Mach number study of supersonic turbulence: The properties of the density field

Abstract

We model driven, compressible, isothermal, turbulence with Mach numbers ranging from the subsonic ($\mathcal{M}\approx 0.65$) to the highly supersonic regime ($\mathcal{M}\approx 16$). The forcing scheme consists both solenoidal (transverse) and compressive (longitudinal) modes in equal parts. We find a relation ${\sigma }_s^2=\mathrm{b}\log (1+{\mathrm{b}}^2{\mathcal{M}}^2)$ between the Mach number and the standard deviation of the logarithmic density with $\mathrm{b}=0.457\pm 0.007$. The density spectra follow $\mathcal{D}(k,\mathcal{M})\propto k^{\zeta (\mathcal{M})}$ with scaling exponents depending on the Mach number. We find $\zeta (\mathcal{M})=\alpha {\mathcal{M}}^{\beta }$ with a coefficient $\alpha$ that varies slightly with resolution, whereas $\beta$ changes systematically. We extrapolate to the limit of infinite resolution and find $\alpha =-1.91\pm 0.01,\beta =-0.30\pm 0.03$. The dependence of the scaling exponent on the Mach number implies a fractal dimension $D=2+0.96{\mathcal{M}}^{-0.30}$. We determine how the scaling parameters depend on the wavenumber and find that the density spectra are slightly curved. This curvature gets more pronounced with increasing Mach number. We propose a physically motivated fitting formula $\mathcal{D}(k)={\mathcal{D}}_0{k}^{\zeta k^{\eta }}$ by using simple scaling arguments. The fit reproduces the spectral behaviour down to scales $k\approx 80$. The density spectrum follows a single power-law $\eta =-0.005\pm 0.01$ in the low Mach number regime and the strongest curvature $\eta =-0.04\pm 0.02$ for the highest Mach number. These values of $\eta$ represent a lower limit, as the curvature increases with resolution.

12 pages, 11 figures, MNRAS, submitted