Fast-ion Distribution Function Research Articles

Particle simulation of bursting Alfvén modes in JT-60U

The results of particle-in-cell simulations of a negative neutral beam heated Alfvén-mode experiment in the Japan Atomic Energy Research Institute Tokamak-60 Upgrade (JT-60U) [H. Ninomiya et al., Fusion Sci. Technol. 42, 7 (2002); A. Kitsunezaki et al., Fusion Sci. Technol. 42, 179 (2002)] are presented. They seem to match quite well the dynamics of the abrupt large-amplitude event (ALE) experimentally observed in the reference JT-60U discharge. The time scale and frequency spread of the ALE are well reproduced too. The issue of the weaker Alfvén fluctuation phase following the ALEs, characterized by fast frequency sweeping modes, is also investigated and an interpretation of the full JT-60U bursting-mode phenomenology is presented. Finally, the simulation tool is exploited by ad hoc synthetic diagnostics on the fast ion distribution function to get a deeper insight into the ALE nonlinear dynamics. The underlying fast-growing energetic particle mode saturates as resonant energetic ions are scattered out of the resonance region and displaced outwards. The radially displaced ions resonate with outer Alfvén modes and enhance their local drive, consistently with the “avalanche” paradigm for mode nonlinear dynamics and energetic ion transports.

Weak effect of ion cyclotron acceleration on rapidly chirping beam-driven instabilities in the National Spherical Torus Experiment

The fast-ion distribution function in the National Spherical Torus Experiment is modified from shot to shot while keeping the total injected power at ∼2 MW. Deuterium beams of different energy and tangency radius are injected into helium L-mode plasmas, producing a rich set of instabilities, including compressional Alfvén eigenmodes, toroidicity-induced Alfvén eigenmodes (TAE), 50–100 kHz instabilities with rapid frequency sweeps or chirps, and strong, low frequency (10–20 kHz) fishbones. The experiment was motivated by a theory that attributes frequency chirping to the formation of holes and clumps in phase-space. In the theory, increasing the effective collision frequency of the fast ions that drive the instability can suppress frequency chirping. In the experiment, high-power (≲3 MW) high harmonic fast wave (HHFW) heating accelerates the fast ions in an attempt to alter the nonlinear dynamics. Steady-frequency TAE modes diminish during the HHFW heating but there is little evidence that frequency chirping is suppressed.

Simulation of fast Alfvén wave interaction with beam ions over a range of cyclotron harmonics in DIII-D tokamak

To study fast Alfvén wave damping on fast ions, a Monte-Carlo code, ORBIT-RF, has been coupled with a 2D full wave code, TORIC4. The ORBIT-RF/TORIC4 combination has been applied to DIII-D experimental conditions to investigate fast wave (FW) heating of injected beam ions over a range of ion cyclotron harmonics. ORBIT-RF using a single dominant toroidal and poloidal Fourier wave number qualitatively reproduces the strong FW–beam interaction at 60 MHz (4ΩD and 5ΩD) and the much weaker interaction at 116 MHz (8ΩD) in DIII-D L-mode plasmas, consistent with experimental observations. The result at 8ΩD differs from linear theory prediction using a Maxwellian to model the fast ion distribution function, suggesting the importance of finite orbit effect, Coulomb collisions for transport across flux surfaces and details of the non-Maxwellian fast ion distribution.

Experimental studies of instabilities and confinement of energetic particles on JET and MAST

In preparation for next step burning plasma devices such as ITER, experimental studies of instabilities and confinement of energetic ions were performed on Joint European Torus (JET) and on Mega-Amper Spherical Tokamak (MAST) with innovative diagnostic techniques, in conventional and shear-reversed plasmas, exploring a wide range of effects for energetic ions. A compendium of recent results testing capabilities of the present-day facilities for burning plasma relevant study is presented in this paper. ‘Alpha tail’ production using 3rd harmonic ion-cyclotron resonance heating (ICRH) of 4He beam ions has been employed on JET for studying 4He of the megaelectronvolt energy range in a ‘neutron-free’ environment. The evolution of ICRH-accelerated ions of 4He with E ⩾ 1.7 MeV and D with E ⩾ 500 keV was assessed from nuclear gamma-ray emission born by the fast ions colliding with Be and C impurities. A simultaneous measurement of spatial profiles of fast 4He and fast D ions relevant to ITER was performed for the first time in positive and strongly reversed magnetic shear discharges. Time-resolved gamma-ray diagnostics for ICRH-accelerated 3He and H minority ions allowed changes in the fast ion distribution function to be assessed in the presence of unstable toroidal Alfvén eigenmodes (TAEs) and sawteeth. A significant decrease of gamma-ray intensity from protons with E ⩾ 5 MeV was detected during the ‘tornado’ modes. This was interpreted as ‘tornado’-induced loss of fast ions with the drift orbit width, Δf, comparable to the minor radius of tokamak a. Experiments performed in the opposite case, Δf/a ≪ 1, for ICRH-accelerated 3He ions with E ⩾ 500 keV, have shown excitation of numerous Alfvén eigenmodes without a significant degradation of the fast ion confinement. The stabilizing effect of fast particles on ‘monster’ sawteeth was experimentally found to fail in low-density plasmas with high power ion cyclotron resonance frequency (ICRF)-heating. The transition from the ‘monster’ to short-period ‘grassy’ sawteeth was investigated with different ICRF phasing, which controls the pinch-effect and radial distribution of ICRF-accelerated ions. Instabilities excited by super-Alfvénic beam ions were investigated on the spherical tokamak MAST. Due to higher values of β and a higher proportion of fast ions on MAST than on JET, a wider variety of modes and nonlinear regimes for the Alfvén instabilities were observed, including the explosive TAE-regimes leading to the formation of hole-clump pairs on the fast ion distribution function. The MAST and START data showed that TAE and chirping modes decrease both in their mode amplitudes and in the number of unstable modes with increasing β.

Non-linear study of fast particle excitation of Alfvén eigenmodes during ICRH

High-power ion–cyclotron resonance heating (ICRH) can produce centrally peaked fast ion distributions with wide non-standard drift orbits exciting Alfvén eigenmodes (AEs). The dynamics of the AE excitation depends not only on the anisotropy and the peaking of the fast ion distribution but also on the decorrelation of the AE interactions and the renewal of the fast ions resonant with the AE by ion–cyclotron interactions. A method of self-consistently including the evolution of the distribution function of fast ions during excitation of AEs and ICRH has been developed and implemented in the SELFO code. Numerical simulations of the AE dynamics and ICRH give a variation of the AE amplitude consistent with the experimentally observed splitting of the mode frequency. The experimentally observed fast damping of the mode as the ICRH is switched off is also evident in the simulations.

A1fven Eigenmodes in Spherical Tokamaks

Electromagnetic instabilities are often excited by fast super-Alfvenic ions produced by neutral beam injection (NBI) in plasmas of the spherical tokamaks START and MAST (toroidal magnetic confinement devices in which the minor a and major R0 radii of the torus are comparable, R0/a = 1.2÷1.8). These instabilities are seen as discrete weakly-damped toroidal and elliptical Alfven eigenmodes (TAEs and EAEs) with frequencies tracing in time the Alfven scaling with the equilibrium magnetic field and plasma density, or as energetic particle modes (EPMs) whose frequencies don't start from TAE-frequency and sweep down in time faster than the equilibrium parameters change. In some discharges the beam drives Alfvenic-type modes that start from the TAE frequency and sweep in both up- and down- directions. Such electromagnetic perturbations are interpreted as `hole-clump' long-living nonlinear fluctuations of the fast ion distribution function predicted by Berk-Breizman-Petviashvili [Phys. Lett. A238 (1998) 408]. It is found on both START and MAST that the Alfven instabilities weaken in their mode amplitude and in the number of unstable modes as the pressure of the thermal plasma increases, in agreement with increased thermal ion Landau damping and the pressure effect on core-localised TAEs.

Error estimation and parameter dependence of the calculation of the fast ion distribution function, temperature, and density using data from the KF1 high energy neutral particle analyzer on Joint European Torus

Joint European Torus high energy neutral particle analyzer measures the flux of fast neutrals originating from the plasma core. From this data, the fast ion distribution function fifast, temperature Ti,⊥fast, and density nifast are derived using knowledge of various plasma parameters and of the cross section for the required atomic processes. In this article, a systematic sensitivity study of the effect of uncertainties in these quantities on the evaluation of the neutral particle analyzer fifast, Ti,⊥fast, and nifast is reported. The dominant parameter affecting nifast is the impurity confinement time and therefore a reasonable estimate of this quantity is necessary to reduce the uncertainties in nifast below 50%. On the other hand, Ti,⊥fast is much less sensitive and can certainly be provided with an accuracy of better than 10%.

Phenomenology of compressional Alfvén eigenmodes

Coherent oscillations with frequency 0.3⩽ω/ωci⩽1, are seen in the National Spherical Torus Experiment [M. Ono, S. M. Kaye, Y.-K. M. Peng et al., Nucl. Fusion 40, 557 (2000)]. This paper presents new data and analysis comparing characteristics of the observed modes to the model of compressional Alfvén eigenmodes (CAE). The toroidal mode number has been measured and is typically between 7<n<9. The polarization of the modes, measured using an array of four Mirnov coils, is found to be compressional. The frequency scaling of the modes agrees with the predictions of a numerical two-dimensional code, but the detailed structure of the spectrum is not captured with the simple model. The fast ion distribution function, as calculated with the beam deposition code in TRANSP [R. V. Budny, Nucl. Fusion 34, 1247 (1994)], is shown to be qualitatively consistent with the constraints of the Doppler-shifted cyclotron resonance drive model. This model also predicts the observed scaling of the low frequency limit for CAE.

Energetic particle mode stability in tokamaks with hollow q-profiles

A thorough analysis of energetic particle modes (EPM) stability and mode structures is presented for tokamaks with hollow q profiles. Focusing on the region near the minimum-q surface, EPM gap modes and resonant EPMs are shown to exist as solutions of the same dispersion relation. By controlling the fast ion distribution function, or, equivalently, their fundamental dynamical properties, a smooth transition between these two classes of modes is obtained within the EPM dispersion relation. When toroidal coupling becomes important, it is demonstrated that EPMs may have either single or double hump radial structures. The local analyses of EPM stability and mode structures near the minimum-q surface are put in the broader framework of EPM stability and EPM induced transport in tokamaks with hollow q profiles and a brief summary is also given of present understanding of such problems based on results of three-dimensional nonlinear hybrid magneto–hydrodynamic–gyrokinetic simulations. Possible implications of present results are discussed in terms of experimental observations and possibilities of designing novel experimental setups to probe, at least conceptually, the complex predictions of theory.

Alpha particle physics in a tokamak burning plasma experiment

Much is known about the behavior of energetic ions in tokamak devices but much remains to be understood. Single-particle effects are well understood and provide a firm basis for extrapolation to a burning plasma. In contrast, collective effects involving fast ions are more poorly understood and extrapolations are unreliable. Collective modes of concern include toroidicity-induced and ellipticity-induced Alfvén eigenmodes, kinetic ballooning modes, and internal kink modes. In addition to these magnetohydrodynamic normal modes, there are also energetic particle modes characterized by strong dependence on the fast-ion distribution function. Although many issues are important areas of study in current experiments, five issues distinguish burning plasma experiments. First, the energetic alphas are not the dominant source of free energy for the instabilities unless the fusion power exceeds the heating power by a factor of 10. Second, the damping of the instabilities depends sensitively on mode coupling to other heavily-damped waves. The magnitude of this coupling is expected to depend on the normalized thermal gyroradius, which is much smaller in a reactor. Third, in a reactor, both the radial extent of the instabilities and the fast-ion orbit contract relative to current experiments, so the fast-ion transport will change. Fourth, when instability occurs, a larger number of modes are unstable, so the mechanism of nonlinear saturation could shift from fast-ion transport to mode coupling. Fifth, because of the extreme sensitivity of energetic particle modes to the distribution function, an isotropic alpha particle distribution function differs from anisotropic fast-ion populations.

Neutral beam heating in the START spherical tokamak

The tight aspect ratios (typically A≈1.4) and low magnetic field of spherical tokamak (ST) plasmas, when combined with densities approaching the Greenwald limit, provide a significant challenge for all currently available auxiliary heating and current drive schemes. NBI heating and current drive are difficult to interpret in sub-megampere machines, as in order to achieve suitable penetration into the plasma core, fast ions have to be highly suprathermal and, as a result of the low magnetic field, can be non-adiabatic (i.e. non-conserving of magnetic moment µ0). The physics of NBI heating in START is discussed. The neutral beam injector deployed on START was clearly successful, having been instrumental in producing a world record tokamak toroidal beta of ≈40%. A fast ion Monte Carlo code (LOCUST) is described that was developed to model non-adiabatic fast ion topologies together with a high level of charge exchange loss and cross-field transport (present in START due to an envelope of high density gas surrounding the plasma). Model predictions compare well with experimental data, collected using a scanning neutral particle analyser, bolometric instruments and equilibrium reconstruction using EFIT. In particular, beta calculations based upon reconstruction of the pressure profile (by combining measurements from Thomson scattering, charge exchange recombination spectroscopy and model predictions for the fast ion distribution function) agree well with beta values calculated using EFIT alone (the routine method for calculation of START beta). These results thus provide increased confidence in the ability of STs to sustain high beta high confinement H mode plasmas and in addition indicate that the injected fast ions in collisional START plasmas evolve mainly due to collisional and charge exchange processes, without driving any significant performance degrading fast particle MHD activity.

Observation of MeV multicharged ions and hot electrons accelerated by a 65-fs laser pulse

The generation of fast particles (both ions and electrons) in a femtosecond laser-produced plasma has been studied experimentally and the possibility of applying particle-in-cell (PIC) codes to the description of the experimental results has been investigated. The energy distribution function of fast ions has been measured directly by means of X-ray spectroscopy. It is shown that this diagnostic can also be used as an indirect method to measure fast electrons inside the plasma. The hot electron distribution in the 0.1–2 MeV energy range was measured directly using a standard electron magnetic spectrometer. The comparison of the experimental results with the numerical simulations show: (1) the electromagnetic PIC-FC code can be successfully applied for simulating plasmas created by the interaction of femtosecond-laser pulses with solid targets and (2) the existence of a transient population of hot electrons with a non-Maxwellian distribution typical of interaction experiments with short-pulse lasers.

Origin of superthermal ion cyclotron emission in tokamaks

It is shown that the key mechanism in the origin of superthermal ion cyclotron emission (ICE) appearing in tokamak fusion experiments is the instability of fast magnetoacoustic waves having very small longitudinal wavenumbers and being excited through toroidicity affected cyclotron resonance with an energetic ion population. This new type of cyclotron resonance leads to an instability growth rate well exceeding the inverse bounce/transit period of fast ions even for a small number of high energy ions. The proposed theory explains the main features of the ICE spectrum observed in JET and the main conclusions are obtained without any assumptions about the explicit form of the fast ion distribution function

Effect of pitch angle scattering on the distribution function of fast fusion products in a tokamak plasma

The results of a three dimensional (3-D) (in constants of motion space) Fokker-Planck simulation of the steady state collisional behaviour of fusion alphas in an axisymmetric TFTR-like tokamak plasma are presented. The distribution function of confined fusion products is shown to be strongly anisotropic and non-uniform in the radial co-ordinate. The influence of pitch angle scattering on distribution function anisotropy is investigated. It is shown that the distribution function of fast ions in the range where the cosine of the pitch angle is small is strongly influenced by this scattering. A qualitative comparison of numerical results and predictions of alpha particle distributions from previous studies is presented

Theory of fast ion transport during sawtooth crashes in tokamaks

An approach is suggested for the description of fast ion transport in tokamaks during sawtooth crashes, which takes into account the orbital motion of the particles and the temporal evolution of the magnetic configuration. The approach is based on a kinetic equation for the distribution function of fast ions derived using different scales with characteristic times of the sawtooth crash and of the fast ion motion. The evolution of the magnetic configuration in the process of the crash is assumed to be known, and its analytical description is suggested for the case of the Kadomtsev type of crash. The derived kinetic equation is analysed for both trapped and circulating particles, and conclusions are drawn concerning redistribution of particles of various energies and pitch angles during crashes. The physical nature of fast ion transport induced by the sawtooth crash is elucidated

Excitation of Alfvén cyclotron instability by charged fusion products in tokamaks

The spectrum of ion cyclotron emission (ICE) observed in tokamak experiments shows narrow peaks at multiples of the edge cyclotron frequency of background ions. A possible mechanism of ICE based on the fast Alfvén Cyclotron Instability (ACI) resonantly excited by high energy charged products (α-particles or protons) is presented here. Two-dimensional eigenmode analysis of ACI mode structure and eigenfrequency are obtained in the large tokamak aspect ratio limit. The ACI is excited via wave-particle resonances in phase space by tapping the fast ion velocity space free energy. The instability growth rates are computed perturbatively from the perturbed fast particle distribution function, which is obtained by integrating the high frequency gyrokinetic equation along the particle orbit. Numerical examples of ACI growth rates are presented for Tokamak Fusion Test Reactor (TFTR) [World Survey of Activities in Controlled Fusion Research [Nuclear Fusion special supplement 1991] (International Atomic Energy Agency, Vienna, 1991)] supershot plasmas. The fast ion distribution function is assumed to be singular in pitch angle near the plasma edge. The results are employed to understand the ICE in Deuterium-Deuterium (DD) and Deuterium-Tritium (DT) tokamak experiments.

Sawtooth oscillations and fast-ion ejection in tokamaks

The authors present a theory of fast-ion transport and fast-ion plasma heating in tokamaks with sawtooth oscillations. The theory is valid for ions with orbit widths that are small compared with the region in which the oscillations occur. It is shown (a) that a sawtooth crash leads to the ejection from the central plasma region of particles with a peaked radial distribution, and (b) that the fast-ion distribution function and the radial plasma heating profile strongly depend on the ratio of the sawtooth oscillation period to the time of Coulomb slowing down of fast ions. The results obtained in this theory are in agreement with neutral injection experiments at JET, where changes in neutron yield have been observed during internal disruptions. However, the comparison of theory and experiment must be regarded as preliminary owing to uncertainties in the database and the simplified theoretical model. The effect of sawtooth oscillations on alpha -particles in a reactor such as ITER is also considered

Quasi-linear theory calculation of ion tails and neutron rates during lower hybrid heating in Alcator-A

The steady-state fast ion distribution function and the resulting neutron rate are calculated for the conditions of the Alcator-A lower hybrid heating experiment from quasi-linear theory. First, ion orbit losses are ignored and the steady-state ion distribution function is calculated. It is found that in this case the experimental neutron rates and neutron rate decay times are consistent with only a small fraction of the incident radio-frequency (RF) power being absorbed in the plasma centre. This absorbed power is centred in the regime 3.5 < n‖ < 4.5. A Monte-Carlo ion simulation technique incorporating ripple orbit losses is then presented which calculates the steady-state ion distribution function in the presence of the RF. The results of this simulation require approximately 50% of the incident RF power to be dissipated in the plasma core in order to obtain agreement with the experimental results. Most of this RF power is lost to ripple-trapped ions. These calculations demonstrate the importance of orbit losses in determining lower hybrid heating efficiencies and additionally provide a technique for calculating them.

Magneto-acoustic cyclotron instability due to neutral beam injection

Instabilities driven by energetic ion beams injected for ion heating in magnetically confined plasmas have attracted considerable interest in connection with both plasma heating and containment. In this paper, the range of fast-ion distribution functions resulting from near-perpendicular neutral injection into Doublet-3 is determined using a numerical Fokker-Planck code and a guiding-centre Monte Carlo code. Guided by these results, we consider the possibility of exciting magnetosonic cyclotron instability due to anisotropic non-Maxwellian energetic ions streaming at an arbitrary angle with respect to the magnetic field, and show the possibility of unstable waves with fairly large growth rates. The threshold condition and the range of physical parameters where such a growth is possible are obtained. We also consider the contribution of magnetically trapped energetic beam particles to the growth of such magneto-acoustic cyclotron waves.

Effect of Neutral Gas and Impurity on Loss of Injected Ions during Slowing Down

Fast ions produced by neutral beam injection are lost mainly by charge exchange with the cold neutral gas and pitch-angle scattering into an orbit loss region. The loss of injected ions due to these processes during slowing down is discussed by obtaining a numerical solution of the fast-ion distribution function. The effect of impurity ions on the containment properties of injected ions is investigated, as well as the effect of charge exchange with the cold neutral gas. The loss of injected ions due to orbit loss is also investigated. Furthermore, an analytical treatment on the enhancement of angular diffusion by impurity ions is also described.