Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By ‘topological information’, we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in $\mathbb{R}^{3}$ of arbitrary topology. We also discuss how using the classic Biot–Savart operator simplifies the expression for helicity and how domain topology affects the physical interpretation of helicity.

The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon–Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density $h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$, where $\boldsymbol{A}$ is the magnetic vector potential and $\boldsymbol{B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field $\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$ and the Godbillon–Vey helicity density $h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$ in general MHD flows in which either $h_{m}=0$ or $h_{m}\neq 0$. A conservation law for $h_{\text{gv}}$ occurs in flows for which $h_{m}=0$. For $h_{m}\neq 0$ the evolution equation for $h_{\text{gv}}$ contains a source term in which $h_{m}$ is coupled to $h_{\text{gv}}$ via the shear tensor of the background flow. The transport equation for $h_{\text{gv}}$ also depends on the electric field potential $\unicode[STIX]{x1D713}$, which is related to the gauge for $\boldsymbol{A}$, which takes its simplest form for the advected $\boldsymbol{A}$ gauge in which $\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$ where $\boldsymbol{u}$ is the fluid velocity. An application of the Godbillon–Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon–Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.

Stellarators are a promising route to steady-state fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating high-dimensional spaces, often necessitating the use of gradient-based methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring $O(N)$ flux computations, where $N$ is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finite-collisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ($O(10^{2})$), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjoint-based optimization of the field strength to obtain minimal bootstrap current on a surface. With adjoint-based derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Furthermore, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.

Analytic treatment is presented of the electrostatic instability of an initially planar electron hole in a plasma of effectively infinite particle magnetization. It is shown that there is an unstable mode consisting of a rigid shift of the hole in the trapping direction. Its low frequency is determined by the real part of the force balance between the Maxwell stress arising from the transverse wavenumber $k$ and the kinematic jetting from the hole’s acceleration. The very low growth rate arises from a delicate balance in the imaginary part of the force between the passing-particle jetting, which is destabilizing, and the resonant response of the trapped particles, which is stabilizing. Nearly universal scalings of the complex frequency and $k$ with hole depth are derived. Particle in cell simulations show that the slow-growing instabilities previously investigated as coupled hole–wave phenomena occur at the predicted frequency, but with growth rates 2 to 4 times greater than the analytic prediction. This higher rate may be caused by a reduced resonant stabilization because of numerical phase-space diffusion in the simulations.

Dipole and stellarator geometries are capable of confining plasmas of arbitrary neutrality, ranging from pure electron plasmas through to quasineutral. The diocotron mode is known to be important in non-neutral plasmas and has been widely studied. However, drift mode dynamics, dominating quasineutral plasmas, has received very little by way of attention in the non-neutral context. Here, we show that non-neutral plasmas can be unstable respect to both density-gradient- and temperature-gradient-driven instabilities. A local shearless slab limit is considered for simplicity. A key feature of non-neutral plasmas is the development of strong electric fields, in this local limit of straight field line geometry, the effect of the corresponding $\boldsymbol{E}\times \boldsymbol{B}$ drift is limited to the Doppler shift of the complex frequency $\unicode[STIX]{x1D714}\rightarrow \unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E}$. However, the breaking of the quasineutrality condition still leads to interesting dynamics in non-neutral plasmas. In this paper we address the behaviour of a number of gyrokinetic modes in electron–ion and electron–positron plasmas with arbitrary degree of neutrality.

Ion-temperature-gradient-driven (ITG) turbulence is compared for two quasi-symmetric (QS) stellarator configurations to determine the relationship between linear growth rates and nonlinear heat fluxes. We focus on the quasi-helically symmetric (QHS) stellarator HSX and the quasi-axisymmetric (QAS) stellarator NCSX. In normalized units, HSX exhibits higher growth rates than NCSX, while heat fluxes in gyro-Bohm units are lower in HSX. These results hold for simulations made with both adiabatic and kinetic electrons. The results show that HSX has a larger number of subdominant modes than NCSX and that eigenmodes are more spatially extended in HSX. We conclude that the consideration of nonlinear physics is necessary to accurately assess the heat flux due to ITG turbulence when comparing QS stellarator equilibria.

In present and future magnetic confined fusion devices with metallic plasma-facing components (PFCs) such as JET-ILW and ITER, the calculation of the plasma composition must account for multiple impurities of a wide range of mass and charge, resolve their poloidal asymmetries and account for different central peakings for various elements. Single measurements of radiation and effective charge are not enough to characterize this complex system and a self-consistent analysis of data from multiple diagnostics is required. This contribution describes a method to calculate the plasma composition simultaneously accounting for contributions of up to two low-Z impurities, and two mid-/high-Z impurities. The analysis stems from methodologies explained in Sertoli et al. (Rev. Sci. Instrum., vol. 89 (11), 2018, 113501), expanded to include more impurities and to coherently analyse multiple diagnostics within the same framework. The example Ne-seeded JET-ILW hybrid discharge reported here shows that Be, Ne, Ni and W are necessary to simultaneously explain the observed soft X-ray emission, the W concentration measured by passive vacuum ultra-violet spectroscopy, the line-of-sight integrated measurement of the effective charge, the observed poloidal asymmetry of the soft X-ray (SXR) emission, the Ne density measured by charge-exchange-recombination spectroscopy and the line-of-sight integrals of the total radiation as measured by bolometry. This consistent picture of the elemental composition enables the calculation of the radial profiles of the effective charge, the dilution and total radiation. For the cases analysed up to now, these are often very different from the typical assumptions presently used when modelling JET-ILW discharges. This will affect, among others, the calculation of neutron rates, current density profile and heat transport. These considerations are of course valid for all present and future magnetic-controlled fusion devices which exhibit multi-material plasma-facing components, including ITER.

Ion motion in a collisionless shock front is affected by macroscopic large-scale weakly varying and microscopic small-scale fast varying magnetic and electric fields. With the increase of the Mach number the role of the microscopic field is expected to become progressively more important. Using a combination of hybrid simulations and test particle analysis, we show that in moderately supercritical shocks macroscopic fields play the main role in ion motion across the shock. Pressure balance across the shock is only weakly broken and non-stationarity is related to the deviations from the total pressure from the constant value.

We present a Vlasov–DArwin numerical code (ViDA) specifically designed to address plasma physics problems, where small-scale high accuracy is requested even during the nonlinear regime to guarantee a clean description of the plasma dynamics at fine spatial scales. The algorithm provides a low-noise description of proton and electron kinetic dynamics, by splitting in time the multi-advection Vlasov equation in phase space. Maxwell equations for the electric and magnetic fields are reorganized according to the Darwin approximation to remove light waves. Several numerical tests show that ViDA successfully reproduces the propagation of linear and nonlinear waves and captures the physics of magnetic reconnection. We also discuss preliminary tests of the parallelization algorithm efficiency, performed at CINECA on the Marconi-KNL cluster. ViDA will allow the running of Eulerian simulations of a non-relativistic fully kinetic collisionless plasma and it is expected to provide relevant insights into important problems of plasma astrophysics such as, for instance, the development of the turbulent cascade at electron scales and the structure and dynamics of electron-scale magnetic reconnection, such as the electron diffusion region.

In hydrodynamic and MHD (magnetohydrodynamic) turbulence, formal expressions for the transfer rates rely on integrals over wavenumber triads $(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ satisfying $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$. As an example $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ denotes the kinetic energy transfer rate to the mode $\boldsymbol{k}$, from the two other modes in the triad, $\boldsymbol{p}$ and $\boldsymbol{q}$. However as noted by Kraichnan (Phys. Rev., vol. 111, 1958, pp. 1747–1747), in $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$, what fraction of the energy transferred to the mode $\boldsymbol{k}$ originated from $\boldsymbol{p}$ and which from $\boldsymbol{q}$ is unknown. Such an expression is thus incongruent with the customary description of turbulence in terms of two-scale energy exchange. Notwithstanding this issue, Dar et al. (Physica D, vol. 157 (3), 2001, pp. 207–225) further decomposed these transfers into separate contributions from $\boldsymbol{p}$-to-$\boldsymbol{k}$ and $\boldsymbol{q}$-to-$\boldsymbol{k}$, thus introducing the concept of mode-to-mode transfers that they applied to MHD turbulence. Doing so, they had to set aside additional transfers circulating within each triad, but failed to calculate them. In the present paper we explain how to derive the complete expressions of the mode-to-mode transfers, including the circulating transfers. We do it for kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic energy, magnetic energy and magnetic helicity in MHD turbulence. We find that the degree of non-uniqueness of the energy transfers derived from the induction equation is a priori higher than the one derived from the Navier–Stokes equations. However, separating the contribution of magnetic advection from magnetic stretching, the energy mode-to-mode transfer rates involving the magnetic field become uniquely defined, in striking contrast to the hydrodynamic case. The magnetic helicity mode-to-mode transfer rate is also found to be uniquely defined, contrary to kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates have the same properties as mode-to-mode transfer rates. Finally calculating the fluxes, we show how the circulating transfers cancel in accordance with conservation laws.

Stellarator configurations with reactor relevant energetic particle losses are constructed by simultaneously optimizing for quasisymmetry and an analytically derived metric ($\unicode[STIX]{x1D6E4}_{c}$), which attempts to align contours of the second adiabatic invariant, $J_{\Vert }$ with magnetic surfaces. Results show that with this optimization scheme it is possible to generate quasihelically symmetric equilibria on the scale of ARIES-CS which completely eliminate all collisionless alpha particle losses within normalized radius $r/a=0.3$. We show that the best performance is obtained by reducing losses at the trapped–passing boundary. Energetic particle transport can be improved even when neoclassical transport, as calculated using the metric $\unicode[STIX]{x1D716}_{\text{eff}}$, is degraded. Several quasihelically symmetric equilibria with different aspect ratios are presented, all with excellent energetic particle confinement.

A spatio-temporal measurement showing the volumetric nature of the phase synchronization of a naturally occurring dust acoustic wave to an external modulation and the relaxation from the driven wave mode back to the naturally occurring wave mode (phase desynchronization) is presented. It is shown that the phase synchronization and desynchronization occur behind a propagating synchronization/desynchronization front that travels at a slower speed than the phase velocity of the wave and that the speed of this synchronization/desynchronization front decreases with increasing neutral gas pressure. It is also observed that volume of the wave that is synchronous depends on the frequency of the external modulation and the neutral gas pressure.

Addition of multispecies impurity ions to the total-f gyrokinetic particle-in-cell code XGCa is reported, including a cross-verification of neoclassical physics against the NEO code. This new version of the neoclassical gyrokinetic code XGCa is used to benchmark and confirm the previous reduced-equation-based prediction that high-$Z$ impurity particles in the Pfirsch–Schlüter regime can exhibit a significant level of up–down poloidal asymmetry, through the large parallel friction force, and thus influence the radial plasma transport significantly. The study is performed in a plasma with weak toroidal rotation. In comparison, when the impurity particles are in the plateau regime, the up–down poloidal asymmetry becomes weak, with the parallel friction force becoming weaker than the parallel viscous force. It is also found that the linearization of the perturbed distribution function, based on the small poloidal asymmetry assumption, can become invalid. Using the numerical data from XGCa, each term in the parallel fluid force-balance equation have been analysed to find that both the main ions and the electrons respond to the poloidal potential variation adiabatically when the high-$Z$ tungsten possesses large poloidal variation.