is similar to that at conventional aspect ratio but with values enhanced relative to those determined from conventional aspect ratio scalings and with a BT dependence. The transport is highly dependent on details of both the flow and magnetic shear. Core turbulence was measured for the first time in an ST through correlation reflectometry. Non-inductive start-up has been explored using PF-only and transient co-axial helicity injection techniques, resulting in up to 140 kA of toroidal current generated by the latter technique. Calculated bootstrap and beam-driven currents have sustained up to 60% of the flat-top plasma current in NBI discharges. Studies of HHFW absorption have indicated parametric decay of the wave and associated edge thermal ion heating. Energetic particle modes, most notably toroidal Alfven eigenmodes and fishbone-like modes result in fast particle losses, and these instabilities may affect fast ion confinement on devices such as ITER. Finally, a variety of techniques has been developed for fuelling and power and particle control.

A major research goal of the national spherical torus experiment is establishing long-pulse, high beta, high confinement operation and its physics basis. This research has been enabled by facility capabilities developed during 2001 and 2002, including neutral beam (up to 7?MW) and high harmonic fast wave (HHFW) heating (up to 6?MW), toroidal fields up to 6?kG, plasma currents up to 1.5?MA, flexible shape control, and wall preparation techniques. These capabilities have enabled the generation of plasmas with of up to 35%. Normalized beta values often exceed the no-wall limit, and studies suggest that passive wall mode stabilization enables this for H mode plasmas with broad pressure profiles. The viability of long, high bootstrap current fraction operations has been established for ELMing H mode plasmas with toroidal beta values in excess of 15% and sustained for several current relaxation times. Improvements in wall conditioning and fuelling are likely contributing to a reduction in H mode power thresholds. Electron thermal conduction is the dominant thermal loss channel in auxiliary heated plasmas examined thus far. HHFW effectively heats electrons, and its acceleration of fast beam ions has been observed. Evidence for HHFW current drive is obtained by comparision of the loop voltage evolution in plasmas with matched density and temperature profiles but varying phases of launched HHFW waves. Studies of emissions from electron Bernstein waves indicate a density scale length dependence of their transmission across the upper hybrid resonance near the plasma edge that is consistent with theoretical predictions. A peak heat flux to the divertor targets of 10?MW?m?2 has been measured in the H mode, with large asymmetries being observed in the power deposition between the inner and outer strike points. Non-inductive plasma startup studies have focused on coaxial helicity injection. With this technique, toroidal currents up to 400?kA have been driven, and studies to assess flux closure and coupling to other current drive techniques have begun.

Research on the spherical torus (or spherical tokamak) (ST) is being pursued to explore the scientific benefits of modifying the field line structure from that in more moderate aspect ratio devices, such as the conventional tokamak. The ST experiments are being conducted in various US research facilities including the MA-class National Spherical Torus Experiment (NSTX) at Princeton, and three medium sized ST research facilities: PEGASUS at University of Wisconsin, HIT-II at University of Washington, and CDX-U at Princeton. In the context of the fusion energy development path being formulated in the US, an ST-based Component Test Facility (CTF) and, ultimately a Demo device, are being discussed. For these, it is essential to develop high performance, steady-state operational scenarios. The relevant scientific issues are energy confinement, MHD stability at high beta (?), non-inductive sustainment, Ohmic-solenoid-free start-up, and power and particle handling. In the confinement area, the NSTX experiments have shown that the confinement can be up to 50% better than the ITER-98-pby2 H-mode scaling, consistent with the requirements for an ST-based CTF and Demo. In NSTX, CTF-relevant average toroidal beta values ?T of up to 35% with a near unity central ?T have been obtained. NSTX will be exploring advanced regimes where ?T up to 40% can be sustained through active stabilization of resistive wall modes. To date, the most successful technique for non-inductive sustainment in NSTX is the high beta poloidal regime, where discharges with a high non-inductive fraction (~60% bootstrap current+NBI current drive) were sustained over the resistive skin time. Research on radio-frequency (RF) based heating and current drive utilizing high harmonic fast wave and electron Bernstein wave is also pursued on NSTX, PEGASUS, and CDX-U. For non-inductive start-up, the coaxial helicity injection, developed in HIT/HIT-II, has been adopted on NSTX to test the method up to Ip ~ 500?kA. In parallel, start-up using a RF current drive and only external poloidal field coils are being developed on NSTX. The area of power and particle handling is expected to be challenging because of the higher power density expected in the ST relative to that in conventional aspect-ratio tokamaks. Due to its promise for power and particle handling, liquid lithium is being studied in CDX-U as a potential plasma-facing surface for a fusion reactor.

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.

NIMROD is a code development project designed to study long-wavelength, low-frequency, nonlinear phenomena in toroidal plasmas with realistic geometry and dynamics. The numerical challenges of solving the fluid equations for a fusion plasma are discussed and our discretization scheme is presented. Simulations of a resistive tearing mode show that time steps much greater than the Alfven time are possible without loss of accuracy. Validation tests of a resistive interchange mode in a shaped equilibrium, a ballooning mode and nonlinear activity in reversed-field pinches are described.

The Korea Superconducting Tokamak Advanced Research, KSTAR, is designed to operate a steady-state, high beta plasma while retaining global magnetohydrodynamic (MHD) stability to establish the scientific and technological basis of an economically attractive fusion reactor. An equilibrium model is established for stability analysis of KSTAR. Reconstructions were performed for the experimental start-up scenario and experimental first plasma operation using the EFIT code. The VALEN code was used to determine the vacuum vessel current distribution. Theoretical high beta equilibria spanning the expected operational range are computed for various profiles including generic L-mode and DIII-D experimental H-mode pressure profiles. Ideal MHD stability calculations of toroidal mode number of unity using the DCON code shows a factor of 2 improvement in the wall-stabilized plasma beta limit at moderate to low plasma internal inductance. The planned stabilization system in KSTAR comprises passive stabilizing plates and actively cooled in-vessel control coils (IVCCs) designed for non-axisymmetric field error correction and stabilization of slow timescale MHD modes including resistive wall modes (RWMs). VALEN analysis using standard proportional gain shows that active stabilization near the ideal wall limit can be reached with feedback using the midplane segment of the IVCC. The RMS power required for control using both white noise and noise taken from NSTX active stabilization experiments is computed for beta near the ideal wall limit. Advanced state-space control algorithms yield a factor of 2 power reduction assuming white noise while remaining robust with respect to variations in plasma beta.

Recent interest in the experimental study of tokamak plasma flow for different magnetic field geometries calls for theoretical understanding of the effects of tokamak magnetic topology changes on the flow. The consequences of total magnetic field reversal and/or X-point reversal on divergence-free plasma flow within magnetic flux surfaces are considered and the results are applied to interpret recent Alcator C-Mod scrape-off layer flow measurements.

This is a large and important work by one of the leading Russian contributors to this subject. It covers topics of central importance to magnetic fusion energy research with a breadth, depth and clarity not found elsewhere. It is intended as a reference book and guide to the original literature for serious practitioners of the field. The book is divided into 8 large parts, listed below, and further subdivided into 31 chapters. The preface states, ``A need was identified to summarize the large amount of information on the topic of instabilities. This information is scattered throughout many papers on certain special subjects on the theory of toroidal-plasma instabilities. This book aims to treat it from a unified point of view.'' This is a good description of the aims and approach. The range of topics is large and the treatment deep and thorough. This is often accomplished by assuming considerable knowledge on the part of the reader, most appropriate for mature researchers in the field. The pace is fast. The requisite background includes extensive knowledge of a wide range of physics, ordinary and partial differential equations, integral equations, differential geometry and special functions. The preface also states, ``The author starts with the fundamental principles, so that no special knowledge of plasma physics is necessary before reading the book. It will therefore be useful for researchers, postgraduates, high-school teachers and students specializing in plasma physics and controlled fusion.'' In the opinion of this reviewer, that may be overly ambitious. The terse approach, characteristic of the whole book, as well as the steep price, would make it rather challenging for such an audience. For example, in the opening chapter, `General results of equilibrium theory', the properties of non-orthogonal, curvilinear co-ordinates are stated tersely without much explanation or derivation, more as a review than a first presentation. It is common for physics students, at least in the USA, to encounter this material first in a course on general relativity, which they might not have taken previously when specializing in plasma physics. While good efforts are made by the author to provide an intuitive understanding of the many analytical results, this is often done with such brevity that a substantial level of maturity is required to comprehend the ideas. Another quote from the preface is, ``The book is based on analytical approaches and should therefore be useful for everybody who is interested in the topic.'' In a field where complex geometry and dynamics and the importance of practical results have required much novel and creative computational work over the past 25 years, there is no mention, no acknowledgment, no hint of its importance. The analytical approach presented here certainly fills an important need, and there is no need for the same work to cover numerical work in depth, but some recognition of the importance of numerical work and its relationship with the analytical side of the theory might have been justified. Despite these shortcomings, this book is a major and welcome addition to the literature on plasma instabilities which I heartily recommend. Contents: 1. Equilibrium of a plasma in toroidal confinement systems; 2. Internal magnetohydrodynamic modes in the cylindrical approximation; 3. Small-scale magnetohydrodynamic instabilities in toroidal confinement systems; 4. Magnetohydrodynamic internal kink modes in toroidal geometry; 5. Magnetohydrodynamic modes in collisionless and neoclassical regimes; 6. Drift-magnetohydrodynamic modes; 7. External kink modes; 8. Alfven eigenmodes and their interaction with high-energy particles; References; Index.