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.