Hierarchical Structure in Nearly Pressureless Flows as a Consequence of Self-similar Statistics

Abstract

A purely statistical origin of hierarchical density structure in pressureless and nearly pressureless turbulent flows without self-gravity is proposed. It is argued that hydrodynamic (Navier-Stokes) flows with rms Mach numbers Mrms ≫ 1, like the cold interstellar medium or some cosmological flows, have a nearly pressureless behavior, except perhaps in a small fraction of the volume. Assuming that in these flows the density field can be considered as a random variable in time with an associated probability distribution function (pdf), a condition on the pdf for the spontaneous development of hierarchical structure is proposed. The condition is essentially a requirement that the joint probability of a succession of n constructively interfering relative density fluctuations of amplitude a be larger than the probability of a single, large fluctuation of the same final amplitude a . An analytic form for this condition is derived, under the assumption that the density pdf is scale- invariant, and three important functional forms of the pdf (a power-law, a lognormal, and an exponential distribution) are tested according to this criterion. It is found that development of hierarchical structure should always be expected for the lognormal and exponential distributions at large values of n and/or a, but is parameter-dependent in the case of the power law. The assumption of a scale-invariant density pdf is motivated (but not rigorously proved) by the scale invariance of the pressureless hydrodynamic equations with negligible self-gravity, and a numerical simulation of a compressible turbulent flow in two dimensions in a transonic regime (M ∼ 1) is used to test this and other secondary assumptions. The density field developed by the simulation clearly exhibits at least three levels of hierarchical nesting. The density statistics, albeit obscured by the limited resolution, which limits the feasible density contrasts, are shown to be consistent with the hypotheses. A discussion on the degree to which the dynamics of two- and three-dimensional flows may differ from one another in the compressible case is given. Finally, a remark is made that the hierarchical structure produced by this mechanism in turbulent compressible flows is transient although statistically stationary. This should be the case of all non-self-gravitating clouds in the cold interstellar medium and of small-scale structures in cosmological flows.