Magnetohydrodynamics and Spectral Theory

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 Alfven 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.