In magnetic confinement fusion devices close to axisymmetry, such as tokamaks, a key element is the winding profile of the magnetic field lines, or its inverse, the safety profile $q=q_{\boldsymbol {B}}$. A corresponding profile, $q_{\boldsymbol {J}}$, can be defined for the current density field lines. Ampère's law relates any mode of current perturbation $\delta \boldsymbol {J}_{m,n}$ with a mode of magnetic perturbation $\delta \boldsymbol {B}_{m,n}$. It is shown that the knowledge of the pair $(q_{\boldsymbol {B}},q_{\boldsymbol {J}})$ allows us then to characterize the resonant, or non-resonant, nature of the modes for both the magnetic and current density field lines. The expression of $q_{\boldsymbol {J}}$ in the flux coordinate is derived. Including this calculation in real-time Grad–Shafranov equilibrium reconstruction codes would yield a comprehensive view of the magnetics. The monitoring of the pair $(q_{\boldsymbol {B}},q_{\boldsymbol {J}})$ would then allow us to investigate the role played by the resonant modes for the current density, that are current filamentary modes, in the plasma small-scale turbulence. By driving the magnetic and current density profiles apart so that the images of $q_{\boldsymbol {B}}$ and $q_{\boldsymbol {J}}$ are disjoint, these filamentary modes would not impact the magnetic field topology, being not associated with magnetic islands but with non-resonant magnetic modes. It remains to be explored to what extent such a configuration, where the spectrum of tiny current density filaments produces a spectrum of magnetic modes that has practically no effect on heat transport, is beneficial.

Hot electrons produced by parametric instabilities can negatively affect the thermonuclear gain of a direct-drive inertial confinement fusion capsule. A Monte Carlo transport is coupled to the hydrodynamics MULTI-IFE code in order to study the hot electron transport in a high-gain target implosion. The thermonuclear energy produced by the directly driven implosion of a reference target drops from 27 MJ without hot electrons to 13.5 kJ when the hot electrons are taken into account. A clear relationship is established between the level of hot electrons produced and the degradation of the thermonuclear yield produced by the target. It is shown that adding a relatively short laser spike (shock ignition) to the main pulse restores the thermonuclear energy released by the implosion. Different shock-ignition windows are presented depending on the duration of the spike. It appears that the longer the duration of the spike, the lower the power required for shock ignition. Finally, a 300 ps-duration spike requires only 50 TW to restore a thermonuclear energy output of 20 MJ.

In this work we consider the problem of optimizing a stellarator subject to hard constraints on the design variables and physics properties of the equilibrium. We survey current numerical methods for handling these constraints, and summarize a number of methods from the wider optimization community that have not been used extensively for stellarator optimization thus far. We demonstrate the utility of new methods of constrained optimization by optimizing a quasi-axisymmetric stellarator for favourable physics properties while preventing strong shaping of the plasma boundary, which can be difficult to create with external current sources.

We have derived a necessary condition for the achievement of breakeven in axisymmetric plasmas with zero toroidal field and confined by a dipole magnetic field (B. Lehnert, Nature, vol. 181, 1958, p. 4605; A. Hasegawa, Comments Plasma Phys. Control. Fusion, vol. 11, no. 3, 1987). Excellent MHD stability, high values of $\beta$ (up to $26$ %) and good confinement properties awaken the interest of private investors after years of neglect due to lack of public funding and competing alternative lines of research like the tokamak. Starting from a requirement of self-consistency between the balances of momentum and energy in a dipole-confined, two-species plasma and assuming a Maxwellian distribution function for ions and electrons, we derive a necessary condition for breakeven. This condition is more stringent than the Lawson criterion because of the lack of a stabilizing toroidal field. For a given current flowing across the toroidal coil internal to the plasma, the crucial factor at stake is the ratio between the radius of the main toroidal coil and the radius of the vacuum chamber.

Microinstabilities drive turbulent fluctuations in inhomogeneous, magnetised plasmas. In the context of magnetic confinement fusion devices, this leads to an enhanced transport of particles, momentum and energy, thereby degrading confinement. In this work, we describe an application of the adjoint method to efficiently determine variations of gyrokinetic linear growth rates on a general set of external parameters in the local $\delta f$-gyrokinetic model. We then offer numerical verification of this approach. When coupled with gradient-based techniques, this methodology can facilitate the optimisation process for the microstability of the confined plasmas across a high-dimensional parameter space. We present a numerical demonstration wherein the ion-temperature-gradient instability growth rate in a tokamak plasma is minimised with respect to flux surface shaping parameters. The adjoint method approach demonstrates a significant computational speed-up compared with a finite-difference gradient calculation.

A systematic theory of the asymptotic expansion of the magnetohydrostatics (MHS) equilibrium in the distance from the magnetic axis is developed to include arbitrary smooth currents near the magnetic axis. Compared with the vacuum and the force-free system, an additional magnetic differential equation must be solved to obtain the pressure-driven currents. It is shown that there exist variables in which the rest of the MHS system closely mimics the vacuum system. Thus, a unified treatment of MHS fields is possible. The mathematical structure of the near-axis expansions to arbitrary order is examined carefully to show that the double-periodicity of physical quantities in a toroidal domain can be satisfied order by order. The essential role played by the leading-order Birkhoff–Gustavson normal form in solving the magnetic differential equations is highlighted. Several explicit examples of vacuum, force-free and MHS equilibrium in different geometries are presented.

The paper presents experimental results from the SMOLA device, which was built in the Budker Institute of Nuclear Physics for the verification of the helical mirror confinement idea. This concept involves active control of axial losses from the confinement zone in an open magnetic trap through the use of multiple mirrors that move in the plasma frame of reference. The discussed experiments focused on determining the cumulative effect of a helical mirror system in combination with a short segment of a stronger magnetic field. Combination of these two methods of axial flow suppression results in higher efficiency compared with each method individually. Different combinations of the mirrors were tested. The most effective flow suppression was observed if the short mirror was placed between the confinement region and the helical mirror. In this configuration, an effective mirror ratio of $R_{{\rm eff}} = 32.6\pm 7.8$ was achieved, along with a more than three-fold increase in plasma density within the confinement region. The possibility of a cumulative effect of different types of magnetic mirrors offers a way to improve the confinement performance of the reactor-grade mirror confinement devices.

A unique field-reversed configuration (FRC) experiment is presently being assembled at the Plasma Technology Research Institute, KFE. It is a compact small-scale FRC device, which uses a set of radio frequency (RF) antennas to produce an internal E × B that drives the electrons for current-drive, in which E is the electric field and B is the magnetic field. This is very similar to the rotating magnetic field (RMF) current-drive, where the horizontal and vertical antennas are driven 90° out of phase. For this device, the RF antennas are arranged differently than the RMF. The RF antennas, being two separate sets, are positioned inside the vacuum chamber. Each set consists of 8 coils, for a total of 16 coils, where 80~100 kHz sine and cosine waveform currents are applied. One set of coils generates a radial B-field, while the other set provides an E-field in the z-direction. As the phase changes, the E and B fields are switched by these two sets. Nevertheless, E × B propagates in the same θ-direction so that this allows the electrons to rotate around the circumference of the device. The FRC device will test wave heating by launching 2.45 GHz microwaves. Also, passive stabilizers are positioned at each end to provide extra stability while preventing tilt instability. The experiment is expected to produce its first plasma in 2025.

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a ‘global-to-local’ approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighbourhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform and quality of quasi-axisymmetry. The database of stellarators, which comprises approximately 200 000 coil sets, is available online and is called QUASR, for ‘quasi-symmetric stellarator repository’.

In a recent publication (Toler et al., J. Plasma Phys., vol. 89, issue 2, 2023, p. 905890210), we demonstrated that for axisymmetric geometries, the Kapur–Rokhlin quadrature rule provided an efficient and high-order accurate method for computing the normal component, on the plasma surface, of the magnetic field due to the toroidal current flowing in the plasma, via the virtual-casing principle. The calculation was indirect, as it required the prior computation of the magnetic vector potential from the virtual-casing principle, followed by the computation of its tangential derivative by Fourier differentiation, to obtain the normal component of the magnetic field. Our approach did not provide the other components of the virtual-casing magnetic field. In this letter, we show that a more direct and more general approach is available for the computation of the virtual-casing magnetic field. The Kapur–Rokhlin quadrature rule accurately calculates the principal value integrals in the expression for all the components of the magnetic field on the plasma boundary, and the numerical error converges at a rate nearly as high as the indirect method we presented previously.

This paper is devoted to experimental studies of plasma confinement with high relative pressure ($\beta$) in the Gas Dynamic Trap (BINP, Novosibirsk). In previous high-$\beta$ confinement studies a maximum local $\beta = 0.6$ was achieved in the fast-ion turning point, contributed to by a beam-driven population of fast ions with an anisotropic distribution function. In this study the axial magnetic field profile was modified to bring the turning points closer to one another, which effectively increased the energy density of plasma and pushed the $\beta$ value higher. Experiments were performed for two non-standard magnetic configurations, where the axial fast-ion confinement region length was reduced by 1.5 and 2 times compared with the standard configuration. The average values of $\langle \beta _{\perp } \rangle$ over the plasma central cross-section were found to be 0.1 and 0.18, respectively, for the two configurations, with the latter value significantly exceeding the $\langle \beta _{\perp } \rangle =0.08$ of the standard configuration, in which the previous record was set. Moreover, halving the fast ion confinement region almost doubled the D–D fusion proton flux from the trap centre compared with the standard configuration. The electron temperature in both new magnetic configurations was only slightly smaller than in the standard configuration. In addition, an effect of Alfvén ion–cyclotron instability (AICI) development on the pressure in the turning points is discussed. Presumably, with some decrease in magnetic field an evolving AICI does not result in considerable pressure axial redistribution, so the pressure maximum is in the turning points’ vicinity despite the instability.

The centrifugal mirror confinement scheme incorporates supersonic rotation of a plasma into a magnetic mirror device. This concept has been shown experimentally to drastically decrease parallel losses and increase plasma stability as compared with prior axisymmetric mirrors. MCTrans++ is a dimensionless (0-D) scoping tool which rapidly models experimental operating points in the Centrifugal Mirror Fusion Experiment (CMFX) at the University of Maryland. In the low-collisionality regime, parallel losses can be modelled analytically. A confining potential is set up that is partially ambipolar and partially centrifugal. Due to the stabilizing effects of flow shear, the perpendicular losses can be modelled as classical. Radiation losses such as bremsstrahlung and cyclotron emission are taken into account. A neutrals model is included, and, in some circumstances, charge-exchange losses are found to exceed all other loss mechanisms. We use the SUNDIALS ARKODE library to solve the underlying equations of this model; the resulting software is suitable for scanning large parameter spaces, and can also be used to model time-dependent phenomena such as a capacitive discharge. MCTrans++ has been used to verify results from prior centrifugal mirrors, create an experimental plan for CMFX and find configurations for future reactor-scale fusion devices.

We present the results of numerical simulation by two-dimensional hybrid particle-in-cell code of high-beta plasma with hot ions in an axisymmetric mirror machine. Two particular effects are discussed: the self-rotating of plasma with Maxwellian ions in regime of diamagnetic confinement and the excitation of axisymmetric magnetosonic waves in a high-beta plasma with sloshing ions.

We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC. Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.

Quasisymmetry (QS), a hidden symmetry of the magnetic field strength, is known to support nested flux surfaces and provide superior particle confinement in stellarators. In this work, we study the ideal magnetohydrodynamic (MHD) equilibrium and stability of high-beta plasma in a large-aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies the problem of three-dimensional quasisymmetric MHD equilibria, which can be reduced to a standard elliptic Grad–Shafranov equation for the flux function. We show that any large-aspect-ratio tokamak, deformed periodically in the vertical direction, is a stellarator with approximate volumetric QS. We discuss exact analytical solutions and numerical benchmarks. Finally, we discuss the ideal ballooning and interchange stability of some of our equilibrium configurations.

We solve the linearized gyrokinetic equation, quasineutrality condition and Ampere's law to obtain the dispersion relation of microtearing modes (MTMs) in collisionless low-$\beta$ toroidal plasmas. Consistent with past studies, we find that MTMs are driven unstable by the electron temperature gradient and that this instability drive is mediated by magnetic drifts. The dispersion relation that we derive can be evaluated numerically very quickly and may prove useful for devising strategies to mitigate MTM instability in fusion devices.

The gyrokinetic theory of the residual flow, in the electrostatic limit, is revisited, with optimized stellarators in mind. We consider general initial conditions for the problem, and identify cases that lead to a non-zonal residual electrostatic potential, i.e. one having a significant component that varies within a flux surface. We investigate the behaviour of the ‘intermediate residual’ in stellarators, a measure of the flow that remains after geodesic acoustic modes have damped away, but before the action of the slower damping that is caused by unconfined particle orbits. The case of a quasi-isodynamic stellarator is identified as having a particularly large such residual, owing to the small orbit width achieved by optimization.

Collisionless charged particle motion and its transport in the two-wire model (TWM) with no axial magnetic fields is investigated numerically. The TWM configuration contains a magnetic X-point, and single particle motions in such a field have two conserved quantities: the total kinetic energy and the base field line value which is a quantity derived from the axial canonical momentum. As gyrating particles travel along the field lines, they may reach near the X-point region where the magnetic moment, the first adiabatic invariant, can be occasionally shifted due to a large gradient of the field. When the magnetic moment becomes large, resulting in a large Larmor radius, particles probabilistically cross the X-point to migrate to the opposite side of the TWM configuration. These phenomena are investigated with single particle simulations. We find that the statistical behaviour of the seemingly chaotic magnetic moment shifts are completely determined by the two aforementioned conserved quantities, and also that there exists a threshold energy, determined by the base field line value, allowing only particles with a higher energy to cross the separatrix and migrate. It is found that the crossing time is distributed exponentially, and that the migration confinement time, which is the average crossing time, is shorter for particles with a base field line closer to the separatrix and a higher energy. We provide an empirical expression, derived with the simulations, for estimating the collisionless migration confinement time.

The paper presents an overview of experimental results of an investigation of different energy loss channels in the gas dynamic trap (GDT), which is a magnetic mirror plasma confinement device in the Budker Institute of Nuclear Physics. Energy losses along magnetic field lines are considered as well as losses onto radial limiters, which restrict the plasma column radius and provide its magnetohydrodynamic stability via the ‘vortex confinement’ mechanism. The losses along the field lines were measured using a set of pyroelectric bolometers on the plasma absorber and the losses onto the limiters were determined with thermistors from their temperature rise. Additionally, the losses due to charge exchange of fast plasma ions on the residual neutral gas in the GDT were measured using a longitudinal array of pyroelectric bolometers mounted on the wall of the central cell. An attempt was made to draw up the energy balance in the GDT in order to identify the predominant loss channels and reduce those losses in the future.

Omnigenity is a desirable property of toroidal magnetic fields that ensures confinement of trapped particles. Confining charged particles is a basic requirement for any fusion power plant design, but it can be difficult to satisfy with the non-axisymmetric magnetic fields used by the stellarator approach. Every ideal magnetohydrodynamic equilibrium previously found to approximate omnigenity has been either axisymmetric, quasi-symmetric or has poloidally closed contours of magnetic field strength $B$. However, general omnigenous equilibria are a much larger design space than these subsets. A new model is presented and employed in the DESC stellarator optimization suite to represent and discover the full parameter space of omnigenous equilibria. Although exact omnigenity aside from quasi-symmetry is impossible, these results reveal that excellent particle confinement can be achieved in practice. Examples far from quasi-symmetry with poloidally, helically and toroidally closed $B$ contours are attained with DESC and shown to have low neoclassical collisional transport and fast particle losses.