Through the local mode analysis of ballooning modes in an L=2/M=10 planar axis heliotron system with an inherenty large Shafranov shift(where L and M are the polarity and toroidal field period of the helical coils, respectively), it has been demonstrated that [N. Nakajima, Phys. Plasmas 3, 4545 and 4556(1996)]:

・The local magnetic shear (which is a stabilizing term for high-mode-number ballooning modes) is related to helicity of the helical coils in the considered vacuum configuration. Its change due to a large Shafranov shift is essentially axisymmetric, i.e., related to toroidicity. This change leads to the disappearance of the (integrated) local magnetic shear on the outer side of torus, even in the region with a stellarator-like global magnetic shear,Ieading to the destabilization of the high-mode-number ballooning modes.

・The local magnetic curvature (which constructs a potentially destabilizing term for high-mode-number ballooning modes together with the pressure gradient)consists of parts due to both toroidicity and helicity of the helical coils,which determines the 3-D properties of the high-mode-number ballooning modes.

In general 3-D MHD equilibria, the eigenvalues ω2 for high-mode-number ballooning modes are functions of the labels of the flux surfaceψ, the magnetic field line α, and the radial wave number θk:ω2 =ω2(ψ, θk, α). Sinceω2 has no α-dependence in axisymmetric systems, the stronger the α-dependence ofω2 is (mainly coming from the helicity part of the local magnetic curvature), the more significant the 3-D properties of ω2 are. The topological properties of the unstable eigenvaluesω2(<0) in (ψ, θk, α) space for the L=2/M=10 planar axis heliotron system are shown that [N. Nakajima, Phys. Plasmas 3, 4556 (1996)]:

・In Mercier unstable cquilibria, there coexist two types of topological level surfaces for ω2in (ψ, θk, α) space. One is a tokamak-like cylindrical level surface with the axis in α direction, the other is a spheroidal level surface inherent to 3-D systems. The spheroidal level surfaces are surrounded by the cylindrical level surfaces. From their relative positional relation, it is clear that modes with spheroidal level surfaces have larger growth rates than those with cylindrical level surfaces.

・In Mercier stable equilibria,only a topologically spheroidal level surface exists. In contrast to Mercier unstable equilibria, this spheroidal level surfaces are surrounded by the level surfaces of stable Toroidicity-induced Alfv Eigenmodes (TAE).

From these results it is conjectured that the global structure of pressure-driven modes has the following properties［N.Nakajima, Phys. Plasmas 3,4556(1996)]:

・Global modes that correspond to modes in the local mode analysis with a cylindrical level surface will be poloidal localized tokamak-like ballooning modes or interchange modes. Effects of the toroidal mode coupling on these modes are weak.

・Global modes that correspond to modes in the local mode analysis with a spheroidal level surface will be ballooning modes inherent to 3-D systems, with quite high poloidal and toroidal mode numbers and localized in both the poloidal and toroidal directions. These modes become to be localized within each toroidal field period of the helical coils, as their typical toroidal mode numbers become higher.

・In Mercier unstable equilibria, Where both cylindrical and spheroidal level surface coexist, tokamak-like ballooning modes or interchange modes appear when their typical toroidal mode numbers are relatively small. As the typical toroidal mode numbers become larger, ballooning modes inherent to 3-D systems appear with larger growth rates.

・In Mercier stable equilibria, where only a spheroidal level surface exists, only ballooning modes inherent to 3-D systems appear.

The purposes of the work are to confirm the above conjecture and to clarify the inherent properties of pressure-driven modes through a global mode analysis in the L=2/M=10 planar axis heliotron system with an inherently large Shafranov shift [J. Chen, N. Nakajima, and M. Okamoto, Global mode analysis of ideal MHD modes in a heliotron/torsatron system: I. Mercier-unstable equilibria].

First the Mercier-unstable equilibria are categorized into two types, namely, toroidicity-dominant Mercier-unstable equilibria and helicity-dominant Mercier-unstable equilibria. This categorization is motivated by the conjecture that tokamak-like ballooning modes or interchange modes exist for relatively small toroidal mode numbers, and is related to the local properties of Mercier-unstable equilibria brought by Shafranov shift. The properties of the vacuum configuration are understood as a straight helical configuration toroidally bended. Since the aspect ratio is relatively large: R0/a=7～8 [here R0 and ａ are the major and minor radii, respectively], the global and local properties of the vacuum configuration are mainly determined by helicity of the helical coils. The properties of the finite-β equilibria are basically understood as a modification of the vacuum configuration by an essentially axisymmetric and inherently large Shafranov shift. As the Sharanov shift becomes larger, the stabilizing term due to the local magnetic shear is more reduced. The toroidicity-dominant Mercier-unstable equilibria are characterized by properties that it is easy for the local magnetic shear to vanish on the outer side of torus, which is brought by a relatively large Shafranov shift. In these equilibria, it is relatively easy for ballooning modes to be destabilized. The helicity-dominant Mercier-unstable equilibria are characterized by properties that it is hard for the local magnetic shear to vanish on the outer side of torus, which is brought by a relatively small Shafranov shift. In these equilibria, it is relatively hard for ballooning modes to be destabilized. Note that, in both types of equilibria, the Shafranov shift, locally reduces (enhances) the unfavorable norma1 magnetic curvature on the outside (inside) of torus, which is another local property due to Shafranov shift.

On the basis of these considerations, the following two types of Mercier-unstable equilibria have been adopted. The toroidicity-dominant Mercier-unstable equilibrium is created with a peaked pressure profile P = P0(1-ψN)2 and β0=5.9%, under the flux conserving condition, i.e., with a specified profile for the rotational transform. The helicity-dominant Mercier-unstable equilibrium is created with a broad pressure profile P = P0(1-ψ2N)2 and β0=4.0%, under the currentless condition.

The global mode analysis are done by CAS3D2MN, a version of CAS3D: Code for Analysis of the MHD Stability of 3-D equilibrium [C. Schwab, Phys. Fluids B 5, 3195 (1993)]. CAS3D have been designed to analyze the global ideal MHD modes of 3-D equilibria based on a formulation of the ideai MHD energy principle with incompressibility and fixed boundary in Boozer coordinate system and the application of Ritz-Galerkin method. In CAS3D2MN, a phase-factor transformation was used in order to save memory and flops.

The inverse iteration with spectral shift is an essential concept in the solution of eigenproblems. It is very efficient if the spectral shift is given to be very close to the desired eigenvalue and the initial vector is chosen to be dominant along the corresponding eigenvector. It is demonstrated in our simulation that convergence will occur after only 3 or 4 steps if the spectral shift itself is a good approximation of the desired eigenvalue and the initial vector has dominant component along the corresponding eigenvector. The left problem is how to guess the spectral shift and give a good initial vector. The spectral shift was calculated by matrix transformation in CAS3D2MN. Since the bandwidth will be destroyed by matrix transformation, the resultant memory and flops will be 0(n2) and 0(n3), respectively. It is shown that the use of matrix transformation is unsuitable, not only because it becomes very expensive in the sense of flops and storage but also the problem size we can deal with is limited by the available computer resources. Here this problem is solved by using the Lanczos algorithm with no re-orthogonalization which keeps the matrix bandwidth from begin to end. The arithmetic operation mainly come from the matrix-vector multiplies and only 3 recently created Lanczos vectors need to be stored. The resultant memory and flops can be controlled to 0(n2) and 0(n3) order. This iteration process is accelerated by an shift-and-invert technique. In the new version CAS3D2MNv1, an efficient initial vector generation is also introduced [J. Chen, N. Nakajima, and M. Okamoto, Comput. Phys. Commun., 113, 1 (1998)].

Since the local magnetic curvature due to helicity has the same period M in the toroidal direction as the toroidal field period of the equilibria, the characteristics of the pressure-driven modes in such Mercier-unstable equilibria dramatically change according to how much the local magnetic shear is reduced (whether the equilibrium is toroidicity-dominant or helicity-dominant) and also according to the relative magnitude of the typical toroidal mode numbers n of the perturbations compared with the toroidal field period M of the equilibria.

In the toroidicity-dominant Mercier-unstable equilibria, the pressure-driven modes change from interchange modes with negligible toroidal mode coupling for low toroidal mode numbers n