Relativistic Fokker-Planck Code Research Articles

A numerical code for the calculation of relativistic electron cyclotron damping with an arbitrary distribution function at an arbitrary harmonic

The relativistic expressions for the anti-Hermitian parts of the relativistic dielectric tensor elements can be expressed as a single integral over the parallel momentum variable, allowing an arbitrary electron distribution function. A computer program has been written for the calculation of this single integral. The numerical results are tested for a relativistic Maxwellian distribution function and agree with analytical expressions for this case. The numerical code is therefore an essential element in a more general validation, evaluation and demonstration of powerful analytical results presented. The computer program is then applied to the calculation of relativistic electron cyclotron harmonic damping at any arbitrary harmonic, for any distorted electron distribution function, distorted for example by an electric field or by RF power sources, as for instance by both electron cyclotron and lower-hybrid waves, as calculated from a relativistic Fokker–Planck code. For generality, we also include the case of relativistic Landau damping, also used to check the code. Program summaryProgram title: DAMPINGCatalogue identifier: AEIS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIS_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 84 108No. of bytes in distributed program, including test data, etc.: 2 581 423Distribution format: tar.gzProgramming language: FORTRANComputer: Any computer or work stationOperating system: Any with a Fortran compilerRAM: 8 Mbytes, using double precision arithmetic, with 400×200 grid points, on an ALPHA serverClassification: 19.8External routines: Requires a link to the NAG libraries for the Bessel functions subroutines.Nature of problem: The computer program calculates generalized expressions for the anti-Hermitian part of the relativistic dielectric tensor in a plasma for an arbitrary distribution function, with comparison to the analytical expressions derived in [1], associated with a relativistic Maxwellian distribution, for any n-th cyclotron harmonic and for the relativistic Landau damping case when n=0. The anti-Hermitian parts of the dielectric tensor are important to know, since they determine the damping or emission of an electromagnetic wave within the plasma medium. New analytical expressions are derived for the relativistic Landau damping and for the electron cyclotron damping including the case n||=1, for the case of an arbitrary distribution function. This is important since studies of the RF heating of tokamak plasmas in a fully self-consistent way involve the simultaneous description of the temporal evolution of the heated species velocity distribution function using a Fokker–Planck code, coupled to the damping of the wave on the evolving distribution function. We can evaluate numerically the degree by which a non-Maxwellian nature of the relativistic distribution function changes the Landau damping and the electron cyclotron damping, with respect to the Maxwellian case. In addition this is also important in the study of electron cyclotron emission, which can be a sensitive indicator of non-thermal electron distributions.Solution method: The relativistic expressions for the anti-Hermitian parts of the dielectric tensor elements can be expressed as a single integral over the parallel momentum variable [1]. This single integral is calculated along the resonance curve, for an arbitrary distribution function calculated from a relativistic Fokker–Planck code. Since Fokker–Planck codes usually evolve the distribution functions using a spherical coordinate system in momentum space, the present code has to first transform the distribution function calculated by the Fokker–Planck code from a spherical coordinate system to a cylindrical coordinate system in momentum space, where the single integral over the parallel momentum variable can be calculated along the resonance curves. In the evaluation of the term in Eq. (8), cubic spline are used for the calculation of the derivatives.Running time: An execution on an ALPHA server 4000 from Digital requires less than 10 s CPU time on a single processor for 400×200 grid points.

Fokker-Planck analysis of energy deposition of laser-produced relativistic electrons in dense plasmas

Evaluation of the energy deposition region of a laser-produced relativistic electron beam in dense core plasma is essential in the fast ignition approach to ICF. Recently, we have developed a two-dimensional relativistic Fokker-Planck code “RFP-2D” for fast electron transport in dense plasmas to investigate the energy deposition profiles. In this paper, we compare the sizes of the energy deposition region derived from RFP-2D and another RFP-type calculation model with stochastic treatments. It is found to be considerably important to take into account the influence of the self-generated electromagnetic field when evaluating the energy deposition region.

Analysis of core plasma heating in fast ignition target by two-dimensional Fokker-Planck-Hydro code

To investigate the core plasma heating in fast ignition targets, we recently developed a two-dimensional relativistic Fokker-Planck code for fast electron transport in dense plasmas. In the present study, we have coupled RFP-2D with a 2D hydrodynamics code for bulk plasma and analyzed the core plasma heating by assuming a condition similar to that in the GXII-PW laser experiments at ILE. It was found that when the source beam electrons enter the dense core region with low (< 1 MeV) average energy and a Super-Gaussian spatial distribution, the core plasma can be heated above 0.8 keV at maximum, which is almost the same temperature as measured in the experiments.

Two-dimensional relativistic Fokker-Planck model for core plasma heating in fast ignition targets

One of the key issues in the fast ignition scheme is a clarification of the imploded dense core heating by laser-produced fast electrons. To investigate the core heating process, a two-dimensional relativistic Fokker-Planck code “RFP-2D” for fast electron behavior in dense core plasmas has been developed. Energy loss of fast electrons due to Coulomb interactions is treated through not only usual short-range binary collisions but also long-range binary collisions, including a collective shielding effect. After describing the physics model, an examination is made of the energy deposition of fast electrons injected into a highly compressed D-T cylindrical plasma. The relative importance of the long-range Coulomb interaction and the influence of a self-generated electromagnetic field on the energy deposition profile are demonstrated.

Analysis of Core Plasma Heating by Relativistic Electrons in Fast Ignition

To investigate core plasma heating in fast ignition, a relativistic Fokker-Planck code for fast electrons is developed in a one-dimensional planar coordinates system. It is found that in dense plasmas, the Joule heating is much smaller than the heating through Coulomb interactions. In the latter energy deposition process, the long-range collective effect is comparable to that of binary electron-electron collisions. Moreover, on the basis of coupled transport-hydrodynamic simulations in one-dimensional planar geometry, the core heating process for an ignition-experiment-grade compressed core (ρR = 0.3 g/cm2) is examined, and a possibility of evaluation of burn history from the neutron spectrum is shown. It is shown that a relatively low energy component (E0 ≤ 1 MeV) of electron beams plays an important role for effective core heating in fast ignition.

K α spectroscopy to study energy transport in ultrahigh-intensity laser produced plasmas

K α line emission from partially ionized material was studied to diagnose energy transport in ultrahigh-intensity laser produced plasmas. Dependence of the emission intensity on initial target conductivity was investigated by using a plastic target (C 2H 3Cl coated with CH) and a metallic target (Al coated with Mg) irradiated with a 1-TW Ti:sapphire laser pulse of 132 fs at 2×10 17 W/cm 2 . Depth of the heated region was deduced from Cl or Al He- α line intensities given as a function of overcoat thickness. No significant difference was observed between the two types of targets. Density–depth product of the hot region was typically 0.1 mg/cm 2 for both targets and the temperature of hot electrons measured by an electron spectrometer was 50 keV . The experimental results were analyzed with an integration calculation with a one-dimensional (1D) relativistic Fokker–Planck code, a 1D radiation hydrodynamics (rad-hydro) simulation, and an atomic–kinetic model developed for ionization-shifted Cl K α lines. The observed transport features are discussed.

Magnetic ripple and the modeling of lower-hybrid current drive in tokamaks

Using ray tracing, a detailed investigation of the lower-hybrid (LH) wave propagation in presence of toroidal magnetic field ripple is presented. The local ray behavior is first depicted for a cylindrical equilibrium periodically modulated along the axial direction. Variations along ray trajectories in the component of the wave vector parallel to the equilibrium magnetic field are observed, with a maximum relative amplitude that is locally of the order of the ripple level. For the full rippled toroidal equilibrium, a similar local behavior is found when the ray trajectory crosses a high ripple region. Despite the modest amplitude of the local ray perturbation, its global influence on ray trajectories may be strong, as a consequence of the combined effects of toroidal and poloidal inhomogeneities. By coupling ray tracing with a one-dimensional relativistic Fokker–Planck code, simulations of LH experiments have been performed for the TORE SUPRA tokamak [Equipe TORE SUPRA, in Proceedings of the 15th Conference on Plasma Physics and Controlled Nuclear Fusion Research, Seville (International Atomic Energy Agency, Vienna, 1995), Vol. 1, p. 105, Paper IAEA-CN-60/A1-5]. It is shown that magnetic ripple may induce significant modifications in the LH power deposition profiles, mainly in the ‘‘few passes’’ regime when the wave makes some, but not many, passes inside the plasma before being absorbed. The effect of magnetic ripple leads then to a broadening of the power deposition profile and a shift towards the center of the plasma, and a better coupling with high energy electrons. This behavior may be explained by an increase in the overall ray stochasticity. Taking into account magnetic ripple in LH simulations, a better agreement is found between numerical predictions and experimental observations.

Analysis of electron cyclotron current drive using neoclassical Fokker–Planck code without bounce-average approximation

A new three-dimensional relativistic Fokker–Planck code (poloidal angle and two dimensions in momentum space) has been developed for the analysis of electron cyclotron current drive (ECCD) in tokamak plasmas. This numerical code takes into consideration trapped electron effects without using the bounce-average approximation. Simulations have been carried out using the code, and the results were compared with those of a bounce-averaged Fokker–Planck code. There are differences in the current drive efficiencies calculated by the two codes, and this difference is shown to be caused by a change in electron velocity parallel to the magnetic field due to the inhomogeneous magnetic field.

Numerical Simulations of Lower Hybrid Current Drive by the (1+2)-Dimensional Fokker-Planck Code Combined with the Ray-Tracing Code

The (1+2) dimensional (1-D in real space and 2-D in momentum space) relativistic Fokker-Planck code combined with the ray-tracing code has been newly developed for analyzing the lower hybrid current drive (LHCD) on tokamak plasmas. This numerical code calculates the 2-D MHD equilibrium, ray-tracing, and Fokker-Planck analyses self-consistently. The simulations have been carried out by using the code, and the results were compared with experiments on the lower hybrid current drive in JT-60. As a result, the obtained simulation results agreed with the experimental results on the current drive efficiency and the current density profile ( or the plasma internal inductance).

Analysis of the lower hybrid current drive by the two-dimensional relativistic Fokker-Planck code combined with the ray-tracing code

The two-dimensional relativistic Fokker-Planck code has been newly developed and combined with the ray-tracing code for analyzing the lower hybrid current drive (LHCD) in Tokamak plasmas. The current drive efficiency evaluated using the parameter of reactor-grade plasmas is about a factor of 1.7-2 times higher than the results obtained from two other analyses (the one-dimensional relativistic Fokker-Planck code combined with the ray-tracing code and the two-dimensional non-relativistic Fokker-Planck code combined with the ray-tracing code). Since the LHCD efficiency enhancement due to the two-dimensional effect and the relativistic effect tends to increase in a high temperature plasma, these effects are important to analyze the lower hybrid current drive in future large Tokamaks.

Analysis of the lower hybrid current drive by the two-dimensional relativistic Fokker-Planck code combined with the ray-tracing code

Analysis of the lower hybrid current drive by the two-dimensional relativistic Fokker-Planck code combined with the ray-tracing code

Electron cyclotron resonance heating in tandem-mirror end cells

A relativistic Fokker–Planck code is used to study the formation of non-Maxwellian electron velocity distributions in the end cell of a tandem mirror from the combined effects of Coulomb collisions and electron cyclotron radio-frequency (rf) heating. Axial electrostatic potential profiles consistent with thermal-barrier operation are assumed which lead to reasonable electron density profiles. The presence of the potentials results in four populations of electrons occupying different regions of phase space, three of which are solved for self-consistently; the fourth is the passing electron distribution from the center cell that is fixed as a Maxwellian. It is found that the second-harmonic barrier rf near the bottom of the magnetic well, together with rf cavity fields, give a good correspondence to the behavior of the hot electrons in the TMX-U experiment [IEEE Trans. Plasma Sci. PS-16, 1 (1988)]. The behavior of the fundamental rf in the plug region part way up the magnetic well shows poor correlation with experimental results.