**5. Dynamics of magnetic relaxation in spheromak configurations**

The relaxation theory as formulated by Taylor (1974) is a variational principle that can not give details on the dynamical aspects of the process. All the considerations we have made regarding the important role of localized reconnection in helicity conservation are only heuristic arguments that try to explain the remarkable success of the theory at predicting the self-organized final state of the plasma.

There are a number of reasons that motivate the study of the dynamics of relaxation. For instance, in the context of spheromak research it is observed that during sustainment the system does not remain at the lowest energy state. Small deviations from the relaxed state as well as the ubiquitous presence of fluctuations are crucial issues that are out of the scope of relaxation theory (Knox et al., 1986);(Willet et al., 1999). In this work we study these aspects using numerical solutions of the non linear resistive MHD equations described in Sec. 3 as an initial and boundary value problem in three spatial dimensions. The nondimensional version described in Sec. 3.5 of these equations is used. The details of the numerical method are not presented here but can be found elsewhere (Garcia-Martinez & Farengo, 2009b).

In this Section we present a study of the dynamics of the kink mode in spheromak configurations. We will focus on the dynamics of systems that are only marginally unstable. Even when this may sound as a rather specific topic we will see that this is a simple setup in which we can study magnetic reconnection and helicity transfer between flux tubes. Firstly, we describe the kink unstable configurations used as initial condition and explain how they can be computed. Secondly, we study the dynamics of the kink instability in several cases and discuss in which cases it leads to a complete relaxation process (as described in the preceding Section) and in which cases the relaxation process is only partial. Thirdly, we introduce the concept of safety factor and resonant surfaces and explain their relevance to the partial relaxation behavior observed in marginally unstable configurations. Then, we analyze in detail the reconnection process that is driven by the dominant kink mode. Finally, we discuss simple models to describe this reconnection process.
