Scaling properties of compressible polytropic turbulence

An exact relation has been derived for homogeneous polytropic turbulence in terms of the two-point fluctuations. Unlike isothermal turbulence, the fluctuating sound speed appears to be a key factor in determining the scaling properties of polytropic turbulence. Three Mach numbers (current, gradient and turbulent Mach number) are defined to characterize the isotropic scaling. At subsonic scale, the Kolmogorov-like -5/3 energy spectrum for ⇢1/3v (obtained for isothermal turbulence) is modified by polytropic contribution. For supersonic turbulence, the source terms must be taken into account which further complicates the scaling and spectral properties. For highly supersonic case, the source terms, however, can be shown to be nearly equal to the source terms in isothermal turbulence thereby leading to a partial simplification. These theoretical results are extremely important for understanding the role of turbulence in the star-formation mechanism in polytropic clouds