Quasilinear treatments are widely used for tokamaks to evaluate radio frequency (rf) heating and current drive. Even though the core of a tokamak plasma is weakly collisional, the solution of the linearized kinetic equation is evaluated using unperturbed collisionless trajectories while often treating successive poloidal circuits of the passing (and trapped) particles as uncorrelated or nearly so. In addition, the most important effect of tokamak geometry, the mirror force, is usually mistreated or ignored when obtaining the solution. These concerning aspects of rf treatments are clarified by considering lower hybrid heating and current drive to illustrate that the electrons in resonance with the applied rf are enclosed by narrow collisional boundary layers, and that tokamak geometry makes it necessary to retain poloidal variation when solving a weakly collisional linearized kinetic equation. Other aspects such as collisional boundary layers at the trapped–passing boundary, cyclotron resonances, and the limitations of quasilinear theory are also considered. The new insights lead to a fundamentally different formulation and interpretation of the solution of the linearized Fokker–Planck equation used for rf quasilinear theory in a tokamak, while retaining many of the features that have contributed to its successful application to rf heating and current drive.

Understanding the partitioning of turbulent energy between ions and electrons in weakly collisional plasmas is crucial for the accurate interpretation of observations and modelling of various astrophysical phenomena. Many such plasmas are ‘imbalanced’, wherein the large-scale energy input is dominated by Alfvénic fluctuations propagating in a single direction. In this paper, we demonstrate that when strongly-magnetised plasma turbulence is imbalanced, nonlinear conservation laws imply the existence of a critical value of the electron plasma beta (the ratio of the thermal to magnetic pressures) that separates two dramatically different types of turbulence in parameter space. For betas below the critical value, the free energy injected on the largest scales is able to undergo a familiar Kolmogorov-type cascade to small scales where it is dissipated, heating electrons. For betas above the critical value, the system forms a ‘helicity barrier’ that prevents the cascade from proceeding past the ion Larmor radius, causing the majority of the injected free energy to be deposited into ion heating. Physically, the helicity barrier results from the inability of the system to adjust to the disparity between the perpendicular-wavenumber scalings of the free energy and generalised helicity below the ion Larmor radius; restoring finite electron inertia can annul, or even reverse, this disparity, giving rise to the aforementioned critical beta. We relate this physics to the ‘dynamic phase alignment’ mechanism (that operates under yet lower beta conditions and in pair plasmas), and characterise various other important features of the helicity barrier, including the nature of the nonlinear wavenumber-space fluxes, dissipation rates, and energy spectra. The existence of such a critical beta has important implications for heating, as it suggests that the dominant recipient of the turbulent energy, ions or electrons, can depend sensitively on the characteristics of the plasma at large scales.

In view of the ultimate goal of producing long-lasting quasi-stationary discharges required for future fusion power stations, the numerical study of steady-state solutions of equations describing the particle and energy balance rightfully gets ample attention. Transient states may, however, differ significantly from the steady state ultimately reached and will – in practice – impact on the actual fate of the discharge. Using brutally simple models, the present paper highlights a number of aspects to illustrate this dynamics. It e.g. shows the different signature of wave and beam heating, potentially giving room to transiently trigger desirable effects that may allow us to better steer a discharge.

The heating rate of plasma electrons induced by external fields or other processes can be used as an experimental tool to measure fundamental plasma properties such as electrical conductivity or electron–ion collision rates. We have developed a technique that can measure electron heating rates in ultracold neutral plasmas (UNPs) with $\sim 10\,\%$ precision while simultaneously referencing the measurement to a calibrated amount of heating. This technique uses a sequence of applied electric fields in four sections: to control the ratio of electrons to ions in the UNP; to provide a time for the application of fields that cause electron heating and subsequent thermalization of the electrons after the application of those fields; to extract electrons from the UNP using a method sensitive to electron temperature that allows the measurement of electron heating; and to extract the remaining electrons to measure the total electron (and therefore ion) number. The primary signal used to measure the heating rate is the measurement of the number of electrons that escape in the third section of the experiment as a larger number of escaping electrons indicates a larger amount of heating. We illustrate the use of this technique by measuring electron heating caused by high-frequency radiofrequency (RF) fields. In addition to the main technique, several subtechniques to calibrate the electron temperature, electron density, amount of heating and applied RF field amplitude were developed as well.

Electron-cyclotron resonance heating (ECRH) is the main heating mechanism in the Wendelstein 7-X (W7-X) stellarator. Although second-harmonic ECRH (X2) has been used routinely for plasma startup, startup at third harmonic (X3) is known to be much more difficult. In this work, we investigate the energy gain of particles during nonlinear wave–particle interaction for conditions relevant to second- and third-harmonic startups in W7-X. We take into account both the beam and the ambient magnetic field inhomogeneities. The latter is shown to significantly increase the mean energy gain resulting from a single wave–particle resonant interaction. In W7-X-like conditions, the improvement in maximum gained energy is up to 4 times the analogous uniform magnetic field case. However, this improvement is not enough to ensure X3 startup. The optimal magnetic field inhomogeneity length scale for average energy gain and start up in W7-X-like conditions is found to be in the range of $1$ to $3\ {\rm km}^{-1}$. A possibility of using multiple beams with neighbouring resonances is also considered. A considerable enhancement of the energy gain is demonstrated.

This study presents novel findings on stochastic electron heating via a random electron cyclotron wave (ECW) in a spherical tokamak. Hard x ray measurements demonstrate the time evolution of hard x ray counts at different energy bands, consistent with predictions from the stochastic heating model. The ECW heating rate shows a positive correlation with applied power, confirming the effectiveness of stochastic heating. Remarkably, the ECW-driven plasma current remains insensitive to ECW incidence angle, consistent with model predictions. The observed stochastic heating of electrons offers potential for exploring innovative non-inductive current drive modes in spherical tokamaks. This research contributes to the understanding of plasma behaviour and motivates the development of new models for non-inductive current drive in fusion devices.

The magnetohydrodynamic stability of plasmas with an anisotropic pressure component is analysed for a low magnetic field configuration of the large helical device. Magnetic equilibria are calculated by the anisotropic Neumann inverse moments equilibrium code, an extension of the three-dimensional variational moments equilibrium code. A modified version of the bi-Maxwellian is used to model the anisotropic particle velocity distribution. Magnetohydrodynamic stability calculations for the $n=1$ mode family are carried out by TERPSICHORE, which has been expanded by the Kruskal–Oberman energy principle. For on-axis particle deposition, the growth rate and plasma displacement show that the parallel dominant plasmas are significantly more stable than isotropic or perpendicular dominant plasmas. For off-axis particle deposition, the growth rate and the Mercier criterion in the peripheral region $\rho =0.9$, show that low field (LF) deposition perpendicular dominant plasmas are most unstable. For the most realistic off-axis deposition profile, it is found that parallel dominant plasmas are most stable for LF deposition, while perpendicular dominant plasmas are most stable for high field deposition. We conclude that, under low magnetic field conditions in the large helical device, tangential neutral beam injection heating has a stabilising influence on the plasma.

In moderately coupled plasmas, a significant fraction of the internal energy resides in electric fields. As these plasmas are heated or compressed, the shifting partition of energy between particles and fields leads to surprising effects, particularly when ions and electrons have different temperatures. In this work, quasi-equations of state (quasi-EOS) are derived for two-temperature moderately coupled plasma in a thermodynamic framework and expressed in a simple form. These quasi-EOS readily yield expressions for correlation heating, in which heating of the electrons causes a rapid increase in ion temperature even in the absence of collisional energy exchange between species. It is also shown that, remarkably, compression of moderately coupled plasma drives a temperature difference between electrons and ions, even when the species start at equal temperatures. These additional channels for ion heating may be relevant in designing ignition schemes for inertial confinement fusion.

The structure of a collisionless shock affects ion motion in the shock front and is affected by the formed ion distribution. In high-Mach-number shocks, a significant fraction of incident ions are reflected by the macroscopic electric and magnetic fields in the shock front. Ions are non-specularly reflected by the combined electric deceleration and magnetic deflection. Here, a first analytical description of the non-specular reflection is presented. The contribution of the increasing magnetic field is evaluated and shown to enhance reflection. The distribution of non-specularly reflected ions ahead of the ramp is calculated and their velocities at the re-entry to the shock are found numerically. Dependence on the angle between the shock normal and the upstream magnetic field vector is illustrated.

Good fast ion confinement in quasi-isodynamic stellarators still has to be demonstrated experimentally. In the absence of fusion alphas, auxiliary heating systems can be used to generate fast ions. For this purpose, W7-X has neutral beam injection and its newly installed ion cyclotron heating system. In practice, both systems can run simultaneously, the synergetic running of both is the focus of the current modelling development. With the upgrades to the SCENIC code described here, we find that, compared with pure neutral beam injection (NBI), the fast ion tail is significantly enhanced when radio frequency (RF) heating is enabled. But the effectiveness of heating and fast ion tail generation strongly depends on the plasma parameters and antenna frequency, which calls for further study, both numerical and experimental.

We study the parametric decay instability of parallel-propagating Alfvén waves in a low-beta plasma using one-dimensional fully kinetic simulations. We focus for the first time on the conversion of the energy stored in the initial Alfvén wave into particle internal energy, and on its partition between particle species. We show that compressible fluctuations generated by the decay of the pump wave into a secondary ion-acoustic mode and a reflected Alfvén wave contribute to the gain of internal energy via two distinct mechanisms. First, the ion-acoustic mode leads nonlinearly to proton trapping and proton phase-space mixing, in agreement with previous work based on hybrid simulations. Second, during the nonlinear stage, a compressible front of the fast type develops at the steepened edge of the backward Alfvén wave leading to a field-aligned proton beam propagating backwards at the Alfvén speed. We find that parametric decay heats preferentially protons, which gain approximately 50 % of the pump wave energy in the form of internal energy. However, we find that electrons are also energized and that they contribute to the total energy balance by gaining 10 % of the pump wave energy. By investigating energy partition and particle heating during parametric decay, our results contribute to the determination of the role of compressible and kinetic effects in wave-driven models of the solar wind.

A collisionless shock is a self-organized structure where fields and particle distributions are mutually adjusted to ensure a stable mass, momentum and energy transfer from the upstream to the downstream region. This adjustment may involve rippling, reformation or whatever else is needed to maintain the shock. The fields inside the shock front are produced due to the motion of charged particles, which is in turn governed by the fields. The overshoot arises due to the deceleration of the ion flow by the increasing magnetic field, so that the drop of the dynamic pressure should be compensated by the increase of the magnetic pressure. The role of the overshoot is to regulate ion reflection, thus properly adjusting the downstream ion temperature and kinetic pressure and also speeding up the collisionless relaxation and reducing the anisotropy of the eventually gyrotropized distributions.

Three machine learning techniques (multilayer perceptron, random forest and Gaussian process) provide fast surrogate models for lower hybrid current drive (LHCD) simulations. A single GENRAY/CQL3D simulation without radial diffusion of fast electrons requires several minutes of wall-clock time to complete, which is acceptable for many purposes, but too slow for integrated modelling and real-time control applications. The machine learning models use a database of more than 16 000 GENRAY/CQL3D simulations for training, validation and testing. Latin hypercube sampling methods ensure that the database covers the range of nine input parameters ($n_{e0}$, $T_{e0}$, $I_p$, $B_t$, $R_0$, $n_{\|}$, $Z_{{\rm eff}}$, $V_{{\rm loop}}$ and $P_{{\rm LHCD}}$) with sufficient density in all regions of parameter space. The surrogate models reduce the inference time from minutes to $\sim$ms with high accuracy across the input parameter space.

We study the nonlinear interaction of three parallel Alfvén wave packets (AWPs) in an initially uniform plasma using 2.5-dimensional particle-in-cell (PIC) numerical simulations. We aim to help to explain the observation of suprathermal electrons by the collision of multiple Alfvén waves in regions where these waves are trapped like the IAR (Ionospheric Alfvén Resonator), Earth radiation belts or coronal magnetic loops. In the context of the acceleration by the parallel Alfvén waves interactions (APAWI) process that has been described by Mottez (Ann. Geophys., vol. 30, issue 1, 2012, pp. 81–95; J. Plasma Phys., vol. 81, issue 1, 2015, p. 325810104), the interaction of two parallel Alfvén waves (AWs) generates longitudinal density modulations and parallel electric fields at the APAWI crossing region that can accelerate particles effectively in the direction of the background magnetic field. Our simulations show that when a third parallel AWP of different initial position arrives at the APAWI crossing region, it gives rise to a strong parallel electron beam ($V \sim 5\text {--}7 V_{Te}$) at longitudinal cavity density gradients. We suggest that velocity drift from an outgoing AW generates interface waves in the transverse direction, which allows propagating waves to develop parallel electric fields by the phase mixing process when $k_{\perp }^{-1}$ of the wavy density gradient (oblique gradient) is in the range of the electron inertial length $c/\omega _{p0}$.

In the ion cyclotron range of frequency (ICRF), the presence of a lower hybrid (LH) resonance can appear in the edge of a tokamak plasma and lead to deleterious edge power depositions. An analytic formula for these losses is derived in the cold plasma approximation and for a slab geometry using an asymptotic approach and an analytical continuation near the LH resonance. The way to minimize these losses in a large machine like ITER is discussed. An internal verification between the power loss computed with the semi-analytical code ANTITER IV for ion cyclotron resonance heating (ICRH) and the analytic result is performed. This allows us to check the precision of the numerical integration of the singular set of cold plasma wave differential equations. The set of cold plasma equations used is general and can be applied in other parameters domain.

Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad–Shafranov equation; such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad–Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad–Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space $H_0^1$ to the Grad–Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad–Shafranov equation are also constructed.

Radio frequency (RF) sheaths occur under a wide variety of conditions when RF waves, material surfaces and plasma coexist. RF sheaths are of special importance in describing the interaction of ion cyclotron range of frequency (ICRF) waves with the boundary plasma in tokamaks, stellarators and other magnetic confinement devices. In this article the basic physics of RF sheaths is discussed in the context of magnetic fusion research. Techniques for modelling RF sheaths, their interaction with RF wave fields and the resulting consequences are highlighted. The article is intended as a guide for the early-career ICRF researcher, but it may equally well serve to provide an overview of basic RF sheath concepts and modelling directions for any interested fusion scientist.

The effect of different density profiles on micron-sized hydrogen plasma spheres is investigated when the plasma gets irradiated with an ultrashort circularly polarized laser. In this study, we show that significant improvement in the characteristics of the accelerated protons viz. maximum proton energy, as well as their monoenergetic behaviour, is possible by using a plasma sphere having a tailored density profile. A linear-shaped density inhomogeneity is introduced in the plasma sphere such that the density is peaked at the centre and gradually dropping outwards. The density gradient is tuned by changing the peak density at the centre. The optimum regime of steepness is found for the maximum energy attained by the protons where the target is opaque enough for the radiation pressure to play its role, however not too opaque to inhibit efficient target heating. A novel Gaussian-shaped density profile is suggested which plays an important role in suppressing the sheath field. With a decreased rear-side field, a visible improvement of the monoenergetic feature of the protons is observed.

In the scrape-off layer and the edge region of a tokamak, the plasma is strongly turbulent and scatters the radiofrequency (RF) electromagnetic waves that propagate through this region. It is important to know the spectral properties of these scattered RF waves, whether used for diagnostics or for heating and current drive. The spectral changes influence the interpretation of the obtained diagnostic data, and the current and heating profiles. A full-wave, three-dimensional (3-D) electromagnetic code ScaRF (see Papadopoulos et al., J. Plasma Phys., vol. 85, issue 3, 2019, 905850309) has been developed for studying the RF wave propagation through turbulent plasma. ScaRF is a finite-difference frequency-domain (FDFD) method used for solving Maxwell's equations. The magnetized plasma is defined through the cold plasma by the anisotropic permittivity tensor. As a result, ScaRF can be used to study the scattering of any cold plasma RF wave. It can also be used for the study of the scattering of electron cyclotron waves in ITER-type and medium-sized tokamaks such as TCV, ASDEX-U and DIII-D. For the case of medium-sized tokamaks, there is experimental evidence that drift waves and rippling modes are present in the edge region (see Ritz et al., Phys. Fluids, vol. 27, issue 12, 1984, pp. 2956–2959). Hence, we have studied the scattering of RF waves by periodic density interfaces (plasma gratings) in the form of a superposition of spatial modes with varying periodicity and random amplitudes (see Papadopoulos et al., J. Plasma Phys., vol. 85, issue 3, 2019, 905850309). The power reflection coefficient (a random variable) is calculated for different realizations of the density interface. In this work, the uncertainty of the power reflection coefficient is rigorously quantified by use of the Polynomial Chaos Expansion (see Xiu & Karniadakis, SIAM J. Sci. Comput., vol. 24, issue 2, 2002, pp. 619–644) method in conjunction with the Smolyak sparse-grid integration (see Papadopoulos et al., Appl. Opt., vol. 57, issue 12, 2018, pp. 3106–3114), which is known as the PCE-SG method. The PCE-SG method is proven to be accurate and more efficient (roughly a 2-orders of magnitude shorter execution time) compared with alternative methods such as the Monte Carlo (MC) approach.

The velocity dependent resonant interaction of particles with applied radiofrequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto (J. Plasma Phys., vol. 86, 2020, 815860302). The behaviour is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). For a wave of parallel wave number ${k_{||}}$ interacting with weakly collisional plasma species of collision frequency $\nu$ and thermal speed ${v_{\textrm{th}}}$, the effective collision frequency becomes of order $\nu {({k_{||}}{v_{th}}/\nu )^{2/3}} \gg \nu $. The narrow boundary layers that arise because of the diffusive nature of the collisions allow a physically meaningful wave–particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behaviour of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave–particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.