This review of scaling theories of magnetohydrodynamic (MHD) turbulence aims to put the developments of the last few years in the context of the canonical time line (from Kolmogorov to Iroshnikov–Kraichnan to Goldreich–Sridhar to Boldyrev). It is argued that Beresnyak's (valid) objection that Boldyrev's alignment theory, at least in its original form, violates the Reduced-MHD rescaling symmetry can be reconciled with alignment if the latter is understood as an intermittency effect. Boldyrev's scalings, a version of which is recovered in this interpretation, and the concept of dynamic alignment (equivalently, local 3D anisotropy) are thus an example of a physical theory of intermittency in a turbulent system. The emergence of aligned structures naturally brings into play reconnection physics and thus the theory of MHD turbulence becomes intertwined with the physics of tearing, current-sheet disruption and plasmoid formation. Recent work on these subjects by Loureiro, Mallet et al. is reviewed and it is argued that we may, as a result, finally have a reasonably complete picture of the MHD turbulent cascade (forced, balanced, and in the presence of a strong mean field) all the way to the dissipation scale. This picture appears to reconcile Beresnyak's advocacy of the Kolmogorov scaling of the dissipation cutoff (as $\mathrm {Re}^{3/4}$) with Boldyrev's aligned cascade. It turns out also that these ideas open the door to some progress in understanding MHD turbulence without a mean field – MHD dynamo – whose saturated state is argued to be controlled by reconnection and to contain, at small scales, a tearing-mediated cascade similar to its strong-mean-field counterpart (this is a new result). On the margins of this core narrative, standard weak-MHD-turbulence theory is argued to require some adjustment – and a new scheme for such an adjustment is proposed – to take account of the determining part that a spontaneously emergent 2D condensate plays in mediating the Alfvén-wave cascade from a weakly interacting state to a strongly turbulent (critically balanced) one. This completes the picture of the MHD cascade at large scales. A number of outstanding issues are surveyed: imbalanced turbulence (for which a new, tentative theory is proposed), residual energy, MHD turbulence at subviscous scales, and decaying MHD turbulence (where there has been dramatic progress recently, and reconnection again turned out to feature prominently). Finally, it is argued that the natural direction of research is now away from the fluid MHD theory and into kinetic territory – and then, possibly, back again. The review lays no claim to objectivity or completeness, focusing on topics and views that the author finds most appealing at the present moment.

We examine the turbulence driven by the ion and electron temperature gradients in selected magnetic configurations of the Wendelstein 7-X (W7-X) stellarator. The inherent flexibility in the configuration space of W7-X enables us to find candidate configurations manifesting low turbulent transport. We follow insights gained by stellarator optimization techniques, in order to identify key geometric features, which are directly related to the ion and electron heat fluxes produced by plasma turbulence. One such a feature is the flux expansion at locations where the curvature is particularly unfavourable. Starting from a configuration routinely used in the W7-X experiment, we end up with an optimized configuration. Based on this equilibrium, we select a configuration from W7-X configuration database with a similar feature as the optimized one. With the help of nonlinear gyrokinetic simulations, we show that the heat flux in this configuration is less stiff than in the initial configuration, both for ion temperature gradient and electron temperature gradient turbulence.

Weakly collisional plasmas are subject to nonlinear relaxation processes, which can operate at rates much faster than the particle collision frequencies. This causes the plasma to respond like a magnetised fluid despite having long particle mean free paths. In this Letter the effective collisional mechanisms are modelled in the plasma kinetic equation to produce density, pressure and magnetic-field responses to compare with spacecraft measurements of the solar wind compressive fluctuations at 1 AU. This enables a measurement of the effective mean free path of the solar wind protons, found to be ${\approx }4 \times 10^{5}$ km, which is approximately $10^{3}$ times shorter than the collisional mean free path. These measurements are shown to support the effective fluid behaviour of the solar wind at scales above the proton gyroradius and demonstrate that effective collision processes alter the thermodynamics and transport of weakly collisional plasmas.

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.

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.

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.

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.

The rate of onset (hazard) of tearing modes is modelled probabilistically using statistical learning algorithms. Axisymmetric energy-density equilibrium fields are taken as raw high-dimensional input features which are reduced with principal component analysis. Signal processing of non-axisymmetric magnetics fluctuation array data provides the target information from which to learn. Model selection, visualization and calibration assessment procedures are detailed. The analysis is deployed at large scale across the DIII-D tokamak database. Standard model selection criteria suggest that the energy-density post-processed feature is a better choice for modelling the onset rate compared to the non-processed equilibrium reconstruction solution. Two example applications of the learned rate function are demonstrated: (i) proximity-to-onset discharge monitoring and (ii) database analysis showing an (expected) observational global trend that the general hazard increases as a plasma performance metric increases. An important connection between the hazard function and its use as a conditional probability generator is reviewed in the Appendix.

We present a fast and accurate data-driven surrogate model for divertor plasma detachment prediction leveraging the latent feature space concept in machine learning research. Our approach involves constructing and training two neural networks: an autoencoder that finds a proper latent space representation (LSR) of plasma state by compressing the multi-modal diagnostic measurements and a forward model using multi-layer perception (MLP) that projects a set of plasma control parameters to its corresponding LSR. By combining the forward model and the decoder network from autoencoder, this new data-driven surrogate model is able to predict a consistent set of diagnostic measurements based on a few plasma control parameters. In order to ensure that the crucial detachment physics is correctly captured, highly efficient 1D UEDGE model is used to generate training and validation data in this study. The benchmark between the data-driven surrogate model and UEDGE simulations shows that our surrogate model is capable of providing accurate detachment prediction (usually within a few per cent relative error margin) but with at least four orders of magnitude speed-up, indicating that performance-wise, it has the potential to facilitate integrated tokamak design and plasma control. Comparing with the widely used two-point model and/or two-point model formatting, the new data-driven model features additional detachment front prediction and can be easily extended to incorporate richer physics. This study demonstrates that the complicated divertor and scrape-off-layer plasma state has a low-dimensional representation in latent space. Understanding plasma dynamics in latent space and utilising this knowledge could open a new path for plasma control in magnetic fusion energy research.

Ion heating in collisionless shocks is non-adiabatic and efficient. The amount of heating and the downstream distributions depend on the shock parameters and on the incident ion distribution. The number of reflected ions and their distribution depend on the detailed shape of the tail of the distribution. In supercritical shocks the reflected ion contribution is significant. Kappa distributed ions are heated more strongly and have a larger fraction of reflected ions than Maxwellian distributed ions with the same upstream temperature and the same shock parameters. For kappa distributions the phase space dips are shallower.

The reflection and transmission of an incident solitary wave with an arbitrary propagation direction due to an interface are investigated in the present paper. It is found that the propagation direction of the transmitted solitary wave depends on not only the propagation direction of the incident solitary wave, but also on the system parameters such as the masses, the number densities of dust particles in two different regions. Dependence of the transmission angle on the plasma parameters and incident angle are given analytically. Moreover, the number and amplitude of transmitted solitary waves and reflected solitary waves are also given when there is only one exact incident solitary wave. Our result has potential application, for example, we can devise an appropriate experiment to measure the differences of the masses and number densities of dust particles between two different regions by using our present results. Furthermore, we can also measure the electric charge of a dust particle by devising an appropriate experiment by using our results.

We develop the formalism of the first-order near-axis expansion of the magnetohydrodynamic equilibrium equations described by Garren & Boozer (Phys. Fluids B, vol. 3, issue 10, 1991, pp. 2805–2821) and Plunk et al. (J. Plasma Phys., vol. 85, issue 6, 2019; J. Plasma Phys., vol. 87, issue 6, 2021) for the case of a quasi-isodynamic, $N$-field-period, stellarator-symmetric, single-well magnetic field equilibrium. The importance of the magnetic axis shape is investigated, and we conclude that control of the curvature and torsion is crucial to obtain omnigenous configurations with finite aspect ratio and low effective ripple, especially for a higher number of field periods. For this reason a method is derived to construct classes of axis shapes with favourable curvature and torsion. Solutions are presented, including a three-field-period configuration constructed at an aspect ratio of $A=20$, with a maximum elongation of $e=3.2$ and an effective ripple under $1\,\%$, which demonstrates that high elongation is not a necessary feature of quasi-isodynamic stellarators.

The prevailing paradigm for plasma turbulence associates a unique stationary state with given equilibrium parameters. We report the discovery of bistable turbulence in a strongly magnetised plasma with a sheared mean flow. Two distinct states, obtained with identical equilibrium parameters in first-principle gyrokinetic simulations, have turbulent fluxes of particles, momentum and energy that differ by an order of magnitude – with the low-transport state agreeing with experimental observations. Occurrences of the two states are regulated by the competition between an externally imposed mean flow shear and ‘zonal’ flows generated by the plasma. With small turbulent amplitudes, zonal flows have little impact, and the mean shear causes turbulence to saturate in a low-transport state. With larger amplitudes, the zonal shear can (partially) oppose the effect of the mean shear, allowing the system to sustain a high-transport state. This poses a new challenge for research that has so far assumed a uniquely defined turbulent state.

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.

We extend our previous work on the two-dimensional (2-D) Dimits transition in ion-scale turbulence (Ivanov et al., J. Plasma Phys., vol. 86, 2020, 855860502) to include variations along the magnetic field. We consider a three-field fluid model for the perturbations of electrostatic potential, ion temperature, and ion parallel flow in a constant-magnetic-curvature geometry without magnetic shear. It is derived in the cold-ion, long-wavelength asymptotic limit of the gyrokinetic theory. Just as in the 2-D model, a low-transport (Dimits) regime exists and is found to be dominated by a quasistatic staircase-like arrangement of strong zonal flows and zonal temperature. This zonal staircase is formed and maintained by a negative turbulent viscosity for the zonal flows. Unlike the 2-D model, the three-dimensional (3-D) one does not suffer from an unphysical blow up beyond the Dimits threshold where the staircase becomes nonlinearly unstable. Instead, a well-defined finite-amplitude saturated state is established. This qualitative difference between the 2-D and 3-D models is due to the appearance of small-scale ‘parasitic’ modes that exist only if we allow perturbations to vary along the magnetic field lines. These modes extract energy from the large-scale perturbations and provide an effective enhancement of large-scale thermal diffusion, thus aiding the energy transfer from large injection scales to small dissipative ones. We show that in our model, the parasitic modes always favour a zonal-flow-dominated state. In fact, a Dimits state with a zonal staircase is achieved regardless of the strength of the linear drive, provided the system is sufficiently extended along the magnetic field and sufficient parallel resolution is provided.

New orbit-averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity-space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS (Velasco et al., J. Comput. Phys., vol. 418, 2020, 109512; Velasco et al., Nucl. Fusion, vol. 61, 2021, 116013). The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt {\nu }$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt {\nu }$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by Calvo et al. (Plasma Phys. Control. Fusion, vol. 59, 2017, 055014). In these stellarators, the $\sqrt {\nu }$ regime exists in the collisionality interval $(a/R) \ln (R/a) \ll \nu _{ii} R a/\rho _i v_{ti} \ll R/a$. In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt {\nu }$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$: $\delta ^{2} |\ln \delta | \ll \nu _{ii} R a/\rho _i v_{ti} \ll 1$. Here, $\nu _{ii}$ is the ion–ion collision frequency, $\rho _i$ and $v_{ti}$ are the ion gyroradius and thermal speed, and $a$ and $R$ are the minor and major radii.

Small-amplitude, symmetry-breaking magnetic field perturbations, including resonant magnetic perturbations (RMPs) and error fields, can profoundly impact plasma properties in both tokamaks and stellarators. In this work, we perform the first comparison between the Stepped Pressure Equilibrium Code (SPEC) (a comparatively fast and efficient equilibrium code based on energy-minimisation principles) and M3D-C$^{1}$ (a high-fidelity albeit computationally expensive initial-value extended-magnetohydro- dynamic (MHD) code) to assess the conditions under which SPEC can be used to model the nonlinear, non-ideal plasma response to an externally applied $(m=2,n=1)$ RMP field in an experimentally relevant geometry. We find that SPEC is able to capture the plasma response in the weakly nonlinear regime – meaning perturbation amplitudes below the threshold for break up of the separatrix and onset of secondary magnetic island formation – when around half of the total toroidal flux is enclosed in the volume containing the $q=2$ resonant surface. The observed dependence of SPEC solutions on input parameters, including toroidal flux and the number of volumes into which the plasma is partitioned, indicates that additional exploration of the underlying Multi-Region Relaxed MHD physics model is needed to constrain the choice of parameters. Nonetheless, this work suggests promising applications of SPEC to optimisation and fusion plasma design.

A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field $\boldsymbol {B_0} = B_0 \hat {\boldsymbol {e}}_\|$. This regime is relevant for scales smaller than the electron inertial length $d_e$. We derive the kinetic equations that describe the three-wave interactions between inertial whistler or kinetic Alfvén waves. We show that for both invariants, energy and momentum, the transfer is anisotropic (axisymmetric) with a direct cascade mainly in the direction perpendicular ($\perp$) to $\boldsymbol {B_0}$. The exact stationary solutions (Kolmogorov–Zakharov spectra) are obtained for which we prove the locality. We also found the Kolmogorov constant $C_K \simeq 8.474$. In the simplest case, the study reveals an energy spectrum in $k_\perp ^{-5/2} k_\|^{-1/2}$ (with k the wavenumber) and a momentum spectrum enslaved to the energy dynamics in $k_\perp ^{-3/2} k_\|^{-1/2}$. These solutions correspond to a magnetic energy spectrum ${\sim }k_\perp ^{-9/2}$, which is steeper than the EMHD prediction made for scales larger than $d_e$. We conclude with a discussion on the application of the theory to space plasmas.

The Vlasov–Fokker–Planck (VFP) equation, a variant of the Boltzmann equation, is frequently used to model the dynamics of laboratory and astrophysical plasmas. A common approach in solving the VFP equation in numerical and analytic studies is to expand the momentum part of the distribution function $f$ (i.e. the particle density in phase-space) in terms of known functions. The Cartesian tensor and spherical harmonic expansion have been widely used, leading to the question of how to convert between the different coefficients of the two expansions. This problem is also familiar in multipole expansions of an electrostatic (or gravitational) potential. The coefficients of the Cartesian tensor expansion of the potential are called (Cartesian) multipole moments and the ones of the spherical harmonic expansion are called spherical multipole moments. In this paper, we investigate the relation between the two kinds of multipole moments and provide a general formalism to convert between them. We subsequently apply this formalism to the coefficients of the expansions of the distribution function $f$. A free, open-source command-line tool which implements this formalism is provided.

Vertical displacement normal modes in shaped tokamak plasmas are studied analytically, based on the reduced ideal-magnetohydrodynamic model. With the help of quadratic forms, and using the appropriate eigenfunction for vertical displacements with toroidal mode number $n=0$ and dominant elliptical-angle mode number $m=1$, a dispersion relation is derived, including the effects of ideal or resistive walls through a single parameter, $D_w(\gamma )$, which is, in general, a function of the complex eigenfrequency $\gamma = -{\rm i}\omega$. For the resistive-wall case, the dispersion relation is cubic in $\gamma$. One root corresponds to the well-known, non-rotating resistive-wall vertical mode, growing on the resistive-wall time scale. The other two roots are weakly damped by wall resistivity, but oscillate with a frequency below the poloidal Alfvén frequency, which makes them immune to continuum damping, but subject to possible instability due to resonant interaction with fast ions.