Lecture Notes | James R. Beattie

Fluid Theory, Turbulence, and Dynamos in Magnetized Plasmas

Wed, 05 Aug 2026 00:00:00 +0000

Contents

Fluid theory, waves, and instabilities.
Conservation laws for magnetized fluids.
Ideal-MHD waves and their restoring forces.
Rayleigh-Taylor instability.
Hydrodynamic turbulence.
Kolmogorov phenomenology, exact energy budgets, and intermittency.
Magnetohydrodynamic turbulence.
Elsasser variables, weak turbulence, critical balance, and dynamic alignment.
Fluid dynamos.
Kinematic and nonlinear small-scale dynamos.
Mean-field theory and large-scale dynamos.
Derivations and mathematical details.

Magnetic dynamos

Mon, 30 Mar 2026 00:00:00 +0000

Contents
Magnetic dynamos.
Goals for this lecture.
The induction equation.
Small-scale dynamo (kinematic only).
Magnetic energy spectrum evolution.
The Kazantsev (1968) k^(3/2) model.
Closure of the third-order correlators.
Diffusion-free regime.
Resistively truncated regime.
Comments.
Large-scale dynamo.
Mean-field induction equation and the EMF.
Exact fluctuation equation and the FOSA closure.
Scale expansion and isotropic reduction.
Derivation of alpha and beta under FOSA.
Why inhomogeneous mean flows are harder.
Inferring transport coefficients from data.
Some simple large-scale dynamos that may or may not exist.
The alpha^2 dynamo.
The alpha Omega dynamo.

Kinetic to Viscous Fluid Model via First-Order Chapman-Enskog Expansion

Sun, 08 Feb 2026 00:00:00 +0000

Contents
Boltzmann equation and the BGK collision operator.
Zeroth-moment (mass conservation).
First-moment (momentum conservation).
Second-moment (energy conservation).
Chapman-Enskog expansion.
Zeroth-order: local equilibrium and Euler equations.
Isotropic pressure.
Vanishing heat flux.
First-order expression for f(1).
The zeroth-order material derivatives.
Partial Derivatives of the Maxwellian.
Moments of f(1): heat flux and pressure corrections.
Relevant first-order fluxes.
Decomposition of f(1) by tensorial parity.
First-order heat flux from odd terms of f(1) and Fourier’s law of heat conduction.
Pressure corrections and bulk viscosity from the isotropic, even terms of f(1).
Shear viscosity from the traceless, even terms of f(1).

Fluid moments and linear waves in ideal magnetohydrodynamics

Sat, 07 Feb 2026 00:00:00 +0000

Contents
Lecture goals.
Boltzmann equation for a monoatomic gas.
Zeroth-moment (mass conservation).
First-moment (momentum conservation).
Second-moment (energy conservation).
Isotropic pressure and moment closure.
Boltzmann equation for a non-relativistic, magnetized plasma.
From Boltzmann equation to an ideal MHD fluid.
Moments of the multi-species Boltzmann equation.
Ideal MHD equations.
The repercussions and assumptions of ideal MHD.
Linearization.
Small perturbations about a homogeneous equilibrium.
Linearization of the ideal MHD equations.
Linearized Continuity Equation.
Linearized Induction Equation.
Linearized Momentum Equation.
Plane-wave solutions.
Constructing the linear system.
Constructing the eigenvalue problem for u-hat.
Rearrange u-hat.
Rearrange rho-hat.
Rearrange B-hat.
Rebuilding u-hat in an appropriate form.
Ideal MHD Eigenmodes.
Alfven mode.
Fast and slow magnetosonic modes.
Limiting cases for propagation angle.
High- and low-beta limits.