The CASTOR3D code is a linear, extended, magnetohydrodynamic stability code that can be applied to tokamak and stellarator equilibria with and without ideal and resistive complex wall and coil structures in the surrounding vacuum region. The code enables the investigation of the effects of resistivity, parallel viscosity, gyroviscosity, ion drift velocity, ExB-velocity, electron diamagnetic drift, parallel heat conductivity and externally driven flows on growth rate, oscillation frequency and mode structure. This review article describes the three-dimensional (3-D) linear stability CASTOR3D code, its versatile application possibilities (e.g. ideal and resistive modes, vertical stability investigations, edge stability limit studies in presence of magnetic perturbation fields, resistive wall modes, etc.) and interesting effects concerning 3-D plasma configurations (e.g. toroidal mode coupling leading to overstable modes, mode locking, critical rotation threshold for locked modes and non-uniform rotation).

In the long-pulse steady-state operation of future advanced tokamaks, intermittent bursts induced by magnetohydrodynamic (MHD) instabilities present a critical challenge to plasma confinement. Therefore, the nonlinear dynamics of intermittent bursts induced by neoclassical double tearing modes in reversed magnetic shear plasmas are systematically investigated using a reduced MHD model in a cylindrical coordinate system. Comparisons between single-helicity (SH) and multihelicity (MH) simulation results reveal that the burst amplitude and average energy of the dominant $m/n=3/1$ mode in the MH case are lower than those in the SH case. This phenomenon is mainly attributed to the redistribution of perturbation energy during the MH nonlinear coupling process: a portion of the magnetic free energy is transferred to the subdominant $m/n=5/2$ mode and higher harmonics. Furthermore, an increased bootstrap current fraction modulates the intermittent bursting characteristics by reducing the ohmic current proportion, thereby weakening the $q$-profile restoration capability. Consequently, when the bootstrap current fraction ($f_{b}$) reaches a certain threshold ($f_{b}\geq 0.2$), the burst amplitude of the dominant mode tends to saturate, and the burst period begins to decrease gradually. Particularly at a high bootstrap current fraction ($f_{b}\geq 0.4$), constrained by the suppressed overall energy level, the MH system maintains only a weak relaxation process, exhibiting greater reductions in burst amplitude and period than the SH case.

We revisit parametric instability (PI) in inhomogeneous plasma using the exact Wentzel–Kramers–Brillouin (WKB) method applied to the coupled daughter-wave equations. Both resonant and non-resonant regimes, as well as forward and backward scattering, are examined. Compared with the previously adopted conventional WKB approach, the exact WKB approach explicitly incorporates boundary conditions and Stokes jumps, providing a more complete description of mode evolution. Consequently, all four PI regimes exhibit a universal daughter-wave amplification factor $|V_r|/\epsilon$, where $V_r$ corresponds to the real part of the Voros multiplier associated with the turning-point pair and $\epsilon$ is the strength of inhomogeneity. The exact WKB method resolves the vanishing-growth prediction for resonant backward scattering PI obtained in the conventional WKB approach, which neglects Stokes jumps. Furthermore, exact WKB analysis shows that the inverse dependence of the amplification factor on inhomogeneity is insensitive to $\epsilon$, provided that the combined wave equation reduces to a Weber-type equation. These results highlight the capability of the exact WKB framework to study PIs beyond the validity conditions of the conventional WKB approach, providing explicit physical insights inaccessible to numerical approaches.

The present study investigates the propagation of power carried by fast magnetosonic waves transmitted beyond the cutoff, utilising a slab model that incorporates non-zero wave poloidal wavenumbers. The plasma density profile is assumed to be an exponential one. The fraction of the transmitted power is analysed as a function of the plasma density decay length, subject to toroidal and poloidal refractive indices, plasma density at the antenna and operating frequency. The numerical analysis of the analytical expressions reveals a substantial sensitivity of the fraction of the transmitted power to the plasma density decay length and poloidal refractive index sign.

Fully kinetic, energy-conserving electrostatic particle-in-cell (PIC) simulations are employed to quantify how geometry and two-dimensional (2-D-$rz$) and three-dimensional (3-D) effects modify confinement, acceleration and cross-field transport by comparing our results with previous 1-D simulations in an open magnetic mirror. The bulk of the simulations are performed using an explicit scheme at a plasma density of approximately $10^{16}\,\mathrm{m}^{-3}$. In addition to being computationally accessible to explicit PIC, this regime has a relatively large Debye length and therefore emphasises charge-separation effects important in 2-D and 3-D simulations. Starting from a 2-D axisymmetric (2-D-$rz$) model and extending to three dimensions, we isolate finite-ion-Larmor-radius (FLR) effects and azimuthal instabilities. In the 2-D-$rz$ case, FLR effects partially decouple ions from magnetised electrons, producing charge separation near the plasma edge and modest radial ion leakage to the wall, while the near-axis density, potential and acceleration profiles remain close to those in the reduced 1-D model. Full 3-D simulations demonstrate azimuthal modes that induce radial currents, broaden radial density profiles and reduce the confining potential and the total axial potential drop. At low ion temperature, a coherent azimuthal mode of low frequency and low mode number $m=2$ is observed. At high ion temperature, fluctuations shift toward higher mode numbers $m\simeq 4$–$5$ and higher frequencies, with a broadband turbulent spectrum. The particle and energy (for ions and electrons) radial and axial losses are studied in detail. We show that for 3-D cases, radial ion losses account for approximately $30\,\%$ of the flux injected from the source region. The radial electric field produced by the endplate biasing introduces azimuthal shear that weakens coherent structures and reduces radial losses. Finally, additional simulations at densities of the order $10^{18}\,\mathrm{m}^{-3}$ have been performed with a semi-implicit electrostatic PIC algorithm.

Electron hole treatments normally match collisionless distribution functions associated with spatially varying electrostatic potential wells containing bound or reflecting electrons to unbound or non-reflecting electrons. When the matching is piecewise continuous at the separatrix between the bound and unbound, the second derivative in velocity space normally diverges, indicating collisions should be retained to remove the singular behaviour. The solubility constraint introduced by the collision operator severely restricts the allowed bound and unbound solutions, as illustrated by solving a model problem for a solitary electron depleted hole. As a result, a narrow collisional boundary layer encloses the separatrix between the bound and barely unbound electrons most sensitive to collisions. The procedure employed does not require resolving the collisional boundary layer, yet self-consistently accounts for the discontinuous behaviour to demonstrate an electron hole can last a long time when collisions are very weak.

Compared with contemporary experimental parameters, a burning plasma device operates at significantly higher densities and temperatures. Furthermore, the magnetic field and device size are several times larger than those of existing tokamaks. Discrepancies of these parameters may influence the linear stability of Alfvén eigenmodes; thus, the extrapolation of energetic particle-driven instabilities in present experiments to burning plasma conditions requires rigorous validation. Starting from a quadratic form of the momentum equation, we employ a perturbative approach to evaluate kinetic effects, specifically the drive from alpha particles and the Landau damping from bulk ions. This framework allows for the determination of the linear growth rate, while including radiative damping and trapped electron collisional damping from analytic formulations. For the core-localised toroidal Alfvén eigenmodes (TAE) and global TAEs, we find that the alpha beta thresholds for TAE instabilities are approximately $1.0\,\%$. By maintaining a constant bulk plasma beta or fusion triple product, we conduct systematic parameter scans to identify the key dependences of the linear stability of TAEs. With increases of background temperature and magnetic field strength, the density and beta of alpha particles increase correspondingly from the consistent parameter variations at a fixed plasma beta or fusion triple product, resulting in enhancement of TAE instabilities. However, other types of parameter scaling with magnetic field may induce the stabilisation of TAEs.