Magnetic Confinement Fusion Plasmas Research Articles

Measurement of the Gamma-Ray-to-Neutron Branching Ratio for the Deuterium-Tritium Reaction in Magnetic Confinement Fusion Plasmas.

At present, magnetic confinement fusion devices rely solely on absolute neutron counting as a direct way of measuring fusion power. Absolute counting of deuterium-tritium gamma rays could provide the secondary neutron-independent technique required for the validation of scientific results and as a licensing tool for future power plants. However, this approach necessitates an accurate determination of the gamma-ray-to-neutron branching ratio. The gamma-ray-to-neutron branching ratio for the deuterium-tritium reaction ^{3}H(^{2}H,γ)^{5}He/^{3}H(^{2}H,n)^{4}He was determined in magnetic confinement fusion plasmas at the Joint European Torus in predominantly deuterium beam heated plasmas. The branching ratio was found to be equal to (2.4±0.5)×10^{-5} over the deuterium energy range of (80±20) keV. This accurate determination of the deuterium-tritium branching ratio paves the way for a direct and neutron-independent measurement of fusion power in magnetic confinement fusion reactors, based on the absolute counting of deuterium-tritium gamma rays.

Enhancement of fusion reactivity under non-Maxwellian distributions: effects of drift-ring-beam, slowing-down, and kappa super-thermal distributions

Non-Maxwellian distributions of particles are commonly observed in fusion studies, especially for magnetic confinement fusion plasmas. The particle distribution has a direct effect on fusion reactivity, which is the focus of this study. We investigate the effects of three types of non-Maxwellian distributions, namely drift-ring-beam, slowing-down, and kappa super-thermal distributions, on the fusion reactivities of D-T (Deuterium-Trillium) and p-B11 (proton-Boron) using a newly developed program, where the enhancement of fusion reactivity relative to the Maxwellian distribution is computed while keeping the total kinetic energy constant. The calculation results show that for the temperature ranges of interest to us, namely 5–50 keV for D-T and 100–500 keV for p-B11, these non-Maxwellian distributions can enhance the fusion reactivities. In the case of the drift-ring-beam distribution, the perpendicular ring beam velocity leads to decreased enhancement in low temperature range and increased enhancement in high temperature range. This effect is favorable for p-B11 fusion reaction and unfavorable for D-T fusion reaction. This is because the important temperature range for fusion reaction significantly overlaps with the high temperature enhancement range, while the important temperature range for fusion significantly overlaps with the low temperature reduction range. In the slowing-down distribution, the birth speed plays a crucial role in both reactions, and increasing birth speed leads to a shift in the enhancement ranges towards lower temperatures, which is beneficial for both reactions. Finally, the kappa super-thermal distribution results in a relatively large enhancement in the low temperature range with a small high energy power-law index . Overall, this study provides insight into the effects of non-Maxwellian distributions on fusion reactivity and highlights potential opportunities for enhancing fusion efficiency.

Lagrangian Chaotic Mixing in a Nonlinear Model of Resistive Drift-wave Turbulence in Tokamak Plasmas

In fusion plasma, numerical simulations are commonly employed to investigate the confinement properties of plasma in the bulk region of tokamaks. The modified Hasegawa-Wakatani (MHW) equations are used to model the behavior of the plasma, which enables us to understand the radial transport in two-dimensional numerical simulations of electrostatic resistive drift-wave turbulence. By utilizing the MHW equations, we have gained insights into the low-to-high confinement (L-H) transitions that occur spontaneously in the plasma when it moves from a low confinement stage, characterized by turbulent flow, to a turbulence-suppressed regime known as zonal flow. To investigate these transitions, we vary the value of a control parameter \(\alpha\), which is related to adiabaticity, in numerical simulations, and observe the transition between the two regimes. This simplified model of L-H transitions can provide valuable information for tokamaks. To identify the Lagrangian coherent structures (LCS) and to better characterize the chaotic mixing during the L-H transition, we computed the finite-time Lyapunov exponent (FTLE) of the calculated velocity field derived from the electrostatic potential. We further compared the statistics of the chaotic mixing of the two regimes. The results of our study offer insights into the turbulent transport processes in magnetic confinement fusion plasmas.

Conceptual design of a GEM (gas electron multiplier) based gas Cherenkov detector for measurement of 17MeV gamma rays from T(D, γ)5He in magnetic confinement fusion plasmas.

The only method for assessing the fusion power throughput of a deuterium-tritium (DT) reactor presently relies on determining the absolute number of 14MeV neutrons produced in the DT plasma. An independent method, developed and investigated during the recent DT campaign at the Joint European Torus, is based on the absolute counting of 17MeV gamma rays produced by the competing T(D, γ)5He reaction that features a very weak branching ratio (about 3-6 × 10-6) when compared to the main T(D,n)4He reaction. The state-of-the-art spectrometer used for gamma-ray measurements in magnetic confinement fusion plasmas is LaBr3(Ce) scintillator detectors, although they require significant neutron shielding to extract a relatively weak gamma-ray signal from a much more abundant neutron field. A better approach relies on a gamma-ray detector that is intrinsically insensitive to neutrons. We have advanced the design of a gamma-ray counter based on the Cherenkov effect for gamma-rays whose energy exceeds 11MeV, optimized to work in the neutron-rich environment of a steady-state, magnetically confined fusion plasma device. The gamma-rays interact with an aluminum window and extract electrons that move into the radiator emitting photons via the Cherenkov effect. Since the Cherenkov light consists of few photons (25 on average) in the far UV band (100-200nm), a pre-amplifier is required to transport the photons to the neutron-shielded location, which may be a few meters away, where the readout elements of the detector, either a silicon or standard photomultiplier tube, are placed. The present work focuses on the development of a scintillating GEM (Gas Electron Multiplier) based pre-amplifier that acts as a Cherenkov photon pre-amplifier and wavelength shifter. This paper presents the result of a set of Garfield++ simulations developed to find the optimal GEM working parameters. A photon gain of 100 is obtained by biasing a single GEM foil to 1kV.

Investigation of ICRF-NBI synergetic heating induced fast ion distribution and transport in EAST tokamak

In magnetic confinement fusion plasmas, radio-frequency wave heating in the ion cyclotron range of frequencies (ICRF) and neutral beam injection (NBI) are two main heating methods. Their synergetic heating has long been a key topic in fusion research. In this work, we clarify the basic principles of ICRF high harmonic heating and the synergetic heating between ICRF and NBI. Then, we perform a series of experiments on EAST tokamak and carry out the corresponding TRANSP simulations. The results indicate that the ICRF-NBI synergetic heating not only significantly increases the plasma parameters (including poloidal beta, plasma stored energy, ion temperature and neutron yield), but also generates a large number of energetic particles and develops an energetic particle tail in its distribution function. For instance, the ICRF third harmonic heating with 1 MW of power can increase the energy of NBI fast ions from 60 to 600 keV. By changing the hydrogen minority concentration, improving the ICRF and NBI heating power, using the on-axis ICRF heating or optimizing the NBI injection angle, the ICRF-NBI synergetic heating effect can be further enhanced, accompanied with an increase of fast ion energy. Moreover, by using the fast ion distribution as input in the orbit tracing code, the transport and loss of energetic particles are calculated. The results show that the initial positions of the lost energetic particles are on the low field side, and their orbits are mainly trapped orbits. The loss of energetic particles is mainly located in the middle and upper plane of the main limiter, ICRF and LH antenna limiters. The lost of these energetic particles are considered as one of the main reasons why hot spots occur on the limiters.

Design of a multi-detector, single line-of-sight, time-of-flight system to measure time-resolved neutron energy spectra.

In the dynamic environment of burning, thermonuclear deuterium-tritium plasmas, diagnosing the time-resolved neutron energy spectrum is of critical importance. Strategies exist for this diagnosis in magnetic confinement fusion plasmas, which presently have a lifetime of ∼1012 longer than inertial confinement fusion (ICF) plasmas. Here, we present a novel concept for a simple, precise, and scale-able diagnostic to measure time-resolved neutron spectra in ICF plasmas. The concept leverages general tomographic reconstruction techniques adapted to time-of-flight parameter space, and then employs an updated Monte Carlo algorithm and National Ignition Facility-relevant constraints to reconstruct the time-evolving neutron energy spectrum. Reconstructed spectra of the primary 14.028MeV nDT peak are in good agreement with the exact synthetic spectra. The technique is also used to reconstruct the time-evolving downscattered spectrum, although the present implementation shows significantly more error.

Machine learning application in complicated burning plasmas for future magnetic fusion exploration

On October 24, 2021, the Chinese government unveiled a guiding document about state-level work to achieve carbon peaking and carbon neutrality goals under a new development philosophy, laying out key targets and measures for upcoming decades. The documment, titled “Working Guidance For Carbon Dioxide Peaking and Carbon Neutrality In Fully And Faithful Implementation Of The New Development Philosophy.” stated that China's carbon dioxide emissions per unit GDP would have dropped by 18% from the level in 2020 by 2025 and by more than 68% from the level in 2005 by 2030. Fusion energy, as an environment-friendly and intrinsically safe energy source with an abundant fuel supply, is considered an effective low-carbon ultimate solution. Fusion research has been conducted for 60 years, and has been demonstrated that magnetic confinement controlled fusion is very effective magnetic confinement fusion research has progressed significantly over the years. Generally, it is believed in the magnetic confinement fusion plasma community that the next pivotal step in magnetic confinement fusion plasma research is to create burning plasma, where the dominant method of heating plasmas is alpha particle self-sustaining heating. The research aims to understand the underlying physics of the confinement, heating, and instability of burning plasma and to explore the technologies associated with the power-producing fusion reactor. In future burning plasmas, multiple mode-number instabilities are very important but complicated physics issues because of the employment of strong power auxiliary heating schemes. Compared with the time-consuming large-scale computer simulation, the machine learning can bring an advantage for the investigations concerning multiple mode-number instabilities, especially in predictions compared to the time-consuming large-scale computer simulation.

Design of Thomson scattering diagnostic system on linear magnetized plasma device

In addition to the magnetic confinement fusion plasma, Thomson scattering has been applied to measure electron density and temperature of low-temperature plasmas. Based on a linear magnetized plasma device, a set of Thomson scattering diagnostic system is designed to diagnose the plasma with and eV. Due to low plasma temperature and density, this diagnostic system needs high spectral resolution and collection efficiency to meet the requirements of electron velocity distribution function measurements. Through the bench test, it is confirmed that the spectral resolution reaches 0.01 nm, and theoretical collection efficiency is high enough to obtain a Thomson scattering spectrum by 1000 accumulations.

Investigation of a Cherenkov-based gamma-ray diagnostic for measurement of 17 MeV gamma-rays from T(D, γ)5He in magnetic confinement fusion plasmas

At present, the only method for assessing the fusion power throughput of a reactor relies on the absolute measurement of 14 MeV neutrons produced in the D-T nuclear reaction. For ITER and DEMO, however, at least another independent measurement of the fusion power is required. The 5He* nucleus produced in the D-T fusion reaction has two de-excitation channels. The most likely is its disintegration in an alpha particle and a neutron, D + T → 5He* → α + n, by means of the nuclear force. There is however also an electromagnetic channel, with a branching ratio ∼10−5, which leads to the emission of a 17 MeV gamma-ray, i.e. D + T → 5He* → 5He + γ. The detection of this gamma-ray emission could serve as an independent method to determine the fusion power. In order to enable 17 MeV gamma-ray measurements, there is need for a detector with some coarse energy discrimination and, most importantly, capable of working in a neutron-rich environment. Conventional inorganic scintillators, such as LaBr3(Ce), have comparable efficiencies to neutrons and gamma-rays and they cannot be used for 17 MeV gamma-ray measurements without significant neutron shielding. In order to overcome this limitation, we here propose the conceptual design of a gamma-ray counter with a variable energy threshold based on the Cherenkov effect and designed to operate in intense neutron fields. The detector geometry has been optimized using Geant4 so to achieve a gamma-ray to neutron efficiency ratio better than 105. The design is based on a gas Cherenkov detector and the photo-sensor is still to investigated.

Modeling of particle transport, neutrals and radiation in magnetically-confined plasmas with Aurora

We present Aurora, an open-source package for particle transport, neutrals and radiation modeling in magnetic confinement fusion plasmas. Aurora’s modern multi-language interface enables simulations of 1.5D impurity transport within high-performance computing frameworks, particularly for the inference of particle transport coefficients. A user-friendly Python library allows simple interaction with atomic rates from the Atomic Data and Atomic Structure database as well as other sources. This enables a range of radiation predictions, both for power balance and spectroscopic analysis. We discuss here the superstaging approximation for complex ions, as a way to group charge states and reduce computational cost, demonstrating its wide applicability within the Aurora forward model and beyond. Aurora also facilitates neutral particle analysis, both from experimental spectroscopic data and other simulation codes. Leveraging Aurora’s capabilities to interface SOLPS-ITER results, we demonstrate that charge exchange is unlikely to affect the total radiated power from the ITER core during high performance operation. Finally, we describe the ImpRad module in the one modeling framework for integrated task framework, developed to enable experimental analysis and transport inferences on multiple devices using Aurora.

Integration of the TESPEL injection system at W7-X

Since impurities determine the plasma performance in magnetically-confined fusion plasmas, analyzing their behavior and controlling their confinement is significantly important for a stable and steady-state operation of fusion devices. A comparison of the impurity transport in the core region of large stellarators, such as Wendelstein 7-X and Large Helical Device (LHD) is quite useful to gain a better understanding of the impurity transport in reactor-relevant stellarator plasmas. Therefore, a Tracer-Encapsulated Solid Pellet (TESPEL) injection system has been newly installed for the operational phase OP1.2b of W7-X. The injector was designed and manufactured by NIFS to achieve a similar TESPEL deposition location as in LHD. Technical guidelines and functional specifications, addressing the specific conditions at W7-X had been implemented in close collaboration between NIFS and IPP. This resulted in a compact 3-stage differential pumping system considering spatial restrictions due to existing and future diagnostics, material requirements and limitations due to the magnetic stray field in the vicinity of the turbo pumps. This contribution reports on the design and the integration of the new TESPEL injection system to W7-X and first achievements during the commissioning of the system.

Interpreting observations of ion cyclotron emission from large helical device plasmas with beam-injected ion populations

Ion cyclotron emission (ICE) is detected from all large toroidal magnetically confined fusion (MCF) plasmas. It is a form of spontaneous suprathermal radiation, whose spectral peak frequencies correspond to sequential cyclotron harmonics of energetic ion species, evaluated at the emission location. In ICE phenomenology, an important parameter is the value of the ratio of energetic ion velocity to the local Alfvén speed . Here we focus on ICE measurements from heliotron-stellarator hydrogen plasmas, heated by energetic proton neutral beam injection (NBI) in the large helical device, for which takes values both larger (super-Alfvénic) and smaller (sub-Alfvénic) than unity. The collective relaxation of the NBI proton population, together with the thermal plasma, is studied using a particle-in-cell (PIC) code. This evolves the Maxwell–Lorentz system of equations for hundreds of millions of kinetic gyro-orbit-resolved ions and fluid electrons, self-consistently with the electric and magnetic fields. For LHD-relevant parameter sets, the spatiotemporal Fourier transforms of the fields yield, in the nonlinear saturated regime, good computational proxies for the observed ICE spectra in both the super-Alfvénic and sub-Alfvénic regimes for NBI protons. At early times in the PIC treatment, the computed growth rates correspond to analytical linear growth rates of the magnetoacoustic cyclotron instability (MCI), which was previously identified to underlie ICE from tokamak plasmas. The spatially localised PIC treatment does not include toroidal magnetic field geometry, nor background gradients in plasma parameters. Its success in simulating ICE spectra from both tokamak and, here, heliotron-stellarator plasmas suggests that the plasma parameters and ion energetic distribution at the emission location largely determine the ICE phenomenology. This is important for the future exploitation of ICE as a diagnostic for energetic ion populations in MCF plasmas. The capability to span the super-Alfvénic and sub-Alfvénic energetic ion regimes is a generic challenge in interpreting MCF plasma physics, and it is encouraging that this first principles computational treatment of ICE has now achieved this.

New Trends in Microwave Imaging Diagnostics and Application to Burning Plasma

Microwave and millimeter-wave (mm-wave)-based diagnostics have proven invaluable in the understanding of magnetic confinement fusion plasma science and have found wide application at major fusion facilities, including DIII-D, EAST, ASDEX-U, HL-2A, KSTAR, and J-TEXT. This trend is expected to continue unabated in future burning plasma devices such as ITER and DEMO. Attractive features include their nonperturbing nature and relatively modest access requirements due to their short wavelength, as well as recent dramatic advances in the state of the technology. In this tutorial, we describe the basic principles of a number of microwave diagnostics measurements emphasizing imaging, recent technology advances including the so-called “system-on-chip (SoC)” technology, and “synthetic” diagnostics which simulate the response of diagnostics under actual experimental scenarios. The application of microwave diagnostics on reactor plasmas benefits immensely from the development of wide-bandgap electronics and advanced optics designs, both of which ameliorate the effects of the harsh environment.

On the synchrotron emission in kinetic simulations of runaway electrons in magnetic confinement fusion plasmas

Developing avoidance or mitigation strategies of runaway electrons (REs) in magnetic confinement fusion (MCF) plasmas is of crucial importance for the safe operation of ITER. In order to develop these strategies, an accurate diagnostic capability that allows good estimates of the RE distribution function in these plasmas is needed. Synchrotron radiation (SR) of RE in MCF, besides of being one of the main damping mechanisms for RE in the high energy relativistic regime, is routinely used in current MCF experiments to infer the parameters of RE energy and pitch angle distribution functions. In the present paper we address the long standing question about what are the relationships between different REs distribution functions and their corresponding synchrotron emission simultaneously including: full-orbit effects, information of the spectral and angular distribution of SR of each electron, and basic geometric optics of a camera. We study the spatial distribution of the SR on the poloidal plane, and the statistical properties of the expected value of the synchrotron spectra of REs. We observe a strong dependence of the synchrotron emission measured by the camera on the pitch angle distribution of runaways, namely we find that crescent shapes of the spatial distribution of the SR as measured by the camera relate to RE distributions with small pitch angles, while ellipse shapes relate to distributions of runaways with larger the pitch angles. A weak dependence of the synchrotron emission measured by the camera with the RE energy, value of the q-profile at the edge, and the chosen range of wavelengths is observed. Furthermore, we find that oversimplifying the angular dependence of the SR changes the shape of the synchrotron spectra, and overestimates its amplitude by approximately 20 times for avalanching runaways and by approximately 60 times for mono-energetic distributions of runaways.

Quantifying Fusion Born Ion Populations in Magnetically Confined Plasmas using Ion Cyclotron Emission.

Ion cyclotron emission (ICE) offers a unique promise as a diagnostic of the fusion born alpha-particle population in magnetically confined plasmas. Pioneering observations from JET and TFTR found that ICE intensity P_{ICE} scales approximately linearly with the measured neutron flux from fusion reactions, and with the inferred concentration, n_{α}/n_{i}, of fusion born alpha particles confined within the plasma. We present fully nonlinear self-consistent kinetic simulations that reproduce this scaling for the first time. This resolves a long-standing question in the physics of fusion alpha-particle confinement and stability in magnetic confinement fusion plasmas. It confirms the magnetoacoustic cyclotron instability as the likely emission mechanism and greatly strengthens the basis for diagnostic exploitation of ICE in future burning plasmas.

NORSE: A solver for the relativistic non-linear Fokker–Planck equation for electrons in a homogeneous plasma

Energetic electrons are of interest in many types of plasmas, however previous modeling of their properties has been restricted to the use of linear Fokker–Planck collision operators or non-relativistic formulations. Here, we describe a fully non-linear kinetic-equation solver, capable of handling large electric-field strengths (compared to the Dreicer field) and relativistic temperatures. This tool allows modeling of the momentum-space dynamics of the electrons in cases where strong departures from Maxwellian distributions may arise. As an example, we consider electron runaway in magnetic-confinement fusion plasmas and describe a transition to electron slide-away at field strengths significantly lower than previously predicted. Program summaryProgram title: NORSEProgram Files doi:http://dx.doi.org/10.17632/86wmgj758w.1Licensing provisions: GPLv3Programming language: MatlabNature of problem: Solves the Fokker–Planck equation for electrons in 2D momentum space in a homogeneous plasma (allowing for magnetization), using a relativistic non-linear electron–electron collision operator. Electric-field acceleration, synchrotron-radiation-reaction losses, as well as heat and particle sources are included. Scenarios with time-dependent plasma parameters can be studied.Solution method: The kinetic equation is represented on a non-uniform 2D finite-difference grid and is evolved using a linearly implicit time-advancement scheme. A mixed finite-difference-Legendre-mode representation is used to obtain the relativistic potentials (analogous to the non-relativistic Rosenbluth potentials) from the distribution.

Multi-energy x-ray detector calibration for T and impurity density (n) measurements of MCF plasmas.

Soft x-ray detection with the new "multi-energy" PILATUS3 detector systems holds promise as a magnetically confined fusion (MCF) plasma diagnostic for ITER and beyond. The measured x-ray brightness can be used to determine impurity concentrations, electron temperatures, ne2Zeff products, and to probe the electron energy distribution. However, in order to be effective, these detectors which are really large arrays of detectors with photon energy gating capabilities must be precisely calibrated for each pixel. The energy-dependence of the detector response of the multi-energy PILATUS3 system with 100 K pixels has been measured at Dectris Laboratory. X-rays emitted from a tube under high voltage bombard various elements such that they emit x-ray lines from Zr-Lα to Ag-Kα between 1.8 and 22.16 keV. Each pixel on the PILATUS3 can be set to a minimum energy threshold in the range from 1.6 to 25 keV. This feature allows a single detector to be sensitive to a variety of x-ray energies, so that it is possible to sample the energy distribution of the x-ray continuum and line-emission. PILATUS3 can be configured for 1D or 2D imaging of MCF plasmas with typical spatial energy and temporal resolution of 1 cm, 0.6 keV, and 5 ms, respectively.

Collisional-radiative model for the visible spectrum of W26+ ions

Plasma diagnostics in magnetic confinement fusion plasmas by using visible spectrum strongly depends on the knowledge of fundamental atomic properties. A detailed collisional-radiative model of W26+ ions has been constructed by considering radiative and electron excitation processes, in which the necessary atomic data had been calculated by relativistic configuration interaction method with the implementation of Flexible Atomic Code. The visible spectrum observed at an electron beam ion trap (EBIT) in Shanghai in the range of 332 nm to 392 nm was reproduced by present calculations. Some transition pairs of which the intensity ratio is sensitive to the electron density were selected as potential candidates of plasma diagnostics. Their electron density dependence is theoretically evaluated for the cases of EBIT plasmas and magnetic confinement fusion plasmas.

Data processing for soft X-ray diagnostics based on GEM detector measurements for fusion plasma imaging

The measurement system based on GEM – Gas Electron Multiplier detector is developed for X-ray diagnostics of magnetic confinement fusion plasmas. The Triple Gas Electron Multiplier (T-GEM) is presented as soft X-ray (SXR) energy and position sensitive detector. The paper is focused on the measurement subject and describes the fundamental data processing to obtain reliable characteristics (histograms) useful for physicists. So, it is the software part of the project between the electronic hardware and physics applications. The project is original and it was developed by the paper authors. Multi-channel measurement system and essential data processing for X-ray energy and position recognition are considered. Several modes of data acquisition determined by hardware and software processing are introduced. Typical measuring issues are deliberated for the enhancement of data quality. The primary version based on 1-D GEM detector was applied for the high-resolution X-ray crystal spectrometer KX1 in the JET tokamak. The current version considers 2-D detector structures initially for the investigation purpose. Two detector structures with single-pixel sensors and multi-pixel (directional) sensors are considered for two-dimensional X-ray imaging. Fundamental output characteristics are presented for one and two dimensional detector structure. Representative results for reference source and tokamak plasma are demonstrated.

Guiding-centre transformation of the radiation–reaction force in a non-uniform magnetic field

In this paper, we present the guiding-centre transformation of the radiation–reaction force of a classical point charge travelling in a non-uniform magnetic field. The transformation is valid as long as the gyroradius of the charged particles is much smaller than the magnetic field non-uniformity length scale, so that the guiding-centre Lie-transform method is applicable. Elimination of the gyromotion time scale from the radiation–reaction force is obtained with the Poisson-bracket formalism originally introduced by Brizard (Phys. Plasmas, vol. 11, 2004, 4429–4438), where it was used to eliminate the fast gyromotion from the Fokker–Planck collision operator. The formalism presented here is applicable to the motion of charged particles in planetary magnetic fields as well as in magnetic confinement fusion plasmas, where the corresponding so-called synchrotron radiation can be detected. Applications of the guiding-centre radiation–reaction force include tracing of charged particle orbits in complex magnetic fields as well as the kinetic description of plasma when the loss of energy and momentum due to radiation plays an important role, e.g. for runaway-electron dynamics in tokamaks.