What are we learning from the relative orientation between density structures and the magnetic field in molecular clouds?

Abstract: We investigate the conditions of ideal magnetohydrodynamic (MHD) turbulence responsible for the relative orientation between density structures, characterized by their gradient, $\vec{\nabla}\rho$, and the magnetic field, $\vec{B}$, in molecular clouds (MCs). For that purpose, we construct an expression for the time evolution of the angle, $\phi$, between $\vec{\nabla}\rho$ and $\vec{B}$ based on the transport equations of MHD turbulence. Using this expression, we find that the configuration where $\vec{\nabla}\rho$ and $\vec{B}$ are mostly parallel, $\cos\phi=1$, and where $\vec{\nabla}\rho$ and $\vec{B}$ are mostly perpendicular, $\cos\phi=0$, constitute attractors, that is, the system tends to evolve towards either of these configurations and they are more represented than others. This fact would explain the predominant alignment or anti-alignment between column density, $N_H$, structures and the projected magnetic field orientation, $\hat{B}\perp$, reported in observations. Additionally, we find that departures from the $\cos\phi=0$ configurations are related to convergent flows, quantified by the divergence of the velocity field, $\vec{\nabla}\cdot\vec{v}$, in the presence of a relatively strong magnetic field. This would explain the observed change in relative orientation between $N_H$-structures and $\hat{B}\perp$ towards MCs, from mostly parallel at low $N_H$ to mostly perpendicular at the highest $N_H$, as the result of the gravitational collapse and/or convergence of flows. Finally, we show that the density threshold that marks the observed change in relative orientation towards MCs, from $N_H$ and $\hat{B}_\perp$ being mostly parallel at low $N_H$ to mostly perpendicular at the highest $N_H$, is related to the magnetic field strength and constitutes a crucial piece of information for determining the role of the magnetic field in the dynamics of MCs.