Machine learning (ML)-driven reduced-order modelling is applied to accelerate steady-state convergence in three-dimensional, nonlinear, flux-driven two-fluid simulations of boundary plasma turbulence. A parametric scan of plasma resistivity, heating and density sources is performed to generate comprehensive datasets across various turbulent regimes for model training and validation. To efficiently manage and interpret these datasets, we apply the proper orthogonal decomposition technique to reduce the dimensionality of key plasma quantities such as plasma density, temperature, electric potential and vorticity. Data-driven models are trained to map physical parameters to low-dimensional representation, enabling the rapid generation of quasi-steady-state plasma profiles. The results demonstrate that density, temperature and electric potential are qualitatively well captured with a relatively low number of bases, whereas vorticity requires a larger number of bases due to its fine spatial structures. A comparison between ML-generated restarts and simulations from scratch demonstrates a significant computational advantage of the ML approach, reducing simulation time by up to a factor of three. This hybrid framework, combining data-driven reduced-order modelling with first-principles simulations, highlights the potential of ML to accelerate plasma turbulence modelling, making high-fidelity simulations more computationally feasible for large-scale fusion devices, such as ITER and DEMO.

The effects of the main parameters of the helicon plasma sources on the volume process of the negative ion production mechanism are investigated. Using COMSOL Multi-Physics software, a helicon plasma source as a source driver of a negative ion source is modelled in three dimensions. In this work, it is considered that the helicon plasma source employs a Nagoya-type antenna at an operational frequency of 13.56 MHz. The influences of the static magnetic field variation, applied radio frequency power and injected gas pressure on electron/plasma density, electron temperature and vibrationally excited molecular density are studied. Variations of the static magnetic field in a range of 0.01–0.08 T, Radio Frequency (RF) power in a range of 800–6000 W and gas pressure range of 0.3–1.5 Pa indicate that the maximum electron (plasma) density is increased in all three cases; nevertheless, the electron temperature and maximum density of the vibrationally excited molecules is increased just by RF power increment. For the pressure of 0.3 Pa, it is found that using a proper coil configuration, the electron density and the vibrationally excited molecular density will be increased without the magnetic field (applied DC power) increment and RF power increment.

The Parker Solar Probe mission has observed near-continuous power in parallel ion cyclotron waves (PICWs) in the young, fast solar wind. These waves are unlikely to be directly produced by the turbulent cascade and are likely born of a local instability; yet, they are observed to both cool – and heat – the plasma. We propose that these observations can be self-consistently explained as the natural consequence of PICWs propagating in the inhomogeneous solar wind after they have been driven unstable. In this work, we argue that strong proton heating by a turbulent cascade of oblique ICWs will result in PICWs being driven unstable in a process known as quasi-linear focusing. Because the power in the turbulent cascade is concentrated at scales above the turbulent transition region, PICWs will be driven unstable within a range of wavenumbers parallel to the background magnetic field, $k_\parallel$, that is bounded from above by $k_{\parallel {\mathrm{P}}}^*$, corresponding to the start of the transition region. As unstable PICWs propagate away from the Sun to regions of lower proton density, their $k_\parallel$, multiplied by the proton inertial length $d_{\mathrm{p}}$, increases. Eventually, $k_\parallel d_{\mathrm{p}}$ of the PICWs becomes larger than $k_{\parallel {\mathrm{P}}}^*d_{\mathrm{p}}$ and the waves damp, heating the solar wind. We call this effect ‘cyclotron breaking’, in analogy with ocean waves breaking on the shore. We then discuss the testable predictions of the theory, including a distinct heating signature in which PICWs cool fast protons and heat slow protons at any given heliocentric distance $r$. Finally, we conjecture that cyclotron breaking can lead to net heating by PICWs if the power emitted as PICWs decreases sufficiently rapidly with $r$ that local emission of PICWs is overwhelmed by the local damping of PICWs generated closer to the Sun.

In solving the Ion Cyclotron Resonance Heating wave equation, plane wave Fourier mode $\exp[ikx]$ base functions are often exploited to produce expressions for the dielectric response when including the effect of finite temperature, requiring locally integrating the equation of motion of charged particles by hand. Standard Fourier analysis notation is used: $x$ is the position, $k$ is the wave vector component in the $x$-direction and $i$ is the imaginary number. In contrast, finite element techniques adopt base functions – typically low - order polynomials – that individually are only non-zero in a small domain. They are a natural and easy approach to capture inhomogeneity effects and readily allow grid refinement to zoom in on regions where this is required. To enable ample realism of kinetic effects while profiting from the rich pool of numerical tools available for solving differential and integro-differential equations relying on finite elements, it is desirable to have a procedure allowing us to profit from both approaches: the detailed physics brought by finite temperature effects – commonly described in terms of Fourier modes – as well as the simplicity from a local polynomial representation. A novel technique is offered to achieve that. It consists of finding the Fourier representation of the localised base functions exploited in finite elements so that the richness of the dielectric response in $k$-space can be accounted for. The resulting equation is assembled just like the finite element method prescribes but the coefficients of the linear local system are assembled differently. The technique allows us to capture finite temperature corrections in both the parallel and perpendicular directions for a dielectric tensor model of choice. The focus in this paper is on the numerical technique while the expressions for the dielectric response are assumed to be known. A few first examples are briefly discussed: the wave equation solutions for a typical minority heating (H)-D JET plasma are provided for (i) an all-FLR model, (ii) the FLR-0 ‘tepid’ equivalent and (iii) a cold plasma model; FLR refers to the Finite Larmor Radius expansion exploited to account for finite temperature corrections.

Plasma in axisymmetric mirror traps is unstable versus flute-like modes if no stabilising measures are taken. Instead of stability it is also possible to aim at suppressing the convective transverse transport generated by unstable modes. A ‘vortex confinement’ scheme of this type is utilised in current operation regimes of the Gas-Dynamic Trap in Novosibirsk Bagryansky et al.(2011 Fusion Sci. Technol., vol. 59, pp. 31–35). The relevant model Beklemishev et al. (2010Fusion Sci. Technol., vol. 57, pp. 351–360) describes the effect as due to nonlinear interaction of the flute modes with the background sheared rotation induced by plasma biasing via end plates and limiters. The rigid $m=1$ mode is saturated only due to current dissipation at the end plates, i.e. the partial line tying. The original model assumes flat radial profiles of plasma density and electron temperature, neglecting possible centrifugal and electron-temperature effects. These sources of instability are added to the original framework using a single scalar forming its hybrid extension. Efficiency of the biasing scheme for nonlinear suppression of flute-like convection is shown to depend primarely on spatial positions of biased facility elements, rather than on additional sources of instability.

The effects of negative triangularity (NT) on boundary plasma turbulence in double-null (DN) configurations are investigated using global, nonlinear, three-dimensional, flux-driven two-fluid simulations. Negative triangularity plasmas exhibit suppressed interchange-driven instabilities, resulting in enhanced confinement and lower fluctuation levels compared with positive triangularity (PT) plasmas. This reduction in interchange instability is associated with the weakening of curvature effects in the unfavourable region, caused by the stretching of magnetic field lines at the outer midplane. The magnetic disconnection between the turbulent low-field side and the quiescent high-field side results in most of the heat flux reaching the DN outer targets. In NT plasmas, the power load on the outer target is reduced, while it increases on the inner target, indicating a reduced in–out power asymmetry compared with PT plasmas. Furthermore, the analysis of power load asymmetry between the upper and lower targets shows that the absolute magnitude of up–down power asymmetry is mitigated in NT plasmas, mainly due to the reduced total power crossing the separatrix. The reduction of interchange instabilities in NT plasmas also affects the blob dynamics. A three-dimensional blob analysis reveals that NT plasmas feature smaller blob sizes and slower propagation velocities. Finally, an analytical scaling law for blob size and velocity that includes plasma shaping effects is derived based on the two-region model and is found to qualitatively capture the trends observed in nonlinear simulations.

Collisional merging experiments on field-reversed configurations (FRCs) in the FRC amplification via translation–collisional merging (FAT-CM) device have suggested that the mirror magnetic field strength influences the final merged plasma state. Analytical models do not account for mirror magnetic fields at the ends of the confinement section. To explore their effect, experiments were conducted at FAT-CM, where the strength of mirror magnetic fields were varied. Increasing the mirror field strength was found to promote relaxation to an FRC with larger radius and shorter axial length. Based on these observations, the dependence of equilibrium states on the mirror field strength and the resulting plasma parameters were investigated using a model that reconstructs fully two-dimensional equilibria primarily constrained by wall-mounted magnetic probe measurements. Consistent with the experimental results, computed equilibria with stronger mirror fields converged to plasma states with a larger radius and shorter length. Further, these results showed a bifurcation of topologies with a closed FRC core for higher mirror field and an open-field high-$\beta$ mirror for lower field. This result agreed with internal magnetic probe data which confirm the FRC-to-high-$\beta$ mirror transition. This demonstrates the importance of employing fully two-dimensional modelling of plasma equilibria.

The two-plasmon decay instability of high-power microwaves used for electron cyclotron resonance heating has been observed experimentally in multiple fusion devices. This type of instability is a nonlinear three-wave interaction that can transfer energy away from the cyclotron harmonics. Analytical models quantify exponential growth rates and power thresholds, but typically evaluate the growth as a spatially averaged gain over the interaction region of the instability. This description effectively excludes field inhomogeneities and noise, leaving their impact on the instability growth uncertain. We assess this assumption by solving the full nonlinear system on a spatial grid. Across all cases considered, we find that the instabilities drive the wave fields toward the spatially averaged behaviour. After a transient period that scales inversely with the instability growth rate, the simulated growth converges to the averaged prediction, indicating that the established models are asymptotically valid.

Runaway electrons (REs), generated during plasma disruptions in tokamaks, pose significant challenges due to the risk of causing damage to the first wall of a device. Understanding the interaction between REs and magnetohydrodynamic (MHD) instabilities is crucial for predicting a safe operation of large future tokamak devices in which RE generation will be drastically enhanced due to the high plasma current. In this work, we introduce a hybrid fluid–kinetic model within the three-dimensional nonlinear MHD code JOREK (Hoelzl et al. 2021 Nucl. Fusion, vol. 61, 065001; 2024 Nucl. Fusion, vol. 64, 112016), treating REs kinetically using a relativistic guiding-centre approach, while describing the background plasma by ansatz-based reduced MHD equations. At first, comprehensive benchmark studies are conducted regarding the two-dimensional equilibrium force balance with $J_{total}= J_{RE}$, and the linear stability of three-dimensional tearing modes (TMs), verifying the accuracy of the model against analytical predictions and other numerical methods, e.g. the full-orbit approach in JOREK and the fluid model in M3D-C1. These benchmark studies build a solid foundation for applying our model to more complex nonlinear scenarios. In this respect, we confirm that the nonlinear saturation of TMs is significantly influenced by the presence of REs. Previous analytical studies (Helander et al. 2007 Phys. Plasmas vol. 14, 122102) suggest that in the case of small $\varDelta ^\prime$, the saturation width of the magnetic island driven by REs is roughly 1.5 times larger than in the otherwise identical Ohmic current scenario. Our simulations are quantitatively in line with this prediction. Moreover, REs alter the energy evolution within the magnetic reconnection process and decouple the bulk plasma and magnetic fields. In summary, RE-driven magnetic reconnection leads to larger magnetic islands but weaker reconnection flows.

Inertial Alfvén waves are thought to accelerate electrons to auroral energies via their parallel electric field in the Earth’s magnetosphere. During active geomagnetic times, it is estimated that a significant percentage of electron precipitation energy into the Earth’s ionosphere can be attributed to these waves. However, self-consistent wave/particle interactions of inertial Alfvén waves with the accelerated electron population are not well understood. We show that recent self-consistent models have a strong nonlinearity in them. A reduced set of equations which describe this nonlinear steepening is derived and shown to agree with drift-kinetic simulations and other published studies. From this reduced set of equations, many properties of the nonlinearity are derived and shown to agree with simulations. This includes the time and length scales and connecting the speed of the wave to the perturbation maximum value.

This work analyses vacuum magnetic field topology in Wendelstein 7-X (W7-X) with respect to changes in the current in the superconducting coils. We develop a fast automated scheme to locate fixed points (such as X- and O-points) and calculate the trace of the Jacobian of the field line map for them ($\mathrm{Tr}(\text{M})$), which represents several important properties of the fixed point. We perform two sets of coil current scans: (i) scans where each coil current is varied individually, using the ‘standard’, ‘high iota’ and ‘low iota’ configurations as starting points; (ii) a scan of over $2\times 10^5$ magnetic configurations in which the coil currents are randomly sampled. In both cases, we constrain the coil currents to the normal range of W7-X. We verify the principal roles of the non-planar, planar and control coils: the non-planar coils establish island chains with a certain phase; the planar coils modify the location of the island chain by both controlling the $\iota$ profile and shifting the configuration ‘inward’ and ‘outward’; the control coil affects the island size and phase. We also find that $\vert (\mathrm{Tr}(\text{M})-2)\vert$ (a quantity closely related to the magnitude of the Greene’s residue) tends to increase with the minor radius of the fixed points, and that $\mathrm{Tr}(\text{M})$ for X- and O-points can be very differently affected by the control coil current. Finally, we show that $\vert (\mathrm{Tr}(\text{M})-2)\vert$ serves as a proxy for island size for internal island chains, which may help identification of suitable experimental candidates.

Neutral atoms recycled from wall interaction interact with confined plasma, thereby refuelling it, most strongly in the region closest to the wall. This occurs near the X-point in diverted configurations, or else near the wall itself in limited configurations. A progression of analytic models is developed for neutral density in the vicinity of a planar or linear source in an ionising domain. First-principles neutral transport simulations with DEGAS2 are used throughout to test the validity and limits of the model when using equivalent sources. The model is further generalised for strong plasma gradients or the inclusion of charge exchange. An important part of the problem of neutral fuelling from recycling is thereby isolated and solved with a closed-form analytic model. A key finding is that charge exchange with the confined plasma can be significantly simplified with a reasonable sacrifice of accuracy by treating it as a loss. The several assumptions inherent to the model (and the simulations with which it is compared) can be adapted according to the particular behaviour of neutrals in the divertor and the manner in which they cross the separatrix.

This study investigates the Lagrangian properties of ion turbulent transport driven by drift-type turbulence in tokamak plasmas focusing on the Cyclone Base Case. Despite the compressible and inhomogeneous nature of Eulerian gyrocentre drifts, numerical simulations with the T3ST code reveal approximate ergodicity, stationarity and time-symmetry. These characteristics are attributed to broad initial phase-space distributions that support ergodic mixing. Moreover, relatively minor constraints on the initial distributions are found to have negligible effects on transport levels.

Cross-field electron transport in partially magnetised plasmas arises from collective, nonlinear instability dynamics that remain only partially understood despite their importance to a wide range of E × B plasma devices. In systems such as Hall thrusters, azimuthal instabilities strongly affect electron confinement and spectral energy distribution, motivating efforts to examine how external modulation may influence these effects. Here, one-and two-dimensional particle-in-cell simulations are employed to investigate how an axially applied oscillatory electric field modifies the instability spectra and the associated cross-field electron transport. The simulations adopt local slab idealisations of an E × B discharge designed to isolate modulation–instability coupling mechanisms and the conclusions should be interpreted within this controlled modelling framework. The simulations show that the plasma response depends sensitively on modulation frequency and amplitude. Notably, modulation near 40 MHz diminishes the amplitude of the electron cyclotron drift instability and reduces axial electron transport by up to 30 %, while modulation near the electron cyclotron frequency leads to spectral broadening and enhanced transport. Bicoherence analysis of the azimuthal electric field fluctuations indicates nonlinear coupling among instability modes, suggesting that modulation reshapes energy pathways, thereby explaining the observed spectral variations. We further show that modulation modifies the phase alignment between azimuthal-electric-field and electron-density fluctuations, in turn directly affecting the observed suppression or amplification of electron transport across modulation regimes. The results provide quantitative evidence of how external modulation can alter instability characteristics in E × B plasmas and point to strategies for controlling electron transport in cross-field plasma technologies, such as Hall thrusters and magnetrons.

Geometrical optics (GO) is widely used for reduced modelling of waves in plasmas, but it fails near reflection points, where it predicts a spurious singularity of the wave amplitude. We show how to avoid this singularity by adopting a different representation of the wave equation. Instead of the physical coordinate $x$ and the wavevector $k$, we use the ray time $\tau$ as the new canonical coordinate and the ray energy $h$ as the associated canonical momentum. To derive the envelope equation in the $\tau$-representation, we construct the Weyl symbol calculus on the $(\tau , h)$ space and show that the corresponding Weyl symbols are related to their $(x, k)$ counterparts by the Airy transform. This allows us to express the coefficients in the envelope equation through the known properties of the original dispersion operator. When necessary, solutions of this equation can be mapped to the $x$-space using a generalised metaplectic transform. However, the field per se might not even be needed in practice. Instead, knowing the corresponding Wigner function usually suffices for linear and quasilinear calculations. As a Weyl symbol itself, the Wigner function can be mapped analytically, using the aforementioned Airy transform. We show that the standard Airy patterns that form in regions where conventional GO fails are successfully reproduced within metaplectic GO (MGO) simply by remapping the field from the $\tau$-space to the $x$-space. An extension to mode-converting waves is also presented. This formulation, which we call generalised MGO, can be particularly useful, for example, for reduced modelling of the O–X conversion in inhomogeneous plasma near the critical density, an effect that is important for fusion applications and also occurs in the ionosphere. Overall, MGO can replace GO for any practical purposes, because it better handles cutoffs and is similar otherwise.

Recently, Alfvénic ion temperature gradient (AITG) modes have been observed in the core plasma on the HL-2A tokamak. Only when electron cyclotron resonance heating (ECRH) and neutral beam injection are simultaneously injected into the deuterium plasma do the AITG modes become unstable. The instability is electromagnetic and localised in the core plasma with an internal transport barrier. Dynamic evolution of AITG modes is greatly affected by the off-axis ECRH. Theoretical analysis suggests that there is a strong dependence of the AITG modes on $\eta _i\simeq \boldsymbol{\nabla }\ln T_i/\boldsymbol{\nabla }\ln n_i$, where $n_i$ is the ion density. It is also found that ECRH can enhances AITG modes by causing a drop of electron density and an increase of $\tau =T_e/T_i$; here $T_e$ and $T_i$ are the electron and ion temperatures, respectively. Besides, high-power ECRH may also change the safety factor or magnetic shear and then contribute to the mitigation of AITG modes. The new findings can not only enrich scientific knowledge for pressure gradient-driven instability, but also be beneficial to active control of core-localised electromagnetic modes in future fusion devices.

A magnetohydrodynamic (MHD) shock is a discontinuity through which plasma flows, while the mass, momentum and energy are conserved, and the entropy increases. In a collisionless system, the entropy is strictly constant along the phase-space trajectory. The MHD required entropy increase is attributed to dissipative processes causing relaxation to local thermodynamic equilibrium. The search for such dissipative processes in the shock observations usually explores variations of the entropy density or entropy per particle across the shock front. We show that both quantities may increase or decrease across the shock front even when the entropy is strictly conserved. Entropy production should be assessed using the entropy flux density.

A conservative formulation of the drift-reduced fluid plasma model is constructed by analytically inverting the implicit relation defining the polarisation velocity as a function of the time derivative of the electric field. The obtained model satisfies exact conservation laws for energy, mass, charge and momentum, in arbitrary magnetic geometry, also when electromagnetic fluctuations are included.