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.
