**7. References**

30 Will-be-set-by-IN-TECH

Then, the flux surface initially placed at *r*0, see Fig. 18 (c), reconnects and ends at *r*<sup>f</sup> , Fig. 18

which expresses the conservation of the area of the shaded regions of Fig. 18 (c) and (d). With

this prediction is much closer than the Kadomtsev's model to the actual final state there is still a significant difference. In what follows we will show that this difference is due to resistive

Relations (55) and (56) express the toroidal flux conservation assuming that it does not decay, i.e. the toroidal fluxes inside *r*<sup>K</sup> and *r*<sup>1</sup> do not change. However, as can be observed in Fig. 18 (a), the toroidal magnetic field is visibly reduced due to resistivity. One way to take into account this resistive decay is to change the reference radius with which we make the

contains at *t* = 0 the same amount of toroidal flux that is contained inside *r*<sup>1</sup> at *t* = 100. If we now compute the reconnected helical flux using Eq. (56) but changing *r*<sup>1</sup> by *r<sup>η</sup>* we obtain

*<sup>h</sup>* , shown by the dashed line of Fig. 18 (b). The agreement with the actual final helical flux is very good and this suggests that the modified model indeed captures the basic physics of the

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

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

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

situation they operate around the kink instability threshold.

*<sup>h</sup>* and Eq. (56) it is possible to compute the reconnected helical flux

*<sup>h</sup>* given by Eq. (56). In particular, we define *r<sup>η</sup>* as the radius of the circle that

*<sup>h</sup>* . This is shown by the green curve of Fig. 18 (b). While

<sup>0</sup> (56)

*r* 2 <sup>f</sup> = *r* 2 <sup>1</sup> − *r* 2

(d), in such a way that

the initial helical flux *ψ*<sup>0</sup>

dissipation.

*ψη*

construction of *ψ*<sup>m</sup>

reconnection process.

**6. Conclusions**

configurations.

predicted by the modified model *ψ*<sup>m</sup>


**1. Introduction** 

**1.1 Field-reversed configuration (FRC)** 

Pulsed High Density FRC Experiments (PHD) (Slough, 2007b).

field of the toroidal plasma current (*I* : *I*>2*Bz*0/

<sup>0</sup> 2 *ze*

as the distance between the two X-points (Armstrong et al., 1981).

the beta value <sup>2</sup>

*p B*

confinement magnetic field pressure (*Bz*<sup>e</sup>

region which acts as a natural diverter.

**5** 

*Nihon University* 

*Japan* 

**MHD Activity in an Extremely** 

A field-reversed configuration (FRC) plasma is extremely high beta confinement system and the only magnetic confinement system with almost 100% of a beta value (Tuszewski, 1988; Steinhauer, 2011). The plasma is confined by the only poloidal magnetic field generated by a self-plasma current. The FRC has several potentials for a fusion energy system. As the one of the candidate for an advanced fusion reactor, for example, D-3He fusion (Momota, 1992), FRC plasma is attractive. Recently, this plasma also has an attraction as target plasmas for an innovative fusion system, Magnetized Target Fusion (MTF) (Taccetti, 2003), Colliding and merging two high- compact toroid (Guo, 2011; Binderbauer, 2010; Slough, 2007a) and

The plasma belongs to a compact toroid system. Here, 'compact' denotes a simply connected geometry, i.e., the absence of a central column. The system consists of a toroidal magnetic confinement system with little or no toroidal magnetic field. The typical magnetic structure of the FRC plasma is shown in Fig. 1. The poloidal confinement field (*Bz*e) consists of the externally applied magnetic field of an external coil (*Bz*0), and the self-generated magnetic

symmetric magnetized plasma, a plasma liner and a simply connected configuration. Then,

field line region in which the high temperature plasma is confined, and an open field line

A scrape-off layer is formed in the open field line region. Two singularities in the magnetic field, i.e., X-points, are formed at the intersections of the symmetric device axis with the separatrix (*B*ze = 0). A null field surface (*B*z = 0) is also formed in the closed separatrix region. The radius (*R*) of the null surface at midplane (minor radius) is 2 *<sup>s</sup> R r* (*r*s: radius of the separatrix at midplane) in the pressure equilibrium state. The separatrix length *l*s is defined

An FRC has three essential geometrical plasma parameters (*S*\*: radial size parameter; *E*: separatrix elongation; and *X*s: normalized separatrix radius), which are related to the physical

), which is the ratio of confined plasma pressure (*p*) to the

2/20), is extremely high. The system has a closed

**High-Beta Compact Toroid** 

Tomohiko Asai and Tsutomu Takahashi

0). The FRC consists of an axially

