Stability of the ‘rigid’ (m = 1$m = 1$) ballooning mode in a mirror axisymmetric trap is studied for the case of oblique neutral beam injection (NBI), which creates an anisotropic population of fast sloshing ions. Since small-scale modes with azimuthal numbers m>1$m>1$ in long thin (paraxial) mirror traps are easily stabilized by finite-Larmor-radius (FLR) effects, suppression of the rigid ballooning and flute modes would mean stabilization of all magnetohydrodynamic (MHD) modes, with the exception of the mirror and firehose disturbances, which are intensively studied in geophysics, but have not yet been identified in mirror traps. Large-scale ballooning mode can, in principle, be suppressed either by the lateral perfectly conducting wall, or by the end MHD anchors such as the cusp, by biased limiters or by a combination of these two methods. The effects of the wall shape, vacuum gap width between the plasma column and the lateral wall, angle of oblique NBI, radial profile of the plasma pressure and axial profile of the vacuum magnetic field are studied. It is confirmed that the lateral conducting wall still creates the upper stability zone, where the ratio \beta$\beta$ of the plasma pressure to the pressure of vacuum magnetic field exceeds the second critical value \beta_{\text{cr2}}$\beta_{\text{cr2}}$, \beta >\beta _{\text {cr2}}$\beta >\beta _{\text {cr2}}$. However, in many cases the upper zone is clamped from above by mirror instability. When the lateral wall is combined with end MHD anchors, a lower stability zone \beta <\beta _{\text {cr}1}$\beta <\beta _{\text {cr}1}$ appears, where \beta$\beta$ is below the first critical value \beta_{\text{cr1}}$\beta_{\text{cr1}}$. These two zones can overlap in the case of a sufficiently smooth radial pressure profile, and/or a sufficiently low mirror ratio and/or a sufficiently narrow vacuum gap between the plasma column and the lateral wall. However, even in this case, the range of permissible values of beta is limited from above by the threshold of mirror instability \beta_{\text{mm}}$\beta_{\text{mm}}$, so that \beta <\beta _{\text {mm}}<1$\beta <\beta _{\text {mm}}<1$, in contrast to the case of transversal NBI, when neutral beams are injected perpendicularly to the magnetic field.

We provide a fundamentally new perspective on subcritical turbulence in plasmas, based on coherent structures, which are obtained and characterised via direct numerical solution. The domains where these coherent states exist appear to be closely connected to the those where related turbulent states can exist, so there may be a deep connection between the stability of these coherent structures and the domain where sustained turbulence is possible. In contrast to previous descriptions of turbulence in terms of a stochastic collection of linear waves, we present a fundamentally nonlinear representation based on more general classes of translating oscillatory nonlinear solutions. In turbulent tokamak plasmas, the transport can often be completely suppressed by introducing a background shear flow, whose amplitude is an important control parameter. As this parameter is decreased below a critical value, radially localised structures appear, becoming larger and more complex, in both gyrokinetic simulations and a simpler fluid model of the plasma. For the fluid model, we directly solve for a particular class of nonlinear solutions, relative periodic orbits, and determine their stability, thus explaining why these isolated structures appear in initial-value simulations. The increase of complexity as the flow shear is reduced is explained by a series of Hopf bifurcations of these nonlinear solutions, which we quantify via stability analysis. In gyrokinetic simulations, we are able to indirectly determine the underlying relative periodic orbits by imposing symmetry conditions on the simulations.

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 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$n$ 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 design of fusion devices is typically based on computationally expensive simulations. This can be alleviated using high aspect ratio models that employ a reduced number of free parameters, especially in the case of stellarator optimization where non-axisymmetric magnetic fields with a large parameter space are optimized to satisfy certain performance criteria. However, optimization is still required to find configurations with properties such as low elongation, high rotational transform, finite beta and good fast particle confinement. In this work, we train a machine learning model to construct configurations with favourable confinement properties by finding a solution to the inverse design problem, that is, obtaining a set of model input parameters for given desired properties. Since the solution of the inverse problem is non-unique, a probabilistic approach, based on mixture density networks, is used. It is shown that optimized configurations can be generated reliably using this method.

Confinement quality in fusion plasma is influenced significantly by the presence of heavy impurities, which can lead to radiative heat loss and reduced confinement. This study explores the clustering of heavy impurity, i.e. tungsten in edge plasma, using high-resolution direct numerical simulations of the Hasegawa–Wakatani equations. We use the Stokes number to quantify the inertia of impurity particles. It is found that particle inertia will cause spatial intermittency in particle distribution and the formation of large-scale structures, i.e. the clustering of particles. The degrees of clustering are influenced by the Stokes number. To quantify these observations, we apply a modified Voronoi tessellation, which assigns specific volumes to impurity particles. By determining time changes of these volumes, we can calculate the impurity velocity divergence, which allows the clustering dynamics to be assessed. To quantify the clustering statistically, several approaches are applied, such as probability density function (PDF) of impurity velocity divergence and joint PDF of volume and divergence.

We solve ‘half’ the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy magnetohydrostatic force balance. This involves determining which hidden symmetries are admissible as quasisymmetries, and then showing explicitly how to construct quasisymmetric magnetic fields given an admissible symmetry. The admissibility conditions take the form of a system of overdetermined nonlinear partial differential equations involving second derivatives of the symmetry's infinitesimal generator.

The linear collisionless plasma response to a zonal-density perturbation in quasisymmetric stellarators is studied, including the geodesic-acoustic-mode oscillations and the Rosenbluth–Hinton residual flow. While the geodesic-acoustic-mode oscillations in quasiaxisymmetric configurations are similar to tokamaks, they become non-existent in quasi-helically symmetric configurations when the effective safety factor in helical-angle coordinates is small. Compared with concentric-circular tokamaks, the Rosenbluth–Hinton residual is also found to be multiplied by a geometric factor \mathcal {C}$\mathcal {C}$ that arises from the flux-surface-averaged classical polarization. Using the near-axis-expansion framework, we derive an analytic expression for \mathcal {C}$\mathcal {C}$, which varies significantly among different configurations. These analytic results are compared with numerical simulation results from the global gyrokinetic particle-in-cell code GTC, and good agreement with the theoretical Rosenbluth–Hinton residual level is achieved when the quasisymmetry error is small enough.

Turbulence self-organization is studied in the flux-driven regime by means of the reduced model Tokam1D. Derived in the electrostatic and isothermal limit but keeping finite electron and ion temperatures, it features two instabilities that are suspected to dominate turbulent transport at the edge of L-mode tokamak plasmas: interchange (a reduced version of the resistive ballooning modes) and collisional drift waves, governed respectively by an effective gravity parameter g$g$ and the adiabaticity parameter C$C$. The usual properties of these two instabilities are recovered in the linear regime. The nonlinear study focuses on the self-organization of collisional drift-wave turbulence at g=0$g=0$. It is found that the energy stored in zonal flows (ZFs) decreases smoothly at small C$C$ due to the reduction of both electric and diamagnetic stresses. Conversely to gradient-driven simulations, no sharp collapse is observed due to the self-consistent evolution of the equilibrium density profile. The ZFs are found to structure into staircases at small and large C$C$. These structures exhibit a rich variety of dynamics but are found to be robust to large perturbations. Their nucleation is found to be critically governed by the phase dynamics. Finally, the staircase structures are lost in the gradient-driven regime, when the system is prevented from storing turbulent energy into the equilibrium density (pressure) profile.

We present two-dimensional global simulations of mitigated and vertically unstable disruptions in ITER in the presence of runaway electrons (REs). An elongated plasma in free-boundary equilibrium is subjected to an artificial thermal quench (TQ) and current profile flattening, followed by one or more massive material injections and a RE avalanche. Scenarios of major disruptions as well as upward and downward vertical displacement events are considered. Results provide important insights into the effects of RE formation, post TQ current profile, injection quantities and timings, and impurity flushout on the overall evolution of disruption and the plasma vertical motion thereof. Interplay between the various effects offers scope for potentially beneficial RE mitigation scenarios.

We construct a description of ion-temperature-gradient (ITG)-driven localised linear modes which retains both wave–particle and magnetic drift resonant effects while capturing the field-line dependence of the electrostatic potential. We exploit the smallness of the magnetic drift and the strong localisation of the mode to resolve the problem with a polynomial–Gaussian expansion in the field-following coordinate. A simple semianalytical formula for the spectrum of the mode is shown to capture long wavelength Landau damping, ion-scale Larmor radius stabilisation, weakening of Larmor radius effects at short wavelengths and magnetic-drift resonant stabilisation. These elements lead to linear spectra with multiple maxima as observed in gyrokinetic simulations in stellarators. Connections to the transition to extended eigenfunctions and those localised by less unfavourable curvature regions (hopping solutions) are also made. The model provides a clear qualitative framework with which to interpret numerically simulated ITG modes’ linear spectra with realistic geometries, despite its limitations for exact quantitative predictions.

The tangential drift of the trapped alpha particles in bounce or transit averaged kinetic treatments of stellarators reverses direction on each flux surface at a particular value of pitch angle. The vanishing of the tangential drift corresponds to a resonance that allows a narrow collisional boundary layer to form due to the presence of pitch angle scattering by the background ions. The alphas in and adjacent to this drift reversal layer are particularly sensitive to collisions because they are in or very close to resonance. As a result, enhanced collisional transport occurs due to the existence of this drift reversal resonance in a nearly quasisymmetric stellarator with a single helicity imperfection. Moreover, the value of the resonant pitch angle for drift reversal on neighbouring flux surfaces varies continuously, with the inner flux surfaces having a larger resonant pitch angle than the outer ones. This pitch angle dependence means phase space ‘tubes’ or ‘pods’ exist that connect the inner flux surfaces to the outer ones. These pods allow collisional radial transport of the alphas to extend over the entire radial cross section. When collisions are finite, but weak, and the single helicity departure from quasisymmetry large enough, the collisionless alpha particle motion remains constrained by collisions as they complete their drift trajectories in phase. In particular, the small radial scales introduced by the radial extent or width of the phase space pods require the retention of the nonlinear radial drift term in the kinetic equation. The associated collisional radial transport is evaluated and found to be significant, but is shown to preferentially remove slower speed alphas without substantially affecting birth alphas.

In this work, we propose a method for optimising stellarator devices to favour the presence of an electron root solution of the radial electric field. Such a solution can help avoid heavy impurity accumulation, improve neoclassical thermal ion confinement and helium ash exhaust, and possibly reduce turbulence. This study shows that an optimisation for such a root is possible in quasi-isodynamic stellarators. Examples are shown for both vacuum and finite plasma pressure configurations.

Upper bounds on the growth of instabilities in gyrokinetic systems have recently been derived by considering the optimal perturbations that maximise the growth of a chosen energy norm. This technique has previously been applied to two-species gyrokinetic systems with fully kinetic ions and electrons. However, in tokamaks and stellarators, the expectation from linear instability analyses is that the most important kinetic electron contribution to ion-scale modes often comes from the trapped electrons, which bounce faster than the time scale upon which instabilities evolve. As a result, a fully kinetic electron response is not required to describe unstable modes in many cases. Here, we apply the optimal mode analysis to a reduced two-species system consisting of fully gyrokinetic ions and bounce-averaged electrons, with the aim of finding a tighter bound on ion-scale instabilities in toroidal geometry. This analysis yields bounds that are greatly reduced in comparison with the earlier two-species result. Moreover, if the energy norm is properly chosen, wave–particle resonance effects can be captured, reproducing the stabilisation of density-gradient-driven instabilities in maximum-J$J$ devices. The optimal mode analysis also reveals that the maximum-J$J$ property has an additional stabilising effect on ion-temperature-gradient-driven instabilities, even in the absence of an electron free energy source. This effect is explained in terms of the concept of mode inertia, making it distinct from other mechanisms.

A quasi-linear reduced transport model is developed from a database of high-\beta$\beta$ electromagnetic nonlinear gyrokinetic simulations performed with spherical tokamak for energy production (STEP) relevant parameters. The quasi-linear model is fully electromagnetic and accounts for the effect of equilibrium flow shear using a novel approach. Its flux predictions are shown to agree quantitatively with predictions from local nonlinear gyrokinetic simulations across a broad range of STEP-relevant local equilibria. This reduced transport model is implemented in the T3D transport solver that is used to perform the first flux-driven simulations for STEP to account for transport from hybrid kinetic ballooning mode turbulence, which dominates over a wide region of the core plasma. Nonlinear gyrokinetic simulations of the final transport steady state from T3D return turbulent fluxes that are consistent with the reduced model, indicating that the quasi-linear model may also be appropriate for describing the transport steady state. Within the assumption considered here, our simulations support the existence of a transport steady state in STEP with a fusion power comparable to that in the burning flat top of the conceptual design, but do not demonstrate how this state can be accessed.

Magnetic reconnection leads to the formation of island-shaped magnetic structure(s). Due to disagreement between theoretical evaluations of the characteristic reconnection time and observations, it is commonly accepted that the collisionality (or resistivity) is too low to explain magnetic reconnection phenomena in fusion plasmas. Thus, magnetic reconnection still raises many open questions. The work presented here aims to improve the fundamental knowledge about ‘the life of a magnetic island’. Here, in the light of the many works of the last 70 years, a new paradigm for understanding magnetic reconnection in fusion plasmas is proposed. The life of a magnetic island (whatever its scale) follows three phases: the origin, the growth and the saturation. The possible physical mechanisms at play in these three phases will be investigated. First, for the island origin, typical time scales in link with magnetic reconnection will be evaluated for three tokamaks of different sizes (TCV, WEST and JET) to verify if magnetic reconnection is such an unexplained phenomenon in fusion plasmas. Second, for the island drive, the richness of possible mechanisms leading to ‘rapid’ magnetic island growth in fusion devices will be presented for small and large scales. Third comes the island saturation step. Results on the prediction of a large island width at saturation are presented and discussed.

Ion cyclotron resonance heating is a versatile heating method that has been demonstrated to be able to efficiently couple power directly to the ions via the fast magnetosonic wave. However, at temperatures relevant for reactor grade devices such as DEMO, electron damping becomes increasingly important. To reduce electron damping, it is possible to use an antenna with a power spectrum dominated by low parallel wavenumbers. Moreover, using an antenna with a unidirectional spectrum, such as a travelling wave array antenna, the parallel wavenumber can be downshifted by mounting the antenna in an elevated position relative to the equatorial plane. This downshift can potentially enhance ion heating as well as fast wave current drive efficiency. Thus, such a system could benefit ion heating during the ramp-up phase and be used for current drive during flat-top operation. To test this principle, both ion heating and current drive have been simulated in a DEMO-like plasma for a few different mounting positions of the antenna using the FEMIC code. We find that moving the antenna off the equatorial plane makes ion heating more efficient for all considered plasma temperatures at the expense of on-axis heating. Moreover, although current drive efficiency is enhanced, electron damping is reduced for lower mode numbers, thus reducing the driven current in this part of the spectrum.

We apply a continuation method to recently optimized stellarator equilibria with excellent quasi-axisymmetry to generate new equilibria with a wide range of rotational transform profiles. Using these equilibria, we investigate how the rotational transform affects fast-particle confinement, the maximum coil–plasma distance, the maximum growth rate in linear gyrokinetic ion-temperature gradient simulations and the ion heat flux in corresponding nonlinear simulations. We find values of two-term quasi-symmetry error comparable to or lower than those of the similar Landreman–Paul (Phys. Rev. Lett., vol. 128, 2022, 035001) configuration for values of the mean rotational transform \bar {\iota }$\bar {\iota }$ between 0.12$0.12$ and 0.75$0.75$. The fast-particle confinement improves with \bar {\iota }$\bar {\iota }$ until \bar {\iota } = 0.73$\bar {\iota } = 0.73$, at which point the degradation in quasi-symmetry outweighs the benefits of further increasing \bar {\iota }$\bar {\iota }$. The required coil–plasma distance only varies by about {\pm }10\,\%${\pm }10\,\%$ for the configurations under consideration, and is between 2.8$2.8$ and 3.3\ \mathrm {m}$3.3\ \mathrm {m}$ when the configuration is scaled up to reactor size (minor radius a=1.7\ \mathrm {m}$a=1.7\ \mathrm {m}$ and volume-averaged magnetic field strength of 5.86\ \mathrm {T}$5.86\ \mathrm {T}$). The maximum growth rate from linear gyrokinetic simulations increases with \bar {\iota }$\bar {\iota }$, but also shifts towards higher k_y$k_y$ values. The maximum linear growth rate is sensitive to the choice of flux tube at rational \bar {\iota }$\bar {\iota }$, but this can be compensated for by taking the maximum over several flux tubes. The corresponding ion heat fluxes from nonlinear simulations display a non-monotonic relation to \bar {\iota }$\bar {\iota }$. Sufficiently large positive shear is destabilizing. This is reflected in both linear growth rates and nonlinear heat fluxes.

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.

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.