We have shown that the Paramagnetic Meissner Effect (PME) is directly associated with pinning, and not necessarily related to the presence of pi-junctions. Through the study of the magnetic properties of two-dimensional Josephson junction arrays (2D-JJA) in the present work we show that, among the systems exhibiting PME, only those with suffciently low dissipation and high capacitance will show dynamics reentrance. The concept of a critical state and its use in the interpretation of AC magnetization data in terms of a critical current density were introduced to derive the magnetic properties of hard type-II superconductors. In the critical state model proposed by Bean, flux lines penetrate into the sample and, due to the presence of disorder they give rise to a steady flux gradient. Here we show that in 2D-JJA this typical picture is valid only in short-range distances. For long-range distances, the picture of uniform flux fronts, as described by a critical state model, breaks down and the penetration of the magnetic field takes place through the growth of magnetic dendrites. De Gennes originally compared the slope of a pile of vortices to a sand-pile, with the slope being proportional to the local magnitude of the critical current. Dynamical properties of the sand-pile problem have attracted new attention since it consists of a marginally stable system displaying self-organized criticality (SOC). In this case, when a superconductor is in the Bean critical state, the addition of vortices occurs by increasing the external magnetic field. This procedure is analogous to the introduction of new grains to a sand-pile and is expected to produce an avalanche of grains of sand (or, equivalently, vortices) of all sizes to maintain a constant gradient in the grain (or, magnetic ux) density. We show in this work strong evidences pointing out that, for some specific conditions, magnetic field penetrates 2D-JJA in ux avalanches.