| Abstract
| - Context. Compressible turbulence influences the dynamics of the interstellar and the intergalactic medium over a vast range of length scales. In numerical simulations, phenomenological subgrid scale (SGS) models are used to describe particular physical processes below the grid scale. In most cases, these models do not cover fluid-dynamical interactions between resolved and unresolved scales, or the employed SGS model is not applicable to turbulence in the highly compressible regime. Aims. 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. Methods. We tested 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 nonlinear 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. Results. 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. Conclusions. 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. By computing the SGS turbulence energy, the treatment of star formation and stellar feedback in galaxy simulations can be improved. Furthermore, we expect that the turbulent pressure on the grid scale affects the stability of gas against gravitational collapse. The influence of small-scale turbulence on emission line broadening, e.g., of O VI, in the intergalactic medium is another potential application.
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