This paper investigates the influence of the driving frequency on the dynamics of a single dust particle in argon radiofrequency discharge. A one-dimensional fluid model is presented and solved in the entire inter-electrode domain using the finite difference method. In order to solve the particle equation of motion, the coefficients describing the amplification and the damping of the dust particle oscillations are analytically calculated around the equilibrium position, these coefficients allow us to find the relation between the plasma and dust parameters. The results obtained cover the discharge characteristics, the charge and the dynamics of the dust particle. It has been found that the driving frequency has a significant effect on not only the discharge properties but also on the damped oscillatory motion and the equilibrium position of the dust particle. Hence, these oscillations become closer to the electrodes with increasing driving frequency whereas the dust equilibrium position becomes relatively farther from the powered electrode when the dust size decreases.

Spatial profiles of impurity emission measurements in the extreme ultraviolet (EUV) spectroscopic range in radiofrequency (RF)-heated discharges are combined with one-dimensional and three-dimensional transport simulations to study the effects of resonant magnetic perturbations (RMPs) on core impurity accumulation at EAST. The amount of impurity line emission mitigation by RMPs appears to be correlated with the ion Z for lithium, carbon, iron and tungsten monitored, i.e. stronger suppression of accumulation for heavier ions. The targeted effect on the most detrimental high-Z impurities suggests a possible advantage using RMPs for impurity control. Profiles of transport coefficients are calculated with the STRAHL one-dimensional impurity transport code, keeping $\nu /D$ fixed and using the measured spatial profiles of $\textrm{F}{\textrm{e}^{20 + }}$, $\textrm{F}{\textrm{e}^{21 + }}$ and $\textrm{F}{\textrm{e}^{22 + }}$ to disentangle the transport coefficients. The iron diffusion coefficient ${D_{\textrm{Fe}}}$ increases from $1.0- 2.0\;{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ to $1.5- 3.0\;{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ from the core region to the edge region $(\rho \gt 0.5)$ after the onset of RMPs. Meanwhile, an inward pinch of iron convective velocity ${\nu _{\textrm{Fe}}}$ decreases in magnitude in the inner core region and increases significantly in the outer confined region, simultaneously contributing to preserving centrally peaked $\textrm{Fe}$ profiles and exhausting the impurities. The ${D_{\textrm{Fe}}}$ and ${\nu _{\textrm{Fe}}}$ variations lead to reduced impurity contents in the plasma. The three-dimensional edge impurity transport code EMC3-EIRENE was also applied for a case of RMP-mitigated high-Z accumulation at EAST and compared to that of low-Z carbon. The exhaust of ${\textrm{C}^{6 + }}$ toward the scrape-off layer accompanying an overall suppression of heavier ${\textrm{W}^{30 + }}$ is observed when using RMPs.

Using recently developed adjoint methods for computing the shape derivatives of functions that depend on magnetohydrodynamic (MHD) equilibria (Antonsen et al., J. Plasma Phys., vol. 85, issue 2, 2019; Paul et al., J. Plasma Phys., vol. 86, issue 1, 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well and quasi-symmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space (${\sim }50\text {--}500$) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume.

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.

Rayleigh–Taylor (RT) and buoyancy-driven (BD) instabilities are driven by gravity in a fluid system with inhomogeneous density. The paper investigates these instabilities for a strongly coupled dusty plasma medium. This medium has been represented here in the framework of the generalized hydrodynamics (GHD) fluid model which treats it as a viscoelastic medium. The incompressible limit of the GHD model is considered here. The RT instability is explored both for gradual and sharp density gradients stratified against gravity. The BD instability is discussed by studying the evolution of a rising bubble (a localized low-density region) and a falling droplet (a localized high-density region) in the presence of gravity. Since both the rising bubble and falling droplet have symmetry in spatial distribution, we observe that a falling droplet process is equivalent to a rising bubble. We also find that both the gravity-driven instabilities get suppressed with increasing coupling strength of the medium. These observations have been illustrated analytically as well as by carrying out two-dimensional nonlinear simulations. Part 2 of this paper is planned to extend the present study of the individual evolution of a bubble and a droplet to their combined evolution in order to understand the interaction between them.

Magnetic wave perturbations are observed in the solar wind and in the vicinity of Earth's bow shock. For such environments, recent work on magnetic pumping with electrons trapped in the magnetic perturbations has demonstrated the possibility of efficient energization of superthermal electrons. Here we also analyse the energization of such energetic electrons for which the transit time through the system is short compared with time scales associated with the magnetic field evolution. In particular, considering an idealized magnetic configuration we show how trapping/detrapping of energetic magnetized electrons can cause effective parallel velocity ($v_{\parallel }$-) diffusion. This parallel diffusion, combined with naturally occurring mechanisms known to cause pitch angle scattering, such as whistler waves, produces enhanced heating rates for magnetic pumping. We find that at low pitch angle scattering rates, the combined mechanism enhances the heating beyond the predictions of the recent theory for magnetic pumping with trapped electrons.

The axisymmetric visco-resistive magnetohydrodynamic steady states allowing flows (i.e. non-vanishing velocity fields) are computed for a toroidal JET-like geometry. It is shown that a spatially inhomogeneous heating of moderate magnitude leads to an increase of typical toroidal speeds with respect to the situation with uniform temperature with identical mean Hartmann numbers. A symmetry argument is introduced to capture the symmetry breaking, induced by the temperature gradient, that produces a net toroidal plasma flow.

A growing body of evidence suggests that the solar wind is powered to a large extent by an Alfvén-wave (AW) energy flux. AWs energize the solar wind via two mechanisms: heating and work. We use high-resolution direct numerical simulations of reflection-driven AW turbulence (RDAWT) in a fast-solar-wind stream emanating from a coronal hole to investigate both mechanisms. In particular, we compute the fraction of the AW power at the coronal base ($P_\textrm {AWb}$) that is transferred to solar-wind particles via heating between the coronal base and heliocentric distance $r$, which we denote by $\chi _{H}(r)$, and the fraction that is transferred via work, which we denote by $\chi _{W}(r)$. We find that $\chi _{W}(r_{A})$ ranges from 0.15 to 0.3, where $r_{A}$ is the Alfvén critical point. This value is small compared with one because the Alfvén speed $v_{A}$ exceeds the outflow velocity $U$ at $r < r_{A}$, so the AWs race through the plasma without doing much work. At $r>r_{A}$, where $v_{A} < U$, the AWs are in an approximate sense ‘stuck to the plasma’, which helps them do pressure work as the plasma expands. However, much of the AW power has dissipated by the time the AWs reach $r=r_{A}$, so the total rate at which AWs do work on the plasma at $r>r_{A}$ is a modest fraction of $P_\textrm {AWb}$. We find that heating is more effective than work at $r < r_{A}$, with $\chi _{H}(r_{A})$ ranging from 0.5 to 0.7. The reason that $\chi _{H} \geq 0.5$ in our simulations is that an appreciable fraction of the local AW power dissipates within each Alfvén-speed scale height in RDAWT, and there are a few Alfvén-speed scale heights between the coronal base and $r_{A}$. A given amount of heating produces more magnetic moment in regions of weaker magnetic field. Thus, paradoxically, the average proton magnetic moment increases robustly with increasing $r$ at $r>r_{A}$, even though the total rate at which AW energy is transferred to particles at $r>r_{A}$ is a small fraction of $P_\textrm {AWb}$.

Ideally, binary-collision algorithms conserve kinetic momentum and energy. In practice, the finite size of collision cells and the finite difference in the particle locations affect the conservation properties. In the present work, we investigate numerically how the accuracy of these algorithms is affected when the size of collision cells is large compared with gradient scale length of the background plasma, a parameter essential in full-$f$ fusion plasma simulations. Additionally, we discuss implications for the conserved quantities in drift-kinetic formulations when fluctuating magnetic and electric fields are present: we suggest how the accuracy of the algorithms could potentially be improved with minor modifications.

Using mixed second-order structure functions, a compact exact law is derived for isothermal compressible Hall magnetohydrodynamic turbulence with the assumptions of statistical homogeneity, time stationarity and infinite kinetic/magnetic Reynolds numbers. The resulting law is written as the sum of a Yaglom-like flux term, with an overall expression strongly reminiscent of the incompressible law, and a pure compressible source. Being mainly a function of the increments, the compact law is Galilean invariant but is dependent on the background magnetic field if one is present. Only the magnetohydrodynamic source term requires multi-spacecraft data to be estimated whereas the other components, which include those introduced by the Hall term, can be fully computed with single-spacecraft data using the Taylor hypothesis. These properties make this compact law more appropriate for analysing both numerical simulations and in situ data gathered in space plasmas, in particular when only single-spacecraft data are available.

The interplay between toroidal drift-wave turbulence and tokamak profiles is investigated using a wave-kinetic description. The coupled system is used to investigate the interplay between marginally stable toroidal drift-wave turbulence and geodesic acoustic modes (GAMs). The coupled system is found to be unstable. Notably, the most unstable mode corresponds to the resonance between the turbulent wave radial group velocity and the GAM phase velocity. For a low-field-side ballooned drift-wave growth, a background flow shear breaks the symmetry between inwards- and outwards-travelling instabilities. Although this turbulence–GAM coupling may not be the primary driver for avalanches in standard core ion temperature gradient simulations, this mechanism is generic and displays many of the expected features, and should be of interest in several other regimes, which include towards the edge or in the presence of energetic particles.

Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman, Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlak et al., Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul, Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paul et al., Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson, Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau, Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

One of the limitations in studying dusty plasmas is that many of the important properties of the dust (like the charge) are directly coupled to the surrounding plasma conditions rather than being determined independently. The application of high-intensity ultraviolet (UV) sources to generate discharging photoelectric currents may provide an avenue for developing methods of controlling dust charge. Careful selection of the parameters of the UV source and dust material may even allow for this to be accomplished with minimal perturbation of the background plasma. The Auburn Magnetized Plasma Research Laboratory (MPRL) has developed a ‘proof-of-concept’ experiment for this controlled photo-discharging of dust; a high-intensity, near-UV source was used to produce large changes in the equilibrium positions of lanthanum hexaboride ($\textrm {LaB}_6$) particles suspended in an argon DC glow discharge with negligible changes in the potential, density and temperature profiles of the background plasma. The shifts in equilibrium position of the dust are consistent with a reduction in dust charge. Video analysis is used to quantify the changes in position, velocity and acceleration of a test particle under the influence of the UV and Langmuir probes are used to measure the effects on the plasma.

Using particle-in-cell numerical simulations with electron–positron pair plasma, we study how the efficiencies of magnetic dissipation and particle acceleration scale with the initial coherence length $\lambda _0$ in relation to the system size $L$ of the two-dimensional ‘Arnold–Beltrami–Childress’ (ABC) magnetic field configurations. Topological constraints on the distribution of magnetic helicity in two-dimensional systems, identified earlier in relativistic force-free simulations, that prevent the high-$(L/\lambda _0)$ configurations from reaching the Taylor state, limit the magnetic dissipation efficiency to about $\epsilon _{\textrm {diss}} \simeq 60\,\%$. We find that the peak growth time scale of the electric energy $\tau _{E,{\textrm {peak}}}$ scales with the characteristic value of initial Alfvén velocity $\beta _{A,{\textrm {ini}}}$ like $\tau _{E,\textrm {peak}} \propto (\lambda _0/L)\beta _{A,{\textrm {ini}}}^{-3}$. The particle energy change is decomposed into non-thermal and thermal parts, with non-thermal energy gain dominant only for high initial magnetisation. The most robust description of the non-thermal high-energy part of the particle distribution is that the power-law index is a linear function of the initial magnetic energy fraction.

The prominence of nulls in reconnection theory is due to the expected singular current density and the indeterminacy of field lines at a magnetic null. Electron inertia changes the implications of both features. Magnetic field lines are distinguishable only when their distance of closest approach exceeds a distance $\varDelta _d$. Electron inertia ensures $\varDelta _d\gtrsim c/\omega _{pe}$. The lines that lie within a magnetic flux tube of radius $\varDelta _d$ at the place where the field strength $B$ is strongest are fundamentally indistinguishable. If the tube, somewhere along its length, encloses a point where $B=0$ vanishes, then distinguishable lines come no closer to the null than $\approx (a^2c/\omega _{pe})^{1/3}$, where $a$ is a characteristic spatial scale of the magnetic field. The behaviour of the magnetic field lines in the presence of nulls is studied for a dipole embedded in a spatially constant magnetic field. In addition to the implications of distinguishability, a constraint on the current density at a null is obtained, and the time required for thin current sheets to arise is derived.

Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi-free-boundary optimization, and the free-boundary (coil) optimization. These are described using the MRxMHD energy functional, the Biot–Savart integral, the coil-penalty functional and the virtual casing integral and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma–coil optimization algorithms, we emphasize the importance of the stability matrix.

In an earlier paper we showed that the combination of azimuthal magnetic fields and super-rotation in Taylor–Couette flows of conducting fluids can be unstable against non-axisymmetric perturbations if the magnetic Prandtl number of the fluid is $\textrm {Pm}\neq 1$. Here we demonstrate that the addition of a weak axial field component allows axisymmetric perturbation patterns for $\textrm {Pm}$ of order unity depending on the boundary conditions. The axisymmetric modes only occur for magnetic Mach numbers (of the azimuthal field) of order unity, while higher values are necessary for the non-axisymmetric modes. The typical growth time of the instability and the characteristic time scale of the axial migration of the axisymmetric mode are long compared with the rotation period, but short compared with the magnetic diffusion time. The modes travel in the positive or negative $z$ direction along the rotation axis depending on the sign of $B_\phi B_z$. We also demonstrate that the azimuthal components of flow and field perturbations travel in phase if $|B_\phi |\gg |B_z|$, independent of the form of the rotation law. Within a short-wave approximation for thin gaps it is also shown (in an appendix) that for ideal fluids the considered helical magnetorotational instability only exists for rotation laws with negative shear.

The global total-$f$ gyrokinetic particle-in-cell code XGC, used to study transport in magnetic fusion plasmas or to couple with a core gyrokinetic code while functioning as an edge gyrokinetic code, implements a five-dimensional continuum grid to perform the dissipative operations, such as plasma collisions, or to exchange the particle distribution function information with a core code. To transfer the distribution function between marker particles and a rectangular two-dimensional velocity-space grid, XGC employs a bilinear mapping. The conservation of particle density and momentum is accurate enough in this bilinear operation, but the error in the particle energy conservation can become undesirably large and cause non-negligible numerical heating in a steep edge pedestal. In the present work we update XGC to use a novel mapping technique, based on the calculation of a pseudo-inverse, to exactly preserve moments up to the order of the discretization space. We describe the details of the implementation and we demonstrate the reduced interpolation error for a tokamak test plasma using first- and second-order elements with the pseudo-inverse method and comparing with the bilinear mapping.

A new algorithm for toroidal flow shear in a linearly implicit, local $\delta f$ gyrokinetic code is described. Unlike the current approach followed by a number of codes, it treats flow shear continuously in time. In the linear gyrokinetic equation, time-dependences arising from the presence of flow shear are decomposed in such a way that they can be treated explicitly in time with no stringent constraint on the time step. Flow shear related time dependences in the nonlinear term are taken into account exactly, and time dependences in the quasineutrality equation are interpolated. Test cases validating the continuous-in-time implementation in the code GS2 are presented. Lastly, nonlinear gyrokinetic simulations of a JET discharge illustrate the differences observed in turbulent transport compared with the usual, discrete-in-time approach. The continuous-in-time approach is shown, in some cases, to produce fluxes that converge to a different value than with the discrete approach. The new approach can also lead to substantial computational savings by requiring radially narrower boxes. At fixed box size, the continuous implementation is only modestly slower than the previous, discrete approach.

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.