Two-dimensional simulations of coronal rain dynamics: I. Model consisting of a vertical magnetic field and an unbounded atmosphere

Context. Coronal rain often comes about as the final product of evaporation and condensation cycles that occur in active regions. Observations show that the condensed plasma falls with an acceleration that is less than that of free fall. Aims. We aim to improve the understanding of the physical mechanisms behind the slower than free-fall motion and the two-stage evolution (an initial phase of acceleration followed by an almost constant velocity phase) detected in coronal rain events. Methods. Using the M ANCHA 3D code, we solve the 2D ideal magnetohydrodynamic equations. We represent the solar corona as an isothermal vertically stratified atmosphere with a uniform vertical magnetic field. We represent the plasma condensation as a density enhancement described by a 2D Gaussian profile. We analyse the temporal evolution of the descending plasma and study its dependence on such parameters as density and magnetic field strength. Results. We confirm previous findings that indicate that the pressure gradient is the main force that opposes the action of gravity and slows down the blob descent, and that larger densities require larger pressure gradients to reach the constant speed phase. We find that the shape of a condensation with a horizontal variation of density is distorted during its fall because the denser parts of the blob fall faster than the lighter ones. This is explained by the fact that the duration of the initial acceleration phase and, therefore, the maximum falling speed attained by the plasma, increases with the ratio of blob to coronal density. We also find that the magnetic field plays a fundamental role in the evolution of the descending condensations. A strong enough magnetic field (greater than 10 G in our simulations) forces each plasma element to follow the path given by a particular field line, which allows for the description of the evolution of each vertical slice of the blob in terms of 1D dynamics, without the influence of the adjacent slices. In addition, under the typical conditions of the coronal rain events, the magnetic field prevents the development of Kelvin-Helmholtz instability.