Particle transport, acceleration and energization are phenomena of major importance for both space and laboratory plasmas. Despite years of study, an accurate theoretical description of these effects is still lacking. Validating models with self-consistent, kinetic simulations represents today a new challenge for the description of weakly collisional, turbulent plasmas. We perform simulations of steady state turbulence in the 2.5-dimensional approximation (three-dimensional fields that depend only on two-dimensional spatial directions). The chosen plasma parameters allow to span different systems, going from the solar corona to the solar wind, from the Earth’s magnetosheath to confinement devices. To describe the ion diffusion we adapted the nonlinear guiding centre (NLGC) theory to the two-dimensional case. Finally, we investigated the local influence of coherent structures on particle energization and acceleration: current sheets play an important role if the ions’ Larmor radii are of the order of the current sheet’s size. This resonance-like process leads to the violation of the magnetic moment conservation, eventually enhancing the velocity-space diffusion.

The nonlinear dynamics of energetic-particle (EP) driven geodesic acoustic modes (EGAM) in tokamaks is investigated, and compared with the beam-plasma system (BPS). The EGAM is studied with the global gyrokinetic (GK) particle-in-cell code ORB5, treating the thermal ions and EP (in this case, fast ions) as GK and neglecting the kinetic effects of the electrons. The wave–particle nonlinearity is only considered in the EGAM nonlinear dynamics. The BPS is studied with a one-dimensional code where the thermal plasma is treated as a linear dielectric, and the EP (in this case, fast electrons) with an N-body Hamiltonian formulation. A one-to-one mapping between the EGAM and the BPS is described. The focus is on understanding and predicting the EP redistribution in phase space. We identify here two distinct regimes for the mapping: in the low-drive regime, the BPS mapping with the EGAM is found to be complete, and in the high-drive regime, the EGAM dynamics and the BPS dynamics are found to differ. The transition is described with the presence of a non-negligible frequency chirping, which affects the EGAM but not the BPS, above the identified drive threshold. The difference can be resolved by adding an ad hoc frequency modification to the BPS model. As a main result, the formula for the prediction of the nonlinear width of the velocity redistribution around the resonance velocity is provided. This article is written as the second of a series of articles (the first being Biancalani et al. (J. Plasma Phys., vol. 83 (6), 2017, 725830602)) on the saturation of EGAMs due to wave–particle nonlinearity.

We explore the growth of large-scale magnetic fields in a shear flow, due to helicity fluctuations with a finite correlation time, through a study of the Kraichnan–Moffatt model of zero-mean stochastic fluctuations of the $\unicode[STIX]{x1D6FC}$ parameter of dynamo theory. We derive a linear integro-differential equation for the evolution of the large-scale magnetic field, using the first-order smoothing approximation and the Galilean invariance of the $\unicode[STIX]{x1D6FC}$-statistics. This enables construction of a model that is non-perturbative in the shearing rate $S$ and the $\unicode[STIX]{x1D6FC}$-correlation time $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$. After a brief review of the salient features of the exactly solvable white-noise limit, we consider the case of small but non-zero $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$. When the large-scale magnetic field varies slowly, the evolution is governed by a partial differential equation. We present modal solutions and conditions for the exponential growth rate of the large-scale magnetic field, whose drivers are the Kraichnan diffusivity, Moffatt drift, shear and a non-zero correlation time. Of particular interest is dynamo action when the $\unicode[STIX]{x1D6FC}$-fluctuations are weak; i.e. when the Kraichnan diffusivity is positive. We show that in the absence of Moffatt drift, shear does not give rise to growing solutions. But shear and Moffatt drift acting together can drive large-scale dynamo action with growth rate $\unicode[STIX]{x1D6FE}\propto |S|$.

The effect of electrostatic microturbulence on fast particles rapidly decreases at high energy, but can be significant at moderate energy. Previous studies found that, in addition to changes in the energetic particle density, this results in non-trivial changes to the equilibrium velocity distribution. These effects have implications for plasma heating and the stability of Alfvén eigenmodes, but make multiscale simulations much more difficult without further approximations. Here, several related analytic model distribution functions are derived from first principles. A single dimensionless parameter characterizes the relative strength of turbulence relative to collisions, and this parameter appears as an exponent in the model distribution functions. Even the most simple of these models reproduces key features of the numerical phase-space transport solution and provides a useful a priori heuristic for determining how strong the effect of turbulence is on the redistribution of energetic particles in toroidal plasmas.

Kinetic ballooning modes (KBMs) are investigated by means of linear electromagnetic gyrokinetic (GK) simulations in the stellarator Wendelstein 7-X (W7-X), for high-$\unicode[STIX]{x1D6FD}$ plasmas, where $\unicode[STIX]{x1D6FD}$ is the ratio of thermal to magnetic plasma pressure. The analysis shows suppression of ion-temperature-gradient (ITG) and trapped particle modes (TEM) by finite-$\unicode[STIX]{x1D6FD}$ effects and destabilization of KBMs at high $\unicode[STIX]{x1D6FD}$. The results are compared with a generic tokamak case. We show that, for large pressure gradients, the frequency of KBMs evaluated by the GENE code is in agreement with the analytical prediction of the diamagnetic modification of the ideal magnetohydrodynamic limit in W7-X general geometry. Thresholds for destabilization of the KBM are predicted for different W7-X equilibrium configurations. We discuss the relation of these thresholds to the ideal magnetohydrodynamic (MHD) stability properties of the corresponding equilibria.

High-energy ions, such as fusion alphas and ions from external heating, can be very sensitive to any non-axisymmetric features in the confining magnetic field due to their collisionless nature. Since understanding the confinement properties of these ions is crucial for ITER (the first fusion reactor currently under construction in Cadarache, France) and beyond, it is of ultimate importance that the predictive simulations are accurate and free of numerical distortions. Adding the third dimension comes at substantial computational cost, calling for new kinds of approaches and computational platforms. In this contribution we discuss what new features, even new physics, the non-axisymmetry brings with it and how one could cope with the ever-increasing demands on both memory and CPU resources. In the end, a few simulation examples with a varying level of non-axisymmetry are given.

Radio frequency (RF) waves are routinely used in tokamak fusion plasmas for plasma heating, current control, as well as in diagnostics. These waves are excited by antenna structures placed near the tokamak’s wall and they have to propagate through a turbulent layer known as the scrape-off layer, before reaching the core plasma (which is their target). This layer exhibits coherent density fluctuations in the form of filaments and blobs. The scattering processes of RF plane waves by a single filament is studied with the assumption that the filament has a cylindrical shape and infinite length. Furthermore, besides the major toroidal component of the externally imposed magnetic field, there is also a small poloidal magnetic field component. Considering also that the cylindrical filament’s axis is not necessarily aligned with the toroidal direction, the total magnetic field is in general neither aligned with the axis of the cylinder nor with the toroidal direction. The investigation concerns the case of electron cyclotron (EC) waves (of frequency $f_{0}=170~\text{GHz}$) for tokamak applications. The study covers a variety of density contrasts between the filament and the ambient plasma, different magnetic field inclinations with respect to the cylinder axis (for the same magnitude of magnetic induction $B=4.5T$) and a wide range of filament radii.

Spacecraft measurements of propagating interplanetary shocks are often interpreted using the ideal magnetohydrodynamic (MHD) model of a planar shock wave travelling with constant velocity $\boldsymbol{V}_{\text{sh}}$ through a spatially uniform plasma. In particular, measurements of the plasma variables upstream and downstream have long been used in conjunction with the Rankine–Hugoniot conditions, also known as the MHD jump conditions, to determine shock velocities and other physical parameters of interplanetary shocks. This procedure is justified only if the shock velocity determined by the MHD jump conditions is unique. In this study the important property of uniqueness is demonstrated for non-perpendicular shocks in MHD media characterized by an isotropic pressure tensor. The primary conclusion is that the shock velocity is uniquely determined by the jump conditions regardless of the type of shock (slow, intermediate or fast). Several new formulas for the shock speed are also derived including one that is independent of the shock normal $\hat{\boldsymbol{n}}$. In principle, the solution technique developed here can be applied to estimate $\boldsymbol{V}_{\text{sh}}$ using solar wind data provided the measurements obey the MHD shock model with sufficient accuracy. That is not its intended purpose, however, and such applications are beyond the scope of this work.

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. These are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.

Models for astrophysical plasmas often have magnetic field lines that leave the boundary rather than closing within the computational domain. Thus, the relative magnetic helicity is frequently used in place of the usual magnetic helicity, so as to restore gauge invariance. We show how to decompose the relative helicity into a relative field-line helicity that is an ideal-magnetohydrodynamic invariant for each individual magnetic field line, and vanishes along any field line where the original field matches the reference field. Physically, this relative field-line helicity is a magnetic flux, whose specific definition depends on the gauge of the reference vector potential on the boundary. We propose a particular ‘minimal’ gauge that depends only on the reference field and minimises this boundary contribution, so as to reveal topological information about the original magnetic field. We illustrate the effect of different gauge choices using the Low–Lou and Titov–Démoulin models of solar active regions. Our numerical code to compute appropriate vector potentials and relative field-line helicity in Cartesian domains is open source and freely available.

Turbulent mass and internal-energy transports in strongly compressible magnetohydrodynamic (MHD) turbulence are investigated in the framework of the multiple-scale direct-interaction approximation, an analytical closure scheme for inhomogeneous turbulence at very high Reynolds numbers. Utilising the analytical representations for the turbulent mass and internal-energy fluxes and their transport coefficients, which are expressed in terms of the correlation and response functions, turbulence models for these fluxes are proposed. In addition to the usual gradient-diffusion transports, cross-diffusion transports mediated by the density variance and the transports along the mean magnetic field mediated by the compressional or dilatational turbulent cross-helicity (velocity–magnetic-field correlation coupled with compressive motions) are shown to arise. These compressibility effects are of fundamental importance since they provide deviations from the usual gradient-diffusion transports. Analogies of the dilatational cross-helicity effects to the magnetoacoustic waves are also argued.

The time-independent projection-operator formalism of Brey et al. (Physica A, vol. 109, 1981, pp. 425–444) for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular non-trivial example, that the Chapman–Enskog methodology employed by Catto & Simakov (CS) (Phys. Plasmas, vol. 11, 2004, pp. 90–102) to calculate the contributions to the parallel viscosity driven by temperature gradients is consistent with formulas previously derived from the two-time formalism by Brey (J. Chem. Phys., vol. 79, 1983, pp. 4585–4598). The work serves to unify previous work on plasma kinetic theory with formalism usually applied to turbulence. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts 1 and 2 of this series provide an introduction to projection-operator methods that should be broadly useful in theoretical plasma physics.

The drift surfaces of minority heated ions differ from flux surfaces due to finite poloidal gyroradius effects. As the minority poloidal gyroradius approaches radial scale lengths in the plasma, the difference between drift and flux surfaces can modify the heating and lead to a symmetric spectrum minority counter-current being driven. In response, a corresponding overall net co-current of comparable size is driven. This beneficial symmetric spectrum current drive in a tokamak is due to the parallel velocity asymmetry in the drift departure from a flux surface. As this new source of driven current is a side effect of minority heating it comes without any additional economic cost to reactor power balance. The symmetric spectrum current driven for near Maxwellian minorities is evaluated by an adjoint method and found to be modest. However, minority heating typically results in strong non-Maxwellian features on minority distributions so it may be possible to drive a significantly larger co-current. A related evaluation is performed for alpha particles in a deuterium minority heated plasma with a tritium majority. The low density of the alphas tends to keep this driven symmetric spectrum current small, but at very high heating levels a significant co-current might be driven. Other mechanisms to drive co-current with a symmetric spectrum are discussed and estimated, including asymmetric electron drag and focusing of the applied minority heating radio frequency fields.

Azimuthal surface waves are eigenmodes of cylindrical plasma–dielectric–metal structures both in the presence of and without an axial static magnetic field. They are actively studied due to possible applications in plasma electronics, nanotechnologies and biomedical diagnostics. Higher radial modes are known to propagate at higher frequencies and shorter wavelengths compared to those of the zeroth mode, a feature which is of interest for practical applications. To gain the advantage of the excitation of higher radial modes of azimuthal surface waves one has first to know their dispersion properties. This paper generalizes the results of earlier papers by including a static axial magnetic field and considering the higher radial modes. The presence of the constant axial magnetic field removes the degeneracy in the wave spectrum with respect to the sign of the azimuthal wavenumber.

Collisionless shocks follow the Rankine–Hugoniot jump conditions to a good approximation. However, for a shock propagating parallel to a magnetic field, magnetohydrodynamics states that the shock properties are independent of the field strength, whereas recent particle-in-cell simulations reveal a significant departure from magnetohydrodynamics behaviour for such shocks in the collisionless regime. This departure is found to be caused by a field-driven anisotropy in the downstream pressure, but the functional dependence of this anisotropy on the field strength is yet to be determined. Here, we present a non-relativistic model of the plasma evolution through the shock front, allowing for a derivation of the downstream anisotropy in terms of the field strength. Our scenario assumes double adiabatic evolution of a pair plasma through the shock front. As a result, the perpendicular temperature is conserved. If the resulting downstream is firehose stable, then the plasma remains in this state. If unstable, it migrates towards the firehose stability threshold. In both cases, the conservation equations, together with the relevant hypothesis made on the temperature, allows a full determination of the downstream anisotropy in terms of the field strength.

Accurate modelling of the interaction between fast electrons and partially ionized atoms is important for evaluating tokamak disruption mitigation schemes based on material injection. This requires accounting for the effect of screening of the impurity nuclei by the cloud of bound electrons. In this paper, we generalize the Fokker–Planck operator in a fully ionized plasma by accounting for the effect of screening. We detail the derivation of this generalized operator, and calculate the effective ion length scales, needed in the components of the collision operator, for a number of ion species commonly appearing in fusion experiments. We show that for high electric fields, the secondary runaway growth rate can be substantially larger than in a fully ionized plasma with the same effective charge, although the growth rate is significantly reduced at near-critical electric fields. Furthermore, by comparison with the Boltzmann collision operator, we show that the Fokker–Planck formalism is accurate even for large impurity content.

Gyrokinetic stability of plasmas in different magnetic geometries is studied numerically using the GENE code. We examine the stability of plasmas, varying the mass ratio between the positive and negative charge carriers, from conventional hydrogen plasmas through to electron–positron plasmas. Stability is studied for prescribed temperature and density gradients in different magnetic geometries: (i) An axisymmetric, circular flux surface, low $\unicode[STIX]{x1D6FD}$ (tokamak) configuration. (ii) A non-axisymmetric quasi-isodynamic (optimised stellarator) configuration using the geometry of the stellarator Wendelstein 7-X. We also present the analytic theory of trapped particle modes in electron–positron plasmas. We found similar behaviour of the growth rate and real frequency compared to previous studies on the tokamak case. We are able to identify two distinct regimes dominated by modes propagating in the electron diamagnetic direction and modes propagating in the ion/positron diamagnetic direction, depending on the mass ratio. In both the tokamak and the stellarator case we observe that the real frequency tends to zero as the mass ratio approaches unity and are able to explain this using gyrokinetic theory.

We investigate analytically and numerically the semi-collisional regime of the plasmoid instability, defined by the inequality $\unicode[STIX]{x1D6FF}_{\text{SP}}\gg \unicode[STIX]{x1D70C}_{s}\gg \unicode[STIX]{x1D6FF}_{\text{in}}$, where $\unicode[STIX]{x1D6FF}_{\text{SP}}$ is the width of a Sweet–Parker current sheet, $\unicode[STIX]{x1D70C}_{s}$ is the ion sound Larmor radius and $\unicode[STIX]{x1D6FF}_{\text{in}}$ is the width of the boundary layer that arises in the plasmoid instability analysis. Theoretically, this regime is predicted to exist if the Lundquist number $S$ and the length of the current sheet $L$ are such that $(L/\unicode[STIX]{x1D70C}_{s})^{14/9}<S<(L/\unicode[STIX]{x1D70C}_{s})^{2}$ (for a sinusoidal-like magnetic configuration; for a Harris-type sheet the lower bound is replaced with $(L/\unicode[STIX]{x1D70C}_{s})^{8/5}$). These bounds are validated numerically by means of simulations using a reduced gyrokinetic model (Zocco & Schekochihin, Phys. Plasmas, vol. 18 (10), 2011, 102309) conducted with the code Viriato. Importantly, this regime is conjectured to allow for plasmoid formation at relatively low, experimentally accessible, values of the Lundquist number. Our simulations obtain plasmoid instability at values of $S$ as low as ${\sim}250$. The simulations do not prescribe a Sweet–Parker sheet; rather, one is formed self-consistently during the nonlinear evolution of the initial tearing mode configuration. This proves that this regime of the plasmoid instability is realizable, at least at the relatively low values of the Lundquist number that are accessible to current dedicated experiments.

With the advent of high power lasers, new opportunities have opened up for simulating astrophysical processes in the laboratory. We show that second-order Fermi acceleration can be directly investigated at the National Ignition Facility, Livermore. This requires measuring the momentum-space diffusion of 3 MeV protons produced within a turbulent plasma generated by a laser. Treating Fermi acceleration as a biased diffusion process, we show analytically that a measurable broadening of the initial proton distribution is then expected for particles exiting the plasma.

A theory of the Bennett pinch is presented for the case of non-relativistic electrons. In this way the basic physics of the phenomenon can be clearly demonstrated. The magnetic force acting on the electrons is transmitted to the positive ions via an electrostatic field.