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

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.