A linearised kinetic equation describing electrostatic perturbations of a Maxwellian equilibrium in a weakly collisional plasma forced by a random source is considered. The problem is treated as a kinetic analogue of the Langevin equation and the corresponding fluctuation-dissipation relations are derived. The kinetic fluctuation-dissipation relation reduces to the standard “fluid” one in the regime where the Landau damping rate is small and the system has no real frequency; in this case the simplest possible Landau-fluid closure of the kinetic equation coincides with the standard Langevin equation. Phase mixing of density fluctuations and emergence of fine scales in velocity space is diagnosed as a constant flux of free energy in Hermite space; the fluctuation-dissipation relations for the perturbations of the distribution function are derived, in the form of a universal expression for the Hermite spectrum of the free energy. Finite-collisionality effects are included. This work is aimed at establishing the simplest fluctuation-dissipation relations for a kinetic plasma, clarifying the connection between Landau and Hermite-space formalisms, and setting a benchmark case for a study of phase mixing in turbulent plasmas.

The high peak brightness of X-ray free-electron lasers (FELs), coupled with X-ray optics enabling the focusing of pulses down to sub-micron spot sizes, provides an attractive route to generating high energy-density systems on femtosecond time scales, via the isochoric heating of solid samples. Once created, the fundamental properties of these plasmas can be studied with unprecedented accuracy and control, providing essential experimental data needed to test and benchmark commonly used theoretical models and assumptions in the study of matter in extreme conditions, as well as to develop new predictive capabilities. Current advances in isochoric heating and spectroscopic plasma studies on X-ray FELs are reviewed and future research directions and opportunities discussed.

Statistical characteristics of plasma fluctuations in the solar wind (SW), the Earth’s magnetosphere and fusion devices are reviewed. The turbulence in all these media has a complicated multiscale structure and exhibits a generalized self-similarity in an extended scale range. The anomalous transport of mass and momentum is intermittent and is carried by sporadic plasma flux bursts with non-Gaussian statistics, long-range correlation and multifractality. Intermittent turbulent transport is characterized by superdiffusion with power law $\langle {\it\delta}x^{2}\rangle \propto {\it\tau}^{{\it\alpha}}$ , ${\it\alpha}\approx 1.2{-}1.8$ . The structure functions in all these plasma environments are well fitted by the log-Poisson model of turbulence. Intermittent plasma turbulence displays universal properties and consists of quasi-1-D singular dissipative structures.

A new turbulent plasma source at the Swarthmore Spheromak Experiment (SSX) facility is described. The MHD wind tunnel configuration employs a magnetized plasma gun to inject high-beta plasma into a large, well-instrumented, vacuum drift region. This provides unique laboratory conditions approaching that in the solar wind: there is no applied background magnetic field in the drift region and has no net axial magnetic flux; the plasma flow speed is on the order of the local sound speed (M ~ 1), so flow energy density is comparable to thermal energy density; and the ratio of thermal to magnetic pressure is of order unity (plasma β ~ 1) so thermal energy density is also comparable to magnetic energy density. Results presented here and referenced within demonstrate the new capabilities and show how the new platform is proving useful for fundamental plasma turbulence studies.

In this paper we review a recently developed approximate method for investigation of dynamics of compressible ellipsoidal figures. Collapse and subsequent behaviour are described by a system of ordinary differential equations for time evolution of semi-axes of a uniformly rotating, three-axis, uniform-density ellipsoid. First, we apply this approach to investigate dynamic stability of non-spherical bodies. We solve the equations that describe, in a simplified way, the Newtonian dynamics of a self-gravitating non-rotating spheroidal body. We find that, after loss of stability, a contraction to a singularity occurs only in a pure spherical collapse, and deviations from spherical symmetry prevent the contraction to the singularity through a stabilizing action of nonlinear non-spherical oscillations. The development of instability leads to the formation of a regularly or chaotically oscillating body, in which dynamical motion prevents the formation of the singularity. We find regions of chaotic and regular pulsations by constructing a Poincaré diagram. A real collapse occurs after damping of the oscillations because of energy losses, shock wave formation or viscosity. We use our approach to investigate approximately the first stages of collapse during the large scale structure formation. The theory of this process started from ideas of Ya. B. Zeldovich, concerning the formation of strongly non-spherical structures during nonlinear stages of the development of gravitational instability, known as ‘Zeldovich’s pancakes’. In this paper the collapse of non-collisional dark matter and the formation of pancake structures are investigated approximately. Violent relaxation, mass and angular momentum losses are taken into account phenomenologically. We estimate an emission of very long gravitational waves during the collapse, and discuss the possibility of gravitational lensing and polarization of the cosmic microwave background by these waves.

In previous galactic dynamo theories of the origin of the magnetic field in our galaxy, the subject of flux-freezing has been omitted. As a consequence, the equation of mass flow has generally also been omitted, particularly in the halo where the galactic gravitational field will operate on the mass flow. In this paper, it has been shown that this neglect could have serious consequences for the results obtained from those galactic dynamo simulations that include the halo. A modification of these dynamo theories is proposed which involves the expulsion of very small pieces of the magnetic field lines, rather than the wholesale expulsion of the complete magnetic lines encapsulated in the previous theories. This expulsion is accomplished by a spike instability that arises from superbubbles when they break out of the galactic disc and their shells fragment. This leads to a cut in the lines of force that still remain in the disc. Subsequently, normal disc turbulence rotates the cut lines and thus dissipates their mean flux, removing them from a role in the dynamo theory. This new process takes a length of time comparable to, but slightly longer than, the previous growth time of the disc dynamo, but avoids the previous difficulties associated with flux freezing and flux expulsion.

We carried out a 3D fully kinetic simulation of Earth's magnetotail magnetic reconnection to study the dynamics of energetic particles. We developed and implemented a new relativistic particle mover in iPIC3D, an implicit Particle-in-Cell code, to correctly model the dynamics of energetic particles. Before the onset of magnetic reconnection, energetic electrons are found localized close to current sheet and accelerated by lower hybrid drift instability. During magnetic reconnection, energetic particles are found in the reconnection region along the x-line and in the separatrices region. The energetic electrons are first present in localized stripes of the separatrices and finally cover all the separatrix surfaces. Along the separatrices, regions with strong electron deceleration are found. In the reconnection region, two categories of electron trajectory are identified. First, part of the electrons are trapped in the reconnection region, bouncing a few times between the outflow jets. Second, part of the electrons pass the reconnection region without being trapped. Different from electrons, energetic ions are localized on the reconnection fronts of the outflow jets.

Ripples in magnetic or electrostatic confinement fields give rise to trapping separatrices, and conventional neoclassical transport theory describes the collisional trapping/detrapping of particles with fractured distribution function. Our experiments and novel theory have now characterized a new kind of neoclassical transport processes arising from chaotic (nominally collisionless) separatrix crossings, which occur due to E × B plasma rotation along θ−ruffled or wave-perturbed separatrices. This chaotic neoclassical transport becomes dominant at low collisionality when the collisional spreading of particle energy during the dynamical period is less than the separatrix energy ruffle.

A gas-filled cylindrical liner z-pinch configuration has been used to drive convergent radiative shock waves into different gases at velocities of 20–50 km s−1. On application of the 1.4 MA, 240 ns rise-time current pulse produced by the Magpie generator at Imperial College London, a series of cylindrically convergent shock waves are sequentially launched into the gas-fill from the inner wall of the liner. This occurs without any bulk motion of the liner wall itself. The timing and trajectories of the shocks are used as a diagnostic tool for understanding the response of the liner z-pinch wall to a large pulsed current. This analysis provides useful data on the liner resistivity, and a means to test equation of state (EOS) and material strength models within MHD simulation codes. In addition to providing information on liner response, the convergent shocks are interesting to study in their own right. The shocks are strong enough for radiation transport to influence the shock wave structure. In particular, we see evidence for both radiative preheating of material ahead of the shockwaves and radiative cooling instabilities in the shocked gas. Some preliminary results from initial gas-filled liner experiments with an applied axial magnetic field are also discussed.

In this paper, we investigate the evolution of braided solar coronal loops. We assume that coronal loops consist of several internal strands which twist and braid about each other. Reconnection between the strands leads to small flares and heating of the loop to x-ray temperatures. Using a method of generating and releasing braid structure similar to a forest fire model, we show that the reconnected field lines evolve to a self-organised critical state. In this state, the frequency distributions of coherent braid sequences as well as flare energies follow power law distributions. We demonstrate how the presence of net helicity in the loop alters the distribution laws.

Fluctuation dynamos are generic to turbulent astrophysical systems. The only analytical model of the fluctuation dynamo, due to Kazantsev, assumes the velocity to be delta-correlated in time. This assumption breaks down for any realistic turbulent flow. We generalize the analytic model of fluctuation dynamos to include the effects of a finite correlation time, ${\it\tau}$ , using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_{L}$ leads to the standard Kazantsev equation in the ${\it\tau}\rightarrow 0$ limit, and extends it to the next order in ${\it\tau}$ . We find that this evolution equation also involves third and fourth spatial derivatives of $M_{L}$ , indicating that the evolution for finite- ${\it\tau}$ will be non-local in general. In the perturbative case of small- ${\it\tau}$ (or small Strouhal number), it can be recast using the Landau–Lifschitz approach, to one with at most second derivatives of $M_{L}$ . Using both a scaling solution and the WKBJ approximation, we show that the dynamo growth rate is reduced when the correlation time is finite. Interestingly, to leading order in ${\it\tau}$ , we show that the magnetic power spectrum preserves the Kazantsev form, $M(k)\propto k^{3/2}$ , in the large- $k$ limit, independent of ${\it\tau}$ .

Immersed boundary methods for computing confined fluid and plasma flows in complex geometries are reviewed. The mathematical principle of the volume penalization technique is described and simple examples for imposing Dirichlet and Neumann boundary conditions in one dimension are given. Applications for fluid and plasma turbulence in two and three space dimensions illustrate the applicability and the efficiency of the method in computing flows in complex geometries, for example in toroidal geometries with asymmetric poloidal cross-sections.

It is often asserted or implicitly assumed, without justification, that the results of two-dimensional investigations of plasma turbulence are applicable to the three-dimensional plasma environments of interest. A projection method is applied to derive two scalar equations that govern the nonlinear evolution of the Alfvénic and pseudo-Alfvénic components of ideal incompressible magnetohydrodynamic (MHD) plasma turbulence. The mathematical form of these equations makes clear the inherently three-dimensional nature of plasma turbulence, enabling an analysis of the nonlinear properties of two-dimensional limits often used to study plasma turbulence. In the anisotropic limit, k⊥ ≫ k∥, that naturally arises in magnetized plasma systems, the perpendicular 2D limit retains the dominant nonlinearities that are mediated only by the Alfvénic fluctuations but lacks the wave physics associated with the linear term that is necessary to capture the anisotropic cascade of turbulent energy. In the in-plane 2D limit, the nonlinear energy transfer is controlled instead by the pseudo-Alfvén waves, with the Alfvén waves relegated to a passive role. In the oblique 2D limit, an unavoidable azimuthal dependence connecting the wavevector components will likely cause artificial azimuthal asymmetries in the resulting turbulent dynamics. Therefore, none of these 2D limits is sufficient to capture fully the rich three-dimensional nonlinear dynamics critical to the evolution of plasma turbulence.

This paper explains how to obtain the distribution function of minority ions in tokamak plasmas using the Monte Carlo method. Since the emphasis is on energetic ions, the guiding-center transformation is outlined, including also the transformation of the collision operator. Even within the guiding-center formalism, the fast particle simulations can still be very CPU intensive and, therefore, we introduce the reader also to the world of high-performance computing. The paper is concluded with a few examples where the presented method has been applied.

Electron–positron (e–p) plasmas are widely thought to be emitted, in the form of ultra-relativistic winds or collimated jets, by some of the most energetic or powerful objects in the Universe, such as black-holes, pulsars, and quasars. These phenomena represent an unmatched astrophysical laboratory to test physics at its limit and, given their immense distance from Earth (some even farther than several billion light years), they also provide a unique window on the very early stages of our Universe. However, due to such gigantic distances, their properties are only inferred from the indirect interpretation of their radiative signatures and from matching numerical models: their generation mechanism and dynamics still pose complicated enigmas to the scientific community. Small-scale reproductions in the laboratory would represent a fundamental step towards a deeper understanding of this exotic state of matter. Here we present recent experimental results concerning the laser-driven production of ultra-relativistic e–p beams. In particular, we focus on the possibility of generating beams that present charge neutrality and that allow for collective effects in their dynamics, necessary ingredients for the testing pair-plasma physics in the laboratory. A brief discussion of the analytical and numerical modelling of the dynamics of these plasmas is also presented in order to provide a summary of the novel plasma physics that can be accessed with these objects. Finally, general considerations on the scalability of laboratory plasmas up to astrophysical scenarios are given.

We consider the development of accurate and efficient numerical methods for the solution of the Vlasov–Landau equation describing a collisional plasma. The methods combine a Lagrangian approach for the Vlasov solver with a fast spectral method for the solution of the Landau operator. To this goal, new modified spectral methods for the Landau integral which are capable to capture correctly the Maxwellian steady state are introduced. A particular care is devoted to the construction of Implicit–Explicit and Exponential Runge–Kutta methods that permit to achieve high-order and efficient time integration of the collisional step. Several numerical tests are reported which show the high accuracy of the numerical schemes here presented.

Zeldovich’s stretch–twist–fold (STF) dynamo provided a breakthrough in conceptual understanding of fast dynamos, including the small-scale fluctuation dynamos. We study the evolution and saturation behaviour of two types of generalized Baker’s map dynamos, which have been used to model Zeldovich’s STF dynamo process. Using such maps allows one to analyse dynamos at much higher magnetic Reynolds numbers $\mathit{Re}_{M}$ as compared to direct numerical simulations. In the two-strip map dynamo there is constant constructive folding, while the four-strip map dynamo also allows the possibility of a destructive reversal of the field. Incorporating a diffusive step parametrized by $\mathit{Re}_{M}$ into the map, we find that the magnetic field $B(x)$ is amplified only above a critical $\mathit{Re}_{M}=R_{\mathit{crit}}\sim 4$ for both types of dynamos. The growing $B(x)$ approaches a shape-invariant eigenfunction independent of initial conditions, whose fine structure increases with increasing $\mathit{Re}_{M}$ . Its power spectrum $M(k)$ displays sharp peaks reflecting the fractal nature of $B(x)$ above the diffusive scale. We explore the saturation of these dynamos in three ways: via a renormalized reduced effective $\mathit{Re}_{M}$ (case I) or due to a decrease in the efficiency of the field amplification by stretching, without changing the map (case IIa), or changing the map (case IIb), and a combination of both effects (case III). For case I, we show that $B(x)$ in the saturated state, for both types of maps, approaches the marginal eigenfunction, which is obtained for $\mathit{Re}_{M}=R_{\mathit{crit}}$ independent of the initial $\mathit{Re}_{M}=R_{M0}$ . On the other hand, in case II, for the two-strip map, we show that $B(x)$ saturates, preserving the structure of the kinematic eigenfunction. Thus the energy is transferred to larger scales in case I but remains at the smallest resistive scales in case II, as can be seen from both $B(x)$ and $M(k)$ . For the four-strip map, $B(x)$ oscillates with time, although with a structure similar to the kinematic eigenfunction. Interestingly, the saturated state in case III shows an intermediate behaviour, with $B(x)$ similar to the kinematic eigenfunction at an intermediate $\mathit{Re}_{M}=R_{\mathit{sat}}$ , with $R_{M0}>R_{\mathit{sat}}>R_{\mathit{crit}}$ . The $R_{\mathit{sat}}$ value is determined by the relative importance of the increased diffusion versus the reduced stretching. These saturation properties are akin to the range of possibilities that have been discussed in the context of fluctuation dynamos.

Wavelet analysis and compression tools are reviewed and different applications for the study of MHD and plasma turbulence are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics based on the wavelet coefficients. We then show how to extract coherent structures out of fully developed turbulent flows using wavelet-based denoising. Finally some multiscale numerical simulation schemes using wavelets are described. Several examples for analysing, compressing and computing one-, two- and three-dimensional turbulent MHD or plasma flows are presented.

All the bodies of the solar system are embedded in the supersonic flux of energetic particles emitted by the Sun. Since the advent of the space age, the models to describe the interaction of this plasma flow with the planets, asteroids, comets etc. have drastically progressed. The possibilities of in situ measurements of the particle distributions and electromagnetic fields have enabled the plasma theories to be tested under astrophysical conditions. Energy transfer from the Sun to the outermost regions of the heliosphere as well as the processes leading to the dissipation of this energy are central questions for heliophysicists. Understanding the dynamics of the particles is thus critical. It is a particularly complicated subject since the medium is (almost) non-collisional. Thus, next to the description of the particles, the development of waves must be considered. Indeed, they participate to the exchange of energy between different species that would not interact otherwise. In other words, waves may play the role of collisions. This paper concentrates on Langmuir waves for their strong links with the electron dynamics. The basic processes of growth and saturation of the Langmuir waves are reviewed to stress their diagnostic capabilities. Then, the characteristics of the waves are described in the several heliophysical contexts: the planetary environments (in particular the ionosphere, the magnetotail and the foreshock) and in the interplanetary medium (in quiescent conditions of the solar wind or during transient events). A particular emphasis is given to results obtained in the last 15 years.

We present a comprehensive survey of various computational methods in CEDRES++ (Couplage Equilibre Diffusion Résistive pour l'Etude des Scénarios) for finding equilibria of toroidal plasma. Our focus is on free-boundary plasma equilibria, where either poloidal field coil currents or the temporal evolution of voltages in poloidal field circuit systems are given data. Centered around a piecewise linear finite element representation of the poloidal flux map, our approach allows in large parts the use of established numerical schemes. The coupling of a finite element method and a boundary element method gives consistent numerical solutions for equilibrium problems in unbounded domains. We formulate a new Newton method for the discretized nonlinear problem to tackle the various nonlinearities, including the free plasma boundary. The Newton method guarantees fast convergence and is the main building block for the inverse equilibrium problems that we can handle in CEDRES++ as well. The inverse problems aim at finding either poloidal field coil currents that ensure a desired shape and position of the plasma or at finding the evolution of the voltages in the poloidal field circuit systems that ensure a prescribed evolution of the plasma shape and position. We provide equilibrium simulations for the tokamaks ITER and WEST to illustrate the performance of CEDRES++ and its application areas.