## Final Activity Report Summary - RECONNECTION (Reconnection of magnetic field lines in weakly collisional plasmas)

The project is devoted to the investigation of theoretical problems related to the physical phenomenon of magnetic reconnection. Such phenomenon consists of a rearrangement of the way small volumes of plasma are connected to each other through magnetic field lines. Magnetic reconnection can take place in both laboratory and astrophysical plasmas and is believed to be responsible for disruptive events such as solar flares and magnetic substorms.

One line of research of the project aimed at a better understanding of the dependence of the nonlinear dynamics of the reconnection process on the value of the parameter beta, defined as the ratio between the plasma pressure and the magnetic pressure exerted by the field perpendicular to the reconnection plane. To this purpose we investigated the two-dimensional four-field model which is valid both for low (i.e. much less than unity) and high values of beta.

We found out that, analogously to previously investigated models for collisionless plasmas, this model can also be cast into a hamiltonian form. By making use of the hamiltonian formulation we derived four independent families of conserved functionals (Casimirs) that generalise the two families previously found in the low-beta case. In particular we found that one of these families of Casimirs depends on an arbitrary function of a Lagrangian invariant, that is a quantity that is preserved in a frame of reference moving with the plasma velocity. The existence of this invariant, which is related to the parallel electron momentum, shows that also in high beta regime there exist quantities whose topology is preserved, as it was the case for the generalised fields derived in the low-beta regime. The knowledge of the Casimirs simplified the search for exact equilibrium solutions of the system and the investigation of their stability properties, which is in progress.

A second branch of research was motivated by the observation of Quasi-Single-Helicity (QSH) states in Reversed Field Pinches (RFP) plasma confinement experiments. These states represent long-lived coherent structures in which the spectrum of the magnetic field possesses a dominant helical mode. A satisfactory theory explaining the appearance of such states is, however, still lacking. We proposed that QSH states can emerge after saturation of a reconnection process generated by a tearing instability.

Our conjecture is that QSH states may form in correspondence to small departures from the force-free states predicted by the classical theory of Taylor for RFPs. In particular we proved by means of an analytical linear stability analysis that if the magnetic field reaches a force-free state in which the ratio between the current density and the magnetic field is a step function of the radial distance from the centre of the cylindrical plasma chamber, then the resulting equilibrium can be unstable to perturbations with the characteristic helicity observed in QSH states. Subsequently we investigated the nonlinear saturation of such tearing instabilities in RFPs by means of a perturbative technique previously adopted for the tokamak configuration. As a result we obtained a relation providing the width of the magnetic island, which appears as consequence of the reconnection process, as a function of the equilibrium parameters.

Finally an investigation of the dynamic accessibility of some exact solutions of the magnetohydrodynamics equations describing stationary reconnection was carried out. The analysis showed that if the initial flow is super-alfvenic these stationary solutions can be accessed as final state of a time dependent evolution of the system regardless of the form of the initial flow. On the contrary if the initial flow is sub-alfvenic then the accessed stationary state depends on the initial value of the plasma vorticity along one separatrix of the flow. Indeed we found out that such quantity is a constant of motion and thus it is preserved during the evolution of the system.

One line of research of the project aimed at a better understanding of the dependence of the nonlinear dynamics of the reconnection process on the value of the parameter beta, defined as the ratio between the plasma pressure and the magnetic pressure exerted by the field perpendicular to the reconnection plane. To this purpose we investigated the two-dimensional four-field model which is valid both for low (i.e. much less than unity) and high values of beta.

We found out that, analogously to previously investigated models for collisionless plasmas, this model can also be cast into a hamiltonian form. By making use of the hamiltonian formulation we derived four independent families of conserved functionals (Casimirs) that generalise the two families previously found in the low-beta case. In particular we found that one of these families of Casimirs depends on an arbitrary function of a Lagrangian invariant, that is a quantity that is preserved in a frame of reference moving with the plasma velocity. The existence of this invariant, which is related to the parallel electron momentum, shows that also in high beta regime there exist quantities whose topology is preserved, as it was the case for the generalised fields derived in the low-beta regime. The knowledge of the Casimirs simplified the search for exact equilibrium solutions of the system and the investigation of their stability properties, which is in progress.

A second branch of research was motivated by the observation of Quasi-Single-Helicity (QSH) states in Reversed Field Pinches (RFP) plasma confinement experiments. These states represent long-lived coherent structures in which the spectrum of the magnetic field possesses a dominant helical mode. A satisfactory theory explaining the appearance of such states is, however, still lacking. We proposed that QSH states can emerge after saturation of a reconnection process generated by a tearing instability.

Our conjecture is that QSH states may form in correspondence to small departures from the force-free states predicted by the classical theory of Taylor for RFPs. In particular we proved by means of an analytical linear stability analysis that if the magnetic field reaches a force-free state in which the ratio between the current density and the magnetic field is a step function of the radial distance from the centre of the cylindrical plasma chamber, then the resulting equilibrium can be unstable to perturbations with the characteristic helicity observed in QSH states. Subsequently we investigated the nonlinear saturation of such tearing instabilities in RFPs by means of a perturbative technique previously adopted for the tokamak configuration. As a result we obtained a relation providing the width of the magnetic island, which appears as consequence of the reconnection process, as a function of the equilibrium parameters.

Finally an investigation of the dynamic accessibility of some exact solutions of the magnetohydrodynamics equations describing stationary reconnection was carried out. The analysis showed that if the initial flow is super-alfvenic these stationary solutions can be accessed as final state of a time dependent evolution of the system regardless of the form of the initial flow. On the contrary if the initial flow is sub-alfvenic then the accessed stationary state depends on the initial value of the plasma vorticity along one separatrix of the flow. Indeed we found out that such quantity is a constant of motion and thus it is preserved during the evolution of the system.