Table of contents

The 2014 joint Varenna-Lausanne international workshop on the theory of fusion plasmas was once more a great meeting. The programme covers a wide variety of topics, namely turbulence, MHD, edge physics and RF wave heating. The broad spectrum of skills involved in this meeting, from fundamental to applied physics, is striking. The works published in this special issue combine mathematics, numerics and physics at various levels - confirming the increasing integration of expertise in our community.

As an incentive to read this cluster, let us mention a few outstanding results. Several papers address fundamental issues in turbulent transport, in particular the dynamics of structures. It is quite remarkable that this subject is now mature enough to propose signatures that can be tested by measurements. Linear and non linear MHD was also at the forefront. Several works illustrate the increasing level of realistic description of a fusion device, in particular by implementing complicated wall geometries. Moreover some noticeable progress has been made in the understanding of reconnection processes in collisionless regimes. The activity on radio-frequency heating and current drive is well represented, driven by the future operation of W7-X, ITER, and DEMO on a longer time scale. Finally the development of innovative numerical techniques, an old tradition of the conference, has driven several nice articles.

The programme committee is traditionally keen in promoting young scientists. A number of senior scientists also attend the meeting on a regular basis, so that the attendance was nicely balanced. We believe that these efforts have been particularly fruitful this year. The number of young (and less young) faces was particularly impressive and this special issue illustrates this feature.

The success of the 2014 edition brings evidence that the joint Varenna-Lausanne is the right place for presenting th

The quality and size of the scientific production is illustrated by the 22 papers which appear in the present volume of Journal of Physics Conference Series - all peer reviewed. Let us mention another set of 19 papers to appear in Plasma Physics and Controlled Fusion. We hope the reader will enjoy this special issue and will find ideas for new bright achievements.

Xavier Garbet, Olivier Sauter

October 23, 2014

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

Electromagnetic waves in the ion-cyclotron (IC) range of frequencies are presently investigated as possible current drive (CD) systems in fusion reactors. Among many physical and technical issues, an accurate description of radio-frequency (RF) power absorption by fusion- born alpha particles is of special importance, since RF heating of these particles is not only detrimental for the CD efficiency, but might worsen the operative conditions by increasing their prompt losses.

The capability of the full-wave TORIC code has been recently augmented to account for RF absorption by fusion-born alpha particles, calculated to all-orders in finite Larmor radius and with a realistic distribution function. Here, we present simulation with TORIC addressing the sensitivity of current drive efficiency on the design of a future reactor, in particular density and temperature profiles, magnetic field intensity, and plasma dimensions. For this purpose, we have investigated possible frequency windows for CD for two proposed versions of the DEMO reactor, namely its pulsed and its more ambitious steady-state design. The important role of the antenna for a realistic estimate of the CD efficiency is pointed out.

First pass absorption of the Lower Hybrid waves in thermonuclear devices like ITER and DEMO is modelled by coupling the ray tracing equations for the wave phase and amplitude with the quasi-linear evolution of the electron distribution function. A system of coupled ordinary differential equations for each Fourier component of the spectrum radiated by the LH antenna is derived and solved when considering both 1D/2D Fokker-Planck model for the electron distribution function. This allows to reconstruct and to evolve the quasi-linear diffusion coefficient consistently with the wave propagation, to calculate the power deposition profile and the amount of current driven by the wave. As usually assumed, the Lower Hybrid Current Drive is not effective in a plasma of a tokamak fusion reactor like ITER or DEMO, because the high electron temperature would enhance the wave absorption and then limit the RF power deposition to the very periphery of the plasma column (near the separatrix). In this work by extensively using this self-consistent modelling for the propagation and absorption of the LH wave, a parametric study on the wave spectrum (and consequently on the antenna design) as spectrum width, peak value, secondary lobes etc. has been performed very accurately. Such a careful investigation aims at controlling the power deposition layer possibly in the external half radius of the plasma, thus providing a valuable aid to the solution of how to control the plasma current profile in a toroidal magnetic configuration. This analysis is useful not only for exploring the possibility of profile control of a pulsed operation reactor, but also in order to reconsider the feasibility of steady state regime.

A minimum model of plasma turbulence in a kinetic framework is presented. It is based on trapped ion turbulence, gyro and bounce averaged, and implemented in the versatile and efficient code TERESA. Zonal flow - streamer interplay are readily shown to be key players that govern the confinement properties of the model. The parameter space of the model is explored with brute force numerics. A generic result is either a streamer dominated pattern with large transport, or a staircase temperature profile with very marked corrugations and quenched transport. A case with off-axis heating is found to exhibit quasiperiodic relaxation events relevant to investigate dynamical turbulence self-organisation.

The formation of plasmoid chains is explored for the first time within the context of the Taylor problem, in which magnetic reconnection is driven by a small amplitude boundary perturbation in a tearing-stable slab plasma equilibrium. Numerical simulations of a magnetohydrodynamical model of the plasma show that for very small plasma resistivity and viscosity, the linear inertial phase is followed by a nonlinear Sweet-Parker evolution, which gives way to a faster reconnection regime characterized by a chain of plasmoids instead of a slower Rutherford phase.

In this work we present a new Eulerian kinetic 1D3V code devoted to the study of plasma-wall interactions and sheath structure. The code solves the Vlasov-Poisson system for two or more kinetic species on a one-dimensional spatial grid between two limiting plates, in the presence of a uniform magnetic field tilted with respect to the normal to the plates. A source-sink term in the Vlasov equation leads to a stationary solution comprising a magnetic and a collisional presheath, as well as the usual Debye sheath in front of the wall. Thanks to a nonuniform spatial grid the sheath structure can be resolved accurately. Several advection schemes are implemented and the code is parallelized. Here, the code is used to illustrate the problems of a stationary sheath-presheath structure in front of a negatively biased wall, with and without a tilted magnetic field.

W7X stellarator 3D equilibrium has been computed with the equilibrium code ANIMEC (Anisotropic Neumann Inverse Moments Equilibrium Code). This equilibrium was used to model ICRH minority heating in ⁴He(H) plasma with the 3D full-wave code LEMan (Low frequency ElectroMagnetic wave propagation). The coupled power spatial distribution is shown for different resonance positions within the range of frequencies foreseen for the ICRH antenna. It is found that for the high mirror equilibrium examined, the antenna frequency can be chosen to optimise the power deposition in the plasma core while limiting the absorption at the edge.

The Ohm's law is modified when turbulent processes are accounted for. Besides an hyper-resistivity, already well known, pinch terms appear in the electron momentum flux. Moreover it appears that turbulence is responsible for a source term in the Ohm's law, called here turbulent current drive. Two terms contribute to this source. The first term is a residual stress in the momentum flux, while the second contribution is an electro-motive force. A non zero average parallel wave number is needed to get a finite source term. Hence a symmetry breaking mechanism must be invoked, as for ion momentum transport. E × B shear flows and turbulence intensity gradients are shown to provide similar contributions. Moreover this source term has to compete with the collision friction term (resistivity). The effect is found to be significant for a large scale turbulence in spite of an unfavorable scaling with the ratio of the electron to ion mass. Turbulent current drive appears to be a weak effect in the plasma core, but could be substantial in the plasma edge where it may produce up to 10 % of the local current density.

The empirical scaling law used to support the projected ITER performance [1, 2] in analysed to determine the control parameters that must be considered in gyrokinetic simulations. The results of such an analysis are contradictory and do not appear to fit with present evidence from gyrokinetic simulations. Analysing the dependence of the correlation length on the ρ_* parameter, we show that local values of the correlation length are governed by the shearing effect of the corrugation patterns, but that coarse graining this value, in time, radially or poloidally does not allow one to recover this match. A comparison to the scaling law must then rely on the scaling properties of the Probability Density Function of the correlation length that is found to exhibit heavy tails with algebraic decay.

In the scrape-off layer (SOL) of tokamaks, the flow acceleration due to the presence of limiter or divertor plates rises the plasma velocity in a sonic regime. These high velocities imply the presence of a strong shear between the SOL and the core of the plasma that can possibly trigger some parallel shear flow instability. The existence of these instabilities, denoted as parallel Kelvin-Helmholtz instability in some works [1, 2] have been investigated theoretically in [3] using a minimal model of electrostatic turbulence composed of a mass density and parallel velocity equations. This work showed that the edge plasma around limiters might indeed be unstable to this type of parallel shear flow instabilities. In this work, we perform 3D simulations of the same simple mathematical model to validate an original finite volume numerical method aimed to the numerical study of edge plasma. This method combines the use of triangular unstructured meshes in the poloidal section and structured meshes in the toroidal direction and is particularly suited to the representation of the real complex geometry of the vacuum chamber of a tokamak.

The numerical results confirm that in agreement with the theoretical expectations as well as with other numerical methods, the sheared flows in the SOL are subject to parallel Kelvin-Helmholtz instabilities. However, the growth rate of these instabilities is low and these computations require both a sufficient spatial resolution and a long simulation time. This makes the simulation of parallel Kelvin-Helmholtz instabilities a demanding benchmark.

A method for calculating the wave field for spatial dispersive media is proposed suitable for FEM. The method is based on operator splitting by separating the induced current and wave field calculations, and solving the system by means of iterations. In order to take into account several coexisting waves with different poloidal mode numbers when calculating the induced current the wave field is decomposed into wavelets, for which the current is calculated assuming the plasma to be weakly non-uniform.

The dynamics of large scale plasma instabilities can be strongly influenced by the mutual interaction with currents flowing in conducting vessel structures. Especially eddy currents caused by time-varying magnetic perturbations and halo currents flowing directly from the plasma into the walls are important.

The relevance of a resistive wall model is directly evident for Resistive Wall Modes (RWMs) or Vertical Displacement Events (VDEs). However, also the linear and non-linear properties of most other large-scale instabilities may be influenced significantly by the interaction with currents in conducting structures near the plasma. The understanding of halo currents arising during disruptions and VDEs, which are a serious concern for ITER as they may lead to strong asymmetric forces on vessel structures, could also benefit strongly from these non-linear modeling capabilities.

Modeling the plasma dynamics and its interaction with wall currents requires solving the magneto-hydrodynamic (MHD) equations in realistic toroidal X-point geometry consistently coupled with a model for the vacuum region and the resistive conducting structures. With this in mind, the non-linear finite element MHD code JOREK [1, 2] has been coupled [3] with the resistive wall code STARWALL [4], which allows us to include the effects of eddy currents in 3D conducting structures in non-linear MHD simulations.

This article summarizes the capabilities of the coupled JOREK-STARWALL system and presents benchmark results as well as first applications to non-linear simulations of RWMs, VDEs, disruptions triggered by massive gas injection, and Quiescent H-Mode. As an outlook, the perspectives for extending the model to halo currents are described.

The presence of impurity species in magnetic confinement fusion devices leads to radiation losses and plasma dilution. Thus it is important to analyze impurity dynamics, and search for means to control them. In stellarator plasmas the neoclassical ambipolar radial electric field often points radially inwards (referred to as the ion root regime), causing impurities to accumulate in the core. This can limit the performance of nonaxisymmetric devices.

In the present work we analyze neoclassical impurity transport in stellarator plasmas using a recently developed continuum drift-kinetic solver, the SFINCS code (the Stellarator Fokker- Planck Iterative Neoclassical Conservative Solver). The study is performed for a case close to the edge of W7-X using the standard configuration magnetic geometry. We investigate the sensitivity of impurity transport to impurity charge, main species density and temperature gradients, as well as ion temperature.

At the studied radial location we find that the neoclassical impurity peaking factor can be very large, particularly for high-Z impurities. The ambipolar radial electric field is in the ion root regime, and impurity accumulation can thus be expected. The accumulation is strengthened by the large main species density and temperature gradients. Moreover we find that the size of the bootstrap current is affected by the value of the plasma effective charge, suggesting that employing a realistic ion composition can be important when calculating the bootstrap current.

When modeling plasma turbulence, two different means of driving the system out of equilibrium are considered. On one hand, flux driven (FD) approaches are based on the idea that no scale separation can be assumed in a turbulent system. On the other hand, gradient driven (GD) approaches rely on the idea that the back-reaction of fluctuations on the mean profiles is not a critical ingredient for turbulence self-organization and saturation. We find that FD and GD systems strongly differ in regimes close to marginal stability. The characteristic non linear upshift is recovered in GD simulations but no comparable behavior is possible in FD case. Discrepancy between the various analysis in terms of diffusion and pinch velocities and between the models is also discussed.

A fully kinetic grid-free model based on a Barnes-Hut tree code is used to selfconsistently simulate a collisionless plasma bounded by two floating walls. The workhorse for simulating such plasma wall transition layers is currently the PIC method. However, the present grid-free formulation provides a powerful independent tool to test it [1] and to possibly extend particle simulations towards collisional regimes in a more internally consistent way. Here, we use the grid-free massively parallel Barnes-Hut tree-code PEPC - a well established tool for simulations of Laser-plasmas and astrophysical applications - to develop a 3D ab initio plasma target interaction model. With our approach an electrostatic sheath naturally builds up within the first couple of Debye lengths close to the wall rather than being imposed as a prescribed boundary condition.

We verified the code using analytic results [2] as well as 1D PIC simulations [3]. The model was then used to investigate the influence of inclined magnetic fields on the plasma material interface. We used the code to study the correlation between the magnetic field angle and the angular distribution of incident particles.

Magnetic X-point configurations in tokamak geometries are critical in determining edge and scrape off layer (SOL) dynamics, and hence particle and heat flux onto plasma facing components. Alternative configurations have been proposed which aim to reduce fluxes to material surfaces, but their performance depends on cross-field transport in the region of the null point which is currently poorly understood. There is therefore a need for theoretical and experimental studies of turbulence in X-point magnetic configurations. In conventional 3D turbulence simulations of tokamaks, a field-aligned coordinate system is used, which introduces numerical instabilities at the null point due to zero volume elements. As a result, X-point dynamics are often interpolated based on nearby flux surfaces, which could exclude relevant physics. Here we simulate X-point configurations in linear geometries using a non-field-aligned coordinate system, and present results of 3D drift-wave turbulence and flow simulations in X- point configurations using an isothermal model which evolves density, vorticity, parallel velocity and parallel current density. Simulations have been performed to explore the feasibility of experimentally studying X-point configurations in linear plasma devices which indicate that even a modest coil set carrying 300A should produce a measurable effect on the driftwave turbulence and plasma profiles near the null region. The energy dynamics of the system are also explored and indicate that an X-point causes ohmic dissipation in higher mode number turbulence.

CASTOR/STARWALL is a code package for linear stability studies consisting of the CASTROR_3DW code, the STARWALL code, and a hybrid version of both codes named CASTOR3D that is currently under development. The latter solves an extended eigenvalue problem consisting of the plasma part defined by the weak form of the perturbed single-fluid MagnetoHydroDynamic (MHD) equations, and a vacuum part which is derived from an energy functional. This new code can describe wall resistivity and plasma inertia simultaneously. Furthermore, besides the so far used straight field line coordinates, additionally, general flux coordinates are implemented which are more appropriate for the description of instabilities located close to the separatrix (e.g. Edge Localized Modes (ELMs)). The formulation of the eigenvalue problem in the new coordinates is not limited to axisymmetric equilibria, so that, finally, stability studies of resistive and rotating, 3D equilibria should be possible. In this paper we will sketch the theory, report on the progress of the code development, and present first results.

Unstable collisional MicroTearing Mode (MTMs) have been found in experiments of high-β Spherical Tokamaks and are believed to be driven by drift resonance of trapped electrons. It has been recently shown that at large aspect ratio, the magnetic drift resonance of highly passing electrons is a minimal mechanism to drive the collisionless MTM unstable. In this work, a preliminary study of inclusion of trapped electrons in large aspect ratio tokamaks indicate that for the reference parameters investigated, the collisionless MTM retain their essential mode structures, while growth rates are only moderately affected.

We consider a simple electromagnetic gyrokinetic model for collisionless plasmas and show that it possesses a Hamiltonian structure. Subsequently, from this model we derive a two-moment gyrofluid model by means of a procedure which guarantees that the resulting gyrofluid model is also Hamiltonian. The first step in the derivation consists of imposing a generic fluid closure in the Poisson bracket of the gyrokinetic model, after expressing such bracket in terms of the gyrofluid moments. The constraint of the Jacobi identity, which every Poisson bracket has to satisfy, selects then what closures can lead to a Hamiltonian gyrofluid system. For the case at hand, it turns out that the only closures (not involving integro/differential operators or an explicit dependence on the spatial coordinates) that lead to a valid Poisson bracket are those for which the second order parallel moment, independently for each species, is proportional to the zero order moment. In particular, if one chooses an isothermal closure based on the equilibrium temperatures and derives accordingly the Hamiltonian of the system from the Hamiltonian of the parent gyrokinetic model, one recovers a known Hamiltonian gyrofluid model for collisionless reconnection. The proposed procedure, in addition to yield a gyrofluid model which automatically conserves the total energy, provides also, through the resulting Poisson bracket, a way to derive further conservation laws of the gyrofluid model, associated with the so called Casimir invariants. We show that a relation exists between Casimir invariants of the gyrofluid model and those of the gyrokinetic parent model. The application of such Hamiltonian derivation procedure to this two-moment gyrofluid model is a first step toward its application to more realistic, higher-order fluid or gyrofluid models for tokamaks. It also extends to the electromagnetic gyrokinetic case, recent applications of the same procedure to Vlasov and drift- kinetic systems.

The dynamics of discrete global modes in a toroidal plasma interacting with an energetic particle distribution is studied, and in particular when the dynamics of the system using the nonlinear and quasilinear descriptions are macroscopically similar. The dynamics can be described with a nonlinear bump-on-tail model in a two-dimensional phase space of particles. A Monte Carlo framework is developed for this model with an included decorrelation of the wave- particle phase, which is used to model extrinsic stochastisation of the wave-particle interactions. From this description, a quasilinear version of the model is also developed, which is described by a diffusive process in energy space due to the added phase decorrelation. Due to the reduced dimensionality of phase space, the quasilinear description is typically less computationally demanding than the nonlinear description. The purpose of the studies is to find conditions when a quasilinear model sufficiently describes the same phenomena of the wave-plasma interactions as a nonlinear model does. Via numerical and theoretical parameter studies, regimes where the two models overlap macroscopically are found. These regimes exist above a given threshold of the strength of the decorrelation, where coherent phase space structures are destroyed on time scales shorter than characteristic time scales of nonlinear particle motion in phase space close to the wave-particle resonance. Specifically for the quasilinear model, a theoretical value of the time scale of quasilinear flattening is derived and numerically verified.

How to apply a reduced model for the turbulent ion heat diffusivity [Nunami M et al. 2013 Phys. Plasmas, 20 092307] derived from a gyrokinetic code to a transport simulation is proposed. The reduced model is given by the function of the linear growth rate of the ion temperature gradient (ITG) mode and the decay time of zonal flows. The ion temperature gradient scale length is chosen for the additional modeling to include the linear growth rate of the ITG mode from a gyrokinetic code. The formula for the zonal flow decay time is derived at the given magnetic field configuration. The calculation of an extremely low computational cost in this article reproduces the results of the reduced model for the turbulent ion heat diffusivity within allowable errors.

The first results on analysis of collisionless driven reconnection with a multihierarchy simulation model are reported. In the multi-hierarchy simulation model, real space in a simulation consists of three parts: a magnetohydrodynamics (MHD) domain to deal with macroscopic dynamics, a particle-in-cell (PIC) domain to solve microscopic kinetic physics from the first principle, and an interface domain to interlock the two domains. By means of multi-hierarchy simulations, the influence of macroscopic dynamics on microscopic physics of magnetic reconnection is investigated. Dynamical behaviors of collisionless reconnection in the PIC domain depend strongly on plasma inflows from the MHD domain. It is found that if the width of an MHD inflow increases as v_w ≲ v_A0, where v_A0 is the Alfvén speed at the upstream boundary, magnetic reconnection has only a single X-point., while in cases of v_w≳2.0v_A0, reconnection with multiple X-points takes place.

Zonal flows (ZF) play a crucial role in regulating ion temperature gradient (ITG) turbulence. In previous global gyrokinetic simulations [1] using the ORB5 code with the adiabatic electron model, it was observed that long-lived ZF structures, leading to a corrugated transport and temperature gradient pattern, could develop in shaped tokamak plasmas much more than in circular shaped plasmas, resulting in reduced transport. These studies are extended to a hybrid electron model in which trapped electrons are kinetic while passing electrons are assumed to have a Boltzmann response for a case dominated by ITG modes. These confirm the results of the fully adiabatic electron model. Simulations done in "gradient-driven" mode, with a Krook-like relaxation towards a given profile, result in non-realistic corrugated heat source/sink profiles. However, after switching off completely the heat source/sink, it is shown that the ZF and transport corrugation remains. Thus the heat source corrugation is merely a consequence, not a cause, of the zonal structures and related radial transport pattern. Considering then core profiles with constant logarithmic gradients and pedestal profiles with linear gradients for L-mode plasmas, as in Ref.[2], we analyze how ITG transport and zonal structures react by independently varying the logarithmic gradients in the core and the linear gradients in the pedestal, using the adiabatic electron model. Results show the presence of large radial zones straddling the core-pedestal transition region. Avalanche-like events propagate over the radial zone at constant speed and repeat with a well defined frequency somewhat below the local geodesic acoustic mode (GAM) frequency. These avalanches are observed on the E × B ZFs, effective heat diffusivity and heat flux, thus a change of gradient in the core affects transport in the pedestal and vice versa. In spite of these non-local effects, attempt is made to characterize transport, and in particular its stiffness, quasi-locally. Global simulation results show that with increased input power the logarithmic gradient in the core is only slightly increased while the linear gradient in the pedestal is substantially enhanced.