Table of contents

The 2010 edition of the joint Varenna–Lausanne workshop on the theory of fusion plasmas was undoubtedly a great success. The programme encompasses a wide variety of topics, namely turbulence, MHD, edge physics and RF wave heating. The present PPCF issue is a collection of 19 outstanding papers, which cover these topics. It follows the publication of 22 refereed contributed papers in Journal of Physics: Conference Series 2010 260. There is no doubt that the production of articles was both abundant and of high scientific quality. This is why the Varenna–Lausanne meeting takes an important place in the landscape of conferences on fusion. Indeed this is the ideal forum for exchanging ideas on theory and modelling, and for substantiating the best results obtained in our field.

The tradition of the meeting is to provide a forum for numerical modelling activities. This custom was clearly respected given the large fraction of papers in this special issue which address this subject. This feature reflects the revolution we have been living through for some years with the fast growth of high-performance computers. It also appears that analytical theory is flourishing. This is important for bringing new ideas and guidance to numerical simulations. Finally, code validation and comparison to experiments are well represented. We believe that this is good news given the complexity of the non-linear physics that is at stake in fusion devices.

Another subject of satisfaction was the presence of many young scientists at the meeting. The encounter between young researchers and senior scientists is certainly a strong point of the Varenna–Lausanne conference.

In conclusion, we anticipate a great success for this special issue of PPCF and we hope that the readers will find therein ideas and inspiration.

This paper describes advances made in the field of energetic-particle physics since the topical review of Alfvén eigenmode observations in toroidal plasmas (Wong 1999 Plasma Phys. Control. Fusion41 R1–R56). The development of plasma confinement scenarios with reversed magnetic shear and significant population of energetic particles, and the development of novel energetic-particle diagnostics were the main milestones in the past decade, and these are the main experimental subjects of this review. The theory of Alfvén cascade eigenmodes in reversed-shear tokamaks and its use in magnetohydrodynamic spectroscopy are presented. Based on experimental observations and nonlinear theory of energetic-particle instabilities in the near-threshold regime, the frequency-sweeping events for spontaneously formed phase-space holes and clumps and the evolution of the fishbone oscillations are described. The multi-mode scenarios of enhanced particle transport are discussed and a brief summary is given of several engaging research topics that are beyond the authors' direct involvement.

We investigate the redistribution of the neutral beam driven current in the presence of small scale turbulence in the ITER steady-state scenario. Gyrokinetic simulations show that anomalous transport of beam ions can be larger than collisional estimates. The impact on the beam driven current in ITER is studied with a single particle following code. The results indicate that the current driven by the 1 MeV neutral beam injection is not significantly redistributed by the microturbulent fields. The numerical investigation shows that a larger impact is expected for lower energy neutral beams.

The control of transport barrier relaxation oscillations by resonant magnetic perturbations (RMPs) is investigated with three-dimensional turbulence simulations of the tokamak edge. It is shown that single harmonics RMPs (single magnetic island chains) stabilize barrier relaxations. In contrast to the control by multiple harmonics RMPs, these perturbations always lead to a degradation of the energy confinement. The convective energy flux associated with the non-axisymmetric plasma equilibrium in the presence of magnetic islands is found to play a key role in the erosion of the transport barrier that leads to the stabilization of the relaxations. This convective flux is studied numerically and analytically. In particular, it is shown that in the presence of a mean shear flow (generating the transport barrier), this convective flux is more important than the radial flux associated with the parallel diffusion along perturbed field lines.

We consider the effects of a finite pedestal radial electric field on ion orbits using a unified approach. We then employ these modified orbit results to retain finite E × B drift departures from flux surfaces in an improved drift-kinetic equation. The procedure allows us to make a clear distinction between transit averages and flux surface averages when solving this kinetic equation. The technique outlined here is intended to clarify and unify recent evaluations of the banana regime decrease and plateau regime alterations in the ion heat diffusivity; the reduction and possible reversal of the poloidal flow in the banana regime, and its augmentation in the plateau regime; the increase in the bootstrap current; and the enhancement of the residual zonal flow regulation of turbulence.

The BOUT++ code is used to simulate edge localized modes (ELMs) in a shifted circle equilibrium. Reduced ideal MHD simulations are first benchmarked against the linear ideal MHD code ELITE, showing good agreement. Diamagnetic drift effects are included finding the expected suppression of high toroidal mode-number modes. Nonlinear simulations are performed, making the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron thermal diffusivity. This allows simulations with realistically high Lundquist numbers (S = 10⁸), finding ELM sizes of 5–10% of the pedestal stored thermal energy. Scans show a strong dependence of the ELM size on resistivity at low Lundquist numbers, with higher resistivity leading to more violent eruptions. At high Lundquist numbers relevant to high-performance discharges, ELM size is independent of resistivity as hyper-resistivity becomes the dominant dissipative effect.

The linear response of a collisionless stellarator plasma to an applied radial electric field is calculated, both analytically and numerically. Unlike in a tokamak, the electric field and associated zonal flow develop oscillations before settling down to a stationary state, the so-called Rosenbluth–Hinton flow residual. These oscillations are caused by locally trapped particles with radially drifting bounce orbits. The particles also cause a kind of Landau damping of the oscillations that depends on the magnetic configuration. The relative importance of geodesic acoustic modes and zonal-flow oscillations therefore varies among different stellarators.

Energy and momentum can be transported across the plasma by waves emitted at one place and absorbed at another. Exchange of momentum and energy between the particles and the waves change the drift orbits, which may give rise to a non-ambipolar particle transport. The main effect of the non-ambipolar transport and quasi-neutrality is a toroidal precession of the trapped particles, which together with the changes in the parallel velocities of the passing resonant particles conserve the toroidal momentum. Non-resonant interactions can give rise to a net change of the local wave number in spatial inhomogeneous plasmas with a resulting force on the medium. Both resonant and non-resonant interactions have to be taken into account in order to have a consistent description of the momentum transported by the waves. The momentum transfer is, in particular, important for waves with short wave length and low frequency, and may explain the enhanced rotation seen in some mode conversion experiments, when the fast magnetosonic wave is converted to an ion-cyclotron wave. The apparent contradiction that the wave momentum may change due to non-resonant wave–particle interactions without changing the energy in geometric optics is explained.

A finite radial temperature gradient has been observed to be maintained within magnetic islands in gyro-kinetic turbulence simulations despite the fast motion along the field, and is related to the trapped particles which do not move along the field around the island. Recent calculations of the interaction of drift-wave turbulence with magnetic islands have shown that turbulence can exist within the separatrix, which in turn allows only the partial flattening of electron temperature profiles. Here we calculate, using a minimal drift-kinetic model, the effect on the bootstrap current in a tokamak. Consequences for the stability of the neoclassical tearing mode are discussed.

In order not only to clarify the difference on the motion of a plasmoid induced by a pellet injection between tokamak and helical plasmas but also to obtain the universal understanding on the motion in torus plasmas, magnetohydrodynamic simulations have been carried out in a tokamak, vacuum toroidal field, reversed field pinch (RFP) like- and Large Helical Device (LHD) configurations. It is found that the plasmoid motion depends on the initial location of the plasmoid in the LHD, whereas the plasmoids drift in the direction opposite to that of the curvature vector in a tokamak, vacuum toroidal field and RFP-like configurations. It is also verified that there are two main forces acting on the plasmoid, and the connection length determines the dominant force.

Minority ion cyclotron resonance heating is studied using the self-consistent numerical model SCENIC. This model includes 3D geometries with full shaping and anisotropic pressure effects, warm contributions to the dielectric tensor and full orbit effects. It evolves the equilibrium, wave field and hot particle distribution function iteratively until a self-consistent solution is found. We will show applications to JET-like two-dimensional equilibria with minority heating scenarios. The effects due to different heating locations on the hot particle distribution function, the hot dielectric tensor and the equilibrium will be studied for symmetric wave injection. Finally, the RF-induced particle pinch is investigated using asymmetric wave injection.

Using the local (flux-tube) version of the Eulerian code GENE (Jenko et al 2000 Phys. Plasmas7 1904), gyrokinetic simulations of microturbulence were carried out considering parameters relevant to electron-internal transport barriers (e-ITBs) in the TCV tokamak (Sauter et al 2005 Phys. Rev. Lett.94 105002), generated under conditions of low or negative shear. For typical density and temperature gradients measured in such barriers, the corresponding simulated fluctuation spectra appears to simultaneously contain longer wavelength trapped electron modes (TEMs, for typically k_⊥ρ_i < 0.5, k_⊥ being the characteristic perpendicular wavenumber and ρ_i the ion Larmor radius) and shorter wavelength ion temperature gradient modes (ITG, k_⊥ρ_i > 0.5). The contributions to the electron particle flux from these two types of modes are, respectively, outward/inward and may cancel each other for experimentally realistic gradients. This mechanism may partly explain the feasibility of e-ITBs. The non-linear simulation results confirm the predictions of a previously developed quasi-linear model (Fable et al2010 Plasma Phys. Control. Fusion52 015007), namely that the stationary condition of zero particle flux is obtained through the competitive contributions of ITG and TEM. A quantitative comparison of the electron heat flux with experimental estimates is presented as well.

Time-dependent effects of a radiofrequency driven current on the magnetic island are investigated by applying a new extended magnetohydrodynamics nonlinear model. New basic problems are pointed out, together with results that shed light on crucial questions for the control of neoclassical tearing modes. An interpretation of the numerical results is found by analogy with the classical Hahm–Kulsrud–Taylor problem.

An improved gyrokinetic electron and fully kinetic ion (GeFi) particle simulation scheme is presented for the investigation of linear collisionless tearing mode instability in a two-dimensional Harris current sheet under a finite guide field B_G and a realistic ion-to-electron mass ratio m_i/m_e. Due to the removal of the rapid electron cyclotron motion while retaining the finite electron Larmor radii, wave–particle interaction, and off-diagonal components of the electron pressure tensor, the GeFi model can be used to investigate the physics of magnetic reconnection with a realistic m_i/m_e in a large-scale current sheet, which in general possesses wave modes ranging from Alfvén waves to lower hybrid/whistler waves, with wave frequency ω < Ω_e, where Ω_e is the electron gyrofrequency. As a necessary step of utilizing the code for magnetic reconnection, the linearized GeFi scheme is benchmarked by comparing the simulation results using a δf method against direct numerical solutions of the tearing-instability eigenmode equations, as well as those obtained analytically via asymptotic matching.

Nonlinear magneto-hydrodynamic (MHD) simulations with the JOREK code may be used to improve our understanding of edge-localized-modes (ELMs) (Huysmans and Czarny 2007 Nucl. Fusion47 659–66, Huysmans et al2009 Plasma Phys. Control. Fusion51 124012, Pamela et al2010 Plasma Phys. Control. Fusion52 075006). These H-mode related instabilities may cause some damage to the tungsten divertor of ITER (Bazylev et al 2007 Phys. Scr.T128 229–33), and it was demonstrated experimentally that the ELM energy losses increase with both machine size and decreasing collisionality (ITER Physics Basis Editors and ITER EDA 1999 Nucl. Fusion39 2175, Loarte et al2003 Plasma Phys. Control. Fusion45 1549–69). In sight of producing simulations of ELMs in ITER, in order to give some predictions of ELM size and divertor heat fluxes in the future device, simulations first need to be quantitatively validated against the experimental data of present machines. This paper presents simulations of ELMs in the JET tokamak for the low-collosionality type-I ELMy H-mode pulse #73569. The simulation results are compared with experimental data to provide a qualitative validation of the simulations. This comparison comprises the dynamics of filaments and divertor heat fluxes, the effect of resistivity and collisionality on ELM energy losses and the observation of ELM precursors prior to the pedestal collapse.

A relatively simple dynamic model of low-frequency plasma convection is proposed to simulate turbulent cross-field transport in the tokamak core. The model is based on a special set of nonlinear weakly dissipative adiabatically reduced MHD-like equations, which self-consistently describe both turbulent plasma fluctuations and the resulting nondiffusive transport. Transient regimes with a fast turn on, turn off, and spatial redistribution of electron-cyclotron heating, as well as regimes with sawtooth oscillations were simulated numerically at time intervals exceeding the plasma energy confinement time. The results obtained demonstrate fast nonlocal responses of the resulting transport processes to the fast changes in external conditions and are in reasonable agreement with experiments in various tokamaks.

During tokamak operation with lower hybrid (LH) power a few per cent of the launched power is absorbed by the scrape-off layer plasma in magnetic flux tubes in front of the LH grill. At strike points of these flux tubes, intense plasma–wall interaction is seen in visible and infrared wavelengths, and local wall damage can occur. The parallel power flux within these 'hot spots' is estimated to be up to about 10 MW m⁻² by infrared imagery. Recent experimental results from retarding field analyzer measurements on Tore Supra as well as JET IR camera measurements have shown the existence of fast electrons as far as a few centimeters from the grill mouth. This finding cannot be explained by the standard theory. We present therefore in this paper a novel theory explaining the fast electron generation in a several cm wide layer in front of the LH grill by taking into account LH wave propagation features closely connected with the blob character of edge turbulence. We demonstrate that the computed power-flows then essentially agree with data from infrared diagnostics. An alternative theoretical explanation considers plasma density modulations due to ponderomotive force effects in front of the LH grill.

Transport analysis of MAST discharges indicates that collisions are an important loss mechanism in the core of a tight aspect ratio tokamak. In the strongly varying equilibrium fields of MAST many of the assumptions of drift kinetic and neoclassical theory (e.g. small plasma inverse aspect ratio and low ratio of toroidal Larmor radius to poloidal Larmor radius) are not met by all particle species and it becomes appropriate to use full orbit analysis to evaluate heat and particle fluxes. Collisional transport of impurity ions (C⁶⁺ and W²⁰⁺) has been studied using a full orbit solver, CUEBIT, to integrate the test-particle dynamics. Electromagnetic fields in MAST plasma have been modelled using the cylindrical and toroidal two-fluid codes CUTIE and CENTORI. A detailed study of the scaling of the test-particle diffusivity with collisionality in the equilibrium field reveals deviations from the standard neoclassical theory, in both the Pfirsch–Schlüter and banana regimes, and difficulties in defining a local diffusivity at low collisionalities. The effect of electric and magnetic fluctuations is also briefly addressed. It is found that field fluctuations enhance the non-diffusive nature of transport. The full orbit analysis presented here predicts levels of transport and confinement times for the examined species broadly consistent with the experimental observations.

A new numerical simulation to evaluate neoclassical toroidal viscosity (NTV) in tokamak configurations with a small perturbation field is developed using the δf Monte Carlo method. The numerical scheme solves the guiding-centre distribution function in non-axisymmetric plasmas according to the drift-kinetic equation, and evaluates the NTV directly from the pressure anisotropy by utilizing the Fourier spectrum expression of the magnetic field in Boozer coordinates. As a first benchmark, the accuracy of the viscosity calculation is demonstrated in a helical configuration of LHD. The convergence of the calculation and dependence of the viscosity on perturbation field amplitude are also tested in a simple tokamak configuration with model perturbation field, which proves the reliability of the simulation. Next, the basic properties of NTV as dependence of the viscosity on collision frequency and magnetic shear are investigated in a multi-helicity perturbation model field and compared with a bounce-averaged analytic formula. It is found that the clear 1/ν and superbanana-plateau dependences cannot be seen in the FORTEC-3D simulation, and the toroidicity of the magnetic field makes a toroidal coupling effect, which enhances NTV if the perturbation has (m, n) and (m ± 1, n) Fourier components simultaneously, where m and n are the poloidal and toroidal numbers of the perturbation field. Local magnetic shear is also found to affect the amplitude of the viscosity.

With a proper choice of a single dimensionless control parameter one describes the transition between subsonic and supersonic flows as a bifurcation. The bifurcation point is characterized by specific properties of the control parameter: the control parameter has a vanishing derivative in space and takes the maximum possible value equal to 1. This method is then applied to the sheath plasma with constant temperatures, allowing one to recover the Bohm boundary condition as well as the location of the point where the bifurcation takes place. This analysis is extended to fronts, rarefaction waves and divertor plasmas. Two cases are found, those where departure from quasineutrality is mandatory to generate a maximum in the variation of the control parameter (sheath and fronts) and those where the physics of the quasineutral plasma can generate such a maximum (rarefaction waves and supersonic flow in divertors). The conditions that are required to recover the Bohm condition, when modelling the wall using the penalization technique, are also addressed and generalized.