Exact relations for energy transfer in self-gravitating isothermal turbulence

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.

Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids

“…This type of equipartition also holds for the linear wave modes of compressible MHD. Following the same argument of [10], we can define the two-point symmetric correlator of total energy in the current case as…”

“…Moreover, unlike the incompressible exact relations, straightforward formulations of compressible counterparts were not unique, indicating that some essential physical constraints are missing. Recently, we derived an exact relation for self-gravitating, isothermal turbulence [10], and showed that proper accounting for the acoustic energy equipartition eliminates single-point contributions from the source S. Furthermore, the new constraint implied that the correlation between the velocity and pressure dilatation does not actually play any role in the total energy cascade process in isothermal HD turbulence which was claimed previously [11]. However, the previous flux-source formulation contained certain terms which could be cast neither as a pure divergence term (or a so-called flux term) nor as a source term.…”

A compact exact law for compressible isothermal Hall magnetohydrodynamic turbulence

“…In addition to the incompressible models, exact laws have also been derived that take compressibility into account both for non-magnetized fluids (Galtier & Banerjee 2011;Banerjee & Galtier 2014;Banerjee & Kritsuk 2017;Ferrand et al 2020) and magnetized plasmas (Banerjee & Galtier 2013;Andrés & Sahraoui 2017;Banerjee & Kritsuk 2018). The use of a compressible description of astrophysical flows is relevant for the solar wind (Banerjee et al 2016;Hadid, Sahraoui & Galtier 2017) but also for highly compressible media, such as the interstellar medium, where supersonic turbulence is expected to play a key role in star-forming structures (Kritsuk et al 2007;Arzoumanian et al 2011;Kritsuk, Wagner & Norman 2013;Federrath 2016).…”

“…In addition to the incompressible models, exact laws have also been derived that take compressibility into account both for non-magnetized fluids (Galtier & Banerjee 2011; Banerjee & Galtier 2014; Banerjee & Kritsuk 2017; Ferrand et al. 2020) and magnetized plasmas (Banerjee & Galtier 2013; Andrés & Sahraoui 2017; Andrés, Galtier & Sahraoui 2018; Banerjee & Kritsuk 2018).…”

Energy Transfer and Spectra in Simulations of Two-Dimensional Compressible Turbulence

“…is an ideal invariant of the isothermal system. Following Ref [2],. we define the spectral densities as cospectra P(a a a, b bb; k) ≡ [ a a a(κ κ κ) · b b b * (κ κ κ) + a a a * (κ κ κ) · b b b(κ κ κ)]δ (k − |κ κ κ|)dκ κ κ/2,with a a a = ρu u u, b b b = u u u in case of the kinetic energy and a = ρ, b = e for the potential energy.…”