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A large number of magnetohydrodynamic (MHD) equilibrium calculations are often required for uncertainty quantification, optimization, and real-time diagnostic information, making MHD equilibrium codes vital to the field of plasma physics. In this paper, we explore a method for solving the Grad–Shafranov equation by using physics-informed neural networks (PINNs). For PINNs, we optimize neural networks by directly minimizing the residual of the partial differential equation as a loss function. We show that PINNs can accurately and effectively solve the Grad–Shafranov equation with several different boundary conditions, making it more flexible than traditional solvers. This method is flexible as it does not require any mesh and basis choice, thereby streamlining the computational process. We also explore the parameter space by varying the size of the model, the learning rate, and boundary conditions to map various tradeoffs such as between reconstruction error and computational speed. Additionally, we introduce a parameterized PINN framework, expanding the input space to include variables such as pressure, aspect ratio, elongation, and triangularity in order to handle a broader range of plasma scenarios within a single network. Parameterized PINNs could be used in future work to solve inverse problems such as shape optimization. [ABSTRACT FROM AUTHOR]

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]

Extensive experimental evidence has shown that the presence of poloidal flow in tokamaks can dramatically improve transport properties. However, theory indicates that poloidal flows are damped by poloidal viscosity, thus necessitating external drivers, such as neutral beam injection or radio frequency heating. In this work, ideal magnetohydrodynamic equilibria are calculated via the FORTRAN code FLOW [Guazzotto et al., Phys. Plasmas 11, 604 (2004)] and a postprocessor is used to estimate the neoclassical poloidal viscosity. The equilibrium inputs, which correspond to intuitive physical quantities, are then numerically optimized to reduce a viscosity figure of merit. We present supersonic equilibria in tokamak geometry with minimized neoclassical poloidal viscosities for various velocity free function inputs, plasma aspect ratios, and collisionality regimes. Benchmarks are made against an analytic theory as well as a classical expression of poloidal viscosity. Numerical confirmation of the analytic theory is obtained in the high aspect ratio and high collisionality limit. Good agreement is also seen near the plasma core and edge, with discrepancies arising in the intermediate region. Outside of these limits, rotation input function profiles are found that provide ∼order of magnitude improvements over the analytic theory, with additional progress being made toward predictions for tokamak-relevant equilibria. [ABSTRACT FROM AUTHOR]

We present an analytical model of the self-consistent equilibrium of a magnetic flux rope, which is obtained in cylindrical geometry. The equilibrium quantities, namely, the azimuthal magnetic field and plasma pressure, are determined in a self-consistent way through the current density, which is derived as a solution of a nonlinear equation. By minimizing the energy functional, it was shown that the constrained equilibrium state is stable. The obtained results are also applicable to the cylindrical tokamak magnetic configurations. It is shown that the analytically predicted radial profiles of equilibrium quantities are in good agreement with the experimental data. [ABSTRACT FROM AUTHOR]

Systematic test particle tracing simulations for runaway electrons (REs) are performed for six post-thermal quench equilibria from DIII-D and ITER, where large scale, kink-like n = 1 (n is the toroidal mode number) magnetohydrodynamic (MHD) instabilities are found. The modeled particle guiding center orbits allow extraction of the effective diffusion–convection coefficients of REs in the presence of large three-dimensional (3D) perturbations up to 10% of the equilibrium toroidal field. With a fixed spatial distribution of the field perturbation, the RE transport coefficients along the plasma radial coordinate track reasonably well with the surface-averaged perturbation level. A substantial variation in the value of the transport coefficients—by three orders of magnitude in most cases, however, occurs with varying launching location of REs along the plasma radius. Large 3D perturbations almost always lead to comparable diffusion and convection processes, meaning that diffusion alone is insufficient to describe the particle motion. At lower (but still high) level of perturbation, the RE convection is found to be dominant over diffusion. A similar observation is made when the perturbation is too strong. In the presence of large perturbation, the dependence of the RE transport on the particle energy is sensitive to the spatial distribution of the perturbation. Based on numerically obtained RE transport coefficients, an analytic fitting model is proposed to quantify the particle diffusion and convection processes due to large MHD events in post-thermal quench plasmas. The model is shown to reasonably well reproduce the direct test particle tracing results for the RE loss fraction and can, thus, be useful for incorporating into other kinetic RE codes in order to simulate the RE beam evolution in the presence of large 3D perturbations. [ABSTRACT FROM AUTHOR]

The equilibrium code, VMEC, is used to study external kinks in low β tokamaks and l = 2 stellarators. The applicability of the code when modeling nonlinear MHD effects is explored in an attempt to understand and predict how the initial saturation of the MHD mode depends on the external rotational transform. It is shown that helicity preserving, free boundary VMEC computations do not converge to a single perturbed solution with increasing spectral resolution. Additional constraints are applied to narrow down the numerical resolution parameters appropriate for physical scans. The dependence of the modeled (4, 1) kink mode on the external rotational transform and field periodicity is then studied. While saturated states can be identified which decrease in amplitude with increasing external rotational transform, alternative bifurcated states are found, which contradict this trend. As a result, it was not possible to use VMEC alone to identify the physical dependency of the nonlinear mode amplitude on the magnetic geometry. Nevertheless, the accuracy of VMEC solutions is demonstrated by showing that the expected toroidal mode coupling is captured in the magnetic energy spectrum for stellarator cases. Comparing with the initial value code, JOREK, the predicted redistribution of poloidal magnetic energy from the vacuum to plasma region in VMEC is shown to be physical. This work is a first step toward using VMEC to study MHD modes in stellarator geometry. [ABSTRACT FROM AUTHOR]

The linear properties of the low-frequency shear Alfvén waves such as those associated with the beta-induced Alfvén eigenmodes (BAEs) and the low-frequency modes observed in reversed-magnetic-shear DIII-D discharges [W. Heidbrink et al., Nucl. Fusion 61, 066031 (2021)] are theoretically investigated and delineated based on the theoretical framework of the general fishbone-like dispersion relation (GFLDR). By adopting representative experimental equilibrium profiles, it is found that, even though both modes are predominantly of Alfvénic polarization, the low-frequency mode is a reactive unstable mode with weak coupling to the energetic particles, while the BAE involves a dissipative instability due to resonant excitation by the energetic ions. Thus, the low-frequency mode is more appropriately called a low-frequency Alfvén mode (LFAM). Moreover, the ascending frequency spectrum patterns of the experimentally observed BAEs and LFAMs can be theoretically reproduced by varying qmin and also be well interpreted based on the GFLDR. The present analysis illustrates the solid predictive capability of the GFLDR and its practical usefulness in enhancing the interpretative capability of both experimental and numerical simulation results. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate a method for constructing the Solov'ev equilibrium with any given 2D shape surrounded by a vacuum region using external poloidal field coils, based on the method previously developed by T. Xu and R. Fitzpatrick [Nuclear Fusion, 59, 064002 (2019)]. The computational domain consists of two parts: the plasma region, where the solution is the same as the Solov'ev solution, and the vacuum region, where the magnetic field generated by external coils as well as plasma current is determined using the Green function method through a matching condition near the separatrix. However, the method is not limited to the Solov'ev equilibrium in particular. The accuracy, efficiency, and robustness of such a scheme suggest that this method may be applied to the 2D shaping of a tokamak plasma with a vacuum region using external coils in general. [ABSTRACT FROM AUTHOR]

A radially adaptive numerical scheme is developed to solve the Grad–Shafranov equation for axisymmetric magnetohydrodynamic equilibrium. A decomposition with independent solutions is employed in the radial direction, and Fourier decomposition is used in the poloidal direction. The independent solutions are then obtained using an adaptive shooting scheme together with the multi-region matching technique in the radial direction. Accordingly, the adaptive toroidal equilibrium (ATEQ) code is constructed for axisymmetric equilibrium studies. The adaptive numerical scheme in the radial direction improves considerably the accuracy of the equilibrium solution. The decomposition with independent solutions effectively reduces the matrix size in solving the magnetohydrodynamic equilibrium problem. The reduction of the matrix size is about an order of magnitude as compared with the conventional radially grid-based numerical schemes. Also, in this ATEQ numerical scheme, no matter how accuracy in the radial direction is imposed, the size of matrices basically does not change. The small matrix size scheme gives ATEQ more flexibility to address the requirement of the number of Fourier components in the poloidal direction in tough equilibrium problems. These two unique features, the adaptive shooting and small matrix size, make ATEQ useful to improve tokamak equilibrium solutions. [ABSTRACT FROM AUTHOR]

New workflows have been developed for predictive modeling of magnetohydrodynamic (MHD) equilibrium in tokamak plasmas. The goal of this work is to predict the MHD equilibrium in tokamak discharges without having measurements of the kinetic profiles. The workflows include a cold start tool, which constructs all the profiles and power flows needed by transport codes; a Grad–Shafranov equilibrium solver; and various codes for the sources and sinks. For validation purposes, a database of DIII-D tokamak discharges has been constructed that is comprised of scans in the plasma current, toroidal magnetic field, and triangularity. Initial efforts focused on developing a workflow utilizing an empirically derived pressure model tuned to DIII-D discharges with monotonic safety factor profiles. This workflow shows good agreement with experimental kinetic equilibrium calculations, but is limited in that it is a single fluid (equal ion and electron temperatures) model and lacks H-mode pedestal predictions. The best agreement with the H-mode database is obtained using a theory-based workflow utilizing pressure profile predictions from a coupled TGLF turbulent transport and EPED pedestal models together with external magnetics and Motional Stark Effect (MSE) data to construct the equilibrium. Here, we obtain an average root mean square error of 5.1% in the safety factor profile when comparing the predicted and experimental kinetic equilibrium. We also find good agreement with the plasma stored energy, internal inductance, and pressure profiles. Including MSE data in the theory-based workflow results in noticeably improved agreement with the q-profiles in high triangularity discharges in comparison with the results obtained with magnetic data only. The predictive equilibrium workflow is expected to have wide applications in experimental planning, between-shot analysis, and reactor studies. [ABSTRACT FROM AUTHOR]

This document is an erratum for an article titled "Grad-Shafranov equilibria via data-free physics informed neural networks" published in the journal Physics of Plasmas. The erratum states that an important reference was accidentally omitted from the original article. The missing reference is a prior work by D. A. Kaltsas and G. N. Throumoulopoulos titled "Neural network tokamak equilibria with incompressible flows." The authors of the erratum emphasize that this reference provides relevant background information for their own work on neural network surrogates for MHD equilibria. [Extracted from the article]

Based on the general expression for the spectral density of equilibrium radiation in a nonrelativistic plasma medium, the high-frequency asymptotics is found for an electron gas accounting for the spatial dispersion of the transverse dielectric permittivity. The necessity of modifying zero oscillations in the presence of a collisionless plasma medium is shown. [ABSTRACT FROM AUTHOR]

Chirping Alfvén modes are considered as potentially harmful for the confinement of energetic particles in burning tokamak plasmas. In fact, by changing their frequency, they are able to extract as much power as possible from these particles, possibly increasing their transport. In this paper, the nonlinear evolution of a single-toroidal-number chirping mode is analyzed by numerical particle simulation. The relevant resonant structures are identified by numerical techniques based on the use of a coordinate system including two constants of motion: the magnetic moment and a suitable function of the initial particle coordinates. The analysis is focused on the dynamics of two different resonant structures in the particle phase space: those yielding the largest drive during the linear and the nonlinear phase, respectively. It is shown that, for each resonant structure, a density-flattening region is formed around the respective resonance radius, with a radial width that increases as the mode amplitude grows. It is delimited by two steepened negative density gradients, drifting inwards and outward. If the mode frequency were constant, phase-space density flattening would quench the resonant-structure drive as the steepened gradients leave the original resonance region. The frequency chirping, however, causes the resonance radius and the resonance region to drift inwards. This drift, along with a relevant increase in the resonance width, delays the moment in which the inner density gradient reaches the inner boundary of the resonance region, leaving it. On the other hand, the island evolves consistently with the resonance radius; as a consequence, the steepened density gradient further moves inward. This process continues as long as it allows to keep the steepened gradient within the resonance region. When this is no longer possible, the resonant structure ceases to be effective in driving the mode. To further extract energy from the particles, the mode has to tap a different resonant structure, possibly making use of additional frequency variations. [ABSTRACT FROM AUTHOR]

The unique high-field side middle single null divertor configuration in the Joint-Texas Experimental Tokamak is attractive in studying detachment, impurity control, L-H transition, and negative triangularity. Its equilibrium configuration with a horizontally elongated (or flattened) poloidal cross section causes position instability and difficulty in equilibrium control. This work disassembles the vertical magnetic field of each group of poloidal field windings and calculates the field decay index (n-index) at different radial positions. It is found that the instability growth time τg of horizontal displacement increases as the equilibrium radial position of plasma moves outward, which is beneficial to the position control. The simulation and experiments verified the effect of equilibrium position on horizontal instability. The equilibrium position slightly outward from the geometric center of the vacuum chamber effectively decreases the horizontal displacement oscillation amplitude and guarantees the controllable horizontal displacement. [ABSTRACT FROM AUTHOR]

The flux fit method is used to self-consistently estimate the power deposition profile and heat transport profiles from temperature measurements originating from perturbative experiments with a modulated source. This Letter improves on this method by addressing the limitations and assumptions. The most crucial improvement is the additional freedom in the source deposition profile. Allowing for a variable central deposition location and height and including a skewness parameter produces deposition profiles more consistent with the measurement data, but still wider than equilibrium ray tracing in two different DIII-D discharges. Moreover, we show that the quality of the estimated deposition profile is key to the accuracy of diffusivity and convectivity estimates, but inversely, the estimated transport parameters hardly affect the quality of the power deposition estimate. Using this method, we show that the power deposition profile estimate is broadened with respect to ray-tracing by about 1.7–1.8 times in two DIII-D discharges. [ABSTRACT FROM AUTHOR]

The electron cross field transport by the induced azimuthal electric field in a Hall thruster exhibits the mobility scaled by 1/B. This study investigates parameters affecting this transport over a Hall thruster's distinct regions, such as the ionization, acceleration, and plume regions. The main focus is on the nonzero equilibrium azimuthal electric field induced by an azimuthally inhomogeneous neutral supply. A fast Fourier transform analysis of the plasma structure reveals that the wavenumber k of the azimuthal plasma structure increases from k = 2, which is the input condition, to k = 4 in the plume region, and that the total axial flux caused by the azimuthal electric field is mainly induced from the structures of the dominant Fourier components. The azimuthal phase relation between plasma potential and density is formed to maximize the axial electron flux at the plume region and starts varying along with other plasma properties as electrons flow toward the acceleration region. The spatial evolution of the effective axial mobility coefficient is extracted, and its regional characteristics are discussed. [ABSTRACT FROM AUTHOR]

We prove the existence of a straight-field-line coordinate system we call generalized Boozer coordinates. This coordinate system exists for magnetic fields with nested toroidal flux surfaces provided ∮ d l / B (j · ∇ ψ) = 0 , where symbols have their usual meaning, and the integral is taken along closed magnetic field lines. All quasisymmetric fields, regardless of their associated form of equilibria, must satisfy this condition. This coordinate system presents itself as a convenient form to describe general quasisymmetric configurations and their properties. Insight can be gained analytically into the difference between strong and weak forms of quasisymmetry, as well as axisymmetry, and the interaction of quasisymmetry with different forms of equilibria. [ABSTRACT FROM AUTHOR]

A method is demonstrated to optimize a stellarator's geometry to eliminate magnetic islands and achieve other desired physics properties at the same time. For many physics quantities that have been used in stellarator optimization, including quasisymmetry, neoclassical transport, and magnetohydrodynamic stability, it is convenient to use a magnetic equilibrium representation that assures the existence of magnetic surfaces. However, this representation hides the possible presence of magnetic islands, which are typically undesirable. To include both surface-based objectives and island widths in a single optimization, two fixed-boundary equilibrium calculations are run at each iteration of the optimization: one that enforces the existence of magnetic surfaces (the Variational Moments Equilibrium Code) [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)] and one that does not (the Stepped Pressure Equilibrium Code) [Hudson et al., Phys. Plasmas 19, 112502 (2012)]. By penalizing the island residues in the objective function, the two magnetic field representations are brought into agreement during the optimization. An example is presented in which, particularly on the surface where quasisymmetry was targeted, quasisymmetry is achieved more accurately than in previously published examples. [ABSTRACT FROM AUTHOR]

Methods for constructing synthetic multidimensional electron hole equilibria without using particle simulation are investigated. Previous approaches have various limitations and approximations that make them unsuitable within the context of expected velocity diffusion near the trapped-passing boundary. An adjustable model of the distribution function is introduced that avoids unphysical singularities there, and yet it is sufficiently tractable analytically to enable prescription of the potential spatial profiles. It is shown why simple models of the charge density as being a function only of potential cannot give solitary multidimensional electron holes, in contradiction of prior suppositions. Fully self-consistent axisymmetric electron holes in the drift-kinetic limit of electron motion (negligible gyro-radius) are constructed and their properties relevant to observational interpretation and finite-gyro-radius theory are discussed. [ABSTRACT FROM AUTHOR]

Toroidal rotation is well known to play significant roles in the edge transport and L–H transition dynamics of tokamaks. Our recent calculation finds that a sufficiently strong localized toroidal rotation can directly bring out the formation of edge pressure pedestal with reversed magnetic shear that is reminiscent of an H-mode plasma, purely through the effects of toroidal rotation on the tokamak MHD equilibrium itself. In particular, the enhanced edge toroidal rotation enables a substantial peaking of the parallel current profile near edge in higher β regimes, which leads to the flattening or reversal of the local q (safety factor) profile. Here the formation of pressure pedestal along with the reversed magnetic shear region is shown to be the natural outcome of the MHD tokamak equilibrium in a self-consistent response to the presence of a localized toroidal rotation typically observed in high confinement-mode (H-mode) or quiescent H-mode. [ABSTRACT FROM AUTHOR]

3D equilibria with an anisotropic pressure component in the large helical device are analyzed with respect to their magnetic axis locations. The anisotropic extension of the 3D equilibrium solver variational moments equilibrium code, anisotropic neumann inverse moments equilibrium code, is used to compute fixed-boundary plasma equilibria based on a bi-Maxwellian distribution function describing the anisotropic particles. Different heating scenarios are assessed by means of parallel and perpendicular pressure anisotropies with different radial anisotropic pressure profiles imposed. A theoretical predicted scaling of the magnetic axis location with the auxiliary parameter βeq as predicted for classical stellarators and heliotrons by Hitchon [Nucl. Fusion 23, 383 (1983)] is found to be applicable to the large helical device in the case of a flat hot-particle profile for parallel or weak perpendicular dominated anisotropies with β ⊥ / β ∥ ≤ 2. For strong perpendicular or non-flat hot-particle profiles, a deviation from the predicted scaling of the magnetic axis location is found. Whereas center-peaked profiles show a stronger shift of the magnetic axis, edge-peaked profiles show no significant change of its radial location. High critical magnetic fields are identified as a necessary condition for strong perpendicular anisotropies. The observed deviations are ascribed to the magnetic field structure and negative pressure gradients. The invalidity of the theoretical predictions in the case of certain configurations is found to be caused by higher-order terms in the pressure components, which are not accounted for by the ordering on which the theory is based. [ABSTRACT FROM AUTHOR]

An improved resonant plasma response model that more accurately captures the physics of the interaction between a tokamak plasma and a resonant magnetic perturbation (RMP) is developed. The model interpolates between the linear and the nonlinear response regimes and takes into account the fact that the slip-frequency is non-zero in the nonlinear regime. The improved model is incorporated into the extended perturbed equilibrium code (EPEC) toroidal asymptotic matching code. The modified EPEC code is used to investigate RMP-induced edge-localized-mode (ELM) suppression in DIII-D H-mode discharge #145380. Somewhat surprisingly, allowing for a finite slip-frequency (i.e., relaxing the so-called no-slip constraint) is found to only slightly facilitate the locking of driven magnetic island chains to the RMP, and, hence, to only slightly facilitate RMP-induced ELM suppression. This is true despite the fact that the nature of non-locked island solutions is radically different when the no-slip constraint is imposed compared to when it is relaxed (in the first case, the widths of the island chains driven at the rational surfaces pulsate, and in the second case, they remain steady). The previously obtained conclusion that the response of a typical H-mode tokamak plasma to an RMP cannot be accurately modeled by linear theory is confirmed. The previously obtained conclusion that the best agreement between theory and observations is achieved by assuming that the natural frequencies of tearing modes, in the absence of the RMP, are determined by the local equilibrium E × B velocity is also confirmed. [ABSTRACT FROM AUTHOR]

We apply the near-axis expansion method for quasisymmetric magnetic fields with anisotropic pressure (developed in Paper I) [E. Rodriguez and A. Bhattacharjee, Phys. Plasmas 28, 012508 (2020)] to construct numerical solutions to circular axis stellarators. The solutions are found to second order in the distance from the axis, not possible in the standard Garren–Boozer construction [D. A. Garren and A. H. Boozer, Phys. Fluids B 3, 2822 (1991)], which assumes magnetostatic equilibria with isotropic pressure. In the limit of zero anisotropy, it is shown that a subset of coefficients can be chosen to avoid the overdetermination problem. [ABSTRACT FROM AUTHOR]

It is well known that the process of construction of quasisymmetric magnetic fields in magnetostatic equilibrium with isotropic pressure suffers from the problem of overdetermination. This has led to the widespread belief that global quasisymmetric solutions are likely not to exist. We develop a general near-axis expansion procedure that does not rely on the assumption of magnetostatic equilibria with isotropic pressure. We then demonstrate that in equilibria with anisotropic pressure, it is possible to circumvent the problem of overdetermination and carry out the power-series solutions to higher order. This suggests, contrary to current belief, that the existence of globally quasisymmetric fields is likely if one relaxes the assumption of magnetostatic equilibria with isotropic pressure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e., bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses a Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a nonvanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist. [ABSTRACT FROM AUTHOR]

The new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f (x) = 0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f (x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume. [ABSTRACT FROM AUTHOR]

This study deals with the equilibrium beta limit for high-beta plasmas of the Large Helical Device using a 3D equilibrium calculation code, HINT, which is an initial value solver based on the relaxation method without an assumption of nested flux surfaces. For the finite beta equilibrium, due to the nonlinear 3D equilibrium response, the flux surface breaking begins, and the magnetic field line becomes stochastic in the peripheral region. However, although the magnetic axis shifts until the conventional theoretical beta limit, the separatrix limiting the equilibrium does not appear in the plasma core. For the high beta equilibrium, breaking of the force balance begins in the strongly stochastized region. To keep the force balance in that region, the pressure gradient decreases and the fixed profile is relaxed. As a result, the volume averaged beta, which is equivalent to the plasma stored energy, saturates, although the peak beta increases. The beta value, where the force balance begins breaking in the inside of the stochastic region, is proposed as a new index of the equilibrium beta limit for the typical heliotron-type stellarator. According to this new index of the equilibrium beta limit, the achieved beta value in the experiment of the Large Helical Device is still lower than the equilibrium beta limit. This suggests that if the heating power can be increased, the achieved beta value may still increase. [ABSTRACT FROM AUTHOR]

A necessary and sufficient set of conditions for a quasisymmetric magnetic field in the form of constraint equations is derived from first principles. Without any assumption regarding the magnetohydrodynamic (MHD) equilibrium of the plasma, conditions for quasisymmetry are constructed starting from the single-particle Lagrangian to the leading order. The conditions presented in the paper are less restrictive than the set recently obtained by Burby et al. ["Some mathematics for quasi-symmetry," arXiv:1912.06468 (2019)], and could facilitate ongoing efforts toward investigating the existence of global quasisymmetric MHD equilibria. It is also shown that quasisymmetry implies the existence of flux surfaces, regardless of whether the field corresponds to an MHD equilibrium. [ABSTRACT FROM AUTHOR]

The formal stability analysis of Eulerian extended magnetohydrodynamics (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and the dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flow under constrained perturbations. The dynamically accessible, second-order variation of the Hamiltonian, which can potentially provide explicit stability criteria for generic equilibria, is also obtained. Moreover, we examine the Lagrangian stability of the general quasineutral two-fluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass, we derive a HMHD energy principle, and in addition, the perturbed induction equation arises from Hamilton's equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables. [ABSTRACT FROM AUTHOR]

An analytic solution to a generalized Grad-Shafranov equation with flow of arbitrary direction is obtained upon adopting the generic linearizing ansatz for the free functions related to the poloidal current, the static pressure, and the electric field. Subsequently, a D-shaped tokamak pertinent equilibrium with sheared flow is constructed using the aforementioned solution. [ABSTRACT FROM AUTHOR]

Exact Solov'ev equilibria for arbitrary plasma cross sections are calculated using a constrained least squares method. The boundary, with or without X-points, can be specified with an arbitrarily large number of constraints to ensure an accurate representation. Thus, the order of the polynomial basis functions in the homogeneous solution of the Grad-Shafranov equation becomes an independent parameter determined only by the accuracy requirements of the overall solution. Examples of exact, highly shaped equilibria are presented. [ABSTRACT FROM AUTHOR]

Linear simulations of toroidal Alfvén eigenmodes (TAEs) driven by energetic particles (EPs) on EAST (Experimental Advanced Superconducting Tokamak) are performed using the hybrid-kinetic MHD model implemented in NIMROD code. The EAST equilibrium is reconstructed using the EFIT code based on the experimental measurement. The "slowing down" distribution is adopted for modeling the equilibrium distribution of the energetic ions from the deuterium neutral beam injection on EAST. The frequency, the dominant poloidal mode number, the radial location and the detailed 2D mode structure of the TAE/RSAE (Reverse Shear Alfvén Eigenmode)/EPM (energetic particle mode) modes are consistent between the eigenvalue analysis and the NIMROD simulation. As the β fraction of EP increases, a transition between even and odd TAEs occurs, along with that between the ballooning and antiballooning mode structures. When the β fraction of EP is close to the transition threshold, both types of TAEs coexist. [ABSTRACT FROM AUTHOR]

As a step toward understanding 3D magnetohydrodynamic (MHD) equilibria, for which smooth solutions may not exist, we develop a simple cylindrical model to investigate the resistive stability of MHD equilibria with alternating regions of constant and nonuniform pressure, producing states with continuous total pressure (i.e., no singular current sheets) but discontinuities in the parallel current density. We examine how the resistive stability characteristics of the model change as we increase the localization of pressure gradients at fixed radii, which approaches a discontinuous pressure profile in the zero-width limit. Equilibria with continuous pressure are found to be unstable to moderate/high-m modes and apparently tend toward ideal instability in some cases. We propose that additional geometric degrees of freedom or symmetry breaking via island formation may increase the parameter space on which equilibria of our model are physically realizable, while preserving the radial localization of pressure gradients. This is consistent with the possibility of realizing, in practice, 3D MHD equilibria which support both continuously nested flux surfaces (where ∇p ≠ 0) and chaotic field regions (where ∇p = 0). [ABSTRACT FROM AUTHOR]

Theoretical and experimental considerations suggest that axisymmetric perturbations that are resonant at the X-point(s) of a magnetic divertor separatrix may play a role in the understanding of Edge Localized Modes in tokamak experiments and their active control via so-called vertical kicks. With this motivation in mind, the first step in the development of an analytical model for resistive axisymmetric X-point modes is presented, i.e., finding an adequate, but at the same time relatively simple analytical magnetohydrodynamic equilibrium for a plasma column with a noncircular cross section bound by a magnetic separatrix. An early example is Gajewski's equilibrium solution [R. Gajewski, Phys. Fluids 15, 70 (1972)], which, however, has the shortcoming that infinite external currents placed at an infinite distance from the X-points produce the elliptical elongation of the plasma column. In this article, Gajewski's solution is extended to the case where external currents are located at a finite distance from the boundary of the plasma current density and the latter is distributed uniformly over a domain bound by a nearly elliptical magnetic flux surface. [ABSTRACT FROM AUTHOR]

The characteristics of propagation of sawtooth periodic and impulsive signals at a transducer are analytically studied in this work. A plasma under consideration is motionless and uniform at equilibrium, and its perturbations are described by a system of ideal magnetohydrodynamic equations. Some generic heating/cooling function, which in turn depends on equilibrium thermodynamic parameters, may destroy adiabaticity of a flow and make the flow acoustically active. Planar waves with the wave vector forming a constant angle θ with the equilibrium straight magnetic strength are considered. This model has been proposed in previous publications listed in the Introduction. Conclusions are drawn for fast and slow magnetoacoustic waves of sawtooth shape and various cases of a nonlinear flow. These nonlinearities occur in accordance with a type of heating/cooling function under consideration. Amplitude and duration of signals are evaluated as functions of a distance from a transducer, θ, plasma-β, and a type of heating-cooling function. In particular, it is observed that the duration of an impulse enlarges infinitely in acoustically stable flows, while on the other hand, it tends to some limiting value in acoustically active flows of plasma. [ABSTRACT FROM AUTHOR]

Using a method developed by Clemente, it is possible to obtain anisotropic magnetohydrodynamic equilibrium in axially symmetric systems, from a previously known solution of the Grad-Schlüter-Shafranov equation. We generalize this method to symmetric systems described by orthogonal as well as nonorthogonal systems of coordinates. Two examples are presented in cylindrical and spherical geometries, for which we give an exact analytic solution of the anisotropic MHD equilibrium. [ABSTRACT FROM AUTHOR]

In this study, we find the points of transition between elliptic and hyperbolic regimes for the axisymmetric extended magnetohydrodynamic (MHD) equilibrium equations. The ellipticity condition is expressed via a single inequality but is more involved than the corresponding two-fluid ones due to the imposition of the quasineutrality condition and is also more complicated than the Hall MHD ellipticity condition, due to electron inertia. In fact, the inclusion of electron inertia is responsible for peculiar results; namely, even the static equilibrium equations can become hyperbolic. [ABSTRACT FROM AUTHOR]

We revisit the Hahm–Kulsrud–Taylor (HKT) problem, a classic prototype problem for studying resonant magnetic perturbations and 3D magnetohydrodynamical equilibria. We employ the boundary-layer techniques developed by Rosenbluth, Dagazian, and Rutherford (RDR) for the internal m = 1 kink instability, while addressing the subtle difference in the matching procedure for the HKT problem. Pedagogically, the essence of RDR's approach becomes more transparent in the reduced slab geometry of the HKT problem. We then compare the boundary-layer solution, which yields a current singularity at the resonant surface, to the numerical solution obtained using a flux-preserving Grad–Shafranov solver. The remarkable agreement between the solutions demonstrates the validity and universality of RDR's approach. In addition, we show that RDR's approach consistently preserves the rotational transform, which hence stays continuous, contrary to a recent claim that RDR's solution contains a discontinuity in the rotational transform. [ABSTRACT FROM AUTHOR]

The SOL width is a parameter of paramount importance in modern tokamaks as it controls the power density deposited at the divertor plates, critical for plasma-facing material survivability. An understanding of the parameters controlling it has consequently long been sought [Connor et al. Nucl. Fusion 39(2), 169 (1999)]. Prior to Chang et al. [Nucl. Fusion 57(11), 116023 (2017)], studies of the tokamak edge have been mostly confined to reduced fluid models and simplified geometries, leaving out important pieces of physics. Here, we analyze the results of a DIII-D simulation performed with the full-f gyrokinetic code XGC1 which includes both turbulence and neoclassical effects in realistic divertor geometry. More specifically, we calculate the particle and heat E × B fluxes along the separatrix, discriminating between equilibrium and turbulent contributions. We find that the density SOL width is impacted almost exclusively by the turbulent electron flux. In this simulation, the level of edge turbulence is regulated by a mechanism that we are only beginning to understand: ∇B-drifts and ion X-point losses at the top and bottom of the machine, along with ion banana orbits at the low field side, result in a complex poloidal potential structure at the separatrix which is the cause of the E × B drift pattern that we observe. Turbulence is being suppressed by the shear flows that this potential generates. At the same time, turbulence, along with increased edge collisionality and electron inertia, can influence the shape of the potential structure by making the electrons non-adiabatic. Moreover, being the only means through which the electrons can lose confinement, it needs to be in a balance with the original direct ion orbit losses to maintain charge neutrality. [ABSTRACT FROM AUTHOR]

The direction of cascades in a two-dimensional model that takes electron inertia and ion sound Larmor radius into account is studied, resulting in analytical expressions for the absolute equilibrium states of the energy and helicities. These states suggest that typically both the energy and magnetic helicity at scales shorter than the electron skin depth have a direct cascade, while at large scales the helicity has an inverse cascade as established earlier for reduced magnetohydrodynamics (MHD). The calculations imply that the introduction of gyro-effects allows for the existence of negative temperature (conjugate to energy) states and the condensation of energy to the large scales. Comparisons between two- and three-dimensional extended MHD models (MHD with two-fluid effects) show qualitative agreement between the two. [ABSTRACT FROM AUTHOR]

A two-field reduced gyrofluid model including electron inertia, ion finite Larmor radius corrections, and parallel magnetic field fluctuations is derived from the model of Brizard [Brizard, Phys. Fluids B 4, 1213 (1992)]. It assumes low βe, where βe indicates the ratio between the equilibrium electron pressure and the magnetic pressure exerted by a strong uniform magnetic guide field, but permits an arbitrary ion-to-electron equilibrium temperature ratio. It is shown to have a noncanonical Hamiltonian structure and provides a convenient framework for studying kinetic Alfvén wave turbulence, from magnetohydrodynamics to sub-de scales (where de holds for the electron skin depth). Magnetic energy spectra are phenomenologically determined within energy and generalized cross-helicity cascades in the perpendicular spectral plane. Arguments based on absolute statistical equilibria are used to predict the direction of the transfers, pointing out that, within the sub-ion range, the generalized cross-helicity could display an inverse cascade if injected at small scales, for example by reconnection processes. [ABSTRACT FROM AUTHOR]

A self-consistent electrostatic odd-symmetric eigenmode (OEM) is analytically found in a solitary type Bernstein-Greene-Kruskal (BGK) equilibrium. The frequency of the OEM is order of the electron bounce frequency and it is spatially odd-symmetric with the scale comparable to that of the solitary BGK equilibrium structure. Such an OEM is consistent with the recent observation from particle-in-cell simulation of the solitary wave [Dokgo et al., Phys. Plasmas 23, 092107 (2016)]. The mode can be driven unstable by trapped electrons within the hole structure of the solitary wave. Such a low frequency, pure electron mode, which may possibly interact resonantly with the ion acoustic mode, provides a possible damping mechanism of the BGK equilibrium. [ABSTRACT FROM AUTHOR]

Due to the X-point singularity, the safety factor tends to infinity as approaching to the last closed flux surface. The numerical treatments of the near X-point behavior become challenging both for equilibrium and stability. The usual solution is to cut off a small fraction of edge region for system stability evaluation or simply use an up-down symmetric equilibrium without X-point as an approximation. In this work, we assess the sensitivity of this type of equilibrium treatments on the stability calculation. It is found that the system stability can depend strongly on the safety factor value (qa) at the edge after the cutting-off. When the edge safety factor value falls in the vicinity of a rational mode number (referred to as the resonant gap), the system becomes quite unstable due to the excitation of the peeling type modes. Instead, when the edge safety factor is outside the resonant gaps, the system is much more stable and the predominant modes become the usual external kink (or ballooning and infernal) type. It is also found that the resonant gaps become smaller and smaller as qa increases. The ideal magnetohydrodynamic peeling ballooning stability diagram is widely used to explain the experimental observations, and the current results indicate that the conventional peeling ballooning stability diagram based on the simplified equilibrium needs to be reexamined. [ABSTRACT FROM AUTHOR]

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electromagnetic surfaces. [ABSTRACT FROM AUTHOR]

A scalar pressure model of solar wind plasma in the presence of Alfvén waves at the coarsegraining scale is discussed. Using a simple sub-grid-scale model, it is shown that the scalar pressure model can be applied to describe the background pressure, because the off-diagonal components of the sub-grid-scale stress tensor disappear in the presence of one-dimensional, sub-grid scale Alfvén waves. Local equilibrium velocity distribution functions corresponding to the scalar pressure including wave dynamics (apparent temperature) are also presented. The specific heat ratio of the apparent temperature is given by the Wentzel-Kramers-Brillouin approximation of the Alfvén waves. The consequence of the local equilibrium velocity distribution functions in the coarsegraining scale model is discussed by using a nonlinear equation set describing envelope-modulated nonlinear Alfvén waves. It is recognized that without sufficient separation between the scale of waves and the scale of integration, artificial changes of the growth rates appear in the solutions of linear growth rates owing to the finite length of the filter function. [ABSTRACT FROM AUTHOR]

Using Vlasov-Maxwell's equations, the spectra of the perpendicular propagating Bernstein wave and Extraordinary wave in ultra-relativistic fully degenerate electron plasma are studied. The equilibrium particle distribution function is assumed to be isotropic Fermian. The analysis of high frequency spectra of the waves is carried out in the weak propagation limit Ω ≫ k.v and in the weak magnetic field limit |x ω - k.v| ≫ Ω and graphically observed. [ABSTRACT FROM AUTHOR]

Two floating sheath solutions with strong electron emission in planar geometry have been proposed, a "space-charge limited" (SCL) sheath and an "inverse" sheath. SCL and inverse models contain different assumptions about conditions outside the sheath (e.g., the velocity of ions entering the sheath). So it is not yet clear whether both sheaths are possible in practice, or only one. Here we treat the global presheath-sheath problem for a plasma produced volumetrically between two planar walls. We show that all equilibrium requirements (a) floating condition, (b) plasma shielding, and (c) presheath force balance, can indeed be satisfied in two different ways when the emission coefficient γ > 1. There is one solution with SCL sheaths and one with inverse sheaths, each with sharply different presheath distributions. As we show for the first time in 1D-1V simulations, a SCL and inverse equilibrium are both possible in plasmas with the same upstream properties (e.g., same N and Te). However, maintaining a true SCL equilibrium requires no ionization or charge exchange collisions in the sheath, or else cold ion accumulation in the SCL's "dip" forces a transition to the inverse. This suggests that only a monotonic inverse type sheath potential should exist at any plasma-facing surface with strong emission, whether be a divertor plate, emissive probe, dust grain, Hall thruster channel wall, sunlit object in space, etc. Nevertheless, SCL sheaths might still be possible if the ions in the dip can escape. Our simulations demonstrate ways in which SCL and inverse regimes might be distinguished experimentally based on large-scale presheath effects, without having to probe inside the sheath. [ABSTRACT FROM AUTHOR]

An exactly solvable model of equilibrium and stability of a gravitationally bound low dust density cloud embedded in background plasma is described. The dust is distributed almost homogeneously within the cloud. The corresponding eigen value equation is solved using the Frobenius method to calculate the complete set of radial eigen modes and corresponding frequencies. The low dust density cloud is shown to be stable. This model can be useful in benchmarking simulation codes and modelling dust density distribution in the interstellar cloud in the HII region. [ABSTRACT FROM AUTHOR]

In the frame of mixed Eulerian-Lagrangian description, a large-perturbation magnetohydrodynamic (MHD) equilibrium system is developed to describe the propagation of nonlinear Alfvén waves in an ideal magnetofluid. The system is only dependent on the reference flow variables and the displacement vector of the fluid particles. Some interesting results are obtained: nonlinear MHD wave speeds are obtained; fixing the equilibrium magnetic field, a large-negative gradient of displacement can generate directional discontinuities; a large-positive pressure perturbation or magnetic perturbation can bring about a high velocity nonlinear Alfvén wave while a large-negative pressure perturbation or magnetic perturbation can bring about a low velocity nonlinear Alfvén wave; a great-negative magnetic perturbation may give birth to counter-propagating Alfvén waves. Our results have a high relevance to such issues as discontinuities, counter-propagating nonlinear Alfvén waves, and ultra-speed Alfvén waves in interplanetary space. [ABSTRACT FROM AUTHOR]