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

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 is proportional to *n*<sup>e</sup> 2/*T*<sup>e</sup> 0.5. Figure 5 (b) - (g) shows a reconstructed tomographic image of the 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

MHD Activity in an Extremely High-Beta Compact Toroid 125

rotational phase from that of the oval plasma boundary, is seen. More specically, this *n* = 1 shift motion and the rotational torque are possible sources of the *n* = 2 mode deformation. This result suggests that the suppression of this shift motion in the formation and

These two global ideal modes driven by rotation—the *n* = 1 wobble and the *n* = 2 rotational instability—have been regularly observed in experiments. These rotational modes have been controlled by applying a straight or helical multipole field (Ohi et al., 1983; Shimamura & Nogi, 1986). The stability criterion (*B*sc (*n* = 2)) of the straight multipole field of an m-pole for the *n* = 2 rotational instability has already been developed, on the basis of MHD equations,

> <sup>0</sup> <sup>1</sup> ( 2) 2 1 *B n sc sr*

(which is twice that of the angular velocity of the plasma column), the mass density of the plasma column, and the magnetic permeability of free space. On the other hand, the stability criterion for the *n* = 1 wobble motion (*B*sc (*n* = 1)) has been derived from the experimental

> <sup>1</sup> ( 1) 2( 1) *B n sc s n r f m*

reduction coefficient which is defined by experiments with different pole numbers and is about 0.3 for the NUCTE FRCs. The ratio of *B*sc (*n* = 1)/*B*sc (*n* = 2) suggests that the amplitude of *n* = 1 mode motion can always be maintained at a low level by the application of *B*sc (*n* =

*m* 

0

n=1 is the angular velocity of the *n* = 1 wobble motion, and *f* is an amplitude

0 are, respectively, the rotational angular velocity of the *n* = 2 deformation

1

, (2)

, (3)

equilibrium phase might impede the growth of *n* = 2 mode deformation.

Fig. 6. Time evolution of (a) toroidal mode intensity, and (b) phase.

**3.1.2 Stability of prolate FRCs with respect to rotational mode** 

by Ishimura (Ishimura, 1984) as

results of NUCTE-III (Fujimoto et al., 2002) as

where *, r*, and

where

2), provided

n=1/*f*<n=2.

latter phase is caused by rotational instability with toroidal mode number of *n* = 2. At the rst stage of deformation, illustrated in Fig. 5 (e), the reconstructed cross section is deformed into an oval shape.

Fig. 5. (a) Time evolution of line integrated electron density, and (b) - (g) reconstructed cross-sectional structure of an FRC.

Generally, this deformation is called 'elliptical deformation.' However, the reconstructed image indicates an internal shift (*n* = 1) in an oval separatrix. In this rotational instability phase, deformation grows due to centrifugal distortion. In the early stage of the growth of instability, the structure of the FRC has a dumbbell-like shape, as shown in Fig. 5 (f). The internal structure at the final stage of the discharge is shown in Fig. 5 (g). The distribution of emissivity shows two clear peaks which orbit around the separatrix axis like binary stars. The analyzed time evolution of mode intensity and phase by the Fourier image transform are shown in Fig. 6. This result indicates that the *n* = 1 shift motion of the plasma column (wobble motion) increases prior to the growth of the *n* = 2 mode. The amplitude of *n* = 1 increases in the equilibrium phase of 20 – 30 s. It is thus apparent that the dominant mode changes to *n* = 2 after a modest peak of *n* = 1. In the tomographic image of this transition region of *n* = 1 to 2 (Fig. 5 (e)), an internal shift of the bright area, which has a different 124 Topics in Magnetohydrodynamics

latter phase is caused by rotational instability with toroidal mode number of *n* = 2. At the rst stage of deformation, illustrated in Fig. 5 (e), the reconstructed cross section is deformed

Fig. 5. (a) Time evolution of line integrated electron density, and (b) - (g) reconstructed

Generally, this deformation is called 'elliptical deformation.' However, the reconstructed image indicates an internal shift (*n* = 1) in an oval separatrix. In this rotational instability phase, deformation grows due to centrifugal distortion. In the early stage of the growth of instability, the structure of the FRC has a dumbbell-like shape, as shown in Fig. 5 (f). The internal structure at the final stage of the discharge is shown in Fig. 5 (g). The distribution of emissivity shows two clear peaks which orbit around the separatrix axis like binary stars. The analyzed time evolution of mode intensity and phase by the Fourier image transform are shown in Fig. 6. This result indicates that the *n* = 1 shift motion of the plasma column (wobble motion) increases prior to the growth of the *n* = 2 mode. The amplitude of *n* = 1 increases in the equilibrium phase of 20 – 30 s. It is thus apparent that the dominant mode changes to *n* = 2 after a modest peak of *n* = 1. In the tomographic image of this transition region of *n* = 1 to 2 (Fig. 5 (e)), an internal shift of the bright area, which has a different

into an oval shape.

cross-sectional structure of an FRC.

rotational phase from that of the oval plasma boundary, is seen. More specically, this *n* = 1 shift motion and the rotational torque are possible sources of the *n* = 2 mode deformation. This result suggests that the suppression of this shift motion in the formation and equilibrium phase might impede the growth of *n* = 2 mode deformation.

Fig. 6. Time evolution of (a) toroidal mode intensity, and (b) phase.

#### **3.1.2 Stability of prolate FRCs with respect to rotational mode**

These two global ideal modes driven by rotation—the *n* = 1 wobble and the *n* = 2 rotational instability—have been regularly observed in experiments. These rotational modes have been controlled by applying a straight or helical multipole field (Ohi et al., 1983; Shimamura & Nogi, 1986). The stability criterion (*B*sc (*n* = 2)) of the straight multipole field of an m-pole for the *n* = 2 rotational instability has already been developed, on the basis of MHD equations, by Ishimura (Ishimura, 1984) as

$$B\_{sc}(n=2) = \frac{1}{2} \sqrt{\frac{\mu\_0 \rho}{m-1}} r\_s \left| \Omega \right| \,\tag{2}$$

where *, r*, and 0 are, respectively, the rotational angular velocity of the *n* = 2 deformation (which is twice that of the angular velocity of the plasma column), the mass density of the plasma column, and the magnetic permeability of free space. On the other hand, the stability criterion for the *n* = 1 wobble motion (*B*sc (*n* = 1)) has been derived from the experimental results of NUCTE-III (Fujimoto et al., 2002) as

0 1 <sup>1</sup> ( 1) 2( 1) *B n sc s n r f m* , (3)

where n=1 is the angular velocity of the *n* = 1 wobble motion, and *f* is an amplitude reduction coefficient which is defined by experiments with different pole numbers and is about 0.3 for the NUCTE FRCs. The ratio of *B*sc (*n* = 1)/*B*sc (*n* = 2) suggests that the amplitude of *n* = 1 mode motion can always be maintained at a low level by the application of *B*sc (*n* = 2), provided n=1/*f*<n=2.

MHD Activity in an Extremely High-Beta Compact Toroid 127

In the FRX-C/LSM, which is a conventional FRTP device using non-tearing reconnection, the stability of FRC plasmas with 1 < *s* < 3.5 and 3 < *E* < 9 (highly kinetic and elongated) was investigated using a Mirnov loop array of 64 external *B* pick-up loops and a soft X-ray end-on camera (Tuszewski et al., 1991). Tilt-like asymmetries (the *n* = 1 axial odd component of *B*) were found, which strongly correlates with FRC confinement. Tilt and other instabilities also appeared with an increase in the bias magnetic field and/or filling pressure (i.e., higher *s* value). An increase in the bias field and filling pressure also coincidently causes strong axial dynamics, which triggers confinement degradation. These experimental results suggest that the tilt-stability condition for kinetic and elongated FRCs is in the range of *s*/*e* < 0.2 - 0.3 (*S*\*/*E*  < 3) (Fig. 7), and becomes *s*/*e* ~ 1 for MHD-like FRC (Fig. 8). Strong axial dynamics during FRC formation results in lesser elongation of the FRC. Therefore it eventually fosters the growth of tilt instability. For an FRC with *s*/*e* ~ 1, the tilt instability grows from small initial perturbations, and becomes large enough to cause major plasma disruptions after 10 - 20 s, which is 3 – 4 times longer than the growth time of instability. In the case of low filling pressures, higher order (*n* = 2 and 3) axially odd asymmetries are also observed. However, the amplitude of these modes is much less than that of the *n* = 1 tilt components. In the case of

higher filling pressures, higher order modes appear earlier and grow vigorously.

summarized in Table 3.

In the LSX, using an improved FRTP formation method with a programmed formation scheme, the correlation between plasma distortions and the confinement properties was investigated (Slough & Hoffman, 1993). A *B* probe array and an end-on soft X-ray camera were employed to determine separatrix movement, which might indicate the existence of lower order modes, such as tilt mode. Experiments were conducted over a large range of *s* (1 < *s* < 8) and no correlation was observed between the quality of confinement and the *B* signal. In fact, the confinement quality correlates more with the shape of the equilibrium radial profile than with *s* . Details of the experimental results are

Fig. 7. Toroidal Fourier analysis of the *B* for good confinement. The Fourier amplitudes and phases are shown as functions of time for (a) even and (b) odd components. The top trace of

(a) is a line-integrated electron density and (b) diamagnetism.

**3.1.3 Stability of prolate FRCs with respect to tilt mode** 

The experimental results of the FIX (FRC injection experiment) with neutral beam injection into FRC plasma formed by FRTP, indicate that the global *n* = 1 mode motion was controlled by neutral beam injection (Asai, et al., 2003). The neutral beam was injected obliquely to the axial direction due to the limited poloidal flux. The stabilization effects of ion rings confined by mirror fields at each end have been noted. Improved confinement properties (e.g., prolonged decay time of plasma volume and increased electron temperature) have also been observed.

TCS (Translation, Confinement and Sustainment) experiments at Washington University indicate that the *n* = 2 mode rotational instability can be controlled by the self-generated toroidal field, which converts from a toroidal into a poloidal field during the capture process of translated FRCs (Guo et al., 2004, 2005). The stabilization effect of the toroidal fields was investigated using the modified energy principle with the magnetic shear effect (Milroy et al., 2008). The following analytic stability criterion was derived as

<sup>0</sup> 0.66 *B r SC <sup>s</sup>* . (4)

This stability criterion is very similar to the one for the multipole field. This formula indicates that a relatively modest toroidal field, which is about 12% in comparison to the external poloidal field, can stabilize the FRC in the case of TCS experiments.

To supply the modest toroidal field to the FRC plasma, a magnetized coaxial plasma gun (MCPG) has been employed in the NUCTE facility (Asai et al., 2010). The MCPG generates a spheromak-like plasmoid which can then travel axially to merge with a pre-existing FRC. Since the MCPG is mounted on-axis and generates a significant helicity, it provides the FRC-relevant version of coaxial helicity injection (CHI) that has been applied to both spheromaks and spherical tokamaks. When CHI is applied, the onset of elliptical deformation of the FRC cross section is delayed until 45 - 50 s from FRC formation, compared to an onset time of 25 s without CHI. Besides delaying instability, MCPG application reduces the toroidal rotation frequency from 67 kHz to 41 kHz. Moreover, the flux decay time is extended from 57 to 67 s. These changes occur despite the quite modest flux content of the plasmoid: ~ 0.05 mWb of poloidal and 0.01 mWb of toroidal flux, compared with the 0.4 mWb of poloidal flux in the pre-formed FRC. The MCPG introduces a different stabilization mechanism, which may be the same as that observed in translated FRCs, because of the existence of modest toroidal flux. The observed global stabilization and confinement improvements suggest that the MCPG can actively control the rotational instability.

 In STX experiments, stabilization effects due to RMF have been observed. The stabilization effects can be attributed to two-fluid effects produced by rotational and ponderomotive forces. In TCS experiments, the stabilization effects of RMF on the *n* = 2 interchange mode and rotational mode have been reported (Guo et al., 2005). The stability criterion of RMF field strength (*B*) was derived as

<sup>0</sup> 1.14 *B r <sup>s</sup>* . (5)

The stability diagram for FRC formed and sustained by the RMF at a different frequency is indicated in Fig. 6 of reference of Guo et al., 2005.

126 Topics in Magnetohydrodynamics

The experimental results of the FIX (FRC injection experiment) with neutral beam injection into FRC plasma formed by FRTP, indicate that the global *n* = 1 mode motion was controlled by neutral beam injection (Asai, et al., 2003). The neutral beam was injected obliquely to the axial direction due to the limited poloidal flux. The stabilization effects of ion rings confined by mirror fields at each end have been noted. Improved confinement properties (e.g., prolonged decay time of plasma volume and increased electron temperature) have also been

TCS (Translation, Confinement and Sustainment) experiments at Washington University indicate that the *n* = 2 mode rotational instability can be controlled by the self-generated toroidal field, which converts from a toroidal into a poloidal field during the capture process of translated FRCs (Guo et al., 2004, 2005). The stabilization effect of the toroidal fields was investigated using the modified energy principle with the magnetic shear effect (Milroy et

> <sup>0</sup> 0.66 *B r SC*

This stability criterion is very similar to the one for the multipole field. This formula indicates that a relatively modest toroidal field, which is about 12% in comparison to the

To supply the modest toroidal field to the FRC plasma, a magnetized coaxial plasma gun (MCPG) has been employed in the NUCTE facility (Asai et al., 2010). The MCPG generates a spheromak-like plasmoid which can then travel axially to merge with a pre-existing FRC. Since the MCPG is mounted on-axis and generates a significant helicity, it provides the FRC-relevant version of coaxial helicity injection (CHI) that has been applied to both spheromaks and spherical tokamaks. When CHI is applied, the onset of elliptical deformation of the FRC cross section is delayed until 45 - 50 s from FRC formation, compared to an onset time of 25 s without CHI. Besides delaying instability, MCPG application reduces the toroidal rotation frequency from 67 kHz to 41 kHz. Moreover, the flux decay time is extended from 57 to 67 s. These changes occur despite the quite modest flux content of the plasmoid: ~ 0.05 mWb of poloidal and 0.01 mWb of toroidal flux, compared with the 0.4 mWb of poloidal flux in the pre-formed FRC. The MCPG introduces a different stabilization mechanism, which may be the same as that observed in translated FRCs, because of the existence of modest toroidal flux. The observed global stabilization and confinement improvements suggest that the MCPG can actively control

 In STX experiments, stabilization effects due to RMF have been observed. The stabilization effects can be attributed to two-fluid effects produced by rotational and ponderomotive forces. In TCS experiments, the stabilization effects of RMF on the *n* = 2 interchange mode and rotational mode have been reported (Guo et al., 2005). The stability criterion of RMF

<sup>0</sup> 1.14 *B r*

 

The stability diagram for FRC formed and sustained by the RMF at a different frequency is

*<sup>s</sup>* . (4)

*<sup>s</sup>* . (5)

al., 2008). The following analytic stability criterion was derived as

external poloidal field, can stabilize the FRC in the case of TCS experiments.

observed.

the rotational instability.

field strength (*B*) was derived as

indicated in Fig. 6 of reference of Guo et al., 2005.

## **3.1.3 Stability of prolate FRCs with respect to tilt mode**

In the FRX-C/LSM, which is a conventional FRTP device using non-tearing reconnection, the stability of FRC plasmas with 1 < *s* < 3.5 and 3 < *E* < 9 (highly kinetic and elongated) was investigated using a Mirnov loop array of 64 external *B* pick-up loops and a soft X-ray end-on camera (Tuszewski et al., 1991). Tilt-like asymmetries (the *n* = 1 axial odd component of *B*) were found, which strongly correlates with FRC confinement. Tilt and other instabilities also appeared with an increase in the bias magnetic field and/or filling pressure (i.e., higher *s* value). An increase in the bias field and filling pressure also coincidently causes strong axial dynamics, which triggers confinement degradation. These experimental results suggest that the tilt-stability condition for kinetic and elongated FRCs is in the range of *s*/*e* < 0.2 - 0.3 (*S*\*/*E*  < 3) (Fig. 7), and becomes *s*/*e* ~ 1 for MHD-like FRC (Fig. 8). Strong axial dynamics during FRC formation results in lesser elongation of the FRC. Therefore it eventually fosters the growth of tilt instability. For an FRC with *s*/*e* ~ 1, the tilt instability grows from small initial perturbations, and becomes large enough to cause major plasma disruptions after 10 - 20 s, which is 3 – 4 times longer than the growth time of instability. In the case of low filling pressures, higher order (*n* = 2 and 3) axially odd asymmetries are also observed. However, the amplitude of these modes is much less than that of the *n* = 1 tilt components. In the case of higher filling pressures, higher order modes appear earlier and grow vigorously.

In the LSX, using an improved FRTP formation method with a programmed formation scheme, the correlation between plasma distortions and the confinement properties was investigated (Slough & Hoffman, 1993). A *B* probe array and an end-on soft X-ray camera were employed to determine separatrix movement, which might indicate the existence of lower order modes, such as tilt mode. Experiments were conducted over a large range of *s* (1 < *s* < 8) and no correlation was observed between the quality of confinement and the *B* signal. In fact, the confinement quality correlates more with the shape of the equilibrium radial profile than with *s* . Details of the experimental results are summarized in Table 3.

Fig. 7. Toroidal Fourier analysis of the *B* for good confinement. The Fourier amplitudes and phases are shown as functions of time for (a) even and (b) odd components. The top trace of (a) is a line-integrated electron density and (b) diamagnetism.

MHD Activity in an Extremely High-Beta Compact Toroid 129

To resolve this discrepancy between MHD predictions and experimental observation, significant progress in the theoretical understanding of FRC stability has been achieved. A host of stabilization effects—for example, the ion FLR effect, the effects of the Hall term and sheared ion flow, resonant particle effects, modern relaxation theory, and two-fluid flowing equilibrium—have been considered in the theoretical studies. Systematic studies of the stability properties of prolate and oblate FRC plasmas have also been presented in a series of

(a) (b) Fig. 9. (a) Time evolution of the amplitudes of different *n*-modes in prolate FRC with S\*=20 and (b) Growth rate of *n* = 1 tilt instability for three elliptical FRC equilibria with E = 4, 6 and 12

shown in Fig. 9 (a) (Belova et al., 2004). Nonlinear saturation of the tilt modes, and growth of the *n* = 2 rotational mode due to ion toroidal spin-up, have been demonstrated. For oblate FRCs, the scaling of the linear growth rate of *n* = 1 internal tilt instability, with the parameter of *S*\*/*E* for elliptical FRC equilibria (*E* = 4, 6, 12), has also been investigated, and is shown in Fig. 9 (b) (Belova et al., 2006a). The growth rate of the *n* = 1 tilt mode is decreased in the range of S\*/E < 3 ~ 4. For oblate FRC plasma, the stabilized region has been found for all *n* = 1 modes (the tilt mode, the radial shift, the interchange mode, and the co-interchange mode), with a closed conducting shell and neutral beam injection (Belova et al. 2006a, 2006b).

(a) (b) Fig. 10. Magnitude of (a) *n* = 1, (b) *n* = 2, (c) *n* = 3 perturbation in (a) *Br* and (b) *Bz*. Hollow symbols are for cases without the center column, and solid symbols for cases with the center

*n*) of different *n*-modes in prolate FRC with *S*\* = 20 is

**3.1.4 Recent progress in theoretical understanding** 

The time evolution of the amplitudes (

column.

Belova's works (Belova et al., 2000, 2001, 2003, 2004, 2006a, 2006b).

Fig. 8. Toroidal Fourier analysis of the *B* for bad confinement. The Fourier amplitudes and phases are shown as functions of time for (a) even and (b) odd components The top trace of (a) is a line-integrated electron density and (b) diamagnetism.


(\*1) 1 3 *s* , (\*2) 3 5 *s* , (\*3) 5 8 *s* , (a) Confinement was not influenced until the mode amplitude was quite large, (b) Poor confinement correlated with non-optimal formation modes that resulted in large-amplitude flutes, (c) All very high-s discharges employed a non-optimal formation sequence, (d) A highly nonlinear flute destroyed the configuration formation

Table 3. Stability properties of FRC in LSX

128 Topics in Magnetohydrodynamics

Fig. 8. Toroidal Fourier analysis of the *B* for bad confinement. The Fourier amplitudes and phases are shown as functions of time for (a) even and (b) odd components The top trace of

(\*1) 1 3 *s* , (\*2) 3 5 *s* , (\*3) 5 8 *s* , (a) Confinement was not influenced until the mode amplitude was quite large, (b) Poor confinement correlated with non-optimal formation modes that resulted in large-amplitude flutes, (c) All very high-s discharges employed a non-optimal formation

sequence, (d) A highly nonlinear flute destroyed the configuration formation

Table 3. Stability properties of FRC in LSX

(a) is a line-integrated electron density and (b) diamagnetism.

#### **3.1.4 Recent progress in theoretical understanding**

To resolve this discrepancy between MHD predictions and experimental observation, significant progress in the theoretical understanding of FRC stability has been achieved. A host of stabilization effects—for example, the ion FLR effect, the effects of the Hall term and sheared ion flow, resonant particle effects, modern relaxation theory, and two-fluid flowing equilibrium—have been considered in the theoretical studies. Systematic studies of the stability properties of prolate and oblate FRC plasmas have also been presented in a series of Belova's works (Belova et al., 2000, 2001, 2003, 2004, 2006a, 2006b).

Fig. 9. (a) Time evolution of the amplitudes of different *n*-modes in prolate FRC with S\*=20 and (b) Growth rate of *n* = 1 tilt instability for three elliptical FRC equilibria with E = 4, 6 and 12

The time evolution of the amplitudes (*n*) of different *n*-modes in prolate FRC with *S*\* = 20 is shown in Fig. 9 (a) (Belova et al., 2004). Nonlinear saturation of the tilt modes, and growth of the *n* = 2 rotational mode due to ion toroidal spin-up, have been demonstrated. For oblate FRCs, the scaling of the linear growth rate of *n* = 1 internal tilt instability, with the parameter of *S*\*/*E* for elliptical FRC equilibria (*E* = 4, 6, 12), has also been investigated, and is shown in Fig. 9 (b) (Belova et al., 2006a). The growth rate of the *n* = 1 tilt mode is decreased in the range of S\*/E < 3 ~ 4. For oblate FRC plasma, the stabilized region has been found for all *n* = 1 modes (the tilt mode, the radial shift, the interchange mode, and the co-interchange mode), with a closed conducting shell and neutral beam injection (Belova et al. 2006a, 2006b).

Fig. 10. Magnitude of (a) *n* = 1, (b) *n* = 2, (c) *n* = 3 perturbation in (a) *Br* and (b) *Bz*. Hollow symbols are for cases without the center column, and solid symbols for cases with the center column.

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