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Timetable (MSIW02)

Large-Scale Computation in Astrophysics

Monday 11th October 2004 to Friday 15th October 2004

Monday 11th October 2004
11:25 to 12:35 Computational MHD: A model problem for widely separated time and space scales (Chair: R Rosner)

The numerical simulaton of the dynamics of magnetized plasmas is among the most challenging problems in computational physics. Strongly magnetized plasmas are characterized widely separated space and time scales, and by extreme anisoptopy. All of these issues affect the design of algorithms. The fundamental mathematical description requires the simultaneous solution of the 6-dimensional kinetic equation along with Maxwell's equations. This is impossible in all but the very simplest cases, so reduced fluid models can be derived by taking velocity moments of the kinetic equation and assuming a closure condition. Different closure assumptions result in different fluid models. MHD is the simplest of these, although by no means universally applicable. MHD appears to be an excellent model for the dynamics of stellar interiors, where the problem reduces to computing large Reynolds' number turbulence. Memory and speed limitations of even the most powerful computer then dictate a further reduction by means of averaging and statistical closures to capture the effect of the sub-grid scale dynamics on the long wave length motions. There is no concensus on the form of these closures. For the case of low density, high temperature, strongly magnetized plasmas, as occur in laboratory fusion experiments, MHD is clearly not a good model on the smallest scales, and the closure problem becomes one of characterizing non-local kinetic effects in a local transport formalism. This is also an unsolved problem, so in both cases it can be said that there is no agreement on what equations to solve. Because of the differing plasma parameters in these 2 cases, different algorithms must be applied. The audience is already familiar with techniques for computing MHD turbulence. Here we will primarily be concerned with methods developed to simulate ther long time scale dynamics of highly magnetized plasmas, as occur in fusion plasmas and the solar corona. Methods of spatial and temporal differencing will be discussed, and examples of the computed dynamics of laboratory and coronal plasmas will be given. Limitations on the scope of simulations for the foreseeable future will be given. Perhaps some of the issues discussed here will also prove to be useful for stellar interiors.

15:30 to 16:40 Planetary convection and dynamos (Chair: N Weiss)

The past decade has seen enormous progress in numerical modelling of planetary dynamos. In this talk I will review some of this work, the numerical techniques that are used, and some of the physics that makes this problem so difficult. I will also compare and contrast the situation in planets versus other astrophysical objects, and try to explain why quite different numerical techniques are often used in the planetary context.

16:40 to 17:05 DNS of anisotropic MHD flows (Chair: N Weiss)

We will present preliminary results of the direct numerical simulation of the incompressible MHD problem. The system is forced at large-scale and an external magnetic field $B_0 \hat{z}$ is applied. Anisotropy is studied in terms of the decomposition group for rotation in $3D$, the so-called $SO(3)$ group. Similarities and differences with the anisotropic pure hydrodynamical case will be discussed.

17:05 to 17:30 Convective instabilities in pre-runaway white dwarfs (Chair: N Weiss)

Studying the evolution of the convective burning process before and during the thermonuclear runaway in a white dwarf is crucial in order to measure the enrichment of the hydrogen envelope by convective overshoot. Recent numerical simulations, that start when the temperature at the base of the envelope is close to 10^8 K, show that in a few hundreds seconds the temperature grows up to 2 10^8 K. At this time the runaway takes place. Our simulations, performed by running a high order of accuracy code, with low numerical viscosity, show that care must be taken in the choice of the initial and boundary conditions. We have observed, in fact, the onset of fast convective instabilities that are driven by boundary effects and affect the dynamics of the pre-runaway phase. We plan, as a next step, to take the initial equilibrium with a peak of temperature close to 10^7 K, that corresponds to earlier and less unstable phase of the white dwarf evolution.

Tuesday 12th October 2004
09:00 to 10:10 Staggered mesh approaches to MHD- and charged-particle simulations of astrophysical turbulence (Chair: DJ Galloway)

I will discuss the computational techniques behind recent modeling of MHD-turbulence in several astrophysical context; e.g. supersonic and super-Alfvenic turbulence in the interstellar medium and molecular clouds, subsonic MHD-turbulent in strongly stratified stellar surface layers, magnetic dissipation in the solar corona, and relativistic turbulence in collisionless shocks.

The computational techniques we have applied in all of these circumstances are related to the more conventional (Godunov-inspired) techniques in much the same way that RISC-technology relates to CISC-technology in the context of CPU-design; we attempt to minimize the number of floating point operations per meshpoint update, and hence to maximize the number of meshpoint updates per CPU per second, while still retaining a high spatial and temporal order of the updates, and the ability to capture and resolve shocks and current sheets.

I will also briefly describe the extension of these techniques to the modeling of relativistic charged particles with a particle-in-mesh (PIC) code, with which it is possible to obtain a grid resolution comparable to reasonably serious MHD-simulations; publications have already appeared based on runs with of the order of 30 million mesh points and a billion particles.

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10:10 to 10:35 NH Brummell ([JILA, Boulder])
Magnetic field transport in turbulent compressible convection (Chair: DJ Galloway)

Magnetic field transport is essential to certain elements of the solar dynamo scenario. It is crucial (a) to move poloidal field from the convection zone to the tachocline relatively quickly, and (b) to transport the magnetic structures created in the tachocline through the convection zone towards the solar surface, for the current dynamo paradigm to work. Here, we examine the competition between magnetic buoyancy and magnetic pumping mechanisms in such situations via high resolution numerical simulations of turbulent compressible penetrative magnetoconvection.

10:35 to 11:00 PLUTO: a modular code for computational astrophysics (Chair: DJ Galloway)

PLUTO is a modular, Godunov-type code intended mainly for astrophysical applications. Written in C, it currently supports classical, relativistic and magneto fluid dynamics modules in curvilinear coordinates in 1, 2 and 3 dimensions. Implementation of the relativistic MHD equations has been recently added. The code is particularly suitable for treating hypersonic flows with strong discontinuities, and several numerical algorithms (TVD, PPM) are available for testing. Source terms include gravity, rotations and optically thin radiative losses. PLUTO works on non-uniform grids and runs either on a single processor or on parallel architectures (using MPI libraries), and has been extensively used on Beowulf clusters (16 and 32 nodes) for 3D relativistic jet applications, accretion on compact objects and accretion disks problems.

11:25 to 12:35 Astrophysical dynamos (Chair: DJ Galloway) INI 1
15:30 to 16:40 Global modelling of solar convection, differential rotation and magnetism (Chair: MRE Proctor) INI 1
16:40 to 17:05 M Browning ([JILA, Boulder])
Simulations of core convection and resulting dynamo action rotating A-type stars (Chair: MRE Proctor)

We present the results of 3--D nonlinear simulations of magnetic dynamo action by core convection within A-type stars of 2 solar masses, at a range of rotation rates. We consider the inner 30% by radius of such stars, with the spherical domain thereby encompassing the convective core and a portion of the surrounding radiative envelope. The compressible Navier-Stokes equations, subject to the anelastic approximation, are solved to examine highly nonlinear flows that span multiple scale heights, exhibit intricate time dependence, and admit magnetic dynamo action. Small initial seed magnetic fields are found to be amplified greatly by the convective and zonal flows, ultimately yielding fields that possess structure on many scales, are strong enough to modify the flows themselves, and persist for as long as we have continued our calculations. The central columns of strikingly slow rotation found in some of our progenitor hydrodynamic simulations continue to be realized in some simulations to a lesser degree, with such differential rotation arising from the redistribution of angular momentum by the nonlinear convection and magnetic fields. We assess the properties of the magnetic fields thus generated, the extent of convective penetration, and the excitation of gravity waves within the radiative envelope, as a number of simulation parameters are varied.

17:05 to 17:30 Mach number dependence of the onset of dynamo action (Chair: MRE Proctor)

The effect of compressibility on the onset of nonhelical turbulent dynamo action is investigated using both direct simulations as well as simulations with shock-capturing viscosities, keeping however the regular magnetic diffusivity. It is found that the critical magnetic Reynolds number for the onset of dynamo activity increases from about 35 in the subsonic regime to about 70 in the supersonic regime. Although the shock structures are sharper in the high resolution direct simulations compared to the low resolution shock-capturing simulations, the magnetic field looks roughly similar in both cases and does not show shock structures. Similarly, the onset of dynamo action is not significantly affected by the shock-capturing viscosity.

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Wednesday 13th October 2004
09:00 to 10:10 R Klein ([Berkeley])
Verification and validation of astrophysical radiation-hydrodynamics (Chair: JM Stone)

The development of sophisticated multi-dimensional self-gravitational radiation hydrodynamic codes in astrophysics has brought important focus onto the verification and the validation of the physical models within the codes as well the integrated codes themselves. In this lecture I shall discuss new methodologies we have developed embodying high order accurate techniques to solve the three-dimensional equations of self-gravitational radiation hydrodynamics using adaptive mesh refinement. I will discuss the astrophysics of strongly coupled radiation hydrodynamic flows in two problems of great current interest; the development of photon bubble instabilities in neutron star atmospheres and accretion disks and the problem of high mass star formation. I will then discuss various approaches to the verification of the equations and the algorithms including the testing of hydrodynamics, radiation transport and coupled radiation hydrodynamics. I shall then describe laboratory experiments we have conducted to validate the component physics of both the radiation and the hydrodynamics in the code and novel work I am engaged in to design an experiment on ultraintense petawatt lasers to simulate for the first time, photon bubbles in the laboratory.

10:10 to 10:35 Modelling of relativistic magnetically dominated plasmas (Chair: JM Stone)

Many phenomena of relativistic astrophysics involve flows of magnetically dominated plasma which are somewhat difficult for analytical and numerical modelling. In this talk we describe three closly related mathematical frameworks used in recent studies of black hole magnetospheres, namely resistive electrodynamics, magnetodynamics, and magnetohydrodynamics, focusing on their advantages and limitations. Particular attention is paid to numerical aspects revealed in recent simulations.

10:35 to 11:00 AA Schekochihin ([Cambridge])
Turbulence in magnetized plasma: do we understand and can we simulate Braginskii viscosity? (Chair: JM Stone)

In low-density high-temperature astrophysical plasmas, e.g., intracluster gas, protogalaxies, the ion Larmor radius is much smaller than the mean free path already for very weak magnetic fields. In this regime, while the magnetic field may not yet be dynamically important, plasma is magnetised and the adiabatic invariant is approximately coserved. This leads to the anisotropisation of the pressure (viscous stress) tensor, so the the viscous dissipation along and across the field is different. When pressure is anisotropic, there appear very fast plasma instabilities. Their growth rates are proportional to the parallel wavenumber of the perturbation, so small-scale fluctuations grow at very small scales and with growth rates that far exceed the rate of strain of the turbulent eddies. The instabilities are not fully suppressed by either Braginskii viscosity or, in the collisionless regime, by the magnetic Landau damping. The stabilisation only occurs at scales somewhat above the ion cyclotron radius, where plasma is no longer perfectly magnetised. I will present a theoretical description of these instabilities both for collisional MHD with Braginskii viscosity and in the collisionless regime. Estimates are made for the effective magnetic cutoff scale. I will also discuss the feasibility of numerical simulations of MHD with Braginskii viscosity and report on some preliminary numerical studies.

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11:25 to 12:35 E Mueller ([Garching])
Supernovae and numerical (R)MHD: challenges and developments (Chair: JM Stone)

After reviewing the current paradigm of core collapse supernova physics, the delayed neutrino driven explosion mechanism, possible effects of magnetic fields on the dynamics of core collapse supernovae are discussed. In order to simulate such effects robust and accurate MHD or RMHD schemes for multi-dimensional supersonic flow are required. To this end some interesting developments in numerical (R)MHD will be addressed. Finally, recent results from magneto-rotational core collapse simulations are presented, and future developments in the field are pointed out.

Thursday 14th October 2004
09:00 to 10:10 Kinetic modeling of magnetic reconnection in space and astrophysical systems (Chair: PH Diamond)

The large scale dynamics of magnetized plasma systems are typically modeled with the MHD equations. However, the MHD description typically breaks down at spatial scales where dissipation is required to either break magnetic field lines, allowing reconnection to occur, or to locally dissipate energy. In the case of magnetic reconnection, the Hall MHD model has been found to accurately reproduce the rates of reconnection determined by kinetic modeling, a consequence of the role of dispersive waves in reconnection. However, critical issues in space and astrophysics remain that require a kinetic description and at the same time have significant consequences for the description of the large-scale dynamics of plasma systems. After reviewing the recent kinetic model of fast reconnection, I will focus on two generic topics, electron heating and kinetic scale turbulence and its role in driving reconnection. Nearly half of the magnetic energy released in solar flares is channeled into energetic electrons and recent observations in the magnetosphere confirm that reconnection can directly drive electrons to near relativistic energies. Simulations reveal that reconnection leads to the formation of extended density cavities that map the magnetic separatrices and support a finite parallel electric field. These cavities act as electron accelerators and as a result of multiple passses through these acceleration cavities electrons quickly reach relativistic energies. In boundary layers of the magnetosphere, where large-scale parallel electric fields are expected from modeling, parallel electric fields take the form of intense, spatially-localized, bipolar structures (electron holes) and double-layers. These are manifestly kinetic nonlinear structures where electrons and ions can directly exchange energy with large scale fields. Simulations of reconnection reveal the self-consistent development of these structures, facilitating the exploration of their role in providing the dissipation required to drive reconnection.

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10:10 to 10:35 3D simulation of emerging flux and magnetic reconnection using the Earth Simulator (Chair: PH Diamond)

The Earth Simulator is a parallel vector supercomputer system of the distributed-memory type, installed at the Earth Simulator Centre, in Yokohama, Japan. It has been at the position of the world's fastest supercomputer since 2002. After some introduction of the Earth Simulator, we present the results of three dimensional MHD simulation of solar emerging flux and magnetic reconnection carried out on the Earth Simulator. In the case of emregence of magnetic sheet, we found that filamentary structure arised due to Rayleigh-Taylor instability. Then many filamentary current sheets were formed in the emerging flux, supporting the iear that the corona are heated by dissipation of small scale current sheets. We also found that magnetic reconnection between the emerging flux and the coronal field occurs in spatially intermittent way. We also present some preliminary result of the case of emerging twisted flux tube and its reconnection with ambient field.

10:35 to 11:00 R Keppens ([Rijnhuizen])
Grid-adaptive simulations of magnetized jet flows (Chair: PH Diamond)

We present high resolution numerical simulations of magnetized plasma jets,modeled by means of the compressible magnetohydrodynamic equations. The computations employ Adaptive Mesh Refinement, which makes it possible to investigate long-term jet dynamics where both large-scale and small-scale effects are at play. We first discuss recent findings for periodic single shear flow layers at moderate Mach numbers (around unity) and large plasma beta values. In such cases, a trend to large scales occurs by continuous pairing/merging between adjacent vortices, simultaneously with the introduction of small-scale features by magnetic reconnection events. The vortices form as a result of Kelvin-Helmholtz unstable shear flow layers, and their coalescence arises from the growth of subharmonic modes at multiple wavelengths of the fastest growing Kelvin-Helmholtz instability. Extensions to 2D jets investigate how varying jet width alters the coalescence process occuring at both edges, e.g. by introducing bachelor coupling between vortices formed at opposing weakly magnetized, close shear layers. Finally, periodic segments of supersonic magnetized jets are simulated in two and three dimensional cases, which are characterized by violent shock-dominated transients.

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11:25 to 12:35 Large scale simulations of astrophysical turbulence (Chair: PH Diamond)

In hydrodynamic and hydromagnetic turbulence simulations, high spatial and temporal accuracy is of the essence. This is why pseudo-spectral methods are to be preferred. However, because of their intrinsic nonlocality, such methods are not well suited for massively parallel machines. In this talk the Pencil Code will be discussed, which uses sixth order finite differences for various types of turbulence simulations at resolutions up to 1024 cubed meshpoints on up to 256 processors: forced turbulence, shear flow turbulence due to the magneto-rotational instability, and convection. For simulations in spheres the physically interesting domain is embedded in a box. Some important results will be discussed and also the issue of code maintenance, development by many people and code validation will be addressed.

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15:30 to 16:40 Non-ideal MHD and beyond (Chair: NE Hurlburt) INI 1
16:40 to 17:05 A numerical scheme for multi-fluid MHD (Chair: NE Hurlburt)

In this talk I shall describe a numerical scheme for multi-fluid hydrodynamics in the limit of small mass densities of the charged particles. The inertia of the charged particles can then be neglected, which makes it possible to write an evolution equation for the magnetic field that can be solved using an implicit scheme. This avoids the severe restriction on the stable timestep that would otherwise arise at high resolution, or when the Hall effect is large. Numerical tests show that the scheme can accurately model steady multi-fluid shock structures both with and without sub-shocks. Although the emphasis is on shocks in molecular clouds, a multi-dimensional version of this code could be applied to any Astrophysical flow in which ambi-polar diffusion or the Hall effect, or both play a significant role.

17:05 to 17:30 A new CT-Godunov MHD algorithm with application to the MRI (Chair: NE Hurlburt)

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 apply this algorithm to the study of the MRI including radiative cooling.

17:30 to 17:55 DS Balsara ([Notre Dame])
Amplification of interstellar magnetic fields by supernova-driven turbulence (Chair: NE Hurlburt)

Several lines of evidence suggest that magnetic fields grow rapidly in protogalactic and galactic environments. However, mean field dynamo theory has always suggested that the magnetic fields grow rather slowly, taking of order a Hubble time to reach observed values. The theoretical difficulties only become worse when the system has a high magnetic Reynold’s number, as is the case for galactic and protogalactic environments. The discrepancy can be reconciled if fast processes for amplifying magnetic field could operate. Following Balsara, Benjamin & Cox (2001), we show that an interstellar medium that is dominated by realistic energy input from supernova explosions will naturally become a strongly turbulent medium with large positive and negative values of the kinetic helicity. Even though the medium is driven by compressible motions, the kinetic energy in this high Mach number flow is mainly concentrated in solenoidal rather than compressible motions. These results stem from the interaction of strong shocks with each other and with the interstellar turbulence they self-consistently generate in our model. Moreover, this interaction also generates large kinetic helicities of either sign. The turbulent flow that we model has two other characteristics of a fast dynamo: magnetic energy growth independent of scale, and with a growth time that is comparable to the eddy turn-over time. This linear phase of growth permits the field to grow rapidly until the magnetic energy reaches about 1% of the kinetic energy. At that stage, other astrophysical processes for producing magnetic fields can take over. Energetics, power spectra, statistics and structures of the turbulent flow are studied here. Shock-turbulence interaction is shown to be a very general mechanism for helicity generation and magnetic field amplification with applicability to damped Ly-a systems, protogalaxies, the Galaxy, starburst galaxies, the inter-cluster medium and molecular clouds.

Friday 15th October 2004
09:00 to 10:10 Simulations of astrophysical jets (Chair: A Ruzmaikin)

Collimated supersonic flows are present in many different astrophysical contexts ranging from Young Stellar Objects to Active Galactic Nuclei. Simulations of these flows have to take into account several different physical effects like, for example, radiative losses in jets from Young Stellar Objects or relativistic effects in jets from Active Galactic Nuclei. These simulations have to address the issues of the jet formation, collimation and propagation and I will give an overview of their main problems and results.

10:10 to 10:35 Short and long term simulations of relativistic magnetized jets (Chair: A Ruzmaikin)

We will present a series of numerical simulations addressed to understand the morphology and dynamics of relativistic, magnetized, axisymmetric jets. Some of the simulations have been specifically set up to follow the long term evolution of extragalactic jets under idealized conditions. The simulations have been done with an extension of the GENESIS code (Aloy et al 1999a} suitable for relativistic magnetohydrodynamcs applications. The code is based on a Godunov-type scheme whose building block is a method of lines. The numerical algorithm can provide up to third order of accuracy and makes use of a constrained transport method in order to keep the divergence--free condition of the magnetic field.

10:35 to 11:00 A Ferrari ([Turin])
AMR simulations of supersonic magnetized jet acceleration from accretion discs (Chair: A Ruzmaikin)

We present a 2.5D magnetohydrodynamic (MHD) simulation of the acceleration of a collimated jet from a magnetized accretion disk employing a MHD code with Adaptive Mesh Refinement (FLASH code -- University of Chicago). Thanks to this tool we can follow the evolution of the system for many dynamical timescales with a high spatial resolution. Assuming an initial condition in which an equilibrium Keplerian disk is threaded by a uniform azimuthal magnetic field, we show how both the accretion inflow and the acceleration of the outflow occur and we present in great detail which are the forces responsible for the launch and the collimation of the jet. Our simulation also shows how the collimating forces due to the self-generated toroidal magnetic field can produce some peculiar knotty features.

11:25 to 12:35 J Hawley ([Virginia])
The magneto-rotational instability (Chair: A Ruzmaikin)

Recent years have witnessed dramatic progress in our understanding of how turbulence arises and transports angular momentum in differentially rotating systems. The key conceptual point is that the combination of a subthermal magnetic field and outwardly decreasing differential rotation rapidly generates magnetohydrodynamical (MHD) turbulence via the magnetorotational instability. With the aid of supercomputers, it is now possible to study accretion disk turbulence at a level comparable to that of stellar convection. This talk will present a review of this instability and the MHD turbulence it generates, and some recent results from fully three-dimensional global disk simulations.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons