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

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2. Hydrodynamic turbulence

2.1. Model description

We first explore driven hydrodynamic turbulence. The three-dimensional numerical simulations on grids with [FORMULA] and [FORMULA] zones and periodic boundary conditions were initialized with an rms Mach number of [FORMULA]. The initial density is uniform and the initial velocity perturbations were drawn from a Gaussian random field, as described by Mac Low (1999). The power spectrum of the perturbations is flat and limited to small wavenumber ranges, [FORMULA], with [FORMULA] and [FORMULA] in Fig. 1. The uniform driver is a simple constant rate of energy input with the distribution pattern fixed.

[FIGURE] Fig. 1. Four simulations of driven hydrodynamic turbulence with an increasing rate of energy input from top to bottom. The distribution of jumps are shown at four times for each simulation, from t=0 (dashed), t=0.5 (dot-dash), t=1.0 (dotted) to t=1.5 (solid). The thin straight line in the lower panel represents an inverse square root law.

The simulations employ a box of length [FORMULA] and a unit of speed, u. The gas is isothermal with a sound speed of [FORMULA]. Hence the sound crossing time is 20 L/u. We take the time unit as [FORMULA]. Assuming a mass m to be contained initially in a cube of side L, the dissipated power is given in units of [FORMULA].

2.2. Shock number distribution

We calculate the one dimensional shock jump function, as discussed and justified in Paper 1. This is the number distribution of the total jump in speed across each converging region along a specific direction. This is written as [FORMULA] where [FORMULA] is the sum of the (negative) velocity gradients (i.e. [FORMULA] across a region being compressed in the x-direction). We employ the jump Mach number in the x-direction [FORMULA] rather than [FORMULA] since this is the parameter relevant to the dynamics. Thus, each bounded region of convergence in the x-direction counts as a single shock and the total jump in [FORMULA] across this region is related to its strength.

Numerically, we scan over the whole simulation grid (x,y,z), recording each shock jump through

[EQUATION]

with the condition that [FORMULA] in the range [FORMULA]. This is then binned as a single shock element. The shock number distribution [FORMULA] is obviously dimensionless.

Note that the shock number depends on the grid size. We find that on doubling the grid size, the shock number increases by a factor of four. This implies that the total shock front area in units of [FORMULA] remains roughly constant. This holds for both driven and decaying turbulence and is an indication that the numerical resolution is sufficient to capture the vast majority of shocks.

2.3. Steady state description

The random Gaussian field at t = 0 rapidly transforms into a shock field (Fig. 1). As can be seen, the shock distribution approaches a steady state, reached by t = 1. The driving energy determines the break in the distribution and the maximum shock speed, but does not influence the distribution of shock speeds.

The driving wavenumber influences the steady state as shown in Fig. 2. There is a moderate dependence of the shock number on the wavenumber, especially at high Mach numbers. This may be partly due to the time involved for the regenerated low wavenumber modes to steepen. Some of the longer wave modes may be damped by interactions with the turbulent field.

[FIGURE] Fig. 2. The shock distribution depends on the wavenumber of the energy input of the driven hydrodynamic turbulence. The straight dotted line has the slope given by Eq. (2).

A power law distribution of shock velocities is uncovered. Note that for high Mach numbers, statistically, we can equate the shock Mach number to the jump Mach number to a good approximation. We find, as shown by the indicated line in the lower box of Fig. 1, an inverse square-root law [FORMULA]. In detail, we find a fit of the form

[EQUATION]

over the power law sections. The break in the power law is found to occur at [FORMULA] given by

[EQUATION]

The k-dependence for [FORMULA] has been estimated by inspecting the energy dissipation diagrams below. The indicated value is consistent with that discernable in Fig. 2.

Integrating Eq. (2), using the limit Eq. (3), yields the remarkable result that the number of shocks is a constant: [FORMULA] for [FORMULA]. That is, when the energy input is low, there are many more weak shocks. As the energy is increased, the number of shocks does not increase. Rather, the shock strengths increase. Hence, the number of shocks reaches a saturation level of about 180,000 on the 1283 grids.

Hence, the total shock surface area is [FORMULA]. This is confirmed directly from the shock counts. We find that the number of jumps does depend weakly on the chosen value for the minimum convergence. We take the case [FORMULA] for illustrative numbers. Taking all converging flow regions, yields 181,000 shocks with 5.17 zones per shock. This gives 46% of the volume occupied by converging regions. Taking instead a minimum convergence of [FORMULA] as a criteria which relates to a steepened wave balanced by artificial viscosity, yields 161,000 regions with an average of 3.95 zones. That is, almost 90% of the converging regions are indeed associated with steepened waves, which occupy a constant 30% of the volume. This number density was also found in decaying turbulence: even though the shocks may weaken and interact, the total number is conserved, and the total shock area is independent of the grid size (Paper 1).

Many predictions and analytical fits to simulations have been published for the velocity gradients within turbulent flows. Here, for supersonic turbulence, we find a Gaussian high-speed tail to the shock distribution provides a rough fit. Fig. 3 displays a combined power-law and Gaussian fit of the form

[EQUATION]

for one case. To test if this is due to the initial Gaussian field we have also run a simulation with an exponential driver and found that a similar tail is still present. Hence the near-Gaussian could be due to the superposition of the initially randomly-located forcing waves, as expected from the Central Limit Theorem. The best fit we find, however, involves a tail of the form [FORMULA],

[EQUATION]

[FIGURE] Fig. 3a and b. Power law fits combined with a a Gaussian tail and b a steep exponential tail of the form [FORMULA] to the shock number distribution for the case [FORMULA] and [FORMULA].

Such steep tails have indeed been commonly found despite Gaussian driving, thought to be due to the more rapid decay of the faster shocks (e.g. Gotoh & Kraichnan 1998).

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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