Heterogeneous Computing Environment Tohline, Cazes & Cohl

1. Motivation

The Astrophysical Problem

From a careful analysis of stellar populations, it has been clear for at least the past fifteen years that the vast majority of stars in our neighborhood of the Galaxy are in binary systems.1 That is to say, well over fifty percent of all stars have a stellar companion about which they orbit. In this sense, our own solar system is an exception because the Sun is not gravitationally bound in an orbit about another star.

Initially it was not clear whether stars preferentially form as binary systems or whether they form as single objects then, at a much later time, become ''captured'' into orbits about one another. But recent studies of the frequency of occurrence of binary stars within newly forming clusters of stars strongly suggest that binary formation is the primary branch of the star-formation process.2 (For evidence that star formation is an ongoing process in our own Milky Way Galaxy, see the recent Hubble Space Telescope images of the Orion Nebula and the Eagle Nebula.)

Although we have gained a general appreciation of how stars form from low-density gas in the interstellar medium of our Galaxy,3 we do not yet understand why stars preferentially form in pairs. It is this question that motivates the research efforts described here.

As Chandrasekhar has reviewed,4 mathematical models of rotating, self-gravitating stars date back 250 years and include the works of Maclaurin, Jacobi, Dedekind, and Reimann. By treating stars as incompressible fluids with uniform rotation (or uniform vorticity), these mathematicians showed analytically that there are allowed equilibrium configurations for stars that are defined precisely by oblate spheroids or triaxial ellipsoids. The accompanying Java applet which builds upon Fig. 15 in Chapter 7 of Chandrasekhar's review,4 illustrates the geometrical and rotational properties of these analytically defined equilibrium stellar structures. In particular, it highlights the properties of a continuous sequence of flatter and flatter spheroids that was first identified by Maclaurin, and two related ellipsoidal sequences defined in separate studies by Jacobi and Dedekind.

Figure 1
Classically, models describing the formation of stars from rotating, interstellar gas clouds have been formulated around such analytically prescribable equilibrium configurations. For example, a large, slowly rotating gas cloud with a relatively small ratio of rotational to gravitational potential energy T/|W| will resemble a Maclaurin spheroid. As it contracts conserving angular momentum and mass, its evolution will proceed along the Maclaurin sequence through progressively flatter configurations of higher T/|W|.

At a sufficiently high T/|W|, one finds that the axisymmetric configuration is no longer the lowest energy state available. Instead, there is an ellipsoidal configuration to which the gas cloud will prefer to evolve. Furthermore, if one follows evolution along a more and more distorted ellipsoidal sequence (such as the Jacobi sequence illustrated in the accompanying Java applet), one finds that eventually other configurations with even higher order surface distortions become energetically favorable. For example, as illustrated here in Fig. 1, there is a ''dumbbell-binary sequence'' that branches smoothly off of the Jacobi ellipsoid sequence. One might imagine, therefore, that binary stars form from the slow contraction of a rapidly rotating gas cloud along the Maclaurin, then Jacobi (or Reimann), then dumbbell-binary sequences. This idea is often referred to as the ''fission hypothesis for binary star formation''5 because it paints a scenario by which a single gas cloud may spontaneously fission into a binary star system.

In reality, the picture is not this clear. Most significantly, detailed work on ellipsoidal figures of equilibrium has only been completed for incompressible fluid systems. It is not at all clear to what extent the results carry over to more realistic structures having compressible equations of state. In order to build more realistic models of rotating stars and examine the relative stability of such structures, we must turn to numerical modeling techniques.

Computational Fluid Dynamic Simulations

Equilibrium models of axisymmetric gas clouds with compressible equations of state and differential (as opposed to uniform) rotation can now be constructed rather routinely using self-consistent-field (SCF) techniques. In particular, over the past decade we have relied heavily on an SCF technique developed by Hachisu6 to construct a wide variety of equilibrium models of rapidly rotating gas clouds that can be thought of as compressible analogues of Maclaurin spheroids. (The interested reader is referred to an online version of Hachisu's SCF technique that we have developed for both instructional7 and application purposes.)

In order to test the fission hypothesis of binary star formation, however, one must do more than simply construct equilibrium models of axisymmetric gas clouds. One must examine whether or not the models are unstable toward the development of nonaxisymmetric structure and, for models which have been found to be unstable, determine whether the nonlinear development of the instability leads to fission. In order to study the nonlinear development of nonaxisymmetric instabilities in rotating gas clouds, we have written a direct numerical simulation (DNS) algorithm to solve the following coupled set of partial differential equations:

Dv = - (1/r)P - F [1]
Dr + r v = 0 [2]
De + P D(1/r) = 0 [3]
2F = 4p Gr [4]

These equations describing the inviscid flow of self-gravitating continuum fluids, together with an appropriate equation of state, relate the time and spatial variation of the fluid velocity v to the pressure P, the mass density r, the specific internal energy e , and the gravitational potential F in a physically consistent fashion.7, 8 By restricting our discussions to physical systems that are governed by this set of equations, we are assuming that no electromagnetic forces act on the fluid (eg., the effects of magnetic fields on an ionized plasma are not considered) and, in the absence of dynamically generated shocks, all compressions and rarefactions are assumed to happen adiabatically.

Our DNS algorithm is patterned after the ZEUS-2D code developed by Stone and Norman9, but it has been extended to handle three-dimensional (3D) fluid systems, and has been written in High-Performance-Fortran (HPF) to execute on a variety of different massively parallel computing platforms. In order to resolve well the structural properties of our 3D flows, we have found it necessary to utilize at least 1283 grid zones. Hence, on parallel computing platforms with 64-bit floating-point processors, each 3D array requires 16 MBytes of storage and a typical simulation demands a minimum of 8 GBytes of RAM. Most of our simulations over the past several years have been performed on an 8K-node MasPar MP-1 in LSU's Concurrent Computing Laboratory for Materials Simulation, and on the Cray T3E at both the San Diego Supercomputer Center and the NAVOCEANO DoD Major Shared Resource Center, although our CFD code's performance has been measured on a variety of different machine architectures.10


In a very real sense, each of our astrophysical fluid simulations generates data that can fill a very large, four-dimensional data array for each of the principal fluid variables. The arrays are 4D because each variable is defined on a three-dimensional spatial lattice as a function of time. Figuring out how to sort through this large amount of data to garner useful physical information about the fluid flow is a major challenge by itself.

When conducting nonlinear stability analyses of rapidly rotating gas clouds in the context of star formation, as described above, we have found the most useful diagnostic tool to be an animation sequence which shows the time-evolution of isodensity surfaces in each evolving cloud. In order to generate such an animation sequence, historically we (as well as other researchers) usually have adopted the following plan:

Although effective, this plan which relegates the data analysis (visualization) to a post-processing task is generally inefficient and puts large demands on data storage.

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