Exact relations for energy transfer in self-gravitating isothermal turbulence

TL;DR

This paper derives an exact relation for energy transfer in self-gravitating isothermal turbulence, revealing the minimal role of pressure dilatation and highlighting the importance of density-gravity correlations in energy dynamics.

Contribution

It introduces a new exact relation for energy transfer in self-gravitating turbulence that accounts for detailed energy equipartition and differs from previous models.

Findings

Pressure dilatation has negligible impact on energy cascade.

Density-gravity correlation significantly influences energy transfer.

The relation is applicable in spectral space for scale-by-scale analysis.

Abstract

Self-gravitating isothermal supersonic turbulence is analyzed in the asymptotic limit of large Reynolds numbers. Based on the inviscid invariance of total energy, an exact relation is derived for homogeneous, (not necessarily isotropic) turbulence. A modified definition for the two-point energy correlation functions is used to comply with the requirement of detailed energy equipartition in the acoustic limit. In contrast to the previous relations (Galtier and Banerjee, Phys. Rev. Lett., 107, 134501, 2011; Banerjee and Galtier, Phys. Rev. E, 87, 013019, 2013), the current exact relation shows that the pressure dilatation terms plays practically no role in the energy cascade. Both the flux and source terms are written in terms of two-point differences. Sources enter the relation in a form of mixed second-order structure functions. Unlike kinetic and thermodynamic potential energy, gravitational contribution is absent from the flux term. An estimate shows that for the isotropic case, the correlation between density and gravitational acceleration may play an important role in modifying the energy transfer in self-gravitating turbulence. The exact relation is also written in an alternative form in terms of two-point correlation functions, which is then used to describe scale-by-scale energy budget in spectral space.