Fokker-Planck Calculations Research Articles

Rapid self-magnetization of laser speckles in plasmas by nonlinear anisotropic instability

Presented here are the first kinetic two-dimensional Vlasov– Fokker–Planck calculations of inertial confinement fusion-related laser–plasma interactions, to include self-consistent magnetic fields, hydrodynamic plasma expansion and anisotropic electron pressure. An underdense plasma, reminiscent of the gas fill of a hohlraum, is heated by a laser speckle with Iλ2=1.0×1015 W cm− 2μm2 and radius w0=5 μm. Inverse bremsstrahlung absorption of the laser and non-local electron transport lead to the development of a collisional analogue of the Weibel electromagnetic instability. The instability is seeded by magnetic fields, generated in an initial period of linear growth due to the anisotropic electron distribution arising in a laser speckle. Using the circular polarization does not generate significant fields. For linear polarization, the field generally saturates when the magnetization is ωτei> 1, and the effective growth rate is similar to the coherence time of typical laser speckles. The presence of these magnetic fluctuations significantly affects the heat fluxes and hydrodynamics in the plasma.

Dynamical evolution of the mass function and radial profile of the Globular Cluster systems of the Milky Way and M87

AbstractEvolution of the mass function (MF) and radial distribution (RD) of the globular cluster (GC) systems of the Milky Way and M87 are calculated using an advanced and realistic Fokker‐Planck (FP) model that considers dynamical friction, disk/bulge shocks, and eccentric cluster orbits. We perform hundreds of FP calculations with different initial cluster conditions, and then search a wide parameter space for the best‐fit initial GC MF and RD that evolves into the observed present‐day GC MF and RD. By allowing both MF and RD of the initial GC system to vary, we find that in case of the Milky Way, our best‐fit models have a higher peak mass for a log‐normal initial MF and a higher cutoff mass for a powerlaw initial MF than previous estimates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamical evolution of the mass function and radial profile of the Galactic globular cluster system

Abstract Evolution of the mass function (MF) and radial distribution (RD) of the Galactic globular cluster (GC) system is calculated using an advanced and a realistic Fokker–Planck (FP) model that considers dynamical friction, disc/bulge shocks and eccentric cluster orbits. We perform hundreds of FP calculations with different initial cluster conditions, and then search a wide-parameter space for the best-fitting initial GC MF and RD that evolves into the observed present-day Galactic GC MF and RD. By allowing both MF and RD of the initial GC system to vary, which is attempted for the first time in the present Letter, we find that our best-fitting models have a higher peak mass for a lognormal initial MF and a higher cut-off mass for a power-law initial MF than previous estimates, but our initial total masses in GCs, MT,i= 1.5–1.8 × 108 M⊙, are comparable to previous results. Significant findings include that our best-fitting lognormal MF shifts downward by 0.35 dex during the period of 13 Gyr, and that our power-law initial MF models well-fit the observed MF and RD only when the initial MF is truncated at ≳105 M⊙. We also find that our results are insensitive to the initial distribution of orbit eccentricity and inclination, but are rather sensitive to the initial concentration of the clusters and to how the initial tidal radius is defined. If the clusters are assumed to be formed at the apocentre while filling the tidal radius there, MT,i can be as high as 6.9 × 108 M⊙, which amounts to ∼75 per cent of the current mass in the stellar halo.

Fokker-Planck modeling of current penetration during electron cyclotron current drive

The current penetration during electron cyclotron current drive (ECCD) on the resistive time scale is studied with a Fokker-Planck simulation, which includes a model for the magnetic diffusion that determines the parallel electric field evolution. The existence of the synergy between the inductive electric field and EC driven current complicates the process of the current penetration and invalidates the standard method of calculation in which Ohm’s law is simply approximated by j−jcd=σE. Here it is proposed to obtain at every time step a self-consistent approximation to the plasma resistivity from the Fokker-Planck code, which is then used in a concurrent calculation of the magnetic diffusion equation in order to obtain the inductive electric field at the next time step. A series of Fokker-Planck calculations including a self-consistent evolution of the inductive electric field has been performed. Both the ECCD power and the electron density have been varied, thus varying the well known nonlinearity parameter for ECCD Prf[MW∕m−3]∕ne2[1019m−3] [R. W. Harvey et al., Phys. Rev. Lett 62, 426 (1989)]. This parameter turns out also to be a good predictor of the synergetic effects. The results are then compared with the standard method of calculations of the current penetration using a transport code. At low values of the Harvey parameter, the standard method is in quantitative agreement with Fokker-Planck calculations. However, at high values of the Harvey parameter, synergy between ECCD and E‖ is found. In the case of cocurrent drive, this synergy leads to the generation of large amounts of nonthermal electrons and a concomitant increase of the electrical conductivity and current penetration time. In the case of countercurrent drive, the ECCD efficiency is suppressed by the synergy with E‖ while only a small amount of nonthermal electrons is produced.

Explanation of the JET n = 0 chirping mode

Persistent rapid up and down frequency chirping modes with a toroidal mode number of zero (n = 0) are observed in the JET tokamak when energetic ions, in the range of several hundred keV, are created by high field side ion cyclotron resonance frequency heating. Fokker–Planck calculations demonstrate that the heating method enables the formation of an energetically inverted ion distribution which supplies the free energy for the ions to excite a mode related to the geodesic acoustic mode. The large frequency shifts of this mode are attributed to the formation of phase space structures whose frequencies, which are locked to an ion orbit bounce resonance frequency, are forced to continually shift so that energetic particle energy can be released to counterbalance the energy dissipation present in the background plasma.

Dynamical Evolution of the Mass Function of the Globular Cluster System from Fokker-Planck Calculations: Preliminary Results

AbstractUsing anisotropic Fokker-Planck models, we calculate the evolution of mass and luminosity functions of the Galactic globular cluster system. Our models include two-body relaxation, binary heating, tidal shocks, dynamical friction, and stellar evolution. We perform Fokker-Planck simulations for a large number of virtual globular clusters and synthesize these results to study the relation between the initial and present GCMFs.

Self-consistent full-wave and Fokker-Planck calculations for ion cyclotron heating in non-Maxwellian plasmas

Magnetically confined plasmas can contain significant concentrations of nonthermal plasma particles arising from fusion reactions, neutral beam injection, and wave-driven diffusion in velocity space. Initial studies in one-dimensional and experimental results show that nonthermal energetic ions can significantly affect wave propagation and heating in the ion cyclotron range of frequencies. In addition, these ions can absorb power at high harmonics of the cyclotron frequency where conventional two-dimensional global-wave models are not valid. In this work, the all-orders global-wave solver AORSA [E. F. Jaeger et al., Phys. Rev. Lett. 90, 195001 (2003)] is generalized to treat non-Maxwellian velocity distributions. Quasilinear diffusion coefficients are derived directly from the wave fields and used to calculate energetic ion velocity distributions with the CQL3D Fokker-Planck code [R. W. Harvey and M. G. McCoy, Proceedings of the IAEA Technical Committee Meeting on Simulation and Modeling of Thermonuclear Plasmas, Montreal, Canada, 1992 (USDOC NTIS Document No. DE93002962)]. For comparison, the quasilinear coefficients can be calculated numerically by integrating the Lorentz force equations along particle orbits. Self-consistency between the wave electric field and resonant ion distribution function is achieved by iterating between the global-wave and Fokker-Planck solutions.

The ‘electron-root’ feature in the WENDELSTEIN-7-AS stellarator with ECRH in O1-mode compared to X2-mode

At the W7-AS stellarator, the neoclassical ‘electron-root’ was found in discharges with electron cyclotron resonance heating (ECRH) in both O1- and X2-mode. The electron heat diffusivity estimated from the power balance is compared with neoclassical predictions with the radial electric field, Er, from the ambipolarity condition taken into account. Strongly positive Er values significantly reducing the electron neoclassical transport coefficients are found in the central region with highly peaked Te-profiles. Furthermore, the ‘electron-root’ feature in the ECE time-traces is analysed in the phase when the ECRH is switched off.The power deposition in phase space is analysed with ray-tracing and bounce-averaged Fokker–Planck calculations. For the O1-mode, the power is almost completely absorbed by passing electrons. These findings are different from the previous X2 results (Maaßberg et al 2000 Phys. Plasmas 7 295) for the magnetic configuration with a significant fraction of ripple-trapped electrons in the ECRH launching plane. For these X2 scenarios, fast ‘convective’ contributions to the radial fluxes had been identified. For the O1-scenarios, a purely ‘diffusive’ neoclassical modelling of the power threshold for obtaining the ‘electron-root’ in the different magnetic configurations is described based on the concept of the ‘effective helical ripple’. Finally, a simulation of the ECRH switch-off by solving the time-dependent electron energy balance is performed. The results show that the ‘electron-root’ feature at W7-AS is consistent with the neoclassical predictions.

Fokker–Planck calculation of spin-torque resistance hysteresis and switching currents

In this article we apply a recently developed Fokker–Planck theory of spin-torque switching rates to the prediction of current hysteresis curves and switching currents. A field dependence of the switching current, which is not predicted by nonthermal theories, is found and compared to experiment. We obtain useful analytic approximations to the exact hysteresis curves and the switching currents, which depend logarithmically on current sweep rate. In addition to switching between the parallel and antiparallel metastable states, we also take into account fluctuations in each of these potential wells, which cause rounding of the hysteresis curves.

Heating and current drive by electron cyclotron waves in JT-60U

The results of studies on heating and current drive by the electron cyclotron (EC) waves in JT-60U are presented. An electron temperature of up to 26 keV, as measured by ECE diagnostics, was achieved by injecting EC waves into the centre of a reversed shear plasma produced by the lower hybrid (LH) waves. The electron temperature Te exceeds 24 keV in the region ρ < 0.3, where ρ is the normalized minor radius. The EC driven current profile was measured at high Te up to 21 keV without using LH waves, and was found to be spatially localized. The ECCD (current drive) efficiency ηCD increased with Te and reached 0.42 × 1019 A W−1 m−2 at Te = 21 keV. The dependence of normalized CD efficiency on deposition location was also studied to optimize the CD efficiency, since the trapped particle effect, which depends strongly on deposition location, is expected to reduce ζ. The effect was detected from a significant decrease in ζ in the lower magnetic field deposition, which is consistent with a linearized Fokker–Planck calculation. The measured CD efficiency ζ increased with electron density ne for the same Te, which showed a stronger dependence on ne compared with the linear calculation. Further off-axis ECCD profile at about half of the minor radius was measured, showing fairly good agreement with linear calculation.

Numerical Analysis of Neoclassical Tearing Mode Stabilization by Electron Cyclotron Current Drive

Neoclassical tearing mode (NTM) stabilization by an electron cyclotron drive (ECCD) has been studied by using the numerical model on the basis of the modified Rutherford equation coupled with the 1.5 D transport code and the EC code. The transport code solves the current diffusion equation, including the EC current profile. The background current modification and the resultant movement of rational surface by ECCD are taken into account. The EC code consists of the ray tracing method and the Fokker-Planck calculation. Undetermined parameters in the modified Rutherford equation are estimated from a comparison with the JT-60U experiments. Sensitivity of stabilization to the EC current location is investigated by simulation. The low EC current and peaked EC current profile mitigates the sensitivity, whereas the high EC current and peaked EC current profile moves the rational surface more largely via background current modification by the EC current and intensifies the sensitivity. The high EC current and broad EC current profile mitigates the sensitivity. The EC current necessary for the full stabilization is studied for ITER parameters. The necessary EC current strongly depends on the parameters of bootstrap current and ECCD terms in the modified Rutherford equation. Necessary ECCD power on ITER is evaluated on the basis of parameters estimated from comparisons with JT-60U experiments. (author)

Toroidal Electric Field Effect and Non-Linear Effect on Electron Cyclotron Current Drive in the JT-60U

Recent progress in the development of electron cyclotron heating⁄current drive systems in regard to power and pulse duration have allowed the extension of two normalized parameters (toroidal electric field Eφ and wave power density p). In the extended regime, the conventional theory requires consideration of a distorted electron distribution function. The first validation of the ECCD theory in the extended regime having large Eφ and p is presented. Linear calculation has a tendency to overestimate the EC driven current, as normalized parameters for Eφ and p increase. While the EC driven current IEC obtained by a linearized Fokker-Planck calculation (1.1 MA) did not agree with the measured EC driven current (0.74 ± 0.06MA), non-linear calculation of the Fokker-Planck equation considering the effect of Eφ (0.76 MA) shows close agreement with the experimental result. Calculations show that the decreasing effect of Eφ on IEC was stronger than the increasing effect of p on IEC in the experiment.

Molecular-dynamic calculation of the relaxation of the electron energy distribution function in a plasma.

A molecular-dynamic (MD) code is used to calculate the temporal evolution of nonequilibrium electron distribution functions in plasmas. To the authors' knowledge, this is the first time that a molecular-dynamic code has been used to treat this problem using a macroscopic number of particles. The code belongs to the class of P3M (particle-particle-particle-mesh) codes. Since the equations solved by the MD code are fundamental, this approach avoids several assumptions that are inherent to alternative methods. For example, the initial energy distribution can be arbitrary, and there is no need to assume a value for the Coulomb logarithm. The advantages of the MD code are illustrated by comparing its results with those of Monte Carlo and Fokker-Planck codes with a set of plasma parameters for which the Fokker-Planck calculation is shown to give incorrect results. As an example, we calculate the relaxation of the electron energy distribution produced by optical field ionization of a mixed plasma containing argon and hydrogen.

Effects of electron trapping and transport on electron cyclotron current drive on DIII-D

Recent experiments on the DIII-D tokamak have focused on determining the effect of trapped particles on the electron cyclotron current drive (ECCD) efficiency. The experimental ECCD efficiency increases as the deposition location is moved towards the inboard midplane or towards smaller minor radius for both co- and counter-injection; the ECCD efficiency also increases with increasing electron density and/or temperature. The experimental ECCD is compared to both the linear theory (TORAY-GA) as well as a quasilinear Fokker–Planck model (CQL3D) and is found to be in better agreement with the more complete Fokker–Planck calculation, especially when the rf power density and/or loop voltage exceed criterion for substantial nonlinear modification of the electron distribution function. The width of the measured ECCD profile is consistent with the theoretically expected width in the absence of radial transport for the current carrying electrons.

ITAL]Hubble Space Telescope[/ITAL] Evidence for an Intermediate-Mass Black Hole in the Globular Cluster M15. II. Kinematic Analysis and Dynamical Modeling

We have used the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope (HST) to obtain high spatial resolution spectroscopy of the central region of the dense globular cluster M15. The observational strategy and data reduction were described in Paper I of this series. Here we analyze the extracted spectra with a cross-correlation technique to determine the line-of-sight velocities of individual stars. Our final STIS velocity sample contains 64 stars, two-thirds of which have their velocity measured for the first time. The new data set triples the number of stars with measured velocities in the central projected R ? 1'' of M15 and doubles the number in the central R ? 2''. We combine our data with existing ground-based data to obtain nonparametric estimates of the radial profiles of the projected rotation velocity, velocity dispersion, and rms velocity ?rms. The results differ from earlier work in the central few arcseconds in that we find that ?rms rises to ~14 km s-1, somewhat higher than the values of 10?12 km s-1 inferred previously from ground-based data. To interpret the results we construct dynamical models based on the Jeans equation for a spherical system. If the velocity distribution is isotropic, then M15 must have a central concentration of nonluminous material. If this is due to a single black hole, then a fit to the full velocity information as function of radius implies that its mass is MBH = (3.9 ? 2.2) ? 103 M?. The existence of intermediate-mass black holes in globular clusters is consistent with several scenarios for globular cluster evolution proposed in the literature. The inferred mass for M15 is consistent with the extrapolation of the relation between MBH and ?rms that has been established for galaxies. Therefore, these results may have important implications for our understanding of the evolution of globular clusters, the growth of black holes, the connection between globular cluster and galaxy formation, and the nature of the recently discovered ultraluminous X-ray sources in nearby galaxies. Instead of a single intermediate-mass black hole, M15 could have a central concentration of dark remnants (e.g., neutron stars) due to mass segregation. However, we argue that the best-fitting Fokker-Planck models that have previously been constructed for M15 do not predict a central mass concentration that is sufficient to explain the observed kinematics. To fit the M15 data without any central dark mass concentration one must assume that the velocity distribution is significantly radially anisotropic near the center, which contradicts predictions from both Fokker-Planck and N-body calculations.

Frequency sweeping of phase space structures

Holes and clumps, spontaneously formed in the nonlinear development of a single mode driven by a weakly destabilizing kinetic distribution function in a dissipative medium, were expected to persist for an appropriate collisional time scale. However, Fokker–Planck calculations for the nonlinear system abruptly break down in the course of the calculation for some initial conditions. It is found that this arises because an adiabatic description of phase space structures at zero collisionality does not necessarily lead to continual adiabatic frequency sweeping. Criteria are found for a class of initial distribution functions that determine whether adiabatic frequency sweeping continues indefinitely, or suddenly terminates. An unresolved bifurcation problem is also observed under certain conditions.

Numerical Fokker–Planck calculations in nonuniform grids

A practical, implicit finite-difference scheme, accurate to second order both in time step and mesh size, is presented to solve nonlinear Fokker–Planck (FP) equations in nonuniform grids. Weighting coefficients are introduced for proper equilibration, and the conditions under which the nonuniformity correction is essential in them to ensure physically meaningful solutions are analyzed. Conservation of particle number and energy is addressed, as well as the constraints on the design of nonuniform grids. The proposed scheme is shown to efficiently handle both linear and weakly nonlinear problems and, in addition, its ability to provide solutions to stronger nonlinear situations is demonstrated. Examples of FP equations (describing Coulomb collisions, Compton scattering, and rf heating and current drive) are solved to illustrate the effectiveness of nonuniform grids in reducing computational effort with no significant deterioration in accuracy, and in increasing the latter at no expense of the former.

Determination of the electron cyclotron current drive profile

Evaluation of the profile of non-inductive current density driven by absorption of electron cyclotron waves (ECCD) using time evolution of the poloidal flux indicated a broader profile than predicted by theory. To determine the nature of this broadening, a 1-1/2 D transport calculation of current density evolution was used to generate the signals which the DIII-D motional Stark effect (MSE) diagnostic would measure in the event that the current density evolution followed the neoclassical Ohm's law with the theoretical ECCD profile. Comparison of the measured MSE data with the simulations indicates the experimental data is consistent with the narrow ECCD profile predicted by theory. The simulations indicate the magnitude of the ECCD is at or above the value found in Fokker–Planck calculations of the ECCD, which includes quasilinear and parallel electric field effects.

The Evolution of Globular Clusters in the Galaxy

We investigate the evolution of globular clusters using N-body calculations and anisotropic Fokker- Planck calculations. The models include a mass spectrum, mass loss due to stellar evolution, and the tidal —eld of the parent galaxy. Recent N-body calculations have revealed a serious discrepancy between the results of N-body calculations and isotropic Fokker-Planck calculations. The main reason for the discrepancy is an oversimpli—ed treatment of the tidal —eld employed in the isotropic Fokker-Planck models. In this paper we perform a series of calculations with anisotropic Fokker-Planck models with a better treatment of the tidal boundary and compare these with N-body calculations. The new tidal boundary condition in our Fokker-Planck model includes one free parameter. We —nd that a single value of this parameter gives satisfactory agreement between the N-body and Fokker-Planck models over a wide range of initial conditions. Using the improved Fokker-Planck model, we carry out an extensive survey of the evolution of globular clusters over a wide range of initial conditions varying the slope of the mass function, the central concentration, and the relaxation time. The evolution of clusters is followed up to the moment of core collapse or the disruption of the clusters in the tidal —eld of the parent galaxy. In general, our model clusters, calculated with the anisotropic Fokker-Planck model with the improved treatment for the tidal boundary, live longer than isotropic models. The diUerence in life- time between the isotropic and anisotropic models is particularly large when the eUect of mass loss via stellar evolution is rather signi—cant. On the other hand, the diUerence is small for relaxation-dominated clusters, which initially have steep mass functions and high central concentrations. Subject headings: galaxies: kinematics and dynamicsgalaxies: star clusters ¨ Galaxy: kinematics and dynamicsglobular clusters: general ¨ methods: numericalopen clusters and associations: general

Modification of the current profile in DIII-D by off-axis electron cyclotron current drive

Localized non-inductive currents due to electron cyclotron wave absorption have been measured on the DIII-D tokamak. Clear evidence of non-inductive currents is seen on the internal magnetic field measurements by motional Stark effect spectroscopy. The magnitude and location of the non-inductive current is evaluated by comparing the total and ohmic current profiles of discharges with and without electron cyclotron wave power. The measured current agrees with Fokker-Planck calculations near the magnetic axis, but exceeds the predicted value as the location of the current drive is moved to the half-radius.