Seminar

Synopsis:

A continuous adjoint method has been developed for obtaining the derivatives of functions of the magneto-hydrodynamic (MHD) equilibrium equations with respect to the shape of the boundary of the domain or the shape of the electro-magnetic coils [1]. This approach is based on the generalization of the self-adjointness of the linearized MHD force operator. The adjoint equation corresponds to a perturbed force balance equation with the addition of a bulk force, rotational transform, or toroidal current perturbation. We numerically demonstrate this approach by adding a small perturbation to the non-linear VMEC [2] solution, obtaining an order $10^2-10^3$ reduction in cost in comparison with a finite difference approach.  Examples are presented for the shape gradient of the rotational transform and vacuum magnetic well, a proxy for MHD stability. The adjoint solution required for the magnetic ripple, a proxy for near-axis quasisymmetry, requires the addition of an anisotropic pressure tensor to the MHD force balance equation. This modification has been implemented in the ANIMEC [3] code. We furthermore demonstrate that this adjoint approach can be applied to compute shape gradients of two important figures of merit [4], the departure from quasisymmetry and the effective ripple in the low-collisionality neoclassical regime, but require the development of new equilibrium solvers. Finally, initial steps toward adjoint solutions with a linearized equilibrium approach will be presented.

[1] Antonsen, T.M., Paul, E.J. & Landreman, M. 2019 Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Journal of Plasma Physics 85 (2).

[2] Hirshman, S.P. & Whitson, J.C. 1983 Steepest descent moment method for three-dimensional magnetohydrodynamic equilibria. Physics of Fluids 26 (12), 3553.

[3] Cooper, W.A., Hirshman, S.P., Merazzi, S. & Gruber, R. 1992 3D magnetohydrodynamic equilibria with anisotropic pressure. Computer Physics Communications 72 (1),1–13.

[4] Paul, E.J., Antonsen, T.M.,  Landreman, M., Cooper, W.A. Submitted to Journal of Plasma Physics.

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We present a mathematical model for the charging simulation of pancake solenoids and propose its fully discrete backward-Euler scheme. The parallel implementation of the scheme is done in Petra-M. The scalability analysis is performed with the iterative and direct solvers for both single turn single pancake and twenty turns double pancakes solenoids. We observe the iterative solver together with the appropriate preconditioner outperforms over the direct solver in both cases.

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In-situ measurements by virtually every spacecraft to date at heliocentric distances above 0.3 Astronomical Units (AU) have found that the turbulent velocity and magnetic fields in the solar wind are predominantly non-compressive fluctuations spanning a broad range of MHD scales. The analysis of these data combined with theory and numerical simulations have helped us shape our understanding of solar wind turbulence beyond 0.3~AU in part due to the validity of the Taylor's Hypothesis (TH), which posits that temporal variation of spacecraft signals is solely due to the spatial variation of a frozen structure passing by the observation point. The Parker Solar Probe (PSP) mission launched last year will explore the solar wind up to seven times closer to the Sun than any previous mission, covering regions where TH is expected to break down. In these regions, a better understanding of the Eulerian space-time correlation is critical for the proper interpretation of time signals from this groundbreaking mission. However, a first-principle derivation of this quantity has remained elusive in turbulence theory due to the statistical closure problem, in which dynamical equations for correlations at order $n$ depend on correlations of order $n+1$. In this talk we will present recent progress in understanding the Eulerian two-time two-point (space-time) correlation for strong and weak incompressible MHD turbulence. In the strong turbulence regime, we propose a model for the space-time correlation that extends Kraichnan's sweeping model for incompressible hydrodynamic (HD) turbulence and validate it against high-resolution numerical simulations. In the weak turbulence regime, an asymptotic wave-turbulence closure is used for the first time to determine the structure of space-time correlations in weak MHD turbulence from the nonlinear equations describing the dynamics.  The wave-turbulence closure that we present may find applications in other weak turbulence regimes found in fluids and plasmas.

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Error fields are predominantly attributed to inevitable coil imperfections. Controlling error fields during coil fabrication and assembly is crucial for stellarators. Excessively tight coil tolerance increases time and cost, and, in part, led to the cancellation of the National Compact Stellarator Experiment and delay of W7-X. In this talk, we develop a Hessian matrix method to rapidly identify important coil deviations. Two of the most common figures of merit, magnetic island size and quasi-symmetry, are analytically differentiated over coil parameters. By extracting the eigenvectors of the Hessian matrix, we can directly identify sensitive coil deviations in the order of the eigenvalues. The new method is applied to the upcoming Chinese First Quasi-axisymmetric Stellarator configuration. Important perturbations that enlarge n/m  =  4/11 islands and deteriorate quasi-axisymmetry of the magnetic field are successfully determined. The results suggest each modular coil should have separate tolerance and some certain perturbation combinations will produce significant error fields. By relaxing unnecessary coil tolerance, this method will hopefully lead to a substantial reduction in time and cost.

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Mirror machines have been researched since the earliest days of the fusion program. Classical axisymmetric mirrors are appealing because of their simple magnetic geometry, but are unable to provide the required confinement and stability for a fusion reactor. In this talk we revisit the axisymmetric mirror, and show that by inducing supersonic rotation in
the confined plasma these problems are cured. We discuss the plasma physics behind these results, and the practicality of inducing such rotation. Throughout, the possibilities for this configuration scaling to a fusion reactor are discussed.

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