Magnetic helicity flux gives information about the topology of a magnetic field passing through a boundary. In solar physics applications, this boundary is the photosphere and magnetic helicity flux has become an important quantity in analysing magnetic fields emerging into the solar atmosphere. In this work we investigate the evolution of magnetic helicity flux in magnetohydrodynamic (MHD) simulations of solar flux emergence. We consider emerging magnetic fields with different topologies and investigate how the magnetic helicity flux patterns correspond to the dynamics of emergence. To investigate how the helicity input is connected to the emergence process, we consider two forms of the helicity flux. The first is the standard form giving topological information weighted by magnetic flux. The second form represents the net winding and can be interpreted as the standard helicity flux less the magnetic flux. Both quantities provide important and distinct information about the structure of the emerging field and these quantities differ significantly for mixed sign helicity fields. A novel aspect of this study is that we account for the varying morphology of the photosphere due to the motion of the dense plasma lifted into the chromosphere. Our results will prove useful for the interpretation of magnetic helicity flux maps in solar observations.

A one-dimensional particle-in-cell/Monte Carlo (PIC/MCC) model of low-pressure capacitive discharges with a large discharge gap is presented in the paper. The results from the model are for the dependence of the plasma parameters on the pressure and on the discharge radius. The study is directed to the heating mechanisms in the discharge. It is shown that the ohmic (Joule) heating in the plasma bulk could act simultaneously with the stochastic heating in the region of the plasma–sheath boundary. In confirmation of the results of the model, experimental results showing qualitatively the same behaviour are presented.

In this paper, the dynamics of electrons emitted by a spherical object when the total charge of the system is constant is studied in detail. In particular, the condition for which the total electron charge presents damped oscillations is deduced rigorously by considering a perturbation with respect to the steady-state solution. The results obtained can be of utility in simulating the expansion of a spherical plasma by separating the ion and electron time scales.

To properly treat the collisional transport of alpha particles due to a weakly rippled tokamak magnetic field the tangential magnetic drift due to its gradient (the $\unicode[STIX]{x1D735}B$ drift) and pitch angle scatter must be retained. Their combination gives rise to a narrow boundary layer in which collisions are able to match the finite trapped response to the ripple to the vanishing passing response of the alphas. Away from this boundary layer collisions are ineffective. There the $\unicode[STIX]{x1D735}B$ drift of the alphas balances the small radial drift of the trapped alphas caused by the ripple. A narrow collisional boundary layer is necessary since this balance does not allow the perturbed trapped alpha distribution function to vanish at the trapped–passing boundary. The solution of this boundary layer problem allows the alpha transport fluxes to be evaluated in a self-consistent manner to obtain meaningful constraints on the ripple allowable in a tokamak fusion reactor. A key result of the analysis is that collisional alpha losses are insensitive to the ripple near the equatorial plane on the outboard side where the ripple is high. As the high field side ripple is normally very small, collisional $\sqrt{\unicode[STIX]{x1D708}}$ ripple transport is unlikely to be a serious issue.

Kinetic treatments of drift tearing modes that match an inner, resonant layer solution to an external magnetohydrodynamic (MHD) solution, characterised by $\unicode[STIX]{x1D6E5}^{\prime }$, can fail to match the ideal MHD boundary condition on the parallel electric field, $E_{\Vert }=0$. In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this, placing the theory on a firm footing. These effects are found to modify the effective critical $\unicode[STIX]{x1D6E5}^{\prime }$ for instability of drift tearing modes, in particular for weak electron temperature gradients. The implications for a realistic hot plasma resonant layer model – involving large ion Larmor radius and semi-collisional electron physics (Connor et al., Plasma Phys. Control. Fusion, vol. 54, 2012, 035003) – are determined.

EDI (enhanced biDimensional pIc) is a two-dimensional (2-D) electrostatic/magnetostatic particle-in-cell (PIC) code designed to optimize plasma based systems. The code is built on an unstructured mesh of triangles, allowing for arbitrary geometries. The PIC core is comprised of a Boris leapfrog scheme that can manage multiple species. Particle tracking locates particles in the mesh, using a fast and simple priority-sorting algorithm. A magnetic field with an arbitrary topology can be imposed to study the magnetized particle dynamics. The electrostatic fields are then computed by solving Poisson’s equation with a a finite element method solver. The latter is an external solver that has been properly modified in order to be integrated into EDI. The major advantage of using an external solver directly incorporated into the EDI structure is its strong flexibility, in fact it is possible to couple together different physical problems (electrostatic, magnetostatic, etc.). EDI is written in C, which allows the rapid development of new modules. A big effort in the development of the code has been made in optimization of the linking efficiency, in order to minimize computational time. Finally, EDI is a multiplatform (Linux, Mac OS X) software.

In magnetized plasma situations where magnetic fields intersect massive conducting boundaries, ‘line-tied’ boundary conditions are often used, analytically and in numerical simulations. For ideal magnetohydrodynamic (MHD) plasmas, these conditions are arrived at given the relatively long time scales for magnetic fields penetrating resistively into good conductors. Under line-tied boundary conditions, numerical simulations often exhibit what could be construed as numerical ‘noise’ emanating from the boundaries. We show here that this ‘noise’ is real. By combining numerical and analytical methods, we highlight the existence of sharp spatial structures near the conductors and confirm the appearance of short wavelength structures riding on long wavelength modes. We conclude that, for numerical fidelity, the short multiscale structures need to be resolved. Generally, the short structure widths scale as the square root of the plasma $\unicode[STIX]{x1D6FD}$.

The shape gradient quantifies the change in some figure of merit resulting from differential perturbations to a shape. Shape gradients can be applied to gradient-based optimization, sensitivity analysis and tolerance calculation. An efficient method for computing the shape gradient for toroidal three-dimensional magnetohydrodynamic (MHD) equilibria is presented. The method is based on the self-adjoint property of the equations for driven perturbations of MHD equilibria and is similar to the Onsager symmetry of transport coefficients. Two versions of the shape gradient are considered. One describes the change in a figure of merit due to an arbitrary displacement of the outer flux surface; the other describes the change in the figure of merit due to the displacement of a coil. The method is implemented for several example figures of merit and compared with direct calculation of the shape gradient. In these examples the adjoint method reduces the number of equilibrium computations by factors of $O(N)$, where $N$ is the number of parameters used to describe the outer flux surface or coil shapes.

The influence of the azimuthal electron drift on anomalous erosion and the sheath profile in a stationary plasma thruster (SPT) is analysed in this article. It is found that the anomalous erosion has a self-organized structure, which is formed by the interaction between the plasma and the ceramic walls. In order to interpret the mechanism of the azimuthal erosion structure, a particle in cell (PIC) model is developed to simulate the azimuthal sheath. The results show that the electron azimuthal Hall drift due to crossed electric and magnetic field plays a key role in the azimuthal erosion evolution process. Electron Hall drift can generate an asymmetric sheath structure and induce azimuthal sheath oscillation. Furthermore, an asymmetric sheath caused by the integrated effect of the azimuthal irregular wall structure and azimuthal Hall drift will result in the azimuthal movement of ions. Based on the sheath simulated results, an erosion model is used to simulate the azimuthal erosion evolution. An asymmetric erosion profile caused by the azimuthal asymmetric ion sputtering is found.

Stellarators are generically small current and low plasma beta devices. Often the construction of vacuum magnetic fields with good magnetic surfaces is the starting point for an equilibrium calculation. Although in cases with some continuous spatial symmetry, flux functions can always be found for vacuum magnetic fields, an analogous function does not, in general, exist in three dimensions. This work examines several simple equilibria and vacuum magnetic field problems with the intent of demonstrating the possibilities and limitations in the construction of such states. Starting with a simple vacuum magnetic field with closed field lines in a topological torus (toroidal shell with a flat metric), we obtain a self-consistent formal perturbation series using the amplitude of the non-symmetric vacuum fields as a small parameter. We show that systems possessing stellarator symmetry allow the construction order by order. We further indicate the significance of stellarator symmetry in the amplitude expansion of the full ideal magnetohydrodynamics (MHD) problem as well. We then investigate the conditions that guarantee neighbouring flux surfaces given the data on one surface, by expanding in the distance from that surface. We show that it is much more difficult to find low shear vacuum fields with surfaces than force-free fields or ideal MHD equilibrium. Finally, we demonstrate the existence of a class of vacuum magnetic fields, analogous to ‘snakes’ observed in tokamaks, which can be expanded to all orders.

The analytical solution in toroidal coordinates of the Grad Shafranov equation has been at the origin of the tokamak breakthrough in the fusion development. Unfortunately, the standard toroidal coordinates have a circular poloidal section, which does not fit the elongated cross-section of the present tokamak experiments. In axisymmetry, the vacuum Grad Shafranov equation coincides with the Laplace equation for the toroidal component of the vector potential. In the present paper the solutions for the Laplace equation and that for the vacuum Grad Shafranov equation are tackled in the elliptical prolate toroidal cap-cyclide coordinates framework. The following report of the geometrical properties and of the metric of these coordinates allows us to work out the analytical solution of both equations in terms of the Wangerin functions.

The dynamics of electron-plasma waves is described at arbitrary collisionality by considering the full Coulomb collision operator. The description is based on a Hermite–Laguerre decomposition of the velocity dependence of the electron distribution function. The damping rate, frequency and eigenmode spectrum of electron-plasma waves are found as functions of the collision frequency and wavelength. A comparison is made between the collisionless Landau damping limit, the Lenard–Bernstein and Dougherty collision operators and the electron–ion collision operator, finding large deviations in the damping rates and eigenmode spectra. A purely damped entropy mode, characteristic of a plasma where pitch-angle scattering effects are dominant with respect to collisionless effects, is shown to emerge numerically, and its dispersion relation is analytically derived. It is shown that such a mode is absent when simplified collision operators are used, and that like-particle collisions strongly influence the damping rate of the entropy mode.

To explain the possible destabilization of a two-dimensional magnetic equilibrium such as the near-Earth magnetotail, we developed a kinetic model describing the resonant interaction of electromagnetic fluctuations and bouncing electrons trapped in the magnetic bottle. A small-$\unicode[STIX]{x1D6FD}$ approximation (i.e., the plasma pressure is lower than the magnetic pressure) is used in agreement with a small field line curvature. The linearized gyro-kinetic Vlasov equation is integrated along the unperturbed particle trajectories, including cyclotron and bounce motions. The dispersion relation for drift-Alfvèn waves is obtained through the plasma quasi-neutrality condition and Ampere’s law for the parallel current. It has been found that for a quasi-dipolar configuration ($\text{L}\sim 8$ corresponds to the set of the Earth’s magnetic field lines, crossing the Earths magnetic equator at 8 Earth radii), unstable electromagnetic modes may develop in the current sheet with a growth rate of the order of a few tenths of a second provided that the typical scale of density gradient slope responsible for the diamagnetic drift effects is over one Earth radius or less. This instability growth rate is large enough to destabilize the current sheet on time scales of 2–4 minutes as observed during substorm onset.

Alpha particle confinement is a serious concern in stellarators and provides strong motivation for optimizing magnetic field configurations. In addition to the collisionless confinement of trapped alphas in stellarators, excessive collisional transport of the trapped alpha particles must be avoided while they tangentially drift due to the magnetic gradient (the $\unicode[STIX]{x1D735}B$ drift). The combination of pitch angle scatter off the background ions and the $\unicode[STIX]{x1D735}B$ drift gives rise to two narrow boundary layers in the trapped region. The first is at the trapped–passing boundary and enables the finite trapped response to be matched to the vanishing passing response of the alphas. The second layer is a region that encompasses the somewhat more deeply trapped alphas with vanishing tangential $\unicode[STIX]{x1D735}B$ drift. Away from (and between) these boundary layers, collisions are ineffective and the alpha $\unicode[STIX]{x1D735}B$ drift simply balances the small radial drift of the trapped alphas. As this balance does not vanish as the trapped–passing boundary is approached, the first collisional boundary layer is necessary and gives rise to $\surd \unicode[STIX]{x1D708}$ transport, with $\unicode[STIX]{x1D708}$ the collision frequency. The vanishing of the tangential drift results in a separate, somewhat wider boundary layer, and significantly stronger superbanana plateau transport that is independent of collisionality. The constraint imposed by the need to avoid significant energy depletion loss in the slowing down tail distribution function sets the allowed departure of a stellarator from an optimal quasisymmetric configuration.

The plasma in the scrape-off layer (SOL) has an important effect on the coupling of the waves in the ion cyclotron range of frequencies (ICRF). The high-frequency B-dot (HFB) probes have been installed to investigate the behaviour of ICRF waves on EAST. The fast and slow ICRF wave field amplitudes are measured and a domain parallel wavenumber is deduced. Results of measurements on a test experimental platform and EAST experiments are presented, which include vacuum and plasma situations. In vacuum it is found that field amplitudes increase linearly with ICRF power. Besides, during plasma operation, field amplitudes measured were decreased by an order of magnitude and no linear relation with power was observed. Fast and slow wave power densities fluctuate strongly with plasma density. The experimental results in the laboratory coincide with the simulation results and allow for validating of the measurement method.