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International audience; The coupling between the poloidal distribution and the collisional flux of impurities can be exploited to derive a simplified analytical model covering toroidal rotation and electrostatic potential asymmetry effects, valid in the Pfirsh-Schlüter regime. This model is compared with earlier works and with the drift-kinetic code NEO, which includes the full Fokker-Planck collision operator. The low computational cost of the model, compared to NEO, is particularly adapted for fast integrated simulation purposes.

This is a correction for 2016 Nuclear Fusion 56, 086004 (2019); International audience; An extended fluid model covering neoclassical physics has been implemented in the XTOR code for Magneto-Hydro-Dynamic computations and is described in [1]. This model allows recovering neoclassical flux average quantities like the bootstrap current and the poloidal ion flow, and in the presence of a magnetic island, it generates a drive for Neoclassical Tearing Modes. The contribution of parallel heat fluxes on the bootstrap current is significant, and it was retained in the simulations presented in this paper. However, we have realized that the closure that is used for these parallel heat fluxes does not meet an important constraint on its spatial distribution, and although this does not change the equilibrium quantities, we do see an impact on the dynamics of the magnetic island. In the following, we propose a different closure that satisfies this constraint, and we also present a slight modification of the neoclassical implementation in the momentum equation. In order to show how these modifications impact the simulations, we present a limited number of examples performed with the new model, with diagnostics tools allowing a better understanding of the physics at play.

The disruptive collapse of the current sustained equilibrium of a tokamak is perhaps the single most serious obstacle on the path toward controlled thermonuclear fusion. The current disruption is generally too fast to be identified early enough and tamed efficiently, and may be associated with a variety of initial perturbing events. However, a common feature of all disruptive events is that they proceed through the onset of magnetohydrodynamic instabilities and field reconnection processes developing magnetic islands, which eventually destroy the magnetic configuration. Therefore the avoidance and control of magnetic reconnection instabilities is of foremost importance and great attention is focused on the promising stabilization techniques based on localized rf power absorption and current drive. Here a short review is proposed of the key aspects of high power rf control schemes (specifically electron cyclotron heating and current drive) for tearing modes, considering also some effects of plasma rotation. From first principles physics considerations, new conditions are presented and discussed to achieve control of the tearing perturbations by means of high power (PEC Pohm) in regimes where strong nonlinear instabilities may be driven, such as secondary island structures, which can blur the detection and limit the control of the instabilities. Here we consider recent work that has motivated the search for the improvement of some traditional control strategies, namely the feedback schemes based on strict phase tracking of the propagating magnetic islands.

A self-consistent fluid model for describing neoclassical tearing modes in global magneto-hydro-dynamic simulations is presented. It is illustrated by its application to a simple toroidal configuration unstable to the (2, 1) tearing mode. The island saturation is verified to increase with the bootstrap current fraction. New features that are specific to this model are evidenced, like the unsteady saturated state of the island, and its deformation to a droplet shape, when the magnetic Prandtl number is not too high. Synthetic diagnostics demonstrate that diamagnetic and neoclassical effects should have in this case a measurable impact on the signature of magnetic islands.

Tearing modes are MagnetoHydroDynamics (MHD) instabilities that reduce the performance of fusion devices. They can however be controlled and suppressed using electron cyclotron current drive (ECCD) as demonstrated in various tokamaks. In this work, simulations of island stabilization by ECCD-driven current have been carried out using the toroidal nonlinear 3D full MHD code XTOR-2F, in which a current source term modeling the ECCD has been implemented. The efficiency parameter, eta(RF), has been computed and its variations with respect to source width and location were also computed. The influence of parameters such as current intensity, source width and position with respect to the island was evaluated and compared to the modified Rutherford equation. We retrieved a good agreement between the simulations and the analytical predictions concerning the variations of control efficiency with source width and position. We also show that the 3D nature of the current source term can lead to the onset of an island if the source term is precisely applied on a rational surface. We report the observation of a flip phenomenon in which the O- and X-points of the island rapidly switch their position in order for the island to take advantage of the current drive to grow.

The crash of monster sawteeth in Tore Supra ion cyclotron resonance heated plasmas is observed to trigger long-lived magneto hydrodynamic (MHD) activity, dominated by a (m = 3, n = 2) magnetic perturbation at the edge. This phenomenon is reminiscent of the triggering of neoclassical tearing modes, although in Tore Supra the MHD activity decays and eventually vanishes. It can be explained by the linear destabilization of the (3, 2) mode as the current sheet developed in the non-linear stage of the internal kink relaxation gets closer to q = 3/2. However, the lifetime of the (3, 2) island is longer than the period of linear instability. We find that the neoclassical drive is essential for explaining the observed lifetime and width of the island, although the overall dynamics is controlled by the relaxation of the current profile on a resistive time scale.

Recent operation of JET with centrally strongly reversed magnetic shear, produced with the help of lower hybrid current drive, has extended the domain in which internal transport barriers (ITBs) can be formed in JET. Performance is frequently limited by magnetohydrodynamic (MHD) instabilities in these reversed shear regimes. The most severe limit is a pressure driven kink mode which leads to a disruption. This disruptive limit is essentially the same in ITB plasmas with low or strongly reversed shear. Unique to the reversed shear regime is a dominantly n = 1 mode, which has multiple harmonics. This mode is a seemingly common limit to performance, in the highest performance plasmas. Also unique to the reversed shear regime are q > 1 sawteeth events, which can in turn trigger n = 1 post-cursor oscillations. In general, these post-cursor oscillations are benign but do provide valuable information on the q-profile. Other instabilities, including 'snakes' at the outer q = 3 surface, are also observed to limit the performance of reversed magnetic shear ITB regimes.

For the first time, data from eight different tokamaks have been combined in an international database for internal transport barriers (ITBs). An analysis of the data for the formation of an ITB with dominant ion heating shows a clear dependence of the threshold power on the minor radius and line-averaged electron density for the formation of ion ITBs. The dependence of ITB formation on the toroidal magnetic field is weak. For the formation of ITBs with dominant electron heating, the database is smaller, but for the threshold power a strong increase with plasma size and a weak toroidal field dependence could also be identified. Based on these results, an expression for the power required to form an ITB is given using global variables only. These results give a basis for the analysis of the database using local values (like magnetic shear) and a detailed comparison with theory-based models.

Tearing modes associated to hollow current profiles are prone to grow in moderate performance plasmas and often constrain the realization of non-inductive discharges in the Tore Supra tokamak, where long pulse duration is performed using Lower Hybrid waves for providing most of the plasma current. The prediction of MHD boundaries in such scenarios is complicated by the importance of diamagnetic effects, combined with curvature stabilization, which determine the stability of these modes. We show that diamagnetic effects, as well as neoclassical forces, are playing a key role in the linear and nonlinear regimes of Double-Tearing Modes on q = 5/3 and q = 2 in these experimental conditions. Detailed comparison with experimental measurements, combined with a scaling in plasma resistivity, give constraints about the experimental equilibrium. Resistive-Interchange Modes destabilized by diamagnetic rotation could also play a role in degrading the energy confinement in the negative magnetic shear region.

This paper presents the results of three-dimensional fluid global simulations of electrostatic ion turbulence in tokamaks with reversed magnetic shear. It is found that a transport barrier appears at the location of magnetic shear reversal. This is due to a rarefaction of resonant surfaces in this region. For the same reason, the barrier is more pronounced when the minimum of the safety factor is a simple rational number. The barrier is broadened by velocity shear effects. It is also found that large-scale transport events hardly cross a transport barrier. Finally, a significant amount of toroidal rotation is generated by the turbulence. This rotation changes its sign at the position of magnetic shear reversal, as expected from a quasi-linear estimate of the Reynolds stresstensor.

The drift wave stability of pellet enhanced performance modes in Tore Supra has been investigated. It is found that these discharges are characterized by a combination of three different stabilizing mechanisms: density peaking, E × B shear and magnetic shear. Drift wave stability appears to be very sensitive to magnetic shear, with the most favourable configuration corresponding to a slightly negative magnetic shear. The conclusions are compatible with the results from JET and TFTR.

Curvature and diamagnetic effects are both known to have an influence on tearing mode dynamics. In this paper, we investigate the impact of these effects on the nonlinear stability and saturation of a (2, 1) island using non-linear two-fluid MHD simulations and we apply our results to Tore Supra experiments, where its behavior is not well understood from the single fluid MHD model. Simulations show that a metastable state induced by diamagnetic effect exists for this mode and that it also produces a reduction of the saturated island size, in presence of toroidal curvature. The mode is found to be nonlinearly destabilized by a seed island and it saturates at a macroscopic level causing a significant confinement degradation. The interpretation of dual states, with either no island on q = 2 or a large one, observed on discharges with high non inductive current source on Tore Supra, is revisited.

Neoclassical physics (here associated to the poloidal variation of the magnetic field strength along field lines in a tokamak) is well known for driving self-generated plasma current and nonlinear magnetic islands associated to it in high performance, ITER relevant plasma discharges. It is demonstrated that the neoclassical friction between a magnetic perturbation and plasma flow already impacts magnetic islands in the linear regime, by inducing a weakening of curvature stabilization for tearing modes. This conclusion holds in particular for regimes where convection is influencing the pressure dynamics, as shown using a simple analytical model and confirmed in full Magneto-Hydro-Dynamics simulations.

The neoclassical viscous stress tensor is implemented in the non-linear MHD code XTOR-2F (Lutjens and Luciani 2010 J. Comput. Phys. 229 8130–43), allowing consistent bi-fluid simulations of MHD modes, including the metastable branch of neoclassical tearing modes (NTMs) (Carrera et al 1986 Phys. Fluids 29 899–902). Equilibrium flows and bootstrap current from the neoclassical theory are formally recovered in this Chew–Goldberger–Low formulation. The non-linear behaviour of the new model is verified on a test case coming from a Tore Supra non-inductive discharge. A NTM threshold that is larger than with the previous model is obtained. This is due to the fact that the velocity is now part of the bootstrap current and that it differs from the theoretical neoclassical value.

Curvature and diamagnetic effects are both recognized to have a stabilizing influence on tearing modes in the linear regime. In this paper, we investigate the impact of these effects on the linear stability of a (2, 1) magnetic island using non-linear two-fluid MHD simulations and we apply our results to Tore Supra experiments where its stability is not well understood from the single fluid MHD model. Simulations show an initial increase of the linear growth rate and then its reduction until full stability as diamagnetic frequency increases. This mechanism is therefore a plausible explanation for experimental observations where the (2, 1) mode was not observed although the single fluid model predicted its growth. Our simulations also show the importance of curvature for an efficient stabilization. A simple analytical model is derived to support the numerical results.

The limit to high performances advanced scenario discharges with qmin above unity is generally set by the (2,1) magneto-hydro-dynamic (MHD) mode in JET. We investigate here the possibility that this mode is a (2,1) neoclassical tearing mode (NTM) by computing the critical island width at which such mode would be unstable, using a non-linear MHD code where the relevant bootstrap current physics is accounted for. We show that the triggering of a (2,1) NTM is consistent with a lowering of the critical island width as the plasma current diffuses towards the centre. This is explained partly by the increase in the magnetic shear at the resonant surface, which weakens the curvature stabilization term, as found in the analytical framework of a generalized Rutherford equation. A comparison with experiment is made in the non-linear regime, showing encouraging results on the dynamics of the confinement degradation and mode structure.

The fully implicit 3D two-fluid MHD code XTOR-2F has been developed to allow an easy implementation of physical models beyond MHD. The numerical method of solution used in XTOR-2F is presented briefly. The code is then applied to the investigation of the long time dynamics of internal kinks, first within the framework of resistive MHD including thermal transport and next adding more refined effects such as ion and electron diamagnetic rotations and separate density and pressure evolutions.

Modes oscillating at the acoustic frequency and identified as beta Alfvén eigenmodes (BAEs) have been observed in Tore-Supra under ion cyclotron resonant heating. In this paper, the linear excitation threshold of these modes, thought to be driven by suprathermal ions, is calculated and compared with Tore-Supra observations. Similar studies of the linear excitation threshold of energetic particles driven modes were carried out previously for toroidal Alfvén eigenmodes or fishbones. In the case of BAEs, the main point is to understand whether the energetic particle drive is able to exceed ion Landau damping, which is expected to be important in the acoustic frequency range. For this, the BAE dispersion relation is computed and simplified in order to derive a tractable excitation criterion suitable for comparison with experiments. The observation of BAEs in Tore-Supra is found to be in agreement with the calculated criterion and confirms the possibility to trigger these modes in the presence of ion Landau damping. Moreover, the conducted analysis clearly puts forward the role of the global tunable parameters which play a role in the BAE excitation (the magnetic field, the density etc), as well as the role of some plasma profiles. In particular, the outcome of a modification of the shear or of the heating localization is found to be non-negligible and it is discussed in the paper.

Attempts to run non-inductive plasma discharges on Tore Supra sometimes fail due to the triggering of magneto-hydro-dynamic (MHD) instabilities that saturate at a large amplitude, producing degraded confinement and loss of wave driven fast electrons (the so-called MHD regime (Maget et al 2005 Nucl. Fusion 45 69–80)). In this paper we investigate the transition to this soft (in the sense of non-disruptive) MHD limit from experimental observations, and compare it with non-linear code predictions. Such a comparison suggests that different non-linear regimes, with periodic relaxations or saturation, are correctly understood. However, successful non-inductive discharges without detectable magnetic island at q = 2 cannot be reproduced if realistic transport coefficients are used in the computation. Additional physics seems mandatory for explaining these discharges, such as diamagnetic effects, that could also justify cases of abrupt transition to the MHD regime.

MHD instabilities driven by fast electrons identified as fishbonelike modes have been detected on Tore Supra during lower hybrid current drive discharges. Direct experimental evidence is reported of a novel feature: the regular redistribution of suprathermal electrons toward external tokamak regions which are correlated to periodic mode frequency jumps. Sharp drops of the electron temperature time trace are factually linked to the cyclical deterioration of the fast electron confinement.

Magnetohydrodynamic (MHD) activity often undermines the realization of fully noninductive plasma discharges in the Tore Supra tokamak [J. Jacquinot, Nucl. Fusion 45, S118 (2005)], by producing large degradation of electron energy confinement in the plasma core and the bifurcation to a regime with permanent MHD activity. The nonlinear evolution of MHD modes in these hollow current density profile discharges is studied with the full-scale three-dimensional MHD code XTOR [K. Lerbinger and J.-F. Luciani, J. Comput. Phys. 97, 444 (1991)] and compared with experimental features. Large confinement degradation is predicted when q(0) is close to 2. This derives either from the full reconnection of an unstable double-tearing mode, or from the coupling between a single tearing mode and adjacent stable modes in a region with reduced magnetic shear.

The experimental programme of Tore Supra is devoted to the study of technology and physics issues associated to long‐duration high performance discharges. This new domain of operation requires simultaneously and in steady state: heat removal capability, particle exhaust, fully non‐inductive current drive, advanced technology integration and real time plasma control. The long discharge allows for addressing new time scale physic such as the wall particle retention and erosion. Moreover, the physics of fully non‐inductive discharges is full of novelty, namely: the MHD stability, the slow spontaneous oscillation of the central electron temperature or the outstanding inward particle pinch.

During fully noninductively driven discharges in the Tore Supra tokamak, large spontaneous oscillations of the core electron temperature (DeltaTe/Te>50%) have been observed for the first time. They occurred during the standard O regime, which is itself characterized by periodic oscillations of much smaller amplitude. The "giant" oscillations appear to involve distinct mechanisms with respect to the O regime and provide a spectacular example of the complex nonlinear interactions between energy confinement, noninductive current sources, and MHD that may occur in a tokamak plasma during steady-state operation.

Improvement (up to a factor of approximately 4) of the electron-cyclotron (EC) current drive efficiency in plasmas sustained by lower-hybrid (LH) current drive has been demonstrated in stationary conditions on the Tore Supra tokamak. This was made possible by feedback controlled discharges at zero loop voltage, constant plasma current, and constant density. This effect, predicted by kinetic theory, results from a favorable interplay of the velocity space diffusions induced by the two waves: the EC wave pulling low-energy electrons out of the Maxwellian bulk, and the LH wave driving them to high parallel velocities.

Recent theoretical work on magnetic reconnection in hot plasma confinement devices is reviewed. The presentation highlights the common aspects of reconnection phenomena, and current research trends are emphasized. Progress in understanding the dynamics of slowly evolving modes of the tearing family, based on advanced analytic techniques and numerical simulation, as well as of faster modes that lead to internal disruptions, is reported.

During noninductively driven discharges in the Tore Supra tokamak, steady sinusoidal oscillations of the central electron temperature, lasting as long as 2 min, have been observed for the first time. Having no helical structure, they cannot be ascribed to any known MHD instability. The most plausible explanation of this new phenomenon is that the plasma current density and the electron temperature evolve as a nonlinearly coupled predator-prey system. This interpretation is supported by the numerical solution of coupled resistive current diffusion and heat transport equations.

The formation of internal transport barriers (ITBs) has been experimentally associated with the presence of rational q surfaces in both JET and ASDEX Upgrade. The triggering mechanisms are related to the occurrence of magneto-hydrodynamic (MHD) instabilities such as mode coupling and fishbone activity. These events could locally modify the poloidal velocity and increase transiently the shearing rate to values comparable with the linear growth rate of ion temperature gradient modes. For JET reversed magnetic shear scenarios, ITB emergence occurs preferentially when the minimum q reaches an integral value. In this case, transport effects localized in the vicinity of zero magnetic shear and close to rational q values may be at the origin of ITB formation. The role of rational q surfaces in ITB triggering stresses the importance of q profile control for an advanced tokamak scenario and could assist in substantially lowering the access power to these scenarios in next step facilities.

Quasistationary operation has been achieved on the Joint European Torus tokamak in internal-transport-barrier (ITB) scenarios, with the discharge time limited only by plant constraints. Full current drive was obtained over all the high performance phase by using lower hybrid current drive. For the first time feedback control on the total pressure and on the electron temperature profile was implemented by using, respectively, the neutral beams and the ion-cyclotron waves. Although impurity accumulation could be a problem in steady state ITBs, these experiments bring some elements to answer to it.

The linear ideal magnetohydrodynamic (MHD) limits of the pedestal in ITER scenarios associated with the preparation and realization of the nominal fusion gain Q?=?10 (inductive scenario at 15?MA/5.3?T, half-field/half-current and intermediate H-mode scenario at 10?MA/3.5?T), as well as the hybrid scenario at 12?MA/5.3?T, are investigated in this work. The accessible part of the MHD stability diagram is determined by computing the bootstrap current and self-consistently evaluating the corresponding pedestal current. This procedure shows that only a small part of peeling?ballooning diagrams is physically accessible. Uncertainties about the foreseen plasma profiles motivate studies evaluating the impact of various parameters on the pedestal limits. We have addressed issues such as the pedestal width, the global performance, pressure peaking, edge current density, internal inductance and plasma shaping. A scaling law for the maximum pedestal pressure in the ITER scenarios is proposed, highlighting that the main dependences are on the plasma current, the edge safety factor, the pedestal width and the internal inductance.

This work presents a comparative study of the experimental observation of MHD instabilities identified as electron fishbone-like modes that appear in plasmas with lower hybrid current drive in FTU and Tore Supra tokamaks. Initially, the mode-induced electronic temperature oscillations measured by electron cyclotron emission were used to study the evolutions of the frequency and position of these modes. In FTU, where fishbones with and without bursting behaviour are observed, it was found that the evolutions of the mode position and frequency follow opposite trend in the two regimes. In Tore Supra, where abrupt changes between modes with different mode structures are often observed, it was found that the mode position evolves continuously and the wavenumbers follow an inverse cascade starting from an m/n = 4/4 mode and finishing in an 1/1 mode. In a second step, the energy of resonant electrons was estimated from the resonant condition of the precession drift frequency. It was found that in Tore Supra the resonant condition does not change during the frequency jumps. The relevance of the correction due to the pitch angle of the resonant electrons and the possible role of the energetic passing particles in the drive of these modes are discussed from the results obtained in both machines.

The non-linear reduced four-field RMHD model in cylindrical geometry was extended to include plasma rotation, neoclassical poloidal viscosity and two fluid diamagnetic effects. Interaction of the static resonant magnetic perturbations (RMPs) with the rotating plasmas in tokamaks was studied. The self-consistent evolution of equilibrium electric field due to RMP penetration is taken into account in the model. It is demonstrated that in the pedestal region with steep pressure gradients, mean flows perpendicular to the magnetic field, which includes and electron diamagnetic components plays an essential role in RMP screening by plasma. Generally, the screening effect increases for lower resistivity, stronger rotation and smaller RMP amplitude. Strong screening of central islands was observed limiting RMP penetration to the narrow region near the separatrix. However, at certain plasma parameters and due to the non-linear evolution of the radial electric field produced by RMPs, the rotation can be compensated by electron diamagnetic rotation locally. In this case, RMPs can penetrate and form magnetic islands. Typical plasma parameters and RMPs spectra on DIII-D, JET and ITER were used in modelling examples presented in the paper.

Real-time control is essential for many aspects of tokamak operation. A key parameter to control is the current profile, since both confinement properties and magneto-hydrodynamic (MHD) activity depend on it in quite a sensitive way. The long pulse capability of the Tore Supra tokamak has allowed a unique type of experiment, where successive stationary states of the safety factor profile, defined by their MHD activity, are established and controlled in real time. Multiple target stationary states could be requested and reached during the main heating phase of a single plasma discharge. Experiments have been carried out featuring (i) control of the presence/absence of sawteeth with varying plasma parameters, (ii) obtaining and sustaining a ‘hot core’ plasma regime without MHD activity and (iii) recovery from a voluntarily triggered deleterious MHD regime. During these experiments, the influence of fast ions on MHD stability could be observed and characterized, as well as indications of an enhanced ‘hot core’ confinement in electron heat transport during quiescent MHD states.

In the Tore Supra tokamak, magnetohydrodynamic instabilities can be destabilized by energetic particles produced by radio-frequency heating systems. This paper reports on experimental observations of two types of energetic particle driven internal modes: the electron fishbone and the beta-induced Alfven eigenmodes (BAEs). The evolutions of these modes in frequency and radial position during core relaxation cycles were determined by their induced electronic temperature and density fluctuations. Electron fishbones were observed in the so-called oscillation regime, while BAEs' observations were performed in the sawtooth regime. The results show that the frequency and radial position of these two instabilities vary substantially during the core relaxation cycles. It indicates that an accurate description of these evolutions must consider the equilibrium profile alterations during the core oscillations.

A new concept of steady-state scenario for tokamak reactors is proposed. It is based on cyclic operations, alternating phases of positive and negative loop voltage with no magnetic flux consumption on average. Localized non-inductive current drive by electron cyclotron waves is used to trigger and sustain an internal transport barrier (ITB), whereas neutral beam current drive is used to periodically recharge the tokamak transformer. The fact of operating in cycles relaxes the hard constraint of simultaneous fusion performance maximization and full non-inductive operation, within the MHD stability limits. Integrated modelling simulations are performed to apply this concept for the ITER steady-state regime. A linear MHD analysis of the instabilities that could appear in this type of scenario is performed, showing that MHD stability would be strongly improved with respect to a steady regime with a strong ITB.

This paper describes what we can learn on the regimes of spontaneous electron temperature oscillations discovered in Tore Supra from the analysis of MHD activity. Since the first observations of this oscillating behaviour of plasma equilibrium, and its interpretation as a predator-prey system involving lower hybrid waves power deposition and electron confinement, analysis of MHD modes has confirmed the reality of safety factor profile oscillations. This points towards the importance of rational values of the safety factor in the transition to transport barriers in reversed magnetic shear plasmas.

International audience; A 6-field reduced-MHD model is derived for plasma dynamics. The new model describes coherently both Ion Temperature Gradient mode and Tearing mode, and includes neoclassical effects. The model allows the construction of an energy-like quantity with a linear pressure contribution that is conserved except for dissipative, finite Larmor radius and neoclassical terms. This model may be used to study the nonlinear interaction between ITG microturbulence and neoclassical tearing mode, which is responsible for large-scale magnetic islands in tokamaks, and opens the way to a coherent description of turbulent impurity transport in magnetic islands.