Equilibrium Of Toroidal Plasma Research Articles

FreeGSNKE: A Python-based dynamic free-boundary toroidal plasma equilibrium solver

We present a Python-based numerical solver for the two-dimensional dynamic plasma equilibrium problem. We model the time evolution of toroidally symmetric free-boundary tokamak plasma equilibria in the presence of the non-linear magnetohydrodynamic coupling with both currents in the “active” poloidal field coils, with assigned applied voltages, and eddy currents in the tokamak passive structures. FreeGSNKE (FreeGS Newton–Krylov Evolutive) builds and expands on the framework provided by the Python package FreeGS (Free boundary Grad–Shafranov). FreeGS solves the static free-boundary Grad–Shafranov (GS) problem, discretized in space using finite differences, by means of Picard iterations. FreeGSNKE introduces: (i) a solver for the static free-boundary GS problem based on the Newton–Krylov (NK) method, with improved stability and convergence properties; (ii) a solver for the linearized dynamic plasma equilibrium problem; and (iii) a solver for the non-linear dynamic problem, based on the NK method. We propose a novel “staggered” solution strategy for the non-linear problem, in which we make use of a set of equivalent formulations of the non-linear dynamic problem we derive. The alternation of NK solution steps in the currents and in the plasma flux lends this strategy an increased resilience to co-linearity and stagnation problems, resulting in favorable convergence properties. FreeGSNKE can be used for any user-defined tokamak geometry and coil configuration. FreeGSNKE's flexibility and ease of use make it a suitably robust control-oriented simulator of plasma magnetic equilibria. FreeGSNKE is entirely written in Python and easily interfaced with Python libraries, which facilitates machine learning based approaches to plasma control.

Equilibria as boundary value problems under Lie transformations

The use of transformations recently gained attention in obtaining invariant solutions to the equilibrium problem of plasma physics. In all of the cases considered, the new solutions were related to a (Generalized) Grad–Shafranov equation. In the same context, the present study focuses on the issue of an axisymmetric, toroidal plasma equilibrium as a boundary value problem associated with new solutions obtained by means of Lie group transformations. It appears that in all the cases examined, only a single infinitesimal generator of the symmetry group permits closed boundary that remains invariant under the transformation. The respective equilibrium, in addition to a peculiar axisymmetric magnetically confined plasma with current hole reaching the axis of symmetry, describes a planet's magnetosphere for low heights.

Effect of runaway electrons on tearing mode stability: with or without favorable curvature stabilization

An MHD-RE hybrid model is developed and implemented into the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681), where the runaway electrons (REs) are treated as a fluid species. The model is utilized to study the tearing mode (TM) stability in a tokamak plasma, with the numerical results compared to that of the analytic theory. The RE effect on the n = 1 TM is systematically investigated in the absence or presence of the favorable average curvature stabilization (the GGJ-effect). Without the GGJ-effect, for a force-free toroidal plasma equilibrium, REs are found to destabilize the TM and induce a finite real frequency to the mode (in a plasma with vanishing equilibrium flow). The RE contribution is found to significantly enhance the toroidal mode coupling. In the presence of the GGJ-effect, for a plasma with finite equilibrium pressure, the MARS-F hybrid model finds that REs destroy the symmetry in the TM spectrum and introduce extra branch of the TM solution due to the eigenvalue bifurcation. This happens because REs lead to coupling to a stable fluid TM mode and destabilize this mode. Despite complicated mode spectrum introduced by REs, we find that the latter play an overall destabilizing role for the TM also in the presence of the GGJ-effect. The MARS-F computed eigenvalues are well fitted with the analytic dispersion relation derived by Liu et al (2020 Phys. Plasmas 27 092507) for the force-free equilibrium. Less satisfactory is the fitting result with a TM dispersion relation that includes both the GGJ-effect and the RE contribution, suggesting further improvement of the proposed dispersion relation.

Axisymmetric equilibria with pressure anisotropy and plasma flow

A generalised Grad–Shafranov equation that governs the equilibrium of an axisymmetric toroidal plasma with anisotropic pressure and incompressible flow of arbitrary direction is derived. This equation includes six free surface functions and recovers known Grad–Shafranov-like equations in the literature as well as the usual static, isotropic one. The form of the generalised equation indicates that pressure anisotropy and flow act additively on equilibrium. In addition, two sets of analytical solutions, an extended Solovev one with a plasma reaching the separatrix and an extended Hernegger–Maschke one for a plasma surrounded by a fixed boundary possessing an X-point, are constructed, particularly in relevance to the ITER and NSTX tokamaks. Furthermore, the impacts both of pressure anisotropy and plasma flow on these equilibria are examined. It turns out that depending on the maximum value and the shape of an anisotropy function, the anisotropy can act either paramagnetically or diamagnetically. Also, in most of the cases considered both the anisotropy and the flow have stronger effects on NSTX equilibria than on ITER ones.

New regime of low ion collisionality in the neoclassical equilibrium of tokamak plasmas

The neoclassical description of an axisymmetric toroidal plasma equilibrium is formulated for an unconventionally low ordering of the collisionality that suits realistic thermonuclear fusion conditions. This requires a drift-kinetic analysis to the second order of the ion Larmor radius, which yields a new contribution to the leading solution for the non-Maxwellian part of the ion distribution function if the equilibrium geometry is not up-down symmetric. An explicit geometrical factor weighs this second Larmor-radius order, low-collisionality effect that modifies the neoclassical ion parallel flow, and the ion contribution to the bootstrap current.

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications

We present a comprehensive survey of various computational methods in CEDRES++ (Couplage Equilibre Diffusion Résistive pour l'Etude des Scénarios) for finding equilibria of toroidal plasma. Our focus is on free-boundary plasma equilibria, where either poloidal field coil currents or the temporal evolution of voltages in poloidal field circuit systems are given data. Centered around a piecewise linear finite element representation of the poloidal flux map, our approach allows in large parts the use of established numerical schemes. The coupling of a finite element method and a boundary element method gives consistent numerical solutions for equilibrium problems in unbounded domains. We formulate a new Newton method for the discretized nonlinear problem to tackle the various nonlinearities, including the free plasma boundary. The Newton method guarantees fast convergence and is the main building block for the inverse equilibrium problems that we can handle in CEDRES++ as well. The inverse problems aim at finding either poloidal field coil currents that ensure a desired shape and position of the plasma or at finding the evolution of the voltages in the poloidal field circuit systems that ensure a prescribed evolution of the plasma shape and position. We provide equilibrium simulations for the tokamaks ITER and WEST to illustrate the performance of CEDRES++ and its application areas.

Free-boundary perturbed MHD equilibria

The concept of perturbed ideal MHD equilibria [Boozer A H and Nuhrenberg C 2006 Phys. Plasmas 13 102501] is employed to study the influence of external error-fields and of small plasma-pressure changes on toroidal plasma equilibria. In tokamak and stellarator free-boundary calculations, benchmarks were successful of the perturbed-equilibrium version of the CAS3D stability code [Nührenberg C et al. 2009 Phys. Rev. Lett. 102 235001] with the ideal MHD equilibrium code NEMEC [Hirshman S P et al. 1986 Comput. Phys. Commun. 43 143].

RETRACTED ARTICLE: Multipole Moments Based Study on Determination of Toroidal Plasma Equilibrium Position and Shift

The multipole moments based technique studied for the determination of toroidal plasma column shift. First, we presented analytical details for using this technique. Then, principle of two different models based on this technique for design and fabrication of a modified Rogowski coil (MRC) and Saddle Sine coil (SSC) will be presented. Because of continuous measurements of magnetic field distribution around the toroidal plasma using the MRC and SSC, this technique is the good method for the determination of toroidal plasma column shift.

Construction of substantially different solutions of an inverse problem for a toroidal plasma equilibrium equation

The paper is devoted to the reconstruction of the current density in toroidal plasma, using experimentally measured data. The toroidal current density is characterized by two functions in the right-hand side of the Grad-Shafranov equation which, together with the poloidal flux, is to be deter-mined. The question about the uniqueness of the solution of the inverse problem was usually not addressed in the numerical methods developed over the past few decades. However, the theoretical study of this question for simplified models showed the possibility of the existence of substantially different solutions. For a correct understanding of the physical properties of a pulse it is necessary to analyze all possible solutions of the inverse problem in its physically correct formulation. This formulation is presented in the paper. A new numerical method for determining all substantially different solutions of the inverse problem is proposed. Examples of the existence of such solutions are constructed for plasma parameters which are close to experimental.

Toroidal equilibria in spherical coordinates

The standard Grad–Shafranov equation for axisymmetric toroidal plasma equilibrium is customary expressed in cylindrical coordinates with toroidal contours, and through which benchmark equilibria are solved. An alternative approach to cast the Grad–Shafranov equation in spherical coordinates is presented. This equation, in spherical coordinates, is examined for toroidal solutions to describe low β Solovev and high β plasma equilibria in terms of elementary functions.

Some theoretical problems of the toroidal plasma equilibrium

Two problems of toroidal magnetic plasma confinement systems are considered. The first one is connected with advanced helical systems. The peculiarity of geometry of these stellarator systems with the improved confinement is discussed. Some additional ways for further optimization are marked too. The second problem is concerned with the island structure in toroidal systems. From the very beginning, the theoretical research on plasma equilibrium in stellarators was based mainly on the assumption that the toroidal magnetic surfaces are nested. It means that there is a unique magnetic axis which serves as the edge of the azimuthal coordinate surface θ = const. To avoid the difficulties with such a coordinate in the case of many magnetic axes that result from islands, the tokamak-like representation of the magnetic field B which does not use the coordinate θ is proposed.

Finite beta effects in quasihelical stellarators

An analytic expansion about a magnetic axis is applied to investigate finite beta effects such as bootstrap current on toroidal plasma equilibria which approximate quasihelical symmetry (QHS). It is found that quasihelically symmetric configurations can be found through the second order in terms of a helical inverse aspect ratio ε, even with the inclusion of a bootstrap current on the magnetic axis. Permitting the on-axis rotational transform to decrease as the on-axis bootstrap current is increased has little effect on the shape of the QHS flux surfaces. However, shifting the position of the magnetic axis in order to maintain the on-axis transform causes a stronger change in the QHS flux surface shape, especially axis shifts in the major radial direction.

Bound for the pressure integral in a toroidal-plasma equilibrium.

A bound for the pressure integral in a toroidal-plasma equilibrium has been studied, invoking an a priori estimate for a solution of the Grad-Shafranov equation. Earlier theories had to use approximate equilibrium solutions to calculate the pressure integral (poloidal beta ratio ${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$), and, hence, fell short of being rigorous estimates. The present theory considers exact solutions and gives a rigorous bound for ${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$.

Two-dimensional analytic theory for toroidal Alfvén eigenmodes

A two-dimensional analysis of the toroidal Alfvén eigenmodes (TAE) is presented, based on an integrodifferential equation describing the shear Alfvén perturbation of a toroidal plasma equilibrium in terms of coupling among the toroidal Alfvén continua with the usual gap structure. Using a method similar to the Van Kampen–Case analysis for the Vlasov equation, exact analytic expressions are derived for the dispersion function and the two-dimensional eigenmode structure. The dispersion function is expressed in terms of Cauchy-type integrals, which explicitly expresses the global character of TAE modes and facilitates the calculation of their damping. The continuum-damped TAE modes are shown to be, in general, not true eigenmodes of the toroidal plasma equilibrium, but rather resonances corresponding to zeros of the analytic continuation of the dispersion function onto unphysical sheets of its Riemann surface. Approximate but explicit expressions for the dispersion relation and the eigenfunction are also obtained in the limit of vanishing inverse aspect ratio.

Formation of the shear layer in toroidal edge plasma

The equilibrium and stability of axisymmetric toroidal edge plasma in the presence of an anomalous ballooninglike transport is studied. The equilibrium density evolves so that the transport rate, D⊥/Ln2 equals the sound transit time, cs/πqR. As a result, confinement is coupled to the local sound velocity such that a reduced sound velocity (larger ion mass) leads to improved confinement. A result of the large asymmetric transport is the generation of nearly sonic parallel flows which allow the system to be unstable to the Stringer spin-up instability. For weak magnetic pumping, a large poloidal rotation of order the poloidal sound speed (acs/qR) is driven, forming a localized shear layer. The direction of rotation corresponds to a negative radial electric field in the closed flux region. For strong magnetic pumping the rotation is equal in magnitude and opposite in direction to the ion diamagnetic drift, the neoclassical result. Particle sources localized in poloidal angle can strongly impact the rotation. A source placed between the bottom and the outside of the torus enhances the rotation while the same source placed between the top and the inside retards the rotation. However, a large enough source at the latter location will drive the rotation in the other direction, reversing the sign of the electric field. The implications of the results for understanding Ohmic, L-mode, and H-mode discharges are discussed.

Magnetic field strength of toroidal plasma equilibria

The confinement and stability properties of toroidal plasma equilibria are largely determined by the strength of the magnetic field expressed in terms of Boozer coordinates. Constraints due to the toroidicity of the configuration and the divergence-free property of the magnetic field limit the freedom in the Taylor series expansion of the magnetic field strength. Such field strength constraints limit the variety of desirable three-dimensional plasma equilibria that are available to the fusion program.

High-Current Runaway Electron Beam in a Tokamak Plasma

An equilibrium of toroidal plasma with a large electron-beam current has been realized using the runaway effect in a tokamak. Reproducible runaway-mode discharges are obtained with pure hydrogen gas by the help of intense titanium flashing which results in a low electron density. The beam current is estimated to be more than a half of the total toroidal current. The equilibrium of this discharge is maintained by a strong vertical field because the beam pressure gives rise to an additional increase in Shafranov Λ. The beam pressure is estimated to be more than 80% of the total pressure. The kinetic energy and the spatial distributions of beam electrons are studied by seeing X-ray emission from a tungsten wire inserted into the plasma. The increase of Shafranov Λ due to beam pressure is enhanced by puffing gas into the discharge.

Toroidal plasma equilibrium with arbitrary current distribution

A new System of co-ordinates is found and a method developed to determine the toroidal equilibrium of plasmas with arbitrary current distribution and plasma cross-section. The method depends on knowledge of the equilibrium of a straight plasma column of similar cross-section and similar current distribution. A large aspect ratio is assumed. By successive approximations, better solutions can be obtained. An explicit formula is presented for the poloidal flux of a nearly circular plasma. This can be written in terms of a function related to the asymmetry of the poloidal field due to toroidality. The method works provided that there is only one magnetic axis.

On the linear stability of a toroidal plasma with resistivity viscosity and Hall current

The purpose of this paper is to study several mathematical questions regarding the linear stability analysis of a toroidal plasma equilibrium on the basis of the magnetohydrodymic equations with finite resistivity, viscosity, and Hall current. We show that the spectrum of the operator in an abstract evolution equation consists of at most finitely many unstable eigenvalues and infinitely many stable eigenvalues, and the set of the eigenfunctions is complete. The existence and uniqueness of a solution to the initial-boundary value problem of the linearized equations is proved. Finally the existence of a toroidal plasma equilibrium subject to a prescribed velocity with small norm on the boundary is established and its linear stability is proved.

A Dissipative Model of Plasma Equilibrium in Toroidal Systems

In order to describe a steady-state plasma equilibrium in tokamaks, stellarators or other non-axisymmetric configurations, the model of ideal MHD with isotropic plasma pressure is widely used: 0 = - ∇p + j x B . The ideal MHD-model of a toroidal plasma equilibrium requires the existence of closed magnetic surfaces. Several numerical codes have been developed in the past to solve the three-dimensional equilibrium problem, but so far no existence theorem for a solution has been proved. Another difficulty is the formation of magnetic islands and field line ergodisation, which can only be described in terms of ideal MHD if the plasma pressure is constant in the ergodic region. In order to describe the formation of magnetic islands and ergodisation of surfaces properly, additional dissipative terms have to be incorporated to allow decoupling of the plasma and magnetic field. In a collisional plasma, viscosity and inelastic collisions introduce such dissipative processes. In the model used a friction term proportional to the velocity v of the plasma is included. Such a term originates from charge exchange interaction of the plasma with a neutral background. With these modifications, the equilibrium problem reduces to a set of quasilinear elliptic equations for the pressure, the electric potential and the magnetic field. The paper deals with an existence theorem based on the Fixed-Point method of Schauder. It can be shown that a self-consistent and unique equilibrium exists if the friction term is large and the plasma pressure is sufficiently low. The essential role of the dissipative terms is to remove the singularities of the ideal MHD model on rational magnetic surfaces. The problem has a strong similarity to Bénard cell convection, and consequently a similar behaviour such as bifurcation and exchange of stability is expected.