MHD Activity in an Extremely High-Beta Compact Toroid

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).


Introduction 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-3 He 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, 2007 a ) and Pulsed High Density FRC Experiments (PHD) (Slough, 2007 b ).
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 (B ze ) consists of the externally applied magnetic field of an external coil (B z0 ), and the self-generated magnetic field of the toroidal plasma current (I  : I  >2B z0 / 0 ).The FRC consists of an axially symmetric magnetized plasma, a plasma liner and a simply connected configuration.Then, the beta value , which is the ratio of confined plasma pressure (p) to the confinement magnetic field pressure (B ze 2 /2 0 ), 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 T e : electron temperature) (Steinhauer, 2011).E = l s /2r 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  0.5x s 2 (Armstrong et al., 1981).
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 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 and Di 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).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 FRCshave 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 larges 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 devices (FRX-C/LSM (Tuszewski et al., 1990(Tuszewski et al., , 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., 1990Tuszewski et al., , 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.

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.
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(Guo et al., , 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 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.
Formation method B e (T)

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 0.03-0.08T, 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 ), 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 estimated by our improved excluded flux method, and the radial profile of bremsstrahlung (proportional to 20 . 5ee nT ).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).

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.

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

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 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 2( 1) 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 .

Topics in Magnetohydrodynamics 126
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(Guo et al., , 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 0 0.66 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 0 1.14 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.

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 svalue).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.

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, 2006 a , 2006 b ).

MHD behavior and stability of oblate FRCs
In the MRX device, the stability of oblate FRC plasma was investigated.The plasma parameters are electron density of 0.5 -2 x 10 20 m -3 , ion temperature of ~18 eV, B z of 300 -200 G, E of 0.65 -0.35, s of 1-3, and sE of 2-5 (Gerhardt et al., 2006).Without a passive stabilizer (a hollow conducting center conductor), tilt and shift instability (n = 1 mode of Fig. 10 (a) and (b)) often appeared.The tilt mode limits the plasma lifetime.The tilt instability can be mitigated by either including a passive stabilizing conductor or forming highly oblate plasmas with a strong mirror field.Without the center column, the growth of the shift mode is reduced, apparently by the large magnetic fields on the outboard side of device.Large perturbations (n = 2 and 3) may still remain after passive stabilization is applied ((b) and (c) of Fig. 10 (a)).These perturbations have the characteristics of co-interchange modes, which have never been observed in conventional oblate FRCs.Such modes cause the early termination of the oblate FRC shape.These co-interchange modes can be stabilized in oblate plasma with a high mirror ratio, and this produces an FRC with maximum configuration lifetime.

Summary
In Table 4, the MHD stability properties of both prolate and oblate FRCs are summarized (Yamada et al., 2007).In oblate FRC plasma, the global mode (n = 1 external tilt and shift mode, co-interchange mode) is unstable.But these modes can be stabilized by employing a close-fitting conducting shell or shaping with a strong external magnetic field.In prolate plasma, the internal tilt mode and co-interchange mode are MHD-unstable.However, these can be stabilized by nonlinear effects, such as FLR, rotation and sheared flow.

Fig. 4 .
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 B t .

Fig. 5 .
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

Fig. 7 .
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.

Fig. 8 .
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.

Fig. 10 .
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 12The 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., 2006 a ).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. 2006 a , 2006 b ).
It is difficult to observe these modes clearly in the experiments.The most destructive rotational mode (n = 2), and wobble (n = 1) mode, are almost always observable in experiments.Active stabilization methods without degradation of c o n f i n e m e n t , s u c h a s C H I , n e e d t o b e developed.

Table 1
. FRC Stability: MHD Theory versus Experimental Observation

Table 2 .
FRC Plasma parameters for various formation methods

Table 3 .
Stability properties of FRC in LSX

Table 4 .
Stability properties of prolate and oblate FRCs www.intechopen.com