عنوان

Magnetohydrodynamics and Spectral Theory

9400925611

9789400925618

b592899

Magnetohydrodynamics and Spectral Theory

[Book]

by Alexander E. Lifschitz.

Dordrecht

Springer Netherlands

1989

(XII, 446 pages).

Developments in electromagnetic theory and applications, 4.

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.

2 The linearized ideal MHO equations. ... ... . 204 3 Spectral problems corresponding to evolutionary problems . . 211 4 Stability of equilibrium configurations and the Energy Principle 215 5 Alternative forms of the plasma potential energy 220 6 Minimization of the potential energy with respect to a parallel displacement ... ... . 222 7 Classification of ideal MHO instabilities . 224 8 The linearized non-ideal MHO equations . 226 Chapter 6. Homogeneous and discretely structured plasma oscillations 229 I Introduction ... ... ... 229 2 Alfven waves in an incompressible ideal plasma 230 3 Cold ideal plasma oscillations. ... 233 4 Compressible hot plasma oscillations 236 5 Finite resistivity effects ... . 239 6 Propagation of waves generated by a local source 240 7 Stratified plasma oscillations ... ... 247 8 Oscillations of a plasma slab ... ... 254 9 Instabilities of an ideal stratified gravitating plasma 256 10 Instabilities of a resistive stratified gravitating plasma. 262 Chapter 7. MHO oscillations of a gravitating plasma slab 265 I Introduction ... ... ... 265 2 Gravitating slab equilibrium ... . . 266 3 Oscillations of a hot compressible plasma slab 267 4 Investigation of the slab stability via the Energy Principle 270 5 On the discrete spectrum of the operator Kk ... 274 6 On the essential spectrum of the operator Kk ... 279 7 On the discrete spectrum embedded in the essential spectrum 282 8 The eigenfunction expansion formula ... ... 285 9 Excitation of plasma oscillations by an external power source . 288 10 The linearized equations governing resistive gravitating plasma slab oscillations ... ... ... ... 290 II Heuristic investigation of resistive instabilities. ... ...

Developments in Electromagnetic Theory and Applications, vol. 4

Magnetohydrodynamics.

Nuclear physics.

Physics.

by Alexander E. Lifschitz.

Alexander E Lifschitz

[Book]

Y