A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren and Boozer's expansion about the magnetic axis (Phys Fluids B 3, 2805 (1991)), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman, Sengupta and Plunk (J Plasma Phys 85, 905850103 (2019)), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer's ideal scaling of quasisymmetry-breaking with the cube of inverse aspect ratio. We also demonstrate a strongly nonaxisymmetric configuration (vacuum $ι> 0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are $< 2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.
Quasisymmetric stellarators are a type of optimized stellarators for which flows are undamped to lowest order in an expansion in the normalized Larmor radius. However, perfect quasisymmetry is impossible. Since large flows may be desirable as a means to reduce turbulent transport, it is important to know when a stellarator can be considered to be sufficiently close to quasisymmetry. The answer to this question depends strongly on the size of the spatial gradients of the deviation from quasisymmetry and on the collisionality regime. Recently, formal criteria for closeness to quasisymmetry have …
A method is demonstrated to optimize a stellarator's geometry to eliminate magnetic islands and achieve other desired physics properties at the same time. For many physics quantities that have been used in stellarator optimization, including quasisymmetry, neoclassical transport, and magnetohydrodynamic stability, it is convenient to use a magnetic equilibrium representation that assures the existence of magnetic surfaces. However, this representation hides the possible presence of magnetic islands, which are typically undesirable. To include both surface-based objectives and island widths in …
Quasisymmetric stellarators are an attractive class of optimised magnetic confinement configurations. The property of quasisymmetry (QS) is in practice limited to be approximate, and thus the construction requires measures that quantify the deviation from the exact property. In this paper we study three measure candidates used in the literature, placing the focus on their origin and the comparison of their forms. The analysis shows clearly the lack of universality in these measures. As these metrics do not directly correspond to any physical property (except when exactly QS), optimisation shou…
Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field $\mathbf{B} = \mathbf{B}_0 + α\mathbf{B}_1$, where $\mathbf{B}_0$ is quasisymmetric, $α\mathbf{B}_1$ is a deviation from quasisymmetry, and $α\ll 1$, the stellarator can rotate at high velocities if $α< ε^{1/2}$, with $ε$ the ion Larmor radius over the characteristic variation length of $\mathbf{B}_0$. The cases in which this result may break down are discussed. If the stellarator is…
Impurity temperature screening is a favorable neoclassical phenomenon involving an outward radial flux of impurity ions from the core of fusion devices. Quasisymmetric magnetic fields are intrinsically ambipolar and give rise to temperature screening for low enough $η^{-1}\equiv d\ln n/d\ln T$. In contrast, neoclassical fluxes in generic stellarators will depend on the radial electric field, which is predicted to be inward for ion-root plasmas, potentially leading to impurity accumulation. Here we examine the impurity particle flux in a number of approximately quasisymmetric stellarator config…
We present a technique that can be used to design stellarators with a high degree of experimental flexibility. For our purposes, flexibility is defined by the range of values the rotational transform can take on the magnetic axis of the vacuum field while maintaining satisfactory quasisymmetry. We show that accounting for configuration flexibility during the modular coil design improves flexibility beyond that attained by previous methods. Careful placement of planar control coils and the incorporation of an integrability objective enhance the quasisymmetry and nested flux surface volume of ea…
Plasma flow is damped in stellarators because they are not intrinsically ambipolar, unlike tokamaks, in which the flux-surface averaged radial electric current vanishes for any value of the radial electric field. Only quasisymmetric stellarators are intrinsically ambipolar, but exact quasisymmetry is impossible to achieve in non-axisymmetric toroidal configurations. By calculating the violation of intrinsic ambipolarity due to deviations from quasisymmetry, one can derive criteria to assess when a stellarator can be considered quasisymmetric in practice, i.e. when the flow damping is weak enou…
Adjoint methods can speed up stellarator optimisation by providing gradient information more efficiently compared to finite-difference evaluations. Adjoint methods are herein applied to vacuum magnetic fields, with objective functions targeting quasisymmetry and a rotational transform value on a surface. To measure quasisymmetry, a novel way of evaluating approximate flux coordinates on a single flux surface without the assumption of a neighbourhood of flux surfaces is proposed. The shape gradients obtained from the adjoint formalism are evaluated numerically and verified against finite-differ…
Quasisymmetry can greatly improve the confinement of energetic particles and thermal plasma in a stellarator. The magnetic field of a quasisymmetric stellarator at high plasma pressure is significantly affected by the bootstrap current, but the computational cost of accurate stellarator bootstrap calculations has precluded use inside optimization. Here, a new efficient method is demonstrated for optimization of quasisymmetric stellarator configurations such that the bootstrap current profile is consistent with the geometry. The approach is based on the fact that all neoclassical phenomena in q…
Optimized stellarator configurations and their analytical properties are obtained using a near-axis expansion approach. Such configurations are associated with good confinement as the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak. This makes them appealing as fusion reactor candidates. Using a direct coordinate approach, where the magnetic field and flux surface functions are found explicitly in terms of the position vector at successive orders in the distance to the axis, the set of ordinary differential equations for first and second ord…
Any viable stellarator reactor will need to be nearly omnigenous, meaning the radial guiding-center drift velocity averages to zero over time for all particles. While omnigenity is easier to achieve than quasisymmetry, we show here that several properties of quasisymmetric plasmas also apply directly or with only minor modification to the larger class of omnigenous plasmas. For example, concise expressions exist for the flow and current, closely resembling those for a tokamak, and these expressions are explicit in that no magnetic differential equations remain. A helicity (M,N) can be defined …
Most quasisymmetric stellarators to date have been designed by numerically optimizing the plasma boundary shape to minimize symmetry-breaking Fourier modes of the magnetic field strength $B$. At high aspect ratio, a faster approach is to directly construct the plasma shape from the equations of quasisymmetry near the magnetic axis derived by Garren and Boozer [Phys Fluids B 3, 2805 (1991)]. Here we show that the core shape and rotational transform of many optimization-based configurations can be accurately described by this direct-construction approach. This consistency supports use of the nea…
A quasisymmetry is a special symmetry that enhances the ability of a magnetic field to trap charged particles. Quasisymmetric magnetic fields may allow the realization of next generation fusion reactors (stellarators) with superior performance when compared with classical (tokamak) designs. Nevertheless, the existence of such magnetic configurations lacks mathematical proof due to the complexity of the governing equations. Here, we prove the existence of weakly quasisymmetric magnetic fields by constructing explicit examples. This result is achieved by a tailored parametrization of both magnet…
Nine stellarator configurations, three quasiaxisymmetric, three quasihelically symmetric and three non-quasisymmetric are scaled to ARIES-CS size and analyzed for energetic particle content. The best performing configurations with regard to energetic particle confinement also perform the best on the neoclassical Γc metric, which attempts to align contours of the second adiabatic invariant with flux surfaces. Quasisymmetric configurations that simultaneously perform well on Γc and quasisymmetry have the best overall confinement, with collisional losses under 3%, approaching the performance of I…
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）
We present a stellarator configuration optimized for a large threshold (``critical gradient'') for the onset of the ion temperature gradient (ITG) driven mode, which achieves the largest critical gradient we have seen in any stellarator. Above this threshold, gyrokinetic simulations show that the configuration has low turbulence levels over an experimentally relevant range of the drive strength. The applied optimization seeks to maximize the drift curvature, leading to enhanced local-shear stabilization of toroidal ITG modes, and the associated turbulence. These benefits are combined with exce…
Designing A Buildable Optimized Stellarator to Confine Electron-Positron Plasmas
Max-Planck-Institut für Plasmaphysik（德国马普等离子体物理研究所）；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）；Department of Physics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校物理系）；SchulerTEC；Department of Mechanical Engineering, McGill University（麦吉尔大学机械工程系）
We have recently demonstrated that by expanding in small distance from the magnetic axis compared to the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the fl…
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 equilib…
Stochastic single-stage stellarator optimization using fixed-boundary equilibria
Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Max-Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；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）；Department of Physics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校物理系）；Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院等离子体与核聚变研究所）
A method is demonstrated to rapidly calculate the shapes and properties of quasi-axisymmetric and quasi-helically symmetric stellarators. In this approach, optimization is applied to the equations of magnetohydrodynamic equilibrium and quasisymmetry, expanded in the small distance from the magnetic axis, as formulated by Garren and Boozer [Phys. Fluids B, 3, 2805 (1991)]. Due to the reduction of the equations by the expansion, the computational cost is significantly reduced, to times on the order of 1 cpu-second, enabling wide and high-resolution scans over parameter space. In contrast to trad…
