Kinetic simulations of collision-less plasmas in open magnetic geometries†

Plasma Source ◽

Computational Effort ◽

Distribution Functions ◽

Ion Distribution ◽

Initial State ◽

Loss Cone

Abstract Laboratory plasmas in open magnetic geometries can be found in many diﬀerent applications such as (1) Scrape-Of-Layer (SOL) and divertor regions in toroidal conﬁnement fusion devices , (2) linear divertor simulators, (3) plasma-based thrusters and (4) magnetic mirrors. A common feature of these plasma systems is the need to resolve, in addition to velocity space, at least one physical dimension (e.g. along ﬂux lines) to capture the relevant physics. In general, this requires a kinetic treatment. Fully kinetic Particle-In-Cell (PIC) simulations can be applied but at the expense of large computational eﬀort. A common way to resolve this is to use a hybrid approach: kinetic ions and ﬂuid electrons. In the present work, the development of a hybrid PIC computational tool suitable for open magnetic geometries is described which includes (1) the eﬀect of non-uniform magnetic ﬁelds, (2) ﬁnite fully-absorbing boundaries for the particles and (3) volumetric particle sources. Analytical expressions for the momentum transport in the paraxial limit are presented with their underlying assumptions and are used to validate the results from the PIC simulations. A general method is described to construct discrete particle distribution functions in state of mirror-equilibrium. This method is used to obtain the initial state for the PIC simulation. Collisionless simulations in a mirror geometry are performed. Results show that the eﬀect of magnetic compression is correctly described and momentum is conserved. The self-consistent electric ﬁeld is calculated and is shown to modify the ion velocity distribution function in manner consistent with analytic theory. Based on this analysis, the ion distribution function is understood in terms of a loss-cone distribution and an isotropic Maxwell-Boltzmann distribution driven by a volumetric plasma source. Finally, inclusion of a Monte-Carlo based Fokker-Planck collision operator is discussed in the context of future work.

The hot hELicon eXperiment (HELIX) and the large experiment on instabilities and anisotropy (LEIA)

10.1017/s0022377814000890 ◽

2014 ◽

Vol 81 (1) ◽

Cited By ~ 13

Author(s):

E. E. Scime ◽

P. A. Keiter ◽

M. M. Balkey ◽

J. L. Kline ◽

X. Sun ◽

...

Keyword(s):

Double Layer ◽

Noble Gas ◽

Plasma Source ◽

Distribution Functions ◽

Double Layers ◽

Driving Frequency ◽

Ion Temperature ◽

Temperature Anisotropy ◽

Ion Distribution

The West Virginia University Hot hELIcon eXperiment (HELIX) provides variable density and ion temperature plasmas, with controllable levels of thermal anisotropy, for space relevant laboratory experiments in the Large Experiment on Instabilities and Anisotropy (LEIA) as well as fundamental studies of helicon source physics in HELIX. Through auxiliary ion heating, the ion temperature anisotropy (T⊥/T∥) is variable from 1 to 20 for parallel plasma beta (β = 8πnkTi∥/B2) values that span the range of 0.0001 to 0.01 in LEIA. The ion velocity distribution function is measured throughout the discharge volume in steady-state and pulsed plasmas with laser induced fluorescence (LIF). The wavelengths of very short wavelength electrostatic fluctuations are measured with a coherent microwave scattering system. Operating at low neutral pressures triggers spontaneous formation of a current-free electric double layer. Ion acceleration through the double layer is detected through LIF. LIF-based velocity space tomography of the accelerated beam provides a two-dimensional mapping of the bulk and beam ion distribution functions. The driving frequency for the m = 1 helical antenna is continuously variable from 8.5 to 16 MHz and frequency dependent variations of the RF coupling to the plasma allow the spontaneously appearing double layers to be turned on and off without modifying the plasma collisionality or magnetic field geometry. Single and multi-species plasmas are created with argon, helium, nitrogen, krypton, and xenon. The noble gas plasmas have steep neutral density gradients, with ionization levels reaching 100% in the core of the plasma source. The large plasma density in the source enables the study of Aflvén waves in the HELIX device.

Magnetosonic solitons in space plasmas: dark or bright solitons?

10.1017/s0022377807006526 ◽

2007 ◽

Vol 73 (6) ◽

pp. 981-992 ◽

Cited By ~ 11

Author(s):

O. A. POKHOTELOV ◽

O.G. ONISHCHENKO ◽

M. A. BALIKHIN ◽

L. STENFLO ◽

P. K. SHUKLA

Keyword(s):

Magnetic Field ◽

Distribution Function ◽

Mhd Equations ◽

Space Plasmas ◽

Ion Distribution ◽

Loss Cone ◽

Solitary Solution ◽

Background Magnetic Field ◽

The Magnetic Field

AbstractThe nonlinear theory of large-amplitude magnetosonic (MS) waves in highβ space plasmas is revisited. It is shown that solitary waves can exist in the form of ‘bright’ or ‘dark’ solitons in which the magnetic field is increased or decreased relative to the background magnetic field. This depends on the shape of the equilibrium ion distribution function. The basic parameter that controls the nonlinear structure is the wave dispersion, which can be either positive or negative. A general dispersion relation for MS waves propagating perpendicularly to the external magnetic field in a plasma with an arbitrary velocity distribution function is derived.It takes into account general plasma equilibria, such as the Dory–Guest–Harris (DGH) or Kennel–Ashour-Abdalla (KA) loss-cone equilibria, as well as distributions with a power-law velocity dependence that can be modelled by κdistributions. It is shown that in a bi-Maxwellian plasma the dispersion is negative, i.e. the phase velocity decreases with an increase of the wavenumber. This means that the solitary solution in this case has the form of a ‘bright’ soliton with the magnetic field increased. On the contrary, in some non-Maxwellian plasmas, such as those with ring-type ion distributions or DGH plasmas, the solitary solution may have the form of a magnetic hole. The results of similar investigations based on nonlinear Hall–MHD equations are reviewed. The relevance of our theoretical results to existing satellite wave observations is outlined.

The velocity distribution function within a shock wave

Journal of Fluid Mechanics ◽

10.1017/s0022112067001557 ◽

1967 ◽

Vol 30 (3) ◽

pp. 479-487 ◽

Cited By ~ 51

Author(s):

G. A. Bird

Keyword(s):

Shock Wave ◽

Distribution Function ◽

Velocity Distribution ◽

Numerical Experiments ◽

Equilibrium Distribution ◽

Distribution Functions ◽

Rigid Sphere ◽

Lateral Velocity ◽

Shock Profile

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.

Loss-cone index in distribution function in a hot magnetized plasma

10.1017/s0022377806004491 ◽

2007 ◽

Vol 73 (2) ◽

pp. 207-214 ◽

Cited By ~ 2

Author(s):

R. P. SINGHAL ◽

A. K. TRIPATHI

Keyword(s):

Distribution Function ◽

Growth Rates ◽

Particle Distribution ◽

Distribution Functions ◽

Magnetized Plasma ◽

Dielectric Tensor ◽

Loss Cone ◽

Bernstein Waves ◽

Cone Index

Abstract.The components of the dielectric tensor for the distribution function given by Leubner and Schupfer have been obtained. The effect of the loss-cone index appearing in the particle distribution function in a hot magnetized plasma has been studied. A case study has been performed to calculate temporal growth rates of Bernstein waves using the distribution function given by Summers and Thorne and Leubner and Schupfer. The effect of the loss-cone index on growth rates is found to be quite different for the two distribution functions.

A study of the mass ratio dependence of the mixed species collision integral for isotropic velocity distribution functions

10.1017/s002237780002643x ◽

1982 ◽

Vol 27 (1) ◽

pp. 135-148 ◽

Cited By ~ 6

Author(s):

A. J. M. Garrett

Keyword(s):

Distribution Function ◽

Velocity Distribution ◽

Particle Distribution ◽

Collision Integral ◽

Distribution Functions ◽

Velocity Space ◽

Particle Distribution Function ◽

Collision Term ◽

Test Species

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

Statistical Inferences of Lift-Off Velocity Distribution Function Form for Saltating Particles

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.108-111.783 ◽

2010 ◽

Vol 108-111 ◽

pp. 783-788

Author(s):

Jian Jun Wu ◽

Li Hong He

Keyword(s):

Distribution Function ◽

Weibull Distribution ◽

Velocity Distribution ◽

Goodness Of Fit ◽

Forward Direction ◽

Distribution Functions ◽

Lift Off ◽

Log Normal

The lift-off velocity distribution of saltating particles, which have been proposed to characterize the dislodgement state of saltating particles, is one of the key issues in the theoretical study of windblown sand transportation. But there were various statistical relations in the early researches. In this paper, the Kolmogorov-Smirnov test for goodness-of-fit is adopted to make an inference of the most probable form of lift-off velocity distribution functions for saltating particles on the basis of the experimental data. The statistical results show that the distribution function of vertical lift-off velocities conforms better to Weibull distribution function than to the normal, log-normal, gamma and exponential ones; while, the distribution function of the absolute values of horizontal lift-off velocities is best described by log-normal distribution in forward direction and Weibull distribution in backward direction, respectively. Finally, two more examples prove to support the above conclusions.

Flowing Granular Materials and the Maxwell-Boltzmann Velocity Distribution

10.1115/imece2000-2001 ◽

2000 ◽

Author(s):

Edward J. Boyle

Keyword(s):

Distribution Function ◽

Velocity Distribution ◽

Hard Sphere ◽

Body Force ◽

Canonical Ensemble ◽

Granular Flows ◽

Distribution Functions ◽

Single Granule

Abstract The single-granule velocity distribution function is shown to be Maxwell-Boltzmann for hard-sphere granular flows at steady-state exhibiting no gradients and absent a body-force. This is accomplished by approximating the two-granule velocity distribution function as the product of two single-granule velocity distribution functions and a correlating function and by applying to a canonical ensemble a function analogous to Boltzmann’s H-function.

The stress tensor in a granular flow at high shear rates

Journal of Fluid Mechanics ◽

10.1017/s0022112081000736 ◽

1981 ◽

Vol 110 ◽

pp. 255-272 ◽

Cited By ~ 325

Author(s):

S. B. Savage ◽

D. J. Jeffrey

Keyword(s):

Distribution Function ◽

Shear Rate ◽

Shear Flow ◽

Stress Tensor ◽

Particle Velocity ◽

Volume Fraction ◽

Particle Diameter ◽

Distribution Functions ◽

Dimensional Parameter

The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R: the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R, the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.

Velocity Distribution Functions and Transport Coefficients of Electron Swarms in Gases in the Presence of Crossed Electric and Magnetic Fields

Australian Journal of Physics ◽

10.1071/ph950557 ◽

1995 ◽

Vol 48 (3) ◽

pp. 557 ◽

Cited By ~ 12

Author(s):

KF Ness

Keyword(s):

Distribution Function ◽

Magnetic Fields ◽

Velocity Distribution ◽

Transport Coefficients ◽

Distribution Functions ◽

Electron Motion ◽

Electric And Magnetic Fields ◽

Spatially Homogeneous

A multi-term solution of the Boltzmann equation is used to calculate the spatially homogeneous velocity distribution function of a dilute swarm of electrons moving through a background of denser neutral molecules in the presence of crossed electric and magnetic fields. As an example, electron motion in methane is considered.

Electromagnetic ion-cyclotron wave growth rates and their variation with velocity distribution function shape

10.1017/s002237780002047x ◽

1977 ◽

Vol 17 (1) ◽

pp. 123-131 ◽

Cited By ~ 59

Author(s):

Abraham Shrauner ◽

W. C. Feldman

Keyword(s):

Distribution Function ◽

Velocity Distribution ◽

Growth Rates ◽

Distribution Functions ◽

Wave Growth ◽

Ion Cyclotron ◽

Velocity Moments ◽

Function Shape

The sensitivity of electromagnetic ion-cyclotron wave growth rates to the details of the shape of proton velocity distribution functions is explored. For this purpose two different forms of bi-Lorentzian for the proton distribution functions were adopted. The growth rates for the two types of bi-Lorentzians and the biMaxwellians for the beam (hot) protons are compared. Although the growth rates for the three shapes depend on the velocity moments of the different velocity distributions in a similar way, their magnitudes were found to vary considerably.