Magnetohydrodynamics and Spectral Theory

Lifschitz, Alexander

doi:10.1007/978-94-009-2561-8

Cited by 161 publications

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“…The spectral properties of operator functions play an important role in mathematical analysis and in many applications [4,13,20]. A classic enclosure of the spectrum is the closure of the numerical range [15].…”

Section: Introductionmentioning

confidence: 99%

Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

Engström

Torshage

2017

Integr. Equ. Oper. Theory

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Abstract. In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.Mathematics Subject Classification. 47J10, 47A56, 47A12.

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Section: Introductionmentioning

confidence: 99%

Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

Engström

Torshage

2017

Integr. Equ. Oper. Theory

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“…Dewar and Glasser [7] argued that this gives rise to a continuous unstable spectrum, with correspondingly singular generalized eigenfunctions. (A more rigorous treatment involves the concept of the essential spectrum and Weyl sequences [13,14]. )…”

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confidence: 99%

Strong “Quantum” Chaos in the Global Ballooning Mode Spectrum of Three-Dimensional Plasmas

2001

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The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a k-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.PACS numbers: 52.35. Py, 52.55.Hc, 05.45.Mt In design studies for new magnetic confinement devices for fusion plasma experiments (e.g. investigations [1,2] leading to the proposed National Compact Stellarator Experiment, NCSX [3]), the maximum pressure that can stably be confined in any proposed magnetic field configuration is routinely estimated by treating the plasma as an ideal magnetohydrodynamic (MHD) fluid. One linearizes about a sequence of equilibrium states with increasing pressure, and studies the spectrum of normal modes (frequency ω) to determine when there is a component with Im ω > 0, signifying instability.Even with the simplification obtained by using the ideal MHD model, the computational task of determining the theoretical stability of a three-dimensional (i.e. nonaxisymmetric) device, such as NCSX or the four currently operating helical axis stellators [4], remains a challenging one.The problem can be posed as a Lagrangian field theory, with the potential term being the energy functional δW [5]. For a static equilibrium, the kinetic energy is quadratic in ω, so that ω 2 is real. Thus instability occurs when ω 2 < 0. There are two main approaches to analyzing the spectrum-local and global. * Permanent address: Research School of Physical Sciences & Engineering, The Australian National University. E-mail: robert.dewar@anu.edu.au.In the local approach, which is used for analytical simplification, one orders the scale length of variation of the eigenfunction across the magnetic field lines to be short compared with equilibrium scale lengths [6]. Both interchange and ballooning stability can be treated by solving the general ballooning equations [7], a system of ordinary differential equations defined on a given magnetic field line.The global (Galerkin) approach is to expand the plasma displacement field in a finite basis set, inserting this ansatz in the Lagrangian to find a matrix eigenvalue representation of the spectral problem. This approach has been implemented for ideal MHD in threedimensional plasmas in two codes, TERPSICHORE [8] and CAS3D [9].Although the Galerkin approach is potentially exact, if one could use a complete, infinite basis set, it is in practice computationally challenging due to the large number of basis functions required to resolve localized instabilities. This leads to very large matrices which must be diagonalize...

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“…Force-free magnetic fields appear as an important class of special solutions of nonlinear equations of magnetohydrodynamics and are intensively studied in different branches of modern physics (see, e.g., [1], [6], [8], [10], [11], [20], [21], [22], [25], [26]). They are characterized by the following pair of equations div B = 0 (13) and…”

Section: Force-free Magnetic Fieldsmentioning

confidence: 99%

Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

Kravchenko¹,

Kravchenko²

2003

J. Phys. A: Math. Gen.

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Magnetohydrodynamics and Spectral Theory

Abstract: Magnetohydrodynamics and spectral theory. (Developments in electromagnetic theory and applications ; 4) Includes index. 1. Magnetohydrodynamics. 2. Spectral theory (Mathematics) I. Title. II. Series.

Cited by 161 publications

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Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

Strong “Quantum” Chaos in the Global Ballooning Mode Spectrum of Three-Dimensional Plasmas

Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

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