Local Maxwellian Distribution Research Articles

A kinetic model for magnetogasdynamics

In this work the equations of ideal magnetogasdynamics are derived based on the introduced local complex Maxwellian distribution function. Using this kinetic model, we obtain the analog of a quasi-dynamic system of equations for magnetogasdynamics, including dissipative processes. The resulting model and the algorithm of its solution have been tested by applying them to a number of well-known problems. The given algorithm can be easily adapted to an architecture of high performance systems with extramassive parallelism.

Particle Ellipsoidal Statistical Bhatnagar–Gross–Krook Approach for Simulation of Hypersonic Shocks

To reduce the computational costs related to direct simulation Monte Carlo computations for low Knudsen number flows, a new particle method, based on the ellipsoidal statistical Bhatnagar–Gross–Krook model of the Boltzmann equation, is investigated. In this method, a fraction of the computational particles in a cell is selected for velocity reassignment from the local Maxwellian distribution based the local collision frequency and an even smaller fraction is selected for internal mode energy reassignment based on the rotational and vibrational relaxation rates. A scaling algorithm is used to decrease the required number of particles per cell and to guarantee that the energy and momentum are conserved in each time step. The accuracy and efficiency of the ellipsoidal statistical Bhatnagar–Gross–Krook method are tested in three different test cases and the results are compared with the direct simulation Monte Carlo solution and experiment as well. The ellipsoidal statistical Bhatnagar–Gross–Krook approach is found to provide good agreement with direct simulation Monte Carlo solutions, however, a factor of 2 speed up compared with direct simulation Monte Carlo is only achieved when the collision frequency is sufficiently high.

A hybrid continuum-particle solver for unsteady rarefied gas flows

The implementation of a hybrid simulation algorithm based on the compressible Navier–Stokes equations and the Direct Simulation Monte Carlo (DSMC) method is discussed, with emphasis on problems of transient nature. The coupling of the two methods at their common interface is achieved by the generation of particles according to the local Maxwellian distribution and the imposition of macroscopic quantities for the continuum solver. Dynamic characteristics of the continuum-particle decomposition scheme lead to a flexible and fully configurable determination of the two regions. The parallelization scheme and the ability to consider arbitrary geometries allow the consideration of the method for non-trivial, practical applications. The validation of the code is performed for two benchmark problems, namely those of shock tube flow and unsteady flow through an orifice.

Reduced Fokker-Planck models for fast particle distribution across a transition layer of disparate plasma temperatures

Across a transition layer of disparate plasma temperatures, the high energy tail of the plasma distribution can have appreciable deviations from the local Maxwellian distribution due to the Knudson layer effect. The Fokker-Planck equation for the tail particle population can be simplified in a series of practically useful limiting cases. The first is the approximation of background Maxwellian distribution for linearizing the collision operator. The second is the supra-thermal particle speed ordering of vTi ≪ v ≪ vTe for the tail ions and vTi ≪ vTe ≪ v for the tail electrons. Keeping both the collisional drag and energy scattering is essential for the collision operator to produce a Maxwellian tail distribution. The Fokker-Planck model for following the tail ion distribution for a given background plasma profile is explicitly worked out for systems of one spatial dimension, in both slab and spherical geometry. A third simplification is an expansion of the tail particle distribution using the spherical harmonics, which are eigenfunctions of the pitch angle scattering operator. This produces a set of coupled Fokker-Planck equations that contain energy-dependent spatial diffusion terms in two coordinates (position and energy), which originate from pitch angle scattering in the original Fokker-Planck equation. It is shown that the well-known diffusive Fokker-Planck model is a poor approximation of the two-mode truncation model, which itself has fundamental deficiency compared with the three-mode truncation model. The cause is the lack of even-symmetry representation in pitch dependence in the two-mode truncation model.

On the applicability of the standard approaches for evaluating a neoclassical radial electric field in a tokamak edge region

The use of the standard approaches for evaluating a neoclassical radial electric field Er, i.e., the Ampere (or gyro-Poisson) equation, requires accurate calculation of the difference between the gyroaveraged electron and ion particle fluxes (or densities). In the core of a tokamak, the nontrivial difference appears only in high-order corrections to a local Maxwellian distribution due to the intrinsic ambipolarity of particle transport. The evaluation of such high-order corrections may be inconsistent with the accuracy of the standard long wavelength gyrokinetic equation (GKE), thus imposing limitations on the applicability of the standard approaches. However, in the edge of a tokamak, charge-exchange collisions with neutrals and prompt ion orbit losses can drive non-intrinsically ambipolar particle fluxes for which a nontrivial (Er-dependent) difference between the electron and ion fluxes appears already in a low order and can be accurately predicted by the long wavelength GKE. The parameter regimes, where the radial electric field dynamics in the tokamak edge region is dominated by the non-intrinsically ambipolar processes, thus allowing for the use of the standard approaches, are discussed.

Collisional and Radiative Processes in Optically Thin Plasmas

Most of our knowledge of the physical processes in distant plasmas is obtained through measurement of the radiation they produce. Here we provide an overview of the main collisional and radiative processes and examples of diagnostics relevant to the microphysical processes in the plasma. Many analyses assume a time-steady plasma with ion populations in equilibrium with the local temperature and Maxwellian distributions of particle velocities, but these assumptions are easily violated in many cases. We consider these departures from equilibrium and possible diagnostics in detail.

Modified unified kinetic scheme for all flow regimes

A modified unified kinetic scheme for the prediction of fluid flow behaviors in all flow regimes is described. The time evolution of macrovariables at the cell interface is calculated with the idea that both free transport and collision mechanisms should be considered. The time evolution of macrovariables is obtained through the conservation constraints. The time evolution of local Maxwellian distribution is obtained directly through the one-to-one mapping from the evolution of macrovariables. These improvements provide more physical realities in flow behaviors and more accurate numerical results in all flow regimes especially in the complex transition flow regime. In addition, the improvement steps introduce no extra computational complexity.

On the excess energy of nonequilibrium plasma

The energy that can be released in plasma due to the onset of instability (the excess plasma energy) is estimated. Three potentially unstable plasma states are considered, namely, plasma with an anisotropic Maxwellian velocity distribution of plasma particles, plasma with a two-beam velocity distribution, and an inhomogeneous plasma in a magnetic field with a local Maxwellian velocity distribution. The excess energy can serve as a measure of the degree to which plasma is nonequilibrium. In particular, this quantity can be used to compare plasmas in different nonequilibrium states.

Maxwell distributions in a model of rough spheres

We consider the Boltzmann equation for the model of rough spherical molecules with both translational and rotational energies. The general form of local Maxwellian distributions for this model is obtained. The main possible types of the corresponding gas flows are selected and analyzed.

Extended hydrodynamic approach to quantum-classical nonequilibrium evolution. I. Theory

A mixed quantum-classical formulation is developed for a quantum subsystem in strong interaction with an N-particle environment, to be treated as classical in the framework of a hydrodynamic representation. Starting from the quantum Liouville equation for the N-particle distribution and the corresponding reduced single-particle distribution, exact quantum hydrodynamic equations are obtained for the momentum moments of the single-particle distribution coupled to a discretized quantum subsystem. The quantum-classical limit is subsequently taken and the resulting hierarchy of equations is further approximated by various closure schemes. These include, in particular, (i) a Grad-Hermite-type closure, (ii) a Gaussian closure at the level of a quantum-classical local Maxwellian distribution, and (iii) a dynamical density functional theory approximation by which the hydrodynamic pressure term is replaced by a free energy functional derivative. The latter limit yields a mixed quantum-classical formulation which has previously been introduced by I. Burghardt and B. Bagchi, Chem. Phys. 134, 343 (2006).

Kinematics of the old stellar population at the Galactic centre

Aims. We aim at a detailed description of the kinematic properties of the old, (several Gyrs) late-type CO-absorption star population among the Galactic centre (GC) cluster stars. This cluster is composed of a central supermassive black hole (Sgr A*) and a selfgravitating system of stars. Understanding its kinematics thus offers the opportunity to understand the dynamical interaction between a central point mass and the surrounding stars in general, especially in view of understanding other galactic nuclei. Methods. We applied AO-assisted, near-infrared imaging and integral-field spectroscopy using the instruments NAOS/CONICA and SINFONI at the VLT. We obtained proper motions for 5445 stars, 3D velocities for 664 stars, and acceleration limits (in the sky plane) for 750 stars. Global kinematic properties were analysed using velocity and velocity dispersion distributions, phase-space maps, twopoint correlation functions, and the Jeans equation. Results. We detect for the first time significant cluster rotation in the sense of the general Galactic rotation in proper motions. Out of the 3D velocity dispersion, we derive an improved statistical parallax for the GC of R0 = 8.07 ± 0.32stat ± 0.13sys kpc. The distribution of 3D stellar speeds can be approximated by local Maxwellian distributions. Kinematic modelling provides deprojected 3D kinematic parameters, including the mass profile of the cluster. We find an upper limit of 4% for the amplitude of fluctuations in the phase-space distribution of the cluster stars compared to a uniform, spherical model cluster. Using upper limits on accelerations, we constrain the minimum line-of-sight distances from the plane of Sgr A* of five stars located within the innermost few (projected) arcsec. The stars within 0.7 �� radius from the star group IRS13E do not co-move with this group, making it unlikely that IRS13E is the core of a substantial star cluster. Overall, the GC late-type cluster is described well as a uniform, isotropic, rotating, dynamically relaxed, phase-mixed system.

MOLECULAR SIMULATION ON DEFORMATION OF SUPERSONIC JET BOUNDARY AND VORTEX SHEDDING BEHIND A FLAT PLATE(Compressible Flow)

The intermolecular collision scheme is developed to obtain an excellent result even if cells are lengthened (more than the local mean free path) in the direct simulation Monte Carlo (DSMC) calculation. In the new scheme, the velocity of one molecule of a collision pair is modified before and after the substantial collision calculation (the core of the collision calculation), assuming that velocity distributions in all flowfield are in local equilibrium with some temperature and flow velocity (local Maxwellian distribution). The new collision scheme is applied to three-dimensional supersonic free jets for a steady problem and a two-dimensional vortex shedding behind a flat plate for an unsteady problem, respectively, and its effect on them is confirmed.

Second-order accurate kinetic schemes for the ultra-relativistic Euler equations

A second-order accurate kinetic scheme for the numerical solution of the relativistic Euler equations is presented. These equations describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic scheme, is based on the well-known fact that the relativistic Euler equations are the moments of the relativistic Boltzmann equation of the kinetic theory of gases when the distribution function is a relativistic Maxwellian. The kinetic scheme consists of two phases, the convection phase (free-flight) and collision phase. The velocity distribution function at the end of the free-flight is the solution of the collisionless transport equation. The collision phase instantaneously relaxes the distribution to the local Maxwellian distribution. The fluid dynamic variables of density, velocity, and internal energy are obtained as moments of the velocity distribution function at the end of the free-flight phase. The scheme presented here is an explicit method and unconditionally stable. The conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. The scheme also satisfies positivity and L 1-stability. The scheme can be easily made into a total variation diminishing method for the distribution function through a suitable choice of the interpolation strategy. In the numerical case studies the results obtained from the first- and second-order kinetic schemes are compared with the first- and second-order upwind and central schemes. We also calculate the experimental order of convergence and numerical L 1-stability of the scheme for smooth initial data.

Global gyrokinetic simulation of ion temperature gradient driven turbulence in plasmas using a canonical Maxwellian distribution

A new gyrokinetic toroidal particle code has been developed to study the ion temperature gradient (ITG) driven turbulence in reactor relevant tokamak parameters. We use a new method based on a canonical Maxwellian distribution FCM(Pφ,ε,μ), which is defined by three constants of motion in the axisymmetric toroidal system—the canonical angular momentum Pφ, the energy ε, and the magnetic moment μ. A quasi-ballooning representation enables linear and nonlinear high-m,n global calculations to be carried out, with a good numerical convergence. Conservation properties are improved by using optimized particle loading. From comprehensive linear global analyses over a wide range of unstable toroidal mode numbers (n = 0–100) in large tokamak parameters (a/ρti = 320–460), it is found that the reversed shear configuration produces an effective stabilizing effect on the ITG mode in the q min region through global effects. In the nonlinear simulation, it is found that the new method based on FCM can simulate a zonal flow damping correctly; and spurious zonal flow oscillations, which are observed in a conventional method based on a local Maxwellian distribution FLM(ψ,ε,μ), do not appear in the nonlinear regime.

Models of two‐dimensional embedded thin current sheets from Vlasov theory

This paper uses the steady state Vlasov theory to obtain an explicit description of the properties of thin current sheets in two space dimensions. The method is general; in particular, it is not restricted to local Maxwellian distribution functions. The treatment includes chaotic particle motion, and the plasma is treated as quasi‐neutral. Such general equilibria are necessary for modeling observed structures in the Earth's magnetotail. In particular, this applies to thin current sheets embedded in wider sheets. Abandoning local Maxwellian distribution functions implies that the electric potential can no longer be transformed to zero such that additional efforts are required to include the electric field. Critical input parameters are the gyroscales of the different particle species and the ion/electron temperature ratio. The results illustrate how finite ion gyroscales affect the current sheet structure. Further aspects that are analyzed include the contributions of electrons and ions to the total current density, Hall currents, the occurrence of significant electric fields and the role of the ion/electron temperature ratio. The necessity of electric potentials in the self‐consistent structure of thin current sheets of the present type may provide a link between the presence or formation of thin current sheets and auroral arcs.

Monte Carlo Simulations of Edge Ion Temperature Profile in ASDEX Upgrade

The possibility of using the low energy channels of neutral particle analyzers to measure edge ion temperature profile in tokamaks has been investigated. The perpendicular velocity distribution of the edge ions has been simulated for an H-mode ASDEX Upgrade discharge using the guiding center orbit following Monte Carlo code ASCOT. The local temperatures extracted from the simulation by assuming a local Maxwellian distribution are within 10% of the background temperature. More simulations with wider radial extent and longer simulation times are required in order to determine if the small discrepancy is due to boundary effects, incomplete filling of the velocity space, or a non-Maxwellian nature of the ion distribution function. When the ion temperature profile is made very steep, the non-Maxwellian nature of the distribution does emerge already in these simulations.

Molecular Flow Analysis in Orifice-Restricted Tube Using Maxwellian Distribution Function.

Free molecular flow through a long conduit with an orifice-restricted exit was analyzed theoretically using a local Maxwellian distribution function. The theoretically derived transmission probabilities for the conduit with the ratio of length to tube diameter greater than 15, and the cross-sectional area ratio of tube and orifice openings greater than 70 agrees well with those computed using a Monte Carlo method.

Nonlinear Debye-Onsager relaxation effect in weakly ionized plasmas

A weakly ionized plasma under the influence of a strong electric field is considered. Supposing a local Maxwellian distribution for the electron momenta the plasma is described by hydrodynamic equations for the pair distribution functions. These equations are solved and the relaxation field is calculated for an arbitrary field strength. It is found that the relaxation effect becomes lower with increasing strength of the electrical field.

Kinetic theory of E0×B0 instabilities in the ionosphere

A self‐consistent kinetic theory of the E0×B0 instabilities in partially ionized, inhomogeneous plasmas with arbitrary values of να/Ωα, where να represents the electron (ion)‐neutral collision frequency and Ωα is the electron (ion) gyrofrequency, is presented. The theory is based on the kinetic equation with a particle number conserving collision term, which allows the particle distribution function to relax toward a local Maxwellian distribution at rest. The method consists of first solving the zero‐order kinetic equation to determine the self‐consistent equilibrium distribution function. The distribution function is shown to accurately represent the plasma equilibrium state, with appropriate Hall and Pedersen drifts that occur in a collisional plasma in the presence of crossed ambient electric and magnetic fields. The linear dispersion relation is then derived from the first‐order kinetic equation, and it can be used to study the E0×B0 instabilities in all altitude regions of the ionosphere in a unified manner, without the need for any a priori knowledge of the different types of particle drifts (Hall and Pedersen drifts) that are responsible for the instabilities in different altitude regions. The present theory therefore provides a more rigorous kinetic description of the E0×B0 instabilities than that obtained from the previously studied kinetic model in which the zero‐order particle drifts are not determined self‐consistently but have to be specified extraneously.

Electron dynamics in two‐ and one‐dimensional oblique supercritical collisionless magnetosonic shocks

Two‐ and one‐dimensional fully electromagnetic, bounded, particle (for both electrons and ions) codes are used in order to study electron dynamics in collisionless magnetosonic shocks propagating in supercritical regime and quasi‐perpendicular direction (90° > θ0 > 45°); θ0 is the angle between the shock normal and the upstream magnetic field. The purpose of the study consists in comparing electrons behavior in one‐dimensional (“pseudo‐oblique”) nonresistive shocks and in two‐dimensional resistive oblique shocks. Resistive effects related to plasma microinstabilities can be self‐consistently included in two‐dimensional particle codes in contrast with one‐dimensional particle codes. Present two‐dimensional results reproduce local electron distribution functions (in particular, downstream “flat tops”) in a self‐consistent way and in good agreement with observational results. On the other hand, one‐dimensional results exhibit either local enlarged Maxwellian distributions with a partial tail, or a flat top distribution according to the particle density n. These results emphasize that (1) the differences observed between one‐ and two‐dimensional codes may be explained in terms of a critical particle density nc used in the one‐dimensional code; (2) the evidence of flat tops in both two‐ and one‐dimensional results (provided that n > nc) proves that the macroscopic potential jump at the shock front is mainly responsible for their formation; (3) microscopic effects (herein related to the self‐consistent cross‐field/field‐aligned currents instabilities) may represent a complementary mechanism for filling the flat top distribution; (4) some relaxation of the unstable electron flat top distribution (T∥/T⊥ ≫ 1) is observed when penetrating further into the downstream region, which means that the main filling mechanisms are localized in the ramp of the shock. Moreover, a detailed study of two‐dimensional results shows that both resistive and nonresistive configurations can be easily distinguished for θ0 ≈ 90°, but not any more for large deviations of θ0 from 90° for which the self‐consistent magnetic field rotates noticeably out of the coplanarity plane at the shock front.