The density variance-Mach number relation in isothermal and non-isothermal adiabatic turbulence

The density variance-Mach number relation of the turbulent interstellar medium is relevant for theoretical models of the star formation rate, efficiency, and the initial mass function of stars. Here we use high-resolution hydrodynamical simulations with grid resolutions of up to 1024³ cells to model compressible turbulence in a regime similar to the observed interstellar medium. We use FYRIS ALPHA, a shock-capturing code employing a high-order Godunov scheme to track large density variations induced by shocks. We investigate the robustness of the standard relation between the logarithmic density variance (σ² _s) and the sonic Mach number (M) of isothermal interstellar turbulence, in the non-isothermal regime. Specifically, we test ideal gases with diatomic molecular (γ = 7/5) and monatomic (γ = 5/3) adiabatic indices. A periodic cube of gas is stirred with purely solenoidal forcing at low wavenumbers, leading to a fully developed turbulent medium. We find that as the gas heats in adiabatic compressions, it evolves along the relationship in the density variance-Mach number plane, but deviates significantly from the standard expression for isothermal gases. Our main result is a new density variance-Mach number relation that takes the adiabatic index into account: σ² _s = ln (1 + b²M^(5γ+1)/3) and provides good fits for bM≲ 1. A theoretical model based on the Rankine-Hugoniot shock jump conditions is derived, σ² _s = ln{1 + (γ + 1)b²M²/[(γ - 1)b²M² + 2]}, and provides good fits also for bM> 1. We conclude that this new relation for adiabatic turbulence may introduce important corrections to the standard relation, if the gas is not isothermal (γ ≠ 1).