Bounce-averaged Fokker-Planck Equation Research Articles

Runaway electron modelling in the self-consistent core European Transport Simulator

Relativistic runaway electrons are a major concern in tokamaks. Although significant theoretical development had been undertaken in recent decades, we still lack a self-consistent simulator that could simultaneously capture all aspects of this phenomenon. The European framework for Integrated Modelling (EU-IM) facilitates the integration of different plasma simulation tools by providing a standard data structure for communication that enables relatively easy integration of different physics codes. A three-level modelling approach was adopted for runaway electron simulations within the EU-IM. Recently, a number of runaway electron modelling modules have been integrated into this framework. The first level of modelling (Runaway Indicator) is limited to the indication if runaway electron generation is possible or likely. The second level (Runaway Fluid) adopts an approach similar to e.g. the GO code, using analytical formulas to estimate changes in the runaway electron current density. The third level is based on the solution of the electron kinetics. One such code is LUKE that can handle the toroidicity-induced effects by solving the bounce-averaged Fokker–Planck equation. Another approach is used in NORSE, which features a fully nonlinear collision operator that makes it capable of simulating major changes in the electron distribution, for example slide-away. Both codes handle the effect of radiation on the runaway distribution. These runaway-electron modelling codes are in different stages of integration into the EU-IM infrastructure, and into the European Transport Simulator (ETS), which is a fully capable modular 1.5D core transport simulator. The ETS with Runaway Fluid was benchmarked to the GO code implementing similar physics. Coherent integration of kinetic solvers requires more effort on the coupling, especially regarding the definition of the boundary between runaway and thermal populations, and on consistent calculation of resistivity. Some of these issues are discussed.

A rapid fast ion Fokker–Planck solver for integrated modelling of tokamaks

The RISK (rapid ion solver for tokamaks) code for simulating the evolution of the distribution function of neutral beam injected ions (NBI) in tokamak plasmas is described. The code has been especially developed for use in integrated modelling frameworks. Within this context, a code needs to be modular, machine independent and fast. RISK fulfils all these conditions. The RISK code solves the bounce averaged Fokker–Planck equation for the species of the injected ions by expanding the distribution function in the eigenfunctions of the collisional pitch angle scattering operator. The velocity dependent coefficient functions are calculated with a finite element solver. Finite orbit width effects are handled by an ad hoc broadening algorithm of the NBI ionization source. In order to assess the validity of the approximations employed in RISK, a comparison with a full orbit following Monte Carlo code is presented. RISK is integrated into the CRONOS transport suite of codes (Artaud et al 2010 Nucl. Fusion 50 043001) and the European integrated modelling (EU-IM) framework (Falchetto et al 2014 Nucl. Fusion 54 043018). The RISK implementation in this platform is discussed and exemplified to show the strength of running simulation codes in a modular and machine independent environment for simulation of fusion plasmas.

Evolution of the plasma sheet electron pitch angle distribution by whistler-mode chorus waves in non-dipole magnetic fields

Abstract. We present a detailed numerical study on the effects of a non-dipole magnetic field on the Earth's plasma sheet electron distribution and its implication for diffuse auroral precipitation. Use of the modified bounce-averaged Fokker-Planck equation developed in the companion paper by Ni et al. (2012) for 2-D non-dipole magnetic fields suggests that we can adopt a numerical scheme similar to that used for a dipole field, but should evaluate bounce-averaged diffusion coefficients and bounce period related terms in non-dipole magnetic fields. Focusing on nightside whistler-mode chorus waves at L = 6, and using various Dungey magnetic models, we calculate and compare of the bounce-averaged diffusion coefficients in each case. Using the Alternative Direction Implicit (ADI) scheme to numerically solve the 2-D Fokker-Planck diffusion equation, we demonstrate that chorus driven resonant scattering causes plasma sheet electrons to be scattered much faster into loss cone in a non-dipole field than a dipole. The electrons subject to such scattering extends to lower energies and higher equatorial pitch angles when the southward interplanetary magnetic field (IMF) increases in the Dungey magnetic model. Furthermore, we find that changes in the diffusion coefficients are the dominant factor responsible for variations in the modeled temporal evolution of plasma sheet electron distribution. Our study demonstrates that the effects of realistic ambient magnetic fields need to be incorporated into both the evaluation of resonant diffusion coefficients and the calculation of Fokker-Planck diffusion equation to understand quantitatively the evolution of plasma sheet electron distribution and the occurrence of diffuse aurora, in particular at L > 5 during geomagnetically disturbed periods when the ambient magnetic field considerably deviates from a magnetic dipole.

Resonant Scattering of Relativistic Outer Zone Electrons by Plasmaspheric Plume Electromagnetic Ion Cyclotron Waves

The bounce-averaged Fokker–Planck equation is solved to study the relativistic electron phase space density (PSD) evolution in the outer radiation belt due to resonant interactions with plasmaspheric plume electromagnetic ion cyclotron (EMIC) waves. It is found that the PSDs of relativistic electrons can be depleted by 1–3 orders of magnitude in 5h, supporting the previous finding that resonant interactions with EMIC waves may account for the frequently observed relativistic electron flux dropouts in the outer radiation belt during the main phase of a storm. The significant precipitation loss of ∼MeV electrons is primarily induced by the EMIC waves in H+ and He+ bands. The rapid remove of highly relativistic electrons (> 5 MeV) is mainly driven by the EMIC waves in O+ band at lower pitch-angles, as well as the EMIC waves in H+ and He+ bands at larger pitch-angles. Moreover, a stronger depletion of relativistic electrons is found to occur over a wider pitch angle range when EMIC waves are centering relatively higher in the band.

Numerical simulations of storm-time outer radiation belt dynamics by wave–particle interactions including cross diffusion

Using the recently developed hybrid finite difference (HFD) code, we solve the two-dimensional bounce-averaged Fokker–Planck equation with cross-pitch-angle-energy diffusion to evaluate the electron phase space density (PSD) evolution driven by multiple wave–particle interactions during storms. Numerical results show that PSDs of ∼ MeV electrons can be depleted by two orders of magnitude at lower pitch-angles during the main phase primarily due to pitch-angle scattering by hiss and electromagnetic ion cyclotron (EMIC) waves, and then enhance by two orders of magnitude compared with the prestorm state during the recovery phase primarily due to acceleration by chorus wave. Furthermore, the effects of the cross terms and various electromagnetic waves are also been identified.

Dynamic Evolution of Outer Radiation Belt Electrons due to Whistler-Mode Chorus

Following our preceding work, we perform a further study on dynamic evolution of energetic electrons in the outer radiation belt L = 4.5 due to a band of whistler-mode chorus frequency distributed over a standard Gaussian spectrum. We solve the 2D bounce-averaged Fokker–Planck equation by allowing incorporation of cross diffusion rates. Numerical results show that whistler-mode chorus can be effective in acceleration of electrons at large pitch angles, and enhance the phase space density for energies of about 1 MeV by a factor of 102 or above in about one day, consistent with observation of significant enhancement in flux of energetic electrons during the recovery phase of a geomagnetic storm. Moreover, neglecting cross diffusion often leads to overestimates of the phase space density evolution at large pitch angle by a factor of 5–10 after one day, with larger errors at smaller pitch angle, suggesting that cross diffusion also plays an important role in wave—particle interaction.

A study of high-energy ions produced by ICRF heating in LHD

This paper reports on the behaviour of high-energy ions created by ion cyclotron range of frequency (ICRF) heating on the Large Helical Device (LHD). In the third experimental campaign conducted in 1999, it was found that minority heating has good heating performance, and high-energy particles were observed. In the fourth campaign in 2000, the temporal behaviour of high-energy ions was investigated in the minority heating regime using turnoff or modulation of ICRF power. The time evolution of the high-energy particle distribution was measured using a natural diamond detector (NDD) and a time-of-flight neutral particle analyser (TOF-NPA). It was found that the count number of higher-energy particles declines faster than that of lower-energy particles after ICRF turnoff. In the modulation experiments, the phase difference of the flux of high-energy particles with respect to the ICRF power modulation increased with energy. These results were explained qualitatively by the Fokker-Planck equation with a simple model. The pitch-angle dependence of the distribution function was also measured in the experiment by changing the line of sight of the TOF-NPA, and an anisotropy of the high-energy tail was found. This anisotropy was reproduced by solving the bounce-averaged Fokker-Planck equation. The second harmonic heating was conducted successfully for the first time in the LHD in high-β plasma, and high-energy particles were also detected in this heating regime.

The influence of trapping effects on neutron production in NBI-heated tokamak plasmas

Fast computer codes for evaluating non-thermal velocity distributions are needed in order to carry out routine analysis of neutron signals in plasmas heated by neutral-beam injection (NBI). By neglecting the magnetic field inhomogeneity along a field line, i.e. neglecting trapped-particle effects, calculations of the velocity distribution can be made relatively rapidly. In this paper, we investigate under which circumstances such an approximation is justified. We solve the bounce-averaged Fokker–Planck equation, which includes particle trapping, by expressing the solution in terms of a series of appropriate eigenfunctions of the pitch-angle scattering operator. The eigenfunctions, eigenvalues and time-dependent coefficient functions are calculated numerically. We use the solution for calculating the effect of particle trapping on neutron production for beam-heated deuterium plasmas.

Temperature anisotropy in a cyclotron resonance heated tokamak plasma and the generation of poloidal electric field

The temperature anisotropy generated by cyclotron resonance heating of tokamak plasmas is calculated and the poloidal equilibrium electric field due to the anisotropy is studied. For the calculation of anisotropic temperatures, a bounce-averaged Fokker–Planck equation with a bi-Maxwellian distribution function of heated particles is solved, assuming a moderate wave power and a constant quasilinear cyclotron resonance diffusion coefficient. The poloidal electrostatic potential variation is found to be proportional to the particle density and the degree of temperature anisotropy of warm species.

A Fokker-Planck Calculation of Bremsstrahlung Emission During Fast Wave Current Drive

Bremsstrahlung emission during fast wave current drive is numerically calculated based on expansion of the electron distribution function in powers of inverse bounce frecluency. The distribution function is determined by solving a bounce-averaged Fokker-Planck equation. The resulting local bremsstrahlung emission shows poloidal-angle dependence. The angular spectrum of the emission along viewing lines is also given and is compared with the experimental results.

Enhancement of current drive efficiency for electron cyclotron waves

The frequency range of an electron cyclotron current drive is explored, in order to enhance the current drive efficiency. Its model is based on a two-dimensional, relativistic, bounce-averaged Fokker-Planck equation. The current drive efficiency is enhanced by a largely Doppler-shifted (up-shifted) frequency, using the relationship Y<or approximately= gamma -1, where Y=l omega ce/ omega (l:harmonic number, omega ce: electron cyclotron frequency, omega : injection frequency) and gamma is the relativistic factor. The behavior of the enhanced efficiency is examined under the toroidal effect as well as the DC electric field effect.

Neoclassical formula for neutral beam current drive

A new analytic formula for the figure of merit for beam current drive j/pb is developed in which neoclassical corrections for both ion current and beam induced electron current are incorporated. Small additional correctors for fast ion energy diffusion and charge exchange loss are also introduced. The present formula for j/pb is based on a circular plasma. The formula is, however, also applicable to a non-circular plasma. With this new formula, the value of j/pb can be evaluated without solution of the bounce averaged Fokker-Planck equation. One of the applications of the formula is to use it in a self-consistent current drive analysis in combination with MHD and/or transport analysis. In the paper, the formula is used to investigate the various parametric dependences of j/pb.

Wave induced ion transport in ICRH tokamak plasmas

The problem of wave induced resonant ion transport in ICRH tokamak plasmas is considered. A diffusion coefficient which describes the transport in the radial dimension in configuration space is first derived and a bounce averaged Fokker-Planck equation formulated. Then, using the method of multiple time-scales, a diffusion equation for the resonant ion density is obtained. The diffusion equation is numerically solved for parameters appropriate to the JET tokamak. The results indicate that for high RF power input and long heating times (t > τs) there is a significant depletion of resonant ions from the centre of the discharge, a reduction in thermonuclear reactivity and a degradation in heating efficiency.

Tokamak ripple transport at arbitrary collision frequency

The bounce averaged Fokker-Planck equation for the distribution function of ripple trapped particles in a tokamak has been solved, for arbitrary collision frequencies, in the 'tokamak' limit in which ripple wells are localized close to the midplane. The equation includes the main terms contributing to collisionless (de)trapping. The solution employs power series expansions for the distribution function in the pitch angle variable k2 and the poloidal angle θ; the series in k2 and θ both terminate. The boundary conditions applied at the trapping/detrapping boundary, that f and ∂f/∂k2 be continuous, become the requirement that in the collisionless limit the derivative with respect to k2 reflect the scale length set by the motion in toroidally blocked orbits. The resulting series solutions reduce to the usual expressions in the high collision frequency limit, but they are considerably lower than the results of previous calculations (which neglect the collisionless detrapping effects), in the low collision frequency limit. Comparison with Monte Carlo calculations for INTOR parameters shows that, in all cases, the analytic results lie somewhat below the numerical results, which is to be expected since banana drift diffusion is also present in the Monte Carlo calculation. However, previous analytic calculations give diffusion coefficients which are much larger than the Monte Carlo results.

Ripple transport in ‘transport-optimized’ stellarators

Transport in ‘transport-optimized’ stellarators at low collision frequencies has been studied using a numerical method of solution of the bounce-averaged Fokker–Planck equation which describes both ripple-trapped and non-ripple-trapped particles in a stellarator. Diffusion rates in ‘transport-optimized’ stellarators had not previously been calculated at collision frequencies which are low enough for the effects of a radial electric field to be important. It was found that the configurations which were optimized for transport at the highest collision frequencies at which ripple-trapped particles exist give improved transport at all lower collision frequencies of interest, and also at higher collision frequencies. Standard stellarators have also been studied since in these cases the results can be compared with existing analytic theories and the transport mechanisms have been clarified as a result. Comparisons with Monte Carlo calculations show excellent agreement, and, although existing analytic methods of solution can strictly only be applied in rather few cases, some extensions of analytic results are discussed which enable us to explain quantitatively the behaviour observed.

A bounce-averaged Fokker-Planck code for stellarator transport

A computer code for solving the bounce-averaged Fokker-Planck equation appropriate to stellarator transport has been developed and its first applications have been made. The code is much faster than the bounce-averaged Monte-Carlo codes, which up to now have provided the most efficient numerical means for studying stellarator transport. Moreover, since the connection to analytic kinetic theory is more direct for the Fokker-Planck approach than for the Monte-Carlo approach, a comparison of theory and numerical experiment is now possible at a considerably more detailed level than previously.

Effects of particle trapping on distortion of the distribution function of RF-heated minority ions in a tokamak plasma

The effect of toroidal geometry upon the distribution function of weakly RF-heated minority ions in a tokamak plasma is considered. An analytic scheme to solve the corresponding bounce-averaged Fokker-Planck equation is developed and explicit solutions are given in the limits of high and low velocities. It is found that trapped-particle effects may significantly reduce the RF-induced anisotropy in the distribution function.

Effects of particle trapping on ion-cyclotron resonance heating in a toroidal plasma

The effect of particle trapping on the velocity space diffusion caused by ion-cyclotron resonance heating is considered. A bounce-averaged Fokker–Planck equation is derived, which determines the distribution function of the heated ions. Trapped particle effects lead to significant changes in the rf-driven diffusion operator as compared to previously used models.

Perturbation Solution of the Bounce-Averaged Fokker-Planck Equation for an Electrostatic Magnetic Mirror

Sloshing ion distributions are a crucial feature in the end cells of recent tandem mirror reactor designs. They provide the ambipolar potentials that confine central ions and often have the function of making the electron thermal barrier with a potential shape that traps enough cold ions at the midplane for the stabilization of loss cone modes.A perturbation method is developed to find solutions of sloshing-ion distributions. This method uses an expansion in the ratio of electrostatic potential to average ion energy to simplify the bounce-averaged Fokker-Planck equation. The zero'th order equation obtained is separated into equations for the angular and velocity-dependent parts of the distribution function. An analytical solution of the angular equation is derived for small charge-exchange to ionization ratios. For any value of this ratio finite element techniques, which provide rapid numerical solutions for parametric studies of sloshing ions, are used to derive the angular and the velocity distribution functions. The density ratio and the ambipolar potential, as functions of axial distance, are computed from the angular distribution function. There is excellent agreement with results from the Lawrence Livermore National Laboratory bounce-averaged Fokker-Planck code with as much as 500 times less CRAY-1 computer time.

Alpha particle loss and energy deposition in tandem mirrors

The bounce averaged Fokker–Planck equation is solved for alpha particles slowing down in a background plasma of elecrons and ions in the central cell of a tandem mirror. The problem is formulated for a general, monotonic magnetic field, and then solved numerically for the square-well case. This gives a distribution function in energy and lambda (magnetic moment/energy). The resulting loss of alpha particles by scattering into the magnetic loss cone and the distribution of alpha-particle energy to ions and electrons are given. For typical tandem mirror reactor parameters (Te ∼30 keV, φ∼100 keV, R∼4) about half of the alpha particles are born in or scatter into the loss cones, but less than 25% of the total alpha particles’ energy is lost through these mechanisms.