Magnetic field fluctuations in anisotropic, supersonic turbulence

The rich structure that we observe in molecular clouds is due to the interplay between strong magnetic fields and supersonic (turbulent) velocity fluctuations. The velocity fluctuations interact with the magnetic field, causing it too to fluctuate. Using numerical simulations, we explore the nature of such magnetic field fluctuations, $\vec{\delta B}$, over a wide range of turbulent Mach numbers, $\mathcal{M} = 2 - 20$ (i.e., from weak to strong compressibility), and Alfv\'en Mach numbers, $\mathcal{M}_{\text{A}0} = 0.1 - 100$ (i.e., from strong to weak magnetic mean fields, $B_0$). We derive a compressible quasi-static fluctuation model from the magnetohydrodynamical (MHD) equations and show that velocity gradients parallel to the mean magnetic field give rise to compressible modes in sub-Alfv\'enic flows, which prevents the flow from becoming two-dimensional, as is the case in incompressible MHD turbulence. We then generalise an analytical model for the magnitude of the magnetic fluctuations to include $\mathcal{M}$, and find $|\vec{\delta B}| = \delta B = c_s\sqrt{\pi\rho_0}\mathcal{M}\mathcal{M}_{\text{A}0}$, where $c_s$ is the sound speed and $\rho_0$ is the mean density of gas. This new relation fits well in the strong $B$-field regime. We go on to study the anisotropy between the perpendicular ($ B_{\perp}$) and parallel ($ B_{\parallel}$) fluctuations and the mean-normalised fluctuations, which we find follow universal scaling relations, invariant of $\mathcal{M}$. We provide a detailed analysis of the morphology for the $\delta B_{\perp}$ and $\delta B_{\parallel}$ probability density functions and find that eddies aligned with $B_0$ cause parallel fluctuations that reduce $B_{\parallel}$ in the most anisotropic simulations. We discuss broadly the implications of our fluctuation models for magnetised gases in the interstellar medium.