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.

Wave interactions with magnetised particles underlie many plasma heating and current drive technologies. Typically, these interactions are modelled by bounce averaging the quasilinear Kennel–Engelmann diffusion tensor over the particle orbit. However, as an object derived in a two-dimensional space, the Kennel–Engelmann tensor does not fully respect the conservation of four-momentum required by the action conservation theorem, since it neglects the absorption of perpendicular momentum. This defect leads to incorrect predictions for the wave-induced cross-field particle transport. Here, we show how this defect can easily be fixed, by extending the tensor from two to four dimensions and matching the form required by four-momentum conservation. The resulting extended tensor, when bounce averaged, recovers the form of the diffusion paths required by action-angle Hamiltonian theory. Importantly, the extended tensor should be easily implementable in Fokker–Planck codes through a mild modification of the existing Kennel–Engelmann tensor.

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.

We present a novel mechanism in which plasma electrons and ions optically acquire angular momentum during local pump depletion of an azimuthally polarised laser, despite the laser carrying none. Using theoretical considerations and multi-dimensional particle-in-cell simulations, we find that this process is enabled by a strong frequency downshift at the gradually eroding laser pulse front. We further show that the angular momentum gained by the plasma electrons is compensated by the ions and by the combined electromagnetic fields of the laser and nonlinear plasma wave. By varying key laser parameters such as phase, frequency and polarisation, we demonstrate that the transverse momentum of high-energy electrons can be effectively controlled.

Oscillations and instabilities in $E\!\times\!B$ plasma discharges play a major role in driving turbulence and anomalous transport. The presence of gradients and applied fields that result in a relative drift between the charged species causes the onset of drift-gradient instabilities. This work puts forward a kinetic approach to the low-frequency stability of $E\!\times\!B$, two-species, partially magnetised plasmas. It presents a local linear analysis of electrostatic modes in the long-wavelength limit, assuming wave propagation perpendicular to the magnetic field. Magnetised electrons are described by a drift-kinetic equation, while unmagnetised ions are described as a cold fluid. This allows for a consistent treatment of electron temperature perturbations in an inhomogeneous plasma that takes into account kinetic resonance effects, changing the threshold conditions for drift instabilities previously found in fluid descriptions. The well-known fluid dispersion relation for the collisionless Simon–Hoh instability is recovered by considering the weak magnetic inhomogeneity limit of our kinetic model and, in addition, a perturbative extension of the instability criterion that includes the gradients of the magnetic field and the electron temperature is obtained.

An efficient novel mechanism of laser pulse focusing with the help of a shaped underdense plasma target immersed in an inhomogeneous magnetic field has been demonstrated. These studies have been carried out with the help of two-dimensional particle-in-cell simulation employing the OSIRIS 4.0 platform. It is shown that the divergent magnetic field profile compresses the electromagnetic wave pulse in the transverse direction. A comparative investigation with plane and lens-shaped plasma geometries has also been conducted to find an optimal configuration for focusing the laser at the desirable location. Furthermore, it is also demonstrated that, when the electron cyclotron resonance layer is placed at a suitable location where the laser is focused, a highly energetic electron beam is generated.

The properties of a collisional magnetized plasma sheath containing non-thermal electrons, multi-component ions (${\text{He}}^{+}$ and $ {\text{Ar}}^{+}$), neutral atoms and negatively charged dust particles are analysed. Using a one-dimensional fluid model, the parametric changes in sheath dynamics are investigated in the presence of nanometre-sized charged dust particles and an oblique magnetic field. The influence of charged dust, ionization, ion–neutral collisions, ion loss and non-thermal electrons on sheath parameters such as ion densities, velocities, electron density and potential is explored through theoretical modelling and numerical analysis. The results indicate that the ion density (${\text{He}}^{+}$ and $ {\text{Ar}}^{+}$) increases throughout the sheath region with rising ionization frequency in the absence of charged dust. However, when charged dust is present, the density of ${\text{He}}^{+}$ ions decreases while the density of $\text{Ar}^{+}$ ions increases, exhibiting a sharp peak near the sheath edge. It is also noted that the increase in ion–neutral collision frequency enhances the density, particularly near the sheath edge. Additionally, the presence of non-thermal electrons initially leads to an increase in ion density near the sheath edge, followed by a decrease within the sheath region. A qualitative explanation of the above phenomena, which occur due to different physical parameters, is provided.

Ion-acoustic waves in a dusty plasma are investigated where it is assumed that the ions follow a Cairns distribution and the electrons are Boltzmann distributed. Two theoretical methods are applied: Sagdeev pseudopotential analysis (SPA) and reductive perturbation theory (RPT). Since SPA incorporates all nonlinearities of the model it is the most accurate but deriving soliton profiles requires numerical integration of Poisson’s equation. By contrast, RPT is a perturbation method which at second order yields the Gardner equation incorporating both the quadratic nonlinearity of the Korteweg–de Vries (KdV) equation and the cubic nonlinearity of the modified KdV equation. For consistency with the perturbation scheme the coefficient of the quadratic term needs to be at least an order of magnitude smaller than the coefficient of the cubic term. Solving the Gardner equation yields an analytic expression of the soliton profile. Selecting an appropriate set of compositional parameters, the soliton solutions obtained from SPA and RPT are analysed and compared.

A theoretical investigation on the space–time evolution of low-frequency dust acoustic waves (DAWs) in opposite polarity dusty plasmas reveals that they undergo phase mixing for arbitrary initial amplitudes, causing them to suffer a gradual loss in coherency. Both positively and negatively charged dynamical dust grains have been considered to coexist in the plasma, in addition to Maxwell–Boltzmann distributed hot electrons and ions. A perturbative analysis of the governing fluid-Maxwell equations leads us to conclude that the competing dynamics of the opposite polarity dust grains is what causes the DAWs to phase mix. An estimate for the phase-mixing time has also been obtained, which has been found to be profoundly influenced by the values of the various plasma parameters, such as the equilibrium densities of the plasma species, the masses of the opposite polarity dust grains and the electron and ion temperatures. The investigation has also been extended to include phase mixing of DAWs in electron-depleted dusty plasmas. The findings of this study are expected to have relevance in various astrophysical and laboratory plasma environments.

A set of experiments were conducted on the LArge Plasma Device (LAPD) at UCLA to test the operational principles of a traveling wave antenna of the comb-line type. This antenna was designed to launch helicon waves (fast waves in the lower hybrid range of frequencies) on DIII-D. With the order-of-magnitude lower static magnetic field on LAPD, the antenna excites waves in a different regime. Whenever fast waves can propagate in LAPD, slow waves are also supported by the plasma so it is necessary to distinguish between the two cold-plasma branches in evaluating the effectiveness of the launcher. The results show that the launcher couples well to fast waves when the plasma supports fast-wave propagation; control of the principal imposed parallel wavenumber can be achieved through varying the launch frequency on the antenna within its bandwidth of operation; and that the launched waves exhibit strong directionality. We also investigate the role of the plasma profile and wave mode on the loading characteristics. Additionally, a comparison with full-wave modeling of the propagating waves is shown using both a cold-plasma model in COMSOL and a hot-plasma model in RFPisa, which obtain similar results in the present regime.

The propagation and absorption of the slow waves in the plasma of the Joint European Torus (JET) tokamak have been investigated by ray tracing. The study aims to obtain a qualitative notion of the penetration into the plasma and absorption of the slow wave excited by the A2 ITER-like antenna. The slow waves are radiated by antennas in the ion cyclotron resonance frequency inverted minority heating or mode conversion heating regimes. It has been discovered that the rays propagate in the toroidal direction over a significant distance, up to 6 × 103 cm, from the antenna. Spreading in the peripheral plasma, mainly between the separatrix and the wall, they slowly shift in the poloidal direction and can reach the divertor region. The change in equilibrium of the JET tokamak has a strong influence on both the propagation and absorption of slow waves. Absorption of the slow waves is caused by ion–electron collisions and Landau damping. In the minority heating regimes, the slow waves are strongly damped in the cyclotron resonance of minority ions even at very low minority density.

Landau’s collisionless result for a weakly damped plasma wave is precisely recovered in a weakly collisional, steady state plasma by treating the physics of the narrow collisional boundary layer associated with the resonant electrons. To recover Landau’s results, the collision frequency must be large enough that islands are unable to form and/or the wave amplitude must be small enough to allow linearization. However, the Landau treatment fails once the collision frequency becomes too weak and/or the wave amplitude too large. Remarkably, Landau’s weakly damped plasma wave results require collisions and are shown to be inappropriate in the collisionless limit for a nonlinear, finite amplitude, steady state wave!

The intense applied lower hybrid electric fields used to drive current in tokamaks can result in the formation of velocity space island structure. When this happens the lower hybrid current drive efficiency can be calculated for a monochromatic wave and is shown to be below the quasilinear level.

We derive a set of simplified equations that can be used for numerical studies of reduced magnetohydrodynamic turbulence within a small patch of the radially expanding solar wind. We allow the box to be either stationary in the Sun’s frame or to be moving at an arbitrary velocity along the background magnetic-field lines, which we take to be approximately radial. We focus in particular on the case in which the box moves at the same speed as outward-propagating Alfvén waves. To aid in the design and optimization of future numerical simulations, we express the equations in terms of scalar potentials and Clebsch coordinates. The equations we derive will be particularly useful for conducting high-resolution numerical simulations of reflection-driven magnetohydrodynamic turbulence in the solar wind, and may also be useful for studying turbulence within other astrophysical outflows.

The phenomenon of focusing of microwave beams in a plasma near a turning-point caustic is discussed by exploiting the analytical solution to the Gaussian beam-tracing equations in the two-dimensional (2-D) linear-layer problem. The location of maximum beam focusing and the beam width at that location are studied in terms of the beam initial conditions. This focusing must be taken into account to interpret Doppler backscattering (DBS) measurements. We find that the filter function that characterises the scattering intensity contribution along the beam path through the plasma is inversely proportional to the beam width, predicting enhanced scattering from the beam focusing region. We show that the DBS signal enhancement for decreasing incident angles between the beam path and the density gradient is due to beam focusing and not due to forward scattering, as was originally proposed by (Gusakov et al., (Plasma Phys. Contr. Fusion, vol. 56, 2014, p. 0250092014, 2017); Plasma Phys. Rep. vol. 43(6), 2017, pp. 605–613). The analytic beam model is used to predict the measurement of the $k_y$ density-fluctuation wavenumber power spectrum via DBS, showing that, in an NSTX-inspired example, the spectral exponent of the turbulent, intermediate-to-high $k_y$ density-fluctuation spectrum might be quantitatively measurable via DBS, but not the spectral peak corresponding to the driving scale of the turbulent cascade.

High-beta magnetised plasmas often exhibit anomalously structured temperature profiles, as seen from galaxy cluster observations and recent experiments. It is well known that when such plasmas are collisionless, temperature gradients along the magnetic field can excite whistler waves that efficiently scatter electrons to limit their heat transport. Only recently has it been shown that parallel temperature gradients can excite whistler waves also in collisional plasmas. Here, we develop a Wigner–Moyal theory for the collisional whistler instability starting from Braginskii-like fluid equations in a slab geometry. This formalism is necessary because, for a large region in parameter space, the fastest-growing whistler waves have wavelengths comparable to the background temperature gradients. We find additional damping terms in the expression for the instability growth rate involving inhomogeneous Nernst advection and resistivity. They (i) enable whistler waves to re-arrange the electron temperature profile via growth, propagation and subsequent dissipation, and (ii) allow non-constant temperature profiles to exist stably. For high-beta plasmas, the marginally stable solutions take the form of a temperature staircase along the magnetic field lines. The electron heat flux can also be suppressed by the Ettingshausen effect when the whistler intensity profile is sufficiently peaked and oriented opposite the background temperature gradient. This mechanism allows cold fronts without magnetic draping, might reduce parallel heat losses in inertial fusion experiments and generally demonstrates that whistler waves can regulate transport even in the collisional limit.

Landau damping is one of the cornerstones of plasma physics. Based on the initial-value approach adopted by Landau in his original derivation of Landau damping, we examine the solutions of the linear Vlasov–Poisson system for different equilibrium distribution functions $f_0(v)$, going beyond the traditional focus on the root with largest imaginary part and investigating the full set of roots that the dispersion relation of the system generally admits. Specifically, we provide analytical insights into the number and the structure of the roots for entire and meromorphic functions $f_0(v)$, such as Maxwellian and $\kappa$ distributions, we discuss the potential issues related to the redefinition of $\partial{f}_0/\partial{v}$ as a complex variable function and we show how different sigmoids affect the root structure associated with non-meromorphic cut-off distribution functions. Finally, based on the comparison of the several root structures considered, we wonder if the multiple roots might hint at a deeper understanding of the Landau damping phenomenon.

The change in direction of the wavevector and group velocity experienced by a wave refracted at the interface of an anisotropic medium in uniform linear motion are determined analytically. These transmission conditions, which are shown to be consistent with the generalised Snell’s law written in the laboratory frame, are then used to examine the effect of motion on waves incident on a magnetised plasma. For an incident wave in the plane perpendicular to the magnetic field the motion is observed to lead to non-negligible deviation of the low-frequency X-mode, as well as to non-symmetrical total reflection angles. These effects are shown to be further complicated when the magnetic field is in the plane formed by the incident wavevector and the medium’s velocity, as the anisotropy now competes with the motion-induced drag. Although obtained in simplified configurations, these results suggest that accounting for motion when modelling plasma wave trajectories could be important under certain conditions, calling for a more detailed quantification of the effect of motion in actual diagnostics and plasma control schemes.

The generation of an autoresonantly phase-locked high-amplitude plasma waves to the chirped beat frequency of two driving lasers is studied in two dimensions using particle-in-cell simulations. The two-dimensional plasma and laser parameters correspond to those that optimized the plasma wave amplitude in one-dimensional simulations. Near the start of autoresonant locking, the two-dimensional simulations appear similar to one-dimensional particle-in-cell results (Luo et al., Phys. Rev. Res., vol. 6, 2024, p. 013338) with plasma wave amplitudes above the Rosenbluth–Liu limit. Later, just below wave breaking, the two-dimensional simulation exhibits a Weibel-like instability and eventually laser beam filamentation. These limit the coherence of the plasma oscillation after the peak plasma wave field is obtained. In spite of the reduction of spatial coherence of the accelerating density structure, the acceleration of self-injected electrons in the case studied remains at $70\,\%$ to $80\,\%$ of that observed in one dimension. Other effects such as plasma wave bowing are discussed.

The linear collisionless plasma response to a zonal-density perturbation in quasisymmetric stellarators is studied, including the geodesic-acoustic-mode oscillations and the Rosenbluth–Hinton residual flow. While the geodesic-acoustic-mode oscillations in quasiaxisymmetric configurations are similar to tokamaks, they become non-existent in quasi-helically symmetric configurations when the effective safety factor in helical-angle coordinates is small. Compared with concentric-circular tokamaks, the Rosenbluth–Hinton residual is also found to be multiplied by a geometric factor $\mathcal {C}$ that arises from the flux-surface-averaged classical polarization. Using the near-axis-expansion framework, we derive an analytic expression for $\mathcal {C}$, which varies significantly among different configurations. These analytic results are compared with numerical simulation results from the global gyrokinetic particle-in-cell code GTC, and good agreement with the theoretical Rosenbluth–Hinton residual level is achieved when the quasisymmetry error is small enough.