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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 1024^3 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 (sigma_s^2) and the sonic Mach number (M) of isothermal interstellar turbulence, in the non-isothermal regime. Specifically, we test ideal gases with diatomic molecular (gamma = 7/5) and monatomic (gamma = 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: sigma_s^2 = ln {1+b^2*M^[(5*gamma+1)/3]} and provides good fits for b*M 1. We conclude that this new relation for adiabatic turbulence may introduce important corrections to the standard relation, if the gas is not isothermal.

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