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 automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.
We derive exact solutions of a linear form of the Grad-Shafranov (GS) equation, including incompressible equilibrium flow, using ansatz-based similarity reduction methods. The linearity of the equilibrium equation allows linear combinations of solutions in order to obtain axisymmetric MHD equilibria with closed and nested magnetic surfaces which are favorable for the effective confinement of laboratory plasmas. In addition, employing the same reduction methods we obtain analytical solutions for several non-linear forms of the GS equation. In this context analytic force-free solutions in both l…
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.
In the above entitled recent publication by Giovanni Lapenta [Phys. Rev. Lett. Vol 90, 135005 (2003) ] it is claimed construction of a new class of solitonlike solutions for the Grad-Shafranov equation in plane geometry. It is proved here that, because of the mathematically erroneous choice \nabla p= |Ψ|^2 Ψ\nabla Ψfor an analytic continuation of the poloidal magnetic flux-function Ψin the complex plane (p is the pressure), the cubic Schrödinger equation considered by the author is irrelevant to the equilibrium problem and the Grad-Shafranov equation.
In this paper, a general equation for the evolution of an axisymmetric magnetic field in a Hall plasma is derived, with an integral similar to the Grad-Shafranov equation. Special solutions arising from curvature -- whistler drift modes that propagate along the electron drift as a Burger's shock, and nonlinear periodic and soliton-like solutions to the generalized Grad-Shafranov integral -- are analyzed. We derive analytical and numerical solutions in a classical electron-ion Hall plasma, in which electrons and ions are the only species in the plasmas. Results may then be applied to the follow…
We employ a conformal mapping transformation to solve a generalized Grad-Shafranov equation with incompressible plasma flow of arbitrary direction and construct particular up-down asymmetric D-shaped and diverted tokamak equilibria. The proposed method can also be employed as an alternative quasi-analytic method to solving two dimensional elliptic partial differential equations.
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…
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…
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.
We review criticality theory as a prelude to consideration of criticality of the Grad-Shafranov equation. Novel criticality conditions of ODEs and PDEs are derived, easily evaluated. The possibility that transport barriers are associated with characteristics of the equilibrium equation is explored. We conjecture that equilibrium criticality permits the appearance of new solution branches: the high confinement branch has high poloidal flux gradient in a diamagnetic region of the plasma. Similarly, criticality may lead to loss of solution, which could be related to MHD instability and/or island …
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.
A new class of soliton-like solutions is derived for the Grad-Shafranov (GS) equations. A mathematical analogy between the GS equation for MHD equilibria and the cubic Schrödinger (CS) equation for non-linear wave propagation forms the basis to derive the new class of solutions. The soliton-like solutions are considered for their possible relevance to astrophysics and solar physics problems. We discuss how a soliton-like solution can be generated by a repetitive process of magnetic arcade stretching and plasmoid formation induced by the differential rotation of the solar photosphere or of an a…
A fast, high-order solver for the Grad-Shafranov equation
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）
In toroidally confined plasmas, the Grad-Shafranov equation, in general a non-linear PDE, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta and the Shafranov shift, then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flu…
A method is proposed to solve the Grad-Shafranov partial differential equation for the poloidal flux function associated with the equilibrium of a plasma magnetically confined in an axisymmetric torus under the assumption that the sources to the equilibrium (the gradient of the plasma pressure and the gradient of half the squared toroidal field function in flux space) are constant, and subjected to the condition of constant value of the poloidal flux at the toroidal boundary. The solution to the equation is written as the sum of the particular solution and a combination of N multipole solution…
The rules to write out any one of the linearly independent functions belonging to the infinite set of those in polynomial form that satisfy the sourceless Grad-Shafranov equation as stated in the toroidal-polar coordinate system are established. It is found that a polynomial solution even in the poloidal angle is given by the product of an integral power of the radial coordinate variable by a complete polynomial of equal degree in this same variable with angular-dependent coefficient functions that are linear combinations of a finite number of Chebyshev polynomials in the cosine of the poloida…
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…
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…
In this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad-Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators ($\nabla$, $\nabla\times$, $\nabla\cdot$) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulat…
Rigid electron rotation of a fully penetrated Rotamak-FRC produces a pressure flux function that is more peaked than the Solov'ev flux function. This paper explores the implications of this peaked pressure flux function, including the isothermal case, which appear when the temperature profile is broader than the density profile, creating both benefits and challenges to a Rotamak-FRC based fusion reactor. In this regime, the density distribution becomes very peaked, enhancing the fusion power. The separatrix has a tendency to become oblate, which can be mitigated by flux conserving current loop…
We describe the evolution of a plasma equilibrium having a toroidal topology in the presence of constant electric resistivity. After outlining the main analytical properties of the solution, we illustrate its physical implications by reproducing the essential features of a scenario for the upcoming Italian experiment Divertor Tokamak Test Facility, with a good degree of accuracy. Although we find the resistive diffusion timescale to be of the order of $10^4\,$s, we observe a macroscopic change in the plasma volume on a timescale of $10^2\,$s, comparable to the foreseen duration of the plasma d…
KAIST（韩国科学技术院）；National Fusion Research Institute（国家核聚变研究所）；Mobiis Co., Ltd.（Mobiis有限公司）
A neural network solving Grad-Shafranov equation constrained with measured magnetic signals to reconstruct magnetic equilibria in real time is developed. Database created to optimize the neural network's free parameters contain off-line EFIT results as the output of the network from $1,118$ KSTAR experimental discharges of two different campaigns. Input data to the network constitute magnetic signals measured by a Rogowski coil (plasma current), magnetic pick-up coils (normal and tangential components of magnetic fields) and flux loops (poloidal magnetic fluxes). The developed neural networks …
Accurate Derivative Evaluation for any Grad-Shafranov Solver
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）
Grad-Shafranov equilibria via data-free physics informed neural networks
Institute for Research in Electronics and Applied Physics, University of Maryland（马里兰大学电子与应用物理研究所）；Courant Institute, New York University（纽约大学柯朗数学科学研究所）；Princeton University（普林斯顿大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
Exact solutions of the equation governing the equilibrium magetohydrodynamic states of an axisymmetric plasma with incompressible flows of arbitrary direction [H. Tasso and G.N.Throumoulopoulos, Phys. Pasmas {\bf 5}, 2378 (1998)] are constructed for toroidal current density profiles peaked on the magnetic axis in connection with the ansatz $S=-ku$, where $S=d/d u [\varrho (dΦ/du)^2]$ ($k$ is a parameter, $u$ labels the magnetic surfaces; $\varrho(u)$ and $Φ(u)$ are the density and the electrostatic potential, respectively). They pertain to either unbounded plasmas of astrophysical concern or b…
An Inverse Grad-Shafranov Neural Network Approach to Tokamak Magnetic Control
MIT Plasma Science and Fusion Center（麻省理工学院等离子体科学与聚变中心）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Google DeepMind（谷歌DeepMind）；Fusionality SA；Commonwealth Fusion Systems（联邦聚变系统）
Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator
Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格兰杰工程学院核、等离子体与放射工程系）；Fusion Energy Division, Oak Ridge National Lab（橡树岭国家实验室聚变能源部）；National Center for Supercomputing Applications（国家超级计算应用中心）；Department of Applied Mechanics, Indian Institute of Technology Delhi（印度理工学院德里分校应用力学系）；Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi（印度理工学院德里分校亚迪人工智能学院）
A theoretical model of the high-beta equilibrium of magnetospheric plasma was constructed by consistently connecting the (anisotropic pressure) Grad-Shafranov equation and the Vlasov equation. The Grad-Shafranov equation was used to determine the axisymmetric magnetic field for a given magnetization current corresponding to a pressure tensor. Given a magnetic field, we determine the distribution function as a specific equilibrium solution of the Vlasov equation, using which we obtain the pressure tensor. We need to find an appropriate class of distribution function for these two equations to b…
VEQ: a fast parametric Grad--Shafranov solver for fixed-boundary tokamak equilibria with flexible source profiles
Dalian University of Technology（大连理工大学）；Beijing VeloAlpha Technology Co., Ltd.（北京维奥阿尔法科技有限公司）；ENN Science and Technology Development Co., Ltd.（新奥科技开发有限公司）；Tianjin University of Technology and Education（天津职业技术师范大学）；Nankai University（南开大学）
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…
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.
Non-axisymmetric stationary magnetic perturbations lead to the formation of homoclinic tangles near the divertor magnetic saddle in tokamak discharges. These tangles intersect the divertor plates in static helical structures that delimit the regions reached by open magnetic field lines reaching the plasma column and leading the charged particles to the strike surfaces by parallel transport. In this article we introduce a non-axisymmetric rotating magnetic perturbation to model the time development of the three-dimensional magnetic field of a single-null DIII-D tokamak discharge developing a ro…
Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad-Shafranov equation by extension from polygonal subdomains
New York University, Courant Institute of Mathematical Sciences（纽约大学 库朗数学科学研究所）；Universidad de Concepción, Department of Mathematical Engineering（康塞普西翁大学 数学工程系）；Universidad de Concepción, Center for Research in Mathematical Engineering（康塞普西翁大学 数学工程研究中心）
What Is the Minimum Number of Parameters Required to Represent Solutions of the Grad-Shafranov Equation?
Hebei Key Laboratory of Compact Fusion（河北紧凑型聚变重点实验室）；ENN Science and Technology Development Co., Ltd.（新奥科技发展有限公司）；School of Science, Tianjin University of Technology and Education（天津职业技术师范大学理学院）
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) …
Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation
ENN Science and Technology Development Co., Ltd.（新奥科技发展有限公司）；Hebei Key Laboratory of Compact Fusion（河北省紧凑型聚变重点实验室）；Xi’an University of Science and Technology（西安科技大学）；Xi’an Jiaotong University（西安交通大学）；Dalian University of Technology（大连理工大学）
