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We investigate the conditions of ideal magnetohydrodynamic (MHD) turbulenceresponsible 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 anexpression for the time evolution of the angle, $\phi$, between$\vec{\nabla}\rho$ and $\vec{B}$ based on the transport equations of MHDturbulence. 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 ofthese configurations and they are more represented than others. This fact wouldexplain 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 thatdepartures from the $\cos\phi=0$ configurations are related to convergentflows, quantified by the divergence of the velocity field,$\vec{\nabla}\cdot\vec{v}$, in the presence of a relatively strong magneticfield. 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 thegravitational collapse and/or convergence of flows. Finally, we show that thedensity threshold that marks the observed change in relative orientationtowards 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 fieldstrength and constitutes a crucial piece of information for determining therole of the magnetic field in the dynamics of MCs.

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