Starting from the assumption that saturation of plasma turbulence driven by temperature-gradient instabilities in fusion plasmas is achieved by a local energy cascade between a long-wavelength outer scale, where energy is injected into the fluctuations, and a small-wavelength dissipation scale, where fluctuation energy is thermalised by particle collisions, we formulate a detailed phenomenological theory for the influence of perpendicular flow shear on magnetised-plasma turbulence. Our theory introduces two distinct regimes, called the weak-shear and strong-shear regimes, each with its own set of scaling laws for the scale and amplitude of the fluctuations and for the level of turbulent heat transport. We discover that the ratio of the typical radial and poloidal wavenumbers of the fluctuations (i.e. their aspect ratio) at the outer scale plays a central role in determining the dependence of the turbulent transport on the imposed flow shear. Our theoretical predictions are found to be in excellent agreement with numerical simulations of two paradigmatic models of fusion-relevant plasma turbulence: (i) an electrostatic fluid model of slab electron-scale turbulence, and (ii) Cyclone-base-case gyrokinetic ion-scale turbulence. Additionally, our theory envisions a potential mechanism for the suppression of electron-scale turbulence by perpendicular ion-scale flows based on the role of the aforementioned aspect ratio of the electron-scale fluctuations.

The selection, design and optimization of a suitable blanket configuration for an advanced high-field stellarator concept is seen as a key feasibility issue and has been incorporated as a vital and necessary part of the Infinity Two fusion pilot plant physics basis. The focus of this work was to identify a baseline blanket which can be rapidly deployed for Infinity Two while also maintaining flexibility and opportunities for higher performing concepts later in development. Results from this analysis indicate that gas-cooled solid breeder designs such as the helium-cooled pebble bed (HCPB) are the most promising concepts, primarily motivated by the neutronics performance at applicable blanket build depths, and the relatively mature technology basis. The lithium lead (PbLi) family of concepts, particularly the dual-cooled lithium lead, offer a compelling alternative to solid blanket concepts as they have synergistic developmental pathways while simultaneously mitigating much of the technical risk of those designs. Homogenized three-dimensional neutronics analysis of the Infinity Two configuration indicates that the HCPB achieves an adequate tritium breeding ratio (TBR) (1.30 which enables sufficient margin at low engineering fidelity), and near appropriate shielding of the magnets (average fast fluence of 1.3 ${\times}$ 10$^{18}$ n cm$^{-2}$ per full-power year). The thermal analysis indicates that reasonably high thermal efficiencies (greater than 30 %) are readily achievable with the HCPB paired with a simple Rankine cycle using reheat. Finally, the tritium fuel cycle analysis for Infinity Two shows viability, with anticipated operational inventories of less than one kilogram (approximately 675 g) and a required TBR (TBR$_{\textrm {req}}$) of less than 1.05 to maintain fuel self-sufficiency (approximately 1.023 for a driver blanket with no inventory doubling). Although further optimization and engineering design are still required, at the physics basis stage all initial targets have been met for the Infinity Two configuration.

We present the first experimental observations of the dust acoustic wave where the wave was observed to propagate in the directions of gravity and magnetic field when these directions were not aligned. The experiments were conducted in the Magnetized Dusty Plasma eXperiment facility using a novel electrode system that allows for the angle between gravity and the magnetic field to be varied in a controlled way. This letter reports on measurements in an rf glow discharge argon plasma environment where the angle between direction of gravity and the magnetic field is 45$^{\circ }$. When there was no applied magnetic field, the wave was observed to propagate in the direction of gravity. However, as the magnetic field increased and the ions transitioned from flowing in the direction of gravity to the direction of the magnetic field, a second wave emerged and two distinct waves were observed to simultaneously propagate, one in the direction of gravity and one in the direction of the magnetic field. As the magnetic field was further increased, the wave that propagated in the direction of gravity was suppressed and the wave was only observed to propagate in the direction of the applied magnetic field. We also observe that the speed and the kinetic temperature of the dust for the mode that propagated in the direction of gravity decreased with increasing magnetic field while the speed and the kinetic temperature of the dust for the mode that propagated in the direction of the magnetic field increased with increasing magnetic field. These measurements suggest that an ion-dust streaming instability is at least partially responsible for the high temperatures that have previously been observed in dusty plasmas when the dust acoustic wave is present.

We develop a phenomenological model of strong imbalanced magnetohydrodynamic (MHD) turbulence that accounts for intermittency and the reflection of Alfvén waves by spatial variations in the Alfvén speed. Our model predicts the slopes of the inertial-range Elsasser power spectra, the scaling exponents of the higher-order Elsasser structure functions and the way in which the parallel (to the magnetic field) length scale of the fluctuations varies with the perpendicular length scale. These predictions agree reasonably well with measurements of solar-wind turbulence from the Parker Solar Probe (PSP). In contrast to previous models of intermittency in balanced MHD turbulence, we find that intermittency in reflection-driven MHD turbulence increases the parallel wavenumbers of the energetically dominant fluctuations at small perpendicular length scales. This, like the PSP measurements with which our model agrees, suggests that turbulence in the solar wind and solar corona may lead to more ion cyclotron heating than previously realized.

The magnetohydrodynamic (MHD) equilibrium and stability properties of the Infinity Two fusion pilot plant baseline plasma physics design are presented. The configuration is a four-field period, aspect ratio $A = 10$ quasi-isodynamic stellarator optimised for excellent confinement at elevated density and high magnetic field $B = 9\,T$. Magnetic surfaces exist in the plasma core in vacuum and retain good equilibrium surface integrity from vacuum to an operational $\beta = 1.6 \,\%$, the ratio of the volume average of the plasma and magnetic pressures, corresponding to $800\ \textrm{MW}$ deuterium–tritium fusion operation. Neoclassical calculations show that a self-consistent bootstrap current of the order of ${\sim} 1\ \textrm{kA}$ slightly increases the rotational transform profile by less than 0.001. The configuration has a magnetic well across its entire radius. From vacuum to the operating point, the configuration exhibits good ballooning stability characteristics, exhibits good Mercier stability across most of its minor radius and it is stable against global low-n MHD instabilities up to $\beta = 3.2\,\%$.

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

The near-axis description of optimised stellarators, at second order in the expansion, provides important information about the field, both of physical and practical importance for stellarator optimisation. It, however, remains relatively underdeveloped for an important class of such stellarators, called quasi-isodynamic (QI). In this paper we develop the theoretical and numerical framework, applying the second-order omnigeneity conditions derived in Rodríguez & Plunk (2023), to make explicit construction of equilibrium solutions. We find that the case of QI stellarators calls for the careful treatment of continuity, smoothness and periodicity of the various functions involved, especially for so-called half-helicity fields, which feature prominently in existing QI designs. The numerical implementation of necessary elements is described, and several examples are constructed and quantitatively verified in detail. This work establishes a basis for further systematic exploration of the space of QI stellarators, and the development of both theoretical and practical tools to facilitate effective optimisation of QI stellarators.

A quasi-linear reduced transport model is developed from a database of high-$\beta$ 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.

Isolated, undamped geodesic-acoustic-mode (GAM) packets have been demonstrated to obey a (focusing) nonlinear Schrödinger equation (NLSE) (E. Poli, Phys. Plasmas, 2021). This equation predicts susceptibility of GAM packets to the modulational instability (MI). The necessary conditions for this instability are analysed analytically and numerically using the NLSE model. The predictions of the NLSE are compared with gyrokinetic simulations performed with the global particle-in-cell code ORB5, where GAM packets are created from initial perturbations of the axisymmetric radial electric field $E_r$. An instability of the GAM packets with respect to modulations is observed both in cases in which an initial perturbation is imposed and when the instability develops spontaneously. However, significant differences in the dynamics of the small scales are discerned between the NLSE and gyrokinetic simulations. These discrepancies are mainly due to the radial dependence of the strength of the nonlinear term, which we do not retain in the solution of the NLSE, and to the damping of higher radial spectral components $k_r$. The damping of the high-$k_r$ components, which develop as a consequence of the nonlinearity, can be understood in terms of Landau damping. The influence of the ion Larmor radius $\rho _i$ as well as the perturbation wavevector $k_\text {pert}$ on this effect is studied. For the parameters considered here the aforementioned damping mechanism hinders the MI process significantly from developing to its full extent and is strong enough to stabilize some of the (according to the undamped NLSE model) unstable wavevectors.

The phenomenon of focusing of microwave beams in a plasma near a turning-point caustic is discussed by exploiting the analytical solution to the Gaussian beam-tracing equations in the two-dimensional (2-D) linear-layer problem. The location of maximum beam focusing and the beam width at that location are studied in terms of the beam initial conditions. This focusing must be taken into account to interpret Doppler backscattering (DBS) measurements. We find that the filter function that characterises the scattering intensity contribution along the beam path through the plasma is inversely proportional to the beam width, predicting enhanced scattering from the beam focusing region. We show that the DBS signal enhancement for decreasing incident angles between the beam path and the density gradient is due to beam focusing and not due to forward scattering, as was originally proposed by (Gusakov et al., (Plasma Phys. Contr. Fusion, vol. 56, 2014, p. 0250092014, 2017); Plasma Phys. Rep. vol. 43(6), 2017, pp. 605–613). The analytic beam model is used to predict the measurement of the $k_y$ density-fluctuation wavenumber power spectrum via DBS, showing that, in an NSTX-inspired example, the spectral exponent of the turbulent, intermediate-to-high $k_y$ density-fluctuation spectrum might be quantitatively measurable via DBS, but not the spectral peak corresponding to the driving scale of the turbulent cascade.

We present a study on the solitons in strongly coupled Yukawa fluids using a simple fluid model (SFE), supplemented by an appropriate equation of state for the medium. The formulation covers a broad range of coupling ($\Gamma$) and screening ($\kappa$) parameters, showing an agreement with the nonlinear quasilocalized charged approximation and generalized hydrodynamic models in the weak screening regime of the solitons in Yukawa media. The results also show a quantitative agreement with the experimentally measured values of the width and Mach number with the normalized amplitude. It has also been observed that the amplitude and width of the soliton in the weak screening limit increase with $\Gamma$ up to $\Gamma \sim 10$, beyond which they remain independent of $\Gamma$ values. Molecular dynamics simulations also confirm that the localization begins to emerge beyond $\Gamma \sim 10$, showing no significant effects on the characteristics of the solitons in Yukawa media. Therefore, the SFE model is capable of predicting the impact of the onset of the localization on the solitons in Yukawa media. Additionally, the amplitude of the soliton increases while its width decreases with $\kappa$ values. The SFE model also explores the possibility of forming refractive soliton structures, whose intensity increases with $\kappa$ values and decreases with $\Gamma$ values.

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 $\bar {\iota }$ between $0.12$ and $0.75$. The fast-particle confinement improves with $\bar {\iota }$ until $\bar {\iota } = 0.73$, at which point the degradation in quasi-symmetry outweighs the benefits of further increasing $\bar {\iota }$. The required coil–plasma distance only varies by about ${\pm }10\,\%$ for the configurations under consideration, and is between $2.8$ and $3.3\ \mathrm {m}$ when the configuration is scaled up to reactor size (minor radius $a=1.7\ \mathrm {m}$ and volume-averaged magnetic field strength of $5.86\ \mathrm {T}$). The maximum growth rate from linear gyrokinetic simulations increases with $\bar {\iota }$, but also shifts towards higher $k_y$ values. The maximum linear growth rate is sensitive to the choice of flux tube at rational $\bar {\iota }$, 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 $\bar {\iota }$. Sufficiently large positive shear is destabilizing. This is reflected in both linear growth rates and nonlinear heat fluxes.

We analyse the collisionless tearing mode instability of a current sheet with a strong shear flow across the layer. The growth rate decreases with increasing shear flow, and is completely stabilised as the shear flow becomes Alfvénic. We also show that, in the presence of strong flow shear, the tearing mode growth rate decreases with increasing background ion-to-electron temperature ratio, the opposite behaviour to the tearing mode without flow shear. We find that even a relatively small flow shear is enough to dramatically alter the scaling behaviour of the mode, because the growth rate is small compared with the shear flow across the ion scales (but large compared with shear flow across the electron scales). Our results may explain the relative absence of reconnection events in the near-Sun Alfvénic solar wind observed recently by NASA’s Parker Solar Probe.

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.

Progress in understanding multi-scale collisionless plasma phenomena requires employing tools which balance computational efficiency and physics fidelity. Collisionless fluid models are able to resolve spatio-temporal scales that are unfeasible with fully kinetic models. However, constructing such models requires truncating the infinite hierarchy of moment equations and supplying an appropriate closure to approximate the unresolved physics. Data-driven methods have recently begun to see increased application to this end, enabling a systematic approach to constructing closures. Here, we use sparse regression to search for heat flux closures for one-dimensional electrostatic plasma phenomena. We examine OSIRIS particle-in-cell simulation data of Landau-damped Langmuir waves and two-stream instabilities. Sparse regression consistently identifies six terms as physically relevant, together regularly accounting for more than 95 % of the variation in the heat flux. We further quantify the relative importance of these terms under various circumstances and examine their dependence on parameters such as thermal speed and growth/damping rate. The results are discussed in the context of previously known collisionless closures and linear collisionless theory.

The regulation of electron heat transport in high-$\beta$, weakly collisional, magnetized plasma is investigated. A temperature gradient oriented along a mean magnetic field can induce a kinetic heat-flux-driven whistler instability (HWI), which back-reacts on the transport by scattering electrons and impeding their flow. Previous analytical and numerical studies have shown that the heat flux for the saturated HWI scales as $\beta _e^{-1}$. These numerical studies, however, had limited scale separation and consequently large fluctuation amplitudes, which calls into question their relevance at astrophysical scales. To this end, we perform a series of particle-in-cell simulations of the HWI across a range of $\beta _e$ and temperature-gradient length scales under two different physical set-ups. The saturated heat flux in all of our simulations follows the expected $\beta _e^{-1}$ scaling, supporting the robustness of the result. We also use our simulation results to develop and implement several methods to construct an effective collision operator for whistler turbulence. The results point to an issue with the standard quasi-linear explanation of HWI saturation, which is analogous to the well-known $90^{\circ }$ scattering problem in the cosmic-ray community. Despite this limitation, the methods developed here can serve as a blueprint for future work seeking to characterize the effective collisionality caused by kinetic instabilities.

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.