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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]

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]

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]

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]

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 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]

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]

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]

Kinetic-MagnetoHydroDynamic hybrid simulations are carried out to investigate fishbone modes excited by fast ions on the Experimental Advanced Superconducting Tokamak. The simulations use realistic equilibrium reconstructed from experiment data with the constraint of the q=1 surface location (q is the safety factor). Anisotropic slowing down distribution is used to model the distribution of the fast ions from neutral beam injection. The resonance condition is used to identify the interaction between the fishbone mode and the fast ions, which shows that the fishbone mode is simultaneously in resonance with the bounce motion of the trapped particles and the transit motion of the passing particles. Both the passing and trapped particles are important in destabilizing the fishbone mode. The simulations show that the mode frequency chirps down as the mode reaches the nonlinear stage, during which there is a substantial flattening of the perpendicular pressure of fast ions, compared with that of the parallel pressure. For passing particles, the resonance remains within the q=1 surface, while, for trapped particles, the resonant location moves out radially during the nonlinear evolution. In addition, parameter scanning is performed to examine the dependence of the linear frequency and growth rate of fishbones on the pressure and injection velocity of fast ions. [ABSTRACT FROM AUTHOR]

The loss-of-equilibrium is a solar eruption mechanism whereby a sudden breakdown of the magnetohydrodynamic force balance in the Sun's corona ejects a massive burst of particles and energy into the heliosphere. Predicting a loss-of-equilibrium, which has more recently been formulated as the torus instability, relies on a detailed understanding of the various forces that hold the pre-eruption magnetic flux rope in equilibrium. Traditionally, idealized analytical force expressions are used to derive simplified eruption criteria that can be compared to solar observations and modeling. What is missing, however, is a validation that these idealized analytical force expressions can be applied to the line-tied, low-aspect-ratio conditions of the corona. In this paper, we address this shortcoming by using a laboratory experiment to study the forces that act on longlived, arched, line-tied magnetic flux ropes. Three key force terms are evaluated over a wide range of experimental conditions: (1) the upward hoop force; (2) the downward strapping force; and (3) the downward toroidal field tension force. First, the laboratory force measurements show that, on average, the three aforementioned force terms cancel to produce a balanced line-tied equilibrium. This finding validates the laboratory force measurement techniques developed here, which were recently used to identify a dynamic toroidal field tension force that can prevent flux rope eruptions [Myers et al., Nature 528, 526 (2015)]. The verification of magnetic force balance also confirms the low-β assumption that the plasma thermal pressure is negligible in these experiments. Next, the measured force terms are directly compared to corresponding analytical expressions. While the measured and analytical forces are found to be well correlated, the low-aspect-ratio, line-tied conditions in the experiment are found to both reduce the measured hoop force and increase the measured tension force with respect to analytical expectations. These two co-directed effects combine to generate laboratory flux rope equilibria at lower altitudes than are predicted analytically. Such considerations are expected to modify the loss-of-equilibrium eruption criteria for analogous flux ropes in the solar corona. [ABSTRACT FROM AUTHOR]

The field reversed configuration (FRC) is one of the candidate plasma targets for the magnetoinertial fusion, and a high temperature FRC can be formed by using the collision-merging technology. Although the merging process and mechanism of FRC are quite complicated, it is thinkable to build a simple model to investigate the macroscopic equilibrium parameters including the density, the temperature and the separatrix volume, which may play an important role in the collisionmerging process of FRC. It is quite interesting that the estimates of the related results based on our simple model are in agreement with the simulation results of a two-dimensional magneto-hydrodynamic code (MFP-2D), which has being developed by our group since the last couple of years, while these results can qualitatively fit the results of C-2 experiments by Tri-alpha energy company. On the other hand, the simple model can be used to investigate how to increase the density of the merged FRC. It is found that the amplification of the density depends on the poloidal fluxincrease factor and the temperature increases with the translation speed of two plasmoids. Published by AIP Publishing. [ABSTRACT FROM AUTHOR]

Experiments in the DIII-D tokamak show that the plasma responds to resonant magnetic perturbations (RMPs) with toroidalmode 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 possible (m = 8, 9, or 10 for n = 2) and (m = 4 or 5 for n = 1). In all cases, the island width is estimated to be 2-3 cm. The Chang-Callen calculated confinement degradation due to the presence of an individual island of this size is 8%-12%, which is close to the 13%-14% measured between the ELMs and suppressed states. This suggests that edge tearing modes may alter the pedestal causing peeling-ballooning stability during RMP induced ELM suppression. [ABSTRACT FROM AUTHOR]

It can be shown that in the presence of a toroidal magnetic field induced by poloidal coils, combined with the electromagnetic field induced by a central solenoid, no static equilibrium is possible within the MHD description, as soon as non-zero resistivity is assumed. The resulting dynamic equilibrium was previously discussed for the case of spatially homogeneous resisitivity. In the present work, it is shown how a spatial inhomogeneity of the viscosity and resisitivity coefficients influences this equilibrium. Parameters in both the stable, tokamak-like regime and unstable, reversed field pinch-like regime are considered. It is shown that, whereas the magnitudes of the velocity and magnetic field fluctuations are strongly modified by the spatial variation of the transport coefficients, the qualitative flow behaviour remains largely unaffected. [ABSTRACT FROM AUTHOR]

The Hahm-Kulsrud (HK) [T. S. Hahm and R. M. Kulsrud, Phys. Fluids 28, 2412 (1985)] solutions for a magnetically sheared plasma slab driven by a resonant periodic boundary perturbation illustrate fully shielded (current sheet) and fully reconnected (magnetic island) responses. On the global scale, reconnection involves solving a magnetohydrodynamic (MHD) equilibrium problem. In systems with a continuous symmetry, such MHD equilibria are typically found by solving the Grad-Shafranov equation, and in slab geometry the elliptic operator in this equation is the 2-D Laplacian. Thus, assuming appropriate pressure and poloidal current profiles, a conformal mapping method can be used to transform one solution into another with different boundary conditions, giving a continuous sequence of solutions in the form of partially reconnected magnetic islands (plasmoids) separated by Syrovatsky current sheets. The two HK solutions appear as special cases. [ABSTRACT FROM AUTHOR]

Error-field penetration in reversed magnetic shear (RMS) configurations is numerically investigated by using a two-dimensional resistive magnetohydrodynamic model in slab geometry. To explore different dynamic processes in locked modes, three equilibrium states are adopted. Stable, marginal, and unstable current profiles for double tearing modes are designed by varying the current intensity between two resonant surfaces separated by a certain distance. Further, the dynamic characteristics of locked modes in the three RMS states are identified, and the relevant physics mechanisms are elucidated. The scaling behavior of critical perturbation value with initial plasma velocity is numerically obtained, which obeys previously established relevant analytical theory in the viscoresistive regime. [ABSTRACT FROM AUTHOR]

Theoretical studies of the plasmoid instability generally assume that the reconnecting magnetic fields are symmetric. We relax this assumption by performing two-dimensional resistive magnetohydrodynamic simulations of the plasmoid instability during asymmetric inflow magnetic reconnection. Magnetic asymmetry modifies the onset, scaling, and dynamics of this instability. Magnetic islands develop preferentially into the weak magnetic field upstream region. Outflow jets from individual X-points impact plasmoids obliquely rather than directly as in the symmetric case. Consequently, deposition of momentum by the outflow jets into the plasmoids is less efficient, the plasmoids develop net vorticity, and shear flow slows down secondary merging between islands. Secondary merging events have asymmetry along both the inflow and outflow directions. Downstream plasma is more turbulent in cases with magnetic asymmetry because islands are able to roll around each other after exiting the current sheet. As in the symmetric case, plasmoid formation facilitates faster reconnection for at least small and moderate magnetic asymmetries. However, when the upstream magnetic field strengths differ by a factor of 4, the reconnection rate plateaus at a lower value than expected from scaling the symmetric results. We perform a parameter study to investigate the onset of the plasmoid instability as a function of magnetic asymmetry and domain size. There exist domain sizes for which symmetric simulations are stable but asymmetric simulations are unstable, suggesting that moderate magnetic asymmetry is somewhat destabilizing. We discuss the implications for plasmoid and flux rope formation in solar eruptions, laboratory reconnection experiments, and space plasmas. The differences between symmetric and asymmetric simulations provide some hints regarding the nature of the three-dimensional plasmoid instability. [ABSTRACT FROM AUTHOR]

Low-frequency, parallel propagating Alfvénic turbulence in collisionless plasmas is theoretically studied. Alfvénic turbulence is derived as an equilibrium state (Beltrami field) in the magnetohydrodynamic equations with the pressure anisotropy and multi-species of ions. It is shown that the conservation of the total 'apparent temperature' corresponds to the Bernoulli law. A simple model of the radially expanding solar wind including Alfvénic turbulence is also discussed. The conversion of the wave energy in the 'apparent temperature' into the 'real temperature' is facilitated with increasing radial distance. [ABSTRACT FROM AUTHOR]

Two reconnection problems are considered. The first problem concerns global physics. The plasma in the global reconnection region is in magnetostatic equilibrium. It is shown that this equilibrium can be uniquely characterized by a set of constraints. During reconnection and independently of the local reconnection physics, these constraints can be uniquely evolved from any initial state. The second problem concerns Petschek reconnection. Petschek's model for fast reconnection, which is governed by resistive MHD equations with constant resistivity is not validated by numerical simulations. Malyshkin et al. [Phys. Plasmas 12, 102920 (2005)], showed that the reason for the discrepancy is that Petschek did not employ Ohm's law throughout the local diffusion region, but only at the X-point. A derivation of Petschek reconnection, including Ohm's law throughout the entire diffusion region, removes the discrepancy. This derivation is based largely on Petschek's original 1964 calculation [in AAS-NASA Symposium on Solar Flares (National Aeronautics and Space Administration, Washington, D.C., 1964), NASA SP50, p. 425]. A useful physical interpretation of the role which Ohm's law plays in the diffusion region is presented. [ABSTRACT FROM AUTHOR]

Coaxial helicity injection in the HIT-II [A. J. Redd et al., Phys. Plasmas 9, 2006 (2002)] spherical torus is modeled with time-dependent resistive magnetohydrodynamics computations run to steady state for conditions without strong relaxation. Laboratory and computed results on injector current and plasma current agree reasonably well as toroidal magnetic field and injector flux are scaled. The scalings are consistent with a dimensional estimate from the Grad-Shafranov equation that provides a new perspective on a previously published model based on a current-sheet equilibrium and the magnetic pressure required for the 'bubble-burst' criterion. Numerical solutions of the Grad-Shafranov equation with an assumed current profile also indicate large qualitative changes as the predicted criterion is crossed. [ABSTRACT FROM AUTHOR]

A theory for localized low-frequency ideal magnetohydrodynamical (MHD) modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal equilibrium plasma flows. The general set of equations describing the coupling of shear Alfvén and slow (sound) modes and defining the continuous spectrum of rotating plasmas in axisymmetric toroidal systems is derived. The equations are applied to study the continuous spectra in large aspect ratio tokamaks. The unstable continuous modes in the case of predominantly poloidal plasma rotation with the angular velocity exceeding the sound frequency are found. Their stabilization by the shear Alfvén coupling effect is studied. [ABSTRACT FROM AUTHOR]

We extend existing 2-D analytical high-β equilibrium solutions obtained by Cowley et al. [Phys. Fluids B 3, 2066 (1991)] and Hsu et al. [Phys. Plasmas 3, 266 (1996)] to the case of toroidally flowing plasma, assuming ideal magnetohydrodynamics and isothermal magnetic surfaces. Invoking the ordering βq2>>[variant_greek_epsilon]2 and Mφ2q2>>[variant_greek_epsilon], we solve the boundary layer problem, re-obtaining the static solution in the zero flow limit. The phenomenon of extreme plasma diamagnetism in the static solution was found to be reduced due to centrifugal drift currents for significant toroidal flow. Example equilibrium calculations are presented using plasma parameters which are approaching attainable values in next generation spherical tokamaks. [ABSTRACT FROM AUTHOR]

Poloidal rotation is routinely observed in present-day tokamak experiments, in particular near the plasma edge and in the high-confinement mode of operation. According to the magnetohydrodynamic (MHD) equilibrium theory [R. Betti and J. P. Freidberg, Phys. Plasmas 7, 2439 (2000)], radial discontinuities form when the poloidal velocity exceeds the poloidal sound speed (or rather, more correctly, the poloidal magneto-slow speed). Two-dimensional compressible magnetohydrodynamic simulations show that the transonic discontinuities develop on a time scale of a plasma poloidal revolution to form an edge density pedestal and a localized velocity shear layer at the pedestal location. While such an MHD pedestal surrounds the entire core, the outboard side of the pedestal is driven by the transonic discontinuity while the inboard side is caused by a poloidal redistribution of the mass. The MHD simulations use a smooth momentum source to drive the poloidal flow. Soon after the flow exceeds the poloidal sound speed, the density pedestal and the velocity shear layer form and persist into a quasi steady state. These results may be relevant to the L-H transition, the early stages of the pedestal and edge transport barrier formation. [ABSTRACT FROM AUTHOR]

The finite-element M3D code [W. Park et al., Phys. Plasmas 6, 1796 (1999)] has been modified to include a free-boundary equilibrium solver with arbitrary toroidal and poloidal flows. With this modification, the M3D code now has the capability to self-consistently model two essential ingredients necessary for equilibrium calculations in the edge region, namely, free-boundary and arbitrary flow. As a free-boundary code, M3D includes the separatrix and scrape-off layer regions in the equilibrium calculation. Poloidal flows in the subsonic, supersonic, and transonic regimes can be calculated with the new version of the M3D code. Calculation results show that the presence of equilibrium flows, in particular those next to the plasma boundary, can considerably influence the position of the X-point and magnetic separatrix shape/location and hence the position of the strike point on the divertor plates. Moreover, it is shown that poloidal flow is not a rigid-body rotation, with the fastest flows occurring on the inboard side of the plasma. A numerical confirmation of the 'de Laval nozzle' model of Betti and Freidberg [R. Betti and J. P. Freidberg, Phys. Plasmas 7, 2439 (2000)] for free-boundary equilibrium calculations is obtained, with the formation of the predicted discontinuities between regions of subsonic and supersonic flows (with respect to the poloidal sound speed). Finally, a detailed comparison between isentropic and isothermal equilibria is presented, showing qualitative analogies and quantitative differences. [ABSTRACT FROM AUTHOR]

We discuss a new family of solutions of the Grad-Shafranov (GS) equation that describes D-shaped toroidal plasma equilibria with sharp gradients at the plasma edge. These solutions have been derived by exploiting the continuous Lie symmetry properties of the GS equation and in particular a special type of 'weak' symmetries. In addition, we review the continuous Lie symmetry properties of the GS equation and present a short but exhaustive survey of the possible choices for the arbitrary flux functions that yield GS equations admitting some continuous Lie symmetry. Particular solutions related to these symmetries are also discussed. [ABSTRACT FROM AUTHOR]

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

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

A cylindrical magnetohydrodynamic model including plasma pressure and longitudinal flow has been employed for the study of resistive wall mode (RWM) in reversed field pinch (RFP) plasmas. In order to validate the model, a careful comparison with the experimental measurements in RFX-mod [P. Sonato et al., Fusion Eng. Des. 66–68, 161 (2003)] on the mode growth rates has been made by well matching the equilibrium parameters F, Θ, and βp. The RWM instability spectrum, which varies with the equilibrium parameters, is also calculated for comparison. The sensitivity of the mode growth rate to the equilibrium parameters is studied in details. It is concluded that the model can provide consistent accuracy in studies of RWM in RFP plasmas. The analysis based on the balance of the potential energy components has been carried out in order to obtain the physical understanding on the mode behavior. [ABSTRACT FROM AUTHOR]

Analyses of plasma behavior often begin with a description of the ideal magnetohydrodynamic equilibrium, this being the simplest model capable of approximating macroscopic force balance. Ideal force balance is when the pressure gradient is supported by the Lorentz force, ∇p=j×B. We discuss the implications of allowing for a chaotic magnetic field on the solutions to this equation. We argue that the solutions are pathological and not suitable for numerical calculations. If the pressure and magnetic field are continuous, the only nontrivial solutions have an uncountable infinity of discontinuities in the pressure gradient and current. The problems arise from the arbitrarily small length scales in the structure of the field, and the consequence of ideal force balance that the pressure is constant along the field-lines, B·∇p=0. A simple method to ameliorate the singularities is to include a small but finite perpendicular diffusion. A self-consistent set of equilibrium equations is described, and some algorithmic approaches aimed at solving these equations are discussed. [ABSTRACT FROM AUTHOR]

A linear benchmark between the linear ideal MHD stability codes ELITE [H. R. Wilson et al., Phys. Plasmas 9, 1277 (2002)], GATO [L. Bernard et al., Comput. Phys. Commun. 24, 377 (1981)], and the extended nonlinear magnetohydrodynamic (MHD) code, NIMROD [C. R. Sovinec et al.., J. Comput. Phys. 195, 355 (2004)] is undertaken for edge-localized (MHD) instabilities. Two ballooning-unstable, shifted-circle tokamak equilibria are compared where the stability characteristics are varied by changing the equilibrium plasma profiles. The equilibria model an H-mode plasma with a pedestal pressure profile and parallel edge currents. For both equilibria, NIMROD accurately reproduces the transition to instability (the marginally unstable mode), as well as the ideal growth spectrum for a large range of toroidal modes (n=1–20). The results use the compressible MHD model and depend on a precise representation of “ideal-like” and “vacuumlike” or “halo” regions within the code. The halo region is modeled by the introduction of a Lundquist-value profile that transitions from a large to a small value at a flux surface location outside of the pedestal region. To model an ideal-like MHD response in the core and a vacuumlike response outside the transition, separate criteria on the plasma and halo Lundquist values are required. For the benchmarked equilibria the critical Lundquist values are 108 and 103 for the ideal-like and halo regions, respectively. Notably, this gives a ratio on the order of 105, which is much larger than experimentally measured values using Te values associated with the top of the pedestal and separatrix. Excellent agreement with ELITE and GATO calculations are made when sharp boundary transitions in the resistivity are used and a small amount of physical dissipation is added for conditions very near and below marginal ideal stability. [ABSTRACT FROM AUTHOR]

The electron magnetohydrodynamics model has been generalized to incorporate relativistic effects. The model is then employed to study the instability associated with sheared electron velocity flow in the relativistic regime. The instability has features similar to the conventional velocity shear driven Kelvin–Helmholtz-like mode [A. Das and P. Kaw, Phys. Plasmas 8, 4518 (2001)] in the weakly relativistic regime. However, in the strongly relativistic regime the instability shows certain distinct characteristics. The threshold value of the wave number is found to be considerably higher than the inverse shear width of the equilibrium velocity profile in this regime. Thus, the unstable domain of the wave-number space is considerably wider in this case. Also the mode does not remain purely growing but acquires a real frequency even for an antisymmetric velocity profile. These features of the mode have been understood by realizing that in the strongly relativistic regime the relativistic mass factor γ0 for the equilibrium has much sharper variations than that of the velocity profile. [ABSTRACT FROM AUTHOR]

A two-dimensional reconnecting current sheet is studied numerically in the magnetohydrodynamic approach. Different simulation setups are employed in order to follow the evolution of the formed current sheet in diverse configurations: two types of initial equilibria, Harris and force-free, two types of boundary conditions, periodic and open, with uniform and nonuniform grid set, respectively. All the simulated cases are found to exhibit qualitatively the same behavior in which a current sheet evolves slowly through a series of quasiequilibria; eventually it fragments and enters a phase of fast impulsive bursty reconnection. In order to gain more insight on the nature and characteristics of the instability taking place, physical characteristics of the simulated current sheet are related to its geometrical properties. At the adopted Lundquist number of S=104 and Reynolds number R=104, the ratio of the length to width (aspect ratio) of the formed current sheet is observed to increase slowly in time up to a maximum value at which it fragments. Moreover, additional turbulence applied to the system is shown to exhibit the same qualitative steps, but with the sooner onset of the fragmentation and at smaller aspect ratio. [ABSTRACT FROM AUTHOR]

In tokamak experiments, equilibrium reconstruction codes are used to calculate the location of the last closed flux surface, to map diagnostic information, and to derive important properties like current density and safety factor. At JET, the equilibrium code EFIT is automatically executed after each discharge. For speed and robustness, intershot EFIT is based on magnetic probe measurements only. As a consequence, the intershot profiles of the safety factor can be wrong for a variety of plasma scenarios. Internal diagnostic information, the pitch angle as measured with the motional stark effect, Faraday rotation angles, as well as pressure profile information can increase the accuracy of the EFIT equilibrium. In this paper, the accuracy of the internal diagnostics at JET and their impact on the EFIT results are discussed in detail. The influence of control parameters like the form of the test functions for ff′ and p′ on the equilibrium is investigated. The qmin from this analysis agrees with information from magnetohydrodynamics analysis (e.g., Alfvén cascades and sawtooth analysis) to within 10%–15%. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of ideal, incompressible Hall magnetohydrodynamics (HMHD) is investigated through classical Gibbs ensemble methods applied to the finite Galerkin representation. The spectral structure of HMHD is derived in a three-dimensional periodic geometry and compared with the MHD case. This provides a general picture of spectral transfer and cascade by the assumption that ideal Galerkin HMHD follows equilibrium statistics as in the case of Euler [U. Frisch et al., J. Fluid Mech. 68, 769 (1975)] and MHD [T. Stribling and W. H. Matthaeus, Phys. Fluids B 2, 1979 (1990)] theories. In HMHD, the equilibrium ensemble is built on the conservation of three quadratic invariants: The total energy, the magnetic helicity, and the generalized helicity. The latter replaces the cross helicity in MHD. In HMHD equilibrium, several differences appear with respect to the MHD case: (i) The generalized helicity (and in a weaker way the energy and the magnetic helicity) tends to condense in the longest wavelength, as in MHD, but also admits the novel feature of spectral enhancement, not a true condensation, at the smallest scales; (ii) equipartition between kinetic and magnetic energy, typical of Alfvénic MHD turbulence, is broken; (iii) modal distributions of energy and helicities show minima due to the presence of the ion skin depth. Ensemble predictions are compared to numerical simulations with a low-order truncation Galerkin spectral code, and good agreement is seen. Implications for general turbulent states are discussed. [ABSTRACT FROM AUTHOR]

A sufficient condition for the linear stability of three dimensional equilibria with incompressible flows parallel to the magnetic field is derived. The condition refers to internal modes and involves physically interpretable terms related to the magnetic shear and the flow shear. [ABSTRACT FROM AUTHOR]

The linear gyrokinetics theory in the axisymmetric configuration is revisited. It is found that the conventional gyrokinetic theory needs to be repaired in order to recover the linear magnetohydrodynamics from the gyrokinetics and to obtain the finite Larmor radius effect on the magnetohydrodynamic modes in an ordering-consistent manner. Two key inclusions are: (1) the solution of the equilibrium gyrokinetic distribution function is carried out to a sufficiently high order; (2) the gyrophase-dependent part of the perturbed distribution function is kept. The new gyrokinetic theory developed in this paper can be used to extend directly the magnetohydrodynamic stability analysis to the gyrokinetic one without invoking the hybrid kinetic-fluid hypothesis. [ABSTRACT FROM AUTHOR]

The method described by Evstatiev et al. [Phys. Plasmas 13, 072902 (2006)] to study the linear stability of line-tied modes in cylindrical geometry is applied to screw pinch equilibria with magnetic shear. The method is based on an expansion in eigenfunctions which depend on radius, and for ideal magnetohydrodynamics (MHD) the inclusion in the expansion of singular eigenfunctions (originating from a continuum) is necessary. The method is also applied to study scaling laws for large cylinder lengths L. It is found that the width of the internal layer of the radial displacement for the line-tied mode scales asymptotically as L-2, consistent with the so-called two-mode approximation. This result is valid in the context of both ideal and resistive MHD and is obtained both analytically and numerically. [ABSTRACT FROM AUTHOR]

A generalized energy principle for finite-pressure, toroidal magnetohydrodynamic (MHD) equilibria in general three-dimensional configurations is proposed. The full set of ideal-MHD constraints is applied only on a discrete set of toroidal magnetic surfaces (invariant tori), which act as barriers against leakage of magnetic flux, helicity, and pressure through chaotic field-line transport. It is argued that a necessary condition for such invariant tori to exist is that they have fixed, irrational rotational transforms. In the toroidal domains bounded by these surfaces, full Taylor relaxation is assumed, thus leading to Beltrami fields ∇×B=λB, where λ is constant within each domain. Two distinct eigenvalue problems for λ arise in this formulation, depending on whether fluxes and helicity are fixed, or boundary rotational transforms. These are studied in cylindrical geometry and in a three-dimensional toroidal region of annular cross section. In the latter case, an application of a residue criterion is used to determine the threshold for connected chaos. [ABSTRACT FROM AUTHOR]

The kinetic stability of the n=1, m=1 internal kink mode is analyzed for realistic equilibria typical of the standard operation scenario of ITER (the International Thermonuclear Experimental Reactor) [ITER Physics Basis Editors, Nucl. Fusion 39, 2137 (1999)]. The kinetic effects modify the inertia and the perturbed potential energy δW of the mode, the two key elements determining the mode stability. Numerical results are obtained for ITER-like equilibria with different q profiles. For moderate magnetic shear within the q=1 surface, the low frequency magnetohydrodynamic (MHD) branch is fully suppressed by the kinetic effects for the expected profiles and parameters up to twice the expected plasma β while the high frequency fishbone branch is found to be destabilized as the plasma β and the radius of the q=1 surface increase. The MHD branch can be destabilized at higher plasma β or larger radii of the q=1 surface only for q profiles with a low magnetic shear within the q=1 surface. [ABSTRACT FROM AUTHOR]

The force balance, or equilibrium, of an axisymmetric toroidal plasma with a scalar pressure can be relatively simply calculated. However, equilibrium calculations present fundamental difficulties when either the geometry or the plasma forces are more complicated. Many of these issues, such as the loss of magnetic surfaces and complicated plasma forces, can be addressed by perturbation theory using ideal magnetohydrodynamics stability codes. This paper outlines the procedure for carrying out such calculations and demonstrates the method by applying it to a problem with a known solution, an axisymmetric tokamak in which the pressure is increased by 15%. [ABSTRACT FROM AUTHOR]

The fractal distribution of charged particles is considered. An example of this distribution is the charged particles that are distributed over the fractal. The fractional integrals are used to describe fractal distribution. These integrals are considered as approximations of integrals on fractals. Typical turbulent media could be of a fractal structure and the corresponding equations should be changed to include the fractal features of the media. The magnetohydrodynamics equations for fractal media are derived from the fractional generalization of integral Maxwell equations and integral hydrodynamics (balance) equations. Possible equilibrium states for these equations are considered. [ABSTRACT FROM AUTHOR]

The stability of ideal magnetohydrodynamic ballooning modes in the presence of sheared flow is investigated for three-dimensional equilibria. Application of ballooning formalism reduces the problem to a partial differential equation in three dimensions that can be solved in the limit of small flow. Analytic calculations demonstrate the stabilizing effect of shear flow. The derived stability criterion generalizes prior work related to axisymmetric equilibrium with sheared toroidal flow. [ABSTRACT FROM AUTHOR]

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

The equilibrium and Lyapunov stability properties for two-dimensional ideal magnetohydrodynamic (MHD) plasmas with incompressible and homogeneous (i.e., constant density) flows are investigated. In the unperturbed steady state, both the velocity and magnetic field are nonzero and have three components in a Cartesian coordinate system with translational symmetry (i.e., one ignorable spatial coordinate). It is proved that (a) the solutions of the ideal MHD steady state equations with incompressible and homogeneous flows in the plane are also valid for equilibria with the axial velocity component being a free flux function and the axial magnetic field component being a constant, (b) the conditions of linearized Lyapunov stability for these MHD flows in the planar case (in which the fields have only two components) are also valid for symmetric equilibria that have a nonplanar velocity field component as well as a nonplanar magnetic field component. On using the method of convexity estimates, nonlinear stability conditions are established. [ABSTRACT FROM AUTHOR]

The magnetohydrodynamic (MHD) plasma model, as modified by the Hall effect, is given a Hamiltonian formulation and its stability properties are studied. It is found that, in most cases, a stable MHD plasma remains stable after the addition of the Hall effect. The most notable exceptions are when the pressure profile decreases with increasing density or when the entropy increases with density. The Hamiltonian structure of the equations enables the derivation of bounds that restrict the location of eigenfrequencies in the complex plane in some cases. The phenomenon of overstability, whereby the real part of a marginally stable eigenfrequency does not vanish, appears to be typical. [ABSTRACT FROM AUTHOR]