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…
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…
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）
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…
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…
In this work, the FALCON code is adopted for illustrating the features of shear Alfvén and continuous spectra in toroidal fusion plasmas. The FALCON codes employs the local Floquet analysis discussed in [Phys. of Plasmas \textbf{26} (8), 082502 (2019)] for computing global structures of continuous spectra in general toroidal geometry. As particular applications, reference equilibria for the Divertor Tokamak Test and ASDEX Upgrade plasmas are considered. In particular, we illustrate the importance of mode polarization for recognizing the physical relevance of the various branches of the continu…
Good magnetic surfaces, as opposed to magnetic islands and chaotic field lines, are generally desirable for stellarators. In previous work, M. Landreman et al. [Phys. of Plasmas 28, 092505 (2021)] showed that equilibria computed by the Stepped-Pressure Equilibrium Code (SPEC) [S. P. Hudson et al., Phys. Plasmas 19, 112502 (2012)] could be optimized for good magnetic surfaces in vacuum. In this paper, we build upon their work to show the first finite-$β$, fixed- and free-boundary optimization of SPEC equilibria for good magnetic surfaces. The objective function is constructed with the Greene's …
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）
A mimetic spectral element solver for the Grad-Shafranov equation
在这项工作中，我们提出了一个强大的和准确的任意阶求解器的固定边界等离子体平衡环轴对称几何形状。为了实现这一点，我们将[56]中提出的模拟谱元素公式应用于Grad-Shafranov方程的求解。这种方法结合了有限体积离散与混合有限元法。这样，离散微分算子（$\nabla$，$\nabla\times$，$\nabla\cdot$）就可以精确地表示出来，并且所有的近似误差都存在于本构关系中。这种提法的结果是一个任意阶的方法，即使在高度弯曲的网格。另外，\reviewerone{toroidal current $J_ϕ$}的积分正好等于极向场在等离子体边界上的边界积分。这个属性可以在平衡和传输求解器之间的耦合中发挥重要作用。所提出的求解器进行了测试的等离子体横截面（光滑和X点），也为广泛的压力和\reviewertwo{toroidal magnetic flux profiles}。平衡精确到机器精度。最佳的代数收敛速度的顺序$(p+1)$和几何收敛速度所示的Soloviev解决方案（包括高Shafranov位移），场反转配置（FRC）的解决方案和spheromak解析解。该方法的鲁棒性证明了非线性测试用例，特别是在平衡解决方案与压力基座。
Relativistic Six-Fluid Rotating-Plasma Force-Balance Equilibria for p-11B Fusion Conditions in Spherical Tori
Beijing ENN Fusion Energy Science and Technology Co., Ltd.（北京恩诺聚变能源科学与技术有限公司）；Beijing Key Laboratory of High Magnetic Field Spherical Torus Fusion Energy（北京市高磁场球环聚变能重点实验室）；Hebei Key Laboratory of Compact Fusion（河北省紧凑型聚变重点实验室）
We prove the existence of a straight-field-line coordinate system we call generalized Boozer coordinates. This coordinate system exists for magnetic fields with nested toroidal flux surfaces} provided $ \oint\mathrm{d}l/B\:(\mathbf{j}\cdot\nablaψ)=0$, where symbols have their usual meaning, and the integral is taken along closed magnetic field lines. All quasisymmetric fields, regardless of their associated form of equilibria, must satisfy this condition. This coordinate system presents itself as a convenient form in which to describe general quasisymmetric configurations and their properties.…
The confinement of Neutral Beam Injection (NBI) particles in the presence of n=3 Resonant Magnetic Perturbations (RMPs) in 15 MA ITER DT plasmas has been studied using full orbit ASCOT simulations. Realistic NBI distribution functions, and 3D wall and equilibria, including the plasma response to the externally applied 3D fields calculated with MARS-F, have been employed. The observed total fast-ion losses depend on the poloidal spectra of the applied n=3 RMP as well as on the absolute toroidal phase of the applied perturbation with respect to the NBI birth distribution. The absolute toroidal p…
