The creation of well-thermalized, hot and dense plasmas is attractive for warm dense matter studies. We investigate collisionally induced energy absorption of an ultraintense and ultrashort laser pulse in a solid copper target using particle-in-cell simulations. We find that, upon irradiation by a $2\times 10^{20}~\text{W}\,\text{cm}^{-2}$ intensity, 60 fs duration, circularly polarized laser pulse, the electrons in the collisional simulation rapidly reach a well-thermalized distribution with ${\sim}3.5~\text{keV}$ temperature, while in the collisionless simulation the absorption is several orders of magnitude weaker. Circular polarization inhibits the generation of suprathermal electrons, while ensuring efficient bulk heating through inverse bremsstrahlung, a mechanism usually overlooked at relativistic laser intensity. An additional simulation, taking account of both collisional and field ionization, yields similar results: the bulk electrons are heated to ${\sim}2.5~\text{keV}$, but with a somewhat lower degree of thermalization than in the pre-set, fixed-ionization case. The collisional absorption mechanism is found to be robust against variations in the laser parameters. At fixed laser pulse energy, increasing the pulse duration rather than the intensity leads to a higher electron temperature.

We study the longitudinal stability of beam–plasma systems in the presence of a density inhomogeneity in the background plasma. Previous works have focused on the non-relativistic regime where hydrodynamical models are used to evolve pre-existing Langmuir waves within inhomogeneous background plasmas. Here, for the first time we study the problem with kinetic equations in a fully relativistic way. We do not assume the existence of Langmuir waves, and we focus on the rate and the mechanism by which waves are excited in such systems from an initial perturbation. We derive the structure of the unstable modes and compute an analytical approximation for their growth rates. Our computation is limited to dilute and cold beams, and shows an excellent agreement with particle-in-cell simulations performed using the SHARP code. We show that, due to such an inhomogeneity, the virulent beam–plasma instabilities in the intergalactic medium are not suppressed but their counterparts in the solar wind can be suppressed as evidenced by propagating type-III solar radio bursts.

Accelerated statistical computation of collisionless fusion alpha particle losses in stellarator configurations is presented based on direct guiding-centre orbit tracing. The approach relies on the combination of recently developed symplectic integrators in canonicalized magnetic flux coordinates and early classification into regular and chaotic orbit types. Only chaotic orbits have to be traced up to the end, as their behaviour is unpredictable. An implementation of this technique is provided in the code SIMPLE (symplectic integration methods for particle loss estimation, Albert et al., 2020b, doi:10.5281/zenodo.3666820). Reliable results were obtained for an ensemble of 1000 orbits in a quasi-isodynamic, a quasi-helical and a quasi-axisymmetric configuration. Overall, a computational speed up of approximately one order of magnitude is achieved compared to direct integration via adaptive Runge–Kutta methods. This reduces run times to the range of typical magnetic equilibrium computations and makes direct alpha particle loss computation adequate for use within a stellarator optimization loop.

We present an energy-conserving discontinuous Galerkin scheme for the full-$f$ electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for $\unicode[STIX]{x2202}A_{\Vert }/\unicode[STIX]{x2202}t$, the inductive component of the parallel electric field, using a generalized Ohm’s law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Ampère cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model of the tokamak scrape-off layer using parameters from the National Spherical Torus Experiment (NSTX). This is the first published nonlinear electromagnetic gyrokinetic simulation on open field lines. Comparisons are made to a corresponding electrostatic simulation.

In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an $\unicode[STIX]{x1D6FC}^{2}$ dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.

The deviation from thermodynamic equilibrium of the ion velocity distribution functions (VDFs), as measured by the Magnetospheric Multiscale (MMS) mission in the Earth’s turbulent magnetosheath, is quantitatively investigated. Making use of the unprecedented high-resolution MMS ion data, and together with Vlasov–Maxwell simulations, this analysis aims at investigating the relationship between deviation from Maxwellian equilibrium and typical plasma parameters. Correlations of the non-Maxwellian features with plasma quantities such as electric fields, ion temperature, current density and ion vorticity are found to be similar in magnetosheath data and numerical experiments, with a poor correlation between distortions of ion VDFs and current density, evidence that questions the occurrence of VDF departure from Maxwellian at the current density peaks. Moreover, strong correlation has been observed with the magnitude of the electric field in the turbulent magnetosheath, while a certain degree of correlation has been found in the numerical simulations and during a magnetopause crossing by MMS. This work could help shed light on the influence of electrostatic waves on the distortion of the ion VDFs in space turbulent plasmas.

Differential forms provide a coordinate-free way to express many quantities and relations in mathematical physics. In particular, they are useful in plasma physics. This tutorial gives a guide so that you can read the plasma physics literature that uses them and apply them yourself.

We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modelling the turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (non-Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the fourth-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the ‘normal’ closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourth-order moment, Landau fluid closures are required.

In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(\unicode[STIX]{x1D701})$, and the associated plasma response function $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})=-Z^{\prime }(\unicode[STIX]{x1D701})/2$. We then consider a one-dimensional (1-D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourth-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1-D closures at higher-order moments than the fourth order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3-D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the fourth-order moment level. In appendix A, we provide comprehensive tables with Padé approximants of $R(\unicode[STIX]{x1D701})$ up to the eighth-pole order, with many given in an analytic form.

The results of a basic electron heat transport experiment using multiple localized heat sources in close proximity and embedded in a large magnetized plasma are presented. The set-up consists of three biased probe-mounted crystal cathodes, arranged in a triangular spatial pattern, that inject low energy electrons along a strong magnetic field into a pre-existing, cold afterglow plasma, forming electron temperature filaments. When the three sources are activated and placed within a few collisionless electron skin depths of each other, a non-azimuthally symmetric wave pattern emerges due to interference of the drift-Alfvén modes that form on each filament’s temperature gradient. Enhanced cross-field transport from chaotic ($\boldsymbol{E}\times \boldsymbol{B}$, where $\boldsymbol{E}$ is the electric field and $\boldsymbol{B}$ the magnetic field) mixing rapidly relaxes the gradients in the inner triangular region of the filaments and leads to growth of a global nonlinear drift-Alfvén mode that is driven by the thermal gradient in the outer region of the triangle. Azimuthal flow shear arising from the emissive cathode sources modifies the linear eigenmode stability and convective pattern. A steady-current model with emissive sheath boundary predicts the plasma potential and shear flow contribution from the sources.

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis (Phys Fluids B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman et al. (J. Plasma Phys., vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform $\unicode[STIX]{x1D704}>0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are ${<}2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

These lecture notes were presented by Allan N. Kaufman in his graduate plasma theory course and a follow-on special topics course (Physics 242A, B, C and Physics 250 at the University of California Berkeley). The notes follow the order of the lectures. The equations and derivations are as Kaufman presented, but the text is a reconstruction of Kaufman’s discussion and commentary. The notes were transcribed by Bruce I. Cohen in 1971 and 1972, and word processed, edited and illustrations added by Cohen in 2017 and 2018. The series of lectures is divided into four major parts: (i) collisionless Vlasov plasmas (linear theory of waves and instabilities with and without an applied magnetic field, Vlasov–Poisson and Vlasov–Maxwell systems, Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) eikonal theory of wave propagation); (ii) nonlinear Vlasov plasmas and miscellaneous topics (the plasma dispersion function, singular solutions of the Vlasov–Poisson system, pulse-response solutions for initial-value problems, Gardner’s stability theorem, gyroresonant effects, nonlinear waves, particle trapping in waves, quasilinear theory, nonlinear three-wave interactions); (iii) plasma collisional and discreteness phenomena (test-particle theory of dynamic friction and wave emission, classical resistivity, extension of test-particle theory to many-particle phenomena and the derivation of the Boltzmann and Lenard–Balescu equations, the Fokker–Planck collision operator, a general scattering theory, nonlinear Landau damping, radiation transport and Dupree’s theory of clumps); (iv) non-uniform plasmas (adiabatic invariance, guiding-centre drifts, hydromagnetic theory, introduction to drift-wave stability theory).

The magnetic presheath is a boundary layer occurring when magnetized plasma is in contact with a wall and the angle $\unicode[STIX]{x1D6FC}$ between the wall and the magnetic field $\boldsymbol{B}$ is oblique. Here, we consider the fusion-relevant case of a shallow-angle, $\unicode[STIX]{x1D6FC}\ll 1$, electron-repelling sheath, with the electron density given by a Boltzmann distribution, valid for $\unicode[STIX]{x1D6FC}/\sqrt{\unicode[STIX]{x1D70F}+1}\gg \sqrt{m_{\text{e}}/m_{\text{i}}}$, where $m_{\text{e}}$ is the electron mass, $m_{\text{i}}$ is the ion mass, $\unicode[STIX]{x1D70F}=T_{\text{i}}/ZT_{\text{e}}$, $T_{\text{e}}$ is the electron temperature, $T_{\text{i}}$ is the ion temperature and $Z$ is the ionic charge state. The thickness of the magnetic presheath is of the order of a few ion sound Larmor radii $\unicode[STIX]{x1D70C}_{\text{s}}=\sqrt{m_{\text{i}}(ZT_{\text{e}}+T_{\text{i}})}/ZeB$, where e is the proton charge and $B=|\boldsymbol{B}|$ is the magnitude of the magnetic field. We study the dependence on $\unicode[STIX]{x1D70F}$ of the electrostatic potential and ion distribution function in the magnetic presheath by using a set of prescribed ion distribution functions at the magnetic presheath entrance, parameterized by $\unicode[STIX]{x1D70F}$. The kinetic model is shown to be asymptotically equivalent to Chodura’s fluid model at small ion temperature, $\unicode[STIX]{x1D70F}\ll 1$, for $|\text{ln}\,\unicode[STIX]{x1D6FC}|>3|\text{ln}\,\unicode[STIX]{x1D70F}|\gg 1$. In this limit, despite the fact that fluid equations give a reasonable approximation to the potential, ion gyro-orbits acquire a spatial extent that occupies a large portion of the magnetic presheath. At large ion temperature, $\unicode[STIX]{x1D70F}\gg 1$, relevant because $T_{\text{i}}$ is measured to be a few times larger than $T_{\text{e}}$ near divertor targets of fusion devices, ions reach the Debye sheath entrance (and subsequently the wall) at a shallow angle whose size is given by $\sqrt{\unicode[STIX]{x1D6FC}}$ or $1/\sqrt{\unicode[STIX]{x1D70F}}$, depending on which is largest.

Stellarator configurations with reactor relevant energetic particle losses are constructed by simultaneously optimizing for quasisymmetry and an analytically derived metric ($\unicode[STIX]{x1D6E4}_{c}$), which attempts to align contours of the second adiabatic invariant, $J_{\Vert }$ with magnetic surfaces. Results show that with this optimization scheme it is possible to generate quasihelically symmetric equilibria on the scale of ARIES-CS which completely eliminate all collisionless alpha particle losses within normalized radius $r/a=0.3$. We show that the best performance is obtained by reducing losses at the trapped–passing boundary. Energetic particle transport can be improved even when neoclassical transport, as calculated using the metric $\unicode[STIX]{x1D716}_{\text{eff}}$, is degraded. Several quasihelically symmetric equilibria with different aspect ratios are presented, all with excellent energetic particle confinement.

Ion motion in a collisionless shock front is affected by macroscopic large-scale weakly varying and microscopic small-scale fast varying magnetic and electric fields. With the increase of the Mach number the role of the microscopic field is expected to become progressively more important. Using a combination of hybrid simulations and test particle analysis, we show that in moderately supercritical shocks macroscopic fields play the main role in ion motion across the shock. Pressure balance across the shock is only weakly broken and non-stationarity is related to the deviations from the total pressure from the constant value.

Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By ‘topological information’, we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in $\mathbb{R}^{3}$ of arbitrary topology. We also discuss how using the classic Biot–Savart operator simplifies the expression for helicity and how domain topology affects the physical interpretation of helicity.

Dipole and stellarator geometries are capable of confining plasmas of arbitrary neutrality, ranging from pure electron plasmas through to quasineutral. The diocotron mode is known to be important in non-neutral plasmas and has been widely studied. However, drift mode dynamics, dominating quasineutral plasmas, has received very little by way of attention in the non-neutral context. Here, we show that non-neutral plasmas can be unstable respect to both density-gradient- and temperature-gradient-driven instabilities. A local shearless slab limit is considered for simplicity. A key feature of non-neutral plasmas is the development of strong electric fields, in this local limit of straight field line geometry, the effect of the corresponding $\boldsymbol{E}\times \boldsymbol{B}$ drift is limited to the Doppler shift of the complex frequency $\unicode[STIX]{x1D714}\rightarrow \unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E}$. However, the breaking of the quasineutrality condition still leads to interesting dynamics in non-neutral plasmas. In this paper we address the behaviour of a number of gyrokinetic modes in electron–ion and electron–positron plasmas with arbitrary degree of neutrality.

These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental contexts disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale magnetohydrodynamic (MHD) dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small- and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

We present three-dimensional direct numerical simulations and an analytic model of reflection-driven magnetohydrodynamic (MHD) turbulence in the solar wind. Our simulations describe transverse, non-compressive MHD fluctuations within a narrow magnetic flux tube that extends from the photosphere, through the chromosphere and corona and out to a heliocentric distance $r$ of 21 solar radii $(R_{\odot })$. We launch outward-propagating ‘$\boldsymbol{z}^{+}$ fluctuations’ into the simulation domain by imposing a randomly evolving photospheric velocity field. As these fluctuations propagate away from the Sun, they undergo partial reflection, producing inward-propagating ‘$\boldsymbol{z}^{-}$ fluctuations’. Counter-propagating fluctuations subsequently interact, causing fluctuation energy to cascade to small scales and dissipate. Our analytic model incorporates dynamic alignment, allows for strongly or weakly turbulent nonlinear interactions and divides the $\boldsymbol{z}^{+}$ fluctuations into two populations with different characteristic radial correlation lengths. The inertial-range power spectra of $\boldsymbol{z}^{+}$ and $\boldsymbol{z}^{-}$ fluctuations in our simulations evolve toward a $k_{\bot }^{-3/2}$ scaling at $r>10R_{\odot }$, where $k_{\bot }$ is the wave-vector component perpendicular to the background magnetic field. In two of our simulations, the $\boldsymbol{z}^{+}$ power spectra are much flatter between the coronal base and $r\simeq 4R_{\odot }$. We argue that these spectral scalings are caused by: (i) high-pass filtering in the upper chromosphere; (ii) the anomalous coherence of inertial-range $\boldsymbol{z}^{-}$ fluctuations in a reference frame propagating outwards with the $\boldsymbol{z}^{+}$ fluctuations; and (iii) the change in the sign of the radial derivative of the Alfvén speed at $r=r_{\text{m}}\simeq 1.7R_{\odot }$, which disrupts this anomalous coherence between $r=r_{\text{m}}$ and $r\simeq 2r_{\text{m}}$. At $r>1.3R_{\odot }$, the turbulent heating rate in our simulations is comparable to the turbulent heating rate in a previously developed solar-wind model that agreed with a number of observational constraints, consistent with the hypothesis that MHD turbulence accounts for much of the heating of the fast solar wind.

Turbulence is commonly observed in nearly collisionless heliospheric plasmas, including the solar wind and corona and the Earth’s magnetosphere. Understanding the collisionless mechanisms responsible for the energy transfer from the turbulent fluctuations to the particles is a frontier in kinetic turbulence research. Collisionless energy transfer from the turbulence to the particles can take place reversibly, resulting in non-thermal energy in the particle velocity distribution functions (VDFs) before eventual collisional thermalization is realized. Exploiting the information contained in the fluctuations in the VDFs is valuable. Here we apply a recently developed method based on VDFs, the field–particle correlation technique, to a $\unicode[STIX]{x1D6FD}=1$, solar-wind-like, low-frequency Alfvénic turbulence simulation with well-resolved phase space to identify the field–particle energy transfer in velocity space. The field–particle correlations reveal that the energy transfer, mediated by the parallel electric field, results in significant structuring of the VDF in the direction parallel to the magnetic field. Fourier modes representing the length scales between the ion and electron gyroradii show that energy transfer is resonant in nature, localized in velocity space to the Landau resonances for each Fourier mode. The energy transfer closely follows the Landau resonant velocities with varying perpendicular wavenumber $k_{\bot }$ and plasma $\unicode[STIX]{x1D6FD}$. This resonant signature, consistent with Landau damping, is observed in all diagnosed Fourier modes that cover the dissipation range of the simulation.