64 research outputs found
A "metric" semi-Lagrangian Vlasov-Poisson solver
We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements
of metric to follow locally the flow and its deformation, allowing one to find
quickly and accurately the initial phase-space position of any test
particle , by expanding at second order the geometry of the motion in the
vicinity of the closest element. It is thus possible to reconstruct accurately
the phase-space distribution function at any time and position by
proper interpolation of initial conditions, following Liouville theorem. When
distorsion of the elements of metric becomes too large, it is necessary to
create new initial conditions along with isotropic elements and repeat the
procedure again until next resampling. To speed up the process, interpolation
of the phase-space distribution is performed at second order during the
transport phase, while third order splines are used at the moments of
remapping. We also show how to compute accurately the region of influence of
each element of metric with the proper percolation scheme. The algorithm is
tested here in the framework of one-dimensional gravitational dynamics but is
implemented in such a way that it can be extended easily to four or
six-dimensional phase-space. It can also be trivially generalised to plasmas.Comment: 32 pages, 14 figures, accepted for publication in Journal of Plasma
Physics, Special issue: The Vlasov equation, from space to laboratory plasma
A solvable model of Vlasov-kinetic plasma turbulence in Fourier-Hermite phase space
A class of simple kinetic systems is considered, described by the 1D
Vlasov-Landau equation with Poisson or Boltzmann electrostatic response and an
energy source. Assuming a stochastic electric field, a solvable model is
constructed for the phase-space turbulence of the particle distribution. The
model is a kinetic analog of the Kraichnan-Batchelor model of chaotic
advection. The solution of the model is found in Fourier-Hermite space and
shows that the free-energy flux from low to high Hermite moments is suppressed,
with phase mixing cancelled on average by anti-phase-mixing (stochastic plasma
echo). This implies that Landau damping is an ineffective route to dissipation
(i.e., to thermalisation of electric energy via velocity space). The full
Fourier-Hermite spectrum is derived. Its asymptotics are at low wave
numbers and high Hermite moments () and at low Hermite
moments and high wave numbers (). These conclusions hold at wave numbers
below a certain cut off (analog of Kolmogorov scale), which increases with the
amplitude of the stochastic electric field and scales as inverse square of the
collision rate. The energy distribution and flows in phase space are a simple
and, therefore, useful example of competition between phase mixing and
nonlinear dynamics in kinetic turbulence, reminiscent of more realistic but
more complicated multi-dimensional systems that have not so far been amenable
to complete analytical solution.Comment: 35 pages, minor edits, final version accepted by JP
ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation
Resolving numerically Vlasov-Poisson equations for initially cold systems can
be reduced to following the evolution of a three-dimensional sheet evolving in
six-dimensional phase-space. We describe a public parallel numerical algorithm
consisting in representing the phase-space sheet with a conforming,
self-adaptive simplicial tessellation of which the vertices follow the
Lagrangian equations of motion. The algorithm is implemented both in six- and
four-dimensional phase-space. Refinement of the tessellation mesh is performed
using the bisection method and a local representation of the phase-space sheet
at second order relying on additional tracers created when needed at runtime.
In order to preserve in the best way the Hamiltonian nature of the system,
refinement is anisotropic and constrained by measurements of local Poincar\'e
invariants. Resolution of Poisson equation is performed using the fast Fourier
method on a regular rectangular grid, similarly to particle in cells codes. To
compute the density projected onto this grid, the intersection of the
tessellation and the grid is calculated using the method of Franklin and
Kankanhalli (1993) generalised to linear order. As preliminary tests of the
code, we study in four dimensional phase-space the evolution of an initially
small patch in a chaotic potential and the cosmological collapse of a
fluctuation composed of two sinusoidal waves. We also perform a "warm" dark
matter simulation in six-dimensional phase-space that we use to check the
parallel scaling of the code.Comment: Code and illustration movies available at:
http://www.vlasix.org/index.php?n=Main.ColDICE - Article submitted to Journal
of Computational Physic
UHECR Acceleration in Dark Matter Filaments of Cosmological Structure Formation
A mechanism for proton acceleration to ~10^21eV is suggested. It may operate
in accretion flows onto thin dark matter filaments of cosmic structure
formation. The flow compresses the ambient magnetic field to strongly increase
and align it with the filament. Particles begin the acceleration by the ExB
drift with the accretion flow. The energy gain in the drift regime is limited
by the conservation of the adiabatic invariant p_perp^2/B. Upon approaching the
filament, the drift turns into the gyro-motion around the filament so that the
particle moves parallel to the azimuthal electric field. In this 'betatron'
regime the acceleration speeds up to rapidly reach the electrodynamic limit
for an accelerator with magnetic field and the orbit radius
(Larmor radius). The periodic orbit becomes unstable and the particle
slings out of the filament to the region of a weak (uncompressed) magnetic
field, which terminates the acceleration.
The mechanism requires pre-acceleration that is likely to occur in structure
formation shocks upstream or nearby the filament accretion flow. Previous
studies identify such shocks as efficient proton accelerators to a firm upper
limit ~10^19.5 eV placed by the catastrophic photo-pion losses. The present
mechanism combines explosive energy gain in its final (betatron) phase with
prompt particle release from the region of strong magnetic field. It is this
combination that allows protons to overcome both the photo-pion and the
synchrotron-Compton losses and therefore attain energy 10^21 eV. A requirement
on accelerator to reach a given E_max placed by the accelerator energy
dissipation \propto E_{max}^{2}/Z_0 due to the finite vacuum impedance Z_0 is
circumvented by the cyclic operation of the accelerator.Comment: 34 pages, 10 figures, to be published in JCA
An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations
In this work we construct a high-order, single-stage, single-step
positivity-preserving method for the compressible Euler equations. Space is
discretized with the finite difference weighted essentially non-oscillatory
(WENO) method. Time is discretized through a Lax-Wendroff procedure that is
constructed from the Picard integral formulation (PIF) of the partial
differential equation. The method can be viewed as a modified flux approach,
where a linear combination of a low- and high-order flux defines the numerical
flux used for a single-step update. The coefficients of the linear combination
are constructed by solving a simple optimization problem at each time step. The
high-order flux itself is constructed through the use of Taylor series and the
Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical
results in one- and two-dimensions are presented
Contour Dynamics Methods
In an early paper on the stability of fluid layers with uniform vorticity Rayleigh (1880) remarks:
"... In such cases, the velocity curve is composed of portions of straight lines which meet each other at finite angles. This state of things may be supposed to be slightly disturbed by bending the surfaces of transition, and the determination of the subsequent motion depends upon that of the form of these surfaces. For co retains its constant value throughout each layer unchanged in the absence of friction, and by a well-known theorem the whole motion depends upon [omega]."
We can now recognize this as essentially the principal of contour dynamics (CD), where [omega] is the uniform vorticity. The theorem referred to is the Biot-Savart law. Nearly a century later Zabusky et al (1979) presented numerical CD calculations of nonlinear vortex patch evolution. Subsequently, owing to its compact form conferring a deceptive simplicity, CD has become a widely used method for the investigation of two-dimensional rotational flow of an incompressible inviscid fluid. The aim of this article is to survey the development, technical details, and vortex-dynamic applications of the CD method in an effort to assess its impact on our understanding of the mechanics of rotational flow in two dimensions at ultrahigh Reynolds numbers.
The study of the dynamics of two- and three-dimensional vortex mechanics by computational methods has been an active research area for more than two decades. Quite apart from many practical applications in the aerodynamics of separated flows, the theoretical and numerical study of vortices in incompressible fluids has been stimulated by the idea that turbulent fluid motion may be viewed as comprising ensembles of more or less coherent laminar vortex structures that interact via relatively simple dynamics and by the appeal of the vorticity equation, which does not contain the fluid pressure. Two-dimensional vortex interactions have been perceived as supposedly relevant to the origins of coherent structures observed experimentally in mixing layers, jets, and wakes, and for models of large-scale atmospheric and oceanic turbulence. Interest has often focused on the limit of infinite Reynolds number, where in the absence of boundaries, the inviscid Euler equations are assumed to properly describe the flow dynamics. The numerous surveys of progress in the study of vorticity and the use of numerical methods applied to vortex mechanics include articles by Saffman & Baker (1979) and Saffman (1981) on inviscid vortex interactions and Aref (1983) on two-dimensional flows. Numerical methods have been surveyed by Chorin (1980), and Leonard (1980, 1985). Caflisch (1988) describes work on the mathematical aspects of the subject. Zabusky (1981), Aref (1983), and Melander et al (1987b) discuss various aspects of CD. The review of Dritschel (1989) gives emphasis to numerical issues in CD and to recent computations with contour surgery.
This article is confined to a discussion of vortices on a two-dimensional surface. We generally follow Saffman & Baker (1979) in matters of definition. In two dimensions a vortex sheet is a line of discontinuity in velocity while a vortex jump is a line of discontinuity in vorticity. We shall, however, use filament to denote a two-dimensional ribbon of vorticity surrounded by fluid with vorticity of different magnitude (which may be zero), rather than the more usual three-dimensional idea of a vortex tube. The ambiguity is unfortunate but is already in the literature. Additionally, a vortex patch is a finite, singly connected area of uniform vorticity while a vortex strip is an infinite strip of uniform vorticity with finite thickness, or equivalently, an infinite filament. Contour Dynamics will refer to the numerical solution of initial value problems for piecewise constant vorticity distributions by the Lagrangian method of calculating the evolution of the vorticity jumps. Such flows are often related to corresponding solutions of the Euler equations that are steady in some translating or rotating frame of reference. These solutions will be called vortex equilibria, and the numerical technique for computing their shapes based on CD is often referred to as contour statics.
The mathematical foundation for the study of vorticity was laid primarily by the well-known investigations of Helmholtz, Kelvin, J. J. Thomson, Love, and others. In our century, efforts to produce numerical simulations of flows governed by the Euler equations have utilized a variety of Eulerian, Lagrangian, and hybrid methods. Among the former are the class of spectral methods that now comprise the prevailing tool for large-scale two- and three-dimensional calculations (see Hussaini & Zang 1987). The Lagrangian methods for two-dimensional flows have been predominantly vortex tracking techniques based on the Helmholtz vorticity laws. The first initial value calculations were those of Rosenhead (193l) and Westwater (1935) who attempted to calculate vortex sheet evolution by the motion of O(10) point vortices. Subsequent efforts by Moore (1974) (see also Moore 1983, 1985) and others to produce more refined computations for vortex sheets have failed for reasons related to the tendency for initially smooth vortex sheet data to produce singularities (Moore 1979). Discrete vortex methods used to study the nonlinear dynamics of vortex patches and layers have included the evolution of assemblies of point vortices by direct summation (e.g. Acton 1976) and the cloud in cell method (Roberts & Christiansen 1972, Christiansen & Zabusky 1973, Aref & Siggia 1980, 1981). For reviews see Leonard (1980) and Aref (1983). These techniques have often been criticized for their lack of accuracy and numerical convergence and because they may be subject to grid scale dispersion. However, many qualitative vortex phenomena observed in nature and in experiments, such as amalgamation events and others still under active investigation (e.g. filamentation) were first simulated numerically with discrete vortices. The contour dynamics approach is attractive because it appears to allow direct access, at least for small times, to the inviscid dynamics for vorticity distributions smoother than those of either point vortices or vortex sheets, while at the same time enabling the mapping of the two-dimensional Euler equations to a one-dimensional Lagrangian form.
In Section 2 we discuss the formulation and numerical implementation of contour dynamics for the Euler equations in two dimensions. Section 3 is concerned with applications to isolated and multiple vortex systems and to vortex layers. An attempt is made to relate this work to calculations of the relevant vortex equilibria and to results obtained with other methods. Axisymmetric contour dynamics and the treatment of the multi-layer model of quasigeostrophic flows are described in Section 4 while Section 5 is devoted to a discussion of the tendency shown by vorticity jumps to undergo the strange and subtle phenomenon of filamentation
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