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We use the 3D magnetohydrodynamic (MHD) code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)] to examine the MHD stability and subsequent evolution of NSTX shot 129169. This discharge had a period with a non-monotonic safety factor profile, q (reversed shear), which was terminated by a MHD event that abruptly lowered the central safety factor, q0, and greatly reduced the peakedness of the pressure profile. We show that the equilibrium just before the MHD event occurred was linearly unstable to many pressure-driven infernal modes. Modes with toroidal mode number n ≥ 3 all had rational surfaces very close to the minimum value of q. However, a non-resonant pressure-driven (1, 1) mode was also present, and this dominated the nonlinear evolution. The final state in the simulation, after the MHD activity subsided, had a reduced and flattened pressure profile and a nearly monotonic q-profile, in qualitative agreement with experimental results. The initial state was also unstable to the resistive interchange criteria in the reversed-shear region, but the final state was stable everywhere. The "double tearing mode" (DTM) does not appear to play a role in the MHD activity of this discharge. In Appendix A, we show that in a torus, the DTM is strongly stabilized by pressure, but it is destabilized in cylindrical geometry (which has been the most extensively analyzed in the literature). [ABSTRACT FROM AUTHOR]

This paper extends the analysis first presented in Jardin et al. [Phys. Rev. Lett. 128, 245001 (2022)] to more thoroughly examine the stability of spherical torus equilibrium to ideal magnetohydrodynamic (MHD) infernal modes and their nonlinear consequences. We demonstrate that in a 3D resistive magnetohydrodynamic (MHD) simulation of a NSTX discharge, anomalous transport can occur due to these instabilities. We generate a family of equilibrium of differing β and use this to show that these instabilities could explain the experimentally observed flattening of the electron temperature profile at modest β. The modes studied in this paper are found to occur with poloidal mode number m and toroidal mode number n when the ratio m/n is in the range of 1.2–1.5, when the central safety factor is in this range or slightly lower, and when the central region has very low magnetic shear. Our analysis gives some insight as to why the unstable linear growth rates are oscillatory functions of the toroidal mode number n. We present a simulation of an initially stable configuration that passes through a stability boundary at a critical β as it is heated. We also show that a particular NSTX discharge is unstable to these modes over a timescale of several hundred ms. We conclude that these modes must be taken into account when performing predictive modeling. An appendix shows that similar modes can be found in R / a = 4 tokamaks for certain q-profiles and β values. [ABSTRACT FROM AUTHOR]

Recently, it has been shown that a vertical displacement event (VDE) can occur in ITER even when the walls are perfect conductors, as a consequence of the current quench (CQ) [A. H. Boozer, Phys. Plasmas 26, 114501 (2019)]. We used the extended-MHD code M3D-C1 with an ITER-like equilibrium and induced a CQ to explore cold VDEs in the limit of perfectly conducting walls, using different wall geometries. In the case of a rectangular first wall with the side walls far away from the plasma, we obtained very good agreement with the analytical model developed by Boozer that considers a top/bottom flat-plates wall. We show that the solution in which the plasma remains at the initial equilibrium position is improved when bringing the side walls closer to the plasma. When approximating the ITER first wall as a perfect conductor, the plasma remains stable at the initial equilibrium position far beyond the value predicted by the flat-plates wall limit. When considering an opposite limit in which only the inner shell of the ITER vacuum vessel acts as a perfect conductor, the plasma is displaced during the CQ, but the edge safety factor remains above 2 longer in the current decay compared to the flat-plates wall limit. In all the simulated cases, the VDE is found to be strongly dependent on the plasma current, in agreement with a similar finding in the flat-plates wall limit, showing an important difference with hot VDEs in which the CQ is not a necessary condition. [ABSTRACT FROM AUTHOR]

The ubiquitous sawtooth phenomena in tokamaks are so named because the central temperature rises slowly and falls rapidly, similar to the blades of a saw. First discovered in 1974, it has so far eluded a theoretical explanation that is widely accepted and consistent with experimental observations. We propose here a new theory for the sawtooth phenomena in auxiliary heated tokamaks, which is motivated by our recent understanding of "magnetic flux pumping." In this theory, the role of the (m , n) = (1 , 1) mode is to generate a dynamo voltage, which keeps the central safety factor, q0, just above 1.0 with low central magnetic shear. When central heating is present, the temperature on axis will increase until at some point, and the configuration abruptly becomes unstable to ideal MHD interchange modes with equal poloidal and toroidal mode numbers, m = n > 1. It is these higher order modes and the localized magnetic stochasticity they produce that cause the sudden crash of the temperature profile, not magnetic reconnection. Long time 3D MHD simulations demonstrate these phenomena, which appear to be consistent with many experimental observations. [ABSTRACT FROM AUTHOR]

A self-regulating magnetic flux pumping mechanism in tokamaks that maintains the core safety factor at q ≈ 1, thus preventing sawteeth, is analyzed in nonlinear 3D magnetohydrodynamic simulations using the M3D-C¹ code. In these simulations, the most important mechanism responsible for the flux pumping is that a saturated (m = 1, n = 1) quasi-interchange instability generates an effective negative loop voltage in the plasma center via a dynamo effect. It is shown that sawtoothing is prevented in the simulations if β is sufficiently high to provide the necessary drive for the (m = 1, n = 1) instability that generates the dynamo loop voltage. The necessary amount of dynamo loop voltage is determined by the tendency of the current density profile to centrally peak which, in our simulations, is controlled by the peakedness of the applied heat source profile. [ABSTRACT FROM AUTHOR]

Free-boundary 3D tokamak equilibria and resistive wall instabilities are calculated using a new resistive wall model in the two-fluid M3D-C1 code. In this model, the resistive wall and surrounding vacuum region are included within the computational domain. This implementation contrasts with the method typically used in fluid codes in which the resistive wall is treated as a boundary condition on the computational domain boundary and has the advantage of maintaining purely local coupling of mesh elements. This new capability is used to simulate perturbed, free-boundary nonaxisymmetric equilibria; the linear evolution of resistive wall modes; and the linear and nonlinear evolution of axisymmetric vertical displacement events (VDEs). Calculated growth rates for a resistive wall mode with arbitrary wall thickness are shown to agree well with the analytic theory. Equilibrium and VDE calculations are performed in diverted tokamak geometry, at physically realistic values of dissipation, and with resistive walls of finite width. Simulations of a VDE disruption extend into the current-quench phase, in which the plasma becomes limited by the first wall, and strong currents are observed to flow in the wall, in the SOL, and from the plasma to the wall. [ABSTRACT FROM AUTHOR]

We demonstrate that in a 3D resistive magnetohydrodynamic simulation, for some parameters it is possible to form a stationary state in a tokamak where a saturated interchange mode in the center of the discharge drives a near helical flow pattern that acts to nonlinearly sustain the configuration by adjusting the central loop voltage through a dynamo action. This could explain the physical mechanism for maintaining stationary nonsawtoothing "hybrid" discharges, often referred to as "flux pumping." [ABSTRACT FROM AUTHOR]

The DK4D code has been written to solve a set of time-dependent, axisymmetric, finite-Larmor-radius drift-kinetic equations (DKEs) for the non-Maxwellian part of the electron and ion distribution functions using the full, linearized Fokker-Planck-Landau collision operator. The plasma is assumed to be in the low- to finite-collisionality regime, as is found in the cores of modern and future magnetic confinement fusion experiments. Each DKE is formulated such that the perturbed distribution function carries no net density, parallel momentum, or kinetic energy. Rather, these quantities are contained within the background Maxwellians and would be evolved by an appropriate set of extended magnetohydrodynamic (MHD) equations. This formulation allows for straight-forward coupling of DK4D to existing extended MHD time evolution codes. DK4D uses a mix of implicit and explicit temporal representations and finite element and spectral spatial representations. These, along with other computational methods used, are discussed extensively. Steady-state benchmarks are then presented comparing the results of DK4D to expected analytic results at low collisionality, qualitatively, and to the Sauter analytic fits for the neoclassical conductivity and bootstrap current, quantitatively. These benchmarks confirm that DK4D is capable of solving for the correct, gyroaveraged distribution function in stationary magnetic equilibria. Furthermore, the results presented demonstrate how the exact drift-kinetic solution varies with collisionality as a function of the magnetic moment and the poloidal angle. [ABSTRACT FROM AUTHOR]

A new code, the Neoclassical Ion-Electron Solver (NIES), has been written to solve for stationary, axisymmetric distribution functions (f) in the conventional banana regime for both ions and electrons using a set of drift-kinetic equations (DKEs) with linearized Fokker-Planck-Landau collision operators. Solvability conditions on the DKEs determine the relevant non-adiabatic pieces of f (called h). We work in a 4D phase space in which ψ defines a flux surface, θ is the poloidal angle, v is the magnitude of the velocity referenced to the mean flow velocity, and λ is the dimensionless magnetic moment parameter. We expand h in finite elements in both v and λ. The Rosenbluth potentials, [uppercase_phi_synonym] and Ψ, which define the integral part of the collision operator, are expanded in Legendre series in cosχ, where χ is the pitch angle, Fourier series in cosθ, and finite elements in v. At each ψ, we solve a block tridiagonal system for hi (independent of fe), then solve another block tridiagonal system for he (dependent on fi). We demonstrate that such a formulation can be accurately and efficiently solved. NIES is coupled to the MHD equilibrium code JSOLVER [J. DeLucia et al., J. Comput. Phys. 37, 183-204 (1980)] allowing us to work with realistic magnetic geometries. The bootstrap current is calculated as a simple moment of the distribution function. Results are benchmarked against the Sauter analytic formulas and can be used as a kinetic closure for an MHD code (e.g., M3D-C1 [S. C. Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)]). [ABSTRACT FROM AUTHOR]

Growth rates of edge localized modes for various benchmark equilibria, including a diverted equilibrium, are calculated using the nonideal fluid code M3D-C1. Growth rates calculated by M3D-C1 in the ideal limit are found to agree with those calculated by ideal magnetohydrodynamics codes. The effects of nonuniform density and resistivity profiles are explored, as well as the sensitivity of growth rates to the position of the ideal vacuum-plasma interface. Growth rates of the diverted equilibrium are found to be particularly sensitive to moving this interface inward from the separatrix, but less sensitive to extending the plasma region beyond the separatrix. The resistivity profile within the plasma is found not to affect growth rates significantly; however, growth rates may be greatly reduced by treating the outer region as a resistive plasma instead of an ideal vacuum. Indeed, it is found that for typical scrape-off layer (SOL) temperatures, the resistive SOL model behaves more like an ideal plasma than a vacuum. [ABSTRACT FROM AUTHOR]

The cylindrical ideal magnetohydrodynamic (MHD) stability problem, including flow and a resistive wall, is cast in the standard mathematical form, ωA·x=B·x, without discretizing the vacuum regions surrounding the plasma. This is accomplished by means of a finite element expansion for the plasma perturbations, by coupling the plasma surface perturbations to the resistive wall using a Green’s function approach, and by expanding the unknown vector, x, to include the perturbed current in the resistive wall as an additional degree of freedom. The ideal MHD resistive wall mode (RWM) can be stabilized when the plasma has a uniform equilibrium flow such that the RWM frequency resonates with the plasma’s Doppler-shifted sound continuum modes. The resonance induces a singularity in the parallel component of the plasma perturbations, which must be adequately resolved. Complete stabilization within the ideal MHD model (i.e., without parallel damping being added) is achieved as the grid spacing in the region of the resonance is extrapolated to 0 step size. [ABSTRACT FROM AUTHOR]

The eigenvalue equations describing a cylindrical ideal magnetohydrodynamics plasma interacting with a thin resistive wall are presented in the standard mathematical form, A·x=λB·x, without discretizing the vacuum regions surrounding the plasma. This is accomplished by using a finite-element basis for the plasma perturbations, and by coupling the plasma surface perturbations to the perturbed electrical current in the wall using a Green’s-function approach. The perturbed wall current introduces a single additional degree of freedom into the system, which, together with an auxiliary variable, u=ωξ, allows the system to take the standard linear form. The standard form allows the use of linear eigenvalue solvers, without additional iterations, to compute the complete spectrum of plasma modes in the presence of a surrounding resistive wall at arbitrary separation. Standard results are recovered in the limits of (i) an infinitely resistive wall (no wall), and (ii) a zero resistance wall (ideal wall). [ABSTRACT FROM AUTHOR]

The sawtooth instability is one of the most fundamental dynamics of an inductive tokamak discharge such as will occur in ITER [R. Aymar et al., Plasma Phys. Controlled Fusion 44, 519 (2002)]. Sawtooth behavior is complex and remains incompletely explained. The Center for Extended MHD Modeling (CEMM) SciDAC project has undertaken an ambitious campaign to model this periodic motion in a small tokamak as accurately as possible using the extended MHD model. Both M3D [W. Park et al., Phys. Plasmas 6, 1796 (1999)] and NIMROD [C. R. Sovinec et al., Phys. Plasmas 10, 1727 (2003)] have been applied to this problem. Preliminary nonlinear MHD results show pronounced stochasticity in the magnetic field following the sawtooth crash but are not yet fully converged. Compared to the MHD model, extended MHD predicts plasma rotation, faster reconnection, and reduced field line stochasticity in the crash aftermath. The multiple time and space scales associated with the reconnection layer and growth time make this an extremely challenging computational problem. However, these calculations are providing useful guidelines to the numerical and physical requirements for more rigorous future studies. [ABSTRACT FROM AUTHOR]

A general coordinate-independent expression for Braginskii’s form of the ion gyroviscosity in the two-dimensional potential field representation is presented, and is implemented in a full two-dimensional, two-fluid extended magnetohydrodynamic (MHD) numerical model. The expression for the gyroviscous force requires no field to be differentiated more than twice, and thus is appropriate for finite elements with first derivatives continuous across element boundaries (C1 finite elements). From the extended MHD model, which includes the full gyroviscous stress, are derived linear dispersion relations of a homogeneous equilibrium and of an inverted-density profile in the presence of gravity. The treatment of the gravitational instability presented here extends previous work on the subject [M. N. Rosenbluth, N. A. Krall, and N. Rostoker, Nucl. Fusion Suppl. 1, 143 (1962); K. V. Roberts and J. B. Taylor, Phys. Rev. Lett. 8, 197 (1962)]. Linear and nonlinear simulations of the gravitational instability are presented. Simulations are shown to agree closely with the derived dispersion relations in the linear regime. The “gyroviscous cancellation” effect is demonstrated, and some limitations of the vvector * approximation are discussed. [ABSTRACT FROM AUTHOR]

Here we describe a technique for solving the four-field extended-magnetohydrodynamic (MHD) equations in two dimensions. The introduction of triangular high-order finite elements with continuous first derivatives (C1 continuity) leads to a compact representation compatible with direct inversion of the associated sparse matrices. The split semi-implicit method is introduced and used to integrate the equations in time, yielding unconditional stability for arbitrary time step. The method is applied to the cylindrical tilt mode problem with the result that a nonzero value of the collisionless ion skin depth will increase the growth rate of that mode. The effect of this parameter on the reconnection rate and geometry of a Harris equilibrium and on the Taylor reconnection problem is also demonstrated. This method forms the basis for a generalization to a full extended-MHD description of the plasma with six, eight, or more scalar fields. [ABSTRACT FROM AUTHOR]

A two-dimensional magnetohydrodynamics (MHD) code is used to investigate the dynamical evolution of driven reconnection in the Magnetic Reconnection Experiment (MRX) [M. Yamada et al., Phys. Plasmas 7, 1781 (2000)]. The initial conditions and dimensionless parameters of the simulation are set to be similar to the experimental values. Many features of the time-evolution of magnetic configurations for both co- and counter-helicity reconnection in MRX are successfully reproduced in the framework of resistive MHD. The resistive MHD model is then augmented by the addition of a "model Hall" term to begin to assess the importance of two-fluid physics in the experiment. The effective decoupling of the ion fluid from the reconnecting magnetic field due to the model Hall term is shown to be important during the early dynamic X-phase of MRX reconnection, while effectively negligible during the late "steady-state" Y-phase, when plasma heating takes place. These results are consistent with the available experimental evidence. Based on simple symmetry considerations, an experiment to directly measure the Hall effect in MRX configuration is proposed and numerical evidence for the expected outcome is given. [ABSTRACT FROM AUTHOR]

Recent experimental and theoretical results have led to two lines of thought regarding the physical processes underlying fast magnetic reconnection. One is based on the traditional Sweet-Parker model but replaces the Spitzer resistivity with an enhanced resistivity caused by electron scattering by ion acoustic turbulence. The other includes the finite gyroradius effects that enter Ohm's law through the Hall and electron pressure gradient terms. A two-dimensional numerical study, conducted with a new implicit parallel two-fluid code, has helped to clarify the similarities and differences in predictions between these two models. The former yields resistivity-dependent reconnection with a thick, moderate-aspect-ratio current sheet. If the sheet thickness is less than or comparable to the ion skin depth, it is verified that the Hall effect will predominate [Shay et al., Geophys. Res. Lett. 26, 2163 (1999)], producing true fast reconnection with a microscopic current sheet of unit aspect ratio and a distorted out-of-plane magnetic field [Mandt et al., Geophys. Res. Lett. 21, 73 (1994)]. [ABSTRACT FROM AUTHOR]

Injection of lower hybrid current drive into the current ramp-up phase of Joint European Torus (JET) discharges has been observed to produce an annular current distribution with a core region of essentially zero current density [Hawkes et al., Phys. Rev. Lett. 87, 115001 (2001)]. Similar "current holes" have been observed in Japan Atomic Energy Research Institute Tokamak 60 Upgrade discharges with off-axis current drive supplied by bootstrap current [T. Fujita et al., Phys. Rev. Lett. 87, 245001 (2001)]. In both cases, the central current does not go negative, although current diffusion calculations indicate that there is sufficient noninductive current drive for this to occur. This is explained by the multilevel 3D code (M3D) nonlinear 2D and 3D resistive magnetohydrodynamic (MHD) simulations in toroidal geometry, which predict that these discharges undergo n = 0 reconnection events -- "axisymmetric sawteeth" -- that redistribute the current to hold its core density near zero. Unlike conventional sawteeth, these events retain the symmetry of the equilibrium, and thus are best viewed as a transient loss of equilibrium caused when an i = 0 rational surface enters the plasma. If the current density profile has a central minimum, this surface will enter on axis; otherwise it will enter off-axis. In the first case, the reconnection is limited to a small region around the axis and clamps the core current at zero. In the second case, more typical of the JET experiments, the core current takes on a finite negative value before the i = 0 surface appears, resulting in discrete periodic axisymmetric sawtooth events with a finite minor radius. Interpretation of the simulation results is given in terms of analytic equilibrium theory, and the relation to conventional sawteeth and to a recent reduced-MHD analysis of this phenomenon in cylindrical geometry [Huysmans et al., Phys. Rev. Lett. 87, 245002 (2001)] is discussed. [ABSTRACT FROM AUTHOR]

Global stability of the oblate (small elongation, E<1) Field-Reversed Configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic (MHD) and hybrid (fluid electrons and kinetic ions) simulations. For every nonzero value of the toroidal mode number n, there are three MHD modes that must be stabilized. For n=1, these are the interchange, the tilt and the radial shift; while for n>1 these are the interchange and two co-interchange modes with different polarization. It is shown that the n=1 tilt mode becomes an external mode when E<1, and it can be effectively stabilized by close-fitting conducting shells, even in the small Larmor radii (MHD) regime. The tilt mode stability improves with increasing oblateness, however at sufficiently small elongations the radial shift mode becomes more unstable than the tilt mode. The interchange mode stability is strongly profile dependent, and all n>=1 interchange modes can be stabilized for a class of pressure profile with separatrix beta larger than 0.035. Our results show that all three n=1 modes can be stabilized in the MHD regime, but the stabilization of the n>1 co-interchange modes still remains an open question. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In a recent paper, Nakayama, Sato, and Matsuoka [Phys. Fluids 31, 630 (1988)] suggested that elliptical cross-section tokamaks with aspect ratio R/a=3.2 and with elongation κ=2.6 are unstable to a splitting (m=2, n=0) instability for plasma β>5%, and that κ≥4.0 plasmas are unstable to a splitting for β≥1%. The magnetohydrodynamic evolution code tsc [J. Comput. Phys. 66, 481 (1986)] indicates, however, that such plasmas are robustly stable with respect to this splitting. In fact, a κ=3.7 plasma with β=23.0% shows no tendency to split. However, the addition of pinching coils at the waist will cause the plasma to split if the current in these coils exceeds a critical value Ic, which decreases with increasing beta. [ABSTRACT FROM AUTHOR]

In recent years, the nonlinear 3D magnetohydrodynamic codes JOREK, M3D-C1, and NIMROD developed the capability of modeling realistic 3D vertical displacement events (VDEs) including resistive walls. In this paper, a comprehensive 3D VDE benchmark is presented between these state-of-the-art codes. The simulated case is based on an experimental NSTX plasma but with a simplified rectangular wall. There are differences between the physics models and numerical methods, and the VDE evolution leads to sensitivities on the initial conditions that cannot be avoided as can be done in edge localized modes (ELM) and sawtooth simulations (due to the non-cyclical nature of VDEs). Nonetheless, the comparison serves to quantify the level of agreement in the relevant quantities used to characterize disruptions, such as the 3D wall forces and energy decay. The results bring confidence regarding the use of the mentioned codes for disruption studies, and they distinguish aspects that are specific to the models used (e.g., reduced vs full MHD models). The simulations show important 3D features for a NSTX plasma, such as the self-consistent evolution of the halo current and the origin of the wall forces. In contrast to other reduced MHD models based on an ordering in the aspect ratio, the ansatz-based JOREK reduced MHD model allows capturing many aspects of the 3D dynamics even in the spherical tokamak limit considered here. [ABSTRACT FROM AUTHOR]

Runaway electrons may be generated in a tokamak during the start up, during normal operation and during a plasma disruption. During a disruption, runaway electrons can be accelerated to high energies, potentially damaging the first wall. To predict the consequences of runaway generation during a disruption, it is necessary to consider resonant interactions of runaways with the bulk plasma. Here we consider the interactions of runaways on low mode number tearing modes. We have developed a fluid runaway electron model for the 3D MHD code M3D-C1 (Jardin et al 2012 J. Comput. Sci. Discovery 6 014002). To benchmark, we have reproduced the MHD linear tearing mode results (with runaway electrons) in a circular cylinder presented in previous analytic studies (Helander et al 2007 Phys. Plasmas 14 104142) and have extended them here with a numerical eigenvalue calculation. We find that the low mode number tearing mode has a rotation caused by the MHD - runaways interaction and the perturbed toroidal current scale length is much smaller with runaways than without and decreases as the runaway speed increases. [ABSTRACT FROM AUTHOR]

Employment of non-inductive plasma start-up techniques would considerably simplify the design of a spherical tokamak fusion reactor. Transient coaxial helicity injection (CHI) is a promising method, expected to scale favorably to next-step reactors. However, the implications of reactor-relevant parameters on the initial breakdown phase for CHI have not yet been considered. Here, we evaluate CHI breakdown in reactor-like configurations using an extension of the Townsend avalanche theory. We find that a CHI electrode concept in which the outer vessel wall is biased to achieve breakdown, while previously successful on NSTX and HIT-II, may exhibit a severe weakness when scaled up to a reactor. On the other hand, concepts which employ localized biasing electrodes such as those used in QUEST would avoid this issue. Assuming that breakdown can be successfully attained, we then apply scaling relationships to predict plasma parameters attainable in the transient CHI discharge. Assuming the use of 1 Wb of injector flux, we find that plasma currents of 1 MA should be achievable. Furthermore, these plasmas are expected to Ohmically self-heat with more than 1 MW of power as they decay, facilitating efficient hand-off to steady-state heating sources. These optimistic scalings are supported by Tokamak Simulation Code simulations. [ABSTRACT FROM AUTHOR]

The main results of an intense vertical displacement event (VDE) modelling activity using the implicit 3D extended MHD code M3D-C1 are presented. A pair of nonlinear 3D simulations are performed using realistic transport coefficients based on the reconstruction of a so-called NSTX frozen VDE where the feedback control was purposely switched off to trigger a vertical instability. The vertical drift phase is solved assuming axisymmetry until the plasma contacts the first wall, at which point the intricate evolution of the plasma, decaying to large extent in force-balance with induced halo/wall currents, is carefully resolved via 3D nonlinear simulations. The faster 2D nonlinear runs allow to assess the sensitivity of the simulations to parameter changes. In the limit of perfectly conducting wall, the expected linear relation between vertical growth rate and wall resistivity is recovered. For intermediate wall resistivities, the halo region contributes to slowing the plasma down, and the characteristic VDE time depends on the choice of halo temperature. The evolution of the current quench and the onset of 3D halo/eddy currents are diagnosed in detail. The 3D simulations highlight a rich structure of toroidal modes, penetrating inwards from edge to core and cascading from high-n to low-n mode numbers. The break-up of flux-surfaces results in a progressive stochastisation of field-lines precipitating the thermalisation of the plasma with the wall. The plasma current then decays rapidly, inducing large currents in the halo region and the wall. Analysis of normal currents flowing in and out of the divertor plate reveals rich time-varying patterns. [ABSTRACT FROM AUTHOR]

Global stability of the field-reversed configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic and hybrid (fluid electron and δf particle ion) simulations. The stabilizing effects of velocity shear and finite ion Larmor radius (FLR) on the n=1 internal tilt mode in the prolate FRCs have been studied. Sheared rotation is found to reduce the growth rate, however a large rotation rate with Mach number of M>=1 is required in order for significant reduction in the instability growth rate to occur. Kinetic effects associated with large thermal ion orbits have been studied for different kinetic equilibria. The simulations show that there is a reduction in the tilt mode growth rate due to FLR effects, but complete linear stability has not been found, even when the thermal ion gyroradius is comparable to the distance between the field null and the separatrix. The instability existing beyond the FLR theory threshold could be due to the resonant interaction of the wave with ions whose Doppler shifted frequency matches the betatron frequency. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Transient coaxial helicity injection (CHI) in the National Spherical Torus Experiment (NSTX) has generated toroidal current on closed-flux surfaces without the use of the central solenoid. When induction from the central solenoid was added, CHI initiated discharges in NSTX achieved 1MA of plasma current using 65% of the solenoid flux of standard induction-only discharges. In addition, the CHI-initiated discharges have lower density and a low normalized internal plasma inductance of 0.35, as desired for achieving advanced scenarios. Transient CHI will be used for non-inductive plasma start-up on the upgrade to NSTX (NSTX-U) that is now under construction. It will have numerous improvements that significantly enhance CHI start-up capability. The TSC code is now starting to be used for full discharge simulation, which includes solenoid-less start-up with CHI and subsequent non-inductive current ramp-up using neutral beams. These results suggest that the increased injector flux capability of NSTX-U should allow CHI start-up at more than the 400 kA level and that the new tangential neutral beam system on NSTX-U should be able to ramp the current to the 1MA levels. [ABSTRACT FROM AUTHOR]

A benchmark exercise for the modeling of vertical displacement events (VDEs) is presented and applied to the 3D nonlinear magnetohydrodynamic codes M3D-C1, JOREK, and NIMROD. The simulations are based on a vertically unstable NSTX equilibrium enclosed by an axisymmetric resistive wall with a rectangular cross section. A linear dependence of the linear VDE growth rates on the resistivity of the wall is recovered for sufficiently large wall conductivity and small temperatures in the open field line region. The benchmark results show good agreement between the VDE growth rates obtained from linear NIMROD and M3D-C1 simulations and from the linear phase of axisymmetric nonlinear JOREK, NIMROD, and M3D-C1 simulations. Axisymmetric nonlinear simulations of a full VDE performed with the three codes are compared, and an excellent agreement is found regarding the plasma location and plasma currents, as well as eddy and halo currents in the wall. [ABSTRACT FROM AUTHOR]

H-mode plasma operation of the Korea Superconducting Tokamak Advanced Research (KSTAR) device has been expanded to significantly surpass the ideal MHD no-wall beta limit. Plasmas with high normalized beta, βN, up to 4.3 have been achieved with reduced plasma internal inductance, li, to near 0.7, exceeding the computed n=1 ideal no-wall limit by a factor of 1.6. Pulse lengths at maximum βN were extended to longer pulses by new, more rapid control. The stability of the observed m/n=2/1 tearing mode that limited the achieved high βN is computed by the M3D-C¹ code, and the effect of sheared toroidal rotation to tearing stability is examined. As a method to affect the mode stability in high βN plasmas, the non-resonant alteration of the rotation profile by non-axisymmetric magnetic fields has been used, enabling a study of the underlying neoclassical toroidal viscosity (NTV) physics and stability dependence on rotation. Non-axisymmetric field spectra were applied using in-vessel control coils (IVCCs) with varied n=2 field configurations to alter the plasma toroidal rotation profile in high beta H-mode plasmas and to analyze their effects on the rotation. The rotation profile was significantly altered with rotation reduced by more than 60% without tearing activity or mode locking. To investigate the physical characteristics and scaling of the measured rotation braking by NTV, changes in the rotation profile are analytically examined in steady state. The expected NTV scaling with the square of the normalized applied field perturbation agrees with the measured profile change δB2.1-2.3. The NTV is also found to scale as Ti2.1-2.4, in general agreement with the low collisionality "1/ν" regime scaling of the NTV theory (TNTV-(1/v) ∝Ti 2.5). [ABSTRACT FROM AUTHOR]

The growth of a tearing mode is described by reduced MHD equations. For a cylindrical equilibrium, tearing mode growth is governed by the modified Rutherford equation, i.e., the nonlinear Δ'(w). For a low beta plasma without external heating, Δ'(w) can be approximately described by two terms, Δ'ql(w), Δ′A(w) [White et al., Phys. Fluids 20, 800 (1977); Phys. Plasmas 22, 022514 (2015)]. In this work, we present a simple method to calculate the quasilinear stability index Δ′ql rigorously, for poloidal mode number m≥2. Δ′ql is derived by solving the outer equation through the Frobenius method. Δ'ql is composed of four terms proportional to: constant Δ'0, w, wlnw, and w 2. Δ′A is proportional to the asymmetry of island that is roughly proportional to w. The sum of Δ′ql and Δ′A is consistent with the more accurate expression calculated perturbatively [Arcis et al., Phys. Plasmas 13, 052305 (2006)]. The reduced MHD equations are also solved numerically through a 3D MHD code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)]. The analytical expression of the perturbed helical flux and the saturated island width agree with the simulation results. It is also confirmed by the simulation that the Δ′A has to be considered in calculating island saturation. [ABSTRACT FROM AUTHOR]

Elimination of the central solenoid would simplify the engineering design of a fusion nuclear science facility and tokamak based devices. The method of transient coaxial helicity injection (CHI) has successfully demonstrated formation of a high-quality closed flux plasma in NSTX and will be used as the front end of the start-up method for a full demonstration of noninductive current start-up, followed by noninductive current ramp-up using neutral beams in the NSTX-U device that is now under construction at the Princeton Plasma Physics Laboratory. CHI is implemented by driving current along open field lines that connect the lower inner and outer divertor plates of a spherical torus. The engineering system requirements and the design of the CHI system on NSTX-U are described. [ABSTRACT FROM PUBLISHER]

In the present paper, the numerical calculation of transonic equilibria, first introduced with the FLOW code in Guazzotto et al. [Phys. Plasmas 11, 604 (2004)], is critically reviewed. In particular, the necessity and effect of imposing explicit jump conditions at the transonic discontinuity are investigated. It is found that 'standard' (low-β, large aspect ratio) transonic equilibria satisfy the correct jump condition with very good approximation even if the jump condition is not explicitly imposed. On the other hand, it is also found that high-β, low aspect ratio equilibria require the correct jump condition to be explicitly imposed. Various numerical approaches are described to modify FLOW to include the jump condition. It is proved that the new methods converge to the correct solution even in extreme cases of very large β, while they agree with the results obtained with the old implementation of FLOW in lower-β equilibria. [ABSTRACT FROM AUTHOR]

Non-inductive steady-state scenarios on ITER will need to operate with internal transport barriers (ITBs) in order to reach adequate fusion gain at typical currents of 9 MA. The large pressure gradients at the location of the internal barrier are conducive to the development of ideal MHD instabilities that may limit the plasma performance and may lead to plasma disruptions. Fully non-inductive scenario simulations with five combinations of heating and current drive sources are presented in this work, with plasma currents in the range 7-10 MA. For each configuration the linear, ideal MHD stability is analysed for variations of the Greenwald fraction and of the pressure peaking factor around the operating point, aiming at defining an operational space for stable, steady-state operations at optimized performance. It is shown that plasmas with lower hybrid heating and current drive maintain the minimum safety factor above 1.5, which is desirable in steady-state operations to avoid neoclassical tearing modes. Operating with moderate ITBs at 2/3 of the minor radius, these plasmas have a minimum safety factor above 2, are ideal MHD stable and reach Q ⋩ 5 operating above the ideal no-wall limit. [ABSTRACT FROM AUTHOR]

The National Spherical Torus Experiment (NSTX) at Princeton University, Princeton, NJ, is designed to assess the potential of the low-aspect-ratio spherical torus concept for magnetic plasma confinement. The plasma has been heated by up to 7 MW of neutral beam injection (NBI) at an injection energy of 100 keV and up to 6 MW of high harmonic fast wave (HHFW) at 30 MHz. NSTX has achieved βT of 32%. A variety of MHD phenomena have been observed to limit β. NSTX has now begun addressing TE scaling, β limits, and current drive issues. During the NBI heating experiments, a broad Ti profile with Ti up to 2 keV, Ti > Te and a large toroidal rotation were observed. Transport analysis suggests that the impurity ions have diffusivities approaching neoclassical. For L-Mode plasmas, TE is up to two times the ITER97L L-Mode scaling and exceeds the ITER98pby2 H-Mode scaling in some cases. Transitions to H-Mode have been observed which result in an approximate doubling of TE after the transition in some conditions. During HHFW heating, Te > Ti and Te up to 3.5 keV were observed. Current drive has been studied using both coaxial helicity injection with up to 390 kA of toroidal current and HHFW. HHFW has produced H-modes with significant bootstrap current fraction at low Ip, high q, and high βp. [ABSTRACT FROM AUTHOR]

Recent operation of the Tokamak Fusion Test Reactor (TFTR) [Plasma Phys. Controlled Nucl. Fusion Research 1, 51 (1986) ] has produced plasma equilibria with values of Γ ≡ β[sub p eq] + l[sub i]/2 as large as 7, ∈β[sub p dia] ≡ 2μ[sub 0] ∈ 〈p[sub ⊥]〉 / 〈〈B[sub p]〉〉² as large as 1.6, and Troyon normalized diamagnetic beta [Plasma Phys. Controlled Fusion 26, 209 ( 1984); Phys. Lett. ll0A, 29 (1985)], β[sub N dia] ≡ 10[sup 8] 〈β[sub t⊥]〉aB[sub o]/I[sub p] as large as 4.7. When ∈β[sub p dia] ... 1.25, a separatrix entered the vacuum chamber, producing a naturally diverted discharge that was sustained for many energy confinement times, τ[sub E]. The largest values of ∈β[sub p] and plasma stored energy were obtained when the plasma current was ramped down prior to neutral beam injection. The measured peak ion and electron temperatures were as large as 24 and 8.5 keV, respectively. Plasma stored energy in excess of 2.5 MJ and τ[sub E] greater than 130 msec were obtained. Confinement times of greater than 3 times that expected from L-mode predictions have been achieved. The fusion power gain Q[sub DD] reached a value of 1.3 × 10[sup -3] in a discharge with I[sub p] = 1 MA and ∈β[sub p dia] = 0.85. A large, sustained negative loop voltage during the steady-state portion of the discharge indicates that a substantial noninductive component of I[sub p] exists in these plasmas. Transport code analysis indicates that the bootstrap current constitutes up to 65% of I[sub p]. Magnetohydrodynamic (MHD) ballooning stability analysis shows that, while these plasmas are near, or at the β[sub p], limit, the pressure gradient in the plasma core is in the first region of stability to high-n modes. [ABSTRACT FROM AUTHOR]

The highly indented plasmas of the PBX-M tokamak experiment [Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1989), Vol. 1, p. 97] have reached plasma regimes of both high volume-averaged beta (β[sub t]), and high-beta poloidal (β[sub p]), and show evidence of the suppression of external surface modes by the passive stabilizing system. Values of β[sub t] up to 4.0 I/aB (% MA/m T) with 'T[sub i] (0) ≈ 4 keV have been obtained. A magnetohydrodynamic analysis of plasmas with β[sub p] = 2.0 indicates that these plasmas are near the threshold of the second stability regime. A value of β[sub t] of 6.8% has been reached with T[sub i] (0) > 5 keV and an indentation of 28%. Control of plasma shape is accomplished with a feedback system that uses a moment expansion about a single equilibrium and is augmented by time-dependent waveforms to redefine plasma shape. Diagnostics to measure the safety factor q have been developed and used to make accurate measurements of q(r) and to verify changes made in q(0). [ABSTRACT FROM AUTHOR]

A numerical analysis was performed of guiding-center (gc) orbits in a variety of spheromak-type force-free, numerically determined equilibria including the normal spherical configurations as well as configurations that are more prolate and oblate, and others having hollow or peaked current profiles. The calculated drift orbits differ significantly from the well-known drift orbits in tokamaks. In particular, the results indicate (i) the existence of particles trapped in separate hemispheres, (ii) the orbit widths d are about proportional to the Larmor radius ρ of the particle (calculated for the field maximum) up to ρ≊0.01–0.02 of the major separatrix dimension, (iii) many orbits, in particular on the outer flux lines, have their maximum normalized width dmax/ρL(ρL= local Larmor radius) away from the midplane of the configuration, and (iv) these maximum normalized widths are about 2–4 over most of the volume for all configurations except for prolate ones with a peaked current profile; in the latter configurations, orbits on outer field surfaces show dmax≤ρL. [ABSTRACT FROM AUTHOR]