Axisymmetric toroidal equilibria with toroidal and poloidal rotations are solved with a specific set of source functions. The two independent solutions are associated to L and H modes. The L/H transition is regarded as a bifurcation from one equilibrium configuration to another, under strong external heating and pellet injection to shape temperature and density profiles. Because of the steep edge gradient of the H solution, large static radial electric field, zonal flow, and improved confinements, come as consequences, not causes, of the H mode.
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.
A simulated annealing (SA) relaxation method is used for calculation of high-beta reduced magnetohydrodynamics (MHD) equilibria in toroidal geometry. The SA method, based on artificial dynamics derived from the MHD Hamiltonian structure, is used to calculate equilibria of large-aspect-ratio and circular-cross-section tokamaks as well as toroidally averaged stellarators. Tokamak equilibria including incompressible poloidal rotations are obtained and the Shafranov shift is seen to increase nearly quadratically in the rotation speed. A mapping procedure between non-rotating and poloidally rotatin…
An analysis of axisymmetric equilibria with arbitrary incompressible flow and finite resistivity is presented. It is shown that with large aspect ratio approximation or vanishing poloidal current, a uniform conductivity profile is consistent with equilibrium flows. Also a comment made on coexistence of both toroidal and poloidal flows in an axisymmetric field-reversed configuration.
By choosing appropriate deformed Maxwellian ion and electron distribution functions depending on the two particle constants of motion, i.e. the energy and toroidal angular momentum, we reduce the Vlasov axisymmetric equilibrium problem for quasineutral plasmas to a transcendental Grad-Shafranov-like equation. This equation is then solved numerically under the Dirichlet boundary condition for an analytically prescribed boundary possessing a lower X-point to construct tokamak equilibria with toroidal sheared ion flow and anisotropic pressure. Depending on the deformation of the distribution func…
We obtain analytic solutions of a generalised Grad-Shafranov equation describing steady states with incompressible plasma flow of arbitrary direction, toroidal current reversal and either nested or non-nested magnetic surfaces. It turns out that the component of the flow velocity non parallel to the magnetic field can result in normal equilibria with central current-reversal, i.e. equilibria with nested magnetic surfaces and monotonically varying pressure profiles.
We present ECOM (Equilibrium solver via COnformal Mapping), a fast and accurate fixed boundary solver for toroidally axisymmetric magnetohydrodynamic equilibria with or without a toroidal flow. ECOM combines conformal mapping and Fourier and integral equation methods on the unit disk to achieve exponential convergence for the poloidal flux function as well as its first and second partial derivatives. As a consequence of its high order accuracy, for dense grids and tokamak-like elongations ECOM computes key quantities such as the safety factor and the magnetic shear with higher accuracy than th…
Axisymmetric equilibria with incompressible flows of arbitrary direction are studied in the framework of magnetohydrodynamics under a variety of physically relevant side conditions. To this end a set of pertinent non-linear ODEs are transformed to quasilinear ones and the respective initial value problem is solved numerically with appropriately determined initial values near the magnetic axis. Several equilibria are then constructed surface by surface. The non field aligned flow results in novel configurations with a single magnetic axis, toroidal shell configurations in which the plasma is co…
Resistive steady states in toroidal magnetohydrodynamics (MHD), where Ohm's law must be taken into account, differ considerably from ideal ones. Only for special (and probably unphysical) resistivity profiles can the Lorentz force, in the static force-balance equation, be expressed as the gradient of a scalar and thus cancel the gradient of a scalar pressure. In general, the Lorentz force has a curl directed so as to generate toroidal vorticity. Here, we calculate, for a collisional, highly viscous magnetofluid, the flows that are required for an axisymmetric toroidal steady state, assuming un…
The equilibrium of a resistive axisymmetric plasma with purely toroidal flow surrounded by a conductor is investigated within the framework of the nonlinear magnetohydrodynamic theory. It is proved that a) the poloidal current density vanishes and b) apart from an idealized case the pressure profile should vanish on the plasma boundary. For the cases of isothermal magnetic surfaces, isentropic magnetic surfaces and magnetic surfaces with constant density, the equilibrium states obey to an elliptic partial differential equation for the poloidal magnetic flux function, which is identical in form…
The equilibrium of an axisymmetric magnetically confined plasma with anisotropic resistivity and toroidal flow is investigated in the framework of magnetohydrodynamics (MHD). The stationary states are determined by an elliptic differential equation for the poloidal magnetic flux function, a Bernoulli equation for the pressure and two relations for the resistivity components parallel and perpendicular to the magnetic field. The equilibrium characteristics of a tokamak with rectangular cross-section are studied by means of eigenfunctions in connection with exact solutions for the cases of "compr…
It is demonstrated that finite-pressure, approximately quasi-axisymmetric stellarator equilibria can be directly constructed (without numerical optimization) via perturbations of given axisymmetric equilibria. The size of such perturbations is measured in two ways, via the fractional external rotation and, alternatively, via the relative magnetic field strength, i.e. the average size of the perturbed magnetic field, divided by the unperturbed field strength. It is found that significant fractional external rotational transform can be generated by quasi-axisymmetric perturbations, with a simila…
We analyze an axisymmetric equilibrium of a plasma endowed with toroidal and poloidal velocity fields, with the aim to characterize the influence of the global motion on the morphology of the magnetic confinement. We construct our configuration assuming that the poloidal velocity field is aligned with the poloidal magnetic field lines and, furthermore, we require that the plasma mass density depend on the magnetic flux function (or equivalently, that the plasma fluid be incompressible). We then derive a sort of Grad-Shafranov equation for such an equilibrium and implement it to tokamak relevan…
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-Masch…
Vlasov equilibria of axisymmetric plasmas with vacuum toroidal magnetic field can be reduced, up to a selection of ions and electrons distributions functions, to a Grad-Shafranov-like equation. Quasineutrality narrow the choice of the distributions functions. In contrast to two-dimensional translationally symmetric equilibria whose electron distribution function consists of a displaced Maxwellian, the toroidal equilibria need deformed Maxwellians. In order to be able to carry through the calculations, this deformation is produced by means of either a Heaviside step function or an exponential f…
Two-dimensional Maxwell-Vlasov equilibria with finite electric fields, axial ("toroidal") plasma flow and isotropic pressure are constructed in plane geometry by using the quasineutrality condition to express the electrostatic potential in terms of the vector potential. Then for Harris-type distribution functions, Ampere's equation becomes of Liouville type and can be solved analytically. As an example, a periodic "cat-eyes" steady state consisting of a row of magnetic islands is presented. The method can be extended to (toroidal) axisymmetric equilibria.
Rotation of tokamak-plasmas, not at the mechanical equilibrium, is investigated utilizing a theorem of thermodynamics, established by Prigogine. This theorem, suitably applied to toroidally confined plasmas, suggests that the global barycentric rotations of the plasma, in the toroidal and poloidal directions, are pure reversible processes. In case of negligible viscosity and by supposing the validity of the balance equation for the internal forces, we show that the plasma, even not in the mechanical equilibrium, may freely rotate as a rigid body in the toroidal direction with an angular freque…
Analytic non-axisymmetric 3D MHD equilibria without stellarator symmetry
Google DeepMind（谷歌DeepMind）；Department of Mechanical and Aerospace Engineering, University of California San Diego（加州大学圣地亚哥分校机械与航空航天工程系）；University of Maryland, College Park（马里兰大学帕克学院）
It is shown that the magnetohydrodynamic equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function $ψ$ coupled with a Bernoulli type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation $Δ^\star ψ=V_c σ$. (Here, $Δ^\star$ is the Grad-Schlüter-Shafranov operator, $σ$ is the conductivity and $V_c$ is the constant toroidal-loop voltage divided by $2 π$)…
Investigation of Toroidal Rotation Effects on Spherical Torus Equilibria using the Fast Spectral Solver VEQ-R
Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams of the Ministry of Education, School of Physics, Dalian University of Technology（大连理工大学物理学院教育部激光、离子和电子束材料改性重点实验室）；ENN Science and Technology Development Co., Ltd.（新奥科技发展有限公司）
Seifert type magnetohydrostatic equilibria in rotationally symmetric toroidal domains
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究委员会数学科学研究所）；Faculty of Physics, University of Bucharest（布加勒斯特大学物理学院）
It is found that the ideal magnetohydrodynamic equilibrium of an axisymmetric gravitating magnetically confined plasma with incompressible flows is governed by a second-order elliptic differential equation for the poloidal magnetic flux function containing five flux functions coupled with a Poisson equation for the gravitation potential, and an algebraic relation for the pressure. This set of equations is amenable to analytic solutions. As an application, the magnetic-dipole static axisymmetric equilibria with vanishing poloidal plasma currents derived recently by Krasheninnikov, Catto, and Ha…
Two-fluid flowing equilibrium configurations of a helicity-driven spherical torus (HD-ST) are numerically determined by using the combination of the finite difference and the boundary element methods. It is found from the numerical results that electron fluids near the central conductor are tied to an external toroidal field and ion fluids are not. The magnetic configurations change from the high-$q$ HD-ST ($q$>1) with paramagnetic toroidal field and low-$β$ (volume average $β$ value, ($<β>\approx$ 2 %) through the helicity-driven spheromak and RFP to the ultra low-$q$ HD-ST (0<q<1) with diagm…
We present a general construction for exact analytic Taylor states in axisymmetric toroidal geometries. In this construction, the Taylor equilibria are fully determined by specifying the aspect ratio, elongation, and triangularity of the desired plasma geometry. For equilibria with a magnetic X-point, the location of the X-point must also be specified. The flexibility and simplicity of these solutions make them useful for verifying the accuracy of numerical solvers and for theoretical studies of Taylor states in laboratory experiments.
The presence of non-axisymmetric perturbations in an axisymmetric magnetic field equilibrium renders the Guiding Center (GC) particle motion non-integrable and may result in particle, energy and momentum redistribution, due to resonance mechanisms. We analyse these perturbations in terms of their spectrum, as observed by the particles in the frame of unperturbed GC motion. We calculate semi-analytically the exact locations and strength of resonant spectral components of multiple perturbations. The presented Orbital Spectrum Analysis (OSA) method is based on an exact Action-Angle transform that…
In this Master thesis we investigate the influence of pressure anisotropy and incompressible flow of arbitrary direction on the equilibrium properties of magnetically confined, axisymmetric toroidal plasmas. The main novel contribution is the derivation of a pertinent generalised Grad-Shafranov equation. 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…
This overview presents a tutorial introduction to the theory of magnetic plasma confinement in toroidal confinement systems with particular emphasis on axisymmetric equilibrium geometries, and tokamaks. The discussion covers three important aspects of plasma physics: Equilibrium, Stability, and Transport. The section on equilibrium will go through an introduction to ideal magnetohydrodynamics, curvilinear system of coordinates, flux coordinates, extensions to axisymmetric equilibrium, Grad-Shafranov Equation (GSE), Green's function formalism, as well as analytical and numerical solutions to GS…
The equations describing the continuous spectrum of shear Alfvén and ion sound waves propagating along magnetic field lines are introduced and solved in the ballooning space for general geometry in the ideal MHD limit. This approach is equivalent to earlier analyses by Chu et al. 1992 [Phys. Fluids B 4, 3713 (1992)] but the present formulation in the ballooning space allows to readily extend it to include gyrokinetic and three-dimensional equilibrium effects. In particular, following Chen and Zonca 2017 [Phys. Plasmas 24, 072511 (2017)], the MHD limit is adopted to illustrate the general metho…
In magnetic confinement fusion devices, the ratio of the plasma pressure to the magnetic field energy, $β$, can become sufficiently large that electromagnetic microinstabilities become unstable, driving turbulence that distorts or reconnects the equilibrium magnetic field. In this paper, a theory is proposed for electromagnetic, electron-driven linear instabilities that have current layers localised to mode-rational surfaces and binormal wavelengths comparable to the ion gyroradius. The model retains axisymmetric toroidal geometry with arbitrary shaping, and consists of orbit-averaged equation…
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 $2π$ in the cylindrical angle $θ$ and $z$. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data is 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 [Ph…
The usage of permanent magnets to shape the confining magnetic field of a stellarator has the potential to reduce or eliminate the need for non-planar coils. As a proof-of-concept for this idea, we have developed a procedure for designing an array of cubic permanent magnets that works in tandem with a set of toroidal-field coils to confine a stellarator plasma. All of the magnets in the design are constrained to have identical geometry and one of three polarization types in order to simplify fabrication while still producing sufficient field accuracy. We present some of the key steps leading t…
A theoretical treatment is presented of kinetic equilibria in accretion discs around compact objects, for cases where the plasma can be considered as collisionless. The plasma is assumed to be axi-symmetric and to be acted on by gravitational and electromagnetic fields; in this paper, the particular case is considered where the magnetic field admits a family of toroidal magnetic surfaces, which are locally mutually-nested and closed. It is pointed out that there exist asymptotic kinetic equilibria represented by generalized bi-Maxwellian distribution functions and characterized by primarily to…
The aim of the present work is to investigate tokamak equilibria with reversed magnetic shear and sheared flow, which may play a role in the formation of internal transport barriers (ITBs), within the framework of two-fluid model. The study is based on exact self-consistent solutions in cylindrical geometry by means of which the impact of the magnetic shear, s, and the "toroidal" (axial) and "poloidal" (azimuthal) ion velocity components on the radial electric field, its shear and the shear of the ExB velocity is examined. For a wide parametric regime of experimental concern it turns out that …
We identify and study new nonlinear axisymmetric equilibria with incompressible flow of arbitrary direction satisfying a generalized Grad Shafranov equation by extending the symmetry analysis presented in [G. Cicogna and F. Pegoraro, Phys. Plasmas Vol. 22, 022520 (2015)]. In particular, we construct a typical tokamak D-shaped equilibrium with peaked toroidal current density, monotonically varying safety factor and sheared electric field.
A new force balance model for the EFIT magnetohydrodynamic equilibrium technique for tokamaks is presented which includes the full toroidal flow and anisotropy changes to the Grad-Shafranov equation. The free functions are poloidal flux functions and all non-linear contributions to the toroidal current density are treated iteratively. The parallel heat flow approximation chosen for the model is that parallel temperature is a flux function and that both parallel and perpendicular pressures may be described using parallel and perpendicular temperatures. This choice for the fluid thermodynamics h…
In this paper, extending the investigation developed in an earlier paper (Cremaschini et al., 2008), we pose the problem of the kinetic description of gravitational Hall-MHD equilibria which may arise in accretion disks (AD) plasmas close to compact objects. When intense EM and gravitational fields, generated by the central object, are present, a convenient approach can be achieved in the context of the Vlasov-Maxwell description. In this paper the investigation is focused primarily on the following two aspects: 1) the formulation of the kinetic treatment of G-Hall-MHD equilibria. Based on the…
The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of he generalized equation are not automatical…
A free-boundary Tokamak Equilibrium Solver (TES), developed for advanced study of tokamak equilibra, is described with two distinctive features. One is a generalized method to resolve the intrinsic axisymmetric instability, which is encountered after all in equilibrium calculation with a free-boundary condition. The other is an extension to deal with a new divertor geometry such as snowflake or X divertors. For validations, the uniqueness of a solution is confirmed by the independence on variations of computational domain, the mathematical correctness and accuracy of equilibrium profiles are c…
We review the evidence for and against the possibility that a strong enough poloidal field stabilizes an axisymmetric magnetostatic field configuration. We show that there does exist a class of resonant MHD waves which produce instability for any value of the ratio of poloidal and toroidal field strength. We argue that recent investigations of the stability of mixed poloidal and toroidal field configurations based on 3-d numerical simulations, can miss this instability because of the very large azimuthal wave numbers involved and its resonant character.
Sawtooth crash in tokamak as a sequence of Multi-region Relaxed MHD equilibria
Nanyang Technological University（南洋理工大学）；Shanghai Jiao Tong University（上海交通大学）；Massachusetts Institute of Technology（麻省理工学院）；the Australian National University（澳大利亚国立大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Australian Nuclear Science and Technology Organisation（澳大利亚核科学与技术组织）
3D free boundary equilibrium computations have recently been used to model external kinks and edge harmonic oscillations (EHOs), comparing with linear MHD stability codes, and nonlinear analytic theory [Kleiner et al, PPCF 61 084005 (2019)]. In this study, results of the VMEC equilibrium code are compared further with nonlinear reduced MHD simulations, using the JOREK code. The purpose of this investigation is to understand the extent to which the modelling approaches agree, and identify the important physical effects which can modify the dynamics. For the simulated external kink, which is dom…
This is a report about a comparison of collisionless simulations on global modes (i.e. low poloidal mode number) with gyrokinetic code NEMORB against analytical theory and other codes. Only axisymmetric modes, i.e. with toroidal mode number n=0, are considered, and flat equilibrium profiles. Benchmarks are performed for GAMs against local analytical theory. In the presence of energetic ions, local benchmarks of NEMORB are performed against semilagrangian gyrokinetic code GYSELA. The models of adiabatic vs trapped-kinetic- vs fully-kinetic-electrons and of electrostatic vs electromagnetic at ve…
The value of the safety factor on the magnetic axis of a finite-beta spheromak is shown to be a function of beta in contrast to what was used in P. M. Bellan, Phys. Plasmas 9, 3050 (2002); this dependence on beta substantially reduces the gradient of the safety factor compared to the previous calculation. The method for generating finite-beta spheromak equilibria is extended to generate equilibria describing toroidal magnetic "bubbles" where the hydrodynamic pressure on the magnetic axis is less than on the toroid surface. This "anti-confinement" configuration can be considered an equilibrium …
Ballooning instabilities are investigated in three-dimensional magnetic toroidal plasma confinement systems with low global magnetic shear. The lack of any continuous symmetry in the plasma equilibrium can lead to these modes being localized along the field lines by a process similar to Anderson localization. This produces a multibranched local eigenvalue dependence, where each branch corresponds to a different unit cell of the extended covering space in which the eigenfunction peak resides. These phenomena are illustrated numerically for the three-field-period heliac H-1, and contrasted with …
The effect of small deviations from a Maxwellian equilibrium on turbulent momentum transport in tokamak plasmas is considered. These non-Maxwellian features, arising from diamagnetic effects, introduce a strong dependence of the radial flux of co-current toroidal angular momentum on collisionality: As the plasma goes from nearly collisionless to weakly collisional, the flux reverses direction from radially inward to outward. This indicates a collisionality-dependent transition from peaked to hollow rotation profiles, consistent with experimental observations of intrinsic rotation.
The Grad-Shafranov equation is solved using spectral elements for tokamak equilibrium with toroidal rotation. The Grad-Shafranov solver builds upon and extends the NIMEQ code [Howell and Sovinec, Comput. Phys. Commun. 185 (2014) 1415] previously developed for static tokamak equilibria. Both geometric and algebraic convergence are achieved as the polynomial degree of the spectral-element basis increases. A new analytical solution to the Grad-Shafranov equation is obtained for Solov'ev equilibrium in presence of rigid toroidal rotation, in addition to a previously obtained analytical solution fo…
Parameter Scan of Multi-Fluid Equilibria in Rotating p-11B Plasmas: Effects on Fusion Power and Bremsstrahlung Losses
Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams of the Ministry of Education, School of Physics, Dalian University of Technology（大连理工大学物理学院激光、离子和电子束材料改性教育部重点实验室）；ENN Science and Technology Development Co., Ltd.（新奥科技发展有限公司）；Beijing VeloAlpha Technology Co., Ltd.（北京唯罗阿尔法科技有限公司）
Charged particle motion in axisymmetric toroidal magnetic fields is analyzed within the context of the canonical Hamiltonian Guiding Center theory. A canonical transformation to variables measuring the drift orbit deviation from a magnetic field line is introduced and an analytical transformation to Action-Angle variables is obtained, under a zero drift width approximation. The latter is used to provide compact formulas for the orbital spectrum of the drift motion, namely the bounce/transit frequencies as well as the bounce/transit averaged toroidal precession and gyration frequencies. These f…
