A Fluid-Dynamical Subgrid Scale Model for Highly Compressible
  Astrophysical Turbulence
  We formulate and implement the Euler equations with SGS dynamics and provide
numerical tests of an SGS turbulence energy model that predicts the turbulent
pressure of unresolved velocity fluctuations and the rate of dissipation for
highly compressible turbulence. We test closures for the turbulence energy
cascade by filtering data from high-resolution simulations of forced isothermal
and adiabatic turbulence. Optimal properties and an excellent correlation are
found for a linear combination of the eddy-viscosity closure that is employed
in LES of weakly compressible turbulence and a term that is non-linear in the
Jacobian matrix of the velocity. Using this mixed closure, the SGS turbulence
energy model is validated in LES of turbulence with stochastic forcing. It is
found that the SGS model satisfies several important requirements: 1. The mean
SGS turbulence energy follows a power law for varying grid scale. 2. The root
mean square (RMS) Mach number of the unresolved velocity fluctuations is
proportional to the RMS Mach number of the resolved turbulence, independent of
the forcing. 3. The rate of dissipation and the turbulence energy flux are
constant. Moreover, we discuss difficulties with direct estimates of the
turbulent pressure and the dissipation rate on the basis of resolved flow
quantities that have recently been proposed. In combination with the energy
injection by stellar feedback and other unresolved processes, the proposed SGS
model is applicable to a variety of problems in computational astrophysics.
Computing the SGS turbulence energy, the treatment of star formation and
stellar feedback in galaxy simulations can be improved. Further, we expect that
the turbulent pressure on the grid scale affects the stability of gas against
gravitational collapse.
