Energetic Particle Distribution Function Research Articles

Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function f 0(r, p) at cylindrical radius r from the jet axis (assumed to lie along the z-axis) is given by convolving the particle momentum spectrum with the Green’s function , which describes the monoenergetic spectrum solution in which as r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for . The flow velocity u = u(r) e z is along the axis of the jet (the z-axis). u is independent of z, and u(r) is a monotonic decreasing function of r. The scattering time in the shear flow region 0 < r < r 2, and , where s > 0 in the region r > r 2 is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r, p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution that particles observed at (r, p) originated from r → ∞ with momentum . The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.

Validation of the current and pressure coupling schemes with nonlinear simulations of TAE and analysis on the linear stability of tearing mode in the presence of energetic particles

Both current and pressure coupling schemes have been adopted in the hybrid kinetic–magnetohydrodynamic code CLT-K recently. Numerical equivalences between these two coupling schemes are strictly verified under different approximations. First, when considering only the perturbed distribution function of energetic particles (EPs), the equivalence can be proved analytically. Second, when both the variations of the magnetic field and the EP distribution function are included, the current and pressure coupling schemes numerically produce the same result in the nonlinear simulations. On this basis, the influences of co-/counter-passing and trapped EPs on the linear stabilities of the m/n = 2/1 tearing mode (TM) have been investigated (where m and n represent the poloidal and toroidal mode numbers, respectively). The results of scanning of EPs show that the co-passing and trapped EPs are found to stabilize the TM, while the counter-passing EPs tend to destabilize the TM. The behind (de)stabilization mechanisms of the TM by EPs are carefully analyzed. Furthermore, after exceeding critical EP betas, the same branch of the high-frequency mode is excited by co-/counter-passing and trapped EPs, which is identified as the m/n = 2/1 energetic particle mode.

Description of global EGAM in the maximum of local frequency during current ramp-up discharges in DIII-D

Energetic-particle-induced geodesic acoustic modes, EGAMs (Fu, Phys. Rev. Let., vol. 101, 2008, pp. 185002), driven by neutral beam injection (NBI), have been observed in many DIII-D tokamak experiments (Nazikian et al., Phys. Rev. Lett., vol. 101, 2008, pp. 185001). This mechanism has been theoretically investigated in (Qiu et al., Plasma Phys. Control. Fusion, vol. 52, 2010, pp. 095003), using a sharp energetic particle distribution function, and in (Qu et al., Plasma Phys. Control. Fusion, vol. 59, 2017, pp. 055018), where the dispersion relation and eigenmode behaviour were obtained for the situation of early beam scenario, that is, for times smaller than the beam slowing down time. In this work, we extend these studies determining the eigenmode for beyond the slowing down time, in a scenario with reverse safety factor $q$ profile, where a small concentration of energetic ions can produce an off-axis maximum in the GAM dispersion relation. The characteristics of EGAM are analytically studied with the drift kinetic equation together with the MHD code NOVA. The toroidal energetic ion transit frequency, coupled with the GAM frequency, produces the maximum in the dispersion relation where the eigenmode can be found. The quantitative correspondence of experimental results with the predictions of the proposed model is analysed.

Orbit tomography of energetic particle distribution functions

Both fast ions and runaway electrons are described by distribution functions, the understanding of which are of critical importance for the success of future fusion devices such as ITER. Typically, energetic particle diagnostics are only sensitive to a limited subsection of the energetic particle phase-space which is often insufficient for model validation. However, previous publications show that multiple measurements of a single spatially localized volume can be used to reconstruct a distribution function of the energetic particle velocity-space by using the diagnostics’ velocity-space weight functions, i.e. velocity-space Tomography. In this work we use the recently formulated orbit weight functions to remove the restriction of spatially localized measurements and present orbit tomography, which is used to reconstruct the 3D phase-space distribution of all energetic particle orbits in the plasma. Through a transformation of the orbit distribution, the full energetic particle distribution function can be determined in the standard {energy, pitch, r, z}-space. We benchmark the technique by reconstructing the fast-ion distribution function of an MHD-quiescent DIII-D discharge using synthetic and experimental FIDA measurements. We also use the method to study the redistribution of fast ions during a sawtooth crash at ASDEX upgrade using FIDA measurements. Finally, a comparison between the orbit tomography and velocity-space tomography is shown.

Fuel flow and stock during deuterium-deuterium start-up of fusion reactor with advanced plasma model

Flow and stock of fuels in a fusion reactor has been investigated from the aspect of system-dynamics. In particular, operational scenario of the start-up only by deuterium has been discussed with an integrated model of burning plasmas including effects of dilution due to helium ash. The start-up only from deuterium has been attracting interests since initial loading of tritium in a fusion reactor is a critical issue because of limited availability of tritium. The present work solves the flow of tritium in a reactor with advanced modeling of plasma compared to a previous work. The temporal evolution of plasma temperature has been solved in an integrated manner of temperature and density profiles, an empirical scaling of energy confinement time, isotope effect of confinement, slowing down and velocity distribution function of energetic particles, dilution due to helium ash, and radiation losses. Performance of a reactor in terms of fusion power as well as tritium breeding has been assessed.

Critical energetic particle distribution in phase space for the Alfvén eigenmode burst with global resonance overlap

Comprehensive computer simulations of the Alfvén eigenmode burst, which is the synchronized sudden growth of multiple Alfvén eigenmodes (AEs) interacting with energetic particles, were conducted with continuous neutral beam injection, collisions, and particle losses. It is found that the energetic-particle distribution in phase space reaches a ‘critical distribution’ with a stairway structure where a resonance overlap triggers the Alfvén eigenmode burst. Before the burst, the gradual growth of the AEs associated with the beam injection broadens the resonant regions in phase space forming the distribution into a stairway shape. When the distribution reaches the ‘critical distribution,’ a resonance overlap triggers multiple resonance overlaps leading to the synchronized growth of AEs and the collapse of the distribution. For another run with the beam deposition power reduced to one-half, the energetic-particle distribution function just before the Alfvén eigenmode burst is close to that for the original beam power. This result indicates that the critical distribution for the Alfvén eigenmode burst is present.

Simulation of toroidicity-induced Alfven eigenmode excited by energetic ions in HL-2A tokamak plasmas

The toroidicity-induced Alfven eigenmode (TAE) excited by energetic (fast) ions was first simulated using the GTC code for HL-2A experiments. The simulation results indicate that the fraction of energetic ions is about 3% in the experiments. The critical density of energetic ions to excite the TAE mode is around 0.023, which is in good agreement with the theoretical expectation of 0.026. The real frequency of TAE with toroidal mode number n = 3 is around 211 kHz and is inversely proportional to the square root of electron density, which is quantitatively in agreement with the experimental observation (Ding et al 2013 Nucl. Fusion 53 043015). The frequency is in resonance with the summations of toroidal precession frequency and transit frequency of passing energetic ions, indicating that resonance with passing energetic ions is dominant for the excitation of the modes in the experiment. It is also checked by the phase space structures of (strength of energetic particle distribution function) that the TAE modes are mainly induced by passing energetic ions in HL-2A. The growth rates of TAE modes increase with fast ion density and its gradient. In addition, lower n TAE modes, such as n = 1 can also be driven by energetic ions when off-axis heating with higher beam energy is employed in the experiment. The width of radial mode structures for lower n modes is usually wider than those for higher n modes. The perpendicular wave vector of the modes and Larmor radius of ions satisfy the relation . Finally, the polarization of the mode shows that the perturbed parallel electric field is much smaller than the electrostatic parallel electric field.

Fast excitation of geodesic acoustic mode by energetic particle beams

A new mechanism for geodesic acoustic mode (GAM) excitation by a not fully slowed down energetic particle (EP) beam is analyzed to explain experimental observations in Large Helical Device. It is shown that the positive velocity space gradient near the lower-energy end of the EP distribution function can strongly drive the GAM unstable. The new features of this EP-induced GAM (EGAM) are: (1) no instability threshold in the pitch angle; (2) the EGAM frequency can be higher than the local GAM frequency; and (3) the instability growth rate is much larger than that driven by a fully slowed down EP beam.

Hole-Clump Pair Creation in the Evolution of Energetic-Particle-Driven Geodesic Acoustic Modes

Nonlinear frequency chirping of the energetic-particle-driven geodesic acoustic mode (EGAM) is investigated using a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic fluid. It is demonstrated in the simulation result that both frequency chirping up and chirping down take place in the nonlinear evolution of the EGAM. It is found that two hole-clump pairs are formed in the energetic particle distribution function in two-dimensional velocity space of pitch angle variable and energy. One pair is formed in the phase space region that destabilizes the instability, while the other is formed in the stabilizing region. The transit frequency of the hole (clump) in the destabilizing region chirps up (down), while in the stabilizing region the hole (clump) chirps down (up). The transit frequencies of particles in the holes and clumps are in good agreement with the chirping EGAM frequency indicating that the particles are kept resonant with the EGAM during the nonlinear frequency chirping. Continuous energy transfer takes place from the destabilizing phase space region to the stabilizing region during the spontaneous frequency chirping of the wave.

Production and diagnosis of energetic particles in FAST

The Fusion Advanced Study Torus (FAST) has been proposed as a possible European satellite facility to study fast-ion physics in deuterium plasmas under conditions relevant to a burning plasma. Energetic minority ions (H or 3He) accelerated by ion cyclotron resonance heating (ICRH), with dimensionless parameters close to those of fusion-born alphas in ITER, can be produced in FAST via 30 MW power ICRH minority heating. This work provides a first assessment of the extent to which the 3He fast-ion population can be diagnosed in FAST with a set of dedicated diagnostics for confined fast particles. Neutron emission spectroscopy (NES), gamma-ray spectroscopy (GRS) and collective Thomson scattering (CTS) diagnostics have been reviewed with a description of the state-of-the-art hardware and a preliminary analysis of the required lines of sight. The results of the analysis, based on numerical simulations of the spatial and energetic particle distribution function of the ICRH-accelerated ions for the standard FAST H-mode scenario, suggest that NES and GRS measurements can provide information on the fast 3He population effective tail temperature, with time resolutions in the range 20–100 ms. The proposed CTS diagnostic can measure the fast-ion parallel and perpendicular temperature with a spatial resolution of 5–10 cm and a time resolution of 10 ms. The paper provides a scientific basis for the prediction of the production and diagnosis of energetic ions in FAST.

Experimental evidence of nonthermal acceleration of relativistic electrons by an intensive laser pulse

Nonthermal acceleration of relativistic electrons is investigated with an intensive laser pulse. An energy distribution function of energetic particles in the universe or cosmic rays is well represented by a power-law spectrum, therefore, nonthermal acceleration is essential to understand the origin of cosmic rays. A possible candidate for the origin of cosmic rays is wakefield acceleration at relativistic astrophysical perpendicular shocks. The wakefield is considered to be excited by large-amplitude precursor light waves in the upstream of the shocks. Substituting an intensive laser pulse for the large amplitude light waves, we performed a model experiment of the shock environments in a laboratory plasma. An intensive laser pulse was propagated in a plasma tube created by imploding a hollow polystyrene cylinder, as the large amplitude light waves propagated in the upstream plasma at an astrophysical shock. Nonthermal electrons were generated, and the energy distribution functions of the electrons have a power-law component with an index of ~2. We described the detailed procedures to obtain the nonthermal components from data obtained by an electron spectrometer.

Minority heating by ICRH: a tool for investigating burning plasma physics in FAST

A combined Fokker–Planck numerical analysis of the quasi-linear plasma–ion-cyclotron (IC) wave interaction and collisional relaxation of minority ion tails created by IC absorption was performed in order to determine the characteristic fast-ion parameters that are necessary for addressing some of the main ITER burning plasma physics issues, e.g. fast-ion transport due to collective mode excitations, cross-scale couplings of micro-turbulence with meso-scale fluctuations due to energetic particles, etc. These investigations refer to actual scenarios of the Fusion Advanced Studies Torus (FAST), a conceptual tokamak design operating with deuterium plasmas in a dimensionless parameter range similar to that of ITER and equipped with IC resonance heating (ICRH) as a main heating scheme. The destabilization and saturation of fast-ion driven Alfvénic modes below and above the energetic particle modes stability threshold are investigated by numerical simulations with the HMGC code, which assumes the anisotropic energetic particle distribution function accelerated by ICRH as input. The results of this study, obtained by integration of different numerical simulation analyses aimed at investigating the various relevant physics, are presented and discussed.

Nonlinear interaction between a resonance-mode (k∥=0) wave and energetic plasma particles

AbstractNonlinear resonant interaction between energetic particles and quasi-electrostatic wave propagating perpendicularly to the ambient magnetic field in a homogeneous plasma is studied in detail, with the main focus on the wave amplification or attenuation caused by resonant interaction. As for k∥ → 0, resonance velocity determined from linear resonance conditions tends to infinity, the interaction under discussion is a completely nonlinear phenomenon. While the wave amplitude may increase or decrease depending on the wave parameters and integral characteristics of the energetic particle distribution function, an essential wave amplification may occur when the wave frequency is close to a multiple of cyclotron harmonic, and when the transversal energy of resonant particles is large enough.

Maintaining the quasi-steady state central current density profile in hybrid discharges

Experimental observations in a number of tokamaks operated in the hybrid regime revealed that the presence of a rotating neoclassical island substantially reduces the occurrence of sawtooth. This possibility of a rotating mangetic island driving counter currents near the plasma centre in present-day tokamaks is studied. Three mechanisms are investigated with two of them driving sufficient current. They rely on establishing an oscillating parallel electric field by the rotating neoclassical island. First is the excitation of an electrostatic side band through diamagnetic and curvature drifts; second is the excitation of the kinetic Alfvén wave at the plasma centre through the polarization drifts. The third mechanism is the ‘prompt’ modification of the energetic particle distribution function by the neoclassical island and was found to be relatively weak. The effect of the energetic particles on the counter-current drive due to modification of the energetic particle distribution function on the long (energetic particle slowing-down) time scale has been simulated using a particle simulation code. A firm conclusion has not yet been drawn. Transport simulations indicate that present experimental observations could be the effect of either a negative current drive in the background plasma or the anomalous transport of the energetic particles.

Diffusive‐Compression Acceleration and Turbulent Diffusion of Cosmic Rays in Quasi‐periodic and Turbulent Flows

Multiple scale perturbation methods are used to study the transport and acceleration of energetic charged particles in quasi-periodic, fluid velocity structures in one, two, or three space dimensions, with spatial period lu, where lu is much less than the diffusion scale length ld = κ0/|u0| and κ0 and u0 are characteristic values of the energetic particle diffusion coefficients and fluid speed, respectively. The particle diffusion tensor K is also allowed to vary periodically on the scale lu. In the case in which the perturbation parameter = lu/ld ≡ u0lu/κ0 is small (0 < 1), the long space and time behavior of the energetic particle distribution function f at lowest order is shown to satisfy a modified Fokker-Planck equation. This equation arises from compatibility conditions imposed on the perturbation equations in order to obtain a consistent perturbation expansion that is free of secular terms. The analysis shows that the particles are accelerated stochastically on the large scale as a result of the divergence ∇ δu of the background fluid velocity perturbation δu. The net acceleration of the particles due to the velocity variations can be described in part by a second-order Fermi-like momentum space diffusion term in the long-scale transport equation obtained by averaging over the short-scale variations. The momentum space diffusion coefficient DT describing the effect depends on the two-point correlation of the fluid velocity divergence ∇ δu at different points in the flow. There is also a further energization term in the long-scale transport equation, corresponding to the work done by the scattering center fluid against the differential cosmic-ray pressure gradient that is modified as a result of the short-scale variations. The convective particle streaming is also modified as a result of the short-scale variations. The analysis shows that the effective spatial diffusion tensor for low-energy particles can be significantly modified as a result of turbulent diffusion, whereas higher energy particles with much larger diffusion tensor elements are not significantly affected by turbulent diffusion. Averaging over a random ensemble of short-scale, quasi-periodic velocity structures generalizes the turbulent transport coefficients obtained by previous authors.

Particle acceleration and coronal mass ejection driven shocks: A theoretical model

There is increasing evidence to suggest that energetic particles observed in “gradual” solar energetic particle (SEP) events are accelerated at shock waves driven out of the corona by coronal mass ejections. Energetic particle abundances suggest too that SEPs are accelerated from in situ solar wind or coronal plasma rather than from high‐temperature flare material. A dynamical time‐dependent model of particle acceleration at a propagating, evolving interplanetary shock is presented here. The theoretical model includes the determination of the particle injection energy (injection here refers to the injection of particles into the diffusive shock acceleration mechanism), the maximum energy of particles accelerated at the shock, energetic particle spectra at all spatial and temporal locations, and the dynamical distribution of particles that escape upstream and downstream from the evolving shock complex. As the shock evolves, energetic particles are trapped downstream of the shock and diffuse slowly away. In the immediate vicinity of the shock, broken power law spectra are predicted for the energetic particle distribution function. The escaping distribution consists primarily of very energetic particles initially with a very hard power law spectrum (harder than that at the shock itself) with a rollover at lower energies. As the shock propagates further into the solar wind, the escaping ion distribution fills in at lower energies, and the overall spectrum remains hard. Downstream of the shock, the shape of the accelerated particle spectrum evolves from a convex, broken power law shape near the shock to a concave spectrum far downstream of the shock. Intensity profiles for particles of different energies are computed, and the relation between arrival times, maximum predicted energies, and shock propagation characteristics are described. These results are of particular importance in the context of predictive space weather studies.

Ambiguity in Determining the Diffusion Coefficient from Shock-Associated Energetic Particle Increases

A numerical solution to the Fisk and Lee (1980) equations for the particle intensity upstream of a corotating interplanetary shock is considered for the November 1991 event observed at Ulysses. A numerically derived parallel diffusion coefficient is available for this region (Quenby et al., 1993), based upon in-situ magnetometer data. Fitting the transport equations solution to the upstream energetic particle distribution function, employing a radial diffusion coefficient К = К0υα r∈, where r and υ are, respectively, radial distance from the Sun and particle velocity, and with К0 fixed from the magnetometer derived coefficient yielded a range of statistically acceptable values of (α, ∈). These ran from (0.5, 0.0) to (1.8, 1.6) along a thin strip of α — ∈ space, hence demonstrating the improbability that the velocity and radial dependence of the particle diffusion can be fixed from such particle and magnetic field data alone.

Analytic Solution for Nonlinear Shock Acceleration in the Bohm Limit

The selfconsistent steady state solution for a strong shock, significantly modified by accelerated particles is obtained on the level of a kinetic description, assuming Bohm-type diffusion. The original problem that is commonly formulated in terms of the diffusion-convection equation for the distribution function of energetic particles, coupled with the thermal plasma through the momentum flux continuity equation, is reduced to a nonlinear integral equation in one variable. Its solution provides selfconsistently both the particle spectrum and the structure of the hydrodynamic flow. A critical system parameter governing the acceleration process is found to be $\Lambda = M^{-3/4}\Lambda_1 $, where $ \Lambda_1 =\eta p_1/mc $, with a suitably normalized injection rate $ \eta $, the Mach number M >> 1, and the cut-off momentum $ p_1 $. We particularly focus on an efficient solution, in which almost all the energy of the flow is converted into a few energetic particles. It was found that (i) for this efficient solution (or, equivalently, for multiple solutions) to exist, the parameter $ \zeta =\eta\sqrt{p_0 p_1}/mc $ must exceed a critical value $\zeta_{cr} \sim 1$ ($p_0$ is the injection momentum), (ii) the total shock compression ratio r increases with M and saturates at a level that scales as $ r \propto \Lambda_1 (iii) the downstream power-law spectrum has the universal index q=3.5 over a broad momentum range. (iv) completely smooth shock transitions do not appear in the steady state kinetic description.

Diffusion model of dechanneling based on the Boltzmann equation: Three regimes of energetic particle motion in crystals

AbstractThe generalized local Boltzmann equation is derived for the distribution function of energetic particles interacting with the thermal vibrations of the lattice. On using the Boltzman collision term the particle energy losses A(ϵ⊥) and the diffusion function B(ϵ⊥) are obtained. The functions A(ϵ⊥) and B(ϵ⊥) have singularities (fracture, peak) connected with the difference of the contributions of the particles moving in the regimes of channeling, quasichanneling, and random motion.

Impact spectropolarimetry in the hydrogen H? line: present atomic data and astrophysical applications

The physics of impact polarization is briefly reviewed in this paper. Methods for derivating the impact spectropolarimetric parameters relevant for the formation of Hα linear polarization are summarized. Spectropolarimetric techniques can provide information on the velocity distribution function of energetic particles. Since the most advanced application of this technique to astrophysics is in solar flare physics, recent results achieved by its application to particles accelerated in solar flares are presented.