Seminar
Recent events from the MagnetoFluid Dynamics Seminar are listed below. To see the complete list of events, visit the seminar's main page, here.
Upcoming Events

Tuesday, November 19, 201911AM, Warren Weaver Hall 905
An adjoint approach for the shape gradients of threedimensional magnetohydrodynamic equilibria
Elizabeth Paul, University of Maryland, College ParkSynopsis:
Stellarators are a class of magnetic confinement devices without continuous toroidal symmetry. The design of modern stellarators often employs gradientbased optimization to navigate the highdimensional spaces used to describe their geometry. However, computing the numerical gradient of a target function with respect to many parameters can be expensive. The adjoint method allows these gradients to be computed at a much lower cost and without the noise associated with finite differences. In addition to gradientbased optimization, the derivatives obtained from the adjoint method are valuable for local sensitivity analysis and tolerance quantification.A continuous adjoint method has been developed for obtaining the derivatives of functions of the magnetohydrodynamic (MHD) equilibrium equations with respect to the shape of the boundary of the domain or the shape of the electromagnetic coils [1]. This approach is based on the generalization of the selfadjointness 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 nonlinear VMEC [2] solution, obtaining an order $10^210^3$ 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 nearaxis 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 lowcollisionality 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 threedimensional magnetic confinement equilibria. Journal of Plasma Physics 85 (2).
[2] Hirshman, S.P. & Whitson, J.C. 1983 Steepest descent moment method for threedimensional 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. 
Tuesday, December 3, 201911AM, Warren Weaver Hall 905
Finite Larmor Radius effects at the Hmode pedestal and their implication of the force free MHD equilibrium
Weili Lee, Princeton Plasma Physics LaboratorySynopsis:

Tuesday, December 10, 201911AM, Warren Weaver Hall 905
Modeling and Scalability Analysis for Pancake Charging Solenoids Using Both Direct and Iterative Solvers
Jaman Mohebujjaman, MIT PSFCSynopsis:
We present a mathematical model for the charging simulation of pancake solenoids and propose its fully discrete backwardEuler scheme. The parallel implementation of the scheme is done in PetraM. 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.
Past Events

Tuesday, November 12, 201911AM, Warren Weaver Hall 905
The Eulerian spacetime correlation of Magnetohydrodynamic (MHD) turbulence and the interpretation of Parker Solar Probe measurements
Jean Perez, Florida TechSynopsis:
Insitu 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 noncompressive 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 spacetime correlation is critical for the proper interpretation of time signals from this groundbreaking mission. However, a firstprinciple 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 twotime twopoint (spacetime) correlation for strong and weak incompressible MHD turbulence. In the strong turbulence regime, we propose a model for the spacetime correlation that extends Kraichnan's sweeping model for incompressible hydrodynamic (HD) turbulence and validate it against highresolution numerical simulations. In the weak turbulence regime, an asymptotic waveturbulence closure is used for the first time to determine the structure of spacetime correlations in weak MHD turbulence from the nonlinear equations describing the dynamics. The waveturbulence closure that we present may find applications in other weak turbulence regimes found in fluids and plasmas.

Tuesday, October 8, 201911AM, Warren Weaver Hall 905
Direct identification of important coil deviations to avoid error fields using the Hessian matrix method
Caoxiang Zhu, Princeton Plasma Physics LaboratorySynopsis:
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 W7X. 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 quasisymmetry, 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 Quasiaxisymmetric Stellarator configuration. Important perturbations that enlarge n/m = 4/11 islands and deteriorate quasiaxisymmetry 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.

Tuesday, October 1, 201911AM, Warren Weaver Hall 905
A Reactor that Spins: The Centrifugally Confined Mirror
Ian Abel, University of Maryland, College ParkSynopsis:
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. 
Tuesday, September 17, 201911AM, Warren Weaver Hall 905
The Role of 3D Geometry on Reducing Turbulent Transport in Stellarators
Benjamin Faber, University of Wisconsin, MadisonSynopsis:
The large space of possible stellarator configurations[1] provides a means to optimize different physics problems by changing the magnetic geometry, which has been most readily achieved in neoclassicaltransportoptimized stellarators such as the quasiisodynamic W7X stellarator and the quasihelically symmetric HSX stellarator. Due to its complicated physical nature and the need to analyze complex numerical simulations, comparatively little progress has been made understanding how magnetic field geometry affects transport due to driftwavedriven microturbulence. However, recent theory and simulation work has indicated the importance the choice of 3D geometry plays on determining microturbulence saturation and subsequent transport levels in stellarators. For IonTemperatureGradientdriven (ITG) turbulence, simulations indicate turbulence saturation in W7X is primarily moderated by nonlinear interactions between turbulent eddies and zonal flows, while in HSX saturation occurs mostly through nonlinear eddyeddy interactions[2]. Detailed analysis of the nonlinear threewave turbulent interactions in an analytic fluid model of ITG turbulence further indicate that for HSX, energy transfer is strongly influenced by threewave nonlinear interactions between unstable modes and stable modes at similar scales[3]. This nonlinear model provides a natural optimization metric, as nonlinear energy transfer is quantified by a threewave interaction time, where the frequencies involved in the interaction time have been modified by the turbulent nonlinearity through a nonlinear closure.Promisingly, HSX possess the flexibility to alter its MHD equilibrium by adding a magnetic well or hill component. Nonlinear gyrokinetic GENE simulations show an ITGdriven turbulence minimum when the magnetic hill is increased, where larger ITGdriven transport is observed with a large magnetic well term. These results are replicated by the nonlinear ITG fluid model, where threewave interaction times peak when the transport is minimized, indicating a more prominent role of stable modes with increasing magnetic hill, while threewave interaction times are smaller with increasing magnetic hill. This picture will be compared against detailed analysis of the spectral gyrokinetic energy transfer, which are obtained with an advanced diagnostic developed to track the evolution of nonlinear energy transfer both between different wavenumbers and different eigenmode that can expose the role of stable modes in turbulence saturation[4]. These results form the motivation for optimization starting from the HSX configuration, focused on increasing threewave interaction times to decrease turbulent transport.[1] M. Landreman, W. Sengupta, and G.G. Plunk, J. Plasma Physics, 85, 90580103 (2019)[2] G.G. Plunk, P. Xanthopoulos, and P. Helander, Phys. Rev. Lett., 119, 105002 (2017)[3] C.C. Hegna, P.W. Terry, and B.J. Faber, Phys. Plasmas, 25, 022511 (2018)[4] G.G. Whelan, M.J. Pueschel, P.W. Terry, J. Citrin, I.J. McKinney, W. Guttenfelder, and H. Doerk., “Saturation and Nonlinear Electromagnetic Stabilization of ITG Turbulence”, submitted to Physics of Plasmas.