A Kolmogorov-like exact relation for compressible polytropic turbulence

Abstract

Compressible hydrodynamic turbulence is studied under the assumption of a polytropic closure. Following Kolmogorov, we derive an exact relation for some two-point correlation functions in the asymptotic limit of a high Reynolds number. The inertial range is characterized by (i) a flux term implying in particular the enthalpy and (ii) a purely compressible term S which may act as a source or a sink for the mean energy transfer rate. At subsonic scales, we predict dimensionally that the isotropic k −5/3 energy spectrum for the density-weighted velocity field (ρ 1/3 v) -previously obtained for isothermal turbulence -is modified by a polytropic contribution, whereas at supersonic scales S may impose another scaling depending on the polytropic index. In both cases, it is shown that the fluctuating sound speed is a key ingredient for understanding polytropic compressible turbulence.

Exact relations for energy transfer in self-gravitating isothermal turbulence

“…This form looks nearly similar to the previously derived exact relations for compressible turbulence [50][51][52]. In order to obtain the concerned form, we combine the above expressions and obtain for R H (and similarly for R ′ H )…”

“…With respect to the previously derived relations of compressible hydrodynamic and magnetohydrodynamic turbulence [50][51][52], here we have modified the definition of the two-point energy correlation function by putting an additional constraint of detailed equipartition (in spectral space) between dilatational kinetic and thermodynamic potential energy in the acoustic limit. Using the modified correlation function, we showed that the resulting relation is indeed much simpler and is free from any pressure-dilatation-velocity correlation.…”

“…The derivation of the exact relation relates to general compressible turbulence whereas the principal physical motivation lies in understanding the role of gravity in cold star-forming clouds where the turbulence is usually supersonic. The derivation follows the methodology previously outlined by Galtier and Banerjee [50], Banerjee and Galtier [51,52]. However, this work revisits the definition of two-point correlation functions which renders the final exact relation simpler and entirely expressible in terms of two-point fluctuations.…”

The density structure and star formation rate of non-isothermal polytropic turbulence

“…What we do know is that turbulence in the interstellar medium is highly compressible and supersonic (Larson 1981;Heyer & Brunt 2004;Roman-Duval et al 2011;Hennebelle & Falgarone 2012), significantly exceeding the complexity of incompressible turbulence (Kolmogorov 1941;Frisch 1995). Supersonic, compressible turbulence is difficult to study analyti-E-mail: christoph.federrath@anu.edu.au † E-mail: supratik.banerjee@uni-koeln.de cally, but some important steps have been taken (Lazarian & Pogosyan 2000;Boldyrev et al 2002;Lazarian & Esquivel 2003;Schmidt et al 2008;Galtier & Banerjee 2011;Aluie 2011Aluie , 2013Banerjee & Galtier 2013, 2014. In order to unravel the statistics and properties of turbulence in detail, however, one must ultimately resort to full 3D computer simulations.…”

Turbulent Density Fluctuations and Proton Heating Rate in the Solar Wind from 9–20 R⊙

“…However, density fluctuations are not explained in this framework, and remain a relative enigma despite noteworthy progress (e.g., Hnat et al (2005); Shaikh & Zank (2010); Banerjee & Galtier (2014)). While most of the data used for solar wind turbulence studies are from in-situ measurements made by near-Earth spacecraft, density fluctuations can often been inferred via remote sensing observations, typically at radio wavelengths.…”