The stellarator as a concept of magnetic confinement fusion requires careful design to confine particles effectively. A design possibility is to equip the magnetic field with a property known as quasisymmetry. Though it is generally believed that a steady-state quasisymmetric equilibrium can only be exact locally (unless the system has a direction of continuous symmetry such as the tokamak), we suggest in this work that a change in the equilibrium paradigm can ameliorate this limitation. We demonstrate that there exists a deep physical connection between quasisymmetry and magnetostatic equilibria with anisotropic pressure, extending beyond the isotropic pressure equilibria commonly considered.
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 derive a sufficient condition for the linear stability of plasma equilibria with incompressible flow parallel to the magnetic field, $\bf B$, constant mass density and anisotropic pressure such that the quantity $σ_d= μ_0(P_\parallel -P_\perp)/B^2$, where $P_\parallel$ ($P_\perp$) is the pressure tensor element parallel (perpendicular) to $\bf B$, remains constant. This condition is applicable to any steady state without geometrical restriction. The condition, generalising the respective condition for MHD equilibria with isotropic pressure and constant density derived in Throumoulopoulos & …
Fusion-graded plasmas are one of the physically complex systems, resulting in continuous establishment of plasma theories for unclarified physical phenomena in order to thoroughly control nuclear fusion reactors. Deep learning has drawn vast attention to this field of controlled fusion plasma to link physical phenomena with control-relevant parameters without a deepened understanding about plasma theories. Albeit, quantifying the uncertainty of deep learning models has been constantly requested due to their fundamental shortage of physical understanding. Thus, a concept of a reliable deep lear…
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…
We present an extension of the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium model that includes pressure anisotropy and general plasma flows. This anisotropic extension to our previous isotropic model is motivated by Sun and Finn's model of relaxed anisotropic magnetohydrodynamic equilibria. We prove that as the number of plasma regions becomes infinite, our anisotropic extension of MRxMHD reduces to anisotropic ideal MHD with flow. The continuously nested flux surface limit of our MRxMHD model is the first variational principle for anisotropic plasma equilibria with general …
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 numerical workflow is developed to explore the viability of running multiple plasma configurations in the ARC fusion pilot plant. Suitable cost functions for various poloidal field coil sets are evaluated based on currents required in the coils, induced stresses, and flexibility in plasma configurations. It is shown, for the first time, that a given set of poloidal field coils can sustain equilibria with both signs of triangularity and that equilibria at Negative Triangularity have comparable coil requirements to those at Positive Triangularity, despite ARC's Positive Triangularity design eq…
Neutral beam injection or ion cyclotron resonance heating induces pressure anisotropy. The axisymmetric plasma equilibrium code HELENA has been upgraded to include anisotropy and toroidal flow. With both analytical and numerical methods, we have studied the determinant factors in anisotropic equilibria and their impact on flux surfaces, magnetic axis shift, the displacement of pressures and density contours from flux surface. With $p_\parallel/p_\perp \approx 1.5$, $p_\perp$ can vary 20% on $s=0.5$ flux surface, in a MAST like equilibrium. We have also re-evaluated the widely applied approxima…
A systematic approach to assess the linear stability of Alfvén eigenmodes in the presence of fusion-born alpha particles is described. Because experimental results for ITER are not available yet, it is not known beforehand which Alfvén eigenmodes will interact more intensively with the alpha-particle population. Therefore, the number of modes that need to be considered in stability assessments becomes quite large and care must be exercised when choosing the numerical tools to work with, which must be fast and efficient. In the presented approach, all possible eigenmodes are first found after i…
It is proved that (a) the solutions of the ideal magnetohydrodynamic equation, which describe the equlibrium states of a cylindrical plasma with purely poloidal flow and arbitrary cross sectional shape [G. N. Throumoulopoulos and G. Pantis, Plasma Phys. and Contr. Fusion 38, 1817 (1996)] are also valid for incompressible equlibrium flows with the axial velocity component being a free surface quantity and (b) for the case of isothermal incompressible equilibria the magnetic surfaces have necessarily circular cross section.
Being motivated by the paper [O. I. Bogoyavlenskij, Phys. Rev. 66, 056410 (2002)] we generalise the symmetry transformations for MHD equilibria with isotropic pressure and incompressible flow parallel to the magnetic field introduced therein in the case of respective CGL equilibria with anisotropic pressure. We find that the geometrical symmetry of the field-aligned equilibria can break by those transformations only when the magnetic field is purely poloidal. In this situation we derive three-dimensional CGL equilibria from given axisymmetric ones. Also, we examine the generic symmetry transfo…
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…
A Variational Framework for Guiding-Center Kinetics, Anisotropic Equilibria, and Quasisymmetry in Stellarators
Courant Institute School of Mathematics, Computing, and Data Science, New York University（纽约大学库朗数学科学研究所）；Department of Astrophysical Sciences, Princeton University（普林斯顿大学天体物理科学系）
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）
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…
High-power-density tokamaks offer a potential solution to design cost-effective fusion devices. One way to achieve high power density is to operate at a high $β$ value (the ratio of thermal to magnetic pressure), i.e., $β\sim 1$. However, a $β\sim 1$ state may be unstable to various pressure- and current-driven instabilities or have unfavorable microstability properties. To explore these possibilities, we generate $β\sim 1$ equilibria and investigate their stability. Initially, we study an analytical technique that was used in the past to generate $β\sim 1$ equilibria and outline its limitatio…
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…
Collisionless physics primarily determines the transport of fusion-born alpha particles in 3D equilibria. Several transport mechanisms have been implicated in stellarator configurations, including stochastic diffusion due to class transitions, ripple trapping, and banana drift-convective orbits. Given the guiding center dynamics in a set of six quasihelical and quasiaxisymmetric equilibria, we perform a classification of trapping states and transport mechanisms. In addition to banana drift convection and ripple transport, we observe substantial non-conservation of the parallel adiabatic invari…
Hamiltonian Particle Dynamics in Fusion Plasmas: Orbital Tomography and Spectrum Analysis for Energy and Momentum Transport under Resonant Non-Axisymmetric Perturbations
It is well known that the process of construction of quasisymmetric magnetic fields in magnetostatic equilibrium with isotropic pressure suffers from the problem of overdetermination. This has led to the widespread belief that global quasisymmetric solutions are likely not to exist. We develop a general near-axis expansion procedure that does not rely on the assumption of magnetostatic equilibria with isotropic pressure. We then demonstrate that in equilibria with anisotropic pressure, it is possible to circumvent the problem of overdetermination and carry out the power-series solutions to hig…
Observations and numerical simulations of laboratory and space plasmas in almost collisionless regimes reveal anisotropic and non-gyrotropic particle distribution functions. We investigate how such states can persist in the presence of a sheared flow. We focus our attention on the pressure tensor equation in a magnetized plasma and derive analytical self-consistent plasma equilibria which exhibit a novel asymmetry with respect to the magnetic field direction. These results are relevant for investigating, within fluid models that retain the full pressure tensor dynamics, plasma configurations w…
Velocity-space anisotropy can significantly modify fusion reactivity. The nature and magnitude of this modification depends on the plasma temperature, as well as the details of how the anisotropy is introduced. For plasmas that are sufficiently cold compared to the peak of the fusion cross-section, anisotropic distributions tend to have higher yields than isotropic distributions with the same thermal energy. At higher temperatures, it is instead isotropic distributions that have the highest yields. However, the details of this behavior depend on exactly how the distribution differs from an iso…
A largely unsolved theoretical issue in controlled fusion research is the consistent \textit{kinetic} treatment of slowly-time varying plasma states occurring in collisionless and magnetized axisymmetric plasmas. The phenomenology may include finite pressure anisotropies as well as strong toroidal and poloidal differential rotation, characteristic of Tokamak plasmas. Despite the fact that physical phenomena occurring in fusion plasmas depend fundamentally on the microscopic particle phase-space dynamics, their consistent kinetic treatment remains still essentially unchalleged to date. The goal…
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）
The plasma equilibrium in a linear trap at $β\approx 1$ (or above the mirror-instability threshold) under the topology-conservation constraint evolves into a kind of diamagnetic "bubble". This can take two forms: either the plasma body greatly expands in radius while containing the same magnetic flux, or, if the plasma radius is limited, the plasma distribution across flux-tubes changes, so that the same cross-section contains a greatly reduced flux. If the magnetic field of the trap is quasi-uniform around its minimum, the bubble can be made roughly cylindrical, with radius much larger than t…
A class of parametric distribution functions has been proposed in [C.DiTroia, Plasma Physics and Controlled Fusion,54,2012] as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic equilibria for Slowing-Down $alpha$ particles and core thermal plasma populations. These EDFs depend on constants of motion (COMs). Assuming an axisymmetric system with no equilibrium electric field, the EDF dep…
We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e. bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that …
We study the thermodynamic equilibrium spectra of the Charney- Hasegawa-Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross-Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction…
It is well known that externally generated resonant magnetic perturbations (RMPs) can form islands in the plasma edge. In turn, large overlapping islands generate stochastic fields, which are believed to play a role in the avoidance and suppression of edge localized modes (ELMs) at DIII-D. However, large coalescing islands can also generate, in the middle of these stochastic regions, KAM surfaces effectively acting as "barriers" against field-line dispersion and, indirectly, particle diffusion. It was predicted in [H. Ali and A. Punjabi, Plasma Phys. Control. Fusion 49 (2007), 1565-1582] that …
It is believed that plasma rotation can affect the transitions to the advanced confinement regimes in tokamaks. In addition, in order to achieve fusion temperatures modern tokamaks rely on auxiliary heating methods. These methods generate pressure anisotropy in the plasma. For incompressible rotation with pressure anisotropy the equilibrium is governed by a Generalized Grad-Shafranov (GGS) equation and a decoupled Bernoulli-type equation for the effective pressure, $\bar{p}=(p_\parallel+p_\perp)/2$, where $p_\parallel$ ($p_\perp$) is the pressure tensor element parallel (perpendicular) to the …
We present the results of three-dimensional kinetic particle-in-cell (PIC) simulations of isotropic periodic relativistically magnetized pair-plasma equilibria known as the ABC fields. We performed several simulations for initial wavenumbers k_ini = 2 or k_ini = 4, different efficiencies of radiative cooling (including radiation reaction from synchrotron and inverse Compton processes), and different mean magnetization values. These equilibria evolve by means of ideal coalescence instability, the saturation of which generates ab initio localized kinetically-thin current layers -- sites of magne…
On shear Alfvén wave-induced energetic ion transport in optimized stellarators
Columbia University（哥伦比亚大学）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Cornell University（康奈尔大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
In this paper, current profiles for plasma discharges on the Mega-Ampere Spherical Tokamak (MAST) are directly calculated from pickup coil, flux loop and Motional-Stark Effect (MSE) observations via methods based in the statistical theory of Bayesian analysis. By representing toroidal plasma current as a series of axisymmetric current beams with rectangular cross-section and inferring the current for each one of these beams, flux-surface geometry and q-profiles are subsequently calculated by elementary application of Biot-Savart's law. The use of this plasma model in the context of Bayesian an…
We apply the near-axis expansion method for quasisymmetric magnetic fields with anisotropic pressure (developed in the companion paper, Part I [E. Rodriguez, A. Bhattacharjee, arXiv:2008.04715 (2020)]) to construct numerical solutions to circular axis stellarators. The solutions are found to second order in the distance from the axis, not possible in the standard Garren-Boozer construction [D. A. Garren and A. H. Boozer, Physics of Fluids B: Plasma Physics, 2822 (1991)], which assumes magnetostatic equilibria with isotropic pressure. In the limit of zero anisotropy, it is shown that a subset o…
A new approach to high pressure magnetically-confined plasmas is necessary to design efficient fusion devices. This paper presents an equilibrium combining two solutions of the Grad-Shafranov equation, which describes the magnetohydrodynamic equilibrium in toroidal geometry. The outer equilibrium is paramagnetic and confines the inner equilibrium, whose strong diamagnetism permits to balance large pressure gradients. The existence of both equilibria in the same volume yields a dual equilibrium structure. Their combination also improves free-boundary mode stability.
In magnetic confinement fusion research, the achievement of high plasma pressure is key to reaching the goal of net energy production. The magnetohydrodynamic (MHD) model is used to self-consistently calculate the effects the plasma pressure induces on the magnetic field used to confine the plasma. Such MHD calculations serve as input for the assessment of a number of important physics questions. The VMEC code is the most widely used to evaluate 3D ideal-MHD equilibria, as prominently present in stellarators. However, considering the computational cost, it is rarely used in large-scale or onli…
The fixed boundary plasma equilibrium basis for a one Gigawatt electric stellarator power plant
Gauss Fusion GmbH（高斯聚变有限公司）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Technical University of Denmark（丹麦技术大学）；Research Instruments GmbH（研究仪器有限公司）；VTT Technical Research Centre of Finland Ltd.（芬兰技术研究中心有限公司）
The non-modal approach for a linearized system differs from a normal mode analysis by following the temporal evolution of some perturbed equilibria, and therefore includes transient effects. We employ a non-modal approach for studying the stability of a bi-Maxwellian magnetized plasma using the Landau fluid model, which we briefly describe. We show that bi-Maxwellian stable equilibria can support transient growth of some physical quantities, and we study how these transients behave when an equilibrium approaches its marginally stable condition. This is relevant to anisotropic plasma, that are …
