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Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Phys. Plasmas, vol. 21, 2014, p. 022515) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by Hameiri (J. Math. Phys., vol. 22, 1981, pp. 2080–2088, § III), to obtain a generalized Grad–Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we can obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

Stellarators are generically small current and low plasma beta devices. Often the construction of vacuum magnetic fields with good magnetic surfaces is the starting point for an equilibrium calculation. Although in cases with some continuous spatial symmetry, flux functions can always be found for vacuum magnetic fields, an analogous function does not, in general, exist in three dimensions. This work examines several simple equilibria and vacuum magnetic field problems with the intent of demonstrating the possibilities and limitations in the construction of such states. Starting with a simple vacuum magnetic field with closed field lines in a topological torus (toroidal shell with a flat metric), we obtain a self-consistent formal perturbation series using the amplitude of the non-symmetric vacuum fields as a small parameter. We show that systems possessing stellarator symmetry allow the construction order by order. We further indicate the significance of stellarator symmetry in the amplitude expansion of the full ideal magnetohydrodynamics (MHD) problem as well. We then investigate the conditions that guarantee neighbouring flux surfaces given the data on one surface, by expanding in the distance from that surface. We show that it is much more difficult to find low shear vacuum fields with surfaces than force-free fields or ideal MHD equilibrium. Finally, we demonstrate the existence of a class of vacuum magnetic fields, analogous to ‘snakes’ observed in tokamaks, which can be expanded to all orders.

Understanding particle drifts in a non-symmetric magnetic field is of primary interest in designing optimized stellarators in order to minimize the neoclassical radial loss of particles. Quasisymmetry and omnigeneity, two distinct properties proposed to ensure radial localization of collisionless trapped particles in stellarators, have been explored almost exclusively for magnetic fields with nested flux surfaces. In this work, we examine radial particle confinement when all field lines are closed. We then study charged particle dynamics in the special case of a non-symmetric vacuum magnetic field with closed field lines obtained recently by Weitzner & Sengupta ( Phys. Plasmas , vol. 27, 2020, 022509). These magnetic fields can be used to construct magnetohydrodynamic equilibria for low pressure. Expanding in the amplitude of the non-symmetric fields, we explicitly evaluate the omnigeneity and quasisymmetry constraints. We show that the magnetic field is omnigeneous in the sense that the drift surfaces coincide with the pressure surfaces. However, it is not quasisymmetric according to the standard definitions.

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys., vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function $\\eta$ such the field strength satisfies $B = B(\\eta )$. Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double-periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady-state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Physics of Plasmas 21, 022515 (2014)) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly-periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by E. Hameiri in (J. Math. Phys. \\textbf{22}, 2080 (1981) Sec. III), to obtain a generalized Grad-Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we are able to obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are consistency conditions necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

This document is the product of a stellarator community workshop, organized by the National Stellarator Coordinating Committee and referred to as Stellcon, that was held in Cambridge, Massachusetts in February 2016, hosted by MIT. The workshop was widely advertised, and was attended by 40 scientists from 12 different institutions including national labs, universities and private industry, as well as a representative from the Department of Energy. The final section of this document describes areas of community wide consensus that were developed as a result of the discussions held at that workshop. Areas where further study would be helpful to generate a consensus path forward for the US stellarator program are also discussed. The program outlined in this document is directly responsive to many of the strategic priorities of FES as articulated in “Fusion Energy Sciences: A Ten-Year Perspective (2015–2025)” [ 1 ]. The natural disruption immunity of the stellarator directly addresses “Elimination of transient events that can be deleterious to toroidal fusion plasma confinement devices” an area of critical importance for the US fusion energy sciences enterprise over the next decade. Another critical area of research “Strengthening our partnerships with international research facilities,” is being significantly advanced on the W7-X stellarator in Germany and serves as a test-bed for development of successful international collaboration on ITER. This report also outlines how materials science as it relates to plasma and fusion sciences, another critical research area, can be carried out effectively in a stellarator. Additionally, significant advances along two of the Research Directions outlined in the report; “Burning Plasma Science: Foundations—Next-generation research capabilities”, and “Burning Plasma Science: Long pulse—Sustainment of Long-Pulse Plasma Equilibria” are proposed.

Non-symmetric vacuum magnetic fields with closed magnetic field lines are of interest in the construction of stellarator equilibria. Beyond the result of D.Lortz (ZAMP \\textbf{21}, 196 (1970)), few results are available. This work presents a closed-form expression for a class of vacuum magnetic fields in a topological torus with closed field lines. We explicitly obtain the invariants of such a field. We finally show that a three-dimensional low beta magnetohydrodynamic (MHD) equilibrium may be constructed in a topological torus starting with these closed line vacuum magnetic fields.

Stellarators are generically small current and low plasma beta (\\(\\beta= p/B^2\\ll1\\)) devices. Often the construction of vacuum magnetic fields with good magnetic surfaces is the starting point for an equilibrium calculation. Although in cases with some continuous spatial symmetry flux functions can always be found for vacuum magnetic fields, an analogous function does not, in general, exist in three dimensions. This work examines several simple equilibria and vacuum magnetic field problems with the intent of demonstrating the possibilities and limitations in the construction of such states. Starting with a simple vacuum magnetic field with closed field lines in a topological torus (toroidal shell with a flat metric), we obtain a self-consistent formal perturbation series using the amplitude of the nonsymmetric vacuum fields as a small parameter. We show that systems possessing stellarator symmetry allow the construction order by order. We further indicate the significance of stellarator symmetry in the amplitude expansion of the full ideal magnetohydrodynamics (MHD) problem as well. We then investigate the conditions that guarantee neighboring flux surfaces given the data on one surface, by expanding in the distance from that surface. We show that it is much more difficult to find low shear vacuum fields with surfaces than force-free fields or ideal MHD equilibrium. Finally, we demonstrate the existence of a class of vacuum magnetic fields, analogous to `snakes' observed in tokamaks, which can be expanded to all orders.

This report summarizes the findings and recommendations of the second Committee of Visitors (COV) whose charge was to review the manner in which the U. S. Department of Energy’s Office of Fusion Energy Science (OFES) manages certain programs under its charter. The specific programs reviewed by this COV involve confinement innovation and basic plasma sciences. The charge letter from the Department of Energy is included as Appendix A.

Quasisymmetry and omnigeneity of an equilibrium magnetic field are two distinct properties proposed to ensure radial localization of collisionless trapped particles in any stellarator. These constraints are incompletely explored, but have stringent restrictions on a magnetic geometry. This work employs an analytic approach to understand the implications of the constraints. The particles move in an intrinsically three dimensional equilibrium whose representation is given by earlier work of Weitzner and its extension here. For deeply trapped particles a local equilibrium expansion around a minimum of the magnetic field strength along a magnetic line suffices. This analytical non-symmetric equilibrium solution, enables explicit representation of the constraints. The results show that it is far easier to satisfy the omnigeneity condition than the quasisymmetry requirement. Correspondingly, there exists a large class of equilibrium close to quasisymmetry that remain omnigeneous while allowing inclusion of error fields, which may destroy quasisymmetry.