Curvature of Magnetic Field Lines in Compressible Magnetized Turbulence: Statistics, Magnetization Predictions, Gradient Curvature, Modes, and Self-gravitating Media

Magnetic field lines in interstellar media have a rich morphology, which could be characterized by geometrical parameters such as curvature and torsion. In this paper we explore the statistical properties of magnetic field line curvature κ in compressible magnetized turbulence. We see that both the mean and standard deviation of magnetic field line curvature obey power-law relations to the magnetization. Moreover, the power-law tail of the curvature probability distribution function is also proportional to the Alfvénic Mach number. We also explore whether the curvature method could be used in the field-tracing velocity gradient technique. In particular, we observe that there is a relation between the mean and standard deviation of the curvature probed by velocity gradients to M_A. Finally, we discuss how curvature is contributed by different MHD modes in interstellar turbulence and suggests that the eigenvectors of MHD modes could be possibly represented by the natural Frenet-Serret frame of the magnetic field lines. We discuss possible theoretical and observational applications of the curvature technique, including the extended understanding on a special length scale that characterizes the importance of magnetic field curvature in driving MHD turbulence, and how it could be potentially used to study a self-gravitating system.