**7. References**


24 Will-be-set-by-IN-TECH

demonstrates a speedup ranging from 8 <sup>−</sup> 15 for a 1282 mesh Chacón (2008a). Another example of a good verification test case in 3D is that of 3D island coalescence (Chacón (2008a)). Reynolds et al. (2006) reported on a 3D ideal MHD problem which models pellet fueling in

In this chapter, we discussed the need for implicit algorithms for resistive magnetohydrodynamics. We highlighted two broad classes of nonlinear methods: Newton-Krylov and nonlinear multigrid. We illustrated two Newton-Krylov approaches for MHD which are essentially very similar in the overall approach, but differed in the preconditioning strategies for expediting the iterative solution steps in the Krylov linear solver stage of the overall method. One preconditioning strategy is based on a "parabolization" approach while the other utilizes the local wave structure of the underlying hyperbolic waves in the MHD PDEs. The literature on the use of nonlinear multigrid for MHD is essentially sparse and therein we focused on a defect-correction approach coupled with a point-wise Gauss-Seidel smoother utilizing a first order upwind approach. Both approaches are valid and have their place, but it is clear that the nonlinear multigrid

In this chapter, we have focused exclusively on methods for single fluid resistive MHD. Future challenges will lie in the area of implicit methods for more complicated extended MHD models with FLR effects, several of which exhibit dispersive wave phenomena such as Whistler, Kinetic Alfvén waves, and gyroviscous waves. These dispersive high frequency waves essentially make the stable explicit time step proportional to the square of the mesh spacing, i.e., Δ*t* ∝ Δ*x*2; and hence the benefit from implicit methods is much more than those for single fluid MHD. Some progress in using Newton Krylov approaches for Hall-MHD has been reported by Chacón & Knoll (2003). However more work is required for general geometry, and inclusion of all dispersive wave families. Research in the area of nonlinear multigrid is essentially unexplored for extended MHD. Another interesting challenge in developing implicit methods for MHD is the combination of JFNK or FAS methods with adaptive mesh refinement (AMR). Some progress towards JFNK with AMR has been reported by Philip et al. (2008) on reduced incompressible MHD in 2D. Combining implicit methods with AMR will help mitigate not only the temporal stiffness issues but also help effectively

Adams, M. F., Samtaney, R. & Brandt, A. (2010). Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics, *J. of Comput. Phys.* 229: 6208–6219. Atlas, I. & Burrage, K. (1994). A high accuracy defect correction multigrid method for the steady incompressible Navier Stokes equations, *J. Comput. Phys.* 114: 227–233. Birn, J. & et al. (2001a). Geospace Environmental Modeling (GEM) magnetic reconnection

approach for MHD is still relatively new and could be further developed.

tokamaks.

**6. Conclusion**

**6.1 Future challenges**

**7. References**

resolve the range of spatial scales in MHD.

challenge, *J. Geophys. Res.* 106: 3715–3719.


**0**

**4**

<sup>1</sup>*France* <sup>2</sup>*Argentina*

Pablo L. Garcia-Martinez

**Dynamics of Magnetic Relaxation in Spheromaks**

The first attempts to get energy from the controlled fusion of two light atoms nuclei date back to the beginning of the fifties of the last century. The crucial difficulty to achieve this goal is that particles need to have a large amount of thermal energy in order to have a significant chance of overcoming the Coulomb repulsion. At such high temperatures the atoms are fully ionized conforming a plasma. Such a hot plasma can not be in contact with solid walls because it will be rapidly cooled down. Two main methods have been developed to confine plasmas: the magnetic confinement and the inertial confinement. Here we are concerned with

Under certain conditions some magnetic configurations studied in the context of plasma confinement become unstable and undergo a process called magnetic (or plasma) relaxation. This process generally causes the system to evolve toward a self-organized state with lower magnetic energy and almost the same magnetic helicity. A key physical mechanism that operates during plasma relaxation is the localized reconnection of magnetic field lines. It was demonstrated that magnetic relaxation can be employed to form and sustain configurations

The theoretical description of magnetic relaxation is given in terms of a variational principle (Taylor, 1974). Despite the remarkable success of this theory to describe the final self-organized state toward which the plasma evolves, it does not provide any information on the dynamics of the plasma during relaxation. Since the process of relaxation always involves fluctuations that degrade plasma confinement it is very important to understand

The dynamics of the fluctuations induced during the relaxation process can be studied in the context the magnetohydrodynamic (MHD) model. In this Chapter, we will study the dynamics of the relaxation in kink unstable spheromak configurations. To that end we will

The rest of the Chapter is organized as follows. In Section 2 we give a general introduction to magnetic confinement of high temperature plasma which is the main motivation of this study. The physical background of this work is the MHD model which is presented in Section 3. In Section 4 we describe the magnetic relaxation theory and its relationship with

solve the time-dependent non-linear MHD equations in three spatial dimensions.

**1. Introduction**

their dynamics.

the magnetic confinement approach.

relevant to magnetic confinement research.

<sup>1</sup>*Laboratoire de Physique des Plasmas, Ecole Polytechnique, Palaiseau cedex* <sup>2</sup>*CONICET and Centro Atómico Bariloche (CNEA), San Carlos de Bariloche*


Trefethen, L. N. & Bau, D. III (1997). *Numerical Linear Algebra*, SIAM.

Trottenberg, U., Oosterlee, C. W. & Schüller, A. (2000). *Multigrid*, Academic Press, London.
