Exact relations for energy transfer in self-gravitating isothermal
  turbulence
  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.
