Fokker-Planck Calculations Research Articles

A unified description of atomic physics for electron Fokker–Planck calculations

Most realistic kinetic calculations for tokamak plasmas are now required to incorporate the effect of partially ionized high-Z elements arising either from uncontrolled influxes of metallic impurities, such as tungsten in high input power regimes or from mitigation of runaway electrons generated after possible major disruptions by massive gas injection. The usual electron–ion Fokker–Planck collision operator must therefore be modified, because all plasma atoms are not entirely ionized, as is the case for light elements. This represents a challenge, in order to perform fast but also accurate calculations, regardless of the type of element present in the plasma, but also their local levels of ionization while covering a wide range of electron energies in a consistent way, from a few keV to tens of MeV in plasmas whose electron temperature may itself vary from 10 eV to several keV. In this context, a unified description of the atomic models is proposed, based on a multi-Yukawa representation of the electrostatic potential calibrated against results obtained by advanced quantum calculations. Besides the possibility to improve the description of inner and outer atomic shells in the determination of the atomic form factor, this model allows one to derive analytical formulations for both elastic and inelastic scattering, which can then be easily incorporated in kinetic calculations. The impact of the number of exponentials in the description of the atomic potential is discussed, and a comparison with simple and advanced atomic models is also performed.

Thermal transport modeling of laser-irradiated spheres

Thermal transport of uniformly laser-irradiated spheres of various materials is investigated computationally. One-dimensional simulations of low- to mid-Z materials (Be, Al, and Cu) are performed to evaluate the impact of nonlocal electron transport on experimental observables under laser intensities of relevance to direct-drive inertial confinement fusion. We compare thermal transport models of different levels of fidelity: flux-limited Spitzer–Harm diffusion, the Schurtz–Nicolai–Busquet (SNB) reduced-order nonlocal model, and a Fokker–Planck description. Spitzer–Harm diffusion with different flux-limiter factors are compared with different implementations of the SNB model in the HYDRA radiation hydrodynamics code. Under the conditions of interest, the peak heat flux in the thermal front with the SNB model shows good agreement with Fokker–Planck calculations, with the largest errors below 10% at 1015 W/cm2 laser intensity. From HYDRA-SNB simulations, two experimentally relevant effects are observed from nonlocal heat transport when compared to flux-limited Spitzer–Harm modeling: coronal temperatures are cooler due to reduced heat fluxes in the expanding plasma and (for mid-Z materials) x-ray emissions are enhanced due to preheating in the dense plasma.

Excitation of Alfvén eigenmodes by fusion-born alpha-particles in D-3He plasmas on JET

Alfvén eigenmode (AE) instabilities driven by alpha-particles have been observed in D-3He fusion experiments on the Joint European Torus (JET) with the ITER-like wall. For the efficient generation of fusion alpha-particles from D-3He fusion reaction, the three-ion radio frequency scenario was used to accelerate the neutral beam injection 100 keV deuterons to higher energies in the core of mixed D-3He plasmas at high concentrations of 3He. A large variety of fast-ion driven magnetohydrodynamic modes were observed, including the elliptical Alfvén eigenmodes (EAEs) with mode numbers n = −1 and axisymmetric modes with n = 0 in the frequency range of EAEs. The simultaneous observation of these modes indicates the presence of rather strong alpha-particle population in the plasma with a ‘bump-on-tail’ shaped velocity distribution. Linear stability analysis and Fokker–Planck calculations support the observations. Experimental evidence of the AEs excitation by fusion-born alpha-particles in the D-3He plasma is provided by neutron and gamma-ray diagnostics as well as fast-ion loss measurements. We discuss an experimental proposal for the planned full-scale D-T plasma experiments on JET based on the physics insights gained from these experiments.

Chemical potential in active systems: predicting phase equilibrium from bulk equations of state?

We derive a microscopic expression for a quantity μ that plays the role of chemical potential of active Brownian particles (ABPs) in a steady state in the absence of vortices. We show that μ consists of (i) an intrinsic chemical potential similar to passive systems, which depends on density and self-propulsion speed, but not on the external potential, (ii) the external potential, and (iii) a newly derived one-body swim potential due to the activity of the particles. Our simulations on ABPs show good agreement with our Fokker–Planck calculations, and confirm that is spatially constant for several inhomogeneous active fluids in their steady states in a planar geometry. Finally, we show that phase coexistence of ABPs with a planar interface satisfies not only mechanical but also diffusive equilibrium. The coexistence can be well-described by equating the bulk chemical potential and bulk pressure obtained from bulk simulations for systems with low activity but requires explicit evaluation of the interfacial contributions at high activity.

Lower hybrid current drive experiments with different launched wave frequencies in the EAST tokamak

EAST has been equipped with two high power lower hybrid current drive (LHCD) systems with operating frequencies of 2.45 GHz and 4.6 GHz. Comparative LHCD experiments with the two different frequencies were performed in the same conditions of plasma for the first time. It was found that current drive (CD) efficiency and plasma heating effect are much better for 4.6 GHz LH waves than for the one with 2.45 GHz. High confinement mode (H-mode) discharges with 4.6 GHz LHCD as the sole auxiliary heating source have been obtained in EAST and the confinement is higher with respect to that produced previously by 2.45 GHz. A combination of ray-tracing and Fokker-Planck calculations by using the C3PO/LUKE codes was performed in order to explain the different experimental observations between the two waves. In addition, the frequency spectral broadening of the two LH wave operating frequencies was surveyed by using a radio frequency probe.

Propagation of the lower hybrid wave in a density fluctuating scrape-off layer (SOL)

The perturbation of the lower hybrid wave (LH) power spectrum by fluctuations of the plasma in the vicinity of the antenna is investigated by solving the full wave equation in a slab geometry using COMSOL Multiphysics®. The numerical model whose generality allows to study the effect of various types of fluctuations, including those with short characteristic wavelengths is validated against a coupling code in quiescent regimes. When electron density fluctuations along the toroidal direction are incorporated in the dielectric tensor over a thin perturbed layer in front of the grill, the power spectrum may be strongly modified from the antenna mouth to the plasma separatrix as the LH wave propagates. The diffraction effect by density fluctuations leads to the appearance of multiple satellite lobes with randomly varying positions and the averaged perturbation is found to be maximum for the Fourier components of the fluctuating spectrum in the vicinity of the launched LH wavelength. This highlights that fast toroidal inhomogeneities with short characteristics length scales in front of the grill may change significantly the initial LH power spectrum used in coupled ray-tracing and Fokker–Planck calculations.

Integrated tokamak modelling with the fast-ion Fokker–Planck solver adapted for transient analyses

Integrated tokamak modelling that enables the simulation of an entire discharge period is indispensable for designing advanced tokamak plasmas. For this purpose, we extend the integrated code TOPICS to make it more suitable for transient analyses in the fast-ion part. The fast-ion Fokker–Planck solver is integrated into TOPICS at the same level as the bulk transport solver so that the time evolutions of the fast ion and the bulk plasma are consistent with each other as well as with the equilibrium magnetic field. The fast-ion solver simultaneously handles neutral beam-injected ions and alpha particles. Parallelisation of the fast-ion solver in addition to its computational lightness owing to a dimensional reduction in the phase space enables transient analyses for long periods in the order of tens of seconds. The fast-ion Fokker–Planck calculation is compared and confirmed to be in good agreement with an orbit following a Monte Carlo calculation. The integrated code is applied to ramp-up simulations for JT-60SA and ITER to confirm its capability and effectiveness in transient analyses. In the integrated simulations, the coupled evolution of the fast ions, plasma profiles, and equilibrium magnetic fields are presented. In addition, the electric acceleration effect on fast ions is shown and discussed.

Globular Clusters within Dark Matter Halos: Case Studies of 47 Tuc, NGC 1851 and M 15

AbstractGlobular clusters (GCs) are known to have a very small amount of or no dark matter (DM). Even if GCs are formed in individual DM halos, they must have lost the majority of the DM through dynamical processes such as mass segregation or tidal stripping. Using Fokker-Planck (FP) calculations, we investigate the dynamical evolution of three Galactic GCs with an assumption that they were formed in mini DM halos. We trace the amount of DM of 47 Tuc, NGC 1851, and M 15, which are a ‘disk/bulge’ cluster, an ‘old halo’ cluster, and a ‘young halo’ cluster, respectively. We find that these three GCs must have initially had insignificant amounts of DM, less than 10 percent of the initial stellar mass of each cluster.

FORMATION AND EVOLUTION OF NUCLEAR STAR CLUSTERS WITH IN SITU STAR FORMATION: NUCLEAR CORES AND AGE SEGREGATION

Nuclear stellar cluster (NSCs) are known to exist around massive black holes (MBHs) in galactic nuclei. Two formation scenarios were suggested for their origin: (1) Build-up of NSCs from consecutive infall of stellar cluster and (2) Continuous in-situ star-formation. Though the cluster-infall scenario has been extensively studied in recent years, the in-situ formation scenario have been hardly explored. Here we use Fokker-Planck (FP) calculations to study the effects of star formation on the build-up of NSCs and its implications for their long term evolution and their resulting structure. We use the FP equation to describe the evolution of several stellar populations, and add appropriate source terms to account for the effects of newly formed stars. We show that continuous star-formation even 1-2 pc away from the MBH can lead to the build-up of an NSC with properties similar to those of the Milky-way NSC. We also find that the general structure of the old stellar population in the NSC with in-situ star-formation could be very similar to the steady-state Bahcall-Wolf cuspy structure. However, its younger stellar population do not yet achieve a steady state. In particular, formed/evolved NSCs with in-situ star-formation contain differential age-segregated stellar populations which are not yet fully mixed. Younger stellar populations formed in the outer regions of the NSC have a cuspy structure towards the NSC outskirts, while showing a core-like distribution inwards; with younger populations having larger core sizes. In principal, such a structure can give rise to an apparent core-like radial distribution of younger (up to 2-3 Gyrs) stars, as observed in the Galactic center. Such an NSC still preserves an underlying stellar cusp of older stars, that can be potentially be missed by current observations of red-giants.

Advances in multi-megawatt lower hybrid technology in support of steady-state tokamak operation

It has been demonstrated that lower hybrid current drive (LHCD) systems play a crucial role for steady-state tokamak operation, owing to their high current drive (CD) efficiency and hence their capability to reduce flux consumption. This paper describes the extensive technology programmes developed for the Tore Supra (France) and the KSTAR (Korea) tokamaks in order to bring continuous wave (CW) LHCD systems into operation. The Tore Supra LHCD generator at 3.7 GHz is fully CW compatible, with RF power PRF = 9.2 MW available at the generator to feed two actively water-cooled launchers. On Tore Supra, the most recent and novel passive active multijunction (PAM) launcher has sustained 2.7 MW (corresponding to its design value of 25 MW m−2 at the launcher mouth) for a 78 s flat-top discharge, with low reflected power even at large plasma-launcher gaps. The fully active multijunction (FAM) launcher has reached 3.8 MW of coupled power (24 MW m−2 at the launcher mouth) with the new TH2103C klystrons. By combining both the PAM and FAM launchers, 950 MJ of energy, using 5.2 MW of LHCD and 1 MW of ICRH (ion cyclotron resonance heating), was injected for 160 s in 2011. The 3.7 GHz CW LHCD system will be a key element within the W (for tungsten) environment in steady-state Tokamak (WEST) project, where the aim is to test ITER technologies for high heat flux components in relevant heat flux density and particle fluence conditions. On KSTAR, a 2 MW LHCD system operating at 5 GHz is under development. Recently the 5 GHz prototype klystron has reached 500 kW/600 s on a matched load, and studies are ongoing to design a PAM launcher. In addition to the studies of technology, a combination of ray-tracing and Fokker–Planck calculations have been performed to evaluate the driven current and the power deposition due to LH waves, and to optimize the N∥ spectrum for the future launcher design. Furthermore, an LHCD system at 5 GHz is being considered for a future upgrade of the ITER Heating and Current Drive systems, with a power capability of 20 MW coupled to the plasma using a PAM launcher. An R&D programme is being conducted at CEA/IRFM to develop a BeO vacuum window which is a safety critical component of the transmission line. In addition, a mock-up of a TE10–TE30 mode converter at 5 GHz, designed for a rectangular transmission line, has been manufactured and successfully tested on Tore Supra at low RF power.

Dynamical modeling of the Arches cluster using Fokker-Planck calculations

AbstractThe Arches cluster is a young, compact, and massive star cluster located in ~30pc away from the Galactic Center in projection. The cluster is located in the extreme environment of the Galactic Center, making it an excellent target for understanding the effects of star-forming environment on the mass function of star clusters. In this study, we estimate the initial condition (mass, concentration parameter, and galactocentric radius) of the Arches cluster by comparing Fokker-Planck calculations with observed velocity dispersion, surface density and mass function data.

Non-linear effects in electron cyclotron current drive applied for the stabilization of neoclassical tearing modes

Due to the smallness of the volumes associated with the flux surfaces around the O-point of a magnetic island, the electron cyclotron power density applied inside the island for the stabilization of neoclassical tearing modes (NTMs) can exceed the threshold for non-linear effects as derived previously by Harvey et al (1989 Phys. Rev. Lett. 62 426). We study the non-linear electron cyclotron current drive (ECCD) efficiency through bounce-averaged, quasi-linear Fokker–Planck calculations in the magnetic geometry as created by the islands. The calculations are performed for the parameters of a typical NTM stabilization experiment on ASDEX Upgrade. A particular feature of these experiments is that the rays of the EC wave beam propagate tangential to the flux surfaces in the power deposition region. The calculations show significant non-linear effects on the ECCD efficiency, when the ECCD power is increased from its experimental value of 1 MW to a larger value of 4 MW. The nonlinear effects are largest in the case of locked islands or when the magnetic island rotation period is longer than the collisional time scale. The non-linear effects result in an overall reduction of the current drive efficiency for this case with absorption of the EC power on the low-field side of the electron cyclotron resonance layer. As a consequence of the non-linear effects, also the stabilizing effect of the ECCD on the island is reduced from linear expectations.

An improved time-dependent nonlocal electron heat-flux model and its verification by laser-driven Al foil acceleration experiment

In hydrodynamics simulation of laser driven systems, the time-dependent nonlocal electron heat-flux models predict the saturation (flux inhibition) and delocalization of the heat-flux automatically. Therefore it avoids commonly used time and space-independent ad hoc flux limiting. Previously proposed analytical nonlocal heat-flux model of Luciani et al. [Phys. Rev. Lett., 51, p-1664, (1983)] which fits the results of numerical Fokker–Planck calculations is simple and straight forward to implement in a fluid code. The proposed expression, however, is a convolution of Spitzer–Harm heat-flux with a delocalization kernel which depends on classical electron collision mean free path. This is rigorously valid for high temperature non-degenerate plasmas. However, in laser driven systems, the energy transport due to electron thermal conduction is important in regions between the critical density and ablation surface where the plasma is mostly degenerate. We have improved this nonlocal heat-flux model by using a wide-range electron collision frequency model valid from warm-dense matter (degenerate plasmas) to fully ionized plasmas. The effect of this improved nonlocal heat-flux model on the free-surface velocity of laser-accelerated Al foils of thickness 2–10 μm is studied by using a two-dimensional radiation hydrodynamics code. The simulated free surface velocities are compared with our experimental results for laser intensities in the range 4 × 1013–3 × 1014 W/cm2. Preliminary analysis shows that the simulation results obtained with improved nonlocal heat-flux model yields better agreement with our experimental values.

Improved dynamical modelling of the Arches cluster

AbstractRecently, Clarkson et al. (2012) measured the intrinsic velocity dispersion of the Arches cluster, a young and massive star cluster in the Galactic center. Using the observed velocity dispersion profile and the surface brightness profile of Espinoza et al. (2009), they estimate the cluster's present-day mass to be ∼ 1.5×104 M⊙ by fitting an isothermal King model. In this study, we trace the best-fit initial mass for the Arches cluster using the same observed data set and also the anisotropic Fokker-Planck calculations for the dynamical evolution.

INITIAL SIZE DISTRIBUTION OF THE GALACTIC GLOBULAR CLUSTER SYSTEM

Despite the importance of their size evolution in understanding the dynamical evolution of globular clusters (GCs) of the Milky Way, studies are rare that focus specifically on this issue. Based on the advanced, realistic Fokker-Planck (FP) approach, we predict theoretically the initial size distribution (SD) of the Galactic GCs along with their initial mass function and radial distribution. Over one thousand FP calculations in a wide parameter space have pinpointed the best-fit initial conditions for the SD, mass function, and radial distribution. Our best-fit model shows that the initial SD of the Galactic GCs is of larger dispersion than today's SD, and that typical projected half-light radius of the initial GCs is ~4.6 pc, which is 1.8 times larger than that of the present-day GCs (~2.5 pc). Their large size signifies greater susceptibility to the Galactic tides: the total mass of destroyed GCs reaches 3-5x10^8 M_sun$, several times larger than the previous estimates. Our result challenges a recent view that the Milky Way GCs were born compact on the sub-pc scale, and rather implies that (1) the initial GCs are generally larger than the typical size of the present-day GCs, (2) the initially large GCs mostly shrink and/or disrupt as a result of the galactic tides, and (3) the initially small GCs expand by two-body relaxation, and later shrink by the galactic tides.

Calculations of lower hybrid current drive in ITER

A detailed study of lower hybrid current drive (LHCD) in ITER is provided, focusing on the wave propagation and current drive mechanisms. A combination of ray-tracing and Fokker–Planck calculations are presented for various plasma scenarios, wave frequency and polarization. The dependence of the driven current and the location of power deposition upon the coupled wave spectrum is systematically determined, in order to set objectives for the antenna design. The respective effects of finite-power levels, magnetic trapping, and detailed antenna spectra are accounted for and quantitatively estimated. The sensitivity of LHCD to density and temperature profiles is calculated. From the simulation results, an optimum value for the parallel index of refraction is proposed as a compromise between efficiency and robustness with respect to those profile variations. The corresponding current drive efficiency is found to be similar for the two frequencies generally considered for ITER, f = 3.7 GHz and f = 5.0 GHz.

Simulation of the neutral beam deposition within integrated tokamak modelling frameworks

The NEMO (NEutral beam MOdelling) code for simulating neutral beam ionization during neutral beam injection (NBI) in tokamak plasmas has been developed for implementation in integrated modelling frameworks and is presented. Integrated modelling of fusion plasmas is becoming increasingly important, both for preparation and analysis of experiments in large devices. Moreover, it should play a crucial role for the design of future fusion reactors. In a modern context, integrated modelling requires codes that are (i) flexible in terms of their interfaces, i.e. can be adapted for different simulation environments, (ii) machine independent; i.e. they should not contain hard coded information on a particular device to be simulated; (iii) optimized for speed of execution, (iv) verified and validated. The NEMO code has been specially designed to meet these requirements. The code is based on the physics concept outlined by Feng et al (1995 Comput. Phys. Commun. 88 161–72) and is a completely modular program: it works with any input NBI geometry and can be coupled to any external Fokker–Planck calculation for evaluating the distribution function of the injected species, i.e. it can provide source terms for both Monte Carlo codes and codes using finite difference/elements methods. The NEMO code has already been integrated with the CRONOS integrated modelling suite (Artaud et al 2010 Nucl. Fusion 50 043001) and the European Integrated Tokamak Modelling Task Force (ITM-TF)b https://www.efda-itm.eu/index.html. The basics of the code are described in this paper along with an illustration of its integration in the ITM-TF simulation platform. A crucial aspect is the verification of the code, the results of benchmarks carried out with other NBI codes for JET and ITER discharges are thereby presented.

Plasma wave simulation based on a versatile finite element method solver

The possibility of plasma wave simulation based on the finite element method (FEM) was explored, and a new full wave simulation code of the lower hybrid (LH) wave, Lower Hybrid wavE Analysis based on FEM (LHEAF), was developed. The efficient iterative solver for the LH wave [O. Meneghini, S. Shiraiwa, and R. Parker, Phys. Plasmas 16, 090701 (2009)] was coupled to one-dimensional Fokker–Planck calculation to obtain the self-consistent distribution function. In LHEAF, seamless handling of the core, scrape off layer, and antenna regions were realized and the wave launching structure were naturally included into the simulation. Simulation results of a Maxwellian tokamak plasma showed good agreement with ray tracing calculations and the TORIC-LH spectral solver [J. C. Wright, P. T. Bonoli, A. E. Schmidt et al., Phys. Plasmas 16, 072502 (2009)]. Compared to spectral domain solvers, the computational requirements are reduced significantly, allowing an Alcator C [M. Porkolab, J. J. Schuss, B. Lloyd et al., Phys. Rev. Lett. 53, 450 (1984)] scale plasma simulation on a desktop computer. Also, the LH full wave simulation of an ITER scale plasma was demonstrated for the first time with moderate increase in the problem size. In addition, the flexibility of the FEM approach has been exploited to address issues of antenna-plasma coupling in the LH and ion cyclotron range of frequencies (ICRF). Techniques using the FEM package for this purpose were validated on the traditional grill antenna and have been applied to the interdigital-line antenna of LH wave. Application to the near field analysis of the new Alcator C-Mod [I. H. Hutchinson, R. Boivin, F. Bombarda et al., Phys. Plasmas 1, 1511 (1994)] ICRF antenna is in progress.

Consequences of finite transport on the effectiveness of ECCD for neoclassical tearing mode stabilization in ITER

One of the main aims of the ITER electron cyclotron resonance heating (ECRH) system, in particular of the Upper Port Launcher, is the control of magnetohydrodynamics instabilities. This control typically requires non-inductively driven currents with a high degree of localization, i.e. with a very narrow profile. A numerical analysis of the effect of the radial diffusion of the EC driven current carrying electrons has been performed in order to estimate the effectiveness of electron cyclotron current drive (ECCD) for neoclassical tearing mode (NTM) stabilization. In particular, Fokker–Planck calculations including radial diffusion for the case of the ITER ECRH Upper Port Launcher are presented. These show a significant decrease in the local current density when radial diffusion at a rate of only 1 m2 s−1 is included and consequently a broadening of the profile with a drop in the predicted efficiency for NTM control. Furthermore, it is shown that a simple formula combining the effect of the radial diffusion and the width of the EC power deposition profile reproduces quite accurately the maximum EC driven current density, which is the more relevant number in determining the NTM suppression figure of merit, for typical ITER parameters.

An assessment of full wave effects on the propagation and absorption of lower hybrid waves

Lower hybrid (LH) waves (Ωci⪡ω⪡Ωce, where Ωi,e≡Zi,eeB/mi,ec) have the attractive property of damping strongly via electron Landau resonance on relatively fast tail electrons and consequently are well-suited to driving current. Established modeling techniques use Wentzel–Kramers–Brillouin (WKB) expansions with self-consistent non-Maxwellian distributions. Higher order WKB expansions have shown some effects on the parallel wave number evolution and consequently on the damping due to diffraction [G. Pereverzev, Nucl. Fusion 32, 1091 (1991)]. A massively parallel version of the TORIC full wave electromagnetic field solver valid in the LH range of frequencies has been developed [J. C. Wright et al., Comm. Comp. Phys. 4, 545 (2008)] and coupled to an electron Fokker–Planck solver CQL3D [R. W. Harvey and M. G. McCoy, in Proceedings of the IAEA Technical Committee Meeting, Montreal, 1992 (IAEA Institute of Physics Publishing, Vienna, 1993), USDOC/NTIS Document No. DE93002962, pp. 489–526] in order to self-consistently evolve nonthermal electron distributions characteristic of LH current drive (LHCD) experiments in devices such as Alcator C-Mod and ITER (B0≈5 T, ne0≈1×1020 m−3). These simulations represent the first ever self-consistent simulations of LHCD utilizing both a full wave and Fokker–Planck calculation in toroidal geometry.