Abstract
Beginning with the fluid-dynamical equations, theoretical approaches to compressible turbulence are outlined in this chapter. After a brief summary of the main results of the Kolmogorov theory of incompressible turbulence, attempts to treat compressible turbulence with a similar theoretical framework are considered. In the case of isothermal gas, which plays an important role for the theory of star formation, an important aspect is the log-normal distribution of density fluctuations. Scaling relations for a strictly self-similar turbulent cascade are empirically found to break down for higher-order statistics. This is explained by the intermittency of turbulence. A heuristic model that was successfully used to predict scaling properties of incompressible and compressible turbulence is based on a hierarchy of dissipative structure with log-Poisson statistics. This leads to a simple two-parameter model for the relative scaling exponents, which can be determined through experimental measurements or numerical simulations. Apart from the general velocity statistics of compressible turbulence, the density structure of self-gravitating turbulence is an unsolved problem. Crucial questions concern the support of the gas against gravity in the highly non-linear regime and the mass distribution of gravitationally unstable clumps.
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Notes
- 1.
This can be understood as a renormalization of the nominally infinite potential for an infinitely extended mass distribution with positive mean density \(\langle \rho \rangle \) to zero for \(\rho =\langle \rho \rangle \).
- 2.
In the fluid dynamics literature, \(\sigma _{ij}\) often includes the pressure term \(P\delta _{ij}\). In the formulation used here, it is a sperate term.
- 3.
In [12], the symbol \(\mathbf f \) is used for the force density. We keep our definition for consistency in this text.
- 4.
In other contexts, the term clump may have a more specific or different meaning.
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Schmidt, W. (2014). Turbulence Theory. In: Numerical Modelling of Astrophysical Turbulence. SpringerBriefs in Astronomy. Springer, Cham. https://doi.org/10.1007/978-3-319-01475-3_1
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