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.

The dissipation of ion-acoustic surface waves propagating in a semi-bounded and collisional plasma which has a boundary with vacuum is theoretically investigated and this result is used for the analysis of edge-relevant plasma simulated by Divertor Plasma Simulator-2 (DiPS-2). The collisional damping of the surface wave is investigated for weakly ionized plasmas by comparing the collisionless Landau damping with the collisional damping as follows: (1) the ratio of ion temperature $({T_i})$ to electron temperature $({T_e})$ should be very small for the weak collisionality $({T_i}/{T_e} \ll 1)$; (2) the effect of collisionless Landau damping is dominant for the small parallel wavenumber, and the decay constant is given as $\gamma \approx{-} \sqrt {\mathrm{\pi }/2} {k_\parallel }{\lambda _{De}}\omega _{pi}^2/{\omega _{pe}}$; and (3) the collisional damping dominates for the large parallel wavenumber, and the decay constant is given as $\gamma \approx{-} {\nu _{in}}/16$, where ${\nu _{in}}$ is the ion–neutral collisional frequency. An experimental simulation of the above theoretical prediction has been done in the argon plasma of DiPS-2, which has the following parameters: plasma density ${n_e} = (\textrm{2--9)} \times \textrm{1}{\textrm{0}^{11}}\;\textrm{c}{\textrm{m}^{ - 3}}$, ${T_e} = 3.7- 3.8\;\textrm{eV}$, ${T_i} = 0.2- 0.3\;\textrm{eV}$ and collision frequency ${\nu _{in}} = 23- 127\;\textrm{kHz}$. Although the wavelength should be specified with the given parameters of DiPS-2, the collisional damping is found to be $\gamma = ( - 0.9\;\textrm{to}\; - 5) \times {10^4}\;\textrm{rad}\;{\textrm{s}^{ - 1}}$ for ${k_\parallel }{\lambda _{De}} = 10$, while the Landau damping is found to be $\gamma = ( - 4\;\textrm{to}\; - 9) \times {10^4}\;\textrm{rad}\;{\textrm{s}^{ - 1}}$ for ${k_\parallel }{\lambda _{De}} = 0.1$.

An equilibrium statistical mechanics theory for the Hasegawa–Mima equations of toroidal plasmas, with canonical constraint on energy and microcanonical constraint on potential enstrophy, is solved exactly as a spherical model. The use of a canonical energy constraint instead of a fixed-energy microcanonical approach is justified by the preference for viewing real plasmas as an open system. A significant consequence of the results obtained from the partition function, free energy and critical temperature, is the condensation into a ground state exhibiting a blob-hole-like structure observed in real plasmas.

A general procedure for understanding plasma behaviour when resonant wave–particle interactions are the sole destabilizing and transport mechanism or only heating and/or current drive source is highlighted without recourse to involved numerical or analytical treatments. These phenomena are characterized by transport that appears to be collisionless even though collisions play a central role in narrow collisional boundary layers. The order of magnitude estimates, which include nonlinear effects, are shown to provide expressions in agreement with the principal results of recent toroidal Alfvén eigenmode (TAE), toroidal magnetic field ripple, and heating and current drive treatments. More importantly, the retention of nonlinearities leads to new estimates of the alpha particle energy diffusivity at saturation for TAE modes, and the ripple threshold at which superbanana plateau evaluations of alpha particle transport are modified by nonlinear radial drift effects. In addition, the estimates indicate when quasilinear descriptions for heating and current drive will begin to fail. The phenomenological procedure demonstrates that in magnetic fusion relevant plasmas, narrow collisional boundary layers must be retained for resonant wave–particle interactions as they enhance the role of collisions, and make stochastic particle motion unlikely to be more important than other nonlinear processes.

The GENE-3D code, the global stellarator version of the established GENE framework, has been extended to an electromagnetic gyrokinetic code. This paper outlines the basic structure of the algorithm, highlighting the treatment of the electromagnetic terms. The numerical implementation is verified against the radially global GENE code in linear and nonlinear tokamak simulations, recovering excellent agreement between both codes. As a first application to stellarator plasmas, linear and nonlinear global simulations with kinetic electrons of ion temperature gradient (ITG) turbulence in Wendelstein 7-X were performed, showing a decrease of ITG activity through the introduction of electromagnetic effects via a finite plasma-$\beta$. The upgrade makes it possible to study a large variety of new physical scenarios, including kinetic electron and electromagnetic effects, reducing the gap between gyrokinetic models and physically realistic systems.

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Phys. Plasmas, vol. 21, 2014, p. 022515) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by Hameiri (J. Math. Phys., vol. 22, 1981, pp. 2080–2088, § III), to obtain a generalized Grad–Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we can obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

The interaction of radio-frequency (RF) waves with edge turbulence modifies the incident wave spectrum, and can significantly affect RF heating and current drive in tokamaks. Previous lower hybrid (LH) scattering models have either used the weak-turbulence approximation, or treated more realistic, filamentary turbulence in the ray tracing limit. In this work, a new model is introduced which retains full-wave effects of RF scattering in filamentary turbulence. First, a Mie-scattering technique models the interaction of an incident wave with a single Gaussian filament. Next, an effective differential scattering width is derived for a statistical ensemble of filaments. Lastly, a Markov chain solves for the transmitted wave spectrum in slab geometry. This model is applied to LH launching for current drive. The resulting wave spectrum is asymmetrically broadened in angular wavenumber space. This asymmetry is not accounted for in previous LH scattering models. The modified wave spectrum is coupled to a ray tracing/Fokker–Planck solver (GENRAY/CQL3D) to study its impact on current drive. The resulting current profile is greatly altered, and there is significant increase in the on-axis current and decrease in the off-axis peaks. This is attributed to a portion of the modified wave spectrum that is strongly dampened on-axis during the first pass.

The tertiary instability is believed to be important for governing magnetised plasma turbulence under conditions of strong zonal flow generation, near marginal stability. In this work, we investigate its role for a collisionless strongly driven fluid model, self-consistently derived as a limit of gyrokinetics. It is found that a region of absolute stability above the linear threshold exists, beyond which significant nonlinear transport rapidly develops. Characteristically, this range exhibits a complex pattern of transient zonal evolution before a stable profile can arise. Nevertheless, the Dimits transition itself is found to coincide with a tertiary instability threshold, so long as linear effects are included. Through a simple and readily extendable procedure, tracing its origin to St-Onge (J. Plasma Phys., vol. 83, issue 05, 2017, 905830504), the stabilising effect of the typical zonal profile can be approximated, and the accompanying reduced mode estimate is found to be in good agreement with nonlinear simulations.

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.

General nonlinear equations describing reversed shear Alfvén eigenmode (RSAE) self-modulation via zero-frequency zonal structure (ZFZS) generation are derived using nonlinear gyrokinetic theory, which are then applied to study the spontaneous ZFZS excitation as well as RSAE nonlinear saturation. It is found that both electrostatic zonal flow and electromagnetic zonal current can be preferentially excited by finite-amplitude RSAE, depending on specific plasma parameters. The modification to local shear Alfvén wave continuum is evaluated using the derived saturation level of zonal current, which is shown to play a comparable role in saturating RSAE with the ZFZS scattering.

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.

In stellarator optimization studies, the boundary of the plasma is usually described by Fourier series that are not unique: several sets of Fourier coefficients describe approximately the same boundary shape. A simple method for eliminating this arbitrariness is proposed and shown to work well in practice.

The derivation and numerical implementation of a linearized version of the gyrokinetic (GK) Coulomb collision operator (Jorge et al., J. Plasma Phys., vol. 85, 2019, 905850604) and of the widely used linearized GK Sugama collision operator (Sugama et al., Phys. Plasmas, vol. 16, 2009, 112503) is reported. An approach based on a Hermite–Laguerre moment expansion of the perturbed gyrocentre distribution function is used, referred to as gyromoment expansion. This approach allows the considering of arbitrary perpendicular wavenumber and expressing the two linearized GK operators as a linear combination of gyromoments where the expansion coefficients are given by closed analytical expressions that depend on the perpendicular wavenumber and on the temperature and mass ratios of the colliding species. The drift-kinetic (DK) limits of the GK linearized Coulomb and Sugama operators are also obtained. Comparisons between the gyromoment approach and the DK Coulomb and GK Sugama operators in the continuum GK code GENE are reported, focusing on the ion-temperature-gradient instability and zonal flow damping, finding an excellent agreement. It is confirmed that stronger collisional damping of the zonal flow residual by the Sugama GK model compared with the GK linearized Coulomb (Pan et al., Phys. Plasmas, vol. 27, 2020, 042307) persists at higher collisionality. Finally, we show that the numerical efficiency of the gyromoment approach increases with collisionality, a desired property for boundary plasma applications.

In order to predict and analyse turbulent transport in tokamaks, it is important to model transport that arises from microinstabilities. For this task, quasilinear codes have been developed that seek to calculate particle, angular momentum and heat fluxes, both quickly and accurately. In this tutorial, we present a derivation of one such code known as QuaLiKiz, a quasilinear gyrokinetic transport code. The goal of this derivation is to provide a self-contained and complete description of the underlying physics and mathematics of QuaLiKiz from first principles. This work serves both as a comprehensive overview of QuaLiKiz specifically as well as an illustration for deriving quasilinear models in general.

We derive the local dispersion relation of energetic-particle-induced geodesic acoustic modes (EGAMs) for both trapped and circulating ion beams with single pitch angle slowing-down and Maxwellian distributions, as well as a bump-on-tail distribution in tokamak plasmas. For slowing-down and Maxwellian particles, the solutions of the local dispersion relation give the spectrum, growth rate and thresholds of excitation as functions of the pitch angle, beam density and frequency of the energetic particles bounce motion. For circulating ions there is only one unstable branch with frequency below the GAM continuum and a threshold of excitation in the pitch angle, for both the slowing-down and single pitch Maxwellian distributions. Trapped particles cause no excitation of a mode for neither slowing-down nor Maxwellian ion beams, but they can excite a mode with a bump-on-tail distribution when the mean velocity of the beam is larger than the threshold and the energetic particle bounce frequency is high enough.

The parallel expansion of a dense, pellet-produced plasmoid is modelled with parameters relevant to pellet fuelling experiments in the Wendelstein7-X stellarator. Good agreement is found between the analytical theory and more detailed modelling. In particular, much of the energy deposited in the pellet by the ambient plasma is transferred to the pellet ions by the ambipolar electric field during the expansion. The validity of the hydrodynamic treatment of the plasmoid and the ambient plasma is discussed.

In the ion cyclotron range of frequencies, electromagnetic surface waves are physically relevant for wave–filament interactions, parasitic edge losses and sheath–plasma waves. They are also important numerically, where non-physical surface waves may occur as side effects of slab-geometry approximations. We give new, completely general, mathematical techniques to construct dispersion relations for electromagnetic surface waves between any two media, isotropic or anisotropic, and first-order corrections for when the material interface is steep but continuous. We discuss numerical issues (localized non-convergence, undesired power generation) that arise in numerical calculations due to the presence of surface waves.

This paper proposes a metric bracket for representing Coulomb collisions in the so-called guiding-centre Vlasov–Maxwell–Landau model. The bracket is manufactured to preserve the same energy and momentum functionals as does the Vlasov–Maxwell part and to simultaneously satisfy a revised version of the H-theorem, where the equilibrium distributions with respect to collisional dynamics are identified as Maxwellians. This is achieved by exploiting the special projective nature of the Landau collision operator and the simple form of the system's momentum functional. A discussion regarding a possible extension of the results to electromagnetic drift-kinetic and gyrokinetic systems is included. We anticipate that energy conservation and entropy dissipation can always be manufactured whereas guaranteeing momentum conservation is a delicate matter yet to be resolved.

The stepped-pressure equilibrium code (SPEC) (Hudson et al., Phys. Plasmas, vol. 19, issue 11, 2012, 112502) is extended to allow the computation of multi-region, relaxed magnetohydrodynamics (MRxMHD) equilibria at prescribed toroidal current profile. Toroidal currents are expressed in the framework of the MRxMHD theory, exhibiting spatial separation between pressure driven and externally driven currents. Additionally, analytical force balance derivatives at constant toroidal current are deployed in order to maintain SPEC's advantageous speed. The newly implemented capability is verified in screw pinch and classical stellarator geometries, and is applied to obtain the equilibrium $\beta$-limit of a classical stellarator without net toroidal currents. This new capability opens the possibility to study the effect of toroidal current on three-dimensional equilibria with the SPEC code.

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.