Magnetohydrodynamics and Spectral Theory [E-Book] / by Alexander E. Lifschitz.
Lifschitz, Alexander E., (author)
Dordrecht : Springer, 1989
XII, 446 p. online resource.
englisch
9789400925618
10.1007/978-94-009-2561-8
Developments in Electromagnetic Theory and Applications ; 4
Full Text
Table of Contents:
  • 1. Historical background and outline of the contents
  • 1 Introduction
  • 2 Historical background
  • 3 Outline of Chapter 2
  • 4 Outline of Chapter 3
  • 5 Outline of Chapter 4
  • 6 Outline of Chapter 5
  • 7 Outline of Chapter 6
  • 8 Outline of Chapter 7
  • 9 Outline of Chapter 8
  • 10 Outline of Chapter 9
  • 2. Preliminaries from the spectral theory of operators
  • 1 Introduction
  • 2 Distribution theory
  • 3 Hilbert spaces
  • 4 Linear functionals and linear operators
  • 5 Linear bounded functionals and linear bounded operators
  • 6 Linear unbounded operators
  • 7 Self-adjoint extensions of symmetric operators
  • 8 Spectral theorem for self-adjoint operators
  • 9 Rigged Hilbert spaces and expansions in terms of generalized eigenvectors of self-adjoint operators
  • 10 The stability investigation via the Energy Principle
  • 11 Preliminaries from the calculus of variations
  • 12 The eigenvalue problem for a system of two first-order ordinary differential equations on a finite segment
  • 3. Basic equations and their main properties
  • 1 Introduction
  • 2 Collisional Maxwell-Boltzmann equations
  • 3 Macroscopic variables describing a plasma
  • 4 The properties of the collisional integral
  • 5 Two-fluid equations
  • 6 Asymptotic simplification of two-fluid equations
  • 7 Single-fluid MHD equations
  • 8 Derivation of MHD equations for conducting fluid
  • 9 Dimensionless parameters
  • 10 Conservative form of the ideal MHD equations and local conservation laws
  • 11 Boundary and initial conditions for MHD equations
  • 12 Global conservation laws and virial theorem
  • 13 Lagrangian form of the ideal MHD equations
  • 4. Magnetohydrodynamical equilibrium
  • 1 Introduction
  • 2 Basic equations
  • 3 On equilibrium of a toroidal plasma with a surface current
  • 4 Toroidal equilibrium configurations with nested magnetic surfaces
  • 5 Natural coordinates
  • 6 Hamada coordinates
  • 7 Equilibrium configurations with closed lines of force
  • 8 Force-free equilibria
  • 9 Structure of the general three-dimensional magnetic field
  • 10 Equilibrium configurations with continuous symmetry
  • 11 Homogeneous and piecewise inhomogeneous equilibria
  • 12 One-dimensional inhomogeneous equilibrium plasma slabs
  • 13 Radially inhomogeneous equilibrium plasma pinches
  • 14 General properties of two-dimensional equilibrium configurations
  • 15 Translationally symmetric equilibrium configurations
  • 16 Axisymmetric equilibrium configurations. The Grad-Shafranov equation
  • 17 Helically symmetrie equilibrium configurations
  • 18 Straight plasma column of elliptic cross-section with homogeneous current
  • 19 Toroidal equilibrium configuration of circular cross-section with quasi-homogeneous current
  • 20 Asymptotic approach to the axisymmetric equilibrium problem
  • 21 The standard tokamak
  • 22 The high-ß tokamak
  • 23 The three-dimensional analogue of the Grad—Shafranov equation
  • 24 Stationary solutions of the resistive MHD equations
  • 5. The linearized MHD equations and their general properties
  • 1 Introduction
  • 2 The linearized ideal MHD equations
  • 3 Spectral problems corresponding to evolutionary problems
  • 4 Stability of equilibrium configurations and the Energy Principle
  • 5 Alternative forms of the plasma potential energy
  • 6 Minimization of the potential energy with respect to a parallel displacement
  • 7 Classification of ideal MHD instabilities
  • 8 The linearized non-ideal MHD equations
  • 6. Homogeneous and discretely structured plasma oscillations
  • 1 Introduction
  • 2 Alfvén waves in an incompressible ideal plasma
  • 3 Cold ideal plasma oscillations
  • 4 Compressible hot plasma oscillations
  • 5 Finite resistivity effects
  • 6 Propagation of waves generated by a local source
  • 7 Stratified plasma oscillations
  • 8 Oscillations of a plasma slab
  • 9 Instabilities of an ideal stratified gravitating plasma
  • 10 Instabilities of a resistive stratified gravitating plasma
  • 7. MHD oscillations of a gravitating plasma slab
  • 1 Introduction
  • 2 Gravitating slab equilibrium
  • 3 Oscillations of a hot compressible plasma slab
  • 4 Investigation of the slab stability via the Energy Principle
  • 5 On the discrete spectrum of the operator Kk
  • 6 On the essential spectrum of the operator Kk
  • 7 On the discrete spectrum embedded in the essential spectrum
  • 8 The eigenfunction expansion formula
  • 9 Excitation of plasma oscillations by an external power source
  • 10 The linearized equations governing resistive gravitating plasma slab oscillations
  • 11 Heuristic investigation of resistive instabilities
  • 12 Analytic evaluation of increments of resistive instabilities
  • 13 Investigation of the slab stability via the Resistive Energy Principle
  • 14 Global resistive interchange quasi-modes
  • 15 On stable resistive modes eigenfrequencies
  • 8. Oscillations of a plasma pinch
  • 1 Introduction
  • 2 General cylindrical equilibrium configuration
  • 3 Cylindrical force operator
  • 4 The Energy Principle for the pinch stability
  • 5 Criterion of the general screw-pinch stability
  • 6 Spectral properties of the cylindrical force operator
  • 7 Secondary heating of a pinch by an external antenna
  • 8 Stability of the straight tokamak
  • 9 Resistive equations governing general screw-pinch oscillations
  • 10 Resonant resistive instabilities of the general screw-pinch
  • 11 Resistive instabilities of the straight tokamak
  • 12 The stable resistive spectrum structure
  • 9. MHD oscillations of two- and three-dimensional configurations
  • 1 Introduction
  • 2 Two- and three-dimensional equilibrium configurations
  • 3 Basic equations
  • 4 The Mercier criterion
  • 5 The ballooning stability criterion
  • 6 Axisymmetric (n = 0) instabilities of the free-boundary plasma
  • 7 Spectral properties of the force operators KIn and KIIn
  • 8 On the essential spectrum of the force operator K
  • 9 Natural oscillations of three-dimensional configurations
  • 10 Natural oscillations of three-dimensional configurations with closed magnetic lines
  • 11 Resonant resistive instabilities of axisymmetric toroidal configurations
  • References.