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  • 11
    Electronic Resource
    Electronic Resource
    Toroidal coupling of ideal magnetohydrodynamic instabilities in tokamak plasmas (1996)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Plasmas 3 (1996), S. 584-592 
    Details
    ISSN: 1089-7674
    Source: AIP Digital Archive
    Topics: Physics
    Notes: A theoretical framework is developed to describe the ideal magnetohydrodynamic (MHD) stability properties of axisymmetric toroidal plasmas. The mode structure is described by a set of poloidal harmonics in configuration space. The energy functional, δW, is then determined by a set of matrix elements that are computed from the interaction integrals between these harmonics. In particular, the formalism may be used to study the stability of finite-n ballooning modes. Using for illustration the s-α equilibrium, salient features of the n(large-closed-square)∞ stability boundary can be deduced from an appropriate choice of test function for these harmonics. The analysis can be extended to include the toroidal coupling of a free-boundary kink eigenfunction to the finite-n ideal ballooning mode. A unified stability condition is derived that describes the external kink mode, a finite-n ballooning mode, and their interaction. The interaction term plays a destabilizing role that lowers the instability threshold of the toroidally coupled mode. These modes may play a role in understanding plasma edge phenomena, L–H physics and edge localized modes (ELMs). © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.871886
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  • 12
    Electronic Resource
    Electronic Resource
    Ideal magnetohydrodynamic stability of the tokamak high-confinement-mode edge region : The 40th annual meeting of the division of plasma physics of the american physical society (1999)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Plasmas 6 (1999), S. 1925-1934 
    Details
    ISSN: 1089-7674
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The ideal magnetohydrodynamic (MHD) stability of the tokamak edge is analyzed, with particular emphasis on radially localized instabilities; it is proposed that these are responsible for edge pressure gradient limits and edge localized modes (ELMS). Data and stability calculations from DIII-D [to appear in Proceedings of the 16th International Conference on Fusion Energy, Yokohama (International Atomic Energy Agency, Vienna, 1998), Paper No. IAEA-F1-CN-69/EX8/1] tokamak equilibria indicate that two types of instability are important: the ballooning mode (driven by pressure gradient) and the peeling mode (driven by current density). The characteristics of these instabilities, and their coupling, are described based on a circular cross-section, large aspect ratio model of the tokamak equilibrium. In addition, preliminary results are presented from an edge MHD stability code which is being developed to analyze general geometry tokamak equilibria; an interpretation of the density threshold to access the high-confinement-mode (H-mode), observed on COMPASS-D [Plasma Phys. Controlled Fusion 38, 1091 (1996)] is provided by these results. Experiments on DIII-D and the stability calculations indicate how to control ELMs by plasma shaping.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.873492
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  • 13
    Electronic Resource
    Electronic Resource
    Collisionless and resistive ballooning stability (1999)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Plasmas 6 (1999), S. 4260-4264 
    Details
    ISSN: 1089-7674
    Source: AIP Digital Archive
    Topics: Physics
    Notes: It has been suggested [Kleva and Guzdar, Phys. Plasmas 6, 116 (1999)] that reconnecting ballooning modes in which electron inertia replaces resistivity in a nonideal magnetohydrodynamic Ohm's law can have substantial growth rates in the low collisionality regime. Numerical calculation, albeit necessarily at unrealistically large values of the collisionless skin depth, showed that strongly growing ballooning modes exist at beta values which are below the ideal beta limit. In order to investigate stability at more realistic values of the skin depth we exploit an analytic approach. As in the case of resistive ballooning modes, we find that inertial ballooning modes are stabilized by favorable average curvature effects at moderate values of ΔB′, the stability index for resistive ballooning. Instability only becomes possible close to the ideal stability boundary (ΔB′→∞) or at unrealistically large values of the toroidal mode number n (e.g., n(approximately-greater-than)102). Another ballooning mode, the collisionless analogue of the Carreras–Diamond mode [Carreras, Diamond, Murakami, Dunlap et al., Phys. Rev. Lett. 50, 503 (1983)] can also be excited at larger values of the collisionless skin depth, but this mode is not valid for realistic parameters in a hot plasma. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.873693
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  • 14
    Electronic Resource
    Electronic Resource
    The mode structure of high-n resistive ballooning modes (1992)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 4 (1992), S. 56-63 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The higher-order corrections (in an n−1/2 expansion) to resistive ballooning theory are analyzed in order to gain information about the radial structure of the Δ'-driven modes. This higher-order theory also predicts that the ballooning phase angle θ0 (which is undetermined in the leading-order theory) must be chosen so as to maximize the value of Δ'. The importance of applying this maximization is illustrated by an analytical calculation of Δ' as a function of θ0 for the s-α model in the limit of small α. It is demonstrated that for this case, one should choose θ0=90° and that the resulting value of Δ' can be very different from that obtained by setting θ0=0, as is frequently imposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.860403
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  • 15
    Electronic Resource
    Electronic Resource
    Stability of toroidicity-induced drift waves in divertor tokamaks (1989)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 1 (1989), S. 1449-1458 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The stability of toroidicity-induced drift waves in a tokamak equilibrium with magnetic separatrix is studied both analytically and numerically. In particular, the task of a proper determination of the complex ballooning parameter θ0 is performed by solving the stationarity condition for the eigenvalue. Results show qualitative dependence on the location of the x point in the meridian plane. Specifically, locating the x point in the equatorial plane, both on the outside and on the inside of the plasma, causes a deepening of the well structure in the potential for the eigenmode, thereby enforcing the inhibition of the shear damping and the marginal stability result obtained in the circular magnetic surfaces case. On the other hand, the location of the x point at the top of the plasma produces a flattening of the well and restores the shear damping, yielding stabilization of the mode. A new quasimarginally stable branch, corresponding to modes localized around the x point, is shown to exist at high values of the separatrix parameter k and x-point location close to the equatorial plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.859201
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  • 16
    Electronic Resource
    Electronic Resource
    Tearing modes in toroidal geometry (1988)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 31 (1988), S. 577-590 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ' calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ') required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.866840
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  • 17
    Electronic Resource
    Electronic Resource
    Resistive ballooning modes in an axisymmetric toroidal plasma with long mean free path (1985)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 28 (1985), S. 2201-2208 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Tokamak devices normally operate at such high temperatures that the resistive fluid description is inappropriate. In particular, the collision frequency may be low enough for trapped particles to exist. However, on account of the high conductivity of such plasmas, one can identify two separate scale lengths when discussing resistive ballooning modes. By describing plasma motion on one of these, the connection length, in terms of kinetic theory the dynamics of trapped particles can be incorporated. On the resistive scale length, this leads to a description in terms of modified fluid equations in which trapped particle effects appear. The resulting equations are analyzed and the presence of trapped particles is found to modify the stability properties qualitatively.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.865272
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  • 18
    Electronic Resource
    Electronic Resource
    Resonant magnetohydrodynamic modes with toroidal coupling. Part II: Ballooning-twisting modes (1991)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 3 (1991), S. 1539-1545 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: This is Part II of a study of resonant perturbations, such as resistive tearing and ballooning modes, in a torus. These are described by marginal ideal magnetohydrodynamic (MHD) equations in the regions between resonant surfaces; matching across these surfaces provides the dispersion relation. Part I [Phys. Fluids B 3, 1532 (1991)] described how all the necessary information from the ideal MHD calculations could be represented by a so-called E matrix. The calculation of this E matrix for tearing modes (even parity in perturbed magnetic field) in a large-aspect-ratio torus was also described. There the toroidal modes comprise coupled cylinder tearing modes and the E matrix is a generalization of the familiar Δ' quantity in a cylinder. In the present paper, resistive ballooning, or twisting modes, which have odd parity in perturbed magnetic field, are discussed. Unlike the tearing modes, these odd-parity modes are intrinsically toroidal and are not directly related to the odd-parity modes in a cylinder. This is evident from the analysis of the high-n limit in ballooning space, where the twisting mode exhibits a singular transition at large aspect ratio when the interchange effect is small (as in a tokamak). Analysis of the high-n limit in coordinate space, rather than ballooning space, clarifies this singular behavior. It also yields a prescription for treating low-n twisting modes and a method for calculating an E matrix for resistive ballooning modes in a large-aspect-ratio tokamak in the limit the interchange term vanishes. The elements of this matrix are given in terms of cylindrical tearing mode solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.859993
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  • 19
    Electronic Resource
    Electronic Resource
    Resonant magnetohydrodynamic modes with toroidal coupling. Part I: Tearing modes (1991)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 3 (1991), S. 1532-1538 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: In a cylindrical plasma, tearing modes can be calculated by asymptotic matching of ideal magnetohydrodynamic (MHD) solutions across a critical layer. This requires a quantity Δ' that represents the "discontinuity'' in the ideal solution across the layer. In a torus, poloidal harmonics are coupled and there are many critical surfaces for each toroidal mode number, and correspondingly many discontinuities Δ'm. The ideal MHD solutions do not then determine the Δm but only a relation between them—described by an "E matrix.'' The calculation of the E matrix for a large-aspect-ratio tokamak is discussed. In a weak-coupling approximation, it is tridiagonal and can be computed from integrals over the uncoupled eigenfunctions or from simple "basis functions'' comprising triplets of coupled poloidal harmonics. This weak-coupling approximation fails if Δ'm is already small for an uncoupled harmonic. An alternative strong-coupling approximation is developed for this case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.859724
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  • 20
    Electronic Resource
    Electronic Resource
    Ballooning modes or Fourier modes in a toroidal plasma? (1987)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 30 (1987), S. 3180-3185 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The relationship between two different descriptions of eigenmodes in a torus is investigated. In one the eigenmodes are similar to Fourier modes in a cylinder and are highly localized near a particular rational surface. In the other they are the so-called ballooning modes that extend over many rational surfaces. Using a model that represents both drift waves and resistive interchanges the transition from one of these structures to the other is investigated. In this simplified model the transition depends on a single parameter which embodies the competition between toroidal coupling of Fourier modes (which enhances ballooning) and variation in frequency of Fourier modes from one rational surface to another (which diminishes ballooning). As the coupling is increased each Fourier mode acquires a sideband on an adjacent rational surface and these sidebands then expand across the radius to form the extended mode described by the conventional ballooning mode approximation. This analysis shows that the ballooning approximation is appropriate for drift waves in a tokamak but not for resistive interchanges in a pinch. In the latter the conventional ballooning effect is negligible but they may nevertheless show a ballooning feature. This is localized near the same rational surface as the primary Fourier mode and so does not lead to a radially extended structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.866493
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