Astron. Astrophys. 362, 333-341 (2000)

1. Introduction

Many structures we come across in the Universe have been shaped by fluid turbulence. In astronomy, we often observe high speed turbulence driven by supersonic ordered motions such as jets, supernova shocks, galactic rotation and stellar winds (e.g. Franco & Carramiñana 1999). Although it has long been thought that astrophysical turbulence provides the best opportunity to investigate supersonic turbulence, the lack of a theory has stunted our attempts to understand the behaviour (von Hoerner 1962). Three dimensional high resolution numerical simulations now provide a method to make real progress. Driven turbulence is explored in this paper and the results compared to a sister study of decaying turbulence (Smith et al. 2000, hereafter Paper 1). Our aim here is to relate the type of turbulence to the properties of the shock waves.

We study here compressible turbulence without thermal conduction. No physical viscosity is modelled, but numerical viscosity remains present, and an artificial viscosity determines the dissipation in regions of strong convergence. Periodic boundary conditions were chosen for the finite difference ZEUS code simulations, fully described by Mac Low et al. (1998). Uniform three dimensional turbulence with an isothermal equation of state, an initially uniform magnetic field and periodic boundary conditions are investigated. The influence of self-gravity is also examined.

Our motivation here is to provide the observer with a means of recognising the type of turbulence from the properties of the generated shock waves. The general energetics of decaying and driven hydrodynamic turbulence have already been computed by Mac Low et al. (1998) and Mac Low (1999), respectively, using ZEUS, a second-order, Eulerian hydrocode. Mac Low et al. (1999) concluded that, since turbulence which is left to decay dissipates rapidly under all conditions, the motions we observe in molecular clouds must be continuously driven. Klessen et al. (2000) extended the hydrodynamic results by calculating self-gravitating models. A smoothed particle hydrodynamics (SPH) code was also employed to confirm the results for both the decaying and driven cases (Mac Low et al. 1998, Klessen et al. 2000). Klessen et al. (2000) have provided the parameter scaling for applications to molecular clouds. Heitsch et al. (2000) present the magnetohydrodynamic extension to the self-gravitating case, and discuss the criteria for gravitational collapse.

Our immediate target is to derive the spectrum of shocks (the Shock Probability Distribution Function) generated by driven turbulence. With this knowledge, we will proceed to predict the spectroscopic properties in a following work. Observed individual bright, sharp features, such as arcs, filaments and sheets, are often interpreted as shock layers within which particles are highly excited (e.g. Eislöffel et al. 2000). Where unresolved, the excitation can still be explored quantitatively by employing spectroscopic methods. The gas excitation then depends on both the physics of shocks as well as the distribution of shock strengths.

Previous studies of compressible turbulence have concentrated on the density and velocity structure of the cold gas rather than the shocks (e.g. Porter et al. 1994; Falgarone et al. 1994, Vázquez-Semadeni et al. 1996, Padoan et al. 1998). This may be appropriate for the interpretation of clouds since, although the Mach number is still high, the shock speeds are too low to produce bright features. The simulations analysed here are also being interpreted by Mac Low & Ossenkopf (2000) in terms of density structure.

The method used to count shocks from grid-based simulations was developed in Paper 1. The one-dimensional counting procedure was verified through a comparison with full three-dimensional integrations of the dissipated energy in Paper 1. This method is appropriate for a ZEUS-type finite difference code for which shock transitions are spread out over a few zones. Here, we also study the shock transitions in all three directions and display the spatial distributions for the energy dissipated through the artificial viscosity in the shocks. We first present the shock jump PDFs and provide analytical fits (Sect. 2). Magnetohydrodynamic simulations (Sect. 3) with Alfven numbers of and are then explored. Note the definition of the Alfvén Mach number , where the Alfvén speed , and is the initial root mean square (rms) velocity.

Which shocks actually dissipate most of the energy? The shock power rather than number density will determine the spectral region in which the system can be best observed. Hence, we determine the energy dissipated as a function of the shock speed in Sect. 4. A large-scale simulation which included self-gravity is then analysed in Sect. 5. We next interpret the results in terms of the dynamical models (Sect. 6).

© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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