Table of contents

We applied a non-integrable drift-kinetic model, valid for large aspect ratio tokamaks, to investigate plasma edge particle transport driven by drift waves. Particle transport is obtained from the chaotic trajectories obtained by numerically integrating the canonical equations of motion, for the total flow formed by the equilibrium sheared flow and few dominant resonant drift waves propagating in the sheared equilibrium magnetic field. Thus, we investigate the transport dependence on the radial profiles of the electric and magnetic fields and show that radial particle transport at the plasma edge can be reduced by properly modifying the electric and magnetic shear profiles. For non-monotonic radial electric fields, we also observe non-twist transport barriers with shearless invariants identified by extremum values of the rotation number profiles of the invariant curves. The observed non-twist barriers are modified by the magnetic shear and persist for magnetic shear variations expected in present tokamaks.

Resonant magnetic perturbations (RMPs) used to control type-I edge-localized modes produce a splitting of the peaks of heat flux profile on the divertor. The additional peaks can appear outside the areas connected to the plasma core by the perturbed field lines due to perpendicular transport. This effect has been studied so far using demanding three-dimensional transport simulations. We present a simple method to quantitatively estimate the heat flux profile that the RMP will produce on the divertor. The method needs only field line tracing to determine the so-called tangle distance function, and knowledge of the heat flux profile on the unperturbed heat flux profile. We find good agreement between this method and a transport simulation.

In order to understand the mechanism by which resonant magnetic perturbations (RMPs) mitigate or suppress edge-localized modes, it is necessary to understand the plasma response to the application of RMPs. TEXTOR's fast movable Mirnov probe can provide direct measurements of the plasma response to RMPs applied using the dynamic ergodic divertor. The effect of toroidal plasma rotation is investigated, and a change in the phase of the plasma response at certain values of rotation is found. Jumps in the phase of the magnetic field are found to occur on resonant surfaces, indicating the formation of screening currents on these surfaces. The first observations of screening currents on multiple surfaces are presented, and the transition from screening to field penetration with increasing strength of the applied RMP field is observed.

The generic analytic formulae of the stable and unstable manifolds near the separatrix and on the divertor plates in tokamaks in the presence of resonant magnetic perturbations are derived. The stable (unstable) manifolds on the divertor plates describe the form of the magnetic footprint boundaries. They are derived using the separatrix mapping. The analytical formulae depend on a few of the equilibrium plasma parameters and the magnetic perturbations.

First experiments on edge-localized mode (ELM) mitigation with the help of ITER-like coils on ASDEX Upgrade are analysed using linear and quasilinear kinetic models to describe the interaction of resonant magnetic field perturbations (RMP) with the plasma. The gyrokinetic derivation of RMP-driven transport coefficients is given in detail. The role of fluid resonances is studied, in particular the role of the resonance associated with the equilibrium electric field reversal point E_r = 0. Like the electron fluid resonance associated with the zero of the total perpendicular electron fluid velocity, the E_r = 0 resonance may lead to enhanced transport due to the reduction of RMP shielding in the pedestal region where the RMP field can even be amplified by this resonance. The conditions on the RMP coil spectrum resulting from the analysis are discussed.

The reversed-field pinch exhibits a strong tendency to self-organize into a helical equilibrium as the plasma current is increased. The helical reversed-field pinch is characterized by reduced magnetic stochasticity and by the formation of electron internal transport barriers. The paper gives an update on recent experimental and modelling work on helical states in RFX-mod (Sonato et al 2003 Fusion Eng. Des.66 161), also discussing similarities with 3D equilibria in tokamaks. The helical equilibrium is modelled with 3D codes developed for stellarators, such as VMEC/V3FIT. The reconstructed safety factor profile has low or reversed magnetic shear in the core, which may be related to transport barrier formation. A significant extension of the RFX-mod database to high current and density confirms the dependence observed before of various helical state properties on macroscopic quantities. Even under conditions where it does not form spontaneously, such as at low current or high density, the 3D magnetic equilibrium can be stimulated and robustly controlled with external fields applied by an extensive set of non axi-symmetric coils. An advanced magnetic feedback algorithm that compensates for error fields induced by eddy currents in the 3D wall structures has been developed. This work stimulated similar experiments in RFX-mod run as a tokamak, where external 3D fields are applied to control a m = 1/n = 1 helical equilibrium.

The effects of applied 3D resonant magnetic perturbations are modelled with and without self-consistent plasma response. The plasma response is calculated using a linear two-fluid model. A synthetic diagnostic is used to simulate soft x-ray (SXR) emission within the steep gradient region of the pedestal, 0.98 > ψ > 0.94. Two methods for simulating the SXR emission given the perturbed fields are considered. In the first method, the emission is assumed to be constant on magnetic field lines, with the emission on each line determined by the penetration depth into the plasma. In the second method, the emission is taken to be a function of the perturbed electron temperature and density calculated by the two-fluid model. It is shown that the latter method is more accurate within the plasma, but is inadequate in the scrape-off layer due to the breakdown of the linearized temperature equation in the two-fluid model. The resulting synthetic emission is compared to measured SXR data, which show helical m = 11 ± 1 displacements around the 11/3 rational surface of sizes up to 5 cm, depending on the poloidal angle. The helical displacements around the 11/3 surface are identified to be directly related to the kink response, caused by amplification of non-resonant components of the magnetic field due to plasma response. The role of different plasma parameters is investigated, but it appears that the electron rotation plays a key role in the formation of screening and resonant amplification, while the kinking appears to be sensitive to the edge current density. It is also hypothesized that the plasma response affects the edge-localized-mode (ELM) stability, i.e. the discharge's operational point relative to the peeling–ballooning stability boundary.

Comprehension of the interactions between tokamak edge plasmas and externally induced resonant magnetic perturbations (RMPs) is an important step in the understanding of the control of edge-localized modes by these RMPs. Such control has been demonstrated experimentally, but previous theoretical investigations have revealed a possible screening of RMPs by a sheared rotation of the plasma. In this work, the penetration of RMPs is investigated via numerical simulations in a reduced magnetohydrodynamic model using the three-dimensional electromagnetic turbulence code EMEDGE3D. In this model, the plasma response to RMPs can be studied in the presence of flux-driven micro-turbulence and a transport barrier induced by sheared plasma rotation. The interplay is, in a first part, studied in a non-turbulent case to deduce a criterion for the penetration in a rotating plasma that is governed by the generation of counter currents. When the plasma is studied in a statistically stationary turbulent state, the self-consistent plasma rotation, governed by Reynolds and Maxwell stresses, leads to a self-organization where RMP penetrates. In a turbulent plasma in the presence of a transport barrier, the RMP harmonic that is resonant at the barrier centre is found to penetrate partially. This partial penetration is sufficient to trigger a local flattening of the pressure gradient that is known to be at the origin of the control of transport barrier relaxations in the present model.

In the reversed-field pinch (RFP) edge, measured transport and flows are strongly influenced by magnetic islands (Vianello 2013 Nucl. Fusion53 073025). In fact, these islands determine a differential radial diffusion of electrons and ions which, interacting with the wall, give rise to a characteristic edge ambipolar potential. Such island structures also arise in tokamak plasmas, when resonant magnetic perturbations (RMPs) are applied for control of edge-localized modes. They impose a characteristic modulation to edge electron density and temperature fields, in close correlation with the local magnetic vacuum topology (Schmitz 2012 Nucl. Fusion52 054001). In order to develop a generic picture of particle transport with magnetic islands located in the plasma edge between RFPs and tokamaks with RMP, test-particle transport simulations are done on TEXTOR with the same tool used in RFX-mod, namely, the guiding-centre code ORBIT (White and Chance 1984 Phys. Fluids27 2455–67). A typical TEXTOR discharge in the (m, n) = (12, 4) configuration is reconstructed and analysed with ORBIT. We use Poincaré and connection length analysis of electrons and ion orbits to analyse the magnetic structure taking into account the different gyro-orbits of both constituents. Density distributions of test ions and electrons are calculated and used to obtain an initial estimate of the plasma potential and radial electric field around the island.

Heat pulse propagation in three-dimensional chaotic magnetic fields is studied by solving numerically the parallel heat transport equation using a Lagrangian Green's function (LG) method. The LG method provides an efficient and accurate technique that circumvents known limitations of finite elements and finite difference methods. The main two problems addressed are (i) the dependence of the radial transport of heat pulses on the level of magnetic field stochasticity (controlled by the amplitude of the magnetic field perturbation, ), and (ii) the role of reversed shear magnetic field configurations on heat pulse propagation. In all the cases considered there are no magnetic flux surfaces. However, the radial transport of heat pulses is observed to depend strongly on due to the presence of high-order magnetic islands and Cantori. These structures act as quasi-transport barriers which can actually preclude the radial penetration of heat pulses within physically relevant time scales. The dependence of the magnetic field connection length, ℓ_B, on is studied in detail. Regions where ℓ_B is large, correlate with regions where the radial propagation of the heat pulse slows down or stops. The decay rate of the temperature maximum, 〈T〉_max(t), the time delay of the temperature response as function of the radius, τ, and the radial heat flux $\langle {{\bit q}\cdot {\hat e}_\psi} \rangle$, are also studied as functions of the magnetic field stochasticity and ℓ_B. In all cases it is observed that the scaling of 〈T〉_max with t transitions from sub-diffusive, 〈T〉_max ∼ t^−1/4, at short times (χ_∥t < 10⁵) to a significantly slower, almost flat scaling at longer times (χ_∥t > 10⁵). A strong dependence on is also observed on τ and $\langle {{\bit q}\cdot {\hat e}_\psi} \rangle$. Even in the case when there are no flux surfaces nor magnetic field islands, reversed shear magnetic field configurations exhibit unique transport properties. The radial propagation of heat pulses in fully chaotic fields considerably slows down in the shear reversal region and, as a result, the delay time of the temperature response in reversed shear configurations is about an order of magnitude longer than the one observed in monotonic q-profiles. The role of separatrix reconnection of resonant modes in the shear reversal region, and the role of shearless Cantori in the observed phenomena are also discussed.

The existence of primary shearless tori is a distinguishing feature of nontwist maps. However, secondary shearless tori have been identified in the phase space of twist maps, in the neighbourhood of peculiar bifurcations of elliptic fixed points. In this paper, we report secondary shearless bifurcations in a twist symplectic map that describes chaotic field lines in tokamaks with an ergodic limiter. We identify the onset of secondary shearless tori, around elliptic fixed points within the island of stability, by examining numerical profiles of the internal rotation number. We present examples of field line transport barriers, associated with secondary shearless tori and their rupture, which reduce the usual magnetic field line escape at the tokamak plasma edge.

Edge localized modes (ELMs) are a concern for future devices, such as ITER, due to the large transient heat loads they generate on the divertor surfaces which could limit the operational lifetime of the device. This paper discusses the application of resonant magnetic perturbations (RMPs) as a mechanism for ELM control on Mega Amp Spherical Tokamak (MAST). Experiments have been performed using an n = 3 toroidal mode number perturbation and measurements of the strike point splitting performed. The measurements have been made using both infrared and visible imaging to measure the heat and particle flux to the divertor. The measured profiles have shown clear splitting in L-mode which compares well with the predication of the splitting location from modelling including the effect of screening. The splitting of the strike point has also been studied as a function of time during the ELM. The splitting varies during the ELM, being the strongest at the time of the peak heat flux and becoming more filamentary at the end of the ELM (200 µs after the peak midplane Dα emission). Variation in the splitting profiles has also been seen, with some ELMs showing clear splitting and others no splitting. A possible explanation of this effect is proposed, and supported by modelling, which concerns the relative phase between the RMP field and the ELM filament location.

Significant changes in the edge localized mode (ELM) crash heat load deposition patterns compared to typical ELMs are seen via infra-red observations during resonant magnetic perturbation experiments at the Joint European Torus (JET). These modifications result from the changed magnetic topology of the plasma, caused by the perturbations. Dependences on the perturbation strength and the edge safety factor are analysed and discussed. A thermoelectric current model shows that current filaments in the plasma edge could explain the observations. This study gives an insight into how the changed magnetic topology affects the peak heat fluxes of ELMs which is crucial for understanding ELM control.

The effect of resonant magnetic perturbations (RMPs) on particle transport is studied in the J-TEXT tokamak. It is found that for the discharges with an existing saturated 2/1 resistive tearing mode (TM), applied RMPs of moderate amplitude lead to a decrease in electron density with a relative amplitude ${\Delta \bar{{n}}_{{\rm e}}}/{\bar{{n}}_{{\rm e}0}}$ ranging from −3% to −10% in the plasma core, and the mode stabilization and electron temperature increase are observed simultaneously in this case. Sufficiently large amplitude of RMPs, however, leads to locked modes and much larger decrease in the electron density as well as in the electron temperature, with ${\Delta \bar{{n}}_{{\rm e}}}/{\bar{{n}}_{{\rm e}0}}\approx -20\%$. For the discharges without 2/1 TMs, applied RMPs cause a relative density decrease ${\Delta \bar{{n}}_{{\rm e}}}/{\bar{{n}}_{{\rm e}0}}\sim -10\%$ (−30%) before (after) field penetration. Using the two-fluid equations and experimental parameters as input, the nonlinear numerical results approximately agree with experimental observations.

In the J-TEXT tokamak, the penetration of resonant magnetic perturbations (RMPs) has been studied by using a set of in-vessel RMP coils. It is found that, once the RMP amplitude exceeds a critical value, the applied RMP can lead to field penetration and excitation of a large locked mode in the tearing-stable plasma. The sawtooth oscillations disappear and the confinement deteriorates significantly accompanied by tearing mode excitation. For the plasma with an initial high frequency tearing mode, the RMP can suppress the tearing mode, and field penetration followed with a further increased RMP. The relationship between the RMP penetration threshold and the electron density has been investigated for tearing-stable plasmas. It is found that the penetration threshold increases with the density and scales proportionally to $\bar{{n}}_{{\rm e}}^{0.5}$ in the ranges of (0.7–2.7) × 10¹⁹ m⁻³. Using the experimental parameters as input, the numerical modelling based on two-fluid equations gives the scaling of $b_{r}^{2.1} \propto \bar{{n}}_{{\rm e}}^{0.57}$, which approximately agrees with the experimental density scaling.

Lobe structures due to the application of resonant magnetic perturbations (RMPs) have been observed using wide-angle imaging of light from He¹⁺ ions in the vicinity of the lower X-point in MAST. The data presented are from lower single-null discharges where RMPs of toroidal mode number, n, of 4 and 6 were applied. It has been found that, above a threshold value, the lobe structures extend radially, linearly with the coil current, both in L-mode and H-mode. It is observed that after the application of the RMP, as the toroidal rotation in the confined plasma decreases, the lobes extend radially. This suggests the plasma is less effectively screening the RMP field. Comparing the imaging data with results from vacuum modelling shows that this technique can accurately predict the number and poloidal location of the lobes, but over-estimates their radial extent. More accurate estimates of the extent of the lobes can be made by accounting for plasma screening of the RMP field. Qualitative agreement between simulation and experiment is found if it is assumed that the RMP penetrates 2% in normalized radius from the last closed flux surface.

The lower hybrid wave (LHW) heating experiments at the Experimental Advanced Superconducting Tokamak (EAST) show a wide range of similarities to effects known from applied resonant magnetic perturbations (RMPs) by in-vessel or external magnetic perturbation coils. These observations suggest a current flow understood to be along scrape-off layer (SOL) field lines; here called helical current filaments (HCFs). For a better understanding of the experimental observations, a model to incorporate the magnetic perturbation of HCFs in the magnetic topology has been developed. Modelled SOL field lines, starting in front of the LHW antenna, show agreement in position and pitch-angle with the experimentally observed radiation belts. The comparison of the pick-up coil signals and the modelled HCFs' perturbation allows for determination of the current strength depending on the filaments' distance from the plasma edge. Agreement of predicted footprint structures with experimentally observed heat load and particle flux profiles at different toroidal angles in the divertor region is found. Based on the modelling results, the idea of LHW-induced RMPs, originating from the experimental observations, is strongly supported.

Kinked saturated m = 1 helical structures are frequently observed in tokamak hybrid plasmas and in reversed field pinches (RFP). These modes occur when an extremum in the safety factor is close to, but necessarily resonant with, a low order rational (typically q_min ≈ 1/1 in tokamaks, and q_max ≈ 1/7 in RFPs). If the exact resonance can be avoided, the essential character of these modes can be modelled assuming ideal nested magnetic flux surfaces. The methods used to characterize these structures include linear and nonlinear ideal magnetohydrodynamic stability calculations, which evaluate the departure from an axisymmetric plasma state, or equilibrium calculations using a 3D equilibrium code. The extent to which these approaches agree in tokamaks and reverse field pinches is investigated, and compared favourably for the first time with an analytic nonlinear treatment that is valid for arbitrary toroidal mode number.

Three-dimensional, ideal MHD tokamak equilibria are computed with the free-boundary NEMEC code, and their stability properties are investigated with the CAS3DN code, which is an ideal, linear MHD code. Numerical problems are discussed in detail by means of a test equilibrium with a large helical core. Further computations are performed for AUG-type equilibria perturbed by resonant magnetic perturbation (RMP) fields, and an ITER scenario taking into account the toroidal field ripple and perturbation fields caused by test blanket modules (TBMs). Moreover, a quantitative measure of the corrugation of the flux surfaces is introduced. It is found that the contour of the corrugation on equilibrium flux surfaces reflect the ideal kink structures of rational-q surfaces, and the periodicity of the external perturbation fields. Both stabilizing and destabilizing effects of the RMP fields, and the influence of the TBMs on the stability properties are observed.

Energetic ions are found to be transported strongly from the core of MAST hybrid-like plasmas during long-lived mode (LLM) magnetohydrodynamic activity. The resulting impact on the neutral beam ion deposition and concurrent current drive is modelled using the guiding-centre approximation in the internal kinked magnetic topology. General coordinate guiding-centre equations are extended for this purpose. It is found that the kinked core spirals around the position of strongest ionization, which remains geometrically centred, so that a large fraction of the population is deposited in the high shear external region where the plasma is almost axisymmetric. Those particles ionized in the low shear region exhibit exotic drift motion due to the strongly non-axisymmetric equilibrium, periodically passing near the magnetic axis and then reflected by the boundary of the kinked equilibrium, which in this respect acts as a confining pinch. Broad agreement is found against experimental measurement of fast ion particle confinement degradation as the MAST LLM amplitude varies.