Ideal Magnetohydrodynamic Equilibria Research Articles

Asymptotic quasisymmetric high-beta three-dimensional magnetohydrodynamic equilibria near axisymmetry

Quasisymmetry (QS), a hidden symmetry of the magnetic field strength, is known to support nested flux surfaces and provide superior particle confinement in stellarators. In this work, we study the ideal magnetohydrodynamic (MHD) equilibrium and stability of high-beta plasma in a large-aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies the problem of three-dimensional quasisymmetric MHD equilibria, which can be reduced to a standard elliptic Grad–Shafranov equation for the flux function. We show that any large-aspect-ratio tokamak, deformed periodically in the vertical direction, is a stellarator with approximate volumetric QS. We discuss exact analytical solutions and numerical benchmarks. Finally, we discuss the ideal ballooning and interchange stability of some of our equilibrium configurations.

Numerical minimization of neoclassical poloidal viscosity for supersonic equilibria in tokamak geometry

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.

Understanding trade-offs in stellarator design with multi-objective optimization

In designing stellarators, any design decision ultimately comes with a trade-off. Improvements in particle confinement, for instance, may increase the burden on engineers to build more complex coils, and the tightening of financial constraints may simplify the design and worsen some aspects of transport. Understanding trade-offs in stellarator designs is critical in designing high performance devices that satisfy the multitude of physical, engineering and financial criteria. In this study, we show how multi-objective optimization (MOO) can be used to investigate trade-offs and develop insight into the role of design parameters. We discuss the basics of MOO, as well as practical solution methods for solving MOO problems. We apply these methods to bring insight into the selection of two common design parameters: the aspect ratio of an ideal magnetohydrodynamic equilibrium and the total length of the electromagnetic coils.

Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch–Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch–Schlüter current density scaling as , where is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to , the scaling relation will lead to a paradox that the Pfirsch–Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm–Kulsrud–Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch–Schlüter current density but also the diamagnetic current density are divergent as . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with . Consequently, the singular current density , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

Erratum: “Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets” [Phys. Plasmas 22, 090704 (2015)

Erratum: “Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets” [Phys. Plasmas <b>22</b>, 090704 (2015)

Time-dependent relaxed magnetohydrodynamics: Inclusion of cross helicity constraint using phase-space action

A phase-space version of the ideal magnetohydrodynamic (MHD) Lagrangian is derived from first principles and shown to give a relabeling transformation when a cross-helicity constraint is added in Hamilton's Action Principle. A new formulation of time-dependent “relaxed” magnetohydrodynamics is derived using microscopic conservation of mass and macroscopic constraints on total magnetic helicity, cross helicity, and entropy under variations of density, pressure, fluid velocity, and magnetic vector potential. This gives Euler–Lagrange equations consistent with previous work on both ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether E+u×B=0 should be a constraint is discussed.

The design and optimisation of tokamak poloidal field systems in the BLUEPRINT framework

Abstract As part of our aim to develop a reactor design framework, we present here our approach to the optimisation of plasma equilibria and tokamak poloidal field systems. A 2-D, free boundary, ideal magneto-hydrodynamic plasma equilibrium solver is described. We use it within the BLUEPRINT reactor design framework to design plasma equilibria and poloidal field coilsets for future fusion reactors in an integrated way. Plasma shape parameters (R0, A, δ, κ) and integral parameters (Ip, βp, li) are controlled for. Constrained optimisation procedures for the positions and currents of the poloidal field coils are presented; these account for current, field, force, and positional constraints on the coils. A pulse length constraint is incorporated in the optimisation procedure in the case of pulsed reactors. We demonstrate our approach on three different reactor concepts: single-null, double-null and negative triangularity tokamaks.

Elimination of current sheets at resonances in three-dimensional toroidal ideal-magnetohydrodynamic equilibria

On the basis of the ideal-magnetohydrodynamic equilibrium code VMEC (Hirshman et al 1986 Comput. Phys. Commun. 43 143) supplemented by transformation of its results into magnetic coordinates, it is shown how current sheets, radially widened by the finite grid size, at rational values of rotational transform in 3d toroidal equilibria can be removed by variation of the equilibrium geometry.

Comparative collisionless alpha particle confinement in stellarator reactors with the XGC gyrokinetic code

Whole volume simulations of stellarators are necessary to address a number of important physics and engineering issues, including turbulent transport prediction and optimising divertor operation. In this work, advances in the development of a whole volume stellarator capability for the global gyrokinetic particle-in-cell code XGC are detailed. A 3D interpolation of the equilibrium magnetic field to the last close flux surface, calculated using the VMEC ideal magnetohydrodynamic equilibrium code, has been implemented. Orbit tracing with the XGC code is verified for the Wendelstein 7-X stellarator by comparison with the BEAMS3D and EUTERPE codes. The XGC stellarator version has then been used to consider collisionless alpha particle confinement in representative quasi-isodynamic and quasi-axisymmetric designs. It has been shown that, at least without further optimisation for energetic particle confinement, the Wendelstein 7-X-like quasi-isodynamic reactor design exhibits better alpha particle confinement.

Low-shear three-dimensional equilibria in a periodic cylinder

We carry out expansions of non-symmetric toroidal ideal magnetohydrodynamic (MHD) equilibria with nested flux surfaces about a periodic cylinder, in which physical quantities are periodic of period$2\unicode[STIX]{x03C0}$in the cylindrical angle$\unicode[STIX]{x1D703}$and$z$. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data are given on the cylindrical flux surface$r=1$. Furthermore, we assume that the magnetic field lines are closed on the lowest-order flux surface, and the magnetic shear is relatively small. We extend earlier work in a flat torus by Weitzner (Phys. Plasmas, vol. 23, 2016, 062512) and demonstrate that a power series expansion can be carried out to all orders using magnetic flux as an expansion parameter. The cylindrical metric introduces certain new features to the expansions compared to the flat torus. However, the basic methodology of dealing with resonance singularities remains the same. The results, even though lacking convergence proofs, once again support the possibility of smooth, low-shear non-symmetric toroidal MHD equilibria.

Dynamics of ideal modes and subsequent ELM crashes in 3D tokamak geometry from external magnetic perturbations

The impact of three-dimensional (3D) tokamak geometry from external magnetic perturbations on edge instabilities has been examined in high confinement mode plasmas with edge localised modes (ELMs) in ASDEX Upgrade. The 3D geometry has been probed using rigidly rotating MP fields. The measured distortions of the plasma boundary are compared to single-fluid ideal magnetohydrodynamic (MHD) equilibria using VMEC and MARS-F applying ideal and resistive MHD, whereas VMEC uses only ideal MHD. Both codes accurately reproduce the measured radial displacements of the edge density and temperature profiles in amplitude, toroidal phase and their dependence on the applied poloidal mode spectrum.The induced 3D geometry distorts the local magnetic shear, which locally reduces the stabilising effect from field-line bending at certain most unstable field lines. Around these field lines, we observe additional stable ideal MHD modes with clear ballooning structure in-between ELMs. After their immediate appearance, they saturate and then grow on timescales of the pedestal pressure recovery. The subsequent ELMs show strongly localised magnetic perturbations of the initial crash and accompanied energetic electrons around the same most unstable field lines. These are strong signatures that filaments at the ELM onset preferentially erupt on these most unstable (‘bad’) field lines with their unfavourable 3D geometry where preceding ballooning modes are observed.

Field-Line Localized Destabilization of Ballooning Modes in Three-Dimensional Tokamaks.

Field-line localized ballooning modes have been observed at the edge of high confinement mode plasmas in ASDEX Upgrade with rotating 3D perturbations induced by an externally applied n=2 error field and during a moderate level of edge localized mode mitigation. The observed ballooning modes are localized to the field lines which experience one of the two zero crossings of the radial flux surface displacement during one rotation period. The localization of the ballooning modes agrees very well with the localization of the largest growth rates from infinite-n ideal ballooning stability calculations using a realistic 3D ideal magnetohydrodynamic equilibrium. This analysis predicts a lower stability with respect to the axisymmetric case. The primary mechanism for the local lower stability is the 3D distortion of the local magnetic shear.

Three dimensional boundary displacement due to stable ideal kink modes excited by external n = 2 magnetic perturbations

In low-collisionality scenarios exhibiting mitigation of edge localized mode (ELMs), stable ideal kink modes at the edge are excited by externally applied magnetic perturbation (MP)-fields. In ASDEX Upgrade these modes can cause three-dimensional (3D) boundary displacements up to the centimeter range. These displacements have been measured using toroidally localized high resolution diagnostics and rigidly rotating MP-fields with various applied poloidal mode spectra. These measurements are compared to non-linear 3D ideal magnetohydrodynamics (MHD) equilibria calculated by VMEC. Comprehensive comparisons have been conducted, which consider for instance plasma movements due to the position control system, attenuation due to internal conductors and changes in the edge pressure profiles.VMEC accurately reproduces the amplitude of the displacement and its dependencies on the applied poloidal mode spectra. Quantitative agreement is found around the low field side (LFS) midplane. The response at the plasma top is qualitatively compared. The measured and predicted displacements at the plasma top maximize when the applied spectra is optimized for ELM-mitigation. The predictions from the vacuum modeling generally fails to describe the displacement at the LFS midplane as well as at the plasma top. When the applied mode spectra is set to maximize the displacement, VMEC and the measurements clearly surpass the predictions from the vacuum modeling by a factor of four. Minor disagreements between VMEC and the measurements are discussed. This study underlines the importance of the stable ideal kink modes at the edge for the 3D boundary displacement in scenarios relevant for ELM-mitigation.

Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows

The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.

Numerical solution of ideal MHD equilibrium via radial basis functions collocation and moving least squares approximation methods

In this study, two different meshfree methods consisting of the Radial Basis Functions (RBFs) and the Moving Least Square Method (MLS) are applied to solve the Grad–Shafranov (GS) equation for the axisymmetric equilibrium of plasma in the tokamak. The validity and the effectiveness of the proposed schemes are studied by several test problems through absolute and Root Mean Squared (RMS) errors. Although, during the past few years, a meshfree method is normally applied in magnetohydrodynamic (MHD) studies to the numerical solution of partial differential equations (PDEs) but to the best of our knowledge, its application in MHD equilibrium of the tokamak plasma investigations is rare. The future more extensive studies regarding this numerical method would definitely have a significant impact on improving tokamak numerical tools.

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calculations. This rescues the possibility of constructing MHD equilibria with current sheets and continuous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface.

Conservation properties and potential systems of vorticity-type equations

Partial differential equations of the form \documentclass[12pt]{minimal}\begin{document}$\mathop {\hbox{\rm div}}{{\bm N}} =0$\end{document}divN=0, \documentclass[12pt]{minimal}\begin{document}${{\bm N}}_t + \mathop {\hbox{\rm curl}}{{\bm M}}=0$\end{document}Nt+curlM=0 involving two vector functions in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 depending on t, x, y, z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4(t,x,y,z)$\end{document}R4(t,x,y,z). The corresponding potential system has a clear physical meaning. For the Maxwell's equations, it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.

Ideal magnetohydrodynamic equilibrium in a non-symmetric topological torus

An alternative representation of an ideal magnetohydrodynamic equilibrium is developed. The representation is a variation of one given by A. Salat, Phys. Plasmas 2, 1652 (1995). The system of equations is used to study the possibility of non-symmetric equilibria in a topological torus, here an approximate rectangular parallelopiped, with periodicity in two of the three rectangular coordinates. An expansion is carried out in the deviation of pressure surfaces from planes. Resonances are manifest in the process. Nonetheless, provided the magnetic shear is small, it is shown that it is possible to select the magnetic fields and flux surfaces in such a manner that no singularities appear on resonant surfaces. One boundary surface of the parallelopiped is not arbitrary but is dependent on the equilibrium in question. A comparison of the solution sets of axisymmetric and non-axisymmetric equilibria suggests that the latter have a wider class of possible boundary shapes but more restrictive rotational transform profiles. No proof of convergence of the series is given.

Non-axisymmetric magneto- hydrodynamic equilibrium in the presence of internal magnetic islands and external magnetic perturbation coils

Non-axisymmetric equilibria arise in DIII-D discharges that are subjected to magnetic perturbation by 3D magnetic coils. But, 3D shaping of the entire plasma, including the boundary, also occurs in the rotating fluid frame of saturated internal magnetic islands (Tobias et al 2013 Plasma Phys. Control. Fusion 55 095006). This is advantageous since internal islands and kink responses that rotate near the fluid velocity of the plasma are easily diagnosed, while static perturbations in the laboratory frame are not. The helicity of the perturbed shape is the same in both rotational frames of reference, making one mode a diagnostic proxy for the other and allowing internal modes to be used as a source of data for comparison to models typically applied to understanding the effect of static perturbations. Discrepancies with ideal magneto-hydrodynamic equilibrium obtained by the IPEC (Park et al 2007 Phys. Plasmas 14 052110) method brings attention to the treatment of plasma displacements near rational surfaces and their relationship to the accessibility of equilibrium states.

SIESTA: A scalable iterative equilibrium solver for toroidal applications

A new solver for rapidly obtaining magnetohydrodynamic (MHD) equilibria in toroidal systems in the presence of islands and stochastic regions is described. It is based on the Kulsrud-Kruskal MHD energy minimization principle. To carry out this minimization, small displacements are made around a convenient set of curvilinear coordinates obtained from a nearby three-dimensional equilibrium that assumes nested surfaces. Because the changes of the magnetic fields and pressure are small, corresponding to small changes in the initial magnetic and kinetic energies, solutions for the linearized perturbations can be used to rapidly and iteratively find lower energy states with magnetic islands. A physics-based preconditioner is developed to accelerate the convergence of the iterative procedure to obtain an ideal MHD equilibrium with broken magnetic surfaces (islands).