  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. A theoretical model based on the
Rankine-Hugoniot shock jump conditions is derived, sigma_s^2 = ln
{1+(gamma+1)*b^2*M^2/[(gamma-1)*b^2*M^2+2]}, and provides good fits also 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.
