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In threedimensional (3D) configurations with vacuum magnetic flux surfaces,the equilibria can be obtained without net toroidal current, where the currentdriven instabilities become unimportant, and only the pressuredriven instabilities need to be intensively studied. The pressuredriven modes consists of interchange modes and ballooning modes,and impose MHD stability β limits. [J.P. Freidberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987] . Interchange modes are basically driven by average unfavorable magnetic curvature. Thus these modes localize on mode rational magnetic field lines and are almost constant along these lines B･∇ ζψ～0.On the other hand, ballooning modes are basically driven by locally unfavorable magnetic curvature, so that they localize on unfavorable magnetic curvature region and change along the magnetic field line B･∇ ζψ≠O. Ballooning modes are considered to be more stringent than interchange modes, whose properties have not yet been clarified in 3D configurations. To study the properties of ballooning modes, one can proceed in two different ways, namely, local mode analysis and global mode analysis. In axisymmetric systems, the global modes can be constructed easily from the results of the local modes analysis. But this is not the case in nonaxisymmetric systems, namely, 3D systems. In fully 3D systems, we can only make some conjectures for global modes from the properties of the local modes.\u003cbr /\u003e 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)]:\u003cbr /\u003e ・The local magnetic shear (which is a stabilizing term for highmodenumber 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 stellaratorlike global magnetic shear,Ieading to the destabilization of the highmodenumber ballooning modes.\u003cbr /\u003e ・The local magnetic curvature (which constructs a potentially destabilizing term for highmodenumber ballooning modes together with the pressure gradient)consists of parts due to both toroidicity and helicity of the helical coils,which determines the 3D properties of the highmodenumber ballooning modes.\u003cbr /\u003e In general 3D MHD equilibria, the eigenvalues ω2 for highmodenumber 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 αdependence in axisymmetric systems, the stronger the αdependence ofω2 is (mainly coming from the helicity part of the local magnetic curvature), the more significant the 3D properties of ω2 are. The topological properties of the unstable eigenvaluesω2(\u003c0) 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)]:\u003cbr /\u003e ・In Mercier unstable cquilibria, there coexist two types of topological level surfaces for ω2in (ψ, θk, α) space. One is a tokamaklike cylindrical level surface with the axis in α direction, the other is a spheroidal level surface inherent to 3D 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.\u003cbr /\u003e ・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 Toroidicityinduced Alfv Eigenmodes (TAE).\u003cbr /\u003e From these results it is conjectured that the global structure of pressuredriven modes has the following properties［N.Nakajima, Phys. Plasmas 3,4556(1996)]:\u003cbr /\u003e ・Global modes that correspond to modes in the local mode analysis with a cylindrical level surface will be poloidal localized tokamaklike ballooning modes or interchange modes. Effects of the toroidal mode coupling on these modes are weak.\u003cbr /\u003e ・Global modes that correspond to modes in the local mode analysis with a spheroidal level surface will be ballooning modes inherent to 3D 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.\u003cbr /\u003e ・In Mercier unstable equilibria, Where both cylindrical and spheroidal level surface coexist, tokamaklike 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 3D systems appear with larger growth rates.\u003cbr /\u003e ・In Mercier stable equilibria, where only a spheroidal level surface exists, only ballooning modes inherent to 3D systems appear.\u003cbr /\u003e The purposes of the work are to confirm the above conjecture and to clarify the inherent properties of pressuredriven 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. Mercierunstable equilibria].\u003cbr /\u003e First the Mercierunstable equilibria are categorized into two types, namely, toroidicitydominant Mercierunstable equilibria and helicitydominant Mercierunstable equilibria. This categorization is motivated by the conjecture that tokamaklike ballooning modes or interchange modes exist for relatively small toroidal mode numbers, and is related to the local properties of Mercierunstable 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 finiteβ 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 toroidicitydominant Mercierunstable 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 helicitydominant Mercierunstable 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.\u003cbr /\u003e On the basis of these considerations, the following two types of Mercierunstable equilibria have been adopted. The toroidicitydominant Mercierunstable equilibrium is created with a peaked pressure profile P = P0(1ψN)2 and β0=5.9%, under the flux conserving condition, i.e., with a specified profile for the rotational transform. The helicitydominant Mercierunstable equilibrium is created with a broad pressure profile P = P0(1ψ2N)2 and β0=4.0%, under the currentless condition.\u003cbr /\u003e The global mode analysis are done by CAS3D2MN, a version of CAS3D: Code for Analysis of the MHD Stability of 3D equilibrium [C. Schwab, Phys. Fluids B 5, 3195 (1993)]. CAS3D have been designed to analyze the global ideal MHD modes of 3D equilibria based on a formulation of the ideai MHD energy principle with incompressibility and fixed boundary in Boozer coordinate system and the application of RitzGalerkin method. In CAS3D2MN, a phasefactor transformation was used in order to save memory and flops.\u003cbr /\u003e 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 reorthogonalization which keeps the matrix bandwidth from begin to end. The arithmetic operation mainly come from the matrixvector 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 shiftandinvert 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)].\u003cbr /\u003e 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 pressuredriven modes in such Mercierunstable equilibria dramatically change according to how much the local magnetic shear is reduced (whether the equilibrium is toroidicitydominant or helicitydominant) 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.\u003cbr /\u003e In the toroidicitydominant Mercierunstable equilibria, the pressuredriven modes change from interchange modes with negligible toroidal mode coupling for low toroidal mode numbers n\u003cM, to tokamaklike poloidally localized ballooning modes with weak toroidal mode coupling for moderate toroiral mode numbers n～M, and finally to both poloidally and toroidally localized ballooning modes purely inherent to 3D systems with strong poloidal and toroidal mode couplings for fairly high toroidal mode numbers n》M. Strong toroidal mode coupling, in cooperation with the poloidal mode coupling, makes the perturbation localize to flux tubes.\u003cbr /\u003e In the helicitydominant Mercierunstable equilibria, the pressuredriven modes change from interchange modes, with negligible toroidal mode coupling for n\u003cM or with weak toroidal mode coupling for n?M, directly to poloidally and toroidally localized ballooning modes purely inherent to 3D systems with strong poloidal and toroidal mode couplings for n》M.\u003cbr /\u003e In the Mercierunstable equilibria, interchange modes with low toroidal mode numbers n\u003cM, experiencing the unfavorable magnetic curvature with its local structure averaged out, occur for both toroidicitydominant and helicitydominant equilibria. For fairly high toroidal mode numbers n》M, the perturbations can feel the fine local structure of the magnetic curvature due to helicity and also the local magnetic shear is reduced more or less in both types of equilibria, and consequently poloidally and toroidally localized ballooning modes inherent to 3D systems are destabilized for both toroidicitydominant and helicitydominant Mercierunstable equilibria. The situation for moderate toroidal mode numbers n?M is diffecrent. The local magnetic shear is more reduced in toroidicitydominant Mercierunstable equilibria than in helicitydominant Mercierunstable equilibria, and also the modes with moderate toroidal mode numbers n～M can not effectively feel the local structure of the normal magnetic curvature due to helicity. Thus, tokamaklike poloidally localized ballooning modes with a weak toroidal mode coupling can be easily destabilized for toroidicitydominant Mercierunstable equilibria, and interchange modes, driven by the average unfavorable magnetic curvature and not experiencing the effect of toroidal mode coupling, can be destabilized for helicitydominant Mercierunstable equilibria. Since the normal magnetic curvature becomes more unfavorable on the inner side than on the outer side of the torus by the Shafranov shift, the interchange modes are localized on the inner side of the torus for both types of equilibria. This type of interchanges mode is antiballooning with respect to the poloidal mode coupling.\u003cbr /\u003e In both types of Mercierunstable equilibria, the pressuredriven modes, i.e., ballooning modes and interchange modes, become more unstable and more localized both on flux tubes and in the radial direction, and have stronger toroidal mode coupling through the normal magnetic curvature due to helicity, as the typical toroidal mode numbers increase. Thus, we can expect that ballooning modes localized in one toroidal field period, as suggested in [N. Nakajima, Phys. Plasmas 3, 4556 (1996)], may occur with very narrower radial extent and larger growth rates, as the typical toroidal mode numbers become larger and larger. All of these properties of the pressuredriven modes in two types of Mercierunstable equilibria are quite consistent with the conjecture from local mode analysis.", "subitem_description_type": "Other"}]}, "item_1_description_18": {"attribute_name": "フォーマット", "attribute_value_mlt": [{"subitem_description": "application/pdf", "subitem_description_type": "Other"}]}, "item_1_description_7": {"attribute_name": "学位記番号", "attribute_value_mlt": [{"subitem_description": "総研大甲第395号", "subitem_description_type": "Other"}]}, "item_1_select_14": {"attribute_name": "所蔵", "attribute_value_mlt": [{"subitem_select_item": "有"}]}, "item_1_select_8": {"attribute_name": "研究科", "attribute_value_mlt": [{"subitem_select_item": "数物科学研究科"}]}, "item_1_select_9": {"attribute_name": "専攻", "attribute_value_mlt": [{"subitem_select_item": "10 核融合科学専攻"}]}, "item_1_text_10": {"attribute_name": "学位授与年度", "attribute_value_mlt": [{"subitem_text_value": "1998"}]}, "item_1_text_20": {"attribute_name": "業務メモ", "attribute_value_mlt": [{"subitem_text_value": "（2018年2月9日）本籍など個人情報の記載がある旧要旨・審査要旨を個人情報のない新しいものに差し替えた。承諾書等未確認。要確認該当項目修正のこと。"}]}, "item_creator": {"attribute_name": "著者", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "CHEN, Jing", "creatorNameLang": "en"}], "nameIdentifiers": [{"nameIdentifier": "8608", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "ファイル情報", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20160217"}], "displaytype": "simple", "download_preview_message": "", "file_order": 0, "filename": "甲395_要旨.pdf", "filesize": [{"value": "711.9 kB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 711900.0, "url": {"label": "要旨・審査要旨 / Abstract, Screening Result", "url": "https://ir.soken.ac.jp/record/477/files/甲395_要旨.pdf"}, "version_id": "09e4ff9174844823b5698e4c5114a0f5"}, {"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20160217"}], "displaytype": "simple", "download_preview_message": "", "file_order": 1, "filename": "甲395_本文.pdf", "filesize": [{"value": "10.2 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 10200000.0, "url": {"label": "本文", "url": "https://ir.soken.ac.jp/record/477/files/甲395_本文.pdf"}, "version_id": "5d72f78d7262424580de8091938eed9c"}]}, "item_language": {"attribute_name": "言語", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"resourcetype": "thesis", "resourceuri": "http://purl.org/coar/resource_type/c_46ec"}]}, "item_title": "Global mode analysis of ideal MHD modes in heliotron system", "item_titles": {"attribute_name": "タイトル", "attribute_value_mlt": [{"subitem_title": "Global mode analysis of ideal MHD modes in heliotron system"}, {"subitem_title": "Global mode analysis of ideal MHD modes in heliotron system", "subitem_title_language": "en"}]}, "item_type_id": "1", "owner": "1", "path": ["12"], "permalink_uri": "https://ir.soken.ac.jp/records/477", "pubdate": {"attribute_name": "公開日", "attribute_value": "20100222"}, "publish_date": "20100222", "publish_status": "0", "recid": "477", "relation": {}, "relation_version_is_last": true, "title": ["Global mode analysis of ideal MHD modes in heliotron system"], "weko_shared_id": 1}
Global mode analysis of ideal MHD modes in heliotron system
https://ir.soken.ac.jp/records/477
https://ir.soken.ac.jp/records/477da673c85cfe74360a1399e3e015aede0
名前 / ファイル  ライセンス  アクション 

要旨・審査要旨 / Abstract, Screening Result (711.9 kB)


本文 (10.2 MB)

Item type  学位論文 / Thesis or Dissertation(1)  

公開日  20100222  
タイトル  
タイトル  Global mode analysis of ideal MHD modes in heliotron system  
タイトル  
言語  en  
タイトル  Global mode analysis of ideal MHD modes in heliotron system  
言語  
言語  eng  
資源タイプ  
資源タイプ識別子  http://purl.org/coar/resource_type/c_46ec  
資源タイプ  thesis  
著者名 
陳, 勁
× 陳, 勁 

フリガナ 
チン, ジン
× チン, ジン 

著者 
CHEN, Jing
× CHEN, Jing 

学位授与機関  
学位授与機関名  総合研究大学院大学  
学位名  
学位名  博士（学術）  
学位記番号  
内容記述タイプ  Other  
内容記述  総研大甲第395号  
研究科  
値  数物科学研究科  
専攻  
値  10 核融合科学専攻  
学位授与年月日  
学位授与年月日  19990324  
学位授与年度  
1998  
要旨  
内容記述タイプ  Other  
内容記述  Ideal magnetohydrodynamics (MHD) epuilibria are subjected to two kind of instabilities, i.e., currentdriven instabilities and pressuredriven instabilities. In threedimensional (3D) configurations with vacuum magnetic flux surfaces,the equilibria can be obtained without net toroidal current, where the currentdriven instabilities become unimportant, and only the pressuredriven instabilities need to be intensively studied. The pressuredriven modes consists of interchange modes and ballooning modes,and impose MHD stability β limits. [J.P. Freidberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987] . Interchange modes are basically driven by average unfavorable magnetic curvature. Thus these modes localize on mode rational magnetic field lines and are almost constant along these lines B･∇ ζψ～0.On the other hand, ballooning modes are basically driven by locally unfavorable magnetic curvature, so that they localize on unfavorable magnetic curvature region and change along the magnetic field line B･∇ ζψ≠O. Ballooning modes are considered to be more stringent than interchange modes, whose properties have not yet been clarified in 3D configurations. To study the properties of ballooning modes, one can proceed in two different ways, namely, local mode analysis and global mode analysis. In axisymmetric systems, the global modes can be constructed easily from the results of the local modes analysis. But this is not the case in nonaxisymmetric systems, namely, 3D systems. In fully 3D systems, we can only make some conjectures for global modes from the properties of the local modes.<br /> 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)]:<br /> ・The local magnetic shear (which is a stabilizing term for highmodenumber 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 stellaratorlike global magnetic shear,Ieading to the destabilization of the highmodenumber ballooning modes.<br /> ・The local magnetic curvature (which constructs a potentially destabilizing term for highmodenumber ballooning modes together with the pressure gradient)consists of parts due to both toroidicity and helicity of the helical coils,which determines the 3D properties of the highmodenumber ballooning modes.<br /> In general 3D MHD equilibria, the eigenvalues ω2 for highmodenumber 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 αdependence in axisymmetric systems, the stronger the αdependence ofω2 is (mainly coming from the helicity part of the local magnetic curvature), the more significant the 3D 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)]:<br /> ・In Mercier unstable cquilibria, there coexist two types of topological level surfaces for ω2in (ψ, θk, α) space. One is a tokamaklike cylindrical level surface with the axis in α direction, the other is a spheroidal level surface inherent to 3D 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.<br /> ・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 Toroidicityinduced Alfv Eigenmodes (TAE).<br /> From these results it is conjectured that the global structure of pressuredriven modes has the following properties［N.Nakajima, Phys. Plasmas 3,4556(1996)]:<br /> ・Global modes that correspond to modes in the local mode analysis with a cylindrical level surface will be poloidal localized tokamaklike ballooning modes or interchange modes. Effects of the toroidal mode coupling on these modes are weak.<br /> ・Global modes that correspond to modes in the local mode analysis with a spheroidal level surface will be ballooning modes inherent to 3D 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.<br /> ・In Mercier unstable equilibria, Where both cylindrical and spheroidal level surface coexist, tokamaklike 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 3D systems appear with larger growth rates.<br /> ・In Mercier stable equilibria, where only a spheroidal level surface exists, only ballooning modes inherent to 3D systems appear.<br /> The purposes of the work are to confirm the above conjecture and to clarify the inherent properties of pressuredriven 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. Mercierunstable equilibria].<br /> First the Mercierunstable equilibria are categorized into two types, namely, toroidicitydominant Mercierunstable equilibria and helicitydominant Mercierunstable equilibria. This categorization is motivated by the conjecture that tokamaklike ballooning modes or interchange modes exist for relatively small toroidal mode numbers, and is related to the local properties of Mercierunstable 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 finiteβ 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 toroidicitydominant Mercierunstable 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 helicitydominant Mercierunstable 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.<br /> On the basis of these considerations, the following two types of Mercierunstable equilibria have been adopted. The toroidicitydominant Mercierunstable equilibrium is created with a peaked pressure profile P = P0(1ψN)2 and β0=5.9%, under the flux conserving condition, i.e., with a specified profile for the rotational transform. The helicitydominant Mercierunstable equilibrium is created with a broad pressure profile P = P0(1ψ2N)2 and β0=4.0%, under the currentless condition.<br /> The global mode analysis are done by CAS3D2MN, a version of CAS3D: Code for Analysis of the MHD Stability of 3D equilibrium [C. Schwab, Phys. Fluids B 5, 3195 (1993)]. CAS3D have been designed to analyze the global ideal MHD modes of 3D equilibria based on a formulation of the ideai MHD energy principle with incompressibility and fixed boundary in Boozer coordinate system and the application of RitzGalerkin method. In CAS3D2MN, a phasefactor transformation was used in order to save memory and flops.<br /> 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 reorthogonalization which keeps the matrix bandwidth from begin to end. The arithmetic operation mainly come from the matrixvector 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 shiftandinvert 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)].<br /> 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 pressuredriven modes in such Mercierunstable equilibria dramatically change according to how much the local magnetic shear is reduced (whether the equilibrium is toroidicitydominant or helicitydominant) 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.<br /> In the toroidicitydominant Mercierunstable equilibria, the pressuredriven modes change from interchange modes with negligible toroidal mode coupling for low toroidal mode numbers n<M, to tokamaklike poloidally localized ballooning modes with weak toroidal mode coupling for moderate toroiral mode numbers n～M, and finally to both poloidally and toroidally localized ballooning modes purely inherent to 3D systems with strong poloidal and toroidal mode couplings for fairly high toroidal mode numbers n》M. Strong toroidal mode coupling, in cooperation with the poloidal mode coupling, makes the perturbation localize to flux tubes.<br /> In the helicitydominant Mercierunstable equilibria, the pressuredriven modes change from interchange modes, with negligible toroidal mode coupling for n<M or with weak toroidal mode coupling for n?M, directly to poloidally and toroidally localized ballooning modes purely inherent to 3D systems with strong poloidal and toroidal mode couplings for n》M.<br /> In the Mercierunstable equilibria, interchange modes with low toroidal mode numbers n<M, experiencing the unfavorable magnetic curvature with its local structure averaged out, occur for both toroidicitydominant and helicitydominant equilibria. For fairly high toroidal mode numbers n》M, the perturbations can feel the fine local structure of the magnetic curvature due to helicity and also the local magnetic shear is reduced more or less in both types of equilibria, and consequently poloidally and toroidally localized ballooning modes inherent to 3D systems are destabilized for both toroidicitydominant and helicitydominant Mercierunstable equilibria. The situation for moderate toroidal mode numbers n?M is diffecrent. The local magnetic shear is more reduced in toroidicitydominant Mercierunstable equilibria than in helicitydominant Mercierunstable equilibria, and also the modes with moderate toroidal mode numbers n～M can not effectively feel the local structure of the normal magnetic curvature due to helicity. Thus, tokamaklike poloidally localized ballooning modes with a weak toroidal mode coupling can be easily destabilized for toroidicitydominant Mercierunstable equilibria, and interchange modes, driven by the average unfavorable magnetic curvature and not experiencing the effect of toroidal mode coupling, can be destabilized for helicitydominant Mercierunstable equilibria. Since the normal magnetic curvature becomes more unfavorable on the inner side than on the outer side of the torus by the Shafranov shift, the interchange modes are localized on the inner side of the torus for both types of equilibria. This type of interchanges mode is antiballooning with respect to the poloidal mode coupling.<br /> In both types of Mercierunstable equilibria, the pressuredriven modes, i.e., ballooning modes and interchange modes, become more unstable and more localized both on flux tubes and in the radial direction, and have stronger toroidal mode coupling through the normal magnetic curvature due to helicity, as the typical toroidal mode numbers increase. Thus, we can expect that ballooning modes localized in one toroidal field period, as suggested in [N. Nakajima, Phys. Plasmas 3, 4556 (1996)], may occur with very narrower radial extent and larger growth rates, as the typical toroidal mode numbers become larger and larger. All of these properties of the pressuredriven modes in two types of Mercierunstable equilibria are quite consistent with the conjecture from local mode analysis.  
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