**6. Conclusions**

In this Chapter we have presented a general picture of the magnetic confinement of high temperature plasmas. This has motivated the introduction of the MHD model which provides an adequate framework to study the macroscopic dynamics of fully ionized plasmas. We have focused our attention on the physical mechanism called plasma relaxation. In particular we have studied the magnetic relaxation process driven by the kink instability in spheromak configurations.

Experiments as well as previous theoretical works showed the existence of a partial relaxation behavior for marginally unstable configurations (they do not evolve toward the minimum energy state). This is in contrast to the well established relaxation theory that states that the plasma should relax to the minimum energy configuration. In this work we have explored these two regimes, namely complete relaxation and partial relaxation, by varying the slope of the initial *λ*(*ψ*) profile. This controls the degree of instability of the initial configuration as well as the position of the rational surface having safety factor equal to one. The relevance of the position of this rational (or resonant) surface to the partial relaxation behavior was discussed. In particular, we showed that in marginally unstable cases this surface is not far from the magnetic axis and the MHD activity during relaxation remains inside this resonant surface (which is no longer resonant after relaxation). These results suggest that the *q* = 1 surface plays a major role in the evolution of spheromaks during sustainment because in that situation they operate around the kink instability threshold.

The analysis of more unstable cases showed that the full relaxation process predicted by the relaxation theory is only achieved when the magnetic fluctuations produce stochastic field line regions of size comparable of that of the whole system. This result clearly indicates that the relaxation theory as formulated by Taylor (1974) is applicable to highly unstable plasmas but it becomes useless to study the operation of configurations near an instability threshold.

The kink instability produces the helical deformation of the flux surfaces near the magnetic axis. This drives the reconnection of the inner flux surfaces with the outer ones. This process has been studied in detail. The reconnection layer has been identified as well as the new structure resulting from the reconnection of the two flux tubes. Taking the low (high) *λ* value of the inner (outer) tube on its inner (outer) side, these crescent shaped structures average the *λ* value inside the *q* = 1 surface. Even when the flux surfaces remain regular during this evolution, the process involves the full reconnection of all the magnetic tubes inside the *q* = 1 surface. This is of course undesired from the point of view of confinement and could partially explain the poor performance of spheromak operation (compared to tokamaks and RFP's). However further studies are required on this topic regarding the coupled dynamics between the kink and the external driving of the system. This could be done by applying appropriate boundary conditions to model the injection of helicity from a source (Garcia-Martinez & Farengo, 2010).

Finally, models for the reconnection process driven by the kink mode were discussed. The Kadomtsev's model was presented and showed to give a poor description of the actual simulation results. A modification to this model that greatly improves the agreement with simulations was proposed. A method to incorporate the correction due to the resistive decay of the configuration was described.
