Most of the visible matter in the Universe is a plasma, that is a dilute gas of electrons, ions,
and neutral particles. Numerical methods are the only viable way of studying the dynamics of astrophysical
plasmas. Using numerical simulations, much progress has been made in recent years in understanding a variety
of important problems, including the structure and evolution of accretion flows around compact objects such
as neutron stars and black holes, and the decay rate and fluctuation statistics of compressible MHD turbulence.
Almost without exception, such advances have used multidimensional MHD codes. However, for many astrophysical
problems, the MHD approximation may not be valid. Examples include the dynamics of very dilute accretion flows,
the dynamics of turbulent plasmas near the energy dissipation scale, or magnetic reconnection. In order to
address fundamental problems in these areas, it will be necessary to move beyond the MHD approximation, and
consider particle kinetics. However, a full time-dependent and multidimensional numerical solution to the
Boltzmann equation is intractable in most circumstances, thus novel methods will be required.
The goal of this workshop is to bring together astrophysicists, plasma physicists, and applied mathematicians
to discuss future developments in numerical methods for astrophysical plasmas. Topics to be covered include:
reviews of the astrophysical problems that motivate future developments, including what we have learned
from current techniques, and why we need new methods,
reviews of modern methods for MHD, including adaptive mesh techniques for multiscale problems, and methods
for non-ideal MHD, and
reviews of modern methods for collisionless plasma dynamics that result from various approximations to
the full collisionless Boltzmann equation. A key ingredient of the workshop is to engage plasma physicists and
applied mathematicians with experience in plasma kinetics in the development of methods suitable for astrophysical plasmas.
Relativistic Shock Waves: Astrophysical Motivation and PIC
Simulations
Jonathan Arons, University of California, Berkeley
Abstract: I summarize the motivations for studying relativistic shocks, from
observations of relativistic winds and jets emerging from compact objects. I survey the properties of
such shocks as known from PIC simulations, for both the magnetized variety with shock dissipation
and structure owing to cyclotron and synchrotron instabilities, and the unmagnetized variety, with
structure and dissipation coming from the Weibel instability. I briefly discuss the relation of these
results to the widespread belief that relativistic shocks create nonthermal particle populations through
diffusive Fermi acceleration.
Successful Particle-in-Cell Methods in Kinetic Plasma Physics
Applications
Zhihong Lin, University of California, Irvine
Abstract: Particle-in-cell (PIC) simulation have proven to be a powerful
computational approach for studying nonlinear physics in both fusion and space plasma applications.
These application areas typically involve nonlinear kinetic effects, disparate spatial-temporal scales,
multiple physical processes, non-local interactions, and complex geometry. In particular, impressive
progress in the massively parallel gyro-kinetic particle simulation capability during the past two
decades have led to significant advances in the fundamental understanding of turbulence and
transport in fusion plasmas. Examples include new paradigms of turbulence self-regulation by zonal
flows and of spectral cascade via non-local interactions in the wave-vector space. In PIC
simulations, all nonlinearities are treated on the same footing and systematically delineated. For
example, while the wave-wave interactions determine the fluctuation characteristics, the turbulent
transport is driven by wave-particle interactions. In addition to reviewing the physics models,
numerical algorithms, and parallel computation for gyro-kinetic turbulence simulation of fusion
plasmas, recent progress and future directions for space plasma applications of gyro-kinetic PIC
simulations will be presented.
On fast reconnection within large-scale magnetofluid
models
Michael Hesse, NASA
Abstract: During the quest to understand the microphysics of magnetic
reconnection, it was discovered that the Hall-term, the manifestation of scale separation between
ions and electrons, also provided a mechanism for reconnection to proceed at rates much higher than
in conventional MHD models. This feature was explained by the dispersion of the Whistler mode,
which permits the stationary, quadrupolar, magnetic field structure near the reconnection region to
always develop the proper gradient scales for reconnection to proceed at Petschek-like rates. MHD
simulations with constant or weakly localized resistivity, however, exhibited a much slower, Sweet
Parker-like behavior. Similar results were found for simulations with magnetic guide fields. Recently,
however, we investigated the idea that a valuable electric field representation for MHD models might
be obtained directly from the ion dynamics. The equality between the electric fields felt by ions on one
and electrons on the other hand implies that the ion equation of motion might yield a larger-scale
description of a nonideal electric field, which should provide larger reconnection rates in MHD models
also. In this presentation, we describe this idea, and present a number of particle-in-cell simulations
for equal ion and electron masses to test it further. In particular, we will show that fast reconnection
indeed results in all cases considered, and we will discuss how the system circumvents the Sweet
Parker limit of the reconnection rate. We emphasize that these results do not question the validity of
the Hall-dynamics in collisionless reconnection – it is essential to provide the proper cross-scale
coupling. The results do provide, however, a mechanism by which it may be possible to obtain fast
reconnection in MHD models also.
Abstract: I discuss the 3D simulations of relativistic collisionless shocks in
electron-positron pair plasmas with the particle-in-cell (PIC) method. The shock structure is mainly
controlled by the shock's magnetization ("sigma" parameter). I will demonstrate how the structure of
the shock varies as a function of sigma for both perpendicular and oblique shocks. At low
magnetizations the shock is mediated mainly by the Weibel instability which generates transient
magnetic fields which can exceed the initial field. At larger magnetizations the shock is dominated by
magnetic reflections. I demonstrate where the transition occurs and argue that it is impossible to
have very low magnetization collisionless shocks in nature (in more than 1 spatial dimension). I
further discuss the acceleration properties of these shocks, and show that higher magnetization
perpendicular shocks do not efficiently accelerate particles in 3D. Among other astrophysical
applications, this poses a restriction on the structure and composition of pulsar wind outflows.
Gyrokinetic Particle Simulation for Magnetic Fusion Plasmas
Wei-li Lee, Princeton University
Abstract: Princeton Plasma Physics Laboratory (PPPL) has a long history in using
Particle-in-Cell (PIC) methods in studying magnetic fusion plasmas and is currently leading a national
laboratory/university collaboration for the investigation of turbulent transport and kinetic-MHD physics
relevant to the International Tokamak Experimental Reactor (ITER). The effort consists of solving the
gyrophased-averaged (gyrokinetic) Vlasov-Maxwell equations on massively parallel computers with
the aid of modern-day parallelization, data management and visualization techniques. The present talk
will briefly review the project as well as the methodology in solving these equations and the evolution
of perturbative particle simulation schemes in reducing the inherent noise problem in PIC codes. The
use of our Global Gyrokinetic Toroidal Code (GTC) for transport physics and the extension of the
code for kinetic-MHD physics will also be discussed.
Thermal conduction in strongly turbulent magnetized
plasmas.
Benjamin Chandran, Universtiy of Iowa
Abstract: Turbulence can affect the diffusion of particles and heat in a number of
ways. This talk will focus on how "tangled" magnetic field lines in a turbulent plasma affect the
trajectories of diffusing particles. I will describe a physical paradigm for fast-particle diffusion and
thermal conduction based on the theory of Rechester and Rosenbluth, as well as analytic and
numerical results on thermal conduction. I will also briefly describe the implications of this work for
the study of galaxy clusters.
Eliot Quataert, University of California, Berkeley
Abstract: I describe several astrophysics problems in which plasma is
macroscopically collisionless (e.g., the solar wind, accretion onto black holes, etc.). In these cases
kinetic MHD (rather than 'regular' MHD) governs the large-scale dynamics of the system. I describe
the stability properties of collisionless plasmas governed by kinetic-MHD, highlighting the differences
relative to the short mean free path limit. I then discuss several of the waves (e.g., cyclotron) and
instabilities (e.g., mirror and firehose) that limit the effective mean free path of particles in collisionless
plasmas, and try to assess to what extent this enforces "MHD-like" dynamics on macroscopically
collisionless plasmas.
Abstract: Recent advances in the direct kinetic simulation of fusion plasma
turbulence now lay the groundwork and provide an enormous impetus for kinetic closure (using
kinetic simulation) of MHD computational models. The topic is lively and is still very much an open
research area. This is because efficient and practical nonlinear MHD and kinetic methods require
subtle underlying orderings, equations and numerical methods (gyrokinetics, gyro-Landau fluid, semi-
implicit, finite-element, particle-in-cell, drift-ordering, etc.) all of which must properly meld together into
one grand simulation. Even on a particular and well-defined MHD problem, (e.g. internal kink instability,
edge-localized modes, tearing modes) knowing whether it is better to use kinetic closure of MHD or
solve the problem directly using kinetics is very much unanswered at this point. In this talk we will
discuss the possible ways to close MHD equations using kinetics, as well as more direct MHD-like
kinetic models. This talk will highlight some recent successes in kinetic-MHD, including modeling of
energetic particle effects in fusion plasmas. We will also discuss recent kinetic and kinetic-MHD
models of tearing mode behavior.
Abstract: I will review the main points of our recent work on small-scale dynamo
and MHD turbulence in media with large magnetic Prandtl number --- with application to interstellar and
intracluster media in mind. The properties of the magnetic fields generated by small-scale dynamo,
the saturation mechanism and the nature of the fully developed state of isotropic MHD turbulence will
be discussed. The key idea is that interactions in isotropic MHD turbulence may be nonlocal in
wavenumber space, so Kolmogorov-style dimensional theories based on the idea of energy cascade
of Alfven-wave packets may not be applicable. I will also discuss briefly the modern state of
observational evidence relevant to this subject. A detailed account of this work is contained in
Schekochihin et al. 2004, ApJ 612, 276.
Abstract: Recent work by Balbus has shown that there exists an instability in
convectively stable atmospheres with weak magnetic fields in the presence of anisotropic thermal
conductivity.1,2 An analogy can be made between this instability and the well-studied magneto-
rotational instability.3 Specifically, in a Keplerian disk with a stable angular momentum gradient
(conserved quantity) the MRI is able to extract energy from the angular velocity gradient (source of
free energy). Analogously, in a convectively stable atmosphere with a monotonically decreasing
entropy gradient (conserved quantity), the MTI is able to extract energy from the temperature gradient
(source of free energy). We simulate this instability by using the Athena magnetohydrodynamics
code with the addition of anisotropic heat conduction.4 We have verified the analytical expression
for the linear behavior of this instability with the computational model. The computational results are
extended to the non-linear regime to examine the astrophysical importance of this instability.
Plasma Simulation Studies using Mutilevel Physics Models
Wonchull Park, Princeton University
Abstract: The M3D (Multilevel 3D) Project carries out plasma simulation studies
using a code package which solves a hierarchy of physics models with increasing realism. The
available physics levels are fluid models: MHD and two-fluids, and hybrid models: Gyrokinetic-
energetic-particle/MHD, and Gyrokinetic-particle-ion/Fluid-electron. The M3D code uses finite element
unstructured meshes on poloidal planes and finite difference in toroidal direction. It runs with good
parallel scaling to more than 1000 processors. The rationale of the project and recent results of
tokamak and stellarator studies will be presented.
Initial Nonlinear Landau-MHD Simulations of Kinetic Effects on
the MRI
Prateek Sharma, Princeton University
Abstract: Preliminary simulations of the collisionless magnetorotational instability
(MRI) will be presented. We have modified the widely used ZEUS code to include anisotropic
pressure. A Landau fluid prescription is used for heat conduction parallel to the magnetic field. The
equations and their numerical implementation will be discussed. Conservation of adiabatic invariants
will lead to pressure anisotropy in a collisionless plasma as the magnetic field amplifies. Implications
of anisotropy on the nonlinear evolution of the MRI will be discussed.
Abstract: Studying the multidimensional, time-dependent and/or highly nonlinear
dynamics of astrophysical plasmas usually requires numerical methods, however developing
accurate and robust methods for compressible MHD is still an active area of research. I will describe
some problems in astrophysics which motivate the development and application of numerical methods
for MHD. Next, I will describe both standard numerical methods used in astrophysics that have
proven to be reliable and robust, as well as recent advances in algorithm development that promise
new and more accurate methods. Finally, I will describe some of what we have learned from
application of the methods.
Successful MHD Simulation Methods in Plasma Physics
Applications
Carl Sovinec, University of Wisconsin, Madison
Abstract: Magnetohydrodynamics is frequently used for assessing the global
force-balance and ideal stability properties of laboratory plasmas. It also provides valuable
information on the nonlinear evolution of macroscopic instabilities—if not quantitatively, then either
qualitatively or as a starting point for multi-fluid and kinetic treatments. The need for numerical
computation arises from strong nonlinear effects and from behavior that is sensitive to geometry.
While the magnetic Mach number is small in magnetically confined plasmas, numerical algorithms must
deal with extreme stiffness and anisotropy in nearly dissipation-free conditions (large Lundquist
number). Presently, several techniques for solving the MHD system with a large separation of time-
scales are being used or are undergoing continued development. The merits of methods described
as 'partially implicit,' 'semi-implicit,' and 'fully implicit' will be discussed and compared. Modeling
extreme anisotropy benefits greatly from high-order spatial representations, as illustrated through
examples from a finite-element application.
High-Accuracy, Implicit Solution of the Extended-MHD Equations
using High-Continuity Finite Elements
Stephen Jardin, Princeton University
Abstract: It has been recognized for some time that it is necessary to go beyond
the simple "resistive MHD" description of the plasma in order to get the correct quantitative results for
the growth and saturation of global dissipative modes in a fusion device. The inclusion of a more
complete "generalized Ohms law" and the off-diagonal terms in the ion pressure tensor introduce
Whistler waves, Kinetic Alfven waves, and gyro-viscous waves, all of which are dispersive and
require special numerical treatment. We have developed a new numerical approach to solving these
Extended-MHD equations using a compact representation that is specifically designed to yield
efficient high-order-of-accuracy, implicit solutions of a general formulation of the compressible
Extended-MHD equations. The representation is based on a triangular finite element with fifth order
accuracy that is constructed to have continuous derivatives across element boundaries, allowing its'
use with systems of equations containing complex spatial derivative operators of up to 4th order. The
final set of equations are solved using the parallel sparse direct solver, SuperLU, which makes linear
solutions exceptionally efficient, since only a one-time LU decomposition is required. The magnetic
and velocity fields are decomposed without loss of generality in in a potential, stream function form.
Subsets of the full set of 6 equations describing unreduced compressible extended MHD yield (1) the
two variable reduced MHD equations, and (2) the 4-field Fitzpatrick-Porcelli equations. Applications
are presented in straight and toroidal geometry showing the accuracy and efficiency of the method in
computing highly anisotropic heat conduction, toroidal equilibrium, and the effect of "two-fluid"
effects on resistive instabilities.
The Magneto-hydrodynamic Richtmyer-Meshkov
Instability
Ravi Samtaney, Princeton University
Abstract: In the past two decades the Richtmyer-Meshkov (RM) instability has
become the subject of extensive experimental, theoretical and computational research due to its
importance in technological applications such as inertial confinement fusion, as well as astrophysical
phenomena such as shock interactions with intersteller clouds. In this talk we will present recent
results from nonlinear simulations of the Richtmyer-Meshkov instability in the presence of a magnetic
field.
The seminar will be divided into three segments. In the first segment, we will present a brief primer on
discontinuities in MHD. In the second segment we will present numerical evidence that the growth of
the Richtmyer-Meshkov instability is suppressed in the presence of a magnetic field. This is due to a
bifurcation which occurs during the refraction of the incident shock on the density interface. The
result is that baroclinically generated vorticity is transported away from the interface to a pair of slow
or intermediate magnetosonic shocks. Consequently, the density interface is devoid of vorticity and
its growth and associated mixing is completely suppressed. We will present analytical results on the
structure of the singular solution in the limit of vanishing magnetic field to the hydrodynamics case.
The third segment on the talk will focus on the numerical method to obtain the aforementioned results.
We will discuss the implementation of an unsplit upwinding method to solve the ideal MHD equations
with adaptive mesh refinement (AMR) using the Chombo framework. The solenoidal property of the
magnetic field is enforced using a projection method which is solved using a multigrid technique.
This work was supported by USDOE Contract no. DE-AC020-76-CH03073. This research used
resources of the National Energy Research Scientific Computing Center, which is supported by the
Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
A new CT-Godunov scheme for MHD with application to the
MRI
Thomas Gardiner, Princeton University
Abstract: In recent years there has been an increased emphasis on applying high
order Godunov-type algorithms to the system of ideal MHD. This is motivated by their strong shock
capturing and their conservation properties which make them ideally suited for use in combination
with adaptive mesh refinement. Such efforts, however, have traditionally met with difficulty owing to
the divergence free constraint on the magnetic field. We describe a new, unsplit MHD Godunov-type
integration algorithm which uses the Constrained Transport approach to ensure the divergence free
character of the magnetic field. The algorithm includes two novel features, 1) the incorporation of
MHD source terms in the PPM-type reconstruction procedure and 2) an upwind CT-algorithm for
combining the Godunov fluxes to calculate the electric fields needed for CT. We present test
calculations comparing this algorithm against previously published results. Finally, we highlight recent
progress in applying this algorithm to problems of astrophysical interest.
Preliminary Results for Adaptive Particle Refinement
Gregory Howes, University of California, Berkeley
Abstract: Adaptive Particle Refinement (APR) is a new scheme providing a
generally adaptive framework for Lagrangian particle methods for numerical simulation of
astrophysical phenomena. The strategy for error estimation and refinement in APR is described. Initial
results of this generally adaptive approach are presented.
An Unsplit Godunov Method for Ideal Magnetohydrodynamic
Simulations of the Interstellar Medium
Robert Crockett, University of California, Berkeley
Abstract: The Interstellar Medium (ISM) can, to good approximation, be treated as
a non-resistive magnetized fluid. In order to accurately simulate this highly turbulent, compressible
fluid requires numerical schemes that faithfully reproduce shocks and other nonlinear structures.
These are handled very well in general by finite-volume Godunov methods. However, certain types
of nonlinearities can cause problems wherein the divergence-free constraint, div.B=0, is not
maintained. This can have deleterious effects, causing incorrect dynamics and field topologies, and
numerical instabilities.
I outline several related Godunov schemes with for ensuring that the effects of not maintaining the
divergence-free condition are minimized, even for highly nonlinear structures. All these schemes are
based on an unsplit, second-order corner-transport upwind method. Our chosen scheme retains the
cell-centering of variables, making extension to adaptive meshes easier. Preliminary results using our
adaptive mesh code, both on relatively simple test problems and on simulations of magnetized
turbulence, are presented.
