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

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6. Interpretation

Turbulence has proven easy to interpret, but the interpretations have been hard to prove. Our aim here is limited to answer: why a universal inverse square root power law? Supersonic turbulence possesses simplifying characteristics which we here employ to understand the shock distribution.

First, it is significant that the flow is dominated by the integrated number and power of the high-speed shocks at the knee in the shock jump distributions. These shocks are generated by the driver and propagate at high speed into the fluid. In contrast, a shock from the power-law section plays a passive rôle: when overtaken by a strong shock, its strength decays according to the speed and density of the oncoming material. Hence the mean lifetime, [FORMULA], of a shock layer, and hence of the shock momentum, is a constant, independent of its own absolute speed, v. Expressed mathematically, for a steady state, this yields

[EQUATION]

from which we find the distribution of absolute shock speeds (i.e. the speed of the layers within the box) is [FORMULA]. This is confirmed from the simulations - Fig. 12 shows that, below the high-speed knee, the distribution of absolute velocities within the shocks is indeed independent of the absolute speed.

[FIGURE] Fig. 12. The remarkable number distribution of shock absolute speeds in uniform driven turbulence for the four times from t=0 to t=1.5. The absolute speed is defined as the average speed in the x-direction within a region of converging flow along the x-axis. So defined, we do not have to contend with the shock direction or with determining the 3-D shock structure, but retain the vital information for modelling.

The relationship between the absolute speed, v, and the jump speed, [FORMULA], for the ensemble of shocks is expected to resemble that of Burger's turbulence (Paper 1, see also Gotoh & Kraichnan 1993). The steepening of these independently-moving shocks is proportional to the absolute speed simply because higher amplitude waves produce stronger shocks, whereas low amplitude waves can only lead to weak jumps. Secondly, the higher amplitudes imply that the velocity gradients are also steeper. Hence the material swept up into a shock is from a wider region which possesses a larger velocity difference. These two linear effects lead to a relationship for the ensemble shock jump and absolute speeds of the form [FORMULA]. This is the essence of mapping closure theory, which then uses this relationship to transform initial Gaussian distributions into exponentials (Chen et al. 1989, Kraichnan 1990). Here, we transform the distribution of absolute shock speeds generated by the uniform driving, to obtain

[EQUATION]

This agrees with the numerical experiments. Note that the result depends on the nature of the driving: uniform driving produces strong shocks locally which then propagate through the ambient medium. In contrast, a white-noise driver would continually create weak small-scale turbulence throughout the space and might thus produce, for example, some other power-law behaviour.

We found here that the high-velocity tail is somewhat steeper than Gaussian. A similar result has been obtained for other types of turbulence, with the same form occurring, [FORMULA], as uncovered here (see Gotoh & Kraichnen 1998). This is due to the Gaussian forcing being modified by the high dissipation rates at these speeds.

The total energy `radiated' in the shocks can be found on integrating Eq. (7) over the jump Mach Number. This yields [FORMULA]. The remaining injected energy is also lost in the shocks where numerical diffusion is, of course, strong. As shown in Table 1 of Paper 1, we expect approximately 0.68 of the energy to be radiated in the shocks. Hence, the description of driven turbulence is fully consistent.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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