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Astron. Astrophys. 353, 339-348 (2000)

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3. Numerics

3.1. Computations

We use simulations of uniform decaying and driven turbulence with and without magnetic fields described by Mac Low et al. (1998) in the decaying case and by Mac Low (1999) in the driven case. These simulations were performed with the astrophysical MHD code ZEUS-3D 1 (Clarke 1994). This is a three-dimensional version of the code described by Stone & Norman (1992a, b) using second-order advection (van Leer 1977), that evolves magnetic fields using constrained transport (Evans & Hawley 1988), modified by upwinding along shear Alfvén characteristics (Hawley & Stone 1995). The code uses a von Neumann artificial viscosity to spread shocks out to thicknesses of three or four zones in order to prevent numerical instability, but contains no other explicit dissipation or resistivity. Structures with sizes close to the grid resolution are subject to the usual numerical dissipation, however.

In this paper, we attempt to use these simulations to derive some of the observable properties of supersonic turbulence. Although our dissipation is clearly greater than the physical value, we can still derive useful results for structure in the flow that does not depend strongly on the details of the behavior at the dissipation scale. Such structure exists in incompressible hydrodynamic turbulence (e.g. Lesieur 1997). In Mac Low et al. (1998) it was shown that the energy decay rate of decaying supersonic hydrodynamic and MHD turbulence was independent of resolution with a resolution study on grids ranging from [FORMULA] to [FORMULA] zones. Because both numerical dissipation and artificial viscosity act across a fixed number of zones, increasing resolution yields decreasing dissipation. The results we describe in this paper suggest that in some cases observable features may be independent enough of resolution, and thus of the strength of dissipation. Despite the limitations of our method we can therefore draw quantitative conclusions. Again, we support this assertion by appealing to resolution studies whenever possible.

The simulations used here were performed on a three-dimensional, uniform, Cartesian grid with side [FORMULA], extending from -1 to 1 with periodic boundary conditions in every direction. For convenience, we have normalized the size of the cube to unity in the analyses described here, so that all length scales are in fractions of the cube size. An isothermal equation of state was used in the computations, with sound speed chosen to be [FORMULA] in arbitrary units. The initial density and, in relevant cases, magnetic field were both initialized uniformly on the grid, with the initial density [FORMULA] and the initial field parallel to the z-axis.

The turbulent flow is initialized with velocity perturbations drawn from a Gaussian random field determined by its power distribution in Fourier space, following the usual procedure. As discussed in detail in Mac Low et al. (1998), it is reasonable to initialize the decaying turbulence runs with a flat spectrum with power from [FORMULA] to [FORMULA] because that will decay quickly to a turbulent state. Note that the dimensionless wavenumber [FORMULA] counts the number of driving wavelengths [FORMULA] in the box. A fixed pattern of Gaussian fluctuations drawn from a field with power only in a narrow band of wavenumbers around some value [FORMULA] offers a very simple approximation to driving by mechanisms that act on that scale. To drive the turbulence, this fixed pattern was normalized to produce a set of perturbations [FORMULA], and at every time step add a velocity field [FORMULA] to the velocity [FORMULA], with the amplitude A now chosen to maintain constant kinetic energy input rate, as described by Mac Low (1999).

3.2. Resolution studies

In Fig. 4 we show how numerical resolution, or equivalently the scale of dissipation, influences the [FORMULA]-variance spectrum that we find from our simulations. We test the influence of the numerical resolution on the structure by comparing a simple hydrodynamic problem of decaying turbulence computed at resolutions from [FORMULA] to [FORMULA], with an initial rms Mach number [FORMULA].

[FIGURE] Fig. 4. [FORMULA]-variance of the density cubes (translated to two-dimensions) for a model of decaying hydrodynamic turbulence at a time [FORMULA], computed using numerical resolutions of [FORMULA], [FORMULA], and [FORMULA] (Model D from Mac Low et al. 1998).

In contrast to the results from Mac Low (1999) which showed little dependence of the energy dissipation rate on the numerical resolution, we find here remarkable differences in the scaling behaviour of the turbulent structures. At small scales we find a very similar decay in the relative structure variations up to scales of about 10 times the pixel size (0.03, 0.06, and 0.1 for the resolutions [FORMULA], [FORMULA], and [FORMULA], respectively) in all three models. This constant length range starting from the pixel scale clearly identifies this decay as an artifact from the simulations which can be attributed to the numerical viscosity acting at the smallest available size scale.

Another very similar behaviour can be observed at the largest lags where the relative structure variations decay for all three simulations on a length scale covering a factor two below half the cube size. This structure reflects the original driving of the turbulence with a maximum wavenumber [FORMULA] that manifests itself in the production of structure on the corresponding length scale. Only for the [FORMULA] cubes we find a range of an approximately self-similar behaviour at intermediate scales that is not yet smoothed out by the influence of numerical viscosity.

Structures larger than at most half the cube size are suppressed by the use of periodicity in the simulations. Together with the viscosity range of about 10 pixels there is only a scale factor about 10, 5 or 3 remaining for the three different resolutions where we can study true structure not influenced by the limiting conditions of the numerical treatment. For the derivation of reliable scaling laws, we must therefore use at least simulations on the [FORMULA] grid. On the other hand we know, however, that the limits of the observations also constrain the scaling factor for structure investigations in observed maps to at most a factor 10 in general Bensch et al. 1999).

Although we have plotted here only the results for a hydrodynamic model there are no essential differences to the resolution dependence when magnetic fields are included as discussed below.

3.3. Statistical variations

Another question concerns the statistical significance of the structure in the simulations. Since each simulation and even each time step provides another structure there is a priori no reason to believe that a statistical measure like the [FORMULA]-variance is about the same for each realization of a given HD/MHD problem.

Restricted by the huge demand for computing power in each simulation we cannot provide a statistically significant analysis of many realizations for each problem. However, we will try to provide some general clues for the uncertainty of the [FORMULA]-variance measured for a certain structure.

A first impression can be obtained from the differences in the three projections of one cube in Fig. 2. Because each projection provides an independent view on the three-dimensional structure their variation can be considered a rough measure for the statistical significance of the [FORMULA]-variance plots. We see that the curves agree well up to lags of about a quarter of the cube size but deviate considerably at larger lags. This is explained by the number of structures contributing to the variations at each scale. Whereas we find many small fluctuations dominating the variance at small scales there is in general only one main structure responsible for the variance at the largest scale. Its different appearance from different directions then produces the uncertainty in the [FORMULA]-variance there.

Looking at the variance determined in three dimensions in Fig. 2 we see however that it provides already a kind of average over the three projected functions. Analyzing the three-dimensional cubes thus removes already part of the statistical variations that could be seen by an observer when looking at the two-dimensional projections only. The statistical uncertainty is reduced for the [FORMULA]-variances determined in three dimensions considered below.

As another estimate for the uncertainty in this case we study the variances for different time steps in the evolution of a continuously driven hydrodynamic model. In the evolution of the simulation different structures are produced which should behave statistically equal since the general process of their formation and destruction remains the same.

Fig. 5 shows four different timesteps in an HD model driven at wavenumber [FORMULA] each separated by 0.75 the box crossing time at the rms velocity. The variations even at larger scales are much less than in Fig. 2. It appears that the [FORMULA]-variance does a good job of characterizing invariant properties of the structure. Only for high accuracy determinations of the slope or the reliable identification of self-similarity ensemble averages should be taken by computing many realizations.

[FIGURE] Fig. 5. Different time steps in the evolution of a driven model (model HE2 from Mac Low 1999 with [FORMULA] driving) which should remain statistically stationary, showing that the [FORMULA]-variation spectra don't vary much due to turbulent evolution. The first time is [FORMULA], and succeeding plots are separated by times of [FORMULA], or 0.75 the crossing time at the rms velocity.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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