Magnetohydrodynamics (MHD)¶

Overview¶

This topic covers advanced magnetohydrodynamics — the physics of electrically conducting fluids interacting with magnetic fields. Building on the MHD foundations in Numerical Simulation (L17-L18) and plasma physics fundamentals (Plasma_Physics), these lessons explore equilibrium theory, stability, magnetic reconnection, turbulence, dynamo action, and astrophysical/fusion applications with comprehensive computational examples.

Prerequisites¶

Plasma_Physics L04-L06 (single particle motion, drifts, adiabatic invariants)
Plasma_Physics L13-L14 (two-fluid model, kinetic-to-MHD derivation)
Numerical_Simulation L17-L18 (ideal MHD equations, 1D MHD numerics)
Mathematical_Methods L05 (vector analysis), L13 (PDE methods)
Python intermediate level (NumPy, SciPy, Matplotlib)

Lesson Plan¶

Equilibrium and Stability (L01-L04)¶

Filename	Difficulty	Key Topics	Notes
01_MHD_Equilibria.md	⭐⭐	Force balance, Z-pinch, θ-pinch, Grad-Shafranov, safety factor, flux surfaces	Equilibrium theory
02_Linear_Stability.md	⭐⭐⭐	Linearized MHD, energy principle (δW), Kruskal-Shafranov, Suydam criterion	Stability framework
03_Pressure_Driven_Instabilities.md	⭐⭐⭐	Rayleigh-Taylor, Parker instability, interchange, ballooning, Mercier	Pressure-driven modes
04_Current_Driven_Instabilities.md	⭐⭐⭐	Kink (m=1), sausage (m=0), tearing mode, NTM, resistive wall mode	Current-driven modes

Magnetic Reconnection (L05-L07)¶

Filename	Difficulty	Key Topics	Notes
05_Reconnection_Theory.md	⭐⭐⭐⭐	Sweet-Parker, Petschek, Hall MHD reconnection, X-point geometry	Theory fundamentals
06_Reconnection_Applications.md	⭐⭐⭐⭐	Solar flares, CME, substorms, sawtooth crashes, island coalescence	Astrophysical/fusion
07_Advanced_Reconnection.md	⭐⭐⭐⭐	Plasmoid instability, turbulent reconnection, guide field, relativistic	Cutting-edge topics

MHD Turbulence and Dynamo (L08-L10)¶

Filename	Difficulty	Key Topics	Notes
08_MHD_Turbulence.md	⭐⭐⭐⭐	IK vs GS95 spectra, Elsässer variables, critical balance, anisotropy	Turbulence theory
09_Dynamo_Theory.md	⭐⭐⭐⭐	Cowling theorem, mean-field theory, α-Ω dynamo, Earth/solar dynamo	Field generation
10_Turbulent_Dynamo.md	⭐⭐⭐⭐	Small-scale (Kazantsev), large-scale dynamo, helicity, DNS/LES	Advanced dynamo

Astrophysical and Fusion Applications (L11-L14)¶

Filename	Difficulty	Key Topics	Notes
11_Solar_MHD.md	⭐⭐⭐	Flux tubes, sunspots, solar dynamo, coronal heating, Parker wind	Solar physics
12_Accretion_Disk_MHD.md	⭐⭐⭐⭐	MRI, angular momentum transport, α-disk, disk winds/jets	Accretion physics
13_Fusion_MHD.md	⭐⭐⭐	Tokamak/stellarator, disruptions, ELM, sawtooth, beta limits	Fusion plasma
14_Space_Weather.md	⭐⭐⭐	Magnetosphere, Dungey cycle, storms, CME propagation, GIC	Space weather

Advanced Computational Methods and Projects (L15-L18)¶

Filename	Difficulty	Key Topics	Notes
15_2D_MHD_Solver.md	⭐⭐⭐⭐	2D finite volume, Constrained Transport, WENO, Orszag-Tang vortex	2D solver
16_Relativistic_MHD.md	⭐⭐⭐⭐	SRMHD, GRMHD basics, relativistic jets, black hole accretion	Relativistic regime
17_Spectral_Methods.md	⭐⭐⭐⭐	Pseudo-spectral, Chebyshev, MHD-PIC hybrid, AMR, SPH-MHD	Advanced methods
18_Projects.md	⭐⭐⭐⭐	Solar flare sim, disruption prediction, spherical dynamo	Three full projects

Recommended Learning Path¶

Equilibrium & Stability (L01-L04)
         │
         ├──→ Reconnection (L05-L07)
         │           │
         │           ▼
         ├──→ Turbulence & Dynamo (L08-L10)
         │           │
         │           ▼
         ├──→ Applications (L11-L14)
         │    Solar, Accretion, Fusion, Space Weather
         │           │
         └───────────┘
                     │
                     ▼
         Advanced Methods & Projects (L15-L18)
         2D Solver, Relativistic, Spectral, Projects

Focused Paths¶

Path	Lessons	Duration
Fusion focus	L01-L04 → L13 → L15	4 weeks
Astrophysics focus	L01-L04 → L05-L07 → L08-L10 → L11-L12 → L15-L16	8 weeks
Computational focus	L01-L02 → L15 → L17 → L18	4 weeks
Full course	L01-L18 in order	12 weeks

Example Code¶

Example code for this topic is available in examples/MHD/.

Total¶

18 lessons (4 equilibrium/stability + 3 reconnection + 3 turbulence/dynamo + 4 applications + 4 advanced/projects)
Difficulty range: ⭐⭐ to ⭐⭐⭐⭐
Languages: Python (primary)
Key libraries: NumPy, SciPy, Matplotlib, Numba (for 2D solvers)

References¶

Textbooks¶

J.P. Freidberg, Ideal MHD (Cambridge, 2014)
D. Biskamp, Nonlinear Magnetohydrodynamics (Cambridge, 1993)
E. Priest, Magnetohydrodynamics of the Sun (Cambridge, 2014)
J.P. Goedbloed, R. Keppens, S. Poedts, Magnetohydrodynamics of Laboratory and Astrophysical Plasmas (Cambridge, 2019)
A. Brandenburg & A. Nordlund, "Astrophysical turbulence modeling" (Rep. Prog. Phys., 2011)

Online¶

NCAR HAO MHD tutorial
Athena++ documentation: https://www.athena-astro.app/