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

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4. Energy dissipation

The increasing number of strong shocks with increasing wavenumber is qualitatively consistent with the finding of Mac Low (1999) that the total energy dissipation rate increases with the driving wavenumber. To discuss the energy dissipation rate, however, we must first also consider the density [FORMULA] within each shock. We have then calculated the power dissipated by artificial viscosity within each shock front, as described in Paper 1. This yields the power dissipated per unit shock speed. We actually calculate in this section the component of the dissipation along the x-axis and the jump Mach number along the x-axis which, given the statistics, represents the true three dimensional `power distribution function'. Numerically, over the whole simulation grid (x,y,z), we identify each compact range [FORMULA] along the x-axis for which [FORMULA]. For each jump we then find the energy dissipated by artificial viscosity as

[EQUATION]

where C measures the number of zones over which artificial viscosity will spread a shock. This is then binned as a single shock element. The shock number distribution [FORMULA] is multiplied by three to account for the energy dissipated in each of the three dimensions. Further details of the method can be found in Paper 1, where its reliability was also verified.

The surface brightness of the shocks, as well as the column density of all the gas within the cube, is displayed in Fig. 6 for a pure hydrodynamic driven example. Note that these are not slices, but we have integrated through the z-direction. Much of the gas has been swept up into a flattened cloud. Many shocks appear sharper when only the x-direction is accounted for since this emphasizes the shocks transverse to the line of sight.

[FIGURE] Fig. 6. Maps of the column and power dissipated from driven turbulence integrated through the cube along the z-direction. The run is D1 of Klessen et al. 2000, at time t = 2, just before self-gravity was switched on. The long wavelength driving ([FORMULA]) generates a large cloud structure visible in the column density, the density projected onto the x-y plane. The energy dissipated in the shocks (1) just in the x-direction, (2) in all directions and (3) in just the strongest shocks, are displayed as indicated. The column density is expressed relative to the average (initial) value, and the power loss per image pixel has been amplified by a factor of 1282.

Dissipating shocks and the density distribution do not correlate very well, although the main cloud contains the main elongated region of dissipation. The most powerful shocks are mainly within the cloud.

We find that the energy dissipation rate also takes on a power law [FORMULA] dependence (Fig. 7). In contrast to the number distribution, it is independent of the driving power input. There is also a wavenumber dependence, as shown in Fig. 8.

[FIGURE] Fig. 7. The rate of energy dissipation from shocks with jump Mach number Mj. The log-log plot demonstrates that power law relation. The 5 curves correspond to the simulations shown in Fig. 1 with increasing input powers dE/dt = (0.1, 0.3, 1, 3, 10).

[FIGURE] Fig. 8. The distributions of the rate of energy dissipation for the three indicated maximum wavenumbers, corresponding to the number distributions shown in Fig. 2.

The total energy dissipated in the power law section (summing the losses for each direction) can be written

[EQUATION]

The error in the power law index of 1.5 is [FORMULA]. The break in the power law is found to occur at [FORMULA] as given by Eq. (3). Integrating Eq. (7), we obtain Eq. (3). Thus we have acquired a self-consistent mathematical description.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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