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DOE OSTI2017

Role of a continuous MHD dynamo in the formation of 3D equilibria in fusion plasmas

Stationary 3D equilibria can form in fusion plasmas via saturation of magnetohydrodynamic (MHD) instabilities or stimulated by external 3D fields. In these cases the current profile is anomalously broad due to magnetic flux pumping produced by the MHD modes. Flux pumping plays an important role in hybrid tokamak plasmas, maintaining the minimum safety factor above unity and thus removing sawteeth. It also enables steady-state hybrid operation, by redistributing non-inductive current driven near the center by electron cyclotron waves. A validated flux pumping model is not yet available, but it would be necessary to extrapolate hybrid operation to future devices. In this work flux pumping physics is investigated for helical core equilibria stimulated by external 3D fields in DIII-D hybrid plasmas. We show that flux pumping can be produced in a continuous way by an MHD dynamo emf. The same effect maintains helical equilibria in reversed-field pinch (RFP) plasmas. The effective MHD dynamo loop voltage is calculated for experimental 3D equilibrium reconstructions, by balancing Ohm’s law over helical flux surfaces, and is consistent with the expected current redistribution. Similar results are also obtained with more sophisticated nonlinear MHD simulations. The same modelling approach is applied to helical RFP states forming spontaneously in RFX-mod as the plasma current is raised above 0.8–1 MA. This comparison allows to identify the underlying physics common to tokamak and RFP: a helical core displacement modulates parallel current density along flux tubes, which requires a helical electrostatic potential to build up, giving rise to a helical MHD dynamo flow.

3D equilibrium reconstruction↗

NASA NTRS1994

Compressional ULF waves in the outer magnetosphere. 2: A case study of Pc 5 type wave activity

In previously published work (Zhu and Kivelson, 1991) the spatial distribution of compressional magnetic pulsations of period 2 - 20 min in the outer magnetosphere was described. In this companion paper, we study some specific compressional events within our data set, seeking to determine the structure of the waves and identifying the wave generation mechanism. We use both the magnetic field and three-dimensional plasma data observed by the International Sun-Earth Explorer (ISEE) 1 and/or 2 spacecraft to characterize eight compressional ultra low frequency (ULF) wave events with frequencies below 8 mHz in the outer magnetosphere. High time resolution plasma data for the event of July 24, 1978, made possible a detailed analysis of the waves. Wave properties specific to the event of July 24, 1978, can be summarized as follows: (1) Partial plasma pressures in the different energy ranges responded to the magnetic field pressure differently. In the low-energy range they oscillated in phase with the magnetic pressure, while oscillations in higher-energy ranges were out-of-phase; (2) Perpendicular wavelengths for the event were determined to be 60,000 and 30,000 km in the radial and azimuthal directions, respectively. Wave properties common to all events can be summarized as follows: (1) Compressional Pc 5 wave activity is correlated with Beta, the ratio of the plasma pressure to the magnetic pressure; the absolute magnitude of the plasma pressure plays a minor role for the wave activity; (2) The magnetic equator is a node of the compressional perturbation of the magnetic field; (3) The criterion for the mirror mode instability is often satisfied near the equator in the outer magnetosphere when the compressional waves are present. We believe these waves are generated by internal magnetohydrodynamic (MHD) instabilities.

Zhu, Xiaoming↗

DOE OSTI2022

Ideal MHD Limited Electron Temperature in Spherical Tokamaks

It is well documented that the central electron temperature in the national spherical torus experiment (NSTX) remains largely unchanged as the external heating power, and hence the normalized volume averaged plasma pressure β increases [Stutman, Phys. Rev. Lett. 102, 115002 (2009)]. Herein we present a hypothesis that low n, pressure driven ideal magnetohydrodynamic (MHD) instabilities that are nondisruptive, can break magnetic surfaces in the central region and thereby flatten the electron temperature profiles. We demonstrate this mechanism in a 3D resistive MHD simulation of a NSTX discharge. By varying the toroidal magnetic field strength, and/or the heating power, we show that there is a critical value of β, above which the central temperature profile no longer peaks on axis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

ARC physics basis–magnetohydrodynamics

ARC is designed to produce 400 ⁢MW of net electricity and prove the commercial feasibility of a fusion power plant. In order to achieve this goal ARC has to operate with optimal core performance in a stationary scenario that minimises wear on the first wall and divertor. This requires avoiding or mitigating magnetohydrodynamic (MHD) instabilities which have the potential to not only degrade the plasma core but also lead to deleterious transient heat loads on plasma facing components. Therefore, this work aims at characterising the MHD stability of the high performance ARC scenario and inform the design of error field correction coils. Firstly, simulations of vertical displacement events show that an in-vessel coil is not needed and instead the poloidal shaping coils can be used to control vertical stability. These simulations also inform the demands on the corresponding coil power supplies. Stability analysis of the ideal kink mode with or without a conducting wall and kinetic effects suggests that the ARC baseline scenario operates deeply in the stable region. Using RDCON, tearing modes at the 𝑚/𝑛 =2/1 and 3/2 surfaces (with poloidal mode number 𝑚, and toroidal mode number 𝑛) are shown to be linearly stable, and including thermal transport effects in the rational surfaces lead to further stabilisation. However, other transient plasma instabilities can seed neoclassical tearing modes (NTMs). The marginally stable width of NTMs in ARC strongly depends on the internal inductance and can fall below 0.1% of the normalised poloidal flux. Furthermore, an empirical cross-machine model of the 𝑛 =1 error field leading to a disruption predicts a critical error field larger than SPARC but smaller than ITER. Three-dimensional coils can be designed with the Generalised Purturbed Equilbium Code based on a simple model that calculates the maximum correctable error field that is limited by the neoclassical toroidal viscosity torque. Broad scans of different coil geometries identify a set of 2 rows of off-midplane coils to be a suitable solution. It is also determined that such a set of three-dimensional coils is capable of correcting 𝑛 =2 error fields to some degree and creating strong enough 𝑛 =2 or 𝑛 =3 edge resonant perturbation fields for the suppression of edge-localised modes at reasonable coil currents. The final design of the first ARC will be further informed by results from SPARC.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2018

Gyrokinetic theory of slab universal modes and the non-existence of the gradient drift coupling (GDC) instability

A gyrokinetic linear stability analysis of a collisionless slab geometry in the local approximation is introduced. We focus on k ∥ =0 universal (or entropy) modes driven by plasma gradients at small and large plasma β. These are small scale non-MHD instabilities with growth rates that typically peak near k ⊥ ρi~1 and vanish in the long wavelength k ⊥ →0 limit. This report also discusses a mode known as the Gradient Drift Coupling (GDC) instability previously reported in the gyrokinetic literature, which has a finite growth rate γ= $\sqrt{β/[2(1+β)]C_{s}/|L_{p}|}$ with C$^{2}_{s}$ =p 0 /ρ 0 for k ⊥ →0 and is universally unstable for 1/L p ≠0. Here, we show that the GDC instability is a spurious, unphysical artifact that erroneously arises due to the failure to respect the total equilibrium pressure balance p 0 +B$^{2}_{0}$ /(8π)=constant, which renders the assumption B$^{\prime}_{0}$ 0 =0 inconsistent if p$^{\prime}_{0}$ ≠0.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2018

A Riccati Solution for the Ideal MHD Plasma Response with Applications to Real-time Stability Control

To effectuate near real-time feedback control of ideal MHD instabilities in a tokamak geometry, a rapid solution for stability analysis is a prerequisite. Toward this end, we reformulate the δW stability method with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the generic tokamak ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD matrix Riccati differential equation (MRDE). Since Riccati equations are prevalent in the control theory literature, such a shift in perspective brings to bear a range of numerical methods that are well-suited to the robust, fast solution of control problems. We discuss the usefulness of Riccati techniques in solving the stiff ODEs often encountered in ideal MHD stability analyses-—for example, in tokamak edge and stellarator physics. We then demonstrate the applicability of such methods to an existing 2D ideal MHD stability code—DCON—enabling its parallel operation in near real-time. Output is shown to match with high accuracy, and with wall-clock time &NestedLessLess; 1s. Such speed may help enable active feedback ideal MHD stability control, especially in tokamak plasmas whose ideal MHD equilibria evolve with inductive timescale τ > 1s-—as in ITER.

Parallel↗

DOE OSTI2026

Toroidal phase of post-disruption runaway electron loss to wall in presence of applied 3-D fields in DIII-D

The toroidal distribution of runaway electrons (REs) striking the centre post in DIII-D post-disruption RE final loss events is simulated utilizing the MARS-F code combined with the REORBIT module, by tracing the guiding-centre drift orbits of test REs in the presence of external (applied) plus internal (intrinsic instability) 3-D fields. To better recover the experimental results, three different equilibria with different safety factor profiles are adopted. Dominant resistive magnetohydrodynamic instabilities are found to be m/n=2/1, 3/2 and 1/1 modes (where m and n are the poloidal and toroidal mode numbers). Externally applied n=1 resonant magnetic perturbations (RMPs), with both even and odd parity configuration, are considered in each equilibrium. Locking of MHD instability to the applied RMPs is simulated by minimizing the total perturbed magnetic energy in the plasma. Here, the simulated toroidal locations, where most REs strike in the presence of an edge m/n= 2/1 instability and the n=1 RMPs in even parity, are found to agree well with experiments. Modeling also captures the measured n = 2 RE loss pattern on the wall for experiments where an edge m/n=3/2 instability is dominant instead of the m/n=2/1 mode. The key overall findings include (a) the measured global RE toroidal impact pattern in experiments can be explained by a locked resistive kink instability, (b) the toroidal phase of the locked mode is well predicted by minimizing the total perturbed magnetic energy, and (c) this edge locked mode determines the toroidal phase of the peak RE wall impact.

Bai, Xue [University of California San Diego, La J↗

DOE OSTI2018

Fast Numerical Solution of the Plasma Response Matrix for Real-time Ideal MHD Control

To help effectuate near real-time feedback control of ideal MHD instabilities in tokamak geometries, a parallelized version of A.H. Glasser’s DCON (Direct Criterion of Newcomb) code is developed. To motivate the numerical implementation, we first solve DCON’s δW formulation with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD Riccati equation. We then describe our adaptation of DCON with numerical methods natural to solutions of the Riccati equation, parallelizing it to enable its operation in near real-time. We replace DCON’s serial integration of perturbed modes—which satisfy a singular Euler- Lagrange equation—with a domain-decomposed integration of state transition matrices. Output is shown to match results from DCON with high accuracy, and with computation time < 1s. Such computational speed may enable active feedback ideal MHD stability control, especially in plasmas whose ideal MHD equilibria evolve with inductive timescale $\tau$ ≳ 1s—as in ITER. Further potential applications of this theory are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Mode spectrum characteristics and onset of the low-shear MHD stability regime

Equilibria with extended regions of weak magnetic shear, including some tokamak scenarios and stellarators, can be susceptible to pressure-driven internal MHD instabilities even though there is no mode rational surface in the plasma. Nonresonant modes, in particular, can have properties that are unattractive for confinement, including displacing substantial volumes of the plasma and leading to more efficient pressure gradient flattening in the nonlinear regime. The onset and linear properties of the low shear stability regime are examined using the initial-value, extended-MHD code M3D-C. For monotonic q-profiles, we demonstrate a clear correlation between the convergents associated with the continued fraction representation of q 0 and the spectrum of unstable modes. Nonresonant modes are observed to be destabilized preferentially to any other resonant instability with the same toroidal mode number when n > 1. Using the observed connection between the spectrum associated with q 0 and the overall equilibrium stability properties, we suggest a technique for reducing the uncertainty on both q 0 and magnetic shear in the core region, obtained either via measurement or through the analysis and reconstruction of experimental results.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1989

Nonlinear twist-kink instability of a coronal loop

Three-dimensional magnetoinductive particle simulations are used to demonstrate that the mechanical twisting motion applied to a magnetized plasma column induces a current aligned to the external magnetic field direction, pinches the plasma and magnetic fields, and stores the energy in poloidal magnetic fields. As the twist motion continues, the field lines locally begin to wrap around the plasma more than one revolution. A strong MHD instability sets in that is a mixture of kink and ballooning modes, releasing the magnetic energy and causing destruction of coherent column structure and flows of turbulent plasma. A similar episode ensues, exhibiting relaxation oscillations. The buildup of poloidal fields and structure and its sudden release driven by the twist motion may be a model for the solar coronal loop dynamics which exhibits a slow energy buildup with some photospheric motion and a sudden energy release by flares.

Zaidman, E. G.↗

DOE OSTI2023

Effects of NBI-induced energetic ions on internal kink stability in HL-2M

A systematic investigation is carried out, studying the effect of the neutral beam injection induced energetic particles (EPs) on the n = 1 (n is the toroidal mode number) internal kink (IK) instability in the HL-2M tokamak, utilizing the MARS-F/K code [Liu et al., Phys. Plasmas 7, 3681 (2000) and 15, 112503 (2008)]. A high-beta sawteething HL-2M scenario, simulated by the TRANSP code [Breslau et al. Computer Software (2018)], is chosen for this study. Compared to the fluid model, non-perturbative magnetohydrodynamic (MHD)-kinetic hybrid computations with MARS-K show a generally stabilization effect on the IK, due to drift kinetic resonances associated with EPs. Further, the bounce resonance of trapped EPs has minor influence on the mode stability. In the absence of the plasma equilibrium flow and with the assumed particle pitch distribution, the transit resonance of co-current (countercurrent) passing EPs destabilizes (stabilizes) the IK. With plasma flow, both co- and countercurrent passing EPs tend to stabilize the mode, but the effect is stronger with the countercurrent particles. These modeling results provide useful guidance for interpreting MHD instabilities in the future high-performance experiments in HL-2M.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Laboratory study of the failed torus mechanism in arched, line-tied, magnetic flux ropes

Coronal mass ejections (CMEs) are some of the most energetic and violent events in our solar system. The prediction and understanding of CMEs are of particular importance due to the impact that they can have on Earth-based satellite systems and, in extreme cases, ground-based electronics. CMEs often occur when long-lived magnetic flux ropes (MFRs) anchored to the solar surface destabilize and erupt away from the Sun. One potential cause for these eruptions is an ideal magnetohydrodynamic (MHD) instability, such as the kink or torus instability. Previous experiments on the magnetic reconnection experiment revealed a class of MFRs that were torus-unstable but kink-stable, which failed to erupt. These “failed-tori” went through a process similar to Taylor relaxation, where the toroidal current was redistributed before the eruption ultimately failed. Herein we have investigated this behavior through additional diagnostics that measure the current distribution at the foot points and the energy distribution before and after an event. These measurements indicate that ideal MHD effects are sufficient to explain the energy distribution changes during failed torus events. This excludes Taylor relaxation as a possible mechanism of current redistribution during an event. A new model that only requires non-ideal effects in a thin layer above the electrodes is presented to explain the observed phenomena. This work broadens our understanding of the stability of MFRs and the mechanism behind the failed torus through the improved prediction of the torus instability and through new diagnostics to measure the energy inventory and current profile at the foot points.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Deep Koopman Neural Network for Analyzing High-Energy-Density Simulations of Electrical Wire Explosions

Megaampere-scale electrical wire experiments (EWEs) provide a platform for studying magnetohydrodynamic (MHD) instability growth in magneto-inertial fusion (MIF) devices. Even when nonlinear simulations of these experiments can digitally reproduce much of the experimentally observed instability growth, interpreting the results and understanding mode growth and evolution can be non-trivial. As a first step toward providing better interpretation of these simulation features, this work investigates the use of a deep neural network that uses Koopman operator theory to analyze the dynamics of pulsed-power-driven explosions of EWEs. This deep neural network is trained on 1-D resistive MHD simulations of EWEs. This neural network learns to transform the nonlinear data into a lower-dimensional representation where the time dynamics are linear. Layers of this neural network are shown to learn features of the simulations, including the locations of shock waves and different physical regimes of the simulation. Using the learned features, the network can compress a time state of the simulation consisting of 5120 data point into a 36-parameter lower-dimensional latent space embedding. Furthermore, these embeddings are shown to be clustered in the latent space by initial radius and time state.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Simulations of saturated MHD activity in the HBT-EP tokamak

The NIMROD code is used to perform simulations of MHD activity in the HBT-EP tokamak, including the effect of a resistive wall. Linear simulations are conducted to scan current and pressure profiles to establish self-consistent baseline equilibria that result in MHD instabilities with low error between experimental diagnostics and synthetic equivalents. A family of such equilibria is found with relatively peaked current and pressure profiles. Nonlinear simulations are performed using the optimal equilibrium, which is linearly unstable to a Resistive Wall Mode—stable in the ideal wall limit and with an Alfvénic growth rate in the no-wall limit. Using resistive wall boundary conditions, nonlinear Hall MHD simulations yield a rotating, saturated n=1 mode similar to the type observed experimentally in HBT-EP discharges. An n=1 mode around the q=2 surface mediates a localized axisymmetric perturbation that flattens the plasma current from a linearly unstable state to a 3D, rotating, stable state. During the saturated period, periodic changes in inter-diagnostic phase lag allow for qualitative estimations of effective, macroscopic transport coefficients used in MHD closure models and could be extended to model post-disruption MHD structure rotation after thermal quenches seen in HBT-EP.

Arnold, David A. (ORCID:0009000059535761)↗

DOE OSTI2020

Interactions Between Microturbulence and Magnetohydrodynamic Instabilities in Magnetized Plasmas (Final Technical Report)

This document details the work accomplished during the period 04/15/2012-01/31/2020 for the collaborative project “Interactions Between Microturbulence and Magnetohydrodynamic Instabilities in Magnetized Plasmas,” and serves as the final report on the University of California, San Diego part of this collaborative project for Department of Energy Office of Science - Fusion Energy Sciences Grant No. DE- SC0007783. This project focused on the development of analytic and computational tools, seeking to better understand the mutual interactions between small-scale temperature-gradient driven microturbulence and macroscopic magnetohydrodynamic (MHD) instabilities, particularly tearing modes, in magnetically confined plasmas relevant for the generation of fusion energy. This project advanced understanding through numerical simulations of different sets of model equations, corresponding to different microturbulence instabilities. Questions addressed included how the magnitudes, spatial structures, and scaling's of turbulent forcing's and drags impact magnetic island dynamics, as well as quantifying the “region of influence” of an island on the microturbulence dynamics. A key finding of the project is that the microturbulence fluctuations and associated cross-field particle and energy fluxes exhibit spatial asymmetries for sufficiently large islands. The results have improved our fundamental understanding of plasma dynamics, motivated new experimental investigations, and will help inform future fusion predictive modeling efforts.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

Self-consistent hybrid neoclassical-magnetohydrodynamic simulations of axisymmetric plasmas

Neoclassical effects (e.g., conductivity reduction and bootstrap currents) have a profound impact on many magnetohydrodynamic (MHD) instabilities in toroidally-confined plasmas, including tearing modes, edge-localized modes, and resistive wall modes. High-fidelity simulations of such phenomena require a multiphysics code that self-consistently couples the kinetic and fluid models. We review a hybrid formulation from the recent literature AB that is appropriate for such studies. In particular, the formulation uses a set of time-dependent drift-kinetic equations (DKEs) to advance the non-Maxwellian part of the electron and ion distribution functions ( fNM ) with linearized Fokker-Planck-Landau collision operators. The form of the DKEs used were derived in a Chapman-Enskog-like fashion, ensuring that fNM carries no density, momentum, or temperature. Rather, these quantities are contained within the background Maxwellian and are evolved by a set of MHD equations which are closed by moments of fNM . We then present two DKE solvers based upon this formulation in axisymmetric toroidal geometries. The Neoclassical Ion-Electron Solver (NIES) solves the steady-state DKEs in the low-collisionality limit. Convergence and benchmark studies are discussed, providing a proof-of-principle that this new formulation can accurately reproduce results from the literature in the limit considered. We then present the DK4D code which evolves the finite-collisionality DKEs time-dependently. Computational methods used and successful benchmarks to other neoclassical models and codes are discussed. Furthermore, we couple DK4D to a reduced, transport-timescale MHD code. The resulting hybrid code is used to simulate the evolution of the current density in a large-aspect-ratio plasma in the presence of several different time-dependent pressure profiles. These simulations demonstrate the self-consistent, dynamic formation of the ohmic and bootstrap currents. In the slowly-evolving plasmas considered, these first-principle simulations are shown to verify a simpler, steady-state neoclassical model that is commonly used in transport codes. Future work will involve coupling DK4D to a spatially three-dimensional, extended MHD code, generalizing DK4D to nonaxisymmetric geometries, and simulating more quickly-evolving and realistic plasmas. The ultimate goal of this work is to perform self-consistent, hybrid simulations of complex tokamak instabilities and calculations of neoclassical toroidal viscosity. A J. J. Ramos, Phys. Plasmas 17 , 082502 (2010). B J. J. Ramos, Phys. Plasmas 18 , 102506 (2011).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2004

Magnetohydrodynamic Instabilities in a Simple Gasdynamic Mirror Propulsion System

The gasdynamic mirror has been proposed as a concept which could form the basis of a highly efficient fusion rocket engine. Gasdynamic mirrors differ from most other mirror type plasma confinement schemes in that they have much larger aspect ratios and operate at somewhat higher plasma densities. To evaluate whether a gasdynamic mirror could indeed confine plasmas in a stable manner for long periods of time, a small scale experimental gasdynamic mirror was built and tested. The objective of this experiment was to determine ranges of mirror ratios and plasma densities over which gasdynamic mirror could maintain stable plasmas. Theoretical analyses indicated that plasma magnetohydrodynamic instabilities were likely to occur during subsonic to supersonic flow transitions in the mirror throat region of the gasdynamic mirror. The experimental evidence based upon data derived from the Langmuir probe measurements seems to confirm this analysis. The assumption that a gasdynamic mirror using a simple mirror geometry could be used as a propulsion system, therefore, appears questionable. Modifications to the simple mirror concept are presented which could mitigate these MHD instabilities.

Emrich, William J., Jr.↗

DOE OSTI2020

Eigenmode analysis of the sheared-flow Z-pinch

Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, but C is order unity and α≈1. It is further demonstrated that even a large value of shear, duz/dr=3vi/a, is not sufficient to provide linear stabilization of the m=1 kink mode for all profiles considered. This result is in contrast to the currently accepted theory predicting stabilization at much lower shear, duz/dr=0.1vi/a, and suggests that the experimentally observed stability cannot be explained within the linear ideal-MHD model.

Angus, J. R. (ORCID:0000000314740002)↗