An important problem in nuclear fusion plasmas is the prediction and control of turbulence which drives the cross-field transport, thus leading to energy loss from the system and deteriorating confinement. Turbulence, being a highly nonlinear and multiscale process, is challenging to theoretically describe and computationally model. Most advanced computational models fall into one of the two categories: fluid or gyro-kinetic. They both come at a high computational cost and cannot be applied for routine simulation of plasma discharge evolution and control. Development of reduced models based on (physics informed) artificial neural networks could potentially fulfil the need for affordable simulations of plasma turbulence. However, the training requires an extensive data base and the obtained models lack extrapolation capability to scenarios not originally encountered during training. This leads to reduced models of limited validity which may not prove adequate for predicting scenarios in future machines. In contrast, we explore a data-driven model discovery approach based on sparse regression to infer governing nonlinear partial differential equations directly from the data. Our input data are generated by simulations of drift-wave turbulence according to the Hasegawa–Wakatani and modified Hasegawa–Wakatani models. Balancing model accuracy and complexity enables the reconstruction of the systems of partial differential equations accurately describing the dynamics simulated in the input data sets. Sparse regression is not data hungry and can be extrapolated to unexplored parameter ranges. We explore and demonstrate the potential of this approach for fusion plasma turbulence modelling. The findings show that the methodology is promising for the development of reduced and computationally efficient turbulence models as well as for existing model cross-validation.

In a magnetised plasma on scales well above ion kinetic scales, any constant-magnitude magnetic field, accompanied by parallel Alfvénic flows, forms a nonlinear solution in an isobaric, constant-density background. These structures, which are also known as spherically polarised Alfvén waves, are observed ubiquitously in the solar wind, presumably created by the growth of small-amplitude fluctuations as they propagate outwards in the corona. Here, we present a computational method to construct such solutions of arbitrary amplitude with general multidimensional structure, and explore some of their properties. The difficulty lies in computing a zero-divergence, constant-magnitude magnetic field, which leaves a single, quasi-free function to define the solution, while requiring strong constraints on any individual component of the field. Motivated by the physical process of wave growth in the solar wind, our method circumvents this issue by starting from low-amplitude Alfvénic fluctuations dominated by a strong mean field, then ‘growing’ magnetic perturbations into the large-amplitude regime. We present example solutions with non-trivial structure in one, two and three dimensions, demonstrating a clear tendency to form very sharp gradients or discontinuities, unless the solution is one-dimensional. As well as being useful as an input for other calculations, particularly the study of parametric decay, our results provide a natural explanation for the extremely sharp field discontinuities observed across magnetic field switchbacks in the low solar wind.

Propagation characteristics (propagation regions and cutoffs) of parallel propagating modes (Langmuir, right- and left-handed circularly polarized waves) are studied for relativistic, weakly relativistic and non-relativistic magnetized electron plasma using the kinetic model. The dispersion relation for parallel propagating modes in relativistic electron plasma is investigated by employing the Maxwell–Boltzmann–J üttner distribution function and the final dispersion relation obtained is more general since no approximation is used. As the integrals in the relativistic dispersion relation cannot be done analytically so these integrals have been solved with the numerical quadrature approach. For $\eta \leq 1$ (ratio of rest mass energy to thermal energy), the increase in the effective mass of electrons will result in a change in the mass-dependent quantities (plasma frequency, electron cyclotron frequency, electron sound velocity, etc.) which in turn significantly affect the propagation characteristics of parallel propagating modes. It is observed that the propagation region for these parallel propagating modes decreases and cutoff points are shifted to lower values when we consider a relativistic plasma environment. Moreover, a low-density and high-temperature plasma is more transparent as compared with a high-density and low-temperature plasma for these modes.

We propose a new method to compute magnetic surfaces that are parametrized in Boozer coordinates for vacuum magnetic fields. We also propose a measure for quasisymmetry on the computed surfaces and use it to design coils that generate a magnetic field that is quasisymmetric on those surfaces. The rotational transform of the field and complexity measures for the coils are also controlled in the design problem. Using an adjoint approach, we are able to obtain analytic derivatives for this optimization problem, yielding an efficient gradient-based algorithm. Starting from an initial coil set that presents nested magnetic surfaces for a large fraction of the volume, our method converges rapidly to coil systems generating fields with excellent quasisymmetry and low particle losses. In particular for low complexity coils, we are able to significantly improve the performance compared with coils obtained from the standard two-stage approach, e.g. reduce losses of fusion-produced alpha particles born at half-radius from $17.7\,\%$ to $6.6\,\%$. We also demonstrate 16-coil configurations with alpha loss ${<}1\,\%$ and neoclassical transport magnitude $\epsilon _{\text {eff}}^{3/2}$ less than approximately $5\times 10^{-9}$.

The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.

We analyse how the turbulent transport of $\boldsymbol {E}\times \boldsymbol {B}$ type in magnetically confined plasmas is affected by the intermittent features of turbulence. The latter are modelled via the non-Gaussian distribution $P(\phi )$ of the turbulent electric potential $\phi$. Our analysis is performed at an analytical level and confirmed numerically using two statistical approaches. We have found that the diffusion is inhibited linearly by intermittency, mainly via the kurtosis of the distribution $P(\phi )$. The associated susceptibility for this linear process is shown to be dependent on the poloidal velocity $V_p$ and on the correlation time $\tau _c$ (or the Kubo number $K_\star$, the ratio between $\tau _c$ and the specific time of flight $\tau _{{\rm fl}}$) with a maximum at $\tau _c\approx \tau _{{\rm fl}}$ ($K_\star \approx 1$). Intermittency does not affect the scaling of diffusion with the Kubo number.

A single-field-period quasi-isodynamic stellarator configuration is presented. This configuration, which resembles a twisted strip, is obtained by the method of direct construction, that is, it is found via an expansion in the distance from the magnetic axis. Its discovery, however, relied on an additional step involving numerical optimization, performed within the space of near-axis configurations defined by a set of adjustable magnetic field parameters. This optimization, completed in 30 s on a single CPU core using the SIMSOPT code, yields a solution with excellent confinement, as measured by the conventional figure of merit for neoclassical transport, effective ripple, at a modest aspect ratio of eight. The optimization parameters that led to this configuration are described, its confinement properties are assessed and a set of magnetic field coils is found. The resulting transport at low collisionality is much smaller than that of W7-X, and the device needs significantly fewer coils because of the reduced number of field periods.

The ideal Chew–Goldberger–Low (CGL) plasma equations, including the double adiabatic conservation laws for the parallel ($p_\parallel$) and perpendicular pressure ($p_\perp$), are investigated using a Lagrangian variational principle. An Euler–Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux ${\boldsymbol {M}}$, the density $\rho$, the entropy variable $\sigma =\rho S$ and the magnetic induction ${\boldsymbol {B}}$. Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, centre of mass and angular momentum. Cross-helicity conservation arises from a fluid relabelling symmetry, and is local or non-local depending on whether the gradient of $S$ is perpendicular to ${\boldsymbol {B}}$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.

Fluctuation measurements reveal the outward electromagnetic energy flux needed to drive the dynamo electromotive force supporting magnetic self-organization in a reversed-field pinch plasma. The radial Poynting flux due to tearing mode fluctuations is measured with an insertable probe during magnetic relaxation. This flux corresponds to transient power levels much larger than the input power and comparable to the global equilibrium magnetic energy transient loss rate. The probe measurements of this flux are roughly as predicted by a simple Poynting's theorem model upon substitution of equilibrium measurement data.

The paper presents experimental results from the SMOLA device that is the first facility with a helical mirror section of the magnetic system. This device was built in the Budker Institute of Nuclear Physics for the verification of the helical mirror confinement idea that is the technique of an active control of axial losses from a confinement zone. Theory predicts that, with rotating plasma, a helical mirror will provide suppression of the axial plasma flow and, simultaneously, density pinching to the axis. Experiments demonstrated the increase in plasma density in the entrance trap by a factor of 1.6 in the helical configuration. The integral axial flux from the transport section drops severalfold. The effective mirror ratio of the helical section was $R_{eff} > 10$. Particle flux returning by the helical mirror section towards the confinement zone was observed. At high corrugation ratios, the axial flux direction is different at the magnetic axis and in the periphery of the plasma in the helical section. All axial fluxes scale linearly with the plasma density, even if the ion mean free path is comparable to the total length of the helical section. Good agreement of the experimental results with theoretical predictions is found.

A new formalism for the derivation of the cnoidal wave solution is presented by introducing a new set of initial conditions and a subacoustic moving frame. The resulting set of solutions is also constructed in a way that ensures the conservation of number density. The solutions are illustrated in a variety of graphical representations, and the effect of the amplitude's magnitude on the cnoidal wave form is presented. Interestingly, it is shown that the wavelength of the solutions decreases with amplitude. In addition, it is shown that the small-amplitude cnoidal wave solutions converge to linear waves in the small-amplitude limit.

Kelvin–Helmholtz (KH) instability plays a significant role in transport and mixing in various media such as hydrodynamic fluids, plasmas, geophysical flows and astrophysical situations. In this paper, we numerically explore this instability for a two-dimensional strongly coupled dusty plasma medium with rotational shear flows. We study this medium using a generalized hydrodynamic fluid model which treats it as a viscoelastic fluid. We consider the specific cases of rotating vorticity with abrupt radial profiles of rotation. In particular, single-circulation and multi-circulation vorticity shell profiles have been chosen. We observe the KH vortices at each circular interface between two relative rotating flows along with a pair of ingoing and outgoing wavefronts of transverse shear waves. Our studies show that due to the interplay between KH vortices and shear waves in the strongly coupled medium, the mixing and transport behaviour are much better than those of standard inviscid hydrodynamic fluids. In the interest of substantiating the mixing and transport behaviour, the generalized hydrodynamic fluid model is extended to include Lagrangian tracer particles. The numerical dispersion of these tracer particles in a flow provides an estimate of the diffusion in such a medium. We present the preliminary observations of tracer distribution (cluster formation) and diffusion (mean square displacement) across the medium.

The Dimits shift, an upshift in the onset of turbulence from the linear instability threshold, caused by self-generated zonal flows, can greatly enhance the performance of magnetic confinement plasma devices. Except in simple cases, using fluid approximations and model magnetic geometries, this phenomenon has proved difficult to understand and quantitatively predict. To bridge the large gap in complexity between simple models and realistic treatment in toroidal magnetic geometries (e.g. tokamaks or stellarators), the present work uses fully gyrokinetic simulations in a Z-pinch geometry to investigate the Dimits shift through the lens of tertiary instability analysis, which describes the emergence of drift waves from a zonally dominated state. Several features of the tertiary instability, previously observed in fluid models, are confirmed to remain. Most significantly, an efficient reduced-mode tertiary model, which previously proved successful in predicting the Dimits shift in a gyrofluid limit (Hallenbert & Plunk, J. Plasma Phys., vol. 87, issue 05, 2021, 905870508), is found to be accurate here, with only slight modifications to account for kinetic effects.

A spacecraft during mission typically switches from chemical propulsion to electric propulsion once it lifts out of the Earth's gravity as the thrust requirement to drive it reduces substantially. Consequently, electric propulsion technology is commonly used for deep space mission, satellite orbit keeping and orbit correction. In the last few decades, the amount of man-made space junk (space debris) has increased enormously and has become a potential danger for space stations, space shuttles and other live satellites. A bi-directional plasma thruster, mounted on a satellite, can be used to remove space debris during satellite operation (Takahashi et al., Sci. Rep., vol. 8, 2018, p. 14417). A directed ion beam ejected from a plasma thruster imparts a net force on space debris to decelerate and facilitates manoeuvring and re-entry of space debris into the Earth's atmosphere where it can burn out. We present a detailed 1D3V PIC-MCC (particle in cell-Monte Carlo collision) simulation of a bi- directional plasma thruster. To this end, a PIC-MCC solver which resolves thruster axial direction and all three velocity dimensions is used to study a magnetic nozzle plasma thruster with both ends open (bi-directional plasma thruster). We show that such a bi-directional plasma thruster can be used to accelerate–decelerate a live satellite and also to remove space debris by altering the magnetic field spatial profile in the plasma expansion region. A detailed study is presented.

The Dougherty model Fokker–Planck operator is extended to describe nonlinear full-$f$ ( f is the distribution function) collisions between multiple species in plasmas. Simple relations for cross-species primitive moments are developed which obey conservation laws, and reproduce familiar velocity and temperature relaxation rates. This treatment of multispecies Dougherty collisions, valid for arbitrary mass ratios, avoids unphysical temperatures and satisfies the $H$-theorem (H is related to the entropy) unlike an analogous Bhatnagar–Gross–Krook operator. Formulas for both a Cartesian velocity space and a gyroaveraged operator are provided for use in Vlasov as well as long-wavelength gyrokinetic models. We present an algorithm for the discontinuous Galerkin discretization of this operator, and provide results from relaxation and Landau damping benchmarks.

Significant variety is observed in spherical crystal x-ray imager (SCXI) data for the stagnated fuel–liner system created in Magnetized Liner Inertial Fusion (MagLIF) experiments conducted at the Sandia National Laboratories Z-facility. As a result, image analysis tasks involving, e.g., region-of-interest selection (i.e. segmentation), background subtraction and image registration have generally required tedious manual treatment leading to increased risk of irreproducibility, lack of uncertainty quantification and smaller-scale studies using only a fraction of available data. We present a convolutional neural network (CNN)-based pipeline to automate much of the image processing workflow. This tool enabled batch preprocessing of an ensemble of $N_{\text {scans}} = 139$ SCXI images across $N_{\text {exp}} = 67$ different experiments for subsequent study. The pipeline begins by segmenting images into the stagnated fuel and background using a CNN trained on synthetic images generated from a geometric model of a physical three-dimensional plasma. The resulting segmentation allows for a rules-based registration. Our approach flexibly handles rarely occurring artifacts through minimal user input and avoids the need for extensive hand labelling and augmentation of our experimental dataset that would be needed to train an end-to-end pipeline. We also fit background pixels using low-degree polynomials, and perform a statistical assessment of the background and noise properties over the entire image database. Our results provide a guide for choices made in statistical inference models using stagnation image data and can be applied in the generation of synthetic datasets with realistic choices of noise statistics and background models used for machine learning tasks in MagLIF data analysis. We anticipate that the method may be readily extended to automate other MagLIF stagnation imaging applications.

We present a study of the linear properties of the ion-temperature gradient (ITG) modes with collisions modelled for the first time by the linearized gyrokinetic (GK) Coulomb collision operator (Frei et al., J. Plasma Phys., vol. 87, issue 5, 2021, 905870501) in the local limit. The study is based on a Hermite–Laguerre polynomial expansion of the perturbed ion distribution function applied to the linearized GK Boltzmann equation, yielding a hierarchy of coupled equations for the expansion coefficients, referred to as gyromoments. We explore the collisionless and high-collisional limits of the gyromoment hierarchy analytically. Parameter scans revealing the dependence of the ITG growth rate on the collisionality modelled using the GK Coulomb operator are reported, showing strong damping at small scales as the collisionality increases and, therefore, the need for a steeper gradient for the ITG onset at high collisionality to overcome the finite Larmor radius (FLR) collisional stabilization. The predictions on the ITG growth rate by the GK Coulomb operator are compared with other collision operator models, such as the Sugama, the Dougherty, as well as the momentum-conserving pitch-angle scattering and the Hirshman–Sigmar–Clarke collision operators derived for the first time in terms of gyromoments. The importance of FLR terms in the collision operators is pointed out by the appearance of a short wavelength ITG branch when collisional FLR terms are neglected, this branch being completely suppressed by FLR collisional effects. Energy diffusion is shown to be important at high collisionality and at small scale lengths. Among the GK collision operators considered in this work, the GK Sugama collision operator yields, in general, the smallest deviation compared with the GK Coulomb collision operator, while the largest deviations are found with the GK Dougherty operator. Convergence studies of the gyromoment method are reported and show that the drifts associated with the gradient and curvature of the magnetic field increase the required number of gyromoments at low collisionality. Nevertheless, the low number of gyromoments necessary for convergence at high collisionality constitutes an attractive numerical and analytical feature of the gyromoment approach to study the plasma dynamics in the boundary of fusion devices.

In the present work we investigate the dynamics of electrons under the action of wave packets of high-frequency electromagnetic carrier waves. When the group velocities of the packets are subluminal, electrons can be efficiently accelerated. We show that the whole process can be described by an accurate ponderomotive canonical formalism that includes relevant extensions of the original ponderomotive approach applied to carriers moving at the speed of light. Single-particle simulations validate our analytical approach and show that extended canonical methods provide better agreement with numerics than previous investigations. In particular, we obtain a precise relationship between the wave amplitude and group velocity for optimum acceleration of initially stationary targets.

The goal of this work is to generate large statistically representative data sets to train machine learning models for disruption prediction provided by data from few existing discharges. Such a comprehensive training database is important to achieve satisfying and reliable prediction results in artificial neural network classifiers. Here, we aim for a robust augmentation of the training database for multivariate time series data using Student $t$ process regression. We apply Student $t$ process regression in a state space formulation via Bayesian filtering to tackle challenges imposed by outliers and noise in the training data set and to reduce the computational complexity. Thus, the method can also be used if the time resolution is high. We use an uncorrelated model for each dimension and impose correlations afterwards via colouring transformations. We demonstrate the efficacy of our approach on plasma diagnostics data of three different disruption classes from the DIII-D tokamak. To evaluate if the distribution of the generated data is similar to the training data, we additionally perform statistical analyses using methods from time series analysis, descriptive statistics and classic machine learning clustering algorithms.

We employ the $\lambda ^{3}$ regime where a near-single-cycle laser pulse is tightly focused, thus providing the highest possible intensity for the minimal energy at a certain laser power. The quantum electrodynamics processes in the course of the interaction of an ultra-intense laser with a solid target are studied via three-dimensional particle-in-cell simulations, revealing the generation of copious $\gamma$-photons and electron–positron pairs. A parametric study of the laser polarisation, target thickness and electron number density shows that a radially polarised laser provides the optimal regime for $\gamma$-photon generation. By varying the laser power in the range of 1 to 300 PW we find the scaling of the laser to $\gamma$-photon energy conversion efficiency. The laser-generated $\gamma$-photon interaction with a high-$Z$ target is further studied using Monte Carlo simulations revealing further electron–positron pair generation and radioactive nuclide creation.