Plasma Physics, Programming, and Penguins

Plasma fluid theory lectures

This is a series of short lectures on plasma physics, and particularly plasma fluid theory. These were produced with the support of the National Research Foundation of Korea and the Ministry of Science and ICT, who kindly made them available as a YouTube playlist. The notes and Jupyter notebooks are listed individually below, or you can download all notes [.tar.gz].

Fluid equations

These videos introduce models for gasses and fluids, which use the same concepts as the more complicated plasma models.

Analytic solutions to the fluid equation

Ideas and tools for solving fluid equations, and finding analytic solutions.

  • Lecture 2a - Equilibrium solutions: Force balance, hydrostatic equilibrium, viscous drag
  • Lecture 2b - Linearisation and waves: Expansion in small parameters, linearisation, and wave dispersion relations
  • Lecture 2c - Buoyancy waves and instabilities: Density stratification, waves and instabilities, dispersion relation
  • Lecture 2d - Rayleigh-Taylor instability: Energy principle, unstable buoyancy waves, boundary layer matching, Atwood number
  • Lecture 2e - Shocks: Conservation laws and discontinuous solutions, Rankine-Hugoniot equations, limits to compression, entropy constraint

Incompressible flows and reduced fluid equations

Simplifying fluid equations by exploiting small parameters. Introduces the concept of vorticity.

  • Lecture 3a - Incompressible flow 1/2: Convective derivative, Mach number, characteristic scales and normalisation, singular limits
  • Lecture 3b - Incompressible flow 2/2: Normalisation, singular limits
  • Lecture 3c - Irrotational flow: Static and dynamic pressure, irrotational flows, vorticity
  • Lecture 3d - Vorticity: Local rotation and vorticity, the vorticity equation, 2D flow

Numerical solutions to the fluid equations

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Introduction to finite difference schemes, and concepts of accuracy and stability.

Fluid equations for plasmas

Extends the ideas and methods from liquids and gasses to plasmas. The derivations in this lecture are (I hope) more intuitive than later derivations, but are less rigorous. Later in the series I'll come back to these topics and go into more detail.

  • Lecture 5a - Collisions in plasmas: Review assumptions in fluid models, typical mean free path values, validity of fluid models
  • Lecture 5b - Ideal MHD: Review neutral fluid (Euler) equations, low frequency electromagnetic forces, assumptions of ideal MHD, some applications
  • Lecture 5c - Plasma fluids from the kinetic equation: Boltzmann equation for plasmas, density and momentum equations
  • Lecture 5d - Single fluid equations: Low frequency approximation, quasineutrality, collisional regime

Equilibrium solutions to ideal MHD

Toroidal systems

Introduces the toroidal devices most widely used in fusion research, particularly the tokamak.

  • Lecture 7a - Toroidal devices: Why toroidal? Particle orbits in a toroidal magnetic field, MHD force balance
  • Lecture 7b - Grad-Shafranov equation: Force balance in an axisymmetric torus, poloidal and toroidal magnetic fields, poloidal flux, flux surfaces
  • Lecture 7c - Shafranov shift: Hoop and tyre forces, need for vertical magnetic field
  • Lecture 7d - Tokamak coils: Toroidal field coils, solenoid, poloidal field coils
  • Lecture 7e - Vertical stability: X-point configurations, plasma elongation, passive and active stabilisation

Toroidal coordinates

A lot of analytic and numerical studies of toroidal plasmas use coordinate systems which are aligned to curved paths and surfaces. This simplifies the analysis, but some ideas from differential geometry are needed and introduced here.

  • Lecture 8a - Coordinate systems: Cylindrical coordinates, transforming vectors and derivatives, tangent and reciprocal basis vectors
  • Lecture 8b - Non-orthogonal coordinates: Advantages of flux coordinates, screw pinch in cylindrical coordinates, aligning coordinates to the magnetic field
  • Lecture 8c - Flux coordinates: Clebsch form of the magnetic field, magnetic flux in a torus, poloidal angle, safety factor and local field pitch

Analytic solutions to the Grad-Shafranov equation

  • Lecture 9a - Large aspect ratio: Toroidal coordinates, large aspect ratio limit of Grad-Shafranov equation
  • Lecture 9b - Shafranov shift: Toroidal corrections to the Grad-shafranov expansion
  • Lecture 9c - Vertical magnetic field: Outer solution in large aspect ratio, matching interior and exterior solutions

Numerical solutions to the Grad-Shafranov equation

These cover the different kinds of numerical tools used to solve the Grad-Shafranov equation. Fixed and free boundary solvers are discussed. Demonstrations are using FreeGS, a code in python which is publicly available.

Linearised single-fluid equations

  • Lecture 11a - Linearisation of Ideal MHD: Small perturbations, linear equations, waves and unstable solutions
  • Lecture 11b - Plasma waves: Waves in homogenous plasma, shear Alfven waves, fast and slow magnetosonic waves
  • Lecture 11c - Normal modes: Derive linear equation for displacements, show that frequency is purely real or imaginary, show that discrete modes are orthogonal
  • Lecture 11d - The energy principle: Testing for plasma stability, intuitive form of energy principle, good and bad curvature

Plasma instabilities

  • Lecture 12a - Types of instabilities: internal and external modes, interchange and ballooning modes, kink modes
  • Lecture 12b - Incompressibility: minimisation of delta-W (variational method), singular surfaces
  • Lecture 13a - Pressure-driven modes: sausage instability, MHD picture, single particle picture
  • Lecture 13b - Ballooning modes: Field line bending, pressure (beta) limits
  • Lecture 13c - Magnetic mirrors: simple mirror, pressure-driven instability, Ioffe bars, baseball and yin-yang coils, tandem mirrors
  • Lecture 13d - Delta-W pinch stability: Surface terms, theta pinch, Z pinches
  • Lecture 14a - Current-driven instabilities: Toroidal pinch, kink modes, stabilisation by walls and toroidal B field, sawteeth, kinks and tearing modes
  • Lecture 14b - Flux surface perturbations: Response of plasma to flux surface movement, cylindrical tearing mode equation, resonant surfaces
  • Lecture 14d - Tearing modes: Magnetic field diffusion, tearing stability index, Rutherford equation
  • Lecture 14e - Resistive Wall Modes: Wall stabilisation, resistive diffusion, mode locking
  • Lecture 15a - The ballooning equation: Ordering for localised modes, lowest order delta-W, Euler equation for perturbations
  • Lecture 15b - The ballooning transform: periodicity

Conserved quantities in nonlinear evolution

Plasma nonlinear behaviour can be very complicated. Conserved quantities provide a way to remove a lot of complication, and can help us to understand the end result, even if we don't understand the details of how the plasma gets there.

  • Lecture 16a - MHD shocks: Conservation laws and discontinuities, parallel (hydrodynamic) shocks, perpendicular shocks, contact discontinuities, tangential discontinuities
  • Lecture 16b - Frozen flux theorem: Proof, magnetic field topology, flux tubes, magnetic field velocity
  • Lecture 16c - Magnetic helicity: Definition, conservation in ideal MHD, cross helicity
  • Lecture 16d - Woltjer's theorem: Minimisation of magnetic energy, force free field
  • Lecture 16e - Taylor relaxation: Toroidal pinches (ZETA), quiescent periods, Taylor's conjecture, solution in large aspect ratio

Dissipation

Real plasmas are not exactly ideal, so these lectures consider in more detail some of these non-ideal effects.

Numerical solution to nonlinear equations

  • Lecture 18a - Numerical methods for nonlinear problems: Summary of issues to consider, conservation properties
  • Lecture 18b - Conservation properties: Directly conserved (fluxes) and indirectly conserved (integration by parts)
  • Lecture 18c - Preserving Div B = 0: Constraints on evolution equations, vector potential, divergence cleaning, constrained transport
  • Lecture 18d - Staggered schemes: Advantages and disadvantages, dispersion relation

Extended MHD, two-fluid equations

  • Lecture 19a - Two-fluid equations: Review derivation from Boltzmann equation, collision operator moments, types of closure
  • Lecture 19c - Generalised Ohm's law: Electron momentum equation, resistivity, Hall terms, finite electron mass terms
  • Lecture 19d - Hall MHD: Magnetic reconnection, plasma dispersive waves, Biermann battery

Linear waves in extended MHD

  • Lecture 20a - Cold plasma wave dispersion: Conductivity tensor, plasma response, refractive index, Languir and whistler waves
  • Lecture 20b - Whistler waves: Observations (radio), group velocity, dispersion relation, nose whistlers
  • Lecture 20c - Helicon waves: Structure of helicon/whistler waves, launching with antennas (Helicon sources), plasma sources
  • Lecture 20d - Electron drift waves: Adiabatic (Boltzmann) response, "Universal" drift wave instability in a slab
  • Lecture 20e - ITG instability: Particle drift picture, extended MHD model, FLR effects, gyroviscous cancellation, dispersion relation

Boundary conditions

Laboratory plasmas at some point have to come into contact with a solid surface or boundary. Close to the wall the difference between electron and ion dynamics becomes important, and the quasineutrality assumption made in the bulk of the plasma is no longer valid.

Numerical methods for extended MHD

  • Lecture 23a - Implicit methods: Time step restrictions, backward Euler implicit method, damping of unresolved waves, semi-implicit and IMEX schemes
  • Lecture 23b - Field aligned coordinates: Non-aligned coordinates, flux tube "ballooning" coordinates, shifted metric method, flux coordinate independent (FCI) scheme

Reduced MHD

  • Lecture 24b - Polarisation drift: Single particle picture, derivation from single particle momentum, derivaion from fluid momentum
  • Lecture 24c - Drift-ordered equations: Simple derivatrion of reduced equations from particle drifts, vorticity equation
  • Lecture 26a - Poisson brackets in Reduced MHD: In field-aligned coordinates, conservation properties, numerical conservation laws, Arakawa brackets

CGL-MHD double-adiabatic closure

  • Lecture 27a - CGL-MHD: Anisotropic pressure, double adiabatic closure
  • Lecture 27b - Derivation of CGL MHD: Collisionless Vlasov equation, outline of derivation
  • Lecture 27c - Firehose and mirror instabilities: CGL MHD dispersion relation, parallel firehose "centrifugal" instability, perpendicular mirror "diamagnetic" instability