The dynamical origin of the magnetic field distributions in compressible turbulence

Magnetohydrodynamical (MHD) simulations of isothermal compressible turbulence report that the density distribution is well described by a lognormal with a variance proportional to the flow's Mach number. The distribution of magnetic field strength also has a lognormal component, but includes long, power-law-like tails. In this work, we use semi-analytical arguments to predict the distributions of density and magnetic field strength in compressible turbulent flows. Specifically, in the Lagrangian description of the continuity and the induction equations, we model the velocity gradients of the turbulent flow as a simple random process, essentially turning these equations into stochastic differential equations. Integrating them leads to a lognormal distribution for the density field and the strength of the magnetic field. The power-law tails in the magnetic field PDF appear when we introduce intermittent shocks due to sampling rare events. Gradually increasing the frequency of these events, essentially going closer to a continuous process, leads to lognormal-like distributions again. The asymmetry is connected to the relative abundance of slow and fast shocks. An overabundance of fast MHD shocks produces a high-value tail, while the contrary produces low-value tails. We propose that the appearance of power-law tails along lognormals in turbulent flows is the signature of the co-existence of continuous, diffusion-like propagation combined with localized, intermittent events.