Extreme-ultraviolet pulses can propagate through ionised solid-density targets, unlike optical pulses and, thus, have the potential to probe the interior of such plasmas on sub-femtosecond timescales. We present a synthetic diagnostic method for solid-density laser-generated plasmas based on the dispersion of an extreme-ultraviolet attosecond probe pulse, in a pump–probe scheme. We demonstrate the theoretical feasibility of this approach through calculating the dispersion of an extreme-ultraviolet probe pulse propagating through a laser-generated plasma. The plasma dynamics is calculated using a particle-in-cell simulation, whereas the dispersion of the probe is calculated with an external pseudo-spectral wave solver, allowing for high accuracy when calculating the dispersion. The application of this method is illustrated on thin-film plastic and aluminium targets irradiated by a high-intensity pump pulse. By comparing the dispersion of the probe pulse at different delays relative to the pump pulse, it is possible to follow the evolution of the plasma as it disintegrates. The high-frequency end of the dispersion provides information on the line-integrated electron density, whereas lower frequencies are more affected by the highest density encountered along the path of the probe. In addition, the presence of thin-film interference could be used to study the evolution of the plasma surface.

First-order-accurate degenerate variational integration (DVI) was introduced in Ellison et al. (Phys. Plasmas, vol. 25, 2018, 052502) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introduce second-order-accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second-order scheme with a preserved two-form by composing a first-order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second-order-accurate DVI schemes. We test these second-order schemes numerically on two systems having non-canonical variables, namely the magnetic field line and guiding centre systems. Variational integration for Hamiltonian systems with non-uniform time steps, in terms of an extended phase space Hamiltonian, is generalized to non-canonical variables. It is shown that preservation of proper degeneracy leads to single-step (one-step) methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second-order-accurate as well as first-order schemes, and can be applied to adapt the time stepping to an error estimate.

In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be cancelled as in the case of the National Compact Stellarator Experiment (NCSX) project (R.L. Orbach, 2008, https://ncsx.pppl.gov/DOE_NCSX_052208.pdf). Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the sample average approximation; and Adam, equipped with a control variate for variance reduction. Numerical simulations performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than $0.1\,\%$ of the cost of previous work, which considered solely local stochastic optimization.

Upper bounds on the growth of free energy in gyrokinetics are derived. These bounds apply to all local gyrokinetic instabilities in the geometry of a flux tube, i.e. a slender volume of plasma aligned with the magnetic field, regardless of the geometry of field, the number of particle species or collisions. The results apply both to linear instabilities and to the nonlinear growth of finite-amplitude fluctuations.

The Weibel instability is investigated theoretically and numerically under three scenarios: counterstreaming electron beams in background plasma, an electron–positron beam and an electron–proton beam in background plasma. These models occur widely in laboratory and astrophysical environments. The Weibel instability growth rates are determined numerically from the corresponding cold-fluid dispersion relations, which are confirmed with two-dimensional particle-in-cell simulations. The maximum growth rates for the counterstreaming beams in background plasma are an order of magnitude smaller than the maximum growth rates for the beams cases in the same range of density ratios and beam energies. The maximum growth rate for the electron–positron beam case is shown to be at most a factor $\sqrt {2}$ greater than the electron–proton beam case with similar dispersion behaviours. A non-monotonic relation is found between the maximum Weibel instability growth rates and the electron–positron beam energy, suggesting that increasing beam energies does not entail an increase in the Weibel instability growth rate.

Magnetohydrodynamic (MHD) simulations of the solar corona have become more popular with the increased availability of computational power. Modern computational plasma codes, relying upon computational fluid dynamics (CFD) methods, allow the coronal features to be resolved using solar surface magnetograms as inputs. These computations are carried out in a full three-dimensional domain and, thus, selection of the correct mesh configuration is essential to save computational resources and enable/speed up convergence. In addition, it has been observed that for MHD simulations close to the hydrostatic equilibrium, spurious numerical artefacts might appear in the solution following the mesh structure, which makes the selection of the grid also a concern for accuracy. The purpose of this paper is to discuss and trade off two main mesh topologies when applied to global solar corona simulations using the unstructured ideal MHD solver from the COOLFluiD platform. The first topology is based on the geodesic polyhedron and the second on $UV$ mapping. Focus is placed on aspects such as mesh adaptability, resolution distribution, resulting spurious numerical fluxes and convergence performance. For this purpose, first a rotating dipole case is investigated, followed by two simulations using real magnetograms from the solar minima (1995) and solar maxima (1999). It is concluded that the most appropriate mesh topology for the simulation depends on several factors, such as the accuracy requirements, the presence of features near the polar regions and/or strong features in the flow field in general. If convergence is of concern and the simulation contains strong dynamics, then grids which are based on the geodesic polyhedron are recommended compared with more conventionally used $UV$-mapped meshes.

We study the effect of negatively charged dust on the magnetic-field-aligned polarisation electrostatic field ($\boldsymbol {E}_{\parallel }$) using Cassini's RPWS/LP in situ measurements during the ‘ring-grazing’ orbits. We derive a general expression for $\boldsymbol {E}_{\parallel }$ and estimate for the first time in situ $\lVert \boldsymbol {E}_{\parallel } \rVert$ (approximately $10^{-5} \, \text {V}\, \text {m}^{-1}$) near the Janus and Epimetheus rings. We further demonstrate that the presence of the negatively charged dust close to the ring plane ($\vert \text {Z} \vert \lesssim 0.11 \, \text {R}_{s}$) amplifies $\lVert \boldsymbol {E}_{\parallel } \rVert$ by at least one order of magnitude and reverses its direction due to the effect of the charged dust gravitational and inertial forces. Such reversal confines the electrons at the magnetic equator within the dusty region, around $0.047 \, \text {R}_{s}$ above the ring plane. Furthermore, we discuss the role of the collision terms, in particular the ion–dust drag force, in amplifying $\boldsymbol {E}_{\parallel }$. These results imply that the charged dust, as small as nanometres in size, can have a significant influence on the plasma transport, in particular ambipolar diffusion along the magnetic field lines, and so their presence must be taken into account when studying such dynamical processes.

The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.

Anisotropic heat conduction in a plasma embedded in a magnetic field with irregular, possibly chaotic, field lines is discussed. If the collisional mean free path exceeds the electron gyroradius, the heat conductivity is much larger along the field lines than across them, and this enhances the transport across a domain where good flux surfaces do not exist. Recognising that anisotropic heat conduction may be cast in a variational form, and by constructing increasingly sophisticated trial functions that are based on invariant and almost-invariant structures under the magnetic field-line flow, bounds are derived on this enhancement and on the temperature variation along the magnetic field. In this way, remarkably accurate approximations for the temperature can be rapidly constructed without solving the diffusion equation, even in the small perpendicular-diffusion limit when the solution for the temperature is dominated by the fractal structure the magnetic field lines.

We present the effective optical potential of the interaction of an electron with an atom in dense semiclassical plasma of noble gases. This potential takes into account the collective screening effect and the quantum mechanical effect of diffraction. The influence of diffraction and screening effects on the characteristics of electron–atom collisions was investigated. Scattering phase shifts decrease with increase of the de Broglie wave. The electron–atom momentum-transfer cross-section at $\lambda _{B}\to 0$ tends to the data obtained earlier with a neglecting of the diffraction effect.

This paper discusses the toroidal localisation of magnetic reconnection during sawtooth crashes. Numerical analysis with realistic heat diffusion coefficients shows that heat distributes itself helically along the torus faster than the temporal resolution of any existing ECE diagnostics. It makes local and global (helically axisymmetric) magnetic reconnection indistinguishable for an observer, while a local crash where the heat stays confined within a finite helical region could be distinguished. Statistical analysis of sawtooth crashes with the ECEI diagnostic is conducted in ASDEX Upgrade. The displacement of the heat within a finite helical region has not been observed. The statistical data supports global magnetic reconnection.

We present a new method for formulating closures that learn from kinetic simulation data. We apply this method to phase mixing in a simple gyrokinetic turbulent system – temperature-gradient-driven turbulence in an unsheared slab. The closure, called the learned multi-mode (LMM) closure, is constructed by, first, extracting an optimal basis from a nonlinear kinetic simulation using singular value decomposition. Subsequent nonlinear fluid simulations are projected onto this basis and the results are used to formulate the closure. We compare the closure with other closures schemes over a broad range of the relevant two-dimensional parameter space (collisionality and gradient drive). We find that the turbulent kinetic system produces phase-mixing rates much lower than the linear expectations, which the LMM closure is capable of capturing. We also compare radial heat fluxes. A Hammett–Perkins closure, generalized to include collisional effects, is quite successful throughout the parameter space, producing ${\sim }14\,\%$ root-mean-square (r.m.s.) error. The LMM closure is also very effective: when trained at three (two) points (in a 35 point parameter grid), the LMM closure produces $8\,\%$ ($12\,\%$) r.m.s. errors. The LMM procedure can be readily generalized to other closure problems.

In fusion devices, the geometry of the confining magnetic field has a significant impact on the instabilities that drive turbulent heat loss. This is especially true of stellarators, where the density-gradient-driven branch of the ‘trapped electron mode’ (TEM) is predicted to be linearly stable if the magnetic field has the maximum-J property, as is very approximately the case in certain magnetic configurations of the Wendelstein 7-X experiment (W7-X). Here we show, using both analytical theory and simulations, that the benefits of the optimisation of W7-X also serve to mitigate ion-temperature-gradient (ITG) modes as long as an electron density gradient is present. We find that the effect indeed carries over to nonlinear numerical simulations, where W7-X has low TEM-driven transport, and reduced ITG turbulence in the presence of a density gradient, giving theoretical support for the existence of enhanced confinement regimes, in the presence of strong density gradients (e.g. hydrogen pellet or neutral beam injection).

Collisionless shocks efficiently convert the energy of the directed ion flow into their thermal energy. Ion distributions change drastically at the magnetized shock crossing. Even in the absence of collisions, ion dynamics within the shock front is non-integrable and gyrophase dependent. The downstream distributions just behind the shock are not gyrotropic but become so quickly due to the kinematic gyrophase mixing even in laminar shocks. During the gyrotropization all information about gyrophases is lost. Here we develop a mapping of upstream and downstream gyrotropic distributions in terms of scattering probabilities at the shock front. An analytical expression for the probability is derived for directly transmitted ions in the narrow shock approximation. The dependence of the probability on the magnetic compression and the cross-shock potential is demonstrated.

Quasisymmetric stellarators are an attractive class of optimised magnetic confinement configurations. The property of quasisymmetry (QS) is in practice limited to be approximate, and thus the construction requires measures that quantify the deviation from the exact property. In this paper we study three measure candidates used in the literature, placing the focus on their origin and a comparison of their forms. The analysis shows clearly the lack of universality in these measures. As these metrics do not directly correspond to any physical property (except when exactly quasisymmetric), optimisation should employ additional physical metrics for guidance. It is suggested that close to QS minima, one should treat QS metrics through inequality constraints so that additional physics metrics dominate optimisation. The impact of different quasisymmetric measures on optimisation is presented through an example, for which the standard metric that weights the asymmetric Fourier modes of the field magnitude appears to perform best.

Given the large anisotropy of transport processes in magnetized plasmas, the magnetic field structure can strongly impact heat diffusion: magnetic surfaces and cantori form barriers to transport while chaotic layers and island structures can degrade confinement. When a small but non-zero amount of perpendicular diffusion is included, the structure of the magnetic field becomes less important, allowing pressure gradients to be supported across chaotic regions and island chains. We introduce a metric for the effective volume over which the local parallel diffusion dominates based on the solution to the anisotropic heat diffusion equation. To validate this metric, we consider model fields with a single island chain and a strongly chaotic layer for which analytic predictions of the relative parallel and perpendicular transport can be made. We also analyse critically chaotic fields produced from different sets of perturbations, highlighting the impact of the mode number spectrum on the heat transport. Our results indicate that this metric coincides with the effective volume of non-integrability in the limit $\kappa _{\perp } \rightarrow 0$, where $\kappa_{\perp}$ is the perpendicular diffusion coefficient. We propose that this metric be used to assess the impact of non-integrability on the heat transport in stellarator equilibria.

Propagation characteristics (propagation regions and cutoffs) of parallel propagating modes (Langmuir, right- and left-handed circularly polarized waves) are studied for relativistic, weakly relativistic and non-relativistic magnetized electron plasma using the kinetic model. The dispersion relation for parallel propagating modes in relativistic electron plasma is investigated by employing the Maxwell–Boltzmann–J üttner distribution function and the final dispersion relation obtained is more general since no approximation is used. As the integrals in the relativistic dispersion relation cannot be done analytically so these integrals have been solved with the numerical quadrature approach. For $\eta \leq 1$ (ratio of rest mass energy to thermal energy), the increase in the effective mass of electrons will result in a change in the mass-dependent quantities (plasma frequency, electron cyclotron frequency, electron sound velocity, etc.) which in turn significantly affect the propagation characteristics of parallel propagating modes. It is observed that the propagation region for these parallel propagating modes decreases and cutoff points are shifted to lower values when we consider a relativistic plasma environment. Moreover, a low-density and high-temperature plasma is more transparent as compared with a high-density and low-temperature plasma for these modes.

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.

Alfvén wave collisions are the primary building blocks of the non-relativistic turbulence that permeates the heliosphere and low- to moderate-energy astrophysical systems. However, many astrophysical systems such as gamma-ray bursts, pulsar and magnetar magnetospheres and active galactic nuclei have relativistic flows or energy densities. To better understand these high-energy systems, we derive reduced relativistic magnetohydrodynamics equations and employ them to examine weak Alfvénic turbulence, dominated by three-wave interactions, in reduced relativistic magnetohydrodynamics, including the force-free, infinitely magnetized limit. We compare both numerical and analytical solutions to demonstrate that many of the findings from non-relativistic weak turbulence are retained in relativistic systems. But, an important distinction in the relativistic limit is the inapplicability of a formally incompressible limit, i.e. there exists finite coupling to the compressible fast mode regardless of the strength of the magnetic field. Since fast modes can propagate across field lines, this mechanism provides a route for energy to escape strongly magnetized systems, e.g. magnetar magnetospheres. However, we find that the fast-Alfvén coupling is diminished in the limit of oblique propagation.