A first-order model is derived for quasisymmetric stellarators where the vacuum field due to coils is dominant, but plasma-current-induced terms are not negligible and can contribute to magnetic differential equations, with $β$ of the order of the ratio of induced to vacuum fields. Under these assumptions, it is proven that the aspect ratio must be large and a simple expression can be obtained for the lowest-order vacuum field. The first-order correction, which involves both vacuum and current-driven fields, is governed by a Grad-Shafranov equation and the requirement that flux surfaces exist. These two equations are not always consistent, and so this model is generally overconstrained, but special solutions exist that satisfy both equations simultaneously. One family of such solutions are the first-order near-axis solutions. Thus, the first-order near-axis model is a subset of the model presented here. Several other solutions outside the scope of the near-axis model are also found. A case study comparing one such solution to a VMEC-generated solution shows good agreement.
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…
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…
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…
Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$. Here instead we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in distance from the magnetic axis, without using optimization. The met…
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…
The stability of the ion-temperature gradient mode in quasisymmetric stellarators is assessed. This is performed using a set of analytical estimates together with linear gyrokinetic simulations. The peak growth rates, their corresponding real frequencies and wave-vectors are identified. A comparison is made between a first-order near-axis expansion model and eleven realistic designs obtained using numerical optimization methods. It is found that while the near-axis expansion is able to replicate the growth rates, real frequencies and perpendicular wave-vector at the inner core (both using simp…
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…
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…
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…
A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented …
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…
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…
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, 9058…
We show that in perfectly quasi-isodynamic magnetic fields, which are generally non-quasisymmetric and which can approximate fields of experimental interest, neoclassical calculations can be carried out analytically more completely than in a general stellarator. Here, we define a quasi-isodynamic field to be one in which the longitudinal adiabatic invariant is a flux function and in which the constant-B contours close poloidally. We first derive several geometric relations among the magnetic field components and the field strength. Using these relations, the forms of the flow and current are o…
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…
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 …
The drift kinetic equation is solved for low density TJ-II plasmas employing slowly varying, time-dependent profiles. This allows to simulate density ramp-up experiments and describe from first principles the formation and physics of the radial electric field shear layer. The main features of the transition are perfectly captured by the calculation, and good quantitative agreement is also found. The results presented here, that should be valid for other non-quasisymmetric stellarators, provide a fundamental explanation for a wealth of experimental observations connected to the shear layer emer…
A new paradigm for rapid stellarator configuration design has been recently demonstrated, in which the shapes of quasisymmetric or omnigenous flux surfaces are computed directly using an expansion in small distance from the magnetic axis. To further develop this approach, here we derive several other quantities of interest that can be rapidly computed from this near-axis expansion. First, the $\nabla\vec{B}$ and $\nabla\nabla\vec{B}$ tensors are computed, which can be used for direct derivative-based optimization of electromagnetic coil shapes to achieve the desired magnetic configuration. Mor…
We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also `nearly' quasisymmetric. The primary idea is, given a desired quasisymmetry direction $ξ$, to change the smooth structure on space so that the vector field $ξ$ is Killing for the new metric and construct $ξ$--symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad-Shafranov equation. If $ξ$ is close to a symmetry of Euc…
Compact quasiaxisymmetric stellarators, a near axisymmetric theory
Princeton University（普林斯顿大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Max Plank Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UK（牛津大学鲁道夫·皮尔斯理论物理中心）
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…
Understanding particle drifts in a non-symmetric magnetic field is of primary interest in designing optimized stellarators 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 that generate nested flux surfaces. In this work, we extend these concepts to the case where all the field lines are closed. We then study charged particle dynamics in the exact non-symmetric vacuum magnetic field with…
