Session CO08: Magnetic Confinement: Stellarator Equilibria & Magnets

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Quasisymmetric (QS) fields have long been a topic of study for stellarators, given that they present an elegant way to realize good particle confinement. It is however generally believed that the construction of globally QS magnetostatic equilibria is probably possible only in the case of true axisymmetry. Such an intuition has been largely built on the original and ensuing work on near axis expansion of magnetic fields\footnote{D. Garren \& A. Boozer, Phys. Fluids B, \textbf{3}(10) (1991)}, which show that the expansion becomes overdetermined at larger distance from the axis. \par In this paper, we follow a different path in constructing quasisymmetric solutions by near-axis expansion than pursued in Ref. 1. Following recent work\footnote{J. Burby, N. Kallinikos \& R. MacKay, arXiv:1912.06468 (2019)}${}^{,}$\footnote{E. Rodr\'{i}guez, P. Helander \& A. Bhattacharjee, Phys. Plasmas, \textbf{27}, 062501 (2020)}, we decouple QS from the particular form of force balance to form a generalised form to the standard near axis expansion for quasisymmetric fields. Specializing in anisotropic plasma equilibrium, we show that, formally, the overdetermined nature of the near-axis expansion may be avoided. This opens the door to the possibility of building QS fields in a global sense.   

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The variational principle of Kruskal and Kulsrud (M. D. Kruskal and R. M. Kulsrud, Phys. Fluids 1, 265, 1958) has long provided the basis for the computation of solutions of 3D ideal magnetostatic equilibria with isotropic pressure. Recently, there has been progress in formulating rigorous conditions of quasisymmetry without relying on assumptions regarding the nature of MHD equilibria. We explore the interplay between quasisymmetry and the variational construction of Kruskal and Kulsrud. In the special case of axisymmetry, the formulation is shown to lead directly to the well-known Grad-Shafranov equation. However, in the more general case of quasi-symmetry, the question of existence of the global extrema of the energy functional with isotropic pressure remains open. Stimulated by recent analytical results (see Rodriguez and Bhattacharjee, this meeting) on MHD equilibria with anisotropic pressure, which avoids the problem of overdetermination in near-axis expansions of quasi-symmetric equilibria with isotropic pressure, we extend the variational principle of Kruskal and Kulsrud to include anisotropic pressure. We argue that a path to the construction of global quasisymmetric MHD equilibria for stellarator optimization might have to include the effects of anisotropic pressure.   

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A direct construction of analytical MHD equilibrium using an expansion in the powers of the distance to the magnetic axis is carried out. The approach developed here makes use of a set of orthogonal coordinates related to the geometry of the axis first derived by Mercier [1]. This reduces the MHD system of equations from a three-dimensional to a one-dimensional one, allowing us to considerably reduce its computational cost and gain greater physical insight into the nature of the magnetic field, even in the presence of chaotic magnetic field lines. The near-axis framework was recently generalized to arbitrary order in Ref. [2], where the analytical forms of the magnetic field, toroidal flux and rotational transform were derived and a numerical solution using a W7-X equilibrium was presented. In here, we focus on new developments using the near-axis approach, such as the construction of first and second order quasisymmetric stellarator shapes [3] and their linear gyrokinetic stability properties. [1] C. Mercier, 1964, Nucl. Fusion 4 (3), 213 [2] R. Jorge, W. Sengupta, M. Landreman, Journal of Plasma Physics 86 (1), 905860106 (2020) [3] R. Jorge, W. Sengupta, M. Landreman, Nuclear Fusion 60 (7), 076021 (2020)   

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The study of the structure of magnetic fields near the axis of a toroidal confinement device is known from the classical works of Solov'ev and Shafranov and Lortz and Nuhrenberg in the 1970s. The magnetic axis is taken to be a closed space curve determined by its local curvature and torsion. In these classic studies, as well as more recent work, it is conventional to assume the existence of magnetic surfaces as would be guaranteed by the magnetostatics equation $J \times B = \nabla p$. We revisit this calculation, without making the assumption of local surfaces, for the case of (near) vacuum fields. It is of interest, that instead of solving for a magnetic potential, $B = \nabla \phi$, one can reformulate the set of equations to find the vector potential $B = \nabla \times A$. At the magnetic axis the local coordinate system is chosen to rotate with the torsion of the axis, and Floquet theory is used to obtain the lowest order, linear behavior of the field lines. It is interesting that even if the local field lines are ``instantaneously'' of hyperbolic character, the one-period flow can be elliptic. A version of Hamiltonian Birkhoff-normal form theory with resonances, adapted to the situation of field-line flows, can be used to construct higher-order terms.   

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We consider a vacuum magnetic field in a three-dimensional box with periodic boundary conditions in y and z. Expanding around the y-z plane, we show that the vacuum field can be made quasisymmetric (QS) provided the lowest order magnetic potential satisfies a real hyperbolic Monge-Ampere like equation of two variables. The nonlinear equation can be solved exactly for a class of problems in the hodograph plane derived from the y and z components of the lowest order vacuum magnetic field. A close analogy can be established between steady, irrotational, compressible fluid dynamics and the QS vacuum problem in the surface expansion analysis. Consistent with recent results that QS implies the existence of flux-surfaces surfaces, we show that the condition of QS allows one to carry out the formal near-surface expansion in the distance from the y-z plane to higher orders without resonances on rational surfaces.   

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The Multiregion Relaxed MHD model has been shown to be successful in the construction of equilibria in 3D configurations. In MRxMHD, the plasma is sliced into sub-volumes separated by ideal interfaces, each undergoes relaxation. The Stepped Pressure Equilibrium Code (SPEC) has been developed to solve MRxMHD equilibria numerically. However, to date, the interfaces in MRxMHD have a degree of arbitrariness: the only requirement is that their rotational transform be sufficiently irrational. We investigate numerical and physical criteria that indicate if a certain interface should be deleted. First, an interface should not be a boundary circle, i.e. a flux surface that has chaos in its neighborhood. This leads to a numerical criterion to compute the analytic width of the interface Fourier harmonics or the Lyapunov exponent in its vicinity. The second method makes use of the pressure jump Hamiltonian (PJH) technique by studying the existence of KAM surfaces in the phase space of PJH. These results have implications for the interface selection in MRxMHD.   

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A variational principle based on a generalization of Taylor's relaxation, referred as~Multi-Region relaxed Magnetohydrodynamics~(MRxMHD), was developed to incorporate both an ideal and resistive MHD equilibrium problem. With a well posed manner, suitable numerical solutions of MRxMHD are constructed using the Stepped Pressure Equilibrium Code (SPEC) [1]. In principle, SPEC could also establish to describe the MRxMHD stability, that of, a plasma equilibrium. A novel theoretical second variation of energy functional so-called Hessian is the guideline, which could predict MHD linear instabilities as a by-product of SPEC equilibrium calculation. We demonstrate a newly implemented Hessian algorithm in SPEC which can predict linear MRxMHD stability. Negative eigenvalues of Hessian predict an instability. Validation of SPEC will be shown for both toroidal and cylindrical geometries, and the numerical results are thoroughly verified against ideal and resistive MHD stability theories. This will open a new pathway to study MHD instabilities in three-dimensional (3D) stellarator geometry. \begin{enumerate} \item Hudson \textit{et.al} Phys. Plasma, \textbf{19}:112502, 2012. \end{enumerate}   

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Modern stellarators have traditionally been designed without analytic derivative information, instead applying gradient-based methods with finite-difference approximations or gradient-free methods. We present the first optimization of fixed-boundary stellarator equilibria with analytic derivatives obtained from an adjoint method. This technique is based on the well-known self-adjointness property of the MHD force operator, which has recently been generalized to allow for perturbations of the rotational transform and the currents outside the confinement region. This self-adjointness property is applied to develop an adjoint method for computing the derivatives of functions that depend on MHD equilibrium solutions, such as the magnetic well and rotational transform, with respect to perturbations of coil shapes or the plasma boundary. The application of this technique provides a reduction of the number of required function evaluations by a factor of $\sim 10^2$, enabling efficient convergence toward the optimum configuration. A discussion of the optimization technique and examples of optimized configurations will be presented.   

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Tight construction tolerances have complicated and slowed down construction of recent fusion experiments, eg. Iter, Wendelstein 7-X, and NCSX, the latter eventually canceled before completion primarily due to problems related to tight construction tolerances. Stochastic programming applied to stellarator coil optimization is able to relax coil tolerances, and, somewhat surprisingly, has also led to coil designs that simultaneously produce a better approximation of the target magnetic field. The stochastic optimization process optimizes a cloud of sample coil sets, each slightly deformed away from each other in shape and position. This optimization indeed finds broader minima compared to the standard (non-stochastic) coil optimization process of a single sample. A large number of samples used during the optimization favorably smoothens out the parameter space, while a small number unfavorably flattens it out. These results indicate that earlier coil-finding algorithms, at least in some cases, would get stuck in local optima which were neither as optimal nor as robust against engineering deviations, as the ones found with this new algorithm. We show new results from applying this optimization approach the recent design of a stellarator DEMO power plant.   

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Optimized superconducting stellarators, such as W7-X, have used low-temperature superconductor technology and conventional machining of support structure to fabricate the required 3D non-planar coils. This technology limits the maximum B-field on-axis to $\leq3$ tesla, and construction is expensive and slow. ARPA-E has recently funded a 2-year project proposed by U. Wisconsin and MIT PSFC that will use high temperature superconducting (YBCO) tape and additive manufacturing (3D metal printing) to fabricate a prototypical 3D non-planar superconducting coil, with the eventual goals of producing higher magnetic field, while reducing the manufacturing cost and schedule. The non-planar coil will be based on HTS technology that has been developed for the SPARC tokamak, modified to accommodate the required non-planar geometry and tight-radius bends. The coil will consist of two multi-turn spiral pancakes, mechanically supported by 3D printed non-planar stainless steel radial plates, with inter-pancake electrical and coolant joints. The coil `diameter' will be $\sim$70 cm, and it will be operated at 77 K, with 40 kA-turns generating a peak field of $\sim$1 tesla on the conductor. Well-diagnosed testing will characterize the critical current, 3D B-field structure, and quench robustness.   

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The development of stellarators that use permanent magnet arrays to shape their confining magnetic fields has been a topic of recent interest, but the requirements for how such magnets must be shaped, manufactured, and assembled remain to be determined. To address these open questions, we have performed a study of geometric concepts for magnet arrays with the aid of the newly developed MAGPIE code. A proposed experiment similar to the National Compact Stellarator Experiment (NCSX) is used as a test case. Two classes of magnet geometry are explored: curved bricks that conform to a regular grid in cylindrical coordinates, and hexahedra that conform to the toroidal plasma geometry. In addition, we test constraints on the magnet polarization. While magnet configurations constrained to be polarized normally to a toroidal surface around the plasma are unable to meet the required magnetic field parameters when subject to physical limitations on the strength of present-day magnets, configurations with unconstrained polarizations are shown to satisfy the physics requirements for a targeted plasma.   

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The nonlinear, extended MHD code NIMROD is employed to simulate self-consistent stellarator behavior at high beta. Finite anisotropic thermal conduction allows for sustained pressure gradients within stochastic regions. The configuration under investigation is an l=2, M=10 torsatron with vacuum rotational transform near unity. Finite-beta plasmas are generated from vacuum fields using a volumetric heating source and temperature dependent resistivity. In sufficiently dissipative regimes, steady-state solutions are obtained which exhibit a conventional equilibrium beta limit. The parametric dependence of the equilibrium beta limit is examined in detail and compared with several reduced models for effective radial transport across stochastic magnetic fields, in the collisional limit. Simulations with less dissipation show signs of interchange-like instabilities which also act to limit the achievable beta, even when simulations only include modes which preserve toroidal stellarator symmetry. Present numerical resources only allow for a preliminary investigation of this behavior, but highlight the importance of ongoing computational development (C. R. Sovinec, C. M. Guilbault, and T. A. Bechtel, this meeting.).   

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Sub-millimeter powder grains of boron and boron nitride are injected for the first time in the Large Helical Device plasma, employing the Impurity Powder Dropper, developed and built by PPPL. Cross- diagnostics measurements show the injected impurities to effectively penetrate into the plasma, as the injection rate and plasma density are varied. The injected impurities provide a supplemental electron source, causing the plasma density and radiated power to increase. For n$_{\mathrm{e,av\thinspace }}$\textless 10$^{\mathrm{19}}$m$^{\mathrm{-3}}$ the powder grains penetrate deeper into the plasma, as they can be less effectively deflected by the plasma flow in the divertor leg, which they have to cross first as they are injected from the top of the machine. In this case, the created boron ions are observed to move outwards from UV spectroscopy and charge exchange measurements, due to the direction of the ambipolar radial electric field, while this is not the case for higher density plasmas. Low density plasmas are therefore better candidates for powder boronization techniques. The experimental observations are supported by numerical results from the codes SFINCS and EMC3-EIRENE coupled with DUSTT.