This paper presents the development of an all-in-one probe to simultaneously measure all components of the generalized Ohm’s law in reversed-field pinch plasmas and tokamaks. The polyhedral configuration of the Mach probe is achieved through the specific arrangement, angle, and depth of the collimator channel apertures drilled into the surface of a hollow boron nitride cylinder encasing it. This probe includes a central Mach probe to assess the ion velocity field in three dimensions. Initial tests at the RELAX and Madison Symmetric Torus machines have confirmed the probe’s effectiveness, revealing an octahedron form similar to a tetrahedron. The probe seems to function correctly and is expected to facilitate the empirical validation of two-fluid equilibria at the periphery of toroidal plasmas.

To better understand the interaction of global tearing modes and microturbulence in the Madison Symmetric Torus (MST) reversed-field pinch (RFP), the global gyrokinetic code \textsc{Gene} is modified to describe global tearing mode instability via a shifted Maxwellian distribution consistent with experimental equilibria. The implementation of the shifted Maxwellian is tested and benchmarked by comparisons with different codes and models. Good agreement is obtained in code-code and code-theory comparisons. Linear stability of tearing modes of a non-reversed MST discharge is studied. A collisionality scan is performed to the lowest order unstable modes ($n=5$, $n=6$) and shown to behave consistently with theoretical scaling. The nonlinear evolution is simulated, and saturation is found to arise from mode coupling and transfer of energy from the most unstable tearing mode to small-scale stable modes mediated by the $m=2$ tearing mode. The work described herein lays the foundation for nonlinear simulation and analysis of the interaction of tearing modes and gyroradius-scale instabilities in RFP plasmas.

Internal fluctuation measurements with Faraday-effect polarimetry in the DIII-D tokamak reveal the onset of a tearing mode with toroidal mode number n = 3 well before it is detected by the sensing coils external to the plasma. This mode appears before the n = 2, 1 modes and is first detected with internal measurements at a lower value of the ideal-wall kink beta limit than is indicated at the time of first detection by the sensing coils. When the mode is first detected, the linear resistive stability parameter, Δ′, indicates marginal stability and continues to do so until later when the mode amplitude begins increasing linearly with time—together suggesting a neoclassical origin for this mode.

Measurements and simulations show that plasma relaxation processes in the reversed field pinch drive and redistribute both magnetic flux and momentum. To examine this relaxation process, a new 3D Mach B-dot probe has been constructed. This probe collects ion saturation currents through six molybdenum electrodes arranged on the flattened vertices of an octahedron made of boron nitride (BN). The ion saturation current flows through configurable voltage dividers for measurement and returns through one of six selectable return electrodes equally spaced along the 12 cm BN probe arm. In addition, the probe arm houses three B-dot magnetic pickup coils in the BN stalk immediately below to the octahedron, to measure the local magnetic field. Inserted in the Madison Symmetric Torus (MST) during deuterium discharges with 220 kA plasma current, density of 0.8 × 1013 cm−3, the probe collects ion saturation currents with sawtooth-like peaks correlated with relaxation events. This compact octahedral design fitting six Mach electrode surfaces within a 1 cm3 cube will enable future multi-point, multi-field probes compatible with the 1.5 in. ports of MST. Such probes will allow for flow circulation, current, and canonical vorticity to be calculated in the center of the finite difference stencil formed by the measurement locations.

Current-carrying, toroidal laboratory plasmas typically cannot be sustained with an electron density above the empirical Greenwald limit. Presented here are tokamak experiments in the Madison Symmetric Torus with a density up to an unprecedented level about 10 times this limit. This is thought to be made possible in part by a thick, stabilizing, conductive wall, and a high-voltage, feedback-controlled power supply driving the plasma current. The radial profile of the toroidal current flattens around twice the limit, without the edge collapse routinely observed in other experiments.

This white paper summarizes major scientific challenges and opportunities in understanding magnetic reconnection and related explosive phenomena as a fundamental plasma process.

A numerical study of magnetohydrodynamics (MHD) and tracer-particle evolution investigates the effects of resonant magnetic perturbations (RMPs) on the confinement of runaway electrons (REs) in tokamak discharges conducted in the Madison Symmetric Torus. In computational results of applying RMPs having a broad toroidal spectrum but a single poloidal harmonic, m = 1 RMP does not suppress REs, whereas m = 3 RMP achieves significant deconfinement but not the complete suppression obtained in the experiment [Munaretto et al., Nuclear Fusion 60, 046024 (2020)]. MHD simulations with the NIMROD code produce sawtooth oscillations, and the associated magnetic reconnection can affect the trajectory of REs starting in the core region. Simulations with m = 3 RMP produce chaotic magnetic topology over the outer region, but the m = 1 RMP produces negligible changes in field topology, relative to applying no RMP. Using snapshots of the MHD simulation fields, full-orbit relativistic electron test particle computations with KORC show ≈50% loss from the m = 3 RMP compared to the 10%–15% loss from the m = 1 RMP. Test particle computations of the m = 3 RMP in the time-evolving MHD simulation fields show correlation between MHD activity and late-time particle losses, but total electron confinement is similar to computations using magnetic-field snapshots.

Tokamak plasmas often exhibit self-organizing behavior in which internal modes shape the toroidal current density profile, a common example being the sawtooth instability. However, such behavior has not been studied in detail for edge safety factor below 2 due to disruptive kink instabilities that typically prevent operation in this regime. Now, steady tokamak plasmas with an edge safety factor down to 0.8 have been created in the Madison Symmetric Torus, where disruptions are prevented due to a thick, conductive wall and a feedback power supply that sustains the plasma current. Internal measurements and nonlinear magnetohydrodynamic modeling reveal a family of safety factor profiles with a central value clamped near unity as the edge safety factor decreases, indicating current profile broadening through a relaxation process. As the safety factor decreases, the magnetic fluctuations become irregular, and the electron energy confinement time decreases.

Novel x-ray detection technology enabled the first profile measurements of the birth and growth dynamics of runaway electrons (REs) at the edge of tokamaks during quiescent RE studies at the Madison Symmetric Torus. The formation of an off-axis RE seed with linear growth rates has been resolved for low energies, a hollow streaming parameter and large electric fields ($E_{\parallel}/E_{D}$) in agreement with theory and simulations. Secondary exponential growth rates have also been spatially resolved for the first time and are consistent with a convective transport of the order of the Ware pinch and energies up to $10^3\times T_{e,0}$. Numerical simulations are shown to reproduce the experimental observations including the off-axis runaway electron generation, radial transport and exponential growth at the core, as well as suppression due to $m=3$ resonant magnetic perturbations.

The first local velocity measurements of helical equilibrium plasmas in the Reversed Field Pinch (RFP) Single Helical Axis (SHAx) state using a Charge Exchange Recombination Spectroscopy (CHERS) diagnostic are presented. Measurements show strong axisymmetric and non-axisymmetric flow, with n = 5 components of flow related to the (m,n)=(1,5) dominant magnetic mode on the order of the axisymmetric flow in certain regions of the plasma, as well as significant n > 5 flow. Flow measurements are compared with NIMROD simulations of visco-resistive, single-fluid MHD in toroidal and cylindrical geometries with limited axial periodicity. Both measurements and the simulation with toroidal geometry show stronger inboard flows relative to the outboard flows, which is attributed to the toroidal geometry of the device. In the experiment, the n = 5 component of flow is phase shifted from the reconnection-like flow pattern observed in the single-fluid simulations, possibly due to decoupling of the ion and electron fluids over much of the plasma. Finally, the strength of the helical angular flow shear relative to the critical shear necessary to disrupt nonlinear coupling between tearing modes is calculated around the helical magnetic axis. The shear in the measured flow is on the order of the theoretical critical threshold needed to nonlinearly decouple modes, but the measurement uncertainty in the gradient of the flow is large.

Reversed field pinch (RFP) plasmas are subject to tearing instability that creates a broad spectrum of magnetic fluctuations. The dominant fluctuations have poloidal and toroidal mode numbers (m,n)=(1,6−10) and can grow to 2–3% of the mean magnetic field. Through nonlinear coupling, this growth culminates in a strong reconnection event and broadening of the magnetic spectrum extending to the ion gyroradius scale. Multiple developments occur during the reconnection stage: ions and electrons are energized, magnetic fluctuation amplitudes increase, plasma flow is halted, and the toroidal magnetic flux increases in a sawtooth-like fashion as the RFP dynamo becomes stronger. Magnetic fluctuations are measured in the plasma edge at multiple radial locations from r/a = 0.75 to 0.96 to assess and characterize the magnetic turbulence. The measured spectrum perpendicular to the mean field, S(k⊥), can be fit to a model spectrum consisting of power-law and exponential component with one free parameter that characterizes dissipation. The measured dissipation is much larger than estimated from classical viscous or resistive dissipation, but it is consistent with a flow damping measurement of anomalous viscosity. The measurements show an evolution of the spectrum during which fluctuation power builds up in the smallest wavenumbers and cascades to the larger wavenumber due to the nonlinear coupling between the linear (m, n) = (1, > 6) and the nonlinear (m, n) = (0, 1) tearing modes.

A multi-energy soft x-ray pinhole camera has been designed, built, and deployed at the Madison Symmetric Torus to aid the study of particle and thermal transport, as well as MHD stability physics. This novel imaging diagnostic technique employs a pixelated x-ray detector in which the lower energy threshold for photon detection can be adjusted independently on each pixel. The detector of choice is a PILATUS3 100 K with a 450 μm thick silicon sensor and nearly 100 000 pixels sensitive to photon energies between 1.6 and 30 keV. An ensemble of cubic spline smoothing functions has been applied to the line-integrated data for each time-frame and energy-range, obtaining a reduced standard-deviation when compared to that dominated by photon-noise. The multi-energy local emissivity profiles are obtained from a 1D matrix-based Abel-inversion procedure. Central values of Te can be obtained by modeling the slope of the continuum radiation from ratios of the inverted radial emissivity profiles over multiple energy ranges with no a priori assumptions of plasma profiles, magnetic field reconstruction constraints, high-density limitations, or need of shot-to-shot reproducibility. In tokamak plasmas, a novel application has recently been tested for early detection, 1D imaging, and study of the birth, exponential growth, and saturation of runaway electrons at energies comparable to 100 × Te,0; thus, early results are also presented.

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Programmable control of the inductive electric field enables advanced operations of reversed-field pinch (RFP) plasmas in the Madison Symmetric Torus (MST) device and further develops the technical basis for ohmically heated fusion RFP plasmas. MST’s poloidal and toroidal magnetic fields ( B p and B t ) can be sourced by programmable power supplies (PPSs) based on integrated-gate bipolar transistors (IGBT). In order to provide real-time simultaneous control of both B p and B t circuits, a time-independent integrated model is developed. The actuators considered for the control are the B p and B t primary currents produced by the PPSs. The control system goal will be tracking two particular demand quantities that can be measured at the plasma surface ( r = a ): the plasma current, I p ∼ B p ( a ), and the RFP reversal parameter, F ∼ B t ( a )/Φ, where Φ is the toroidal flux in the plasma. The edge safety factor, q ( a ) ∝ B t ( a ), tends to track F but not identically. To understand the responses of I p and F to the actuators and to enable systematic design of control algorithms, dedicated experiments are run in which the actuators are modulated, and a linearized dynamic data-driven model is generated using a system identification method. We perform a series of initial real-time experiments to test the designed feedback controllers and validate the derived model predictions. The feedback controllers show systematic improvements over simpler feedforward controllers.