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A new fluid model for runaway electron simulation based on fluid description is introduced and implemented in the magnetohydrodynamics code M3D-C1, which includes self-consistent interactions between plasma and runaway electrons. The model utilizes the method of characteristics to solve the continuity equation for the runaway electron density with large convection speed, and uses a modified Boris algorithm for pseudo particle pushing. The model was employed to simulate magnetohydrodynamics instabilities happening in a runaway electron final loss event in the DIII-D tokamak. Nonlinear simulation reveals that a large fraction of runaway electrons get lost to the wall when kink instabilities are excited and form stochastic field lines in the outer region of the plasma. Plasma current converts from runaway electron current to Ohmic current, and get pinched at the magnetic axis. Given the good agreement with experiment, the simulation model provides a reliable tool to study macroscopic plasma instabilities in existence of runaway electron current, and can be used to support future studies of runaway electron mitigation strategies in ITER., 26 pages, 16 figures

Self-consistent simulations of neoclassical and electrostatic turbulent transport in a DIII-D H-mode edge plasma under resonant magnetic perturbations (RMPs) have been performed using the global total-f gyrokinetic particle-in-cell code x-point gyrokinetic code (XGC), in order to study density pump-out and electron heat confinement. The RMP field is imported from the extended magneto-hydrodynamics code M3D-C1, taking into account the linear two-fluid plasma response. With both neoclassical and turbulence physics considered together, the XGC simulation reproduces two key features of experimentally observed edge transport under RMPs: increased radial particle transport in the pedestal region that is sufficient to account for the experimental pump-out rate and suppression of the electron heat flux in the steepest part of the edge pedestal. In the simulation, the density fluctuation amplitude of modes moving in the electron diamagnetic direction increases due to interaction with RMPs in the pedestal shoulder and outward, while the electron temperature fluctuation amplitude decreases.

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, q(0), 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.

The capability to model the nonlinear magnetohydrodynamic (MHD) evolution of stellarator plasmas is developed by extending the M3D-$C^1$ code to allow non-axisymmetric domain geometry. We introduce a set of logical coordinates, in which the computational domain is axisymmetric, to utilize the existing finite-element framework of M3D-$C^1$. A $C^1$ coordinate mapping connects the logical domain to the non-axisymmetric physical domain, where we use the M3D-$C^1$ extended MHD models essentially without modifications. We present several numerical verifications on the implementation of this approach, including simulations of the heating, destabilization, and equilibration of stellarator plasmas with strongly anisotropic thermal conductivity, and of the relaxation of stellarator equilibria to integrable and non-integrable magnetic field configurations in realistic geometries., Comment: 8 pages, 6 figures

Nonresonant internal modes can be difficult to anticipate as there is no resonant surface in the plasma. However, equilibria that are unstable to multiple nonresonant magnetohydrodynamic (MHD) modes may be more prone to global loss of confinement since these instabilities generate spatially extended linear displacements, potentially enhancing magnetic field line chaos via nonlinear interactions. Here, we successfully predict the unstable nonresonant pressure-driven modes for equilibria with zero shear in the plasma core, irrational q on axis, and a central pressure gradient, which is consistent with pre-crash profiles in sawtoothing tokamak plasmas in the large-aspect-ratio limit. A criterion for identifying nonresonant modes most likely to be unstable is developed from the convergents of the continued fraction representation of q0. A higher-order analysis of the standard Energy Principle reveals the conditions under which these modes are expected to dominate. Linear growth rate spectra, as a function of toroidal mode number (up to n = 30), calculated using the initial-value extended-MHD code, M3D-C1, recover the characteristic dependence observed for ideal infernal modes. Nonresonant modes have also been invoked in some ideal sawtooth crash models. This work provides a mechanism to predict the mode numbers of infernal modes and, potentially, the width of some post-sawtooth-crash profiles.

The presence of error fields in a tokamak device is inevitable, so it is of fundamental importance to understand their impact on the operation of the device and up to what level they can be tolerated. In this paper a prediction of the impact on the magnetic footprints on the divertor plates due to the misalignment of NSTX-U equilibrium coils is presented. Resistive MHD simulations are used to predict the magnetic field perturbations experienced by the field lines due to the presence of error fields. A linear relation between the magnitude of the misalignment and the area of the magnetic footprints is found, as well as an inverse proportionality between the size of a magnetic footprint and the plasma penetration of the field lines composing the footprint. This study is intended as a starting point in the process of investigating divertor footprints associated with the plasma response to intrinsic error fields with resistive MHD codes. A basic interpretation of the results suggests that large footprints resulting from equilibrium coils misalignment may be desirable, to some extent, to decrease the peak heat load on the divertor components. A more complete interpretation of the results presented here will involve a number of complex physics issues, for example pedestal transport and stability, along with the role of various scrape-off layer and divertor processes, as well as experimental data to confirm the predictions.

An explanation is provided for the disruptive instability in diverted tokamaks when the safety factor$q$at the 95 % poloidal flux surface,$q_{95}$, is driven below 2.0. The instability is a resistive kink counterpart to the current-driven ideal mode that traditionally explained the corresponding disruption in limited cross-sections (Shafranov,Sov. Phys. Tech. Phys., vol. 15, 1970, p. 175) when$q_{edge}$, the safety factor at the outermost closed flux surface, lies just below a rational value$m/n$. Experimentally, external kink modes are observed in limiter configurations as the current in a tokamak is ramped up and$q_{edge}$decreases through successive rational surfaces. For$q_{edge}, the instability is always encountered and is highly disruptive. However, diverted plasmas, in which$q_{edge}$is formally infinite in the magnetohydrodynamic (MHD) model, have presented a longstanding difficulty since the theory would predict stability, yet, the disruptive limit occurs in practice when$q_{95}$, reaches 2. It is shown from numerical calculations that a resistive kink mode is linearly destabilized by the rapidly increasing resistivity at the plasma edge when$q_{95}, but$q_{edge}\gg 2$. The resistive kink behaves much like the ideal kink with predominantly kink or interchange parity and no real sign of a tearing component. However, the growth rates scale with a fractional power of the resistivity near the$q=2$surface. The results have a direct bearing on the conventional edge cutoff procedures used in most ideal MHD codes, as well as implications for ITER and for future reactor options.

New evidence indicates that there is significant 3D variation in density fluctuations near the boundary of weakly 3D tokamak plasmas when resonant magnetic perturbations are applied to suppress transient edge instabilities. The increase in fluctuations is concomitant with an increase in the measured density gradient, suggesting that this toroidally localized gradient increase could be a mechanism for turbulence destabilization in localized flux tubes. Two-fluid magnetohydrodynamic simulations find that, although changes to the magnetic field topology are small, there is a significant 3D variation of the density gradient within the flux surfaces that is extended along field lines. This modeling agrees qualitatively with the measurements. The observed gradient and fluctuation asymmetries are proposed as a mechanism by which global profile gradients in the pedestal could be relaxed due to a local change in the 3D equilibrium. These processes may play an important role in pedestal and scrape-off layer transport in ITER and other future tokamak devices with small applied 3D fields.

Small 3D perturbations to the magnetic field in DIII-D ( δB/B∼2×10−4) result in large modulations of density fluctuation amplitudes in the pedestal, which are shown using Doppler backscattering measurements to vary by a factor of 2. Helical perturbations of equilibrium density within flux surfaces have previously been observed in the pedestal of DIII-D plasmas when 3D fields are applied and were correlated with density fluctuation asymmetries in the pedestal. These intra-surface density and pressure variations are shown through two fluid MHD modeling studies using the M3D-C1 code to be due to the misalignment of the density and temperature equilibrium iso-surfaces in the pedestal region. This modeling demonstrates that the phase shift between the two iso-surfaces corresponds to the diamagnetic direction of the two species, with the mass density surfaces shifted in the ion diamagnetic direction relative to the temperature and magnetic flux iso-surfaces. The resulting pedestal density, potential, and turbulence asymmetries within flux surfaces near the separatrix may be at least partially responsible for several poorly understood phenomena that occur with the application of 3D fields in tokamaks, including density pump out and the increase in power required to transition from L- to H-mode.Small 3D perturbations to the magnetic field in DIII-D ( δB/B∼2×10−4) result in large modulations of density fluctuation amplitudes in the pedestal, which are shown using Doppler backscattering measurements to vary by a factor of 2. Helical perturbations of equilibrium density within flux surfaces have previously been observed in the pedestal of DIII-D plasmas when 3D fields are applied and were correlated with density fluctuation asymmetries in the pedestal. These intra-surface density and pressure variations are shown through two fluid MHD modeling studies using the M3D-C1 code to be due to the misalignment of the density and temperature equilibrium iso-surfaces in the pedestal region. This modeling demonstrates that the phase shift between the two iso-surfaces corresponds to the diamagnetic direction of the two species, with the mass density surfaces shifted in the ion diamagnetic direction relative to the temperature and magnetic flux iso-surfaces. The resulting pedestal density, potential, and turbul...

M3D-C^1 is an implicit, high-order finite element code for the solution of the time-dependent nonlinear two-fluid magnetohydrodynamic equations [S.C. Jardin, J. Breslau, N. Ferraro, A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions, J. Comp. Phys. 226 (2) (2007) 2146-2174]. This code has now been extended to allow computations in toroidal geometry. Improvements to the spatial integration and time-stepping algorithms are discussed. Steady-states of a resistive two-fluid model, self-consistently including flows, anisotropic viscosity (including gyroviscosity) and heat flux, are calculated for diverted plasmas in geometries typical of the National Spherical Torus Experiment (NSTX) [M. Ono et al., Exploration of spherical torus physics in the NSTX device, Nucl. Fusion 40 (3Y) (2000) 557-561]. These states are found by time-integrating the dynamical equations until the steady-state is reached, and are therefore stationary or statistically steady on both magnetohydrodynamic and transport time-scales. Resistively driven cross-surface flows are found to be in close agreement with Pfirsch-Schluter theory. Poloidally varying toroidal flows are in agreement with comparable calculations [A.Y. Aydemir, Shear flows at the tokamak edge and their interaction with edge-localized modes, Phys. Plasmas 14]. New effects on core toroidal rotation due to gyroviscosity and a local particle source are observed.

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."

The Magnetic Nozzle Experiment (MNX) is a linear magnetized helicon-heated plasma device, with applications to advanced spacecraft-propulsion methods and solar-corona physics. This paper reviews ion and electron energy distributions measured in MNX with laser-induced fluorescence (LIF) and probes, respectively. Ions, cold and highly collisional in the main MNX region, are accelerated along a uniform magnetic field to sonic then supersonic speeds as they exit the main region through either mechanical or magnetic apertures. A sharp decrease in density downstream of the aperture(s) helps effect a transition from collisional to collisionless plasma. The electrons in the downstream region have an average energy somewhat higher than that in the main region. From LIF ion-velocity measurements, we find upstream of the aperture a presheath of strength Deltaphips=mrTe, where mrTe is the electron temperature in the main region, and length ~3 cm, comparable to the ion-neutral mean-free-path; immediately downstream of the aperture is an electrostatic double layer of strength DeltaphiDL=3-10 mrTe and length 0.3-0.6 cm, 30-600lambdaD. The existence of a small, ca. 0.1%, superthermal electron population with average energy ~10 mrTe is inferred from considerations of spectroscopic line ratios, floating potentials, and Langmuir probe data. The superthermal electrons are suggested to be the source for the large DeltaphiDL

Using a two-dimensional two-fluid numerical model with finite electron inertia, we have observed turbulence to occur in simulations of plasmas undergoing magnetic reconnection when the plasma β is sufficiently small. This turbulence is caused by secondary instabilities, which are excited by large gradients in the out-of-plane current. These gradients spontaneously form due to the dynamics of ion force balance across the layer, and first arise in the outflow regions downstream from the magnetic x point. This turbulence is not observed to have a substantial effect on the rate of reconnection.

Measurements show fast ion losses correlated with applied three-dimensional (3D) fields in a variety of plasmas ranging from L-mode to resonant magnetic perturbation (RMP) edge localized mode (ELM) suppressed H-mode discharges. In DIII-D L-mode discharges with a slowly rotating n = 2 magnetic perturbation, scintillator detector loss signals synchronized with the applied fields are observed to decay within one poloidal transit time after beam turnoff indicating they arise predominantly from prompt loss orbits. Full orbit following using M3D-C1 calculations of the perturbed fields and kinetic profiles reproduce many features of the measured losses and points to the importance of the applied 3D field phase with respect to the beam injection location in determining the overall impact on prompt beam ion loss. Modeling of these results includes a self-consistent calculation of the 3D perturbed beam ion birth profiles and scrape-off-layer ionization, a factor found to be essential to reproducing the experimental measurements. Extension of the simulations to full slowing down timescales, including fueling and the effects of drag and pitch angle scattering, show the applied n = 3 RMPs in ELM suppressed H-mode plasmas can induce a significant loss of energetic particles from the core. With the applied n = 3 fields, up to 8.4% of the injected beam power is predicted to be lost, compared to 2.7% with axisymmetric fields only. These fast ions, originating from minor radii ρ > 0.7, are predicted to be primarily passing particles lost to the divertor region, consistent with wide field-of-view infrared periscope measurements of wall heating in n = 3 RMP ELM suppressed plasmas. Edge fast ion Dα (FIDA) measurements also confirm a large change in edge fast ion profile due to the n = 3 fields, where the effect was isolated by using short 50ms RMP-off periods during which ELM suppression was maintained yet the fast ion profile was allowed to recover. The role of resonances between fast ion drift motion and the applied 3D fields in the context of selectively targeting regions of fast ion phase space is also discussed.

The National Spherical Torus Experiment (NSTX) is currently being upgraded to operate at twice the toroidal field and plasma current (up to 1 T and 2 MA), with a second, more tangentially aimed neutral beam (NB) for current and rotation control, allowing for pulse lengths up to 5 s. Recent NSTX physics analyses have addressed topics that will allow NSTX-Upgrade to achieve the research goals critical to a Fusion Nuclear Science Facility. These include producing stable, 100% non-inductive operation in high-performance plasmas, assessing plasma–material interface (PMI) solutions to handle the high heat loads expected in the next-step devices and exploring the unique spherical torus (ST) parameter regimes to advance predictive capability. Non-inductive operation and current profile control in NSTX-U will be facilitated by co-axial helicity injection (CHI) as well as radio frequency (RF) and NB heating. CHI studies using NIMROD indicate that the reconnection process is consistent with the 2D Sweet–Parker theory. Full-wave AORSA simulations show that RF power losses in the scrape-off layer (SOL) increase significantly for both NSTX and NSTX-U when the launched waves propagate in the SOL. Toroidal Alfven eigenmode avalanches and higher frequency Alfven eigenmodes can affect NB-driven current through energy loss and redistribution of fast ions. The inclusion of rotation and kinetic resonances, which depend on collisionality, is necessary for predicting experimental stability thresholds of fast growing ideal wall and resistive wall modes. Neutral beams and neoclassical toroidal viscosity generated from applied 3D fields can be used as actuators to produce rotation profiles optimized for global stability. DEGAS-2 has been used to study the dependence of gas penetration on SOL temperatures and densities for the MGI system being implemented on the Upgrade for disruption mitigation. PMI studies have focused on the effect of ELMs and 3D fields on plasma detachment and heat flux handling. Simulations indicate that snowflake and impurity seeded radiative divertors are candidates for heat flux mitigation in NSTX-U. Studies of lithium evaporation on graphite surfaces indicate that lithium increases oxygen surface concentrations on graphite, and deuterium–oxygen affinity, which increases deuterium pumping and reduces recycling. In situ and test-stand experiments of lithiated graphite and molybdenum indicate temperature-enhanced sputtering, although that test-stand studies also show the potential for heat flux reduction through lithium vapour shielding. Non-linear gyro kinetic simulations have indicated that ion transport can be enhanced by a shear-flow instability, and that non-local effects are necessary to explain the observed rapid changes in plasma turbulence. Predictive simulations have shown agreement between a microtearing-based reduced transport model and the measured electron temperatures in a microtearing unstable regime. Two Alfven eigenmode-driven fast ion transport models have been developed and successfully benchmarked against NSTX data. Upgrade construction is moving on schedule with initial physics research operation of NSTX-U planned for mid-2015.

Recent experimental observations have found turbulent fluctuation structures that are non-axisymmetric in a tokamak with applied 3D fields. In this paper, two fluid resistive effects are shown to produce changes relevant to turbulent transport in the modeled 3D magnetohydrodynamic (MHD) equilibrium of tokamak pedestals with these 3D fields applied. Ideal MHD models are insufficient to reproduce the relevant effects. By calculating the ideal 3D equilibrium using the VMEC code, the geometric shaping parameters that determine linear turbulence stability, including the normal curvature and local magnetic shear, are shown to be only weakly modified by applied 3D fields in the DIII-D tokamak. These ideal MHD effects are therefore not sufficient to explain the observed changes to fluctuations and transport. Using the M3D-C1 code to model the 3D equilibrium, density is shown to be redistributed on flux surfaces in the pedestal when resistive two fluid effects are included, while islands are screened by rotation in this region. The redistribution of density results in density and pressure gradient scale lengths that vary within pedestal flux surfaces between different helically localized flux tubes. This would produce different drive terms for trapped electron mode and kinetic ballooning mode turbulence, the latter of which is expected to be the limiting factor for pedestal pressure gradients in DIII-D.

New experimental studies and modelling of the coherent edge harmonic oscillation (EHO), which regulates the conventional Quiescent H-mode (QH-mode) edge, validate the proposed hypothesis of edge rotational shear in destabilizing the low-n kink-peeling mode as the additional drive mechanism for the EHO. The observed minimum edge E × B shear required for the EHO decreases linearly with pedestal collisionality , which is favorable for operating QH-mode in machines with low collisionality and low rotation such as ITER. In addition, the QH-mode regime in DIII-D has recently been found to bifurcate into a new 'wide-pedestal' state at low torque in double-null shaped plasmas, characterized by increased pedestal height, width and thermal energy confinement (Burrell 2016 Phys. Plasmas 23 056103, Chen 2017 Nucl. Fusion 57 022007). This potentially provides an alternate path for achieving high performance ELM-stable operation at low torque, in addition to the low-torque QH-mode sustained with applied 3D fields. Multi-branch low-k and intermediate-k turbulences are observed in the 'wide-pedestal'. New experiments support the hypothesis that the decreased edge E × B shear enables destabilization of broadband turbulence, which relaxes edge pressure gradients, improves peeling-ballooning stability and allows a wider and thus higher pedestal. The ability to accurately predict the critical E × B shear for EHO and maintain high performance QH-mode at low torque is an essential requirement for projecting QH-mode operation to ITER and future machines.

The 3D fluid-plasma edge transport code EMC3-EIRENE is used to test several magnetic field models with and without plasma response against DIII-D experimental data for even and odd-parity n = 3 magnetic field perturbations. The field models include ideal and extended MHD equilibria, and the vacuum approximation. Plasma response is required to reduce the stochasticity in the pedestal region for even-parity fields, however too much screening suppresses the measured splitting of the downstream T e profile. Odd-parity perturbations result in weak tearing and only small additional peaks in the downstream measurements. In this case plasma response is required to increase the size of the lobe structure. No single model is able to simultaneously reproduce the upstream and downstream characteristics for both odd and even-parity perturbations.

In order to understand the effect of rotation on the response of a plasma to three-dimensional magnetic perturbations, we perform a systematic scan of the zero-crossing of the rotation profile in a DIII-D ITER-similar shape equilibrium using linear, time-independent modeling with the M3D-C1 extended magnetohydrodynamics code. We confirm that the local resonant magnetic field generally increases as the rotation decreases at a rational surface. Multiple peaks in the resonant field are observed near rational surfaces, however, and the maximum resonant field does not always correspond to zero rotation at the surface. Furthermore, we show that non-resonant current can be driven at zero-crossings not aligned with rational surfaces if there is sufficient shear in the rotation profile there, leading to amplification of near-resonant Fourier harmonics of the perturbed magnetic field and a decrease in the far-off-resonant harmonics. The quasilinear electromagnetic torque induced by this non-resonant plasma response provides drive to flatten the rotation, possibly allowing for increased transport in the pedestal by the destabilization of turbulent modes. In addition, this torque acts to drive the rotation zero-crossing to dynamically stable points near rational surfaces, which would allow for increased resonant penetration. By one or both of these mechanisms, this torque may play an important role in bifurcations into suppression of edge-localized modes. Finally, we discuss how these changes to the plasma response could be detected by tokamak diagnostics. In particular, we show that the changes to the resonant field discussed here have a significant impact on the external perturbed magnetic field, which should be observable by magnetic sensors on the high-field side of tokamaks but not on the low-field side. In addition, TRIP3D-MAFOT simulations show that none of the changes to the plasma response described here substantially affects the divertor footprint structure.

Phase-space time-resolved measurements of fast-ion losses induced by edge localized modes (ELMs) and ELM mitigation coils have been obtained in the ASDEX Upgrade tokamak by means of multiple fast-ion loss detectors (FILDs). Filament-like bursts of fast-ion losses are measured during ELMs by several FILDs at different toroidal and poloidal positions. Externally applied magnetic perturbations (MPs) have little effect on plasma profiles, including fast-ions, in high collisionality plasmas with mitigated ELMs. A strong impact on plasma density, rotation and fast-ions is observed, however, in low density/collisionality and q(95) plasmas with externally applied MPs. During the mitigation/suppression of type-I ELMs by externally applied MPs, the large fast-ion bursts observed during ELMs are replaced by a steady loss of fast-ions with a broad-band frequency and an amplitude of up to an order of magnitude higher than the neutral beam injection (NBI) prompt loss signal without MPs. Multiple FILD measurements at different positions, indicate that the fast-ion losses due to static 3D fields are localized on certain parts of the first wall rather than being toroidally/poloidally homogeneously distributed. Measured fast-ion losses show a broad energy and pitch-angle range and are typically on banana orbits that explore the entire pedestal/scrape-off-layer (SOL). Infra-red measurements are used to estimate the heat load associated with the MP-induced fast-ion losses. The heat load on the FILD detector head and surrounding wall can be up to six times higher with MPs than without 3D fields. When 3D fields are applied and density pump-out is observed, an enhancement of the fast-ion content in the plasma is typically measured by fast-ion D-alpha (FIDA) spectroscopy. The lower density during the MP phase also leads to a deeper beam deposition with an inward radial displacement of approximate to 2 cm in the maximum of the beam emission. Orbit simulations are used to test different models for 3D field equilibrium reconstruction including vacuum representation, the free boundary NEMEC code and the two-fluid M3D-C1 code which account for the plasma response. Guiding center simulations predict the maximum level of losses, approximate to 2.6%, with NEMEC 3D equilibrium. Full orbit simulations overestimate the level of losses in 3D vacuum fields with approximate to 15% of lost NBI ions.

We diagnose local properties of magnetic reconnection during a sawtooth crash employing the three-dimensional toroidal, extended-magnetohydrodynamic (MHD) code M3D-C1. To do so, we sample simulation data in the plane in which reconnection occurs, the plane perpendicular to the helical mode at the q = 1 surface, where m and n are the poloidal and toroidal mode numbers and q is the safety factor. We study the nonlinear evolution of a particular test equilibrium in a non-reduced field representation using both resistive-MHD and extended-MHD models. We find growth rates for the extended-MHD reconnection process exhibit a nonlinear acceleration and greatly exceed that of the resistive-MHD model, as is expected from previous experimental, theoretical, and computational work. We compare the properties of reconnection in the two simulations, revealing the reconnecting current sheets are locally different in the two models and we present the first observation of the quadrupole out-of-plane Hall magnetic field that appears during extended-MHD reconnection in a 3D toroidal simulation (but not in resistive-MHD). We also explore the dependence on toroidal angle of the properties of reconnection as viewed in the plane perpendicular to the helical magnetic field, finding qualitative and quantitative effects due to changes in the symmetry of the reconnection process. This study is potentially important for a wide range of magnetically confined fusion applications, from confirming simulations with extended-MHD effects are sufficiently resolved to describe reconnection, to quantifying local reconnection rates for purposes of understanding and predicting transport, not only at the q = 1 rational surface for sawteeth, but also at higher order rational surfaces that play a role in disruptions and edge-confinement degradation.

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), ΔA′(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 Δql′ rigorously, for poloidal mode number m≥2. Δql′ 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 w2. ΔA′ is proportional to the asymmetry of island that is roughly proportional to w. The sum of Δql′ and ΔA′ 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. Dis...

Observations on the National Spherical Torus Experiment (NSTX) indicate that externally applied non-axisymmetric magnetic perturbations (MP) can reduce the amplitude of toroidal Alfven eigenmodes (TAE) and global Alfven eigenmodes (GAE) in response to pulsed n = 3 non-resonant fields. From full-orbit following Monte Carlo simulations with the one- and two-fluid resistive MHD plasma response to the magnetic perturbation included, it was found that in response to MP pulses the fast-ion losses increased and the fast-ion drive for the GAEs was reduced. The MP did not affect the fast-ion drive for the TAEs significantly but the Alfven continuum at the plasma edge was found to be altered due to the toroidal symmetry breaking which leads to coupling of different toroidal harmonics. The TAE gap was reduced at the edge creating enhanced continuum damping of the global TAEs, which is consistent with the observations. The results suggest that optimized non-axisymmetric MP might be exploited to control and mitigate Alfven instabilities by tailoring the fast-ion distribution function and/or continuum structure.

In the quiescent H-mode (QH-mode) regime, edge harmonic oscillations (EHOs) play an important role in avoiding transient edge localized mode (ELM) power fluxes by providing benign and continuous edge particle transport. A detailed theoretical, experimental and modeling comparison has been made of low-n (n ⩽ 5) EHO in DIII-D QH-mode plasmas. The calculated linear eigenmode structure from the extended magentoohydrodynamics (MHD) code M3D-C1 matches closely the coherent EHO properties from external magnetics data and internal measurements using the ECE, BES, ECE-Imaging and microwave imaging reflectometer (MIR) diagnostics, as well as the kink/peeling mode properties found by the ideal MHD code ELITE. Numerical investigations indicate that the low-n EHO-like solutions from M3D-C1 are destabilized by rotation and/or rotational shear while high-n modes are stabilized. This effect is independent of the rotation direction, suggesting that EHOs can be destabilized in principle with rotation in either direction. The modeling results are consistent with observations of EHO, support the proposed theory of the EHO as a low-n kink/peeling mode destabilized by edge E × B rotational shear, and improve our understanding and confidence in creating and sustaining QH-mode in present and future devices.

Recent experiments on DIII-D have demonstrated that having a toroidally-monochromatic spectral content of edge-resonant magnetic perturbations (RMPs) is not a necessary condition for suppression of edge localized modes (ELMs). Robust ELM suppression has been reproducibly obtained on DIII-D during experiments in which various non-axisymmetric coil loops were turned off pseudo-randomly producing a variety of n=1 ?> , n= 2 ?> , and n= 3 ?> spectral contributions. It was shown that RMP ELM suppression could be achieved with as few as 5 out of 12 internal coil loops (I-coils) on DIII-D at similar coil currents and with good plasma confinement. Linear MHD plasma response (m3dc1, ipec, mars) and vacuum (surfmn, trip3d) modelling have been performed in order to understand the effects of the perturbation spectrum on the plasma response and ELM suppression. The results suggest that reduction of the dominant n= 3 ?> perturbation field is compensated by increased n= 2 ?> field in the plasma that may lead to RMP ELM suppression at lower levels of n= 3 ?> perturbative magnetic flux from the I-coils. These results provide additional confidence that ITER may be capable of RMP ELM suppression in the event of multiple internal coil failures.

The linear dispersion relation for the magnetorotational instability (MRI) is derived including finite Larmor radius (FLR) effects. In particular, the Braginskii form of the ion gyroviscosity, which represents the first-order FLR corrections to the two-fluid equations, is retained. It is shown that FLR effects are the most important effects in the limit of weak magnetic fields, and are much more important than the Hall effect when beta >> 1, where beta is the ratio of the ion thermal pressure to the magnetic pressure. FLR effects may completely stabilize even MRI modes having wavelengths much greater than the ion Larmor radius. Some implications for astrophysical accretion disks are discussed., 7 pages, 2 figures. Accepted for publication in ApJ

Detailed measurements of the plasma’s response to applied magnetic perturbations provide experimental evidence that the form of three-dimensional (3D) tokamak equilibria, with toroidal mode number n = 1, is determined by multiple stable kink modes at high-pressure. For pressures greater than the ideal magnetohydrodynamic (MHD) stability limit, as calculated without a stabilizing wall, the 3D structure transitions in a way that is qualitatively predicted by an extended MHD model that includes kinetic wave-particle interactions. These changes in poloidal mode structure are correlated with the proximity of rotation profiles to thermal ion bounce and the precession drift frequencies suggesting that these kinetic resonances are modifying the relative amplitudes of the stable modes. These results imply that each kink may eventually be independently controlled.

Experiments in the DIII-D tokamak show that the plasma responds to resonant magnetic perturbations (RMPs) with toroidal mode numbers of n = 2 and n = 3 without field line reconnection, consistent with resistive magnetohydrodynamic predictions, while a strong nonlinear bifurcation is apparent when edge localized modes (ELMs) are suppressed. The magnetic response associated with this bifurcation is localized to the high field side of the machine and exhibits a dominant n = 1 component despite the application of a constant amplitude, slowly toroidally rotating, n = 2 applied field. The n = 1 mode is born locked to the vacuum vessel wall, while the n = 2 mode is entrained to the rotating field. Based on these magnetic response measurements and Thomson scattering measurements of flattening of the electron temperature profile, it is likely that these modes are magnetic island chains near the H-mode pedestal. The reduction in ∇Te occurs near the q = 4 and 5 rational surfaces, suggesting five unique islands are p...

Calculations of the plasma response to applied non-axisymmetric fields in several DIII-D discharges show that predicted displacements depend strongly on the edge current density. This result is found using both a linear two-fluid-MHD model (M3D-C1) and a nonlinear ideal-MHD model (VMEC). Furthermore, it is observed that the probability of a discharge being edge localized mode (ELM)-suppressed is most closely related to the edge current density, as opposed to the pressure gradient. It is found that discharges with a stronger kink response are closer to the peeling–ballooning stability limit in ELITE simulations and eventually cross into the unstable region, causing ELMs to reappear. Thus for effective ELM suppression, the RMP has to prevent the plasma from generating a large kink response, associated with ELM instability. Experimental observations are in agreement with the finding; discharges which have a strong kink response in the MHD simulations show ELMs or ELM mitigation during the RMP phase of the experiment, while discharges with a small kink response in the MHD simulations are fully ELM suppressed in the experiment by the applied resonant magnetic perturbation. The results are cross-checked against modeled 3D ideal MHD equilibria using the VMEC code. The procedure of constructing optimal 3D equilibria for diverted H-mode discharges using VMEC is presented. Kink displacements in VMEC are found to scale with the edge current density, similar to M3D-C1, but the displacements are smaller. A direct correlation in the flux surface displacements to the bootstrap current is shown.

DIII-D experiments using new detailed magnetic diagnostics show that linear, ideal magnetohydrodynamics (MHD) theory quantitatively describes the magnetic structure (as measured externally) of three-dimensional (3D) equilibria resulting from applied fields with toroidal mode number n = 1, while a nonlinear solution to ideal MHD force balance, using the VMEC code, requires the inclusion of n ≥ 1 to achieve similar agreement. These tests are carried out near ITER baseline parameters, providing a validated basis on which to exploit 3D fields for plasma control development. Scans of the applied poloidal spectrum and edge safety factor confirm that low-pressure, n = 1 non-axisymmetric tokamak equilibria are determined by a single, dominant, stable eigenmode. However, at higher beta, near the ideal kink mode stability limit in the absence of a conducting wall, the qualitative features of the 3D structure are observed to vary in a way that is not captured by ideal MHD.

One modeling framework for integrated tasks (OMFIT) is a comprehensive integrated modeling framework which has been developed to enable physics codes to interact in complicated workflows, and support scientists at all stages of the modeling cycle. The OMFIT development follows a unique bottom-up approach, where the framework design and capabilities organically evolve to support progressive integration of the components that are required to accomplish physics goals of increasing complexity. OMFIT provides a workflow for easily generating full kinetic equilibrium reconstructions that are constrained by magnetic and motional Stark effect measurements, and kinetic profile information that includes fast-ion pressure modeled by a transport code. It was found that magnetic measurements can be used to quantify the amount of anomalous fast-ion diffusion that is present in DIII-D discharges, and provide an estimate that is consistent with what would be needed for transport simulations to match the measured neutron rates. OMFIT was used to streamline edge-stability analyses, and evaluate the effect of resonant magnetic perturbation (RMP) on the pedestal stability, which have been found to be consistent with the experimental observations. The development of a five-dimensional numerical fluid model for estimating the effects of the interaction between magnetohydrodynamic (MHD) and microturbulence, and its systematic verification against analytic models was also supported by the framework. OMFIT was used for optimizing an innovative high-harmonic fast wave system proposed for DIII-D. For a parallel refractive index , the conditions for strong electron-Landau damping were found to be independent of launched and poloidal angle. OMFIT has been the platform of choice for developing a neural-network based approach to efficiently perform a non-linear multivariate regression of local transport fluxes as a function of local dimensionless parameters. Transport predictions for thousands of DIII-D discharges showed excellent agreement with the power balance calculations across the whole plasma radius and over a broad range of operating regimes. Concerning predictive transport simulations, the framework made possible the design and automation of a workflow that enables self-consistent predictions of kinetic profiles and the plasma equilibrium. It is found that the feedback between the transport fluxes and plasma equilibrium can significantly affect the kinetic profiles predictions. Such a rich set of results provide tangible evidence of how bottom-up approaches can potentially provide a fast track to integrated modeling solutions that are functional, cost-effective, and in sync with the research effort of the community.

In comparing equilibrium solutions for a DIII-D shot that is amenable to analysis by both stellarator and tokamak three-dimensional (3D) equilibrium codes, a significant disagreement has been seen between solutions of the VMEC stellarator equilibrium code and solutions of tokamak perturbative 3D equilibrium codes. The source of that disagreement has been investigated, and that investigation has led to new insights into the domain of validity of the different equilibrium calculations, and to a finding that the manner in which localized screening currents at low order rational surfaces are handled can affect global properties of the equilibrium solution. The perturbative treatment has been found to break down at surprisingly small perturbation amplitudes due to overlap of the calculated perturbed flux surfaces, and that treatment is not valid in the pedestal region of the DIII-D shot studied. The perturbative treatment is valid, however, further into the interior of the plasma, and flux surface overlap does not account for the disagreement investigated here. Calculated equilibrium solutions for simple model cases and comparison of the 3D equilibrium solutions with those of other codes indicate that the disagreement arises from a difference in handling of localized currents at low order rational surfaces, with such currents being absent in VMEC and present in the perturbative codes. The significant differences in the global equilibrium solutions associated with the presence or absence of very localized screening currents at rational surfaces suggests that it may be possible to extract information about localized currents from appropriate measurements of global equilibrium plasma properties. That would require improved diagnostic capability on the high field side of the tokamak plasma, a region difficult to access with diagnostics.

The linear response profiles for the 3D perturbed magnetic fields, currents, ion velocities, plasma density, pressures, and electric potential from low-n external resonant magnetic field perturbations (RMPs) are obtained from the collisional two-fluid M3D-C1 code [N. M. Ferraro and S. C. Jardin, J. Comput. Phys. 228, 7742 (2009)]. A newly developed post-processing RMPtran code computes the resulting quasilinear E×B and magnetic (J×B) radial transport flows with respect to the unperturbed flux surfaces in all channels. RMPtran simulations focus on ion (center of mass) particle and transient non-ambipolar current flows, as well as the toroidal angular momentum flow. The paper attempts to delineate the RMP transport mechanisms that might be responsible for the RMP density pump-out seen in DIII-D [M. A. Mahdavi and J. L. Luxon, Fusion Sci. Technol. 48, 2 (2005)]. Experimentally, the starting high toroidal rotation does not brake to a significantly lower rotation after the pump-out suggesting that convective and E×B transport mechanisms dominate. The direct J×B torque from the transient non-ambipolar radial current expected to accelerate plasma rotation is shown to cancel much of the Maxwell stress J×B torque expected to brake the plasma rotation. The dominant E×B Reynolds stress accelerates rotation at the top of the pedestal while braking rotation further down the pedestal.

Odd-parity rotating magnetic fields (RMFo) applied to mirror-configuration plasmas have produced average electron energies exceeding 200 eV at line-averaged electron densities of approximately 10(12) cm-3. These plasmas, sustained for over 10(3)tauAlfven, have low Coulomb collisionality, vc* triple bond L/lambdaC approximately 10(-3), where lambdaC is the Coulomb scattering mean free path and L is the plasma's characteristic half length. Divertors allow reduction of the electron-neutral collision frequency to values where the RMFo coupling indicates full penetration of the RMFo to the major axis.

Recent observations on DIII-D have advanced the understanding of plasma response to applied resonant magnetic perturbations (RMPs) in both H-mode and L-mode plasmas. Three distinct 3D features localized in minor radius are imaged via filtered soft x-ray emission: (i) the formation of lobes extending from the unperturbed separatrix in the X-point region at the plasma boundary, (ii) helical kink-like perturbations in the steep-gradient region inside the separatrix, and (iii) amplified islands in the core of a low-rotation L-mode plasma. These measurements are used to test and to validate plasma response models, which are crucial for providing predictive capability of edge-localized mode control. In particular, vacuum and two-fluid resistive magnetohydrodynamic (MHD) responses are tested in the regions of these measurements. At the plasma boundary in H-mode discharges with n = 3 RMPs applied, measurements compare well to vacuum-field calculations that predict lobe structures. Yet in the steep-gradient region...

Numerical modelling of the plasma boundary position and its displacement due to external magnetic perturbations in DIII-D low-collisionality H-mode discharges is presented. The results of the vacuum model are compared to the experimental measurements for boundary displacements including Thomson scattering electron temperature Te, charge exchange recombination spectroscopy, beam emission spectroscopy, soft x-ray, and divertor Langmuir probe measurements. Magnetically perturbed discharges with toroidal mode number n = 2 and n = 3 are studied. It is shown that the vacuum model predictions agree well with the measurements above and below the midplane, and disagree at the outer midplane in discharges where significant kink amplification is present. The role of the plasma response is studied using the two-fluid MHD code M3D-C1, and the results are compared to the vacuum model showing that the plasma response model underestimates the boundary displacements.

In this paper we introduce a new indirect method for identifying the radial extent of the stochastic layer due to applying resonant magnetic perturbations (RMPs) in H-mode plasmas by measuring the spin-up of the plasma near the separatrix. This spin-up is a predicted consequence of enhanced electron loss, due to magnetic stochastization (Kaveeva et al 2008 Nucl. Fusion 48 075003). We find that in DIII-D H-mode plasmas with n = 3 RMPs applied for edge localized mode suppression, the stochastic layer is limited to the outer 5% region in normalized magnetic flux, ΨN. This is in contrast to vacuum modelling predictions where this layer can penetrate up to 20% in ΨN. Theoretical predictions of a stochastic radial electric field, Er component exceed the experimental measurements by about a factor 3 close to the separatrix, suggesting that the outer region of the plasma is weakly stochastic. Linear response calculations with M3D-C1, a resistive two-fluid model, show that in this outer 5% region, plasma response often reduces the resonant magnetic field components by 67% or more in comparison with vacuum calculations. These results for DIII-D are in reasonable agreement with results from the MAST tokamak, where the magnetic field perturbation from vacuum field calculations needed to be reduced by 75% for agreement with experimental measurements of the x-point lobe structures (Kirk et al 2012 Phys. Rev. Lett. 108 255003).

The effects of applied 3D resonant magnetic perturbations are modelled with and without self-consistent plasma response. The plasma response is calculated using a linear two-fluid model. A synthetic diagnostic is used to simulate soft x-ray (SXR) emission within the steep gradient region of the pedestal, 0.98 > ψ > 0.94. Two methods for simulating the SXR emission given the perturbed fields are considered. In the first method, the emission is assumed to be constant on magnetic field lines, with the emission on each line determined by the penetration depth into the plasma. In the second method, the emission is taken to be a function of the perturbed electron temperature and density calculated by the two-fluid model. It is shown that the latter method is more accurate within the plasma, but is inadequate in the scrape-off layer due to the breakdown of the linearized temperature equation in the two-fluid model. The resulting synthetic emission is compared to measured SXR data, which show helical m = 11 ± 1 displacements around the 11/3 rational surface of sizes up to 5 cm, depending on the poloidal angle. The helical displacements around the 11/3 surface are identified to be directly related to the kink response, caused by amplification of non-resonant components of the magnetic field due to plasma response. The role of different plasma parameters is investigated, but it appears that the electron rotation plays a key role in the formation of screening and resonant amplification, while the kinking appears to be sensitive to the edge current density. It is also hypothesized that the plasma response affects the edge-localized-mode (ELM) stability, i.e. the discharge's operational point relative to the peeling–ballooning stability boundary.

The three-dimensional plasma boundary displacements induced by applied non-axisymmetric magnetic perturbations have been measured in ASDEX Upgrade, DIII-D, JET, MAST and NSTX. The displacements arising from applied resonant magnetic perturbations (RMPs) are measured up to +/- 5% of the minor radius in present-day machines. Good agreement can be found between different experimental measurements and a range of models-be it vacuum field line tracing, ideal three-dimensional MHD equilibrium modelling, or nonlinear plasma amplification. The agreement of the various experimental measurements with the different predictions from these models is presented, and the regions of applicability of each discussed. The measured displacement of the outboard boundary from various machines is found to correlate approximately linearly with the applied resonant field predicted by vacuum modelling (though it should be emphasized that one should not infer that vacuum modelling accurately predicts the displacement inside the plasma). The RMP-induced displacements foreseen in ITER are expected to lie within the range of those predicted by the different models, meaning less than +/- 1.75% (+/- 3.5 cm) of the minor radius in the H-mode baseline and less than +/- 2.5% (+/- 5 cm) in a 9MA plasma. Whilst a displacement of 7 cm peak-to-peak in the baseline scenario is marginally acceptable from both a plasma control and heat loading perspective, it is important that ITER adopts a plasma control system which can account for a three-dimensional boundary corrugation to avoid an n = 0 correction which would otherwise locally exacerbate the displacement caused by the applied fields.

Type-I edge-localized modes (ELMs) have been suppressed in DIII-D (Luxon et al 2003 Nucl. Fusion 43 1813) H-mode discharges with a H98Y2 confinement factor near 1.0 using magnetic perturbations (MPs) with dominant toroidal mode number n = 2. This expands access to the ELM-suppressed regime, which was previously attainable in DIII-D only with n = 3 fields. ELM suppression is obtained with two rows of internal coils for 1.8 s with normalized beta of 1.9 and average triangularity of 0.53, corresponding to a scaled version of ITER scenario 2 at an ITER relevant electron collisionality of 0.2. The applied field reduces the pedestal pressure and edge current via the density without degrading the edge thermal transport barrier. ELITE calculations find that the resulting profiles are stable to intermediate-n peeling–ballooning modes. ELM suppression is found within different ranges of q95 depending on the coil configuration used to generate the MP. The edge safety factors associated with suppression do not correspond to those that maximize the pitch-resonant components of the applied vacuum field. Instead, ELM suppression is correlated with an increase in the amplification of kink-resonant components of the calculated ideal MHD plasma response field.

Linear, two-fluid, resistive modelling of the plasma response to applied non-axisymmetric fields shows significant displacement of edge temperature and density profiles. The calculated displacements, often of 2 cm or more in H-mode pedestals with parameters appropriate to DIII-D, are due to the helical distortions resulting from stable edge modes being driven to finite amplitude by the applied fields. In many cases, these displacements are greater in magnitude, and different in phase, than the distortions of the separatrix manifolds predicted from vacuum modelling. Comparison of these results with experimental measurements from Thomson scattering and soft x-ray imaging finds good quantitative agreement. In these experiments, the phase of the applied non-axisymmetric magnetic field was flipped or rotated in order to probe the non-axisymmetric features of the response. The poloidal structures measured by x-ray imaging show clear indications of a helical response, as opposed to simply a change in the axisymmetric transport. Inclusion of two-fluid effects and rotation are found to be important in obtaining quantitative agreement with Thomson scattering data. Modelling shows screening of islands in the H-mode pedestal, but island penetration near the top of the pedestal where the electron rotation vanishes in plasmas with co-current rotation. Enhanced transport due to these islands may provide a mechanism for maintaining the pedestal width below the stability threshold of edge-localized modes. For typical DIII-D parameters, it is shown that the linear approximation is often near or beyond the limit of validity in the H-mode edge; however, the general agreement with experimental measurements indicates that these linear results nevertheless maintain good predictive value for profile displacements.

With the installation of non-axisymmetric coil systems on major tokamaks for the purpose of studying the prospects of ELM-free operation, understanding the plasma response to the applied fields is a crucial issue. Application of different response models, using standard tools, to DIII-D discharges with applied non-axisymmetric fields from internal coils, is shown to yield qualitatively different results. The plasma response can be treated as an initial value problem, following the system dynamically from an initial unperturbed state, or from a nearby perturbed equilibrium approach, and using both linear and nonlinear models [A. D. Turnbull, Nucl. Fusion 52, 054016 (2012)]. Criteria are discussed under which each of the approaches can yield a valid response. In the DIII-D cases studied, these criteria show a breakdown in the linear theory despite the small 10 3 relative magnitude of the applied magnetic field perturbations in this case. For nonlinear dynamical evolution simulations to reach a saturated nonlinear steady state, appropriate damping mechanisms need to be provided for each normal mode comprising the response. Other issues arise in the technical construction of perturbed flux surfaces from a displacement and from the presence of near nullspace normal modes. For the nearby equilibrium approach, in the absence of a full 3D equilibrium reconstruction with a controlled comparison, constraints relating the 2D system profiles to the final profiles in the 3D system also need to be imposed to assure accessibility. The magnetic helicity profile has been proposed as an appropriate input to a 3D equilibrium calculation and tests of this show the anticipated qualitative behavior.

Imaging beam emission spectroscopy has been used to study the displacement of the plasma boundary in ELMing H-mode discharges with a 10 Hz rotating n = 2 external magnetic field perturbation in DIII-D. The rotating magnetic field creates a helical displacement of the beam emission profile of ∼2 cm on the low-field-side (LFS) midplane which rotates with the applied resonant magnetic perturbation. This shift in the beam emission profile is due primarily to a shift in the electron density profile, which is independently measured to be 1.9 cm on the LFS midplane. These boundary displacements exceed calculations for the displacement of the stable and unstable manifolds formed by the interaction of the magnetic perturbation with the divertor separatrix by a factor of 4–5, suggesting that the vacuum field model does not correctly model the effect of the magnetic perturbations even near the separatrix. The measured displacements are suggestive of a non-resonant kink response.

The M3D-C1 (Breslau et al 2009 Phys. Plasmas 16 092503) code is designed for performing three-dimensional nonlinear magnetohydrodynamics (MHD) calculations of a tokamak plasma that span the timescales associated with ideal and resistive stability as well as parallel and perpendicular transport. This requires a scalable fully implicit time advance where the time step is not limited by a Courant condition based on the MHD wave velocities or plasma flow but is chosen instead to accurately and efficiently resolve the physics. In order to accomplish this, we make use of several techniques to improve the effective condition number of the implicit matrix equation that is solved every time step. The split time advance known as the differential approximation (Caramana 1991 J. Comput. Phys. 96 484) reduces the size of the matrix and improves its diagonal structure. A particular choice of velocity variables and annihilation operators approximately splits the large matrix into three sub-matrices, each with a much improved condition number. A final block-Jacobi preconditioner further dramatically improves the condition number of the final matrix, allowing it to converge in a Krylov solver (GMRES) with a small number of iterations. As an example, we have performed transport timescale simulations of a tokamak plasma that periodically undergoes sawtooth oscillations (Von Goeler et al 1974 Phys. Rev. Lett. 33 1201). We specify the transport coefficients and apply a 'current controller' that adjusts the boundary loop-voltage to keep the total plasma current fixed. The short-time plasma response depends on the initial conditions, but the long-time behavior depends only on the transport coefficients and the boundary conditions applied.

The zero-frequency linear plasma response to static applied non-axisymmetric fields is calculated using a resistive two-fluid model in diverted, toroidal geometry. Within this model, the effects on the plasma response of resistivity, rotation, differential ion and electron velocity, and dissipation are explored. Rotation is generally found to inhibit the formation of islands in the plasma, in qualitative agreement with theoretical results. When two-fluid effects are included, it is found that the penetration of the non-axisymmetric fields is generally greatest when the part of the electron rotation perpendicular to the equilibrium magnetic field is small at the mode-rational surface. Strong rotation shear in the edge is found to enhance the plasma response there. The entire plasma, including the separatrix and scrape-off layer, is included in the computational domain.

The discrete symmetries of time-dependent, non-axisymmetric, two-fluid magnetohydrodynamics (MHD) equations are considered. Solutions of the time-independent, axisymmetric, ideal-MHD equations remain solutions under reversals of the toroidal field, current density, or rotation. Introducing non-axisymmetry, resistivity, and two-fluid effects into the equations each break different symmetries. Symmetry groups for solutions possessing up-down spatial symmetry, and the effect of flipping the magnetic geometry across the horizontal midplane, are also considered. It is shown that poloidal velocity may or may not be reversed under a simultaneous reversal of the toroidal field and a vertical reflection of the magnetic geometry, depending on the symmetry of the velocity. Because the symmetry groups of ideal, resistive, and two-fluid MHD are distinct, it may be possible to ascertain the dominant physical mechanism of various phenomena through empirical observations of symmetry-breaking. These results, which hold fo...

A new approach for solving the 3D MHD equations in a strongly magnetized toroidal plasma is presented which uses high-order 2D finite elements with C1 continuity. The vector fields use a physics-based decomposition. An efficient implicit time advance separates the velocity and field advance. ITAPS (SCOREC) adaptivity software and TOPS solvers are used.

In a published Physical Review Letter (Reiman 2007 Phys. Rev. Lett. 99 135007), it was shown that axisymmetric (or vertical) stability can be improved by placing a set of parallelogram coils above and below the plasma oriented at an angle to the constant toroidal planes. The physics of this stabilization can be understood as providing an effective additional positive stability index. The original work was based on a simplified model of a straight tokamak and is not straightforwardly applicable to a finite aspect ratio, strongly shaped plasma such as in DIII-D. Numerical calculations were performed in a real DIII-D -like configuration to provide a proof of principal that 3-D fields can, in fact raise the elongation limits as predicted. A four field period trapezioid-shaped coil set was developed in toroidal geometry and 3D equilibria were computed using trapezium coil currents of 10 k A , 100 k A , and 500 k A . The ideal magnetohydrodynamics growth rates were computed as a function of the conformal wall position for the n = 0 symmetry-preserving family. The results show an insignificant relative improvement in the stabilizing wall location for the two lower coil current cases, of the order of 10−3 and less. In contrast, the marginal wall position is increased by 7% as the coil current is increased to 500 k A , confirming the main prediction from the original study in a real geometry case. In DIII-D the shift in marginal wall position of 7% would correspond to being able to move the existing wall outward by 5 to 10 cm. While the predicted effect on the axisymmetric stability is real, it appears to require higher coil currents than could be provided in an upgrade to existing facilities. Additional optimization over the pitch of the coils, the number of field periods and the coil positions, as well as plasma parameters, such as the internal inductivity ℓ i , β , and q 95 would mitigate this but seem unlikely to change the conclusion.