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

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

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 field of the toroidal plasma current (*I* : *I*>2*Bz*0/0). The FRC consists of an axially symmetric magnetized plasma, a plasma liner and a simply connected configuration. Then, the beta value <sup>2</sup> <sup>0</sup> 2 *ze p B* ), which is the ratio of confined plasma pressure (*p*) to the confinement magnetic field pressure (*Bz*<sup>e</sup> 2/20), is extremely high. The system has a closed field line region in which the high temperature plasma is confined, and an open field line region which acts as a natural diverter.

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

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

MHD Activity in an Extremely High-Beta Compact Toroid 119

rotation has not yet been completely understood. But, given rotation, the condition for instability has been fairly well understood. The stability threshold was expressed in terms of a parameter=/Di, where andDi are rotation angular frequency and ion diamagnetic drift frequency. The threshold for n=1 mode was =1 and in n=2 mode could also grow for a greater than 1.2 -1.4, for zero bias limit (Freidberg & Pearlstein, 1978). For FRC plasma, a

**Character Mode Name Experimental** 

**(Ballooning) No** 

**(Tilt) Seldom** 

**Axial Tearing Yes (always?)** 

**Co-interchange**

**Observation** 

**(n=1,2,3)** 

**(high-** *<sup>s</sup>* **)** 

**Mode** 

**Radial**

**0 1 Axial Roman candle No 1 0 Sausage Shift Interchange Often, occasional** 

**>1 1 Axial Tilt No** 

**>1 0 Flute Interchange Often, always** 

**1 1 Radial Wobble Yes (occasional) 2 1 Radial n=2 Yes (always) >2 1 Radial n>2 Yes (often, high-** *<sup>s</sup>* **)** 

These two global ideal modes driven by rotation—the *n* = 1 wobble, of little concern since it saturates at low amplitude, and the *n* = 2 rotational instability, which destroys most FRCs have been regularly observed experimentally. These rotational modes (*n* = 1, 2) have been controlled by applying a straight or helical multi-pole field (Ohi et al., 1983; Shimamura & Nogi, 1986; Fujimoto et al., 2002). Higher-order (*n* > 2) rotational modes have often been observed in large- *s* FRC experiments, with *s* in the region of 3 < *s* < 8 (Slough & Hoffman, 1993). FRC plasma with a higher *s* value behaves as a MHD plasma and with low *s* one becomes more kinematically. According to the several theoretical works, FRC plasmas have been predicted to be unstable because of a bad curvature of a closed confinement field. Various local and global non-rotating ideal MHD modes are listed in Table 1, and it is worth noting that stable FRC plasma is impregnable against these low *n*modes. The m/n=1/1 tilt mode instability is thought to be most dangerous. The stability of prolate and oblate FRC plasma has been investigated experimentally on several FRTP

**1 1 Radial Sideways shift** 

Table 1. FRC Stability: MHD Theory versus Experimental Observation

**0 Interchange Only lowest order** 

similar threshold is ~1.3-1.5 (Seyler, 1979).

**1. Local Ideal Mode** 

**2. Global Mode No rotating** 

**Rotating** 

**3. Resistive Mode** 

**m (poloidal mode)** 

**1,2 Axial or** 

**0 2 Radial and** 

properties of the FRC plasma. *S*\*=*r*s/(*c*/pi) is defined as the ratio of the separatrix radius to the ion skin depth (*c*/pi). Here, *r*s, *c* and pi are the separatrix radius, speed of light and ion plasma frequency, respectively. Two other radial size parameters, 2 *S r s i* and *sr s i <sup>R</sup> s rdr r* , are sometimes used, where i and io are the local ion Larmor-radius and the reference ion Larmor-radius, respectively, based on the external magnetic field *B*ze. These parameters indicate the importance of the two-fluid (ion and electron fluid) and finite-Larmorradius effects; for example, \* *S S* 1.3 and 5 *<sup>s</sup> s XS* under under *T*i ~ 2*T*e (*T*i: ion temperature; *T*e: electron temperature) (Steinhauer, 2011). *E* = *l*s/2*r*s is defined as the ratio of the separatrix radius to the diameter, and indicates the elongation of the separatrix, which is different from that of a tokamak system. It is known that this elongation affects the global plasma stability of an FRC. Oblate and prolate FRC plasmas are usually categorized as 0 < *E* < 1 and *E* > 1 FRCs, respectively. *X*s = *r*s/*r*w is defined as the ratio of the separatrix to the confinement coil radius. This normalized radius has a strong relation to the poloidal flux (p) of an elongated (prolate) FRC plasma, and to the FRC confinement time scaling. The average beta value can be written as 1 0.5*xs* <sup>2</sup> (Armstrong et al., 1981).

Fig. 1. A field-reversed configuration plasma formed by the field-reversed theta-pinch (FRTP) method.

The FRC topology is similar to an elongated, low-aspect-ratio, toroidal version of the Zpinch, as shown in Fig. 1. Since the FRC plasma has no toroidal field, and no center conductor, theoretical studies predict that FRC plasma is unstable with respect to an MHD mode with low toroidal mode number. The principal instabilities of the FRC, predicted by magnetohydrodynamics (MHD) theory, are listed in Table 1 (Tuszewski, 1988; Slough & Hoffman, 1993). Here, the tokomak nomenclature has been adopted, with *n* and *m* being the toroial and poloidal mode numbers, respectively.

The plasma current of FRC, at just after formation, is primarily carried by electrons. On the other hand, ions are approximately at rest. However, in most of the FRC experiments, ions soon begin to rotate to a diamagnetic direction. The rotation speed often reaches to a supersonic level. Instabilities driven by the Rotational mode is appeared. The origin of 118 Topics in Magnetohydrodynamics

pi) is defined as the ratio of the separatrix radius to

pi are the separatrix radius, speed of light and ion

io are the local ion Larmor-radius and the

p) of an elongated (prolate)

can be

and

plasma frequency, respectively. Two other radial size parameters, 2 *S r s i*

i and 

reference ion Larmor-radius, respectively, based on the external magnetic field *B*ze. These parameters indicate the importance of the two-fluid (ion and electron fluid) and finite-Larmorradius effects; for example, \* *S S* 1.3 and 5 *<sup>s</sup> s XS* under under *T*i ~ 2*T*e (*T*i: ion temperature; *T*e: electron temperature) (Steinhauer, 2011). *E* = *l*s/2*r*s is defined as the ratio of the separatrix radius to the diameter, and indicates the elongation of the separatrix, which is different from that of a tokamak system. It is known that this elongation affects the global plasma stability of an FRC. Oblate and prolate FRC plasmas are usually categorized as 0 < *E* < 1 and *E* > 1 FRCs, respectively. *X*s = *r*s/*r*w is defined as the ratio of the separatrix to the confinement coil radius.

pi). Here, *r*s, *c* and

This normalized radius has a strong relation to the poloidal flux (

FRC plasma, and to the FRC confinement time scaling. The average beta value

Fig. 1. A field-reversed configuration plasma formed by the field-reversed theta-pinch

The FRC topology is similar to an elongated, low-aspect-ratio, toroidal version of the Zpinch, as shown in Fig. 1. Since the FRC plasma has no toroidal field, and no center conductor, theoretical studies predict that FRC plasma is unstable with respect to an MHD mode with low toroidal mode number. The principal instabilities of the FRC, predicted by magnetohydrodynamics (MHD) theory, are listed in Table 1 (Tuszewski, 1988; Slough & Hoffman, 1993). Here, the tokomak nomenclature has been adopted, with *n* and *m* being the

The plasma current of FRC, at just after formation, is primarily carried by electrons. On the other hand, ions are approximately at rest. However, in most of the FRC experiments, ions soon begin to rotate to a diamagnetic direction. The rotation speed often reaches to a supersonic level. Instabilities driven by the Rotational mode is appeared. The origin of

<sup>2</sup> (Armstrong et al., 1981).

properties of the FRC plasma. *S*\*=*r*s/(*c*/

, are sometimes used, where

the ion skin depth (*c*/

*sr s i <sup>R</sup> s rdr r*

written as

(FRTP) method.

toroial and poloidal mode numbers, respectively.

1 0.5*xs*

rotation has not yet been completely understood. But, given rotation, the condition for instability has been fairly well understood. The stability threshold was expressed in terms of a parameter=/Di, where andDi are rotation angular frequency and ion diamagnetic drift frequency. The threshold for n=1 mode was =1 and in n=2 mode could also grow for a greater than 1.2 -1.4, for zero bias limit (Freidberg & Pearlstein, 1978). For FRC plasma, a similar threshold is ~1.3-1.5 (Seyler, 1979).


Table 1. FRC Stability: MHD Theory versus Experimental Observation

These two global ideal modes driven by rotation—the *n* = 1 wobble, of little concern since it saturates at low amplitude, and the *n* = 2 rotational instability, which destroys most FRCs have been regularly observed experimentally. These rotational modes (*n* = 1, 2) have been controlled by applying a straight or helical multi-pole field (Ohi et al., 1983; Shimamura & Nogi, 1986; Fujimoto et al., 2002). Higher-order (*n* > 2) rotational modes have often been observed in large- *s* FRC experiments, with *s* in the region of 3 < *s* < 8 (Slough & Hoffman, 1993). FRC plasma with a higher *s* value behaves as a MHD plasma and with low *s* one becomes more kinematically. According to the several theoretical works, FRC plasmas have been predicted to be unstable because of a bad curvature of a closed confinement field. Various local and global non-rotating ideal MHD modes are listed in Table 1, and it is worth noting that stable FRC plasma is impregnable against these low *n*modes. The m/n=1/1 tilt mode instability is thought to be most dangerous. The stability of prolate and oblate FRC plasma has been investigated experimentally on several FRTP

MHD Activity in an Extremely High-Beta Compact Toroid 121

FRC lifetime has been prolonged to the order of several ms, and the confinement properties have also been improved [34]. The plasma parameters and lifetimes of FRCs formed by the

Experimental and theoretical studies of FRC stability have mainly focused on elongated (*E* > 1) and oblate (0 < *E* < 1) FRC plasmas, formed by the FRTP (Slough & Hoffman, (1993), Fujimoto et al., 2002; Tuszewski, et al., 1990; Tuszewski et al., 1991; Kumashiro et al.,1993; Asai et al., 2006) and CHSW methods (Yamada et al., 1990; Ono et al., 1993; Gerhardt et al. 2008), respectively. Details of the two methods are introduced in the next

> np ( 10<sup>20</sup> *m*3 )

0.45 0.007-0.17 0.02-

0.5 2 0.02-

0.5 1 0.02-

The schematic of a typical field-reversed theta-pinch device, NUCTE-III (Nihon University Compact Torus Experiment 3), is shown in Fig. 2 (Asai et al., 2006). A transparent fused silica glass discharge tube lies in a cylindrical one-turn coil. The tube is filled with a working gas (usually hydrogen or deuterium gas) by static filling or gas puffing, and then a *z*-discharge or

which is embedded in the reversed-bias field of 0.03-0.08 T, produced by 2 mF of the bias bank. A main bank of 67.5 F rapidly reverses the magnetic field in the discharge tube (rising time of 4 s). The circuit of the main bank is crowbarred on reaching the maximum current, and resistively decays with a decay time of 120 s. A thin current sheet is initially formed around the inner wall of the discharge tube by an inductive electric field ( 0.5 *E rdB dt*

shields the plasma from the rising forward field. The rising field works as a 'magnetic piston' to implode the plasma radially. At both ends of the coil, the reversed-bias field is reconnected with the forward field, and a closed magnetic structure is created. The tension formed due to the magnetic curvature produces a shock-like axial contraction. Then the radial and axial dynamics rapidly dissipate within about 20 s, and the FRC plasma reaches an equilibrium/quiescent phase. Figure 3 shows the separatrix and the equi-magnetic surface

Ti (keV)

0.24 5-500 0.1-15 0.4-12 0.03-

p (mWb)

0.5 0.4-5 0.2-0.5 1.5 0.2-0.4 5-8 0.8

0.4 1 0.5-0.6 12 ~1.0 5 1

0.2 <sup>10</sup> 0.075-

0.2 2-3 0.35-


life

0.2 1-10 1-2.5 1-3 -

0.1

0.6

(ms) <sup>E</sup>*s*

0.4 2.5-10 0.5-

0.35- 0.65 1-3

0.35- 0.65 1-3

*<sup>z</sup>* ), and

5

above methods are summarized in Table 2.

(T)

0.01- 0.06

0.025

**2.1 Field-reversed theta-pinch method (FRTP)** 

rs (m)

0.12-

0.03-

Table 2. FRC Plasma parameters for various formation methods

section.

Formation method Be

FRTP+translation-

FRTP+collision-

Spheromak-

trapping

FRTP 0.3-2 0.05-

merging 1.0-1.1 0.3-

Spheromak-merging 0.2-0.3 0.4-

merging+CS 0.2-0.3 0.4-

RMF 0.006-

inductive theta-discharge (

devices (FRX-C/LSM (Tuszewski et al., 1990, 1991), LSX, (Slough & Hoffman, 1993), NUCTE-III (Kumashiro et al., 1993; Asai et al., 2006; Ikeyama et al., 2008), etc.), and on spheromak-merging facilities such as the TS-3 (Ono et al., 1993) and MRX (Gerhardt et al., 2006). In addition, the stability of prolate and oblate FRCs has been analyzed by means of visible and x-ray photography with an end-on camera (Slough & Hoffman, 1993; Tuszewski et al., 1991), computer tomography reconstruction of the visible emission profile (Asai et al., 2006), as well as mode analysis of the external *B*-magnetic probe array (Mirnov coil array) (Slough & Hoffman, 1993; Tuszewski et al., 1990, 1991; Kumashiro et al., 1993; Ikeyama et al., 2008) and the internal magnetic probe array (Ono et al., 1993; Gerhardt et al., 2006).

In the following sections, the formation methods for FRC plasma (Section 2) and the stability of FRC plasma (Section 3) are described based on these experimental results and some theoretical studies.
