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In hot laboratory plasmas, internal transport barriers (ITBs) have recently been observed, localized in the radial profile `around' rational values of the winding number ω(r) = 1/q(r). Such barriers are obviously related to the perturbed magnetic structure, described by a 1+1/2 Hamiltonian in presence of a perturbation. From the point of view of non-linear Hamiltonian dynamical systems [1], this experimental result appears highly paradoxical since rational q-values generally correspond to the less robust tori.

We have studied the appearance of chaos of toroidal magnetic lines by a discrete area-preserving map named `tokamap' [2]. By increasing the perturbation, we have observed in a wide chaotic sea the destruction of the last confining Kolmogorov–Arnold–Moser surfaces, broken and transformed into permeable Cantor sets (Cantori). The flux across a Cantorus has been computed by using refined mathematical techniques due to MacKay, Mather and Aubry. We have proved that the ITB observed in the tokamap is actually composed [3] of two permeable Cantori with `noble' values of ω (in the definition of Percival), each Cantorus forming a partial (permeable) barrier inhibiting the magnetic line motion.

More generally, between the dominant chains of rational islands q = m/(m−1), the most resistant barriers between q = (m+2)/(m+1) and (m+1)/m have been checked (Greene, MacKay and Stark) to be localized on the `most irrational' numbers in these Farey intervals, i.e. on the noble numbers N(1,m)≡1+[1/(m+1/G)] (where G is the Golden number), defined by their continuous fraction expansion N(i,m) = [i,m,(1)^∞].

In conclusion, the study of the tokamap mapping allowed us to predict on mathematical basis that ITB can occur in tokamak plasmas not only `around' rational magnetic surfaces but more precisely on nobleq−values of irrational surfaces, and to localize them by the Fibonacci series of their convergents.

The work presented in the following cluster of papers was performed during 2000-2001 on the JET device, now operated under the European Fusion Development Agreement (EFDA). Though not exhaustive, it consists of most of the experimental material and the main conclusions drawn from the activity of the so-called `EFDA-JET task force S2', in charge of developing Advanced Tokamak Regimes on this tokamak. It appeared important to us that collecting and presenting the material under a cluster of papers would focus the attention of the reader on the crucial role played by the plasma current density profile on the triggering, behaviour, performance, sustainment, internal and edge stability of internal transport barriers (ITBs). These observations allowed us to make significant progress towards a credible regime of advanced tokamak operation. For the first time, ITBs were real-time controlled on pressure profile characteristics and then sustained over pulse lengths that were limited only by the hardware capability.

On behalf of the very large number of authors of this cluster, acknowledgments are due to all the contributors to the EFDA-JET work programme, in particular those involved in the task force S2, and the non-European collaborators, as well as to UKAEA, operator of the EFDA-JET facilities, to the EFDA Close Support Unit in Culham (UK), and to the Plasma Physics and Controlled Fusion Editorial Board and Referees who immediately supported our publication initiative. My special thanks then go to Jean Jacquinot, Claude Gormezano and Robert Wolf for multiple discussions and advice on the management of the task force, to Jérôme Pamela and Franck Briscoe who are fully responsible for the success of the EFDA-JET experiment, and to Jill Membrey who managed the Editorial process.

Alain Bécoulet on behalf of the contributors to the EFDA-JET work programme

The fusion performance of JET plasmas can be enhanced by the generation of internal transport barriers. The influence of theq-profile shape in the local and global plasma performance has been investigated in cases where the core magnetic shear ranges from small and positive to large and negative. Internal barriers extending to large plasma radii can be effective in raising the global performance of the plasma. It is found that such barriers tend to be generated more easily if the q-profile contains a region of negative magnetic shear. The formation is favoured by neutral beam injection compared with ion cyclotron resonance heating in scenarios where the two systems are used together. The minimum power level required to observe a local transport reduction is significantly lower than the value at which very steep pressure gradients can be achieved. This results in a practical threshold in the power to access a regime of high plasma performance that is sensitive to the q-profile shape.

Quasi-steady operation has been achieved at JET in the high-confinement regime with internal transport barriers (ITBs). The ITB has been maintained up to 11 s. This duration, much larger than the energy confinement time, is already approaching a current resistive time. The high-performance phase is limited only by plant constraints. The radial profiles of the thermal electron and ion pressures have steep gradients typically at mid-plasma radius. A large fraction of non-inductive current (above 80%) is sustained throughout the high-performance phase with a poloidal beta exceeding unity. The safety factor profile plays an important role in sustaining the ITB characteristics. In this regime where the self-generated bootstrap current (up to 1.0 MA) represents 50% of the total current, the resistive evolution of the non-monotonic q-profile is slowed down by using off-axis lower-hybrid current drive.

We present the results of recent experiments related to real-time control of internal transport barriers (ITBs) in JET. Using a simple criterion to characterize the ITB existence, location and strength, we have successfully controlled for the first time the radial electron temperature profile within the ITB. The dimensionless variable used in the real-time algorithm - ratio of the ion gyro-radius to the local gradient scale length of the electron temperature - is a measure of the normalized electron temperature gradient and characterizes satisfactorily the main ITB features with a relatively low computational cost. We show several examples of control of this variable in various experimental conditions of toroidal field and plasma current, using different heating systems as control actuators. We also present a double-loop feedback scheme where both the global neutron rate from D-D reactions and the ITB strength are controlled simultaneously. In this case the ITB is sustained in a fully non-inductive current drive regime during several seconds. With the proposed control method, disruptions are avoided by holding the plasma performance at a prescribed target and this opens the route towards stationary operation of tokamak plasmas with ITBs. Initial results suggest that the additional control of the current profile is an important issue for achieving steady-state operation, in particular in the triggering and the sustainment of the ITB.

In JET discharges where lower hybrid heating and current drive (LHCD) is applied early during the current ramp, a region of the plasma with zero current density is formed near the axis. At the boundary of this region the current density is large and B_θ increases rapidly over a small distance. In the central region the safety factor, q, is effectively infinite, but this falls steeply in the boundary region. Outside the boundary region q reaches a minimum, where the magnetic shear s≡r/q (dq/dr) becomes zero. The formation of this region of zero current is dependent on both the heating and the current drive effects of the LHCD. When LHCD is switched off the current profile begins to relax towards the resistive peaked current distribution of fully inductive tokamak operation. If LHCD is not used in the current rise then these current profiles are not established. Although the physical mechanism exists to drive the central plasma current below zero, in most cases it appears to be prevented from going negative. At least one MHD mechanism has been identified which could be responsible for this. The presence of the zero central current is closely linked to the periodic relaxation events seen in these discharges. In these discharges, internal transport barriers have been observed with additional heating powers substantially below the values required to obtain barriers in monotonic q profile cases.

Injection of lower hybrid heating and current drive into the current ramp-up phase of JET discharges can produce extremely reversed q-profiles characterized by a core region of very small or zero current density (within motional Stark effect diagnostic measurement errors) and q_min>1. T_e-profiles show sawtooth-like collapses and the presence of an internal transport barrier. Accurate equilibrium reconstructions of these discharges are obtained using the ESC code, which was recently extended to allow equilibrium reconstructions in which a free boundary solver determines the plasma boundary and a fixed boundary solver provides the magnetic geometry and current density profile. The core current density does not appear to go negative, although current diffusion calculations indicate that sufficient non-inductive current drive to cause this is present. This is explained by nonlinear resistive MHD simulations in toroidal geometry which predict that these discharges undergo n = 0 reconnection events (axisymmetric sawteeth) that redistribute the current to hold the core current density near zero.

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.

By applying lower hybrid current drive (LHCD) for strong electron heating and off-axis current drive starting very early in the discharge, an electron internal transport barrier (eITB) can be generated. The barrier is formed just inside the location of minimum q, and slowly moves inward with this location. During the current flat-top the barrier can be sustained during many seconds, either with continued LHCD, or by ion cyclotron resonance heating. In this paper both scenarios are analysed and compared.

Low-frequency/long-wavelength density turbulence is found to be reduced within the volume enclosed by an electron thermal internal transport barrier generated with lower hybrid microwave heating in the JET tokamak. The turbulence reduction coincides with a region (typically r/a<0.4) of negative, or reversed magnetic shear s and reduced electron thermal diffusion χ_e. In the tokamak edge region (r/a>0.8), the turbulence amplitude is also reduced coinciding with large positive magnetic shear s>1. Estimates of the core turbulence wavelengths (λ_⊥> 0.1 m) are not in agreement with expectations from electron temperature gradient instabilities.

Transport calculations illustrate that the lower hybrid current drive (LHCD) and off-axis electron cyclotron current drive (ECCD) are the only preheating methods that can create a wide, deeply reversed q-profile, i.e. large negative magnetic shear, on the JET tokamak. Off-axis neutral beam injection (NBI) and off-axis ion cyclotron resonance heating (ICRH) preheating yields a weakly reversed q-profile (small negative magnetic shear), whereas NBI and ICRH on-axis heating as well as ohmic preheating produce a monotonic q-profile in the preheating phase. Here, on-axis power deposition and current drive refers to heating and current drive at or close to magnetic axis and correspondingly, off-axis refers to heating and current drive deposited typically around the half minor radius (r/a = 0.3-0.6). The results on LHCD, ICRH and ohmic preheating have been verified in the recent JET experiments. The current drive efficiency scan shows that in the case of LHCD, ECCD and off-axis NBI, the driven current is absolutely crucial to obtain a reversed q-profile and to modify the current profile evolution drastically in the preheating phase. Taking into account only the direct electron heating effect, LHCD does not create a reversed q-profile. The timing scans indicate that the radial location of q_min at the end of the preheating phase is generally quite insensitive to the start time of the preheating, once started 0-2 s after the plasma initiation if the method relies upon the driven current. On the other hand, methods relying only upon electron heating are very sensitive to that. In both cases, the magnitude of the negative magnetic shear, however, seems to be very sensitive to the start time of the preheating.

The ASDEX Upgrade advanced tokamak scenario with central q close to 1 has been reproduced on JET. For almost identical q profiles, the comparative analysis does show similar features like the fishbone activity and the current profile evolution. In JET, transport analyses indicates that an internal transport barrier (ITB) has been produced. Gradient length criterions based on the ion temperature gradient turbulence stabilization are used to characterize the ITBs in both devices. The trigger of ITBs is associated with rational surfaces in both devices although the underlying physics for this triggering seems different. This experiment has the prospect to get closer to identity experiments between the two tokamaks.

The transport, flow shear, and linear growth rates of microturbulence are studied for a JET plasma with high central qin which an internal transport barrier (ITB) forms and grows to a large radius. The linear microturbulence growth rates of the fastest growing (most unstable) toroidal modes with high toroidal mode number are calculated using the GS2 and FULL gyrokinetic codes. These linear growth rates, γ_lin, are large, but the flow-shearing rates, γ_E×B (dominated by the toroidal rotation contribution), are also comparably large when and where the ITB exists.

We present a method for parametrizing the Zeeman effect in hydrogen-like systems in high-temperature plasmas, where the fine-structure is completely unresolved. The method is based on the observation that the different polarization components behave collectively like separate entities, with simple relations. The entire Zeeman pattern can then be reduced to just three components, whose dependence on the magnetic field and the temperature can be described by only three numerical parameters. This makes it possible to include the influence of the Zeeman effect directly into ion temperature diagnostic procedures with minimal increase in the required computational effort and without the need for pre-calculated correction factors. We have tabulated such parametrizations—which are accurate for a wide range of fields and temperatures, even for cases when the total line-shape is no longer Gaussian—for 44 commonly studied hydrogen-like transitions. The effects of non-statistical population distribution in the upper sub-levels are briefly discussed, and we also note a temperature-dependent wavelength shift of the centre positions of the transitions.

Fast electrons generated via the interaction of ultra-intense laser pulses with a solid target can produce multi-MeV ions from laser-induced plasmas. These fast ions can be used for various applications ranging from the ion implantation to the stimulation of nuclear reactions. The most important point here is the efficiency of production of such fast ions. We analyse in detail, with the help of an analytical model and particle-in-cell simulations, the most efficient acceleration mechanisms including the ponderomotive force driving and acceleration by the shock wave, and compare the electrostatic ion acceleration at the front side and at the rear side of a foil target. We also determine the optimal plasma density distribution shaped by the laser pre-pulse.

The neutron emission spectrum from d+t→α+n reactions has been measured as a means to study the plasma response to radio frequency (RF) power coupled to hydrogen and deuteron minority components (through fundamental and second harmonic, respectively) in a tritium discharge at JET. The spectrum was measured with the magnetic proton recoil spectrometer and was analysed in terms of two spectral components due to thermal (TH) and high-energy (HE) deuterons interacting with the bulk ion population of thermal tritons. The results were used to derive information on the deuteron population in terms of temperatures (T_TH and T_HE) as well as corresponding particle and kinetic energy densities of the plasma; the bulk ion temperature (T_i = T_TH) was determined both before (with Ohmic heating only) and during the RF pulse. Similar information on protons was derived from other measurements in order to estimate the different RF effects on protons and deuterons. This paper illustrates qualitatively the type of empirical ion kinetic information that can be obtained from neutron emission spectroscopy; the data serves as a basis for comparison with results of predictive and interpretative models on RF effects in plasmas.

At temperatures below 15 000 K, the major pathway for the electron impact dissociation of H₂ is through excitation to the b ³Σ_u⁺ excited electronic state. Total cross sections and energy differential cross sections for threshold energies as a function of vibrational states (v) for H₂v = 0–4, D₂v = 0–6 and T₂v = 0–7 are calculated. The rates of dissociation as a function of electron temperature for each state are parametrized. Near-threshold rates are shown to be so critically dependent on the vibrational level that dissociation from very high-lying vibrational levels must be included in calculations of the rate at local thermal equilibrium (LTE) even at low temperature. An adapted version of the extrapolation procedure of Stibbe and Tennyson (1999 Astrophys. J.513 L147) is used to approximate the rates for all of the higher vibrational levels, which are then used to calculate the LTE rate. The LTE rate is an order of magnitude greater than the v = 0 rate. Calculations of energy differential cross sections suggest that impact dissociation of vibrationally excited molecules could be the source of low energy H atoms observed in tokamak plasmas.

This paper presents the results of studies of fast ion emission from the multiply charged high-Z number plasma generated using the PALS high-energy iodine laser system (⩽1.2 kJ, 0.4 ns) at the PALS Research Center in Prague. The properties of the emitted ion streams were investigated using ion collectors located at various angles with respect to the target normal and an electrostatic energy analyser. The x-ray emission from the plasma was measured using semiconductor detectors. Different groups of ions (slow, thermal and fast) were observed in the ion collector signals. Ion current densities higher than 80 mA cm⁻² at ∼1 m from the target were demonstrated. The charge velocity distribution, ion current density and angular distribution of ion charge emission, as well as total charge and average ion energy were obtained from these signals. Using the electrostatic ion-energy analyser, the emission of highly charged heavy ions (Ta⁵²⁺, Ag³⁸⁺) with energies up to 7 MeV for Ta ions was demonstrated. The dependence of ion stream parameters on the experimental conditions is discussed. We also report the results of preliminary experiments on the direct implantation of laser-produced ions into various materials.

An advanced high confinement mode scenario with normalized beta (β_N) up to 3.5 in stationary conditions (up to 20 energy confinement times) and β_N = 3.8 transiently, has been obtained in ASDEX Upgrade. For transport analysis of high β_N discharges, two types of simulations have been performed with the automated system for transport analysis code. One is the simulation of the current density profile with experimental data assuming neoclassical electrical conductivity. The other is the simulation of the energy transport and the current density profile using the Weiland transport model (ion temperature gradient and trapped electron mode limit the temperature gradient length). The comparison of the simulations of the q-profiles and temperature profiles with the experimental profiles shows good agreement and demonstrates that for this plasmas regime the temperature profiles are stiff (no internal transport barrier) as in conventional H-mode discharges. The contribution of the Ohmic, bootstrap and neutral beam driven current to the total current density profiles is calculated. The sum of the non-inductively driven plasma current is in the range 50–57% of the total plasma current. Simulations with the Weiland model are performed to investigate the influence of neoclassical tearing modes, which limit confinement and achievable β_N in this scenario, by comparing with the measured kinetic data. In addition, the high β_N discharges are compared to the improved H-mode discharges in ASDEX Upgrade with respect to the profile stiffness, MHD activity and experimental conditions.

Massive negative ions (C₆₀⁻) are locally produced in a collisionless magnetized-plasma (Q-machine plasma) column for the purpose of investigating a new aspect of negative ion plasma. It is experimentally found that time-averaged plasma profiles along the magnetic field are uniquely modified around the production region of massive negative ions, and a transition from stationary to dynamical plasma states takes place depending on a production rate of massive negative ions. In the dynamical state, solitary pulses with negative potential are spontaneously excited. An electrostatic PIC simulation which is based on Q-machine configuration is simultaneously performed in order to clarify effects of the local production of massive negative ions on the plasma structure. It is worth noting that in case of high production rate solitary pulses are intermittently excited and play a role in the rapid ejection of massive negative ions from the system.

Cases where three kinds of fluctuations having the different typical scale lengths coexist are analysed, and the statistical theory of strong turbulence in inhomogeneous plasmas is developed. Statistical nonlinear interactions between fluctuations are kept in the analysis as the renormalized drag, statistical noise and the averaged drive. The nonlinear interplay through them induces a quenching or suppressing effect, even if all the modes are unstable when they are analysed independently. Variety in mode appearance takes place: one mode quenches the other two modes, or one mode is quenched by the other two modes, etc. The bifurcation of turbulence is analysed and a phase diagram is drawn. Phase diagrams with cusp-type catastrophe and butterfly-type catastrophe are obtained. The subcritical bifurcation is possible to occur through the nonlinear interplay, even though each one is supercritical turbulence when analysed independently. Analysis reveals that the nonlinear stability boundary (marginal point) and the amplitude of each mode may substantially shift from the conventional results of independent analyses.

The study of (generally weak) relativistic damping of waves in plasmas is not new, but the principal analytical results have been obtained through a series of approximations leading to expressions for the dielectric tensor elements in terms of weakly relativistic plasma dispersion functions, where μ = m_ec²/kT_e≫1. If one focuses only on the case with propagation perpendicular to the magnetic field, however, where there is no damping at all without relativistic effects, it is possible to reduce the exact calculation of each dielectric tensor element to one or two integrals without sums or approximation. The exact results are compared with two approximate methods that do not involve a numerical integration, one being the weakly relativistic appproximation and the other a new extension to moderately relativistic plasmas.

The coupling of lower hybrid waves to the plasma is a crucial issue for efficient current drive in tokamaks. In this work, the coupling problem is attacked with a new tool in this context: an electromagnetic particle-in-cell (PIC) code, XOOPIC (Verboncoeur J P et al 1995 Comp. Phys. Comm.87 199). A model for a grill with 32 waveguides is constructed using perfectly conducting walls. The wave propagation in the waveguides and the coupling to plasma is followed. The wave–plasma interaction is studied and the evolution of the launched spectrum is resolved in the near-field of the grill. The reflection coefficients in the individual waveguides are determined.

Observations of internal core plasma structural behaviour during the magnetohydrodynamic (MHD) destabilization of the central cell plasmas are carried out by the use of our developed semiconductor x-ray detector arrays installed in both central cell and anchor regions of the GAMMA 10 tandem mirror. In the present paper, it is found from the developed x-ray diagnostics that the bulk plasmas rotate without a change in its shape and structure with an E×B velocity during the destabilization. The onset of the off-axis rotation is identified to be closely related to a scaling of the MHD stability boundary (i.e. the anchor beta requirements for stabilizing central cell hot ion plasmas). These data confirm pressure driven interchange instability in tandem mirror plasmas, and reveal the rigid rotational bulk plasma structure as the first demonstrated interior plasma property during the destabilization.

The lack of axisymmetry in stellarators guarantees that in general magnetic islands and chaotic magnetic field lines will exist. As particle transport is strongly tied to the magnetic field lines, magnetic islands and chaotic field lines result in poor plasma confinement. For stellarators to be feasible candidates for fusion power stations it is essential that, to a good approximation, the magnetic field lines lie on nested flux-surfaces, and the suppression of magnetic islands is a critical issue for stellarator coil design, particularly for small aspect ratio devices.

A procedure for modifying stellarator coil designs to eliminate magnetic islands in free-boundary full-pressure magnetohydrodynamic equilibria is presented. Islands may be removed from coil–plasma free-boundary equilibria by making small changes to the coil geometry and also by variation of trim coil currents. A plasma and coil design relevant to the National Compact Stellarator Experiment is used to illustrate the technique.

The Shafranov shift is derived from the two-dimensional profile of x-ray intensity measured with a soft x-ray CCD camera in large helical device. The accuracy of the measurement of the magnetic axis is 3% of the Shafranov shift at high β. The measured Shafranov shift increases linearly up to 280±3 mm, which is 47% of the minor radius as the volume averaged β ⟨β_dia⟩ measured with a diamagnetic loop is increased up to 2.6%. The Shafranov shift measured with a soft x-ray CCD camera agrees with that calculated with the three-dimensional equilibrium code VMEC. The reduction of the Shafranov shift due to the higher harmonics of the Fourier components of the magnetic field is also observed.

A spectroscopic study of VUV emission from injected and intrinsic low-Z impurities has been carried out for Tore Supra ergodic divertor (ED) plasmas. Analysis of plasmas in which a nearby limiter effectively provides a spatially localized source of recycled impurities provides information illuminating the dynamical processes of impurity penetration in ED plasmas. The profile evolution behaviour is found to be consistent with an interpretation which identifies both ED-influenced edge confinement and scrape-off layer and edge phenomena provided by the localized source. Preferentially increased edge impurity transport is a beneficial aspect of the ED.

The carbon concentration in the plasma of the stellarator experiment Wendelstein 7-AS is measured in the framework of the first island divertor campaign to examine divertor impurity control. Dedicated plasma density and electron cyclotron resonance heating power scans are performed and the influence of both the magnetic configuration and boronization of the vessel are investigated. A new diagnostic system is employed to measure C⁶⁺ densities via charge-exchange collisions with the neutral high-energy Li beam in the plasma edge and the core region. Carbon concentrations with respect to electron density are derived by using the density evaluation of the Li(2p) light profile. The results obtained during a period of approximately 2500 shots demonstrate that boronization reduces the carbon core concentration by a factor of three to a base level of 0.5–1%, which is not affected by further boronizations. The carbon concentration in island divertor configurations is found to be far below those obtained in a limiter configuration, which is attributed to a more than a factor two better impurity screening. The analysis of the carbon influx by spectroscopic measurements of C¹⁺ radiation suggests that the carbon core concentration in discharges of different densities and heating power is adequately described using the electron density at the last closed flux surface normalized to the line-averaged electron density as a scaling parameter. This simple global approach underlines the important role of the balance of impurity sources and impurity dilution. The effect of changes in impurity confinement on the measured carbon concentrations is also discussed.