1. Introduction
The existence of the MHD stable high beta core plasma lead the way for the realization of economic fusion power systems. The LHD experimental results of achieving average beta value 5% without the beta collapse suggests the possibility of the helical equilib- riums with ultrahigh beta MHD stable core plasmas. MHD stability of plasma is determined by the MHD potential energy(= W ), which is the sum of plasma thermal energy(= W T ) and the magnetic filed energy(= WB), The integration domain is extended not only to the plasma volume but also to the outer region of external coils. The diamagnetic current reduces the exter- nally applied magnetic field(= Bext) at the plasma volume(= V p ). The pressure equilibrium relation and the definition of the plasma beta value(= β ) lead the expression for variation of the MHD potential energy δW as follows,
where P 0 and B 0 are the pressure and magnetic field in the plasma region, and β ≡ P 0
Therefore, the diamagnetic current cannot conduct to the MHD stable configuration as shown in eq.(2). On the other hand, the toroidal current driven by plasma pressure (the bootstrap current) produces author’s e-mail: wata@toki-fs.jp magnetic field outside the plasma column, which di- minish the magnetic field outside of external coils. When the MHD potential energy reduction
is caused by the cancellation of the magnetic field, the bootstrap transitions to MHD stable high beta equilibrium become possible. Bootstrap current is usually created by the ba- nana orbit particles, which comes from the toroidal effect. The density gradient of banana particles pro- vides the bootstrap current[1].  In a tokamak the boot- strap current J bs is simply given by
where ε = α/R is the inverse aspect ratio and ρ is the minor radius of the torus. R.J.Bickerton, J.W.Connor and J.B.Taylor have pointed out that in low collision frequency regime, the enhanced diffusion is associated with the currents which flow parallel to the magnetic field; not only are these current much larger than the Pfirsch-Sclüter currents, but they are also unidi- rectional. They have proposed a bootstrap tokamak which operates in steady state by diffusion driven net toroidal current without any external driving force [2]. In the straight helical system, we have ε = 0, and then J bs described in eq.(4) reduce to zero. However, the bootstrap currents described in the paper have still finite value even in a straight helical system. Pfirsch-Sclüter currents are derived from the fol- lowing relation
which shows the condition of the no charge accumula- tion induced by the the imbalance of the net outward force due to the larger surface area at the larger ma- jor radius. For the case of usual axisymmetric system with concentric magnetic surfaces with the rotational transform ι ,
and can be integrated as
where C is the integration constant of the differential equation (7). If we set C = 0, the expression eq.(8) is the expression for the Pfirsch-Sclüter currents Jps [3],
The integration constant C is determined from not eq.(5) but from the force balance condition in the ρ direction. In axisymmetric torus, MHD equation ∇P = J × B determines the plasma current J as follows [4].
Then the pressure driven equilibrium current, the first term of the RHS of the eq.(10), is expressed as where we have
The expression eq.(11) determines the integration con- stant C as
where we have used the expression for the rotational transform eq.(6). The first term of the RHS of eq.(14) is net toroidal currents similar to the diamagnetic cur- rent produced by the poloidal magnetic field Bo and the plasma pressure gradient in the ρ direction. Since this currents produce magnetic field mainly outside the plasma column without producing the magnetic field inside the plasma column, we call this pressure driven toroidal net currents is also one of the boot- strap currents. The second term of the RHS of term eq.(14) prevent the net charge accumulation due to the imbalance of the net outward force. This expres- sion is just same to the current profile given by the second term of the eq.(11), but the factor 2 smaller compared to the traditional expression of the Pfirsch- Sclüter currents eq.(9). The net charge accumulation du to the pressure imbalance is partially prevented by the pressure driven toroidal net current. In the paper, we have developed a new numerical scheme to solve the equilibrium based on the Biot- Savart' law. Since this scheme can compute the mag- netic field at the region including inside and outside of helical coils, MHD potential energy W can be cal- culated, simultaneously. We have analyzed the equilibrium and the stabil- ity of straight helical systems, to verify based on the first principle that the LHD type helical magnetic field configuration can sustain MHD stable high beta core plasmas. Straight helical coil system and the rotating he- lical coordinate system are explained in §2. New nu- merical scheme to solve MHD equilibrium and plasma model are explained in §3. To analyze the role of bootstrap current for the MHD stability, a new cri- terion for the MHD potential energy is developed in §4. Bootstrap transition from peaked pressure profile to flat-top pressure profile is analyzed in §5. In §6, it is shown that small value of the helical pitch parameters is favorable for the high beta plasma confinement. §7 is devoted to the summary and discussions.
2. Straight Helical System
Fig. 1 Straight helical coils composed of 3 layers(Hc- among these three layers. Straight helical system has symmetry and MHD equations are possible to be solved without approxi- mations. Figure 1 shows an example of straight he- lical coils and the rotating helical coordinate system (X,Y,(), which rotates in synchronization with them. X , Y and ζ are for the directions of the long axis, short axis and magnetic axis of magnetic surfaces, respec- tively.  Under the rotating helical coordinate system, ζ dependency disappears in a helical magnetic field. An important index of a helical coil system is a helical pitch parameter(= γ), which is defined by the product of the radius of current distribution center(= a c ) and the axial wave number of the helical coil system(= k ),
We have assumed in the following analysis that the size and the configuration of the helical coils are similar to those of the LHD[5]. Table 1 shows typical examples of the coil currents and of the magnetic field parameters.

3.
Equilibrium  Analysis
Ideal MHD equation ∇P = J × B can be in- tegrated along the magnetic field B and the plasma current J. Arbitrary functions P(¥) and I(¥) are in- troduced and plasma current is expressed as follows,
where P(Ψ) is the plasma pressure distribution. The first term of the right-hand side of eq.(16) is the driven current term, which is independent to the plasma pressure.  X and Y components of the second term of eq.(16) are the diamagnetic currents, which pro- duce magnetic field on the inside of the plasma col- umn, mainly.  The ζ components of the second term of eq.(16) is the bootstrap currents, which produce magnetic field on the outside of the plasma column, mainly. Magnetic field B , vector potential A and the mag- netic flux function Ψ can be calculated by Biot-Savart 7 law as follows,
where Jc(r') is the current density of helical coils. Because the flux function Ψ is a linear function of the vector potential A , Ψ is given by the sum of contribution from the plasma current(= Ψ P (Ψ)) and contribution from the helical coil currents(= Ψ c ), = Ψ ρ (Ψ) + Ψ ρ
Plasma equilibrium computations is reduced to solve eq.(20) self-consistently for Ψ.  We have confirmed that relaxation scheme and Newton iteration method are possible to solve eq.(20), treating the separatrix position of Ψ as an unknown parameters. In the following, we assume that the driven cur- rent is not present ( I ′ (Ψ) = 0). The pressure profile Р(Ψ) is assumed to be
where β ax, P 1 , P 2 , P 3 and D are some constants. B ax and Ψ s are the magnetic field on the magnetic axis and the value of Ψ at the separatrix, respectively. When P 1 = 1 and P 2 = P 3 = 0, the pressure pro- file become a peaked profile and the bootstrap currents distribute in a similar profile, also. When P 1 = 0, the pressure profile become a flat-top profile and the boot- strap currents become zero on the magnetic axis. The bootstrap currents distribute in a surface current type profile.
Stability  Analysis
4.
Fig. 2 The field intensity distribution( B c : up) and the magnetic flux function(Ψ: down), in a helical coil. Stability analysis is based on the energy conser- vation law for the ideal MHD equations[6]. where v, ρ and E are fluid velocity, mass density and electric field, respectively. The conservation lows eq.(22) shows that W,
is the MHD potential energy, and W minimum config- uration is an MHD stable equilibrium.  To analyze the role of the bootstrap current to the MHD stability, the integral domain of eq.(23) should be extended to the region outside of the helical coils.  For the computation of the MHD potential energy W , we have developed a magnetic field calculation code which can compute accurately the field intensity inside the helical coils. Numerical example of field intensity distributions in- side the helical coil is shown in Fig.(2). MHD stability of an equilibrium is evaluated by the amount of the decreased MHD potential energy δW ,
where Bvac is the vacuum magnetic field and W 0 is the MHD potential energy of the vacuum state.  When δW < 0, transition to vacuum state is energetically- prohibited and beta collapse of core plasma will not occur.  Core plasma develops to a minimum state of W under the imposed constraints.
5. Bootstrap
Transition
to
Flat-Top
Pressure Profile
In first, we compare the equilibrium and its sta- bility, for the case of peaked pressure profile and for the case of flat-top pressure profile. The helical pitch parameter is set to one of the standard value of the LHD (γ = 1.2538).  Pressure distributions for both cases are assumed to be the following form,
Pressure profiles and the bootstrap current distribu- tions under the vacuum magnetic field are shown in Fig.3. We have calculated variations of MHD potential energy( = δW) and beta value at the magnetic axis (= βax), increasing the stored energy W T (Fig. 4). Peaked pressure profile can reach stably to high beta equilibrium with relatively small amount of stored energy, until the decrease of the MHD poten- tial energy reverse ( W T ~ 0.77 MJ/m).  When heating input is increased much more, peaked pressure profile equilibrium become unstable. Flat-top pressure pro- file, on the other hand, is more stable, and can increase the stored energy almost 3 times of the level of peaked profile case ( WT ≈ 2.12 MJ/m). Bootstrap transition from peaked pressure profile to flat-top pressure profile will be occur, when suffi- cient heating input is present.
6.  Helical Pitch Parameter Dependency
of High Beta Equilibrium
Helical pitch parameter, γ defined by eq.(15) is an important index of a helical coil system.  We study the pitch parameter dependency of high beta equilibrium and field intensity | B | along the long axis X of the magnetic surface.  (a) and (b) are for the case of peaked and flat-top pressure profiles, respectively. in the case of flat-top pressure profile given by eq.(26).
Numerical results are summarized in Fig.5. When γ is large, the size of the magnetic surface become large, since the axial magnetic field( = B c ) is increased relatively. However, the ability of MHD sta- bility decreases as suggested by eq.(2). This tendency is shown by the plot of MHD potential energy δW  for the case of γ = 1.3817 in Fig.5. When γ is small, the size of the magnetic surface become small, since the axial magnetic field decreased relatively. However, the ability of MHD stability in- creases since the role of bootstrap current is increased. This tendency is shown by the plot of MHD potential energy δW for the case of γ = 1.1221 in Fig.5. Structure of vacuum magnetic field is shown in Fig.6 (a) and (b) for the case of γ = 1.1221.  Clear- ance between helical coils and the last closed flux sur- face are very wide, though the plasma volume is small (Vlcfs ~ 18.6m³: reduced value for the LHD). Next, we show the MHD stable ultrahigh beta equilibrium, which has the minimum MHD potential energy in Fig.6 (c) and (d), sustained in this mag- netic field (γ = 1.1221).  We have confirmed that the size of the magnetic surface become very large (Vicfs ~ 45.1m3: reduced value for the LHD). Mag- netic field intensity is very much reduced in the outer region of helical coils by sustaining of high beta plasma (compare Fig.6(b) and (d)).  The reduction of total magnetic field energy decreases the MHD potential energy and realize the sustainment of MHD stable ul- trahigh beta plasma. Beta value on the magnetic axis reach to βax ~ 176%. Figure 7 shows profiles of equilibrium quantities along the long axis of magnetic surface for the case of βα x = 176%. Figure 7(b) shows that steep pressure gradient is primarily sustained by the magnetic well, shown by the graph of the specific volume U . Remaining pe- ripheral pressure gradient should be sustained stably by high magnetic shear, shown by the graph of the rotational transform ι/2π.
7. Summary and Discussions
We have developed a new numerical scheme to solve MHD equilibrium using Biot-Savart' law. This scheme can compute the equilibrium magnetic field of inner and outer region of the magnetic surface. Fur- thermore, we have developed a new numerical scheme to compute the MHD potential energy, which can eval- uate the role of bootstrap current for the MHD sta¬ bility of core plasma. We have concluded that LHD type magnetic field configuration, which is produced by continuous wind- ing helical coils and without the toroidal coil, can sustain MHD stable ultrahigh beta plasma. The MHD potential energy can be decreased by the pres- sure driven toroidal current (bootstrap current), even though plasma thermal energy increased. Since the MHD potential energy decreases according to the in- crease of the heating power input to plasma, MHD stable bootstrap transition to higher beta plasmas oc¬ cur by ramp-up of heating power input. By the comparison of the MHD potential energy, it is suggested that peaked pressure profile equilibrium make a bootstrap transition to the flat-top pressure profile equilibrium, and small value of the helical pitch parameter is more favorable for the high beta plasma sustainment. Ultrahigh beta core plasmas discussed in the pa- per have excellent features for the reactor system. As shown in Fig.  6(a,c), there is no drastic changes in diverter-leg of ultrahigh beta core plasma and vac- uum magnetic field. Moreover, magnetic field in core plasma region retain almost the 60% of the value of the vacuum magnetic field, even for the case of β ax = 176% (see Fig.7(a)). This will be very much fa- vorable for the alpha particle heating of core plasma. In addition, small magnetic surface volume for the vacuum magnetic field shown in Fig.6(a) may also be favorable for reactor systems, because the relatively small heating power unit become sufficient for the fusion ignition.  After the ignition, high level fusion power output is expected because of large plasma vol- ume of ultrahigh beta plasma. In the present paper, we have analyzed a straight helical system, because of the simplicity. In this case, MHD equilibrium can be solved without any approx- imation.  For the axisymmetric toroidal case, equi¬ librium of β ax = ∞ solutions have been numerically obtained[7], for the case of no toroidal magnetic field. In toroidal helical systems chaotic field line region appears in surroundings of the magnetic surface.  In this case, magnetic flux function is expressed by the adiabatic invariant of lines of force[8]. The flux function for the magnetic field B and for the plasma current J , are reduced to the following[8],
where p and r are pitch number and usual radial co- ordinate, respectively, and ( X, Y, φ ) is the toroidal ro- tating helical coordinate system[9]. In magnetic surface region, adiabatic invariants 〈Ψ〉 and 〈I〉 can be calculated.
• \ represents the average along the lines of where force.  Numerical example of the adiabatic invariant 〈Ψ〉 is shown in Fig.8. lations,
Then the MHD equation ∇P = × B gives the fol- lowing expression.
Because the plasma current J is given as a func- tion of the magnetic vector potential A , it seems that the equilibrium calculation of the toroidal helical sys- tem is possible by similar method written in §3. The plasma volume is determined by the adia- baticity of lines of force of peripheral region of mag- netic surface. The adiabaticity is controllable by the vertical field coils currents configuration. This physi- cal situation is same for the vacuum or high beta mag- netic configuration.  In the LHD vacuum magnetic field, magnetic surfaces accompanied with very thin chaotic field line layers are confirmed by appropriate choices of the vertical field coils currents. Numerical demonstration and validation of its re- sults are future tasks. This work is performed with the support and under the auspices of the NIFS Collaborative Re- search Program NIFS08KDBD006, NIFS09KTBL010 and NIFS08KNXN144. [1] K.Miyamoto, Plasma Physics and Controlled Fusion, (Springer-Verlag, 2005) p. 303. Bickerton et al, Nature Phys. Sci. 229 , 110 (1971). [2] [3] K.Miyamoto, Plasma Physics and Controlled Fusion, (Springer-Verlag, 2005) p. 71. [4] K.Miyamoto, Plasma Physics and Controlled Fusion, (Springer-Verlag, 2005) p. 64. [5] A. Iiyoshi et al Nucl. Fusion 39 , 1245 (1999). G. Bateman, MHD Instabilities (MIT Press, Cam- [6] bridge, Massachusetts, and London, England, 1980) p. 38. [7] T. Watanabe et al J. Advanced Science 19 , 74 (2008), in Japanese. [8] T. Watanabe et al Nucl. Fusion 46 , 291 (2006). [9] Watanabe T. and Akao H. J. Plasma Fusion Res. 73 , 186 (1997), in Japanese.