Potential form of the MHD equations

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Table of contents

1 Introduction 
1.1 Motivation
1.2 Outline of this thesis
1.3 Magnetohydrodynamic equations
1.3.1 Maxwell equations
1.3.2 Navier-Stokes equations
1.3.3 Magnetohydrodynamics equations
1.4 Dynamo effect
1.4.1 Concluding remarks
I System description 
2 The von Kármán flow 
2.1 Brief overview of studies
2.1.1 History
2.1.2 Stability analysis
2.1.3 Turbulence
2.2 Mathematical model
2.2.1 System description
2.2.2 Dimensionless equations
2.2.3 Equations of magnetohydrodynamics
3 Poloidal-toroidal decomposition 
3.1 Motivation
3.2 Poloidal-toroidal decomposition and the gauge freedom
3.3 Potential form of the MHD equations
3.4 Compatibility condition
3.4.1 Equivalence of potential and primitive variable formulation
3.4.2 Hydrodynamic compatibility condition
3.4.3 Magnetic compatibility condition
3.5 Hydrodynamic boundary conditions
3.6 Magnetic boundary conditions
3.6.1 General case
3.6.2 Conductor/vacuum configuration
3.7 Discussion
3.7.1 Advantages/disadvantages
3.7.2 Concluding remarks
II Numerical method – Spectral solver 
4 Spectral discretization 
4.1 Introduction
4.1.1 Local methods
4.1.2 Spectral precision
4.1.3 Advantages and limitations of spectral methods
4.2 Azimuthal direction
4.3 Axial direction
4.4 Radial direction
4.4.1 Regularity condition
4.4.2 Regularization of an arbitrary spectral basis
4.4.3 Regular basis of radial polynomials
4.4.4 Differential operators
4.5 Discretization in 3D
4.6 Boundary condition regularization
4.6.1 Overview of singularity treatment techniques
4.6.2 Boundary velocity regularization
5 Spectral solver 
5.1 Introduction
5.2 Poisson solver
5.2.1 τ method – one dimension
5.2.2 Partial diagonalization method – two/three dimensions
5.2.3 Solving the Helmholtz equation – τ-method in radial direction
5.3 High order PDE solver
5.3.1 2D Navier-Stokes in streamfunction formulation
5.3.2 Multi-Poisson solver
5.4 Influence matrix
5.4.1 Green function method
5.4.2 Discrete Green functions method – influence matrix
5.4.3 Towards an invertible influence matrix
5.5 Tests – Stokes problem in 2D
5.5.1 Polynomial solutions
5.5.2 Non-polynomial case
5.6 Towards an MHD solver
5.6.1 External solution in vacuum
5.6.2 Continuity conditions
5.6.3 Multi-Poisson solver for induction equation
5.6.4 Influence matrix for the magnetic problem
5.6.5 Elimination of external solution
5.7 Concluding remarks
6 Stability/Validation 
6.1 Nonlinear term
6.1.1 Evaluation of −u × w
6.1.2 Regularity of the nonlinear term
6.2 Time integration
6.3 Tests
6.3.1 Axisymmetric rotor-stator configuration
6.3.2 First instability in 3D
6.4 Spectral convergence
6.5 Parallelization
6.6 Concluding remarks
7 Applications & perspectives 
7.1 Turbulent bifurcation
7.1.1 The effect of blades
7.1.2 Preliminary results
7.2 Axisymmetric turbulence
7.2.1 Theoretical framework
7.2.2 Experimental confirmations
7.2.3 Numerical results
7.2.4 Imposed axisymmetry
8 Concluding remarks 
A Recursive formulas for radial polynomials 
B Remarks concerningMPI parallelization