**2. Formation methods for FRC plasma**

An FRC is formed through a violent formation process dominated by self-organization. In this process, plasma pressure is built up, coinciding with the formation of reversed magnetic structure and plasma current drive within the diffusion time. The beta value of the FRC at the magnetic axis is infinite, and the volume-averaged beta value <> is nearly equal to one.

The plasma current density *j*(r), the confinement magnetic field *B*ze, the coil magnetic field *B*z0, and the formed plasma pressure *p*(*r*), satisfy the following condition:

$$\begin{aligned} I\_{\mathcal{A}} &= \int\_{0}^{r\_{\ast}} j\_{\mathcal{A}}(r) dr \ge 2B\_{z0}/\mu\_{0} \\ \nabla p(r) &= \frac{\partial p(r)}{\partial r} = j\_{\mathcal{A}}(r)B\_{z}(r) \end{aligned} \tag{1}$$

FRC plasma is traditionally formed by the field-reversed theta-pinch (FRTP) method (Armstrong et al., 1981). Since the 1990's, a variety of alternative formation methods have emerged (Ono, Y et al., 1993; Gerhardt et al. 2008; Slough & Miller, 2000; Knight & Jones, 1990; A. Hoffman et al., 2002; Guo et al., 2007; Logan et al., 1976; Davis, 1976; Greenly et al., 1986; Schamiloglu et. al., 1993). At the same time, the FRTP-based formation method has been significantly improved (Slough et al., 1989; Hoffman et al., 1993; Pietrzyk et al., 1987; Pierce et al., 1995; Guo et al., 2004, 2005; Asai et al., 2000; Binderbauer et al., 2010; Guo, et al., 2011). Recently initiated new formation methods include (1) counter-helicity spheromakmerging (CHSM) (Yamada et al., 1990; Ono et al., 1993; Gerhardt et al. 2008), (2) rotating magnetic field (RMF) (Slough & Miller, 2000; Knight & Jones, 1990; A. Hoffman et al., 2002; Guo et al., 2007), (3) field-reversed mirror configuration (FRM) driven by neutral beam injection (NBI) (Logan et al., 1976), relativistic electron beam (REB) (Davis, 1976), and intense light ion beam (ILIB) injections (Greenly et al., 1986; Schamiloglu et. al., 1993).

Improved FRTP methods have also introduced low inductive voltage and program formation by FRTP (Slough et al., 1989; Hoffman et al., 1993), the coaxial slow source (CSS) (Pietrzyk et al., 1987; Pierce et al., 1995), translation-trapping formation by FRTP (Guo et al., 2004, 2005; Asai et al., 2000), and collision FRC merging by FRTP (Binderbauer et al., 2010; Guo et al., 2011). Through these innovative new methods and improved FRTP methods, the 120 Topics in Magnetohydrodynamics

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

An FRC is formed through a violent formation process dominated by self-organization. In this process, plasma pressure is built up, coinciding with the formation of reversed magnetic structure and plasma current drive within the diffusion time. The beta value of the FRC at

The plasma current density *j*(r), the confinement magnetic field *B*ze, the coil magnetic field

0 0 <sup>0</sup> () 2 ( ) () () ()

*p r <sup>p</sup> <sup>r</sup> <sup>j</sup> rB r r*

FRC plasma is traditionally formed by the field-reversed theta-pinch (FRTP) method (Armstrong et al., 1981). Since the 1990's, a variety of alternative formation methods have emerged (Ono, Y et al., 1993; Gerhardt et al. 2008; Slough & Miller, 2000; Knight & Jones, 1990; A. Hoffman et al., 2002; Guo et al., 2007; Logan et al., 1976; Davis, 1976; Greenly et al., 1986; Schamiloglu et. al., 1993). At the same time, the FRTP-based formation method has been significantly improved (Slough et al., 1989; Hoffman et al., 1993; Pietrzyk et al., 1987; Pierce et al., 1995; Guo et al., 2004, 2005; Asai et al., 2000; Binderbauer et al., 2010; Guo, et al., 2011). Recently initiated new formation methods include (1) counter-helicity spheromakmerging (CHSM) (Yamada et al., 1990; Ono et al., 1993; Gerhardt et al. 2008), (2) rotating magnetic field (RMF) (Slough & Miller, 2000; Knight & Jones, 1990; A. Hoffman et al., 2002; Guo et al., 2007), (3) field-reversed mirror configuration (FRM) driven by neutral beam injection (NBI) (Logan et al., 1976), relativistic electron beam (REB) (Davis, 1976), and intense light ion beam (ILIB) injections (Greenly et al., 1986; Schamiloglu et. al., 1993).

Improved FRTP methods have also introduced low inductive voltage and program formation by FRTP (Slough et al., 1989; Hoffman et al., 1993), the coaxial slow source (CSS) (Pietrzyk et al., 1987; Pierce et al., 1995), translation-trapping formation by FRTP (Guo et al., 2004, 2005; Asai et al., 2000), and collision FRC merging by FRTP (Binderbauer et al., 2010; Guo et al., 2011). Through these innovative new methods and improved FRTP methods, the

*z*

*z*

> is nearly equal to one.

. (1)

theoretical studies.

**2. Formation methods for FRC plasma** 

the magnetic axis is infinite, and the volume-averaged beta value <

*B*z0, and the formed plasma pressure *p*(*r*), satisfy the following condition:

*sr*

*I j r dr B*

 

 

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 above methods are summarized in Table 2.

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 section.


Table 2. FRC Plasma parameters for various formation methods
