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

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 inductive theta-discharge (-discharge) generates a pre-ionized plasma of the working gas, 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 <sup>z</sup>* ), and 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

MHD Activity in an Extremely High-Beta Compact Toroid 123

In the Tokyo Spheromak 3 (TS-3) experiment, merging spheromaks are formed by z discharge (Ono et al., 1993, 1997). The Swarthmore Spheromak Experiment (SSX) utilizes the coaxial plasma gun method (Cohen et al., 2003). The Tokyo Sheromak 4 (TS-4) and the Magnetic Reconnection Experiment (MRX) employ the flux core method (Gerhardt et al., 2006). The TS-3 device is illustrated in Fig. 4 (Ono et al., 1997). The time evolutions of the magnetic surface of the poloidal and toroidal fields, the toroidal flow, and the ion

Fig. 4. (a) TS-3 merging device and 2D contours of poloidal flux surface and toroidal magnetic field on the R-Z plane; radial profiles of (b) ion toroidal velocity *V*, and (c) ion temperature (*T*i), on the midplane, during the counter helicity merging of two spheromaks with equal but opposing *B*t. The red and blue colors indicate the positive and negative

The global deformation of the internal structure of an FRC, and its time evolution, were investigated by means of an optical diagnostic system (Takahashi et al., 2004), combined with tomographic reconstruction, in the NUCTE facility (Asai et al., 2006). Fourier image transform was applied to the reconstructed image, and the correlation of global modes with *n* = 1 and 2 was investigated. The typical plasma parameters are separatrix radius of 0.06 m, separatrix length of 0.8 m, electron density of 2.5 x 10-20 m-3, total temperature of 270 eV, particle confinement time of 80 s, and *s* -value of 1.9. Figure 5 shows the time evolution of the 2D emissivity profile of bremsstrahlung of 550 nm. Here, the intensity of bremsstrahlung

cross-sectional structure at each phase indicated in the time history of line integrated electron density, measured along the y-axis (Fig. 5 (a)). Figure 5 (b) shows the emissivity structure 1 s after application of the main compression field. We can see that the radial compression has started at the chamber wall. The following radial compression phase is shown in Fig. 5 (c). The circular boundary of the bright area indicates azimuthally uniform compression. After the formation phase, the equilibrium phase, with a circular crosssectional structure, lasts approximately 20 s (Fig. 5 (d)). The oscillation observed in the

0.5. Figure 5 (b) - (g) shows a reconstructed tomographic image of the

temperature, are also shown in the figure.

amplitudes of *Bt*.

is proportional to *n*<sup>e</sup>

**3. MHD behavior of FRCs** 

**3.1 MHD behavior of prolate FRCs** 

2/*T*<sup>e</sup>

**3.1.1 General picture of prolate FRC MHD behavior** 

estimated by our improved excluded flux method, and the radial profile of bremsstrahlung (proportional to 2 0.5 *n T e e* ). An FRC plasma with a separatrix radius of 0.055 m and a length of 0.8 m is formed at about 20 s, and is isolated from the discharge tube (*r*t = 0.13 m).

Fig. 3. (a) Time evolution of equi-magnetic surface of FRC at formation phase, and (b) pressure profile at equilibrium phase.

#### **2.2 Counter-helicoty spheromak-merging method (CHSM)**

A spheromak also belongs to the family of compact toroids. The plasma has a toroidal field nearly equal to the poloidal field. The spheromak is formed by various means, such as a simultaneous axial and discharge (*z*- discharge), a coaxial plasma gun, or a toroidal flux core containing both toroidal and poloidal winding (Ono et al., 1993; Gerhardt et al., 2006; Yamada et al., 1990; Ono et al., 1993; Gerhardt et al., 2008).

Two spheromaks, with a common geometric axis and opposite helicity (opposite toroidal fields) of equal value, are separately formed, and merge to form the FRC. If the respective helicity values of the spheromaks differ significantly, the merged plasma remains a spheromak (Ono et al., 1993; Yamada et al., 1990). This formation method naturally forms an oblate FRC, in contrast to the prolate FRC formed by the FRTP method.

122 Topics in Magnetohydrodynamics

estimated by our improved excluded flux method, and the radial profile of bremsstrahlung (proportional to 2 0.5 *n T e e* ). An FRC plasma with a separatrix radius of 0.055 m and a length of

(a) (b) Fig. 2. (a) Schematic diagram of NUCTE-III and (b) a typical waveform of magnetic field on

(a) (b) Fig. 3. (a) Time evolution of equi-magnetic surface of FRC at formation phase, and

A spheromak also belongs to the family of compact toroids. The plasma has a toroidal field nearly equal to the poloidal field. The spheromak is formed by various means, such as a

core containing both toroidal and poloidal winding (Ono et al., 1993; Gerhardt et al., 2006;

Two spheromaks, with a common geometric axis and opposite helicity (opposite toroidal fields) of equal value, are separately formed, and merge to form the FRC. If the respective helicity values of the spheromaks differ significantly, the merged plasma remains a spheromak (Ono et al., 1993; Yamada et al., 1990). This formation method naturally forms an

discharge), a coaxial plasma gun, or a toroidal flux

0.8 m is formed at about 20 s, and is isolated from the discharge tube (*r*t = 0.13 m).

FRTP method.method.

(b) pressure profile at equilibrium phase.

simultaneous axial and

**2.2 Counter-helicoty spheromak-merging method (CHSM)** 

discharge (*z*-

oblate FRC, in contrast to the prolate FRC formed by the FRTP method.

Yamada et al., 1990; Ono et al., 1993; Gerhardt et al., 2008).

In the Tokyo Spheromak 3 (TS-3) experiment, merging spheromaks are formed by z discharge (Ono et al., 1993, 1997). The Swarthmore Spheromak Experiment (SSX) utilizes the coaxial plasma gun method (Cohen et al., 2003). The Tokyo Sheromak 4 (TS-4) and the Magnetic Reconnection Experiment (MRX) employ the flux core method (Gerhardt et al., 2006). The TS-3 device is illustrated in Fig. 4 (Ono et al., 1997). The time evolutions of the magnetic surface of the poloidal and toroidal fields, the toroidal flow, and the ion temperature, are also shown in the figure.

Fig. 4. (a) TS-3 merging device and 2D contours of poloidal flux surface and toroidal magnetic field on the R-Z plane; radial profiles of (b) ion toroidal velocity *V*, and (c) ion temperature (*T*i), on the midplane, during the counter helicity merging of two spheromaks with equal but opposing *B*t. The red and blue colors indicate the positive and negative amplitudes of *Bt*.
