A surrogate model of the runaway electron avalanche growth rate in a magnetic fusion plasma is developed. This is accomplished by employing a physics-informed neural network (PINN) to learn the parametric solution of the adjoint to the relativistic Fokker–Planck equation. The resulting PINN is able to evaluate the runaway probability function across a broad range of parameters in the absence of any synthetic or experimental data. This surrogate of the adjoint relativistic Fokker–Planck equation is then used to infer the avalanche growth rate as a function of the electric field, synchrotron radiation and effective charge. Predictions of the avalanche PINN are compared against first principle calculations of the avalanche growth rate with excellent agreement observed across a broad range of parameters.

When simulating runaway electron dynamics in tokamak disruptions, fluid models with lower numerical cost are often preferred to more accurate kinetic models. The aim of this work is to compare fluid and kinetic simulations of a large variety of different disruption scenarios in ITER. We consider both non-activated and activated scenarios; for the latter, we derive and implement kinetic sources for the Compton scattering and tritium beta decay runaway electron generation mechanisms in our simulation tool Dream (Hoppe et al., Comput. Phys. Commun., vol. 268, 2021, 108098). To achieve a diverse set of disruption scenarios, Bayesian optimization is used to explore a range of massive material injection densities for deuterium and neon. The cost function is designed to distinguish between successful and unsuccessful disruption mitigation based on the runaway current, current quench time and transported fraction of the heat loss. In the non-activated scenarios, we find that fluid and kinetic disruption simulations can have significantly different runaway electron dynamics, due to an overestimation of the runaway seed by the fluid model. The primary cause of this is that the fluid hot-tail generation model neglects superthermal electron transport losses during the thermal quench. In the activated scenarios, the fluid and kinetic models give similar predictions, which can be explained by the significant influence of the activated sources on the runaway dynamics and the seed.

The explanation of the mechanisms of anomalous resistance and diffusion of plasma, as well as the generation of fast particles, is given from a unified position on the basis of experimental results obtained in plasma in a magnetically isolated diode of an electron accelerator with a microsecond pulse duration. It is shown that oscillations of the potential and current in plasma with current occur by charge separation and the formation of electric domains with a strong field. Electric domains are generated by means of runaway electrons resulting from the redistribution of the current density and its channelling in the current-carrying plasma. An increase in plasma resistance occurs due to a decrease in the number of electrons in the current-conducting state, since some of the plasma electrons pass into layers of excess negative charge of electric domains. The speed of movement of electrons in the composition of the electric domain is much lower than the velocity of electrons in the conducting state. The nucleation of domains is accompanied by microwave emission. Quasi-neutrals in whole electrical domains are injected into the region of the anode wall. The destruction of domains leads to the appearance of plasma channels between the anode and cathode plasma. The transverse electromagnetic waves generated during the nucleation of domains capture charged particles and inject them in a direction that is perpendicular to the insulating longitudinal magnetic field. Plasma turbulence occurs due to charge separation and the generation of electrical domains that create channels between the plasma and the chamber wall, as well as generating fluxes of fast particles.

A Bayesian optimization framework is used to investigate scenarios for disruptions mitigated with combined deuterium and neon injection in ITER. The optimization cost function takes into account limits on the maximum runaway current, the transported fraction of the heat loss and the current quench time. The aim is to explore the dependence of the cost function on injected densities, and provide insights into the behaviour of the disruption dynamics for representative scenarios. The simulations are conducted using the numerical framework Dream (Disruption Runaway Electron Analysis Model). We show that, irrespective of the quantities of the material deposition, multi-megaampere runaway currents will be produced in the deuterium–tritium phase of operations, even in the optimal scenarios. However, the severity of the outcome can be influenced by tailoring the radial profile of the injected material; in particular, if the injected neon is deposited at the edge region it leads to a significant reduction of both the final runaway current and the transported heat losses. The Bayesian approach allows us to map the parameter space efficiently, with more accuracy in favourable parameter regions, thereby providing us with information about the robustness of the optima.

Plasma-terminating disruptions in future fusion reactors may result in conversion of the initial current to a relativistic runaway electron beam. Validated predictive tools are required to optimise the scenarios and mitigation actuators to avoid the excessive damage that can be caused by such events. Many of the simulation tools applied in fusion energy research require the user to specify input parameters that are not constrained by the available experimental information. The conventional approach, where an expert modeller calibrates these input parameters based on domain knowledge, is prone to lead to an intractable validation challenge without systematic uncertainty quantification. Bayesian inference algorithms offer a promising alternative approach that naturally includes uncertainty quantification and is less subject to user bias in choosing the input parameters. The main challenge in using these methods is the computational cost of simulating enough samples to construct the posterior distributions for the uncertain input parameters. This challenge can be overcome by combining probabilistic surrogate modelling, such as Gaussian process regression, with Bayesian optimisation, which can reduce the number of required simulations by several orders of magnitude. Here, we implement this type of Bayesian optimisation framework for a model for analysis of disruption runaway electrons, and explore for simulations of current quench in a JET plasma discharge with an argon induced disruption. We use this proof-of-principle framework to explore the optimum input parameters with uncertainties in optimisation tasks ranging from one to seven dimensions. The relevant Python codes that are used in the analysis are available via https://github.com/aejarvin/BO_FOR_RE_SIMULATIONS/.

Understanding generation and mitigation of runaway electrons in disruptions is important for the safe operation of future tokamaks. In this paper we investigate the runaway dynamics in reactor-scale spherical tokamaks, focusing on a compact nominal design with a plasma current of 21 megaamperes (MA), 1.8 T magnetic field on axis and major radius of approximately 3 m. We study both the severity of runaway generation during unmitigated disruptions, and the effect that typical mitigation schemes based on massive material injection have on runaway production. The study is conducted using the numerical framework dream (Disruption Runaway Electron Analysis Model). We find that, in many cases, mitigation strategies are necessary to prevent the runaway current from reaching multi-MA levels. Our results indicate that, with a suitably chosen deuterium–neon mixture for mitigation, it is possible to achieve a tolerable runaway current and ohmic current evolution. However, this does not account for the runaway source due to wall activation, which has been found to severely limit successful mitigation at conventional aspect ratios, but whose definition requires a more complete wall specification. Furthermore, the majority of the thermal energy loss is found to happen through radial transport rather than radiation, which poses a risk of unacceptable localised heat loads.

Recently, the collisionless pitch-angle scattering for relativistic runaway electrons (REs) in toroidal geometries such as tokamaks was discovered through a full orbit simulation approach (Liu et al., Nucl. Fusion, vol. 56, 2016, p. 064002), and it was then theoretically investigated that a new expression for the magnetic moment, including the second-order corrections, could essentially reproduce the so-called collisionless pitch-angle scattering process (Liu et al., Nucl. Fusion, vol. 58, 2018, p. 106018). In this paper, with synchrotron radiation, extensive numerical verification of the validity of the high-order guiding-centre theory is given for simulations involving REs by incorporating such an expression for the magnetic moment into our particle tracing code. A high-order guiding-centre simulation approach with synchrotron radiation (HGSA) is applied. Synchrotron radiation plays an essential role in the life cycle of REs. The energy of REs first increases and then becomes saturated until the electric field acceleration is balanced by the radiation dissipation. Unfortunately, the process cannot be simulated accurately with the standard guiding-centre model, i.e. the first-order guiding-centre model. Remarkably, it is found that the HGSA can effectively produce the fundamental process of REs. Since the time scale of the energy saturation of REs is close to seconds, the computational cost becomes significant. In order to save costs, it is necessary to estimate the time of energy saturation. An analytical estimate is derived for the time it takes for synchrotron drag to balance an accelerating electric field and the provided formula has been numerically verified. Test calculations reveal that HGSA is favourable for exploiting the dynamics of REs in tokamak plasmas.

The safe operation of tokamak reactors requires a reliable modelling capability of disruptions, and in particular the spatio-temporal dynamics of associated runaway electron currents. In a disruption, instabilities can break up magnetic surfaces into chaotic field line regions, causing current profile relaxation, as well as a rapid radial transport of heat and particles. Using a mean-field helicity transport model implemented in the disruption runaway modelling framework Dream, we calculate the dynamics of runaway electrons in the presence of current relaxation events. In scenarios where flux surfaces remain intact in parts of the plasma, a skin current is induced at the boundary of the intact magnetic field region. This skin current region becomes an important centre concerning the subsequent dynamics: it may turn into a hot ohmic current channel, or a sizeable radially localized runaway beam, depending on the heat transport. If the intact region is in the plasma edge, runaway generation in the countercurrent direction can occur, which may develop into a sizeable reverse runaway beam. Even when the current relaxation extends to the entire plasma, the final runaway current density profile can be significantly affected, as the induced electric field is reduced in the core and increased in the edge, thereby shifting the centre of runaway generation towards the edge.

Tokamak start-up is characterized by low electron densities and strong electric fields, in order to quickly raise the plasma current and temperature, allowing the plasma to fully ionize and magnetic flux surfaces to form. Such conditions are ideal for the formation of superthermal electrons, which may reduce the efficiency of ohmic heating and prevent the formation of a healthy thermal fusion plasma. This is of particular concern in ITER where engineering limitations put restrictions on the allowable electric fields and limit the prefill densities during start-up. In this study, we present a new 0D burn-through simulation tool called STREAM (STart-up Runaway Electron Analysis Model), which self-consistently evolves the plasma density, temperature and electric field, while accounting for the generation and loss of relativistic runaway electrons. After verifying the burn-through model, we investigate conditions under which runaway electrons can form during tokamak start-up as well as their effects on the plasma initiation. We find that Dreicer generation plays a crucial role in determining whether a discharge becomes runaway-dominated or not, and that a large number of runaway electrons could limit the ohmic heating of the plasma, thus preventing successful burn-through or further ramp-up of the plasma current. The runaway generation can be suppressed by raising the density via gas fuelling, but only if done sufficiently early. Otherwise a large runaway seed may have already been built up, which can avalanche even at relatively low electric fields and high densities.

The formation of a substantial postdisruption runaway electron current in ASDEX Upgrade material injection experiments is determined by avalanche multiplication of a small seed population of runaway electrons. For the investigation of these scenarios, the runaway electron description of the coupled 1.5-D transport solvers ASTRA-STRAHL is amended by a fluid model describing electron runaway caused by the hot-tail mechanism. Applied in simulations of combined background plasma evolution, material injection and runaway electron generation in ASDEX Upgrade discharge #33108, both the Dreicer and hot-tail mechanism for electron runaway produce only ${\sim }$ 3 kA of runaway current. In colder plasmas with core electron temperatures $T_\textrm {e,c}$ below 9 keV, the postdisruption runaway current is predicted to be insensitive to the initial temperature, in agreement with experimental observations. Yet in hotter plasmas with $T_\textrm {e,c}$ above 10 keV, hot-tail runaway can be increased by up to an order of magnitude, contributing considerably to the total postdisruption runaway current. In ASDEX Upgrade high-temperature runaway experiments, however, no runaway current is observed at the end of the disruption, despite favourable conditions for both primary and secondary runaway.

The electron runaway phenomenon in plasmas depends sensitively on the momentum- space dynamics. However, efficient simulation of the global evolution of systems involving runaway electrons typically requires a reduced fluid description. This is needed, for example, in the design of essential runaway mitigation methods for tokamaks. In this paper, we present a method to include the effect of momentum-dependent spatial transport in the runaway avalanche growth rate. We quantify the reduction of the growth rate in the presence of electron diffusion in stochastic magnetic fields and show that the spatial transport can raise the effective critical electric field. Using a perturbative approach, we derive a set of equations that allows treatment of the effect of spatial transport on runaway dynamics in the presence of radial variation in plasma parameters. This is then used to demonstrate the effect of spatial transport in current quench simulations for ITER-like plasmas with massive material injection. We find that in scenarios with sufficiently slow current quench, owing to moderate impurity and deuterium injection, the presence of magnetic perturbations reduces the final runaway current considerably. Perturbations localised at the edge are not effective in suppressing the runaways, unless the runaway generation is off-axis, in which case they may lead to formation of strong current sheets at the interface of the confined and perturbed regions.

Synchrotron radiation images from runaway electrons (REs) in an ASDEX Upgrade discharge disrupted by argon injection are analysed using the synchrotron diagnostic tool Soft and coupled fluid-kinetic simulations. We show that the evolution of the runaway distribution is well described by an initial hot-tail seed population, which is accelerated to energies between 25–50 MeV during the current quench, together with an avalanche runaway tail which has an exponentially decreasing energy spectrum. We find that, although the avalanche component carries the vast majority of the current, it is the high-energy seed remnant that dominates synchrotron emission. With insights from the fluid-kinetic simulations, an analytic model for the evolution of the runaway seed component is developed and used to reconstruct the radial density profile of the RE beam. The analysis shows that the observed change of the synchrotron pattern from circular to crescent shape is caused by a rapid redistribution of the radial profile of the runaway density.

SPARC is being designed to operate with a normalized beta of $\beta _N=1.0$, a normalized density of $n_G=0.37$ and a safety factor of $q_{95}\approx 3.4$, providing a comfortable margin to their respective disruption limits. Further, a low beta poloidal $\beta _p=0.19$ at the safety factor $q=2$ surface reduces the drive for neoclassical tearing modes, which together with a frozen-in classically stable current profile might allow access to a robustly tearing-free operating space. Although the inherent stability is expected to reduce the frequency of disruptions, the disruption loading is comparable to and in some cases higher than that of ITER. The machine is being designed to withstand the predicted unmitigated axisymmetric halo current forces up to 50 MN and similarly large loads from eddy currents forced to flow poloidally in the vacuum vessel. Runaway electron (RE) simulations using GO+CODE show high flattop-to-RE current conversions in the absence of seed losses, although NIMROD modelling predicts losses of ${\sim }80$ %; self-consistent modelling is ongoing. A passive RE mitigation coil designed to drive stochastic RE losses is being considered and COMSOL modelling predicts peak normalized fields at the plasma of order $10^{-2}$ that rises linearly with a change in the plasma current. Massive material injection is planned to reduce the disruption loading. A data-driven approach to predict an oncoming disruption and trigger mitigation is discussed.

A runaway avalanche can result in a conversion of the initial plasma current into a relativistic electron beam in high-current tokamak disruptions. We investigate the effect of massive material injection of deuterium–noble gas mixtures on the coupled dynamics of runaway generation, resistive diffusion of the electric field and temperature evolution during disruptions in the deuterium–tritium phase of ITER operations. We explore the dynamics over a wide range of injected concentrations and find substantial runaway currents, unless the current quench time is intolerably long. The reason is that the cooling associated with the injected material leads to high induced electric fields that, in combination with a significant recombination of hydrogen isotopes, leads to a large avalanche generation. Balancing Ohmic heating and radiation losses provides qualitative insights into the dynamics; however, an accurate modelling of the temperature evolution based on energy balance appears crucial for quantitative predictions.

Massive material injection has been proposed as a way to mitigate the formation of a beam of relativistic runaway electrons that may result from a disruption in tokamak plasmas. In this paper we analyse runaway generation observed in eleven ASDEX Upgrade discharges where disruption was triggered using massive gas injection. We present numerical simulations in scenarios characteristic of on-axis plasma conditions, constrained by experimental observations, using a description of the runaway dynamics with a self-consistent electric field and temperature evolution in two-dimensional momentum space and zero-dimensional real space. We describe the evolution of the electron distribution function during the disruption, and show that the runaway seed generation is dominated by hot-tail in all of the simulated discharges. We reproduce the observed dependence of the current dissipation rate on the amount of injected argon during the runaway plateau phase. Our simulations also indicate that above a threshold amount of injected argon, the current density after the current quench depends strongly on the argon densities. This trend is not observed in the experiments, which suggests that effects not captured by zero-dimensional kinetic modelling – such as runaway seed transport – are also important.

Plasma terminating disruptions in tokamaks may result in relativistic runaway electron beams with potentially serious consequences for future devices with large plasma currents. In this paper, we investigate the effect of plasma elongation on the coupled dynamics of runaway generation and resistive diffusion of the electric field. We find that elongated plasmas are less likely to produce large runaway currents, partly due to the lower induced electric fields associated with larger plasmas, and partly due to direct shaping effects, which mainly lead to a reduction in the runaway avalanche gain.

Integrated modelling of electron runaway requires computationally expensive kinetic models that are self-consistently coupled to the evolution of the background plasma parameters. The computational expense can be reduced by using parameterized runaway generation rates rather than solving the full kinetic problem. However, currently available generation rates neglect several important effects; in particular, they are not valid in the presence of partially ionized impurities. In this work, we construct a multilayer neural network for the Dreicer runaway generation rate which is trained on data obtained from kinetic simulations performed for a wide range of plasma parameters and impurities. The neural network accurately reproduces the Dreicer runaway generation rate obtained by the kinetic solver. By implementing it in a fluid runaway-electron modelling tool, we show that the improved generation rates lead to significant differences in the self-consistent runaway dynamics as compared to the results using the previously available formulas for the runaway generation rate.

Accurate modelling of the interaction between fast electrons and partially ionized atoms is important for evaluating tokamak disruption mitigation schemes based on material injection. This requires accounting for the effect of screening of the impurity nuclei by the cloud of bound electrons. In this paper, we generalize the Fokker–Planck operator in a fully ionized plasma by accounting for the effect of screening. We detail the derivation of this generalized operator, and calculate the effective ion length scales, needed in the components of the collision operator, for a number of ion species commonly appearing in fusion experiments. We show that for high electric fields, the secondary runaway growth rate can be substantially larger than in a fully ionized plasma with the same effective charge, although the growth rate is significantly reduced at near-critical electric fields. Furthermore, by comparison with the Boltzmann collision operator, we show that the Fokker–Planck formalism is accurate even for large impurity content.

Sufficiently strong electric fields in plasmas can accelerate charged particles to relativistic energies. In this paper we describe the dynamics of positrons accelerated in such electric fields, and calculate the fraction of created positrons that become runaway accelerated, along with the amount of radiation that they emit. We derive an analytical formula that shows the relative importance of the different positron production processes, and show that, above a certain threshold electric field, the pair production by photons is lower than that by collisions. We furthermore present analytical and numerical solutions to the positron kinetic equation; these are applied to calculate the fraction of positrons that become accelerated or thermalized, which enters into rate equations that describe the evolution of the density of the slow and fast positron populations. Finally, to indicate operational parameters required for positron detection during runaway in tokamak discharges, we give expressions for the parameter dependencies of detected annihilation radiation compared to bremsstrahlung detected at an angle perpendicular to the direction of runaway acceleration. Using the full leading-order pair-production cross-section, we demonstrate that previous related work has overestimated the collisional pair production by at least a factor of four.

Large-angle Coulomb collisions lead to an avalanching generation of runaway electrons in a plasma. We present the first fully conservative large-angle collision operator, derived from the relativistic Boltzmann operator. The relation to previous models for large-angle collisions is investigated, and their validity assessed. We present a form of the generalized collision operator which is suitable for implementation in a numerical kinetic equation solver, and demonstrate the effect on the runaway-electron growth rate. Finally we consider the reverse avalanche effect, where runaways are slowed down by large-angle collisions, and show that the choice of operator is important if the electric field is close to the avalanche threshold.