The above question may be answered as follows. A typical longitudinal plasma wave for which the collision frequency, $\nu$, is low relative to the angular wave frequency, $\omega$, only interacts with the degree of freedom, $D$, related to motion parallel to the wave vector. This gives an adiabatic coefficient corresponding to $D = 1$, i.e. 3. At high $\nu /\omega$, relevant to neutral gases, collisions cause even longitudinal sound waves to interact with all active $D$, yielding an adiabatic coefficient of $(D+2)/D$. We present a minimal example illustrating the above transition based on linear analysis of a non-relativistic, isotropic, homogeneous, Maxwellian one-component system with a Bhatnagar–Gross–Krook collision operator. Macroscopic forces, mainly included in plasma physics, are essential for the transition at low $\nu /\omega$. Additionally, the tensor nature of pressure and collision operators satisfying mass, momentum and energy conservation must be invoked to obtain the correct response. Our analysis yields a polytropic index at arbitrary $\omega$ and wavenumbers, which reduces to the adiabatic coefficient in the corresponding limit. An adiabatic wave response is found in two distinct regimes. If the wave phase velocity far exceeds the thermal particle speed, an adiabatic response occurs at any $\nu /\omega$ (plasma wave case). Alternatively, for mean free paths much shorter than the wavelength, an adiabatic response can be obtained even at phase velocities comparable with the thermal particle speed (sound wave case). The analysis finally demonstrates the occurrence of Landau damping in neutral gases with arbitrary $D$ at low $\nu /\omega$.

We report an unanticipated electron dynamics in a classical setting of a uniform magnetic field, a parallel electric field and a right-handed circularly polarised wave (R-wave). The setting admits a natural trajectory that a particle accelerated by the electric field reaches a Doppler–shifted cyclotron resonance and becomes trapped in the resonance space. Remarkably, once it becomes resonantly trapped, the electron undergoes reversal of parallel acceleration together with perpendicular energisation, despite the parallel electric field remaining constant. This counterintuitive behaviour has important implications for particle scattering in various laboratory and space plasmas. Applied to fusion devices, particle-in-cell simulations show that an externally injected R-wave can act as a firewall suppressing further runaway-electron acceleration.

The requirement for large-scale global simulations of plasma is an ongoing challenge in both space and laboratory plasma physics. Any simulation based on a fluid model inherently requires a closure relation for the high-order plasma moments. This review compiles and analyses the recent surge of machine learning (ML) approaches developing improved plasma closure models capable of capturing kinetic phenomena within plasma fluid models. We survey two methodological families: neural network surrogates (from multi-layer perceptrons to Fourier neural operators, the latter recently reproducing both linear and nonlinear Landau damping online within a fluid solver) and equation discovery methods such as sparse regression; and organise the studies by whether they are tested offline against reference data or online within a time-evolving solver. We outline the challenges associated with ML closures, including off-diagonal pressure-tensor accuracy, generalisation beyond the training distribution and stable integration into large-scale simulations, and the directions future research might take to address them.

Strong gradient regions in tokamaks such as the pedestal or internal transport barriers are regions of reduced turbulence where neoclassical transport can play a dominant role. In pedestals, gradient lengths comparable to the ion poloidal gyroradius have been measured. Standard neoclassical theory can miss important strong gradient effects in these regions because it assumes that the gradient length scales of density, temperature and potential are larger than the ion poloidal gyroradius. We extend plateau regime neoclassical theory into regions of gradients of the order of the ion poloidal gyroradius to capture strong gradient effects on transport processes in the pedestal and internal transport barriers. The fundamental idea behind our new framework is to keep a scale separation between the orbit widths and the gradient length scales by performing a large aspect ratio expansion. In the plateau regime, strong gradients cause poloidal variation that is in–out as well as up–down asymmetric. We study two different test cases assuming either radial force balance or the absence of turbulence and show that strong gradient effects can enhance or reduce standard neoclassical theory predictions in the plateau regime in strong gradient regions.

This paper revisits the onset of high-harmonic fast-wave (HHFW) propagation in the scrape-off layer (SOL) plasma of the NSTX/NSTX-U spherical tokamak, motivated by past HHFW heating and current drive experiments and modelling. Previously, the fast-wave propagation in the SOL was correlated with the opening (suppression) of the fast-wave right-hand cutoff layer in front of the antenna. In this work, the SOL propagation is interpreted as a geometric waveguide mode guided by the SOL geometry, whose radial width can be comparable to the wave’s perpendicular wavelength in the HHFW regime. A two-dimensional circular model is first employed using full-wave solvers to characterise the poloidal eigenmode structures and to clarify their relationship to annulus resonance. By progressively adding a tokamak-like magnetic-field configuration, starting from a uniform axial field, the effects of magnetic-field gradients and pitch on the SOL waveguide mode are characterised. The poloidal mode numbers supported by the SOL plasma agree well with analytic estimates, indicating that an anisotropic plasma in a bounded geometry selectively supports and amplifies the resonant poloidal mode number. Additionally, two-dimensional axisymmetric NSTX-U simulations demonstrate that the SOL eigenmode features identified in the circular model persist in the experimentally relevant configurations. A control approach based on lengthening the wave perpendicular wavelength relative to the SOL width, including a higher magnetic (B)-field operation, is discussed. The analysis here shows the key role of a bounded geometry in interpreting HHFW eigenmode coupling and propagation in the NSTX-U SOL plasma.

Vlasov–Poisson–Fokker–Planck (VPFP) simulations of large-amplitude electron plasma waves, where the bounce frequency is much larger than the collision frequency, $\omega _B \gg \nu _{\textit{ee}}$, show that the evolution of these waves exhibits three phases: (i) a short-lived trapping phase during which collisional effects are minimal; (ii) a long-lived detrapping phase during which collisional effects are most influential; (iii) a short-lived Landau damping phase where the effect of collisions becomes minimal again. While the dispersion relation during the trapping and Landau damping phase is well known, the wave behaviour during the detrapping phase is not as well understood. The simulations show that during the detrapping phase, the interplay between weak electron–electron collisions and strong wave–electron interactions results in an increasing frequency shift further from the linear root, $\omega _{\text{EPW}}$. At the conclusion of the detrapping phase, the distribution function is nearly Maxwellian, the frequency shift rapidly diminishes and the wave damps at a larger rate than the Landau damping rate. Empirical fits to the damping rates, frequency shift enhancement rate and the lifetime of the plasma waves are provided as functions of collision frequency, wavenumber and wave amplitude.

The magnetic confinement topologies potentially capable of macro-stability with core plasma pressure greatly exceeding the vacuum magnetic field pressure (or ‘$\beta \rightarrow \infty$’) are discussed. An emphasis is placed on application to magnetic fusion energy (MFE) given the perceived advantages. The particle loss rate from highly diamagnetic plasmas in open-field topologies is quantified, including a generalised model for the gas-dynamic regime. It is concluded that, regardless of the topology type, attractive fusion energy gain from such a system is impossible unless (i) the scrape-off-layer can be made much thinner than one thermal ion gyroradius through ambipolar effects, (ii) sufficient stability exists in the stochastic phase space of certain topologies so as to increase the average number of ion orbits before loss by $\sim$1000 times over the isotropic prediction or (iii) electrostatic or centrifugal effects are used to recover adiabatic confinement. Outside of MFE, high-$\beta$ topologies offer many alternative applications in plasma science. The required magnet bore to limit perpendicular particle loss in a point cusp is estimated. Additionally, the unique spherical multipole topology is reviewed and comments are made on the use of cusp topologies for ‘electromagnetic’ confinement efforts.

One of the more interesting features of kinetic descriptions of plasmas are resonant wave particle interactions. These often appear to be collisionless as the collision frequency does not appear in key results even though collisions are actually vital whenever the Coulomb logarithm is large. This deceptive behaviour normally occurs when a resonance is resolved by causality or by inserting a small Krook collision frequency in a linearised kinetic equation to resolve a singularity. However, the actual behaviour of the resonance is more complicated as a narrow collisional boundary layer is responsible for resolving the singular behaviour. The most remarkable example is linear Landau damping, which is normally viewed as a collisionless process. However, an evaluation by Zakharov & Karpman (1963 Sov. Phys. JETP vol. 16, pp. 351-357) proved it to be a collisional process for which collisions can never be ignored for a finite wave amplitude in the collisionless limit. This realisation implies that a small tail bump on a Maxwellian will have a growth or damping rate sensitive to collisions for a finite amplitude monochromatic plasma wave in the very weak collisionality limit. This behaviour is verified here by retaining collisions in both the island dominated limit, as well as the linear, collisional boundary layer dominated regime.

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 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.