Magnetic topology plays an important role in the global dynamics of high temperature plasmas. Within the ideal MHD plasma description, two plasma elements that are initially connected by a magnetic field line remain connected at any subsequent time. This condition introduces a topological linking between plasma elements that is preserved during the ideal plasma evolution. Magnetic linking constraints the plasma dynamics by making configurations with lower magnetic energy, but different topological linking, inaccessible. Magnetic field line reconnection partially removes these constrains by allowing the field lines to decouple locally from the plasma motion and to reknit in a different net of connections. In collisionless magnetic field line reconnection the decoupling between the magnetic field and the plasma motion occurs because of the current limitation due to the finite electron inertia (in the fluid limit) or to thermal effects (in the kinetic plasma description). However, in the absence of dissipation, the plasma response both in the fluid and in the kinetic electron treatment admits generalized linking conditions that in a two-dimensional configuration are preserved during the process of magnetic reconnection in the form of Lagrangian invariants.
Here we compare the analytical and numerical results obtained recently [1,2] in the study of the nonlinear development of magnetic reconnection in the fluid and in the drift-kinetic limits of the electron response and establish a clear link between these two regimes by showing that the (two) fluid Lagrangian invariants and the (infinite number of) drift-kinetic Lagrangian invariants evolve in time in an analogous fashion: in both cases the growth and saturation of the magnetic island is accompanied by their spatial mixing in the reconnection plane. In particular we show that in the cold electron fluid limit the pattern of current layers formed within the magnetic island in the nonlinear phase of the reconnection process is subject to the onset of a secondary instability of the Kelvin Helmholtz type which leads to a turbulent redistribution of the current layers and to the development of long lived fluid vortices inside the magnetic island.
 E. Cafaro, et al., Phys. Rev. Lett, 80, 4430 (1998); D. Grasso, et al., Phys. Rev.Lett., 86, 5051 (2001); D. Del Sarto, et al., Phys. Rev. Lett., 91, 235001 (2003).  T. Liseikina, et al., Phys. Plasmas, in press (2004).