Arbitrary Velocity Distribution Research Articles

Directivity of horns mounted in finite enclosures: A multimodal formulation.

The beamwidth is a primary directivity metric for the design of a constant directivity horn. To date, investigations on this property have predominantly been restricted to the half-space radiation or idealized geometries. This paper examines the beamwidth behavior of an axisymmetric horn mounted in a finite cylindrical enclosure by proposing an elegant multimodal solution to the far-field directivity pattern. The variation of beamwidth is examined for the frequency, dimensions of the enclosure, and shape of the horn. At low frequencies, a fitted model is proposed to precisely depict the intrinsic beam narrowing governed by the enclosure diffraction. The asymptotic behavior of the beamwidth is explored as the flange width increases. In the high-frequency range, the horn profile is a determinant of the directivity characteristics. We report the possibility of extending the bandwidth of a constant directivity horn by leveraging the enclosure diffraction effects. The proposed analytical method is highly accurate and much faster than the finite element method for wideband analysis. It allows for an arbitrary velocity distribution at the mouth of the horn and incorporates idealized flange configurations such as an infinite baffle, a zero-thickness closed baffle, and an infinitely long enclosure as limit cases.

Coupling multi-fluid dynamics equipped with Landau closures to the particle-in-cell method

The particle-in-cell (PIC) method is successfully used to study magnetized plasmas. However, this requires large computational costs and limits simulations to short physical run times and often to set-ups of less than three spatial dimensions. Traditionally, this is circumvented either via hybrid-PIC methods (adopting massless electrons) or via magneto-hydrodynamic-PIC methods (modelling the background plasma as a single charge-neutral magneto-hydrodynamical fluid). Because both methods preclude modelling important plasma-kinetic effects, we introduce a new fluid-PIC code that couples a fully explicit and charge-conserving multi-fluid solver to the PIC code SHARP through a current-coupling scheme and solve the full set of Maxwell's equations. This avoids simplifications typically adopted for Ohm's law and enables us to fully resolve the electron temporal and spatial scales while retaining the versatility of initializing any number of ion, electron or neutral species with arbitrary velocity distributions. The fluid solver includes closures emulating Landau damping so that we can account for this important kinetic process in our fluid species. Our fluid-PIC code is second-order accurate in space and time. The code is successfully validated against several test problems, including the stability and accuracy of shocks and the dispersion relation and damping rates of waves in unmagnetized and magnetized plasmas. It also matches growth rates and saturation levels of the gyro-scale and intermediate-scale instabilities driven by drifting charged particles in magnetized thermal background plasmas in comparison with linear theory and PIC simulations. This new fluid-SHARP code is specially designed for studying high-energy cosmic rays interacting with thermal plasmas over macroscopic time scales.

On the analytical description of the nonlinear stage of the Weibel instability in collisionless anisotropic plasma

On the basis of the energy invariants for the Weibel instability in a collisionless non-relativistic plasma, an analytical relation is obtained between the instantaneous values of the space-averaged energy density of the magnetic field, its dominant wavenumber and the average plasma anisotropy parameter. The relation is valid for arbitrary particle velocity distribution functions, including ones that vary with time at the nonlinear stage of instability. It is obtained under the assumption that the plasma is homogeneous along a certain axis, determined, for example, by an external magnetic field, and taking into account only the modes with wavevectors orthogonal to this axis. We give an estimate of the maximum magnetic field energy achievable during the Weibel instability and show that its ratio to the initial longitudinal energy of particles, even for a large initial plasma anisotropy parameter, cannot exceed the value approximately equal to 0.2. The obtained analytical relation is verified using two-dimensional particle-in-cell simulations.

A simple and fast approach for computing the fusion reactivities with arbitrary ion velocity distributions

Calculating fusion reactivity involves a complex six-dimensional integral of the fusion cross section and ion velocity distributions of two reactants. We demonstrate a simple Monte Carlo approach that efficiently computes this integral for arbitrary ion velocity distributions with a time complexity of O(N), where N is the number of samples. This approach generates random numbers that satisfy the reactant velocity distributions. In cases where these numbers are not readily available, we propose using Gaussian random numbers with weighted factors. For cases where only a small number of N samples are available, a O(N2) method can be used. We benchmarked this approach against analytical results for drift bi-Maxwellian distributions and provided examples of drift ring beam and slowing down distributions. Our results show that the error can be less than 1% with N∼104 samples for our standard approach.

Plane-Parallel Laminar Flow of Viscous Fluid in the Transition Zone of the Inlet Section

A study was conducted to analyze how hydrodynamic parameters change in the entrance region of plane-parallel flow under stationary flow conditions, with an initial arbitrary distribution of velocities in the entrance section. This study was based on boundary layer equations, and a boundary problem was formed under the conditions of plane-parallel flow. The boundary conditions were chosen to reflect the pattern of arbitrary velocity distribution in the entrance section. A general solution of the approximating Navier-Stokes equations is provided, which depends on the initial conditions and the Reynolds number. The boundary conditions are established based on the nature of the motion, and the boundary value problem is described. A method for integrating the boundary value problem has been developed, and regularities for the change in velocities along the length of the inlet section have been obtained for both constant and parabolic velocity distributions in the entrance sections. Analytical solutions have been derived to obtain patterns of velocity and pressure changes in any given flow direction. Through computer analysis, velocity change patterns in various sections along the inlet transition area have been constructed, allowing for the determination of fluid velocity at any point on the section and an estimation of the length of the transition area. These proposed solutions provide a framework for accurately constructing individual units of hydromechanical equipment.

Wide-binary Stars Formed in the Turbulent Interstellar Medium

The ubiquitous interstellar turbulence regulates star formation and the scaling relations between the initial velocity differences and the initial separations of stars. We propose that the formation of wide binaries with initial separations r in the range ∼103 au ≲ r ≲ 105 au is a natural consequence of star formation in the turbulent interstellar medium. With the decrease of r, the mean turbulent relative velocity v tur between a pair of stars decreases, while the largest velocity v bon at which they still may be gravitationally bound increases. When v tur < v bon, a wide binary can form. In this formation scenario, we derive the eccentricity distribution p(e) of wide binaries for an arbitrary relative velocity distribution. By adopting a turbulent velocity distribution, we find that wide binaries at a given initial separation generally exhibit a superthermal p(e), irrespective of the exact turbulent velocity scaling. This provides a natural explanation for the observed superthermal p(e) of the wide binaries in the solar neighborhood.

Lamb-type solution and properties of unsteady Stokes equations

We derive the general solution of the unsteady Stokes equations in unbounded viscous fluid exterior to a sphere. The solution is an expansion in vector spherical harmonics and it uses the decomposition originally introduced by Lamb. The coefficients of the expansion are uniquely determined by the arbitrary velocity distribution at the spherical boundary. The proposed solution can be applied, for example, to construct transient flow around a spherical ``squirmer'', to study particle-particle and particle-wall interactions in oscillatory flows and in other applications.

Kinetic theory of instability in the interaction of an electron beam and plasma with an arbitrary anisotropic electron velocity distribution function

Based on the kinetic approach, this work investigates the stability of the system consisting of a fast electron beam and a dense plasma at an arbitrary (anisotropic) electron velocity distribution function. It is shown that during the interaction of a fast electron beam with a cold plasma, both the conditions for losing stability and the growth rate of disturbances do not depend on the form of the electron distribution function (EDF) of a plasma and are determined only by the ratio of the electron beam energy to the mean energy in a plasma. With an increase in the mean electron energy in the plasma, it becomes necessary to take into account the following energy moments of the EDF. It was found that the plasma anisotropy has a significant effect on both the stability loss conditions and the growth rate. The physical reason for this effect is the shift in the plasma frequency due to the Doppler effect caused by the plasma anisotropy in the coordinate system moving along with the beam. Other findings include a region of anomalous dispersion of the electron beam–plasma system and regions of negative group velocity of perturbations in such system. Physical interpretations are proposed for all the observed effects.

Identification of slow-wave ion cyclotron emission on JT-60U

Ion cyclotron emissions (ICEs) are generally considered to be emissions of fast waves driven by velocity-space instabilities due to fast ions. However, on JT-60U, measured properties of ICEs observed at frequencies lower than the bulk ion cyclotron frequency (L-ICEs) do not match the fast-wave dispersion relation, indicating the possibility of L-ICEs being slow-wave emissions. In this paper, we show that L-ICE observation on JT-60U can be explained in terms of a slow-wave emission driven by fast ions, i.e. a slow-wave ICE. To investigate whether the slow wave can be driven by fast ions, its linear growth rate has been calculated in a typical discharge using a wave dispersion code. This wave dispersion code is capable of performing calculations with an arbitrary spatially-localized velocity distribution. For the growth rate calculation, we have directly used fast ion velocity distributions evaluated by an orbit following Monte-Carlo simulation. It is found that negative-ion-source neutral beam (N-NB) injected fast ions can destabilize the slow wave. Its frequency and wavenumber at high linear growth rates are close to experimental observations of L-ICE. In addition, observed L-ICE frequencies agree reasonably well with frequencies based on the slow-wave dispersion relation at measured toroidal wavenumbers in several discharges. Moreover, the observed frequencies also agree with the Doppler-shifted cyclotron resonance frequencies for the N-NB injected ion. Therefore, L-ICE in JT-60U is identified as a slow-wave ICE driven by the N-NB injected fast ions and distinguished from other ICEs through a measurement of its frequency and wavenumber.

Survival probability of a run-and-tumble particle in the presence of a drift

We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate γ, the (new) velocity of the particle is drawn randomly from a distribution W(v). We study the survival probability S(x, t) of a particle starting from x ⩾ 0 up to time t and obtain an explicit expression for its double Laplace transform (with respect to both x and t) for an arbitrary velocity distribution W(v), not necessarily symmetric. This result is obtained as a consequence of Spitzer’s formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity ±v 0, the so-called persistent random walk (PRW), and in the presence of a constant drift μ and obtain an explicit expression for S(x, t), for which we present more detailed results. Depending on the drift μ, we find a rich variety of behaviors for S(x, t), leading to three distinct cases: (i) subcritical drift −v 0 < μ < v 0, (ii) supercritical drift μ < −v 0 and (iii) critical drift μ = −v 0. In these three cases, we obtain exact analytical expressions for the survival probability S(x, t) and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.

Viscous Slip MHD Flow over a Moving Sheet with an Arbitrary Surface Velocity

The magnetohydrodynamic (MHD) flow induced by a stretching or shrinking sheet under slip conditions is studied.Analytical solutions based on the boundary layer assumption are obtained in a closed form and can be applied to a flow configuration with any arbitrary velocity distributions. Seven typical sheet velocity profiles are employed as illustrating examples. The solutions to the slip MHD flow are derived from the general solution and discussed in detail. Different from self-similar boundary layer flows, the flows studied in this work have solutions in explicit analytical forms. However, the current flows require special mass transfer at the wall, which is determined by the moving velocity of the sheet. The effects of the slip parameter, the mass transfer at the wall, and the magnetic field on the flow are also demonstrated.

Incoherent Scattering of Electromagnetic Waves by Langmuir Fluctuations Trapped in a Plasma Density Well

Incoherent scattering of a probing wave by Langmuir fluctuations trapped and enhanced near a local minimum of the electron density (plasma density well) in plasma with a parabolic density profile is considered. Steady-state amplitudes of fluctuations are calculated for arbitrary velocity distribution functions of plasma particles with allowance for electron collisions. It is shown that quasi-periodic oscillations with two characteristic scales can be present in the spectrum of the plasma line. The smaller scale is due to the wellknown effect of discretization of the spectrum of Langmuir fluctuations in a plasma density well. The larger scale is associated with the generation of scattered waves in two spatial regions and subsequent interference of these waves at the exit from the density well. Oscillations with this scale are more stable under unsteady plasma conditions and can be more often observed in experiments. The results of this work can be used to experimentally determine the plasma parameters, such as the electron collision frequency and the size and lifetime of the plasma density well.

Full-wave simulations of ICRF heating regimes in toroidal plasma with non-Maxwellian distribution functions

At the power levels required for significant heating and current drive in magnetically-confined toroidal plasma, modification of the particle distribution function from a Maxwellian shape is likely (Stix 1975 Nucl. Fusion 15 737), with consequent changes in wave propagation and in the location and amount of absorption. In order to study these effects computationally, both the finite-Larmor-radius and the high-harmonic fast wave (HHFW), versions of the full-wave, hot-plasma toroidal simulation code TORIC (Brambilla 1999 Plasma Phys. Control. Fusion 41 1 and Brambilla 2002 Plasma Phys. Control. Fusion 44 2423), have been extended to allow the prescription of arbitrary velocity distributions of the form . For hydrogen (H) minority heating of a deuterium (D) plasma with anisotropic Maxwellian H distributions, the fractional H absorption varies significantly with changes in parallel temperature but is essentially independent of perpendicular temperature. On the other hand, for HHFW regime with anisotropic Maxwellian fast ion distribution, the fractional beam ion absorption varies mainly with changes in the perpendicular temperature. The evaluation of the wave-field and power absorption, through the full wave solver, with the ion distribution function provided by either a Monte-Carlo particle and Fokker–Planck codes is also examined for Alcator C-Mod and NSTX plasmas. Non-Maxwellian effects generally tend to increase the absorption with respect to the equivalent Maxwellian distribution.

On a problem of optimal control of the laminar boundary layer in incompressible flow

In this paper, it is shown that the optimal injection velocity in the plane critical point is the finite one at the arbitrary velocity distribution on the outer edge of the boundary layer.

Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

We consider super-diffusive Lévy walks in dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d > 1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

Analytical solutions for steady two-dimensional suspended sediment transport in channels with arbitrary advection velocity and eddy diffusivity distributions

ABSTRACTAn analytical solution approach is proposed for the steady two-dimensional suspended sediment transport in channels using the generalized integral transform technique (GITT). Previous analytical solutions are limited by advection velocity distribution, eddy diffusivity distribution, or boundary conditions. The contribution of the new approach is to eliminate these limitations. It solves the suspended sediment transport equation with arbitrary advection velocity and eddy diffusivity distributions. The new solution approach also considers realistic boundary conditions at the free surface and the bed. Compared with previous analytical solutions as well as numerical solutions from a high resolution spectral code, the new approach is of high accuracy. It greatly extends the domain of analytical solutions for suspended sediment transport problems.

Evaluation of the spatial impulse response of planar ultrasonic radiators

A method for evaluating the spatial impulse response for a planar source having arbitrary velocity and delay distribution is presented. The time-domain approach is based on the Rayleigh integral formulation for the velocity potential. To obtain the velocity potential one must evaluate and superpose the contribution from each elemental element of the radiator. This step entails the accumulation of the amplitude contribution from each elemental source arriving at the observation point at a prescribed time. The problem of sorting of these arrivals is examined in this work and procedures for mapping the algorithm for implementation on general-purpose graphics processing units are presented.

Undulator Radiation of Dense Plane Electron Bunches

A simple and general self-consistent theory of coherent spontaneous undulator radiation from electron bunches has been developed. The approach is based on the model of a moving and radiating plane layer filled with oscillating relativistic electrons (gas of radiating and interacting 1-D particles). Using simple expressions for the field of a charged plane makes it easy to consider both Coulomb and radiation interaction of the electrons in bunches with arbitrary particle trajectories, energy chirp, and velocity and density distributions. The application of the developed theory for estimations of a compact radiation source in the terahertz frequency range has demonstrated the possibility of efficiency enhancement and narrowing of the radiation spectrum by means of using bunches with decreasing electron density.

The effects of solar wind velocity distributions on the refilling of the lunar wake: ARTEMIS observations and comparisons to one‐dimensional theory

AbstractThe lunar plasma wake refills from all directions, with processes operating both parallel and perpendicular to the magnetic field. The resulting wake structure depends sensitively on the properties of the flowing plasma, including the form of the ion and electron velocity distributions. In this manuscript, we discuss theoretical approximations for the refilling of the lunar wake along the magnetic field. While an often‐used treatment for the parallel refilling assumes cold ions, one can derive solutions for arbitrary ion velocity distributions. Similarly, though the most tractable theory utilizes Maxwellian electrons, one can derive solutions for other types of distributions. We discuss the theoretical framework for various one‐dimensional solutions, spanning the full range from cold‐ion theories to gas‐dynamic solutions, and utilizing both Maxwellian and kappa electron distributions. We compare these solutions to ARTEMIS observations of the lunar wake, for time periods with appropriate plasma parameters. We also present cases that reveal the inherent limitations of one‐dimensional approximations, including those related to electron anisotropies and those related to perpendicular processes associated with both fluid flow and ion gyro‐motion.

Harmonics in the dark-matter sky: directional detection in the Fourier-Bessel basis

Details about the velocity distribution of weakly interacting massive particle (WIMP) dark matter in our galaxy may be revealed by nuclear-recoil detectors with directional sensitivity. Previous studies have assumed that the velocity distribution takes a simple functional form characterized by a small number of parameters. More recent work has shown that basis-function expansions may allow for more general parameterization; such an approach has been considered for both the one-dimensional speed and momentum distributions, and also for three-dimensional velocity distributions obeying certain equilibrium conditions. In this work, I extend this basis-function approach to allow for arbitrary velocity distributions by working in the Fourier-Bessel basis, deriving an analytic expression for the directional recoil spectrum. Such an approach is completely general, and may be useful if the velocity distribution is too complex to be characterized by simple functional forms or is not completely virialized. Results concerning the three-dimensional Radon transform of the Fourier-Bessel basis functions may be of general interest for tomographic applications.