We report on an analytical and numerical study of the dynamics of a three-dimensional array of identical magnetic flux tubes in the reduced-magnetohydrodynamic description of the plasma. We propose that the long-time evolution of this system is dictated by flux-tube mergers, and that such mergers are dynamically constrained by the conservation of the pertinent (ideal) invariants, viz. the magnetic potential and axial fluxes of each tube. We also propose that in the direction perpendicular to the merging plane, flux tubes evolve in a critically balanced fashion. These notions allow us to construct an analytical model for how quantities such as the magnetic energy and the energy-containing scale evolve as functions of time. Of particular importance is the conclusion that, like its two-dimensional counterpart, this system exhibits an inverse transfer of magnetic energy that terminates only at the system scale. We perform direct numerical simulations that confirm these predictions and reveal other interesting aspects of the evolution of the system. We find, for example, that the early time evolution is characterized by a sharp decay of the initial magnetic energy, which we attribute to the ubiquitous formation of current sheets. We also show that a quantitatively similar inverse transfer of magnetic energy is observed when the initial condition is a random, small-scale magnetic seed field.

Self-amplified spontaneous emission (SASE) free electron laser (FEL) devices have disclosed an unexpected interplay between the laser intensity growth and regions of the electron bunch of the order of the coherence length. They are usually identified with the bunch slice and contribute to the laser dynamics with their own characteristics. The dynamical effects inducing geometrical and phase space misalignment of bunch slice in X-ray operating FELs can be traced back to a plethora of phenomena, both in the Linac accelerating section or inside the beam transport optic magnet. They are responsible for spoiling of the beam projected qualities and, if not corrected properly, induce an increase of the saturation length and a decreasing of the output power. We discuss the inclusion of these effects in models employing scaling formulae.

A Hamiltonian and action principle formalism for deriving three-dimensional gyroviscous magnetohydrodynamic models is presented. The uniqueness of the approach in constructing the gyroviscous tensor from first principles and its ability to explain the origin of the gyromap and the gyroviscous terms are highlighted. The procedure allows for the specification of free functions, which can be used to generate a wide range of gyroviscous models. Through the process of reduction, the noncanonical Hamiltonian bracket is obtained and briefly analysed.

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.

The core toroidal plasma intrinsic rotation has been studied by experiments and simulations in the Joint Texas Experimental Tokamak (J-TEXT). The direction of core intrinsic rotation in the J-TEXT plasma is counter-current. As the plasma density ramps up, the rotation velocity increases in the counter-current direction. By comparing four different electron densities, linear local gyrokinetic simulations have been performed by the Gyrokinetic Electromagnetic Numerical Experiment code for the first time on J-TEXT. It is found that the most dominant turbulence is the ion temperature gradient at $0.2a$, where $a$ is the minor radius of the plasma and this is unchanged during the plasma density ramp up. By scanning the radial wave vectors, it is found that the residual stress term reverses from negative to positive when the plasma density exceeds a certain threshold. The pinch term is larger than the residual stress term at all four electron densities, which means that the pinch term is always dominant in the core of a J-TEXT plasma.

Analytical solutions of the Grad–Shafranov equilibrium equation in simply connected plasma configurations, comprised of toroidal magnetic surfaces and open surfaces connected to electrodes, are reviewed and generalised. The Grad–Shafranov equation is linearised introducing assumptions on plasma current and pressure, which preserve regularity of solutions on the symmetry axis, as required for a simply connected geometry. Particular solutions are found by separation of variables both in cylindrical coordinates and in spherical ones. Equilibria that model local or global features of PROTO-SPHERA plasmas are constructed by combining a few particular solutions.

In this work, the FALCON code is adopted for illustrating the features of shear Alfvén and sound continuous spectra in toroidal fusion plasmas. The FALCON codes employ the local Floquet analysis discussed in (Phys. Plasmas, vol. 26, issue 8, 2019, 082502) for computing global structures of continuous spectra in general toroidal geometry. As particular applications, reference equilibria for the divertor tokamak test and ASDEX Upgrade plasmas are considered. In particular, we illustrate the importance of mode polarization for recognizing the physical relevance of the various branches of the continuous spectra in the ideal magnetohydrodynamics limit. We also analyse the effect of plasma compression and the validity of the slow sound approximation.

This work presents updates in the diagnostics systems, magnetohydrodynamics (MHD) calculations and simulations of microwave heating scenarios of the small modular Stellarator of Costa Rica 1 (SCR-1). Similarly, the design of a flexible bolometer and magnetic diagnostics (a set of Mirnov coils, Rogowski coils and two diamagnetic loops) are introduced. Furthermore, new MHD equilibrium calculations for the plasma of the SCR-1 device were performed using the VMEC code including the poloidal cross-section of the magnetic flux surfaces at different toroidal positions, profiles of the rotational transform, magnetic well, magnetic shear and total magnetic field norm. Charged particle orbits in vacuum magnetic field were computed by the magnetic field solver BS-SOLCTRA (Vargas et al. In 27th IAEA Fusion Energy Conference (FEC 2018), 2018. IAEA). A visualization framework was implemented using Paraview (Solano-Piedra et al. In 23rd IAEA Technical Meeting on the Research Using Small Fusion Devices (23rd TM RUSFD), 2017) and compared with magnetic mapping results (Coto-Vílchez et al. In 16th Latin American Workshop on Plasma Physics (LAWPP), 2017, pp. 43–46). Additionally, simulations of microwave heating scenarios were performed by the IPF-FDMC full-wave code. These simulations calculate the conversion of the ordinary waves to extraordinary waves and allow us to identify the location where the conversion takes place. Finally, the microwave heating scenarios for the $330^{\circ }$ toroidal position are presented. The microwave heating scenarios showed that the O–X–B mode conversion is around 12–14 %. It was possible to identify the spatial zone where the conversion takes place (upper hybrid frequency).

A careful examination of the effects of collisions on resonant wave–particle interactions leads to an alternate interpretation and deeper understanding of the quasilinear operator originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388) for collisionless, magnetized plasmas, and widely used to model radio frequency heating and current drive. The resonant and nearly resonant particles are particularly sensitive to collisions that scatter them out of and into resonance, as for Landau damping as shown by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). As a result, the resonant particle–wave interactions occur in the centre of a narrow collisional boundary when the collision frequency $\unicode[STIX]{x1D708}$ is very small compared to the wave frequency $\unicode[STIX]{x1D714}$. The diffusive nature of the pitch angle scattering combined with the wave–particle resonance condition enhances the collision frequency by $(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708})^{2/3}\gg 1$, resulting in an effective resonant particle collisional interaction time of $\unicode[STIX]{x1D70F}_{\text{int}}\sim (\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714})^{2/3}/\unicode[STIX]{x1D708}\ll 1/\unicode[STIX]{x1D708}$. A collisional boundary layer analysis generalizes the standard quasilinear operator to a form that is fully consistent with Kennel–Englemann, but allows replacing the delta function appearing in the diffusivity with a simple integral (having the appropriate delta function limit) retaining the new physics associated with the narrow boundary layer, while preserving the entropy production principle. The limitations of the collisional boundary layer treatment are also estimated, and indicate that substantial departures from Maxwellian are not permitted.

Following up on earlier work which demonstrated an improved numerical stellarator coil design optimization performance by the use of stochastic optimization (Lobsien et al., Nucl. Fusion, vol. 58 (10), 2018, 106013), it is demonstrated here that significant further improvements can be made – lower field errors and improved robustness – for a Wendelstein 7-X test case. This is done by increasing the sample size and applying fully three-dimensional perturbations, but most importantly, by changing the design sequence in which the optimization targets are applied: optimization for field error is conducted first, with coil shape penalties only added to the objective function at a later step in the design process. A robust, feasible coil configuration with a local maximum field error of 3.66 % and an average field error of 0.95 % is achieved here, as compared to a maximum local field error of 6.08 % and average field error of 1.56 % found in our earlier work. These new results are compared to those found without stochastic optimization using the FOCUS and ONSET suites. The relationship between local minima in the optimization space and coil shape penalties is also discussed.

The shape gradient is a local sensitivity function defined on the surface of an object which provides the change in a characteristic quantity, or figure of merit, associated with a perturbation to the shape of the object. The shape gradient can be used for gradient-based optimization, sensitivity analysis and tolerance calculations. However, it is generally expensive to compute from finite-difference derivatives for shapes that are described by many parameters, as is the case for typical stellarator geometry. In an accompanying work (Antonsen, Paul & Landreman J. Plasma Phys., vol. 85 (2), 2019), generalized self-adjointness relations are obtained for magnetohydrodynamic (MHD) equilibria. These describe the relation between perturbed equilibria due to changes in the rotational transform or toroidal current profiles, displacements of the plasma boundary, modifications of currents in the vacuum region or the addition of bulk forces. These are applied to efficiently compute the shape gradient of functions of MHD equilibria with an adjoint approach. In this way, the shape derivative with respect to any perturbation applied to the plasma boundary or coil shapes can be computed with only one additional MHD equilibrium solution. We demonstrate that this approach is applicable for several figures of merit of interest for stellarator configuration optimization: the magnetic well, the magnetic ripple on axis, the departure from quasisymmetry, the effective ripple in the low-collisionality $1/\unicode[STIX]{x1D708}$ regime $(\unicode[STIX]{x1D716}_{\text{eff}}^{3/2})$ (Nemov et al. Phys. Plasmas, vol. 6 (12), 1999, pp. 4622–4632) and several finite-collisionality neoclassical quantities. Numerical verification of this method is demonstrated for the magnetic well figure of merit with the VMEC code (Hirshman & Whitson Phys. Fluids, vol. 26 (12), 1983, p. 3553) and for the magnetic ripple with modification of the ANIMEC code (Cooper et al. Comput. Phys. Commun., vol. 72 (1), 1992, pp. 1–13). Comparisons with the direct approach demonstrate that, in order to obtain agreement within several per cent, the adjoint approach provides a factor of $O(10^{3})$ in computational savings.

An ion cyclotron resonance heating (ICRH) antenna system must launch radio frequency (RF) power with a wavenumber spectrum which maximizes the coupling to the plasma. It should also ensure good absorption while minimizing the wave interaction with the plasma edge. Such interactions lead to impurity release, whose effect has been measured far from the antenna location (Klepper et al. 2013; Wukitch et al. 2017; Perkins et al. 2019) and can involve the entire scrape-off layer. In the normal heating scenario, for which the frequency of the waves launched by the antenna is larger than the ion cyclotron frequency of the majority ions $\omega > \omega _{\textrm {ci},\textrm {maj}}$, release of impurities due to ICRH can be affected by minimizing the low $|k_{\parallel }| < k_0$ power spectrum components of the antenna. Impurity release can be the result of low central absorption of the waves or power transfer from the fast to the slow wave due to the presence of a confluence in the plasma edge. In ASDEX Upgrade (AUG), a reduction of heavy impurity release by ICRH in the plasma was qualitatively well correlated to the parallel electric field and RF currents flowing around the antenna (Bobkov et al. 2017). In this article, we first show a correlation between the reduction in impurity release by ICRH in AUG and the rejection of the low $|k_{\parallel }| < k_0$ region of the antenna power spectrum. We show that the same correlation holds for results obtained in the Alcator C-Mod tokamak. Finally, using this idea, we reproduce ICRH induced impurity release behaviour in a not yet published experiments of JET, and make predictions for ITER and DEMO.

High-resolution spectra emitted by laboratory plasmas provide invaluable diagnostic tools for the measurement of plasma properties. To be implemented, they require a large amount of atomic data and transition rates, which are available in several spectral codes. In this paper we present a new feature added to the CHIANTI code, which allows us to calculate the Zeeman splitting of spectral lines in the presence of a magnetic field with known intensity and orientation. When combined with the CHIANTI database and software to calculate level populations and line emissivities, this new feature returns the emissivities in all four Stokes parameters, that can be utilized for the measurement of the magnetic field inside laboratory plasma chambers, along with other plasma parameters. This new feature can be applied to the analysis of the emission of laboratory plasmas created in different devices.

A magnetic field configuration of an island divertor for a quasi-axisymmetric stellarator (CFQS) is proposed. The configuration incorporates large islands surrounding the core confinement region. The interface between the core region and the peripheral region of the island divertor is a clear magnetic separatrix similar to a tokamak divertor. The structure of divertor magnetic field lines is very regular without stochasticity and the connection length is sufficiently long for good divertor performance. Such a divertor configuration is produced in the magnetic field configuration for the CFQS device, which is now under construction in China.

The modelling of edge carbon transport and emission on EAST tokamak under resonant magnetic perturbation (RMP) fields has been conducted with the three-dimensional edge transport code EMC3-EIRENE. The measured vertical distribution of CVI emission by the extreme ultraviolet spectrometer system for the perturbed case shows a reduction in the CVI emission by 20 % compared to the equilibrium case. The chord-integrated CVI emission can be reconstructed by EMC3-EIRENE modelling, which presents an increase in the CVI emission with RMP fields. The discrepancy between experiments and simulations has been investigated by parameter study to examine the sensitivity of the simulation results on the edge plasma conditions and the impurity perpendicular transport. It is found that the variation of edge plasma conditions for the equilibrium case cannot resolve the discrepancy in the CVI emission between simulations and measurements. The simulations with enhanced impurity perpendicular transport coefficient allows a reasonable agreement with the measured reduction of CVI emission.

Density functional theory (DFT) is a powerful theoretical tool widely used in such diverse fields as computational condensed-matter physics, atomic physics and quantum chemistry. DFT establishes that a system of $N$ interacting electrons can be described uniquely by its single-particle density $n(\boldsymbol{r})$, instead of the $N$-body wave function, yielding an enormous gain in terms of computational speed and memory storage space. Here, we use time-dependent DFT to show that a classical collisionless plasma can always, in principle, be described by a set of fluid equations for the single-particle density and current. The results of DFT guarantee that an exact closure relation, fully reproducing the Vlasov dynamics, necessarily exists, although it may be complicated (non-local in space and time, for instance) and difficult to obtain in practice. This goes against the common wisdom in plasma physics that the Vlasov and fluid descriptions are mutually incompatible, with the latter inevitably missing some ‘purely kinetic’ effects.

We rescale the generalized Ohm’s law and consider the limits that imply the electric field which moves with an ideal (inviscid and perfectly conducting) plasma be proportional to the time rate of change of the current density. Therefore, we show that those limits are satisfied by a sufficiently low electron number density, $n_{\text{e}}$. We also show that the electron–ion collision frequency, $\unicode[STIX]{x1D708}_{\text{ei}}$, is much smaller than the ion cyclotron frequency, $\unicode[STIX]{x1D714}_{\text{ci}}$. The combination of that condition with the Lawson criterion for a typical deuterium–tritium fusion in ITER reveals a lower bound for the geometric mean of the confinement time, $\unicode[STIX]{x1D70F}_{\text{C}}$, and collision interval, $\sqrt{\unicode[STIX]{x1D70F}_{\text{C}}/\unicode[STIX]{x1D708}_{\text{ei}}}\gg 10^{-5}~\text{s}$. For that reaction, we estimate that $n_{\text{e}}\sim 10^{19}~\text{m}^{-3}$, and contrast typical parameters of our fully ionized gas with those of warm, hot and thermonuclear plasmas. When, in addition, $n_{\text{e}}$ varies slowly in time and weakly in space, we generalize Alfvén’s theorem, by showing that the frozen-in condition holds true for an effective magnetic field, which depends on a finite electron skin depth. We perform a (divergenceless) helical perturbation on an axisymmetric equilibrium, to derive a dispersion relation in the cylindrical tokamak limit, and, subsequently, apply our analytical formulation to the peaked model, which assumes a logarithmic derivative profile for the poloidal component of the equilibrium magnetic field. In that formulation, the definition of the safety factor in terms of the effective field yields a shift in the magnetic surfaces. We find that the instability peak may triple that predicted on neglect of a finite electron mass. We also find that inertial effects may centuple the radius of the stable cylindrical column.

A novel derivation of the parallel ion velocity, and the bootstrap and Pfirsch–Schlüter currents in an imperfectly optimized (that is, almost omnigenous) stellarator magnetic field, $\boldsymbol{B}$, is presented that somewhat more generally recovers expressions completely consistent with previous analytic results. However, it is also shown that, when the conventional radially local form of the drift kinetic equation is employed, the flow velocity and the bootstrap current acquire a spurious contribution proportional to $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708}$, where $\unicode[STIX]{x1D714}$ denotes the $\boldsymbol{E}\times \boldsymbol{B}$ rotation frequency (due to the radial electric field $\boldsymbol{E}$) and $\unicode[STIX]{x1D708}$ the collision frequency. This contribution is particularly large in the $\sqrt{\unicode[STIX]{x1D708}}$ regime and at smaller collisionalities, where $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708}\gtrsim 1$, and is presumably present in most numerical calculations, but it disappears if a more accurate drift kinetic equation is used.

The evolution of a linearly polarized, long-wavelength Alfvén wave propagating in a collisionless magnetized plasma with a sheared parallel-directed velocity flow is here studied by means of two-dimensional hybrid Vlasov–Maxwell (HVM) simulations. The unperturbed sheared flow has been represented by an exact solution of the HVM set of equations of (Malara et al., Phys. Rev. E, vol. 97, 2018, 053212), thus avoiding spurious oscillations that would arise from the non-stationary initial state and inevitably affect the dynamics of the system. We have considered the evolution of both a small and a moderate amplitude Alfvén wave, in order to separate linear wave–shear flow couplings from kinetic effects, the latter being more relevant for larger wave amplitudes. The phase mixing generated by the shear flow modifies the initial perturbation, leading to the formation of small-scale transverse fluctuations at scales comparable with the proton inertial length/Larmor radius. By analysing both the polarization and group velocity of perturbations in the shear regions, we identify them as kinetic Alfvén waves (KAWs). In the moderate amplitude run, kinetic effects distort the proton distribution function in the shear region. This leads to the formation of a proton beam, at the Alfvén speed and parallel to the magnetic field. Such a feature, due to the parallel electric field associated with KAWs, positively compares with solar wind observations of suprathermal ion populations, suggesting that it may be related to the presence of ion-scale KAW-like fluctuations.