Steady, helical perturbations known as ‘density snakes’ with poloidal and toroidal mode numbers $m=1$, $n=1$ have been studied in several tokamak experiments. These three-dimensional, helical states are interesting due to their stability and persistence, including their coexistence with the sawtooth cycle. Presented here are studies of density snakes in tokamak plasmas in the Madison Symmetric Torus (MST) device. They are diagnosed using an 11-chord interferometer, internal and edge magnetic coils and impurity ion spectroscopy. Compared with observations in other tokamak plasmas, snakes in MST form with relatively high resistivity and low edge safety factor, $ q(a) \geqslant 2.2$, which moves the $q=1$ resonant surface outward in radius and probably forms a large magnetic island. As a result, the density perturbation associated with the snake is larger, the structure occupies a broader span of minor radius and the snakes are somewhat less stable. The helical structure and distribution of snake events are characterized, including whether they are best described as ideal or resistive kink modes. Finally, an analysis of their perturbation or destruction during sawtooth crashes is given.

Landau damping is one of the cornerstones of plasma physics. Based on the initial-value approach adopted by Landau in his original derivation of Landau damping, we examine the solutions of the linear Vlasov–Poisson system for different equilibrium distribution functions $f_0(v)$, going beyond the traditional focus on the root with largest imaginary part and investigating the full set of roots that the dispersion relation of the system generally admits. Specifically, we provide analytical insights into the number and the structure of the roots for entire and meromorphic functions $f_0(v)$, such as Maxwellian and $\kappa$ distributions, we discuss the potential issues related to the redefinition of $\partial{f}_0/\partial{v}$ as a complex variable function and we show how different sigmoids affect the root structure associated with non-meromorphic cut-off distribution functions. Finally, based on the comparison of the several root structures considered, we wonder if the multiple roots might hint at a deeper understanding of the Landau damping phenomenon.

This study investigates the impact of local thermal non-equilibrium on the stability analysis of partially ionized plasma within a porous medium. The plasma, heated from below, is enclosed by various combinations of bounding surfaces. Both nonlinear (via the energy method) and linear (utilizing the normal mode analysis method) analyses are performed. Eigenvalue problems for both analyses are formulated and solved using the Galerkin method. The study also explores the effects of compressibility, medium permeability and magnetic fields on system stability. The collisional frequency among plasma components and the thermal diffusivity ratio significantly influence energy decay. The results reveal that the Rayleigh–Darcy number is identical for both nonlinear and linear analyses, thus eliminating the possibility of a subcritical region and confirming global stability. The principle of exchange of stabilities is validated, indicating the absence of oscillatory convection modes. Medium permeability, heat-transfer coefficient and compressibility delay the onset of convection, demonstrating stabilizing effects. Conversely, the porosity-modified conductivity ratio hastens the convection process, indicating destabilizing effects. Rigid–rigid bounding surfaces are found to be thermally more stable for confining the partially ionized plasma. Additionally, the magnetic field exerts a stabilizing influence.

When analysing stellarator configurations, it is common to perform an asymptotic expansion about the magnetic axis. This so-called near-axis expansion is convenient for the same reason asymptotic expansions often are, namely, it reduces the dimension of the problem. This leads to convenient and quickly computed expressions of physical quantities, such as quasisymmetry and stability criteria, which can be used to gain further insight. However, it has been repeatedly found that the expansion diverges at high orders in the distance from axis, limiting the physics the expansion can describe. In this paper, we show that the near-axis expansion diverges in vacuum due to ill-posedness and that it can be regularised to improve its convergence. Then, using realistic stellarator coil sets, we demonstrate numerical convergence of the vacuum magnetic field and flux surfaces to the true values as the order increases. We numerically find that the regularisation improves the solutions of the near-axis expansion under perturbation, and we demonstrate that the radius of convergence of the vacuum near-axis expansion is correlated with the distance from the axis to the coils.

High-beta magnetised plasmas often exhibit anomalously structured temperature profiles, as seen from galaxy cluster observations and recent experiments. It is well known that when such plasmas are collisionless, temperature gradients along the magnetic field can excite whistler waves that efficiently scatter electrons to limit their heat transport. Only recently has it been shown that parallel temperature gradients can excite whistler waves also in collisional plasmas. Here, we develop a Wigner–Moyal theory for the collisional whistler instability starting from Braginskii-like fluid equations in a slab geometry. This formalism is necessary because, for a large region in parameter space, the fastest-growing whistler waves have wavelengths comparable to the background temperature gradients. We find additional damping terms in the expression for the instability growth rate involving inhomogeneous Nernst advection and resistivity. They (i) enable whistler waves to re-arrange the electron temperature profile via growth, propagation and subsequent dissipation, and (ii) allow non-constant temperature profiles to exist stably. For high-beta plasmas, the marginally stable solutions take the form of a temperature staircase along the magnetic field lines. The electron heat flux can also be suppressed by the Ettingshausen effect when the whistler intensity profile is sufficiently peaked and oriented opposite the background temperature gradient. This mechanism allows cold fronts without magnetic draping, might reduce parallel heat losses in inertial fusion experiments and generally demonstrates that whistler waves can regulate transport even in the collisional limit.

We investigate the global stability properties of an electron–positron pair plasma in the linear regime. The plasma is confined by the magnetic field of an infinitely long wire. This configuration is the large-aspect-ratio limit of the levitated dipole experiment of the APEX collaboration. The stability is governed by the diocotron mode and the interchange mode. The diocotron mode dominates in the case of a cold, non-neutral plasma. For specific density profiles we find analytic solutions. We derive a necessary condition for instability and find unstable solutions if the plasma forms a thin shell around the wire. Solutions for arbitrary density profiles with finite temperature are obtained numerically. We find that finite-temperature effects stabilise the diocotron mode. The interchange mode, on the other hand, dominates if the plasma is neutral and has a finite temperature. This mode becomes unstable for a steep-enough density gradient, that is aligned with the gradient of the magnetic field strength and is stabilised by the equilibrium $E\times B$ drift of a non-neutral plasma.

The change in direction of the wavevector and group velocity experienced by a wave refracted at the interface of an anisotropic medium in uniform linear motion are determined analytically. These transmission conditions, which are shown to be consistent with the generalised Snell’s law written in the laboratory frame, are then used to examine the effect of motion on waves incident on a magnetised plasma. For an incident wave in the plane perpendicular to the magnetic field the motion is observed to lead to non-negligible deviation of the low-frequency X-mode, as well as to non-symmetrical total reflection angles. These effects are shown to be further complicated when the magnetic field is in the plane formed by the incident wavevector and the medium’s velocity, as the anisotropy now competes with the motion-induced drag. Although obtained in simplified configurations, these results suggest that accounting for motion when modelling plasma wave trajectories could be important under certain conditions, calling for a more detailed quantification of the effect of motion in actual diagnostics and plasma control schemes.

This paper presents a theoretical model of plasma equilibrium in the diamagnetic confinement mode in an axisymmetric mirror device with neutral beam injection. The hot ionic component is described within the framework of the kinetic theory, since the Larmor radius of the injected ions appears to be comparable to or even larger than the characteristic scale of the magnetic field inhomogeneity. The electron drag of the hot ions is taken into account, while the angular scattering of the hot ions due to Coulomb collisions is neglected. The background warm plasma, on the contrary, is considered to be in local thermal equilibrium, i.e. has a Maxwellian distribution function and is described in terms of magnetohydrodynamics. The density of the hot ions is assumed to be negligible compared with that of the warm plasma. Both the conventional gas-dynamic loss and the non-adiabatic loss specific to the diamagnetic confinement mode are taken into account. In this work, we do not consider the effects of the warm plasma rotation as well as the inhomogeneity of the electrostatic potential. A self-consistent theoretical model of the plasma equilibrium is constructed. In the case of the cylindrical bubble, this model is reduced to a simpler one. The numerical solutions in the limit of a thin transition layer of the diamagnetic bubble are found. Examples of the equilibria corresponding to the gas-dynamic multiple-mirror trap device are considered.

Even if the magnetic field in a stellarator is integrable, phase-space integrability for energetic particle guiding-center trajectories is not guaranteed. Both trapped and passing particle trajectories can experience convective losses, caused by wide phase-space island formation, and diffusive losses, caused by phase-space island overlap. By locating trajectories that are closed in the angle coordinate but not necessarily closed in the radial coordinate, we can quantify the magnitude of the perturbation that results in island formation. We characterize island width and island overlap in quasihelical (QH) and quasiaxisymmetric (QA) equilibria with finite plasma pressure $\beta$ for both trapped and passing energetic particles. For trapped particles in QH, low-shear toroidal precession frequency profiles near zero result in wide island formation. While QA transit frequencies do not cross through the zero resonance, we observe that island overlap is more likely since higher shear results in the crossing of more low-order resonances.

The intense applied lower hybrid electric fields used to drive current in tokamaks can result in the formation of velocity space island structure. When this happens the lower hybrid current drive efficiency can be calculated for a monochromatic wave and is shown to be below the quasilinear level.

Kinetic theory of particles near resonances is a current topic of discussion in plasma physics and astrophysics. We extend this discussion to the kinetic theory of the interaction between alpha particles (energetic particles predicted to exist in large quantities in next-generation fusion experiments) and a neoclassical tearing mode (NTM), a resistively-driven perturbation which sometimes exists in a tokamak. We develop a quasilinear treatment of the interaction between alpha particles and an NTM, showing why an NTM can be a source of significant passing alpha particle transport in tokamaks. The limitations on quasilinear theory constrain our theory's applicability to small amplitude NTMs, highlighting the importance of nonlinear studies.

Accurate modelling of runaway electron generation and losses during tokamak disruptions is crucial for the development of reactor-scale tokamak devices. In this paper, we present a reduced model for runaway electron losses due to flux surface scrape-off caused by the vertical motion of the plasma. The model is made compatible with computationally inexpensive one-dimensional models averaging over a fixed flux-surface geometry, by formulating it as a loss term outside an estimated time-varying minor radius of the last closed flux surface. We then implement this model in the disruption modelling tool DREAM and demonstrate its impact on selected scenarios relevant for ITER. Our results indicate that scrape-off losses may be crucial for making complete runaway avoidance possible even in a $15\,\rm MA$ DT H-mode ITER scenario. The results are however sensitive to the details of the runaway electron generation and phenomena affecting the current density profile, such as the current profile relaxation at the beginning of the disruption.

We provide an assessment of the Infinity Two fusion pilot plant (FPP) baseline plasma physics design. Infinity Two is a four-field period, aspect ratio $A = 10$, quasi-isodynamic stellarator with improved confinement appealing to a max-$J$ approach, elevated plasma density and high magnetic fields ($ \langle B\rangle = 9$ T). Here $J$ denotes the second adiabatic invariant. At the envisioned operating point ($800$ MW deuterium-tritium (DT) fusion), the configuration has robust magnetic surfaces based on magnetohydrodynamic (MHD) equilibrium calculations and is stable to both local and global MHD instabilities. The configuration has excellent confinement properties with small neoclassical transport and low bootstrap current ($|I_{bootstrap}| \sim 2$ kA). Calculations of collisional alpha-particle confinement in a DT FPP scenario show small energy losses to the first wall (${\lt}1.5 \,\%$) and stable energetic particle/Alfvén eigenmodes at high ion density. Low turbulent transport is produced using a combination of density profile control consistent with pellet fueling and reduced stiffness to turbulent transport via three-dimensional shaping. Transport simulations with the T3D-GX-SFINCS code suite with self-consistent turbulent and neoclassical transport predict that the DT fusion power$P_{{fus}}=800$ MW operating point is attainable with high fusion gain ($Q=40$) at volume-averaged electron densities $n_e\approx 2 \times 10^{20}$ m$^{-3}$, below the Sudo density limit. Additional transport calculations show that an ignited ($Q=\infty$) solution is available at slightly higher density ($2.2 \times 10^{20}$ m$^{-3}$) with $P_{{fus}}=1.5$ GW. The magnetic configuration is defined by a magnetic coil set with sufficient room for an island divertor, shielding and blanket solutions with tritium breeding ratios (TBR) above unity. An optimistic estimate for the gas-cooled solid breeder designed helium-cooled pebble bed is TBR $\sim 1.3$. Infinity Two satisfies the physics requirements of a stellarator fusion pilot plant.

This paper presents a novel approach for simulating plasma instabilities in tokamak plasmas using the piecewise field-aligned finite element method in combination with the particle-in-cell method. Our method traditionally aligns the computational grid, but defines the basis functions in piecewise field-aligned coordinates to avoid grid deformation while naturally representing the field-aligned mode structures. This scheme is formulated and implemented numerically. It also applies to the unstructured triangular meshes in principle. We have conducted linear benchmark tests, which agree well with previous results and traditional schemes. Furthermore, multiple-$n$ simulations are also carried out as a proof of principle, demonstrating the efficiency of this scheme in nonlinear turbulence simulations within the framework of the finite element method.

The decay of a turbulent magnetic field is slower with helicity than without. Furthermore, the magnetic correlation length grows faster for a helical than a non-helical field. Both helical and non-helical decay laws involve conserved quantities: the mean magnetic helicity density and the Hosking integral. Using direct numerical simulations in a triply periodic domain, we show quantitatively that in the fractionally helical case the mean magnetic energy density and correlation length are approximately given by the maximum of the values for the purely helical and purely non-helical cases. The time of switchover from one to the other decay law can be obtained on dimensional grounds and is approximately given by $I_{H}^{1/2}I_{M}^{-3/2}$, where $I_{H}$ is the Hosking integral and $I_{M}$ is the mean magnetic helicity density. An earlier approach based on the decay time is found to agree with our new result and suggests that the Hosking integral exceeds naive estimates by the square of the same resistivity-dependent factor by which also the turbulent decay time exceeds the Alfvén time. In the presence of an applied magnetic field, the mean magnetic helicity density is known to be not conserved, and we show that then also the Hosking integral is not conserved.

The bootstrap current in stellarators can be presented as a sum of a collisionless value given by the Shaing–Callen asymptotic formula and an off-set current, which non-trivially depends on plasma collisionality and radial electric field. Using NEO-2 modeling, analytical estimates and semi-analytical studies with the help of a propagator method, it is shown that the off-set current in the $1/\nu$ regime does not converge with decreasing collisionality $\nu _\ast$ but rather shows oscillations over $\log \nu _\ast$ with an amplitude of the order of the bootstrap current in an equivalent tokamak. The convergence to the Shaing–Callen limit appears in regimes with significant orbit precession, in particular, due to a finite radial electric field, where the off-set current decreases as $\nu _\ast ^{3/5}$. The off-set current strongly increases in case of nearly aligned magnetic field maxima on the field line where it diverges as $\nu _\ast ^{-1/2}$ in the $1/\nu$ regime and saturates due to the precession at a level exceeding the equivalent tokamak value by ${v_E^\ast }^{-1/2}$, where $v_E^\ast$ is the perpendicular Mach number. The latter off-set, however, can be minimized by further aligning the local magnetic field maxima and by fulfilling an extra integral condition of “equivalent ripples” for the magnetic field. A criterion for the accuracy of this alignment and of ripple equivalence is derived. In addition, the possibility of the bootstrap effect at the magnetic axis caused by the above off-set is also discussed.

We analyse distributions of the spatial scales of coherent intermittent structures – current sheets – obtained from fully kinetic, two-dimensional simulations of relativistic turbulence in a collisionless pair plasma using unsupervised machine-learning data dissection. We find that the distribution functions of sheet length $\ell$ (longest scale of the analysed structure in the direction perpendicular to the dominant guide field) and curvature $r_c$ (radius of a circle fitted to the structures) can be well-approximated by power-law distributions, indicating self-similarity of the structures. The distribution for the sheet width $w$ (shortest scale of the structure) peaks at the kinetic scales and decays exponentially at larger values. The data shows little or no correlation between $w$ and $\ell$, as expected from theoretical considerations. The typical $r_c$ depends linearly on $\ell$, which indicates that the sheets all have a similar curvature relative to their sizes. We find a weak correlation between $r_c$ and $w$. Our results can be used to inform realistic magnetohydrodynamic subgrid models for plasma turbulence in high-energy astrophysics.

It is common wisdom that collisionless shocks become non-planar and non-stationary at sufficiently high Mach numbers. Whatever the shock structure, the upstream and downstream fluxes of the mass, momentum and energy should be equal. At low Mach numbers, these conservation laws are satisfied when the shock front is planar and stationary. When this becomes impossible, inhomogeneity and time dependence, presumably in the form of rippling, develop. In this study, we show that the shock structure changes as a kind of ‘phase transition’ when the Mach number is increased while other shock parameters are kept constant.

A charged particle in a suitably strong magnetic field spirals along the field lines while slowly drifting transversely. This note provides a brief derivation of an effective Lagrangian formulation for the guiding-centre approximation that captures this dynamics without resolving the gyro motion. It also explains how the effective Lagrangian may, for special magnetic fields, admit a ‘quasi-symmetry’ which can give rise to a conserved quantity helpful for plasma confinement in fields lacking a geometric isometry. The aim of this note is to offer a pedagogical introduction and some perspectives on this well-established subject.

We investigate a model of turbulent magnetic reconnection introduced by (Higashimori, Yokoi and Hoshino 2013 Phys. Rev. Lett. 110, 255001) and show that the classic two-dimensional, steady-state Sweet–Parker and Petschek reconnection solutions are supported. We present evidence that these are the only two steady-state reconnection solutions, and we determine the criterion for their selection. Sweet–Parker reconnection occurs when there is no growth in turbulent energy, whereas Petschek reconnection occurs when the current density in the reconnecting current sheet is able to surpass a critical value, allowing for the growth of turbulent energy that creates the diffusion region. Further, we show that the Petschek solutions are self-similar, depending on the value of the turbulent time scale, and produce a universal steady reconnection rate. The self-consistent development of Petschek reconnection through turbulence, within the model, is an example of fast and steady magnetic reconnection without an explicit need for the collisionless terms in an extended Ohm’s law.