BOUT++ is a 3D nonlinear finite-difference plasma simulation code, capable of solving quite general systems of Partial Differential Equations (PDEs), but targeted particularly on studies of the edge region of tokamak plasmas. BOUT++ is publicly available, and has been adopted by a growing number of researchers worldwide. Here we present improvements which have been made to the code since its original release, both in terms of structure and its capabilities. Some recent applications of these methods are reviewed, and areas of active development are discussed. We also present algorithms and tools which have been developed to enable creation of inputs from analytic expressions and experimental data, and for processing and visualisation of output results. This includes a new tool Hypnotoad for the creation of meshes from experimental equilibria. Algorithms have been implemented in BOUT++ to solve a range of linear algebraic problems encountered in the simulation of reduced Magnetohydrodynamics (MHD) and gyro-fluid models: A preconditioning scheme is presented which enables the plasma potential to be calculated efficiently using iterative methods supplied by the PETSc library (the Portable, Extensible Toolkit for Scientific Computation) (Balay et al. 2014), without invoking the Boussinesq approximation. Scaling studies are also performed of a linear solver used as part of physics-based preconditioning to accelerate the convergence of implicit time-integration schemes.

In this and the accompanying paper, the problem of the maximally achievable elongation ${\it\kappa}$ in a tokamak is investigated. The work represents an extension of many earlier studies, which were often focused on determining ${\it\kappa}$ limits due to (i) natural elongation in a simple applied pure vertical field or (ii) axisymmetric stability in the presence of a perfectly conducting wall. The extension investigated here includes the effect of the vertical stability feedback system which actually sets the maximum practical elongation limit in a real experiment. A basic resistive wall stability parameter, ${\it\gamma}{\it\tau}_{w}$, is introduced to model the feedback system which although simple in appearance actually captures the essence of the feedback system. Elongation limits in the presence of feedback are then determined by calculating the maximum ${\it\kappa}$ against $n=0$ resistive wall modes for fixed ${\it\gamma}{\it\tau}_{w}$. The results are obtained by means of a general formulation culminating in a variational principle which is particularly amenable to numerical analysis. The principle is valid for arbitrary profiles but simplifies significantly for the Solov’ev profiles, effectively reducing the 2-D stability problem into a 1-D problem. The accompanying paper provides the numerical results and leads to a sharp answer of ‘how much elongation is too much’?

This paper is devoted to study the dynamics of a discharge lamp in different position. As an example of application, we chose the mercury lamp. For this, we realized a three-dimensional model, steady state. After the validation of this model, we used it to reproduce the influence of some parameters that have appeared on major transport phenomena of mass and energy in studying the lamp operating in different positions. Indeed, the pressure and the orientation of the lamp are modified. The effect of convective transport and the accumulation of mercury behind the electrodes are studied.

Clusters of galaxies are embedded in halos of optically thin, gravitationally stratified, weakly magnetized plasma at the system’s virial temperature. Owing to radiative cooling and anisotropic heat conduction, such intracluster medium (ICM) is subject to local instabilities, which are combinations of the thermal, magnetothermal and heat-flux-driven buoyancy instabilities. If the ICM rotates significantly, its stability properties are substantially modified and, in particular, also the magnetorotational instability (MRI) can play an important role. We study simple models of rotating cool-core clusters and we demonstrate that the MRI can be the dominant instability over significant portions of the clusters, with possible implications for the dynamics and evolution of the cool cores. Our results give further motivation for measuring the rotation of the ICM with future X-ray missions such as ASTRO-H and ATHENA.

Superbanana and superbanana plateau transport processes are critical to plasma confinement in tokamaks with broken symmetry. The transport is caused by the superbanana resonance, which occurs at a pitch angle that makes the toroidal drift speed vanish, i.e. the tips of the superbananas. The physics consequences of the resonance on the symmetry breaking induced toroidal momentum damping and on the energetic alpha particle transport have been demonstrated using large aspect ratio expansion. Here, the existing theory for the superbanana and superbanana plateau transport is extended for finite aspect ratio tokamaks with broken symmetry. The effects of finite plasma β, and magnetic field shear are naturally included. Here, β is the ratio of the thermal plasma pressure to the magnetic field pressure. The explicit expressions for the transport fluxes in these regimes in terms of the equilibrium quantities are presented. It is shown that the main effects are to modify the resonance function G(k) and the expression for the pitch angle parameter k in the existing theory.

The Weibel/filamentation instability is known to play a key role in the physics of weakly magnetized collisionless shock waves. From the point of view of high energy astrophysics, this instability also plays a crucial role because its development in the shock precursor populates the downstream with a small-scale magneto-static turbulence which shapes the acceleration and radiative processes of suprathermal particles. The present work discusses the physics of the dissipation of this Weibel-generated turbulence downstream of relativistic collisionless shock waves. It calculates explicitly the first-order nonlinear terms associated to the diffusive nature of the particle trajectories. These corrections are found to systematically increase the damping rate, assuming that the scattering length remains larger than the coherence length of the magnetic fluctuations. The relevance of such corrections is discussed in a broader astrophysical perspective, in particular regarding the physics of the external relativistic shock wave of a gamma-ray burst.

The analytic theory presented in Paper I is converted into a form convenient for numerical analysis. A fast and accurate code has been written using this numerical formulation. The results are presented by first defining a reference set of physical parameters based on experimental data from high performance discharges. Scaling relations of maximum achievable elongation (${\it\kappa}_{max}$) versus inverse aspect ratio (${\it\varepsilon}$) are obtained numerically for various values of poloidal beta (${\it\beta}_{p}$), wall radius $(b/a)$ and feedback capability parameter (${\it\gamma}{\it\tau}_{w}$) in ranges near the reference values. It is also shown that each value of ${\it\kappa}_{max}$ occurs at a corresponding value of optimized triangularity (${\it\delta}$), whose scaling is also determined as a function of ${\it\varepsilon}$. The results show that the theoretical predictions of ${\it\kappa}_{max}$ are slightly higher than experimental observations for high performance discharges, as measured by high average pressure. The theoretical ${\it\delta}$ values are noticeably lower. We suggest that the explanation is associated with the observation that high performance involves not only $n=0$ MHD stability, but also $n\geqslant 1$ MHD modes described by ${\it\beta}_{N}$ in the Troyon limit and transport as characterized by ${\it\tau}_{E}$. Operation away from the $n=0$ MHD optimum may still lead to higher performance if there are more than compensatory gains in ${\it\beta}_{N}$ and ${\it\tau}_{E}$. Unfortunately, while the empirical scaling of ${\it\beta}_{N}$ and ${\it\tau}_{E}$ with the elongation (${\it\kappa}$) has been determined, the dependence on ${\it\delta}$ has still not been quantified. This information is needed in order to perform more accurate overall optimizations in future experimental designs.

In this paper, we focus on the numerical investigation of the vircator with a controlling emission from a field-emission cathode. The external harmonic signal is added to the accelerating electric field in the beam formation region and effects on the beam emission process leading to the electron emission modulation. As a consequence, the beam is injected into the drift chamber of the vircator being density-modulated. The strong influence of the modulation parameters (modulation depth and frequency) on the characteristics of virtual cathode oscillations has been discovered. We have shown that the tuning of the modulation frequency to the harmonics of the basic frequency of virtual cathode oscillations leads to the considerable power increase of its higher harmonics in the output spectrum.

Modelling the nucleation and growth of nanoparticles in liquid-phase laser ablation is very important to optimize and control the size and the structure of nanoparticles. However, the detailed formation process of nanoparticles after laser ablation is still unclear. In the present study we investigated for the first time the kinetic growth of nanoparticles synthesized by laser ablation in water, emphasizing the leading role of the plasma medium and in particular the electrostatic agglomeration due to the charging of the nanoparticle in the plasma plume. The importance of the confining role of the liquid medium on the plasma plume is revealed, showing how an isothermal expansion is able to produce smaller nanoparticles compared to an adiabatic cooling.

The experiments revealed that ball lightning is a self-confining quasi-neutral in a whole plasma system that rotates around its axis. Ball lightning has a structure of a spherical electric domain, consisting of a kernel with excess negative charge and an external spherical layer with excess positive charge. The excess of charges of one sort and the lack of charges of the other sort in the kernel or in the external spherical layer significantly reduces the possibility of electron capture by means of an electric field, created by the nearest ions and leads to a drastic slowdown of recombination process. Direct proof has been obtained that inside of ball lightning – in an external spherical layer that rotates around the axis – there is a circular current of sub-relativistic particles. This current creates and maintains its own poloidal magnetic field of ball lightning, i.e. it carries out the function of magnetic dynamo. The kernel of ball lightning is situated in a region with minimum values of induction of the magnetic field. The inequality of positive and negative charges in elements of ball lightning also significantly reduces losses of the charged plasma on bremsstrahlung. Ball lightning generation occurs in a plasmic vortex. The ball lightning energy in the region of its generation significantly differs from the ball lightning energy, which is drifting in space. The axial component of kinetic energy of particles slightly exceeds 100 keV and the rotational component of the ions energy is a bit greater than 1 MeV. Ball lightning is ‘embedded’ in atmosphere autonomous accelerator of charged particles of a cyclotron type due to self-generation of strong crossed electric and magnetic fields. A discussion of the conditions of stability and long-term existence of ball lightning is given.

Plasma confinement is based on the use of nested toroidal magnetic surfaces. In axisymmetric configurations the nestedness is provided by the existence of a flux function describing the magnetic surfaces explicitly. In the case of a three-dimensional magnetic field, the nested surfaces represent an exception. More typically, magnetic islands are formed on the resonant surfaces. The islands could degrade the plasma performance. The rigorous condition for the existence of nested surfaces without islands was formulated by Hamada (Nucl. Fusion, vol. 2, 1962, pp. 23–37) but was not implemented directly into numerical codes used, for example, for designing the stellarator configurations. This paper introduces a method of implementation of the Hamada principle in numerical algorithms. The proposed approach allows for simple linearized equilibrium equations (LEE) and potentially very efficient three-dimensional calculations of nested equilibria.

We start from a Mattor–Parker system and its generalization to include diffusion and derive the Random Phase equations. It is shown that the same type of fluid closure holds in the coherent and turbulent regimes. This is due to the fact that the Random Phase levels (1/I1 = 1/I2 + 1/I3), where Ij is the intensity of wave packet ‘j’, are attractors for the wave dynamics both in the coherent and incoherent cases. Focus here is on the wave dynamics with phase velocities varying due to nonlinear frequency shifts. Thus a Maxwellian distribution function is kept in all cases.

The properties of nonlinear electron-acoustic rogue waves have been investigated in an unmagnetized collisionless four-component plasma system consisting of a cold electron fluid, Maxwellian hot electrons, an electron beam and stationary ions. It is found that the basic set of fluid equations is reduced to a nonlinear Schrodinger equation. The dependence of rogue wave profiles and the associated electric field on the carrier wave number, normalized density of hot electron and electron beam, relative cold electron temperature and relative beam temperature are discussed. The results of the present investigation may be applicable in auroral zone plasma.

A theory of electrodeless electric propulsion systems (EEPS) based on the use of the solenoid magnetic field and the rotating electromagnetic field produced by antennas is developed, which includes a study of the plasma acceleration by the Radio Frequency (RF) field and the concomitant thrust. It was assumed that the frequency of the RF field exceeds the lower hybrid frequency but is much less than the electron gyrofrequency. Relations for the thrust are obtained and analyzed. It is shown that thrust gain is significant only when the RF-induced drift velocity well exceeds the fluid velocity of the injected plasma. It is revealed that the curvature of the magnetic field lines and the plasma acceleration in the region outside the solenoid are the factors which can considerably increase the thrust. On the other hand, it is found that the axial inhomogeneity of the plasma and some other factors are unfavorable for the thrust. The obtained results can be used for the optimization of particular experiments aimed to create a new thruster for long-time space missions.

The understanding of plasma disruptions in tokamaks and predictions of their effects require realistic simulations of electric current excitation in three-dimensional vessel structures by the plasma touching the walls. As discovered at JET in 1996 (Litunovski JET Internal Report contract no. JQ5/11961, 1995; Noll et al., Proceedings of the 19th Symposium on Fusion Technology, Lisbon (ed. C. Varandas & F. Serra), vol. 1, 1996, p. 751. Elsevier) the wall-touching kink modes are frequently excited during vertical displacement events and cause large sideways forces on the vacuum vessel which are difficult to withstand in large tokamaks. In disruptions, the sharing of electric current between the plasma and the wall plays an important role in plasma dynamics and determines the amplitude and localization of the sideways force (Riccardo et al., Nucl. Fusion, vol. 40, 2000, p. 1805; Riccardo & Walker, Plasma Phys. Control. Fusion, vol. 42, 2000, p. 29; Zakharov, Phys. Plasmas, vol. 15, 2008, 062507; Riccardo et al., Nucl. Fusion, vol. 49, 2009, 055012; Bachmann et al., Fusion Engng Des., vol. 86, 2011, pp. 1915–1919). This paper describes a flat triangle representation of the electric circuits of a thin conducting wall of arbitrary three-dimensional geometry. Implemented into the shell simulation code (SHL) and the source sink current code (SSC), this model is suitable for modelling the electric currents excited in the wall inductively and through current sharing with the plasma.

Recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams. These diagrams are constructed using four pivotal dimensionless parameters: the Lundquist number $S$ , the magnetic Prandtl number $P_{m}$ , the amplitude of the boundary perturbation $\hat{{\it\Psi}}_{0}$ , and the perturbation wave number $\hat{k}$ . This new representation highlights the parameter regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution. Contrary to previously proposed phase diagrams, the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of $S$ and $P_{m}$ , depending on the parameters that characterize the external drive, which have not been considered until now. These features are crucial to understanding the onset and evolution of magnetic reconnection in diverse physical systems.

Cross sections in the very low energy range are also represented by the modified effective-range theory (MERT) for low-energy electron scattering from the rare gas (argon). Simulations using published (theoretical) phase shifts indicate that extended versions of the standard effective-range theory with four adjustable parameters are required to give an adequate description of the phase shifts for argon. A four-parameter MERT fit gives a good representation of a recent electron–argon (e-Ar) total cross section experiment at energies less than 10.0 eV. Cross section Q(l) (E) for collision in dilute gases is given for any order l. Here Q(l) (E) are presented for l = 1. . .6. We present calculations for the elastic cross sections for electron scattering from argon. The improvement in the agreement between our theoretical calculations and the experimental measurements in the case of argon in scattering calculations are showed. Differential scattering experiments have been performed for the systems e-Ar in the energy range E = 0–10 eV and the angular range θ = 0–20° using a crossed-beam arrangement. Differential and integrated cross sections for the elastic scattering of low- and intermediate-energy (0–50 eV) electrons by argon atoms are calculated. For each impact energy, the phase shifts of the lower partial waves are obtained exactly by numerical integration of the radial equation. Transport coefficients of argon plasma are requested exactly, which is why we calculated the average collision cross sections for s = 1. . .11, l = 1. . .6.

A new equation for the kinetics of nonlinear conversion of Langmuir waves to electromagnetic waves is developed. Based on this, the former vision of Langmuir turbulence energy thermalization via stochastic plasma electron acceleration (Erofeev 2010 J. Fusion Energy29, 337) is adapted to weakly turbulent plasmas in which this three-wave process occurs. Respective analysis of wave energy dissipation is extended to account for previously unrecognized terms in the kinetics of plasma electrons. It is stated that, cumulatively, these terms do not lead to substantial changes in the picture of Langmuir turbulence dissipation in the corresponding nonlinear plasma.

The properties of dust acoustic and drift waves are investigated in a charge varying magnetized dusty plasma. The plasma is composed of non-thermal electrons and ions with dynamic dust particles. The mathematical expression which describes the dust charge fluctuation is obtained using ${\it\kappa}$-distribution for both the electrons and ions. A dispersion relation is derived and analysed numerically by choosing space plasma parameters. It is found that the inclusion of variable dust charge along with the non-thermal effects of electrons and ions significantly affect linear/nonlinear properties of the dust acoustic and dust drift waves. The effects of different physical parameters including spectral index (${\it\kappa}$), dust charge number ($Z_{d}$), electron density ($n_{e}$) and ion temperature ($T_{i}$) on the wave dispersion and instability are presented. It is found that the presence of the non-thermal electron and ion populations reduce the growth rate of the instability which arises due to the dust charging effect. In addition, the nonlinear vortex solutions are also obtained. For illustration, the results are analysed by using the dusty plasma parameters of Saturn’s magnetosphere.

The transverse dynamics of a pure electron plasma confined in a Penning–Malmberg trap is investigated, taking advantage of two-dimensional particle-in-cell numerical simulations. The evolution of the electron plasma turbulence is studied by means of a wavelet-based multiresolution analysis. In particular, a modified recursive denoising algorithm is developed to separate coherent and incoherent (not necessarily homogeneous) components of the flow. A set of simulations have been carried out changing systematically the radii of an initial annular density distribution of the electrons. The results of the multiresolution analysis indicate that the initial density configuration may have a considerable effect on the evolution of turbulence. Even very small initial density fluctuations can lead to quite different final states, especially in the presence of multiple active diocotron modes characterized by similar growth rates.