These notes are based on lectures given by one of us (J.B.T.) at the University of Texas in Austin in 1991. Part I concerns some basic features of plasma confinement by magnetic fields as an introduction to an account of plasma relaxation in Part II. Part III discusses confinement by magnetic mirrors, especially minimum- $B$ systems. It also includes a general discussion of adiabatic invariants and of the principle of maximal ordering in perturbation theory. Part IV is devoted to the analysis of perturbations in toroidal plasmas and the stability of ballooning modes.

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al., Astrophys. J. Suppl. Ser., vol. 182, 2009, pp. 310–377) to the case where the mean distribution function of the plasma is pressure-anisotropic and different ion species are allowed to drift with respect to each other – a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas such as the intracluster medium. Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g. the Alfvén ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. Beyond these order-unity corrections, the main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvénic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvénic cascade is fluid, satisfying RMHD equations (with the Alfvén speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping (and for a bi-Maxwellian plasma splits into three independent collisionless cascades). Secondly, the organising principle of this turbulence is elucidated in the form of a conservation law for the appropriately generalised kinetic free energy. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses, and that these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics.

The Wisconsin Plasma Astrophysics Laboratory (WiPAL) is a flexible user facility designed to study a range of astrophysically relevant plasma processes as well as novel geometries that mimic astrophysical systems. A multi-cusp magnetic bucket constructed from strong samarium cobalt permanent magnets now confines a $10~\text{m}^{3}$ , fully ionized, magnetic-field-free plasma in a spherical geometry. Plasma parameters of $T_{e}\approx 5$ to $20~\text{eV}$ and $n_{e}\approx 10^{11}$ to $5\times 10^{12}~\text{cm}^{-3}$ provide an ideal testbed for a range of astrophysical experiments, including self-exciting dynamos, collisionless magnetic reconnection, jet stability, stellar winds and more. This article describes the capabilities of WiPAL, along with several experiments, in both operating and planning stages, that illustrate the range of possibilities for future users.

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.

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.

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 ${\it\tau}$ .

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.

We derive equations for the mean entropy and the mean internal energy in low-Mach-number temperature stratified turbulence (i.e. for turbulent convection or stably stratified turbulence), and show that turbulent flux of entropy is given by $\boldsymbol{F}_{s}=\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}$ , where $\overline{{\it\rho}}$ is the mean fluid density, $s$ is fluctuation of entropy and overbars denote averaging over an ensemble of turbulent velocity fields, $\boldsymbol{u}$ . We demonstrate that the turbulent flux of entropy is different from the turbulent convective flux, $\boldsymbol{F}_{c}=\overline{T}\,\overline{{\it\rho}}\,\overline{\boldsymbol{u}s}$ , of the fluid internal energy, where $\overline{T}$ is the mean fluid temperature. This turbulent convective flux is well-known in the astrophysical and geophysical literature, and it cannot be used as a turbulent flux in the equation for the mean entropy. This result is exact for low-Mach-number temperature stratified turbulence and is independent of the model used. We also derive equations for the velocity–entropy correlation, $\overline{\boldsymbol{u}s}$ , in the limits of small and large Péclet numbers, using the quasi-linear approach and the spectral ${\it\tau}$ approximation, respectively. This study is important in view of different applications to astrophysical and geophysical temperature stratified turbulence.

Accretion disc theory is less developed than stellar evolution theory although a similarly mature phenomenological picture is ultimately desired. While the interplay of theory and numerical simulations has amplified community awareness of the role of magnetic fields in angular momentum transport, there remains a long term challenge to incorporate the insights gained from simulations into improving practical models for comparison with observations. What has been learned from simulations that can lead to improvements beyond SS73 in practical models? Here, we emphasize the need to incorporate the role of non-local transport more precisely. To show where large-scale transport would fit into the theoretical framework and how it is currently missing, we review why the wonderfully practical approach of Shakura & Sunyaev (Astron. Astrophys., vol. 24, 1973, pp. 337–355, SS73) is necessarily a mean field theory, and one which does not include large-scale transport. Observations of coronae and jets, combined with the interpretation of results from shearing box simulations, of the magnetorotational instability (MRI) suggest that a significant fraction of disc transport is indeed non-local. We show that the Maxwell stresses in saturation are dominated by large-scale contributions and that the physics of MRI transport is not fully captured by a viscosity. We also clarify the standard physical interpretation of the MRI as it applies to shearing boxes. Computational limitations have so far focused most attention toward local simulations, but the next generation of global simulations should help to inform improved mean field theories. Mean field accretion theory and mean field dynamo theory should in fact be unified into a single theory that predicts the time evolution of spectra and luminosity from separate disc, corona and outflow contributions. Finally, we note that any mean field theory, including that of SS73, has a finite predictive precision that needs to be quantified when comparing the predictions to observations.

The particle-in-cell (PIC) algorithm is the most popular method for the discretisation of the general 6D Vlasov–Maxwell problem and it is widely used also for the simulation of the 5D gyrokinetic equations. The method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields. In this review we derive a Monte Carlo PIC finite-element model starting from a gyrokinetic discrete Lagrangian. The variations of the Lagrangian are used to obtain the time-continuous equations of motion for the particles and the finite-element approximation of the field equations. The Noether theorem for the semi-discretised system implies a certain number of conservation properties for the final set of equations. Moreover, the PIC method can be interpreted as a probabilistic Monte Carlo like method, consisting of calculating integrals of the continuous distribution function using a finite set of discrete markers. The nonlinear interactions along with numerical errors introduce random effects after some time. Therefore, the same tools for error analysis and error reduction used in Monte Carlo numerical methods can be applied to PIC simulations.

Firstly, a brief overview will be given on different models that are able to describe the behaviour of wave propagation as a function of specific frequency ranges. Each range corresponds to different heating systems, namely, 20–100 MHz for the ion cyclotron resonant heating, 2–20 GHz for lower-hybrid heating or current drive, and 100–250 GHz for electron cyclotron resonant heating or current drive systems. The specification of every system will be explained in detail, including the typical set of equations and the assumptions needed to describe the properties of these heating or current drive systems, as well as their specific domains of validity. In these descriptions, special attention will be paid to the boundary conditions. A review of specific physical problems associated with the wave heating systems will also be provided. The review will detail the role of simulation in answering questions that arise from experiments on magnetized plasma devices devoted to fusion. A few examples that will be covered are the impact of edge turbulence on wave propagation and its consequences on heating system performance, the effects of fast particles and ponderomotive effects, among others. A study that is more focused on radio-frequency sheath effects will also be discussed. It shows that such simulations require very sophisticated tools to gain a partial understanding of the observations undertaken in dedicated experiments. To conclude this review, an overview will be given about the requirements and progress necessary to obtain relevant predictive simulation tools able to describe the wave heating systems used in fusion devices.

Based on the analysis of data from the numerous dedicated experiments on plasma disruptions in the TEXTOR tokamak the mechanisms of the formation of runaway electron (RE) beams and their losses are proposed. The plasma disruption is caused by a strong stochastic magnetic field formed due to nonlinearly excited low-mode-number magneto-hydro-dynamics (MHD) modes. It is hypothesized that the RE beam is formed in the central plasma region confined by an intact magnetic surface due to the acceleration of electrons by the inductive toroidal electric field. In the case of plasmas with the safety factor $q(0)<1$ the most stable RE beams are formed by the outermost intact magnetic surface located between the magnetic surface $q=1$ and the closest low-order rational surface $q=m/n>1~(q=5/4,q=4/3,\dots )$. The thermal quench (TQ) time caused by the fast electron transport in a stochastic magnetic field is calculated using the collisional transport model. The current quench (CQ) stage is due to the particle transport in a stochastic magnetic field. The RE beam current is modelled as a sum of a toroidally symmetric part and a small-amplitude helical current with a predominant $m/n=1/1$ component. The REs are lost due to two effects: (i) by outward drift of electrons in a toroidal electric field until they touch the wall and (ii) by the formation of a stochastic layer of REs at the beam edge. Such a stochastic layer for high-energy REs is formed in the presence of the $m/n=1/1$ MHD mode. It has a mixed topological structure with a stochastic region open to the wall. The effect of external resonant magnetic perturbations on RE loss is discussed. A possible cause of the sudden MHD signals accompanied by RE bursts is explained by the redistribution of runaway current during the resonant interaction of high-energetic electron orbits with the $m/n=1/1$ MHD mode.

Runaway electrons, which are generated in a plasma where the induced electric field exceeds a certain critical value, can reach very high energies in the MeV range. For such energetic electrons, radiative losses will contribute significantly to the momentum space dynamics. Under certain conditions, due to radiative momentum losses, a non-monotonic feature – a ‘bump’ – can form in the runaway electron tail, creating a potential for bump-on-tail-type instabilities to arise. Here, we study the conditions for the existence of the bump. We derive an analytical threshold condition for bump appearance and give an approximate expression for the minimum energy at which the bump can appear. Numerical calculations are performed to support the analytical derivations.

The transport of energetic electrons is sensitive to magnetic perturbations. By using three-dimensional numerical simulation of test particle drift orbits we show that the transport of untrapped electrons through an open region with magnetic perturbations cannot be described by a diffusive process. Based on our test particle simulations, we propose a model that leads to an exponential loss of particles.

In this paper, we present the guiding-centre transformation of the radiation–reaction force of a classical point charge travelling in a non-uniform magnetic field. The transformation is valid as long as the gyroradius of the charged particles is much smaller than the magnetic field non-uniformity length scale, so that the guiding-centre Lie-transform method is applicable. Elimination of the gyromotion time scale from the radiation–reaction force is obtained with the Poisson-bracket formalism originally introduced by Brizard (Phys. Plasmas, vol. 11, 2004, 4429–4438), where it was used to eliminate the fast gyromotion from the Fokker–Planck collision operator. The formalism presented here is applicable to the motion of charged particles in planetary magnetic fields as well as in magnetic confinement fusion plasmas, where the corresponding so-called synchrotron radiation can be detected. Applications of the guiding-centre radiation–reaction force include tracing of charged particle orbits in complex magnetic fields as well as the kinetic description of plasma when the loss of energy and momentum due to radiation plays an important role, e.g. for runaway-electron dynamics in tokamaks.

High-intensity laser–solid interactions generate relativistic electrons, as well as high-energy (multi-MeV) ions and x-rays. The directionality, spectra and total number of electrons that escape a target-foil is dependent on the absorption, transport and rear-side sheath conditions. Measuring the electrons escaping the target will aid in improving our understanding of these absorption processes and the rear-surface sheath fields that retard the escaping electrons and accelerate ions via the target normal sheath acceleration (TNSA) mechanism. A comprehensive Geant4 study was performed to help analyse measurements made with a wrap-around diagnostic that surrounds the target and uses differential filtering with a FUJI-film image plate detector. The contribution of secondary sources such as x-rays and protons to the measured signal have been taken into account to aid in the retrieval of the electron signal. Angular and spectral data from a high-intensity laser–solid interaction are presented and accompanied by simulations. The total number of emitted electrons has been measured as $2.6\times 10^{13}$ with an estimated total energy of $12\pm 1~\text{J}$ from a $100~{\rm\mu}\text{m}$ Cu target with 140 J of incident laser energy during a $4\times 10^{20}~\text{W}~\text{cm}^{-2}$ interaction.

For ITER-relevant runaway electron studies, such as suppression, mitigation, termination and/or control of a runaway beam, it is important to obtain the runaway electrons after the disruption. In this paper we report on the first discharges achieved with a post-disruptive runaway electron beam, termed a ‘runaway plateau’, in the COMPASS tokamak. The runaway plateau is produced by a massive gas injection of argon. Almost all of the disruptions with runaway electron plateaus occurred during the plasma current ramp-up phase. The Ar injection discharges with and without a runaway plateau were compared for various parameters. Parametrisation of the discharges shows that the COMPASS disruptions fulfil the range of parameters important for runaway plateau occurrence. These parameters include electron density, electric field, disruption speed, effective safety factor, and the maximum current quench electric field. In addition to these typical parameters, the plasma current value just before the massive gas injection proved to be surprisingly important.

Disruptions with runaway electron generation have been deliberately induced by injection of argon using a disruption mitigation valve. A second disruption mitigation valve has been utilised to inject varying amounts of helium after a short time delay. No generation of runaway electrons has been observed when more than a critical amount of helium has been injected no later than 5 ms after the triggering of the first valve. The required amount of helium for suppression of runaway electron generation is up to one order of magnitude lower than the critical density according to Connor & Hastie (1975) and Rosenbluth & Putvinski (1997).

We analyse, both analytically and numerically, a two-dimensional six-field fluid model for collisionless magnetic reconnection, accounting for temperature and heat flux fluctuations along the direction of the magnetic guide field. We show that the model possesses a Hamiltonian structure with a non-canonical Poisson bracket. This bracket is characterized by the presence of six infinite families of Casimirs, associated with Lagrangian invariants. This reveals that the model can be reformulated as a system of advection equations, thus generalizing previous results obtained for Hamiltonian isothermal fluid models for reconnection. Numerical simulations indicate that the presence of heat flux and temperature fluctuations yields slightly larger growth rates and similar saturated island amplitudes, with respect to the isothermal models. For values of the sonic Larmor radius much smaller than the electron skin depth, heat flux fluctuations tend to be suppressed and temperature fluctuations follow density fluctuations. Increasing the sonic Larmor radius results in an increasing fraction of magnetic energy converted into heat flux, at the expense of temperature fluctuations. In particular, heat flux fluctuations tend to become relevant along the magnetic island separatrices. The qualitative structures associated with the electron field variables are also reinterpreted in terms of the rotation of the Lagrangian invariants of the system.

Recent results of three astrophysically relevant experiments at Caltech are summarized. In the first experiment magnetohydrodynamically driven plasma jets simulate astrophysical jets that undergo a kink instability. Lateral acceleration of the kinking jet spawns a Rayleigh–Taylor instability, which in turn spawns a magnetic reconnection. Particle heating and a burst of waves are observed in association with the reconnection. The second experiment uses a slightly different setup to produce an expanding arched plasma loop which is similar to a solar corona loop. It is shown that the plasma in this loop results from jets originating from the electrodes. The possibility of a transition from slow to fast expansion as a result of the expanding loop breaking free of an externally imposed strapping magnetic field is investigated. The third and completely different experiment creates a weakly ionized plasma with liquid nitrogen cooled electrodes. Water vapour injected into this plasma forms water ice grains that in general are ellipsoidal and not spheroidal. The water ice grains can become quite long (up to several hundred microns) and self-organize so that they are evenly spaced and vertically aligned.