AMATH 866: Magnetohydrodynamics and Plasma Physics
Estimated study time: 2 hr 5 min
Table of contents
These notes synthesize material from P.A. Davidson’s An Introduction to Magnetohydrodynamics, H. Goedbloed, R. Keppens, and S. Poedts’s Magnetohydrodynamics of Laboratory and Astrophysical Plasmas, D. Biskamp’s Nonlinear Magnetohydrodynamics, J.P. Freidberg’s Ideal MHD, and supplementary material from Cambridge DAMTP MHD notes (Proctor, Tobias), Princeton Plasma Physics Laboratory lecture notes, and Oxford Mathematical Geoscience materials.
Chapter 1: Electromagnetic Foundations
Magnetohydrodynamics describes the behaviour of electrically conducting fluids — liquid metals, stellar interiors, accretion discs, the solar wind — in the presence of magnetic fields. Before we can couple electromagnetism to fluid mechanics, we must understand which parts of Maxwell’s equations survive in the MHD limit and why. This chapter develops the electromagnetic foundations: the pre-Maxwell equations appropriate for slowly varying fields, the induction equation that governs the evolution of the magnetic field, and the celebrated frozen-in flux theorem of Alfvén.
1.1 Maxwell’s Equations in Matter
We begin with the full Maxwell equations in a medium characterised by free charge density \(\rho_f\), free current density \(\mathbf{J}\), electric permittivity \(\varepsilon_0\), and magnetic permeability \(\mu_0\):
\[ \nabla \cdot \mathbf{E} = \frac{\rho_f}{\varepsilon_0}, \quad \nabla \cdot \mathbf{B} = 0, \]\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}. \]The justification for dropping the displacement current is a scale analysis. If the fluid moves with characteristic velocity \(U\) and the fields vary on a length scale \(L\), the ratio of the displacement current to the conduction current is of order \((U/c)^2\), where \(c = 1/\sqrt{\mu_0 \varepsilon_0}\) is the speed of light. For any non-relativistic flow, \(U \ll c\), and this ratio is negligible. The pre-Maxwell equations are thus the natural electromagnetic framework for MHD. Note that \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J}\) immediately implies \(\nabla \cdot \mathbf{J} = 0\): currents are solenoidal, and there is no accumulation of free charge on MHD time scales.
1.2 Ohm’s Law for Conducting Fluids
In a stationary conductor, Ohm’s law is simply \(\mathbf{J} = \sigma \mathbf{E}\), where \(\sigma\) is the electrical conductivity. In a moving conductor, however, the relevant electric field is that seen in the frame co-moving with the fluid. For non-relativistic flows, the Lorentz transformation gives the electric field in the fluid frame as \(\mathbf{E}' = \mathbf{E} + \mathbf{u} \times \mathbf{B}\), where \(\mathbf{u}\) is the fluid velocity.
In the limit \(\sigma \to \infty\) (ideal MHD), this reduces to \(\mathbf{E} + \mathbf{u} \times \mathbf{B} = \mathbf{0}\).
This is the simplest form of Ohm’s law appropriate for MHD. More complete versions include the Hall term \(\mathbf{J} \times \mathbf{B}/(ne)\), electron pressure gradients, and electron inertia, but these corrections are important only at scales comparable to the ion skin depth or ion Larmor radius. For the bulk of classical MHD, the scalar conductivity form suffices.
where \(T\) is the electron temperature in eV and \(\ln\Lambda \approx 10\text{--}20\) is the Coulomb logarithm. Note the strong temperature dependence: hot plasmas are excellent conductors. The solar corona at \(T \sim 10^6\;\text{K}\) has \(\sigma \sim 10^6\;\text{S}\,\text{m}^{-1}\), comparable to copper, while the Earth’s liquid iron core at \(T \sim 5000\;\text{K}\) has \(\sigma \sim 10^6\;\text{S}\,\text{m}^{-1}\) (from metallic conduction rather than the Spitzer formula). The corresponding magnetic diffusivities are \(\eta \sim 1\;\text{m}^2\,\text{s}^{-1}\) in both cases.
1.3 The Magnetic Induction Equation
The central equation of MHD electrodynamics is obtained by eliminating \(\mathbf{E}\) and \(\mathbf{J}\) from the pre-Maxwell equations using Ohm’s law.
Using \(\mathbf{J} = \nabla \times \mathbf{B}/\mu_0\) and writing \(\eta = 1/(\mu_0 \sigma)\):
\[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) - \eta \nabla \times (\nabla \times \mathbf{B}). \]Since \(\nabla \cdot \mathbf{B} = 0\), we have \(\nabla \times (\nabla \times \mathbf{B}) = -\nabla^2 \mathbf{B}\), giving
\[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}. \quad \square \]The induction equation is the magnetic analogue of the vorticity equation in fluid dynamics. The first term on the right represents advection and stretching of field lines by the flow; the second term represents ohmic diffusion. The relative importance of these two effects is measured by a single dimensionless number.
1.4 The Magnetic Reynolds Number
where \(U\) and \(L\) are characteristic velocity and length scales, and \(\eta = 1/(\mu_0 \sigma)\) is the magnetic diffusivity.
When \(\mathrm{Rm} \gg 1\), advection dominates over diffusion and the magnetic field is effectively “frozen” into the fluid — this is the ideal MHD regime. Astrophysical plasmas almost always have \(\mathrm{Rm} \gg 1\): for the solar corona, \(\mathrm{Rm} \sim 10^{12}\); for the Earth’s liquid iron core, \(\mathrm{Rm} \sim 10^2\text{--}10^3\). By contrast, laboratory liquid metals typically have \(\mathrm{Rm} \sim 1\text{--}10\), making them useful testbeds for resistive MHD phenomena.
1.5 The Frozen-In Flux Theorem
The most profound consequence of the ideal MHD limit is Alfvén’s theorem, proved by Hannes Alfvén in 1942. It states that in a perfectly conducting fluid, magnetic flux through any material surface is conserved. This means that field lines move with the fluid as if they were “frozen in.”
Since \(\nabla \cdot \mathbf{B} = 0\) and the ideal induction equation gives \(\partial \mathbf{B}/\partial t = \nabla \times (\mathbf{u} \times \mathbf{B})\), we have
\[ \frac{d}{dt}\int_{S(t)} \mathbf{B} \cdot d\mathbf{S} = \int_S \left[\nabla \times (\mathbf{u} \times \mathbf{B}) + \nabla \times (\mathbf{B} \times \mathbf{u})\right] \cdot d\mathbf{S} = 0, \]since \(\mathbf{u} \times \mathbf{B} = -\mathbf{B} \times \mathbf{u}\). \(\square\)
Alfvén’s theorem is the magnetic analogue of Kelvin’s circulation theorem in inviscid fluid dynamics. It has far-reaching consequences: magnetic field lines can be stretched, compressed, and distorted by fluid motions, but they cannot be broken or reconnected. This topological constraint is what makes ideal MHD so powerful — and what makes its breakdown (magnetic reconnection, Chapter 6) so physically important.
An important corollary of the frozen-in theorem is that the magnetic field can be amplified by stretching motions. If a fluid element is elongated by a factor \(\ell\) along a field line while its cross-sectional area decreases to conserve volume, the field strength increases by the same factor \(\ell\). This stretching mechanism is the basis of the dynamo process (Chapter 7) and explains why turbulent conducting fluids tend to amplify magnetic fields.
1.6 The MHD Approximation: Summary
To summarise, the MHD approximation rests on three pillars: (i) the flow is non-relativistic (\(U \ll c\)), so the displacement current can be dropped; (ii) the plasma is quasi-neutral, so charge separation effects are negligible on the length and time scales of interest; and (iii) the conductivity is high enough (or the scales large enough) that magnetic diffusion can often be neglected in a first approximation. Under these conditions, the electromagnetic sector reduces to the induction equation, and the only coupling back to the fluid is through the Lorentz force \(\mathbf{J} \times \mathbf{B}\), which we develop in the next chapter.
Chapter 2: The MHD Equations
With the electromagnetic foundations in place, we now derive the full system of magnetohydrodynamic equations by coupling the induction equation to the Navier-Stokes equations via the Lorentz force. The resulting system describes the self-consistent co-evolution of velocity and magnetic fields in a conducting fluid. We discuss the physical meaning of magnetic pressure and tension, the distinction between ideal and resistive MHD, and the elegant Elsasser variable formulation.
2.1 The Lorentz Force
The Lorentz force per unit volume on a conducting fluid is \(\mathbf{F}_L = \mathbf{J} \times \mathbf{B}\). Using the pre-Maxwell relation \(\mathbf{J} = \nabla \times \mathbf{B}/\mu_0\), we can rewrite this in a form that reveals the underlying physics.
The first term represents magnetic tension (a restoring force along curved field lines) and the second represents the gradient of magnetic pressure \(p_m = B^2/(2\mu_0)\).
The magnetic tension term \((\mathbf{B} \cdot \nabla)\mathbf{B}/\mu_0\) acts like the tension in an elastic string: if field lines are curved, the tension acts to straighten them, producing a restoring force. This is the mechanism behind Alfvén waves. The magnetic pressure term \(\nabla(B^2/(2\mu_0))\) acts isotropically and supplements the gas pressure. The total pressure in MHD is therefore \(p + B^2/(2\mu_0)\).
When \(\beta \ll 1\), magnetic forces dominate (as in the solar corona); when \(\beta \gg 1\), the magnetic field is dynamically weak (as in stellar interiors).
It is often useful to express the Lorentz force in terms of the Maxwell stress tensor, which makes explicit the analogy between magnetic forces and elastic stresses.
The Lorentz force per unit volume is \((\mathbf{J} \times \mathbf{B})_i = \partial M_{ij}/\partial x_j\). The diagonal elements show an isotropic magnetic pressure \(-B^2/(2\mu_0)\) supplemented by a tension \(B_i^2/\mu_0\) along the field direction.
The stress tensor picture reveals that magnetic field lines behave like elastic strings under tension \(B^2/\mu_0\) embedded in a medium with isotropic magnetic pressure \(B^2/(2\mu_0)\). The net effect is an anisotropic stress: tension along the field and pressure perpendicular to it. This anisotropy is the root cause of the rich wave spectrum of MHD and the anisotropic nature of MHD turbulence.
2.2 The Full MHD System
Combining the Navier-Stokes equations with the Lorentz force and the induction equation, we obtain the full resistive MHD system. For a compressible fluid of density \(\rho\), velocity \(\mathbf{u}\), pressure \(p\), and magnetic field \(\mathbf{B}\):
together with an equation of state (e.g., \(p = \rho R T / \bar{m}\) for an ideal gas, or an adiabatic closure \(d(p\rho^{-\gamma})/dt = 0\)).
This system of equations is the starting point for nearly all of classical MHD. Note that the magnetic field equation involves no time derivative of \(\mathbf{E}\) — the electric field has been eliminated entirely. The system is closed once we specify an energy equation or equation of state.
2.3 Ideal MHD
Ideal MHD conserves total energy, magnetic helicity, and cross-helicity.
Ideal MHD is the workhorse of astrophysical fluid dynamics. It is appropriate whenever \(\mathrm{Rm} \gg 1\) and viscous effects are negligible on the scales of interest. The ideal system is hyperbolic, admitting wave-like solutions (Chapter 3), and its conservation laws provide powerful constraints on the dynamics. The hyperbolic character means that information propagates at finite speeds (the Alfvén and magnetosonic speeds), and discontinuous solutions (shocks, contact discontinuities, tangential discontinuities) can form from smooth initial data. The Rankine-Hugoniot jump conditions for MHD shocks generalise those of ordinary gas dynamics and admit a richer taxonomy: fast shocks, slow shocks, intermediate (rotational) discontinuities, and switch-on/switch-off shocks.
2.4 Elsasser Variables
A beautiful symmetry of incompressible MHD emerges when we introduce the Elsasser variables, named after Walter Elsasser, who used them in his pioneering work on the geodynamo in the 1950s.
For incompressible ideal MHD with a uniform background field \(\mathbf{B}_0\), the equations become
\[ \frac{\partial \mathbf{z}^\pm}{\partial t} \mp (\mathbf{v}_A \cdot \nabla)\mathbf{z}^\pm + (\mathbf{z}^\mp \cdot \nabla)\mathbf{z}^\pm = -\nabla P, \]where \(\mathbf{v}_A = \mathbf{B}_0/\sqrt{\mu_0 \rho}\) is the Alfvén velocity and \(P\) is the total pressure divided by \(\rho\).
The Elsasser formulation shows that MHD can be viewed as two counter-propagating wave packets, \(\mathbf{z}^+\) and \(\mathbf{z}^-\), which interact nonlinearly only when they collide. In the absence of nonlinear coupling, each wave packet propagates undistorted along the background field — a remarkable simplification that underlies the theory of MHD turbulence (Iroshnikov 1964, Kraichnan 1965).
2.5 Energy Equation and Conservation Laws
The total energy of the MHD system has three contributions: kinetic, thermal, and magnetic.
satisfies a conservation law
\[ \frac{\partial e}{\partial t} + \nabla \cdot \left[\left(\frac{1}{2}\rho u^2 + \frac{\gamma p}{\gamma - 1}\right)\mathbf{u} + \frac{1}{\mu_0}\mathbf{B} \times (\mathbf{u} \times \mathbf{B})\right] = 0. \]The magnetic contribution to the energy flux, \(\mathbf{B} \times (\mathbf{u} \times \mathbf{B})/\mu_0\), is the Poynting flux in the MHD approximation. It represents the transport of magnetic energy by the flow.
In addition to energy, ideal MHD conserves two further quadratic invariants that play a central role in MHD turbulence theory.
It measures the correlation between velocity and magnetic field fluctuations and is conserved in ideal, incompressible MHD. A state with maximal cross-helicity (\(\mathbf{u} = \pm \mathbf{B}/\sqrt{\mu_0\rho}\)) corresponds to a pure Alfvén wave and is an exact nonlinear solution.
2.6 Boundary Conditions and Dimensional Analysis
At a perfectly conducting wall, the normal component of velocity vanishes (\(\mathbf{u} \cdot \hat{\mathbf{n}} = 0\)) and the magnetic field must satisfy \(\mathbf{B} \cdot \hat{\mathbf{n}} = 0\) (no normal flux through the conductor) plus continuity of the tangential electric field. At a free boundary between plasma and vacuum, we require continuity of total pressure and continuity of the normal component of \(\mathbf{B}\).
Chapter 3: MHD Waves
One of the most striking features of MHD is its support for wave-like disturbances that have no counterpart in ordinary hydrodynamics. The magnetic tension in curved field lines acts as a restoring force, giving rise to Alfvén waves — transverse waves propagating along field lines at the Alfvén speed. The coupling of magnetic and gas pressure produces two additional families: fast and slow magnetosonic waves. This chapter derives these wave modes from the linearised MHD equations and explores their propagation characteristics.
3.1 Linearization About a Uniform Equilibrium
Consider a uniform equilibrium with constant density \(\rho_0\), pressure \(p_0\), zero velocity, and a uniform magnetic field \(\mathbf{B}_0 = B_0 \hat{\mathbf{z}}\). We perturb all quantities:
\[ \rho = \rho_0 + \rho_1, \quad \mathbf{u} = \mathbf{u}_1, \quad \mathbf{B} = \mathbf{B}_0 + \mathbf{B}_1, \quad p = p_0 + p_1, \]with \(|\rho_1| \ll \rho_0\), etc. Substituting into the ideal, compressible MHD equations and retaining only first-order terms yields the linearised system:
\[ \rho_0 \frac{\partial \mathbf{u}_1}{\partial t} = -\nabla p_1 + \frac{1}{\mu_0}(\nabla \times \mathbf{B}_1) \times \mathbf{B}_0, \]\[ \frac{\partial \mathbf{B}_1}{\partial t} = \nabla \times (\mathbf{u}_1 \times \mathbf{B}_0), \]\[ \frac{\partial \rho_1}{\partial t} + \rho_0 \nabla \cdot \mathbf{u}_1 = 0, \quad p_1 = c_s^2 \rho_1, \]where \(c_s = \sqrt{\gamma p_0/\rho_0}\) is the adiabatic sound speed.
3.2 Alfvén Waves
Hannes Alfvén predicted in 1942 that a conducting fluid threaded by a magnetic field should support transverse waves propagating along the field lines. This prediction, initially met with scepticism (Alfvén recounted that both Fermi and Teller initially disbelieved him), was later confirmed experimentally and earned Alfvén the 1970 Nobel Prize in Physics.
The quantity \(v_A\) is the Alfvén speed.
which is the wave equation with phase speed \(v_A = B_0/\sqrt{\mu_0 \rho_0}\). \(\square\)
Alfvén waves are the magnetic analogue of waves on a plucked string: the magnetic field lines play the role of the string, with tension \(B_0^2/\mu_0\) and mass per unit length \(\rho_0\). The perturbation is purely transverse and incompressible — there is no density or pressure fluctuation. The velocity and magnetic field perturbations are related by the Walén relation: \(\mathbf{u}_1 = \mp \mathbf{B}_1/\sqrt{\mu_0 \rho_0}\).
3.3 Fast and Slow Magnetosonic Waves
When compressibility is included, the linearised MHD equations support two additional wave modes that couple magnetic and gas pressure. For a plane wave with wavevector \(\mathbf{k}\) making angle \(\theta\) with \(\mathbf{B}_0\), the general dispersion relation is obtained by seeking solutions proportional to \(e^{i(\mathbf{k} \cdot \mathbf{x} - \omega t)}\).
where \(\theta\) is the angle between \(\mathbf{k}\) and \(\mathbf{B}_0\). The \(+\) sign gives the fast magnetosonic wave, the \(-\) sign gives the slow magnetosonic wave, and the intermediate Alfvén wave has \(v_\text{ph}^2 = v_A^2 \cos^2\theta\).
The three MHD wave modes satisfy the ordering \(v_{\text{slow}} \leq v_A |\cos\theta| \leq v_{\text{fast}}\) for all angles \(\theta\). At \(\theta = 0\) (propagation along the field), the fast wave degenerates to whichever of \(v_A\) or \(c_s\) is larger, the slow wave to whichever is smaller, and the Alfvén wave to \(v_A\). At \(\theta = \pi/2\) (propagation perpendicular to the field), the slow wave speed vanishes, the Alfvén wave speed vanishes, and the fast wave propagates at \(\sqrt{v_A^2 + c_s^2}\).
The Friedrichs diagram reveals the profoundly anisotropic nature of MHD wave propagation. The Alfvén wave propagates only along the field and has zero phase speed perpendicular to it. The fast wave is nearly isotropic (its speed varies relatively little with \(\theta\)), while the slow wave is confined to a narrow cone around the field direction.
3.4 Wave Propagation in Stratified and Rotating MHD
When the background is not uniform — for instance, in a gravitationally stratified atmosphere with a magnetic field — the wave analysis becomes considerably richer. Gravity introduces buoyancy oscillations (internal gravity waves, frequency \(N\)), rotation introduces inertial oscillations (frequency \(f\)), and the magnetic field introduces Alfvén and magnetosonic modes.
where \(k_h\) is the horizontal wavenumber perpendicular to the field. These magneto-gravity waves are important in the solar tachocline and in the stably stratified layers of planetary interiors.
3.5 Phase Mixing and Resonant Absorption
In an inhomogeneous medium, the Alfvén speed varies across the magnetic field. Neighbouring field lines therefore oscillate at different frequencies, causing an initially coherent disturbance to develop increasingly fine transverse structure — a process called phase mixing.
Phase mixing is closely related to resonant absorption, where an incoming fast magnetosonic wave encounters a surface where its frequency matches the local Alfvén frequency. Energy accumulates at this resonant layer and is converted to Alfvén waves, which then phase-mix and dissipate. This mechanism is a leading candidate for heating the solar corona.
Chapter 4: MHD Equilibria
Before we can study the stability or dynamics of a magnetised plasma, we must first understand the equilibrium configurations that the plasma can adopt. The equilibrium condition in MHD is a balance between the pressure gradient, the magnetic (Lorentz) force, and possibly gravity. This chapter develops the theory of MHD equilibria, from the general force-balance condition to the Grad-Shafranov equation for axisymmetric fusion devices, to force-free fields and the topological constraints imposed by magnetic helicity.
4.1 Force Balance
In a static MHD equilibrium with no flow (\(\mathbf{u} = \mathbf{0}\)), the momentum equation reduces to a balance of forces.
together with \(\nabla \cdot \mathbf{B} = 0\). Equivalently,
\[ \nabla\left(p + \frac{B^2}{2\mu_0}\right) = \frac{1}{\mu_0}(\mathbf{B} \cdot \nabla)\mathbf{B}. \]The force balance has an immediate geometric consequence: taking the dot product with \(\mathbf{B}\) gives \(\mathbf{B} \cdot \nabla p = 0\), meaning that the pressure is constant along field lines. Similarly, dotting with \(\mathbf{J}\) gives \(\mathbf{J} \cdot \nabla p = 0\): pressure is also constant along current lines. Therefore, surfaces of constant pressure are simultaneously magnetic surfaces (surfaces to which field lines are tangent) and current surfaces. This is a remarkable geometric constraint: the field lines and current lines must both be tangent to the same family of nested surfaces, and the pressure serves as a label for these surfaces.
In three dimensions, the requirement that both \(\mathbf{B}\) and \(\mathbf{J}\) lie in surfaces of constant \(p\) is extremely restrictive. In two-dimensional or axisymmetric geometries, it can always be satisfied, but in fully three-dimensional configurations, the existence of such nested surfaces is generically violated — a fact closely related to the KAM theorem of Hamiltonian mechanics, since the field-line equations have Hamiltonian structure.
4.2 Flux Functions and the Grad-Shafranov Equation
For axisymmetric equilibria, which are the basis of tokamak design, the magnetic field can be expressed in terms of a single scalar flux function. This reduction was independently derived by Harold Grad and Vitalii Shafranov in 1958.
where \(\psi(R,Z)\) is the poloidal flux function. The surfaces \(\psi = \text{const}\) are the magnetic surfaces.
Substituting this representation into the force balance equation and using the axisymmetry to show that both \(p\) and \(RB_\phi\) must be functions of \(\psi\) alone, one obtains a single, nonlinear, elliptic PDE for \(\psi\).
where \(p(\psi)\) is the pressure and \(F(\psi) = RB_\phi\) is the poloidal current function. The functions \(p(\psi)\) and \(F(\psi)\) are free and must be specified as input.
The Grad-Shafranov equation is the fundamental equation of tokamak equilibrium theory. Given appropriate choices for \(p(\psi)\) and \(F(\psi)\) and boundary conditions (e.g., \(\psi\) prescribed on a conducting wall or at a magnetic separatrix), it determines the shape of the magnetic surfaces and hence the equilibrium configuration. The equation is nonlinear in general (since \(p(\psi)\) and \(F(\psi)\) are arbitrary functions), and analytic solutions exist only for special choices of these profiles. Numerical solution of the Grad-Shafranov equation is a routine step in tokamak design, performed by codes such as EFIT, CHEASE, and HELENA.
4.3 Force-Free Fields
In astrophysical contexts, particularly in the solar corona and magnetosphere, the gas pressure is often much smaller than the magnetic pressure (\(\beta \ll 1\)). In this limit, the equilibrium condition becomes \(\mathbf{J} \times \mathbf{B} = \mathbf{0}\), meaning the current density is everywhere parallel to the magnetic field.
where \(\alpha\) is a scalar function. If \(\alpha\) is constant, the field is linear force-free; if \(\alpha = 0\), the field is potential (current-free).
Taking the divergence of the force-free equation and using \(\nabla \cdot \mathbf{B} = 0\), we obtain \(\mathbf{B} \cdot \nabla\alpha = 0\): the function \(\alpha\) is constant along field lines. For a linear force-free field with constant \(\alpha\), the equation \(\nabla \times \mathbf{B} = \alpha \mathbf{B}\) can be solved by Fourier methods and yields the Chandrasekhar-Kendall representation used in solar physics.
where \(J_0\) and \(J_1\) are Bessel functions of the first kind. This solution describes a magnetic flux rope with helical field lines and is widely used to model coronal mass ejections and interplanetary flux ropes. The field reverses direction at the first zero of \(J_0\), a feature exploited in the reversed-field pinch.
4.4 Magnetic Helicity and Woltjer’s Theorem
Magnetic helicity is a topological invariant of ideal MHD that measures the linkage and twist of magnetic field lines. Its conservation places powerful constraints on the relaxation of magnetic configurations.
It is gauge-invariant (under the boundary condition) and is an ideal invariant: \(dH/dt = 0\) when \(\eta = 0\).
The conservation of helicity in ideal MHD was proved by Woltjer (1958), who also showed that the minimum-energy state at fixed helicity is a linear force-free field. This result was later extended by J.B. Taylor to resistive plasmas in a profound insight that underpins relaxation theory.
where \(\alpha\) is a constant (the Lagrange multiplier associated with the helicity constraint).
Since \(\delta\mathbf{B} = \nabla \times \delta\mathbf{A}\) is arbitrary, we require \(\nabla \times \mathbf{B} = \alpha\mathbf{B}\), a linear force-free field. \(\square\)
4.5 Taylor Relaxation
J.B. Taylor (1974) made the bold conjecture that in a turbulent, weakly resistive plasma, magnetic energy is rapidly dissipated by reconnection while helicity is approximately conserved (being an ideal invariant with slow resistive decay). The end state of this relaxation is therefore the Woltjer minimum-energy state — a linear force-free field — regardless of the initial configuration.
Chapter 5: MHD Stability
An equilibrium exists, but is it physically realisable? The answer depends on stability: a small perturbation from equilibrium may either oscillate (stability) or grow exponentially (instability). MHD stability theory is one of the most developed branches of plasma physics, driven in large part by the need to understand and control instabilities in magnetic confinement fusion devices. This chapter covers normal mode analysis, the powerful energy principle, and the major MHD instabilities.
5.1 Normal Mode Analysis
The most direct approach to linear stability is to linearise the ideal MHD equations about an equilibrium and seek normal-mode solutions proportional to \(e^{-i\omega t}\). If all eigenfrequencies \(\omega\) are real, the equilibrium is stable; if any \(\omega\) has a positive imaginary part, the equilibrium is unstable. The self-adjoint structure of the ideal MHD force operator, first elucidated by Bernstein, Frieman, Kruskal, and Kulsrud (1958) in one of the foundational papers of plasma physics, guarantees that the eigenvalues \(\omega^2\) are real — so instabilities grow purely exponentially, without oscillation.
For an equilibrium with \(\mathbf{u}_0 = \mathbf{0}\), a displacement field \(\boldsymbol{\xi}(\mathbf{x},t)\) describes the perturbation: the fluid element initially at \(\mathbf{x}\) moves to \(\mathbf{x} + \boldsymbol{\xi}\). The linearised momentum equation can be written as
\[ \rho_0 \frac{\partial^2 \boldsymbol{\xi}}{\partial t^2} = \mathbf{F}(\boldsymbol{\xi}), \]where \(\mathbf{F}\) is the force operator, a linear operator acting on \(\boldsymbol{\xi}\).
where \(\mathbf{Q} = \nabla \times (\boldsymbol{\xi} \times \mathbf{B}_0)\) is the perturbed magnetic field.
A fundamental property of this operator, proved by Bernstein, Frieman, Kruskal, and Kulsrud in their landmark 1958 paper, is that it is self-adjoint.
5.2 The Energy Principle
for all admissible displacements \(\boldsymbol{\xi}\).
The energy principle reduces stability analysis to a variational problem: the equilibrium is stable if and only if no trial function \(\boldsymbol{\xi}\) can make \(\delta W\) negative. This is immensely powerful because one does not need to solve the full eigenvalue problem — one only needs to check the sign of a quadratic form.
The potential energy \(\delta W\) can be decomposed into several physically interpretable terms. For a plasma with vacuum region:
\[ \delta W = \delta W_p + \delta W_s + \delta W_v, \]where \(\delta W_p\) is the plasma contribution, \(\delta W_s\) the surface contribution, and \(\delta W_v\) the vacuum contribution.
The first two terms are always stabilising (field line bending and compression), while the third and fourth can be destabilising (pressure-driven and current-driven instabilities).
This decomposition is the basis for understanding all MHD instabilities in terms of the free energy sources that drive them.
where \(q(r) = rB_z/(RB_\theta)\) is the safety factor and primes denote radial derivatives. The first term represents the stabilising effect of magnetic shear and the second the destabilising effect of the pressure gradient.
Suydam’s criterion is the cylindrical analogue of the Rayleigh criterion for convective stability: it provides a necessary (but not sufficient) condition. Violation of Suydam’s criterion at any radius guarantees instability, while satisfaction everywhere is necessary but not sufficient for stability. A sufficient condition requires additionally that the global energy principle be satisfied.
5.3 Interchange Instabilities
The interchange instability is the MHD analogue of the Rayleigh-Taylor instability in hydrodynamics. It occurs when the pressure gradient opposes the curvature of the magnetic field lines — physically, when dense plasma is supported by a magnetic field whose curvature is concave toward the plasma.
Unfavourable curvature (pressure gradient and curvature pointing in the same direction) is destabilising.
5.4 Kink Instabilities
The kink instability is a current-driven instability that occurs in cylindrical or toroidal plasma columns when the current (or equivalently the magnetic shear) exceeds a critical threshold. It is the most dangerous macroscopic instability in tokamaks.
is unstable to the external kink mode with \(m = 1\), \(n = 1\) if \(q(a) < 1\). The condition \(q > 1\) everywhere is necessary (but not sufficient) for stability in a tokamak.
The safety factor \(q\) measures how many times a field line winds around the torus the long way (toroidally) for each circuit the short way (poloidally). The Kruskal-Shafranov limit \(q > 1\) sets a fundamental upper bound on the toroidal current in a tokamak, which in turn limits the poloidal field and hence the achievable plasma pressure (through the Troyon limit \(\beta \lesssim \beta_N I/(aB)\)).
5.5 Ballooning Modes
Ballooning modes are pressure-driven instabilities that are localised on the outboard (unfavourable curvature) side of a toroidal device. They were first analysed by Connor, Hastie, and Taylor (1978) and represent the toroidal analogue of interchange modes modified by magnetic shear.
The ballooning stability limit provides the most stringent constraint on the achievable plasma pressure in tokamaks. It depends sensitively on the magnetic shear profile, the pressure gradient, and the shape of the magnetic surfaces (triangularity, elongation). Optimising these parameters to maximize the stable pressure limit is one of the central goals of tokamak design.
where \(\beta_N \approx 2.8\text{--}3.5\) is the normalised beta limit, \(I_p\) is the plasma current, \(a\) is the minor radius, and \(B_T\) is the toroidal field. For ITER parameters (\(I_p = 15\;\text{MA}\), \(a = 2\;\text{m}\), \(B_T = 5.3\;\text{T}\)), this gives \(\beta_{\max} \approx 2.5\%\). Pushing beyond this limit through optimised shaping and current profiles is an active area of advanced tokamak research.
5.6 Rayleigh-Taylor and Kelvin-Helmholtz in MHD
The classical hydrodynamic instabilities of Rayleigh-Taylor (heavy fluid over light) and Kelvin-Helmholtz (velocity shear at an interface) are both modified by the presence of a magnetic field.
where \(k = |\mathbf{k}|\). The magnetic field stabilises modes with \(k_x \neq 0\) through tension, but modes with \(k_x = 0\) (interchange modes perpendicular to the field) remain unstable.
The magnetic field thus provides a partial stabilisation: it suppresses modes that must bend the field lines but is powerless against modes that can interchange flux tubes without bending. This anisotropic stabilisation is a recurring theme in MHD. The physical picture is clear: perturbations that bend the field lines must do work against the magnetic tension \(B^2/\mu_0\), and when this energy cost exceeds the gravitational energy released by the interchange, the mode is stabilised. But perturbations aligned with the field lines slip through without bending them, extracting gravitational energy unopposed.
5.7 Kelvin-Helmholtz Instability with Magnetic Field
The Kelvin-Helmholtz instability arises at a shear layer between two fluid streams moving at different velocities. In MHD, a magnetic field component along the flow direction can suppress this instability through the tension force.
The magnetic tension term \((\mathbf{k} \cdot \mathbf{B}_0)^2\) stabilises the instability when the field is strong enough. Complete stabilisation occurs when the Alfvén speed exceeds half the velocity jump, which explains why strong magnetic fields can suppress turbulent mixing at shear layers in astrophysical jets and magnetospheric boundaries.
Chapter 6: Magnetic Reconnection
In ideal MHD, the frozen-in flux theorem prohibits changes in the topology of magnetic field lines: field lines that are initially connected remain connected for all time. Yet observations of solar flares, magnetospheric substorms, and sawtooth crashes in tokamaks demonstrate that topological restructuring of the magnetic field does occur, often explosively, releasing vast amounts of stored magnetic energy. The process by which this happens is called magnetic reconnection. It requires some departure from ideal MHD — resistivity, electron inertia, or kinetic effects — but its consequences are felt on macroscopic scales far exceeding the small dissipation region.
6.1 The Reconnection Problem
The fundamental problem of reconnection theory is one of scale separation. Consider two regions of oppositely directed magnetic field brought together by a flow. At the boundary, a thin current sheet forms. Inside this sheet, the magnetic field reverses direction and the frozen-in constraint must break down, allowing field lines to “reconnect” with new partners. The rate at which this happens determines the energy release rate.
where \(L\) is the macroscopic length scale. In astrophysical plasmas, \(S\) is enormous: \(S \sim 10^{10}\text{--}10^{14}\) in the solar corona.
The challenge of reconnection theory is to explain how a process that occurs in a tiny resistive layer (of width \(\delta \ll L\)) can produce reconnection rates that are fast enough to account for the observed energy release times. Simple resistive models predict rates that scale as negative powers of \(S\) and are far too slow for astrophysical applications.
6.2 The Sweet-Parker Model
The first quantitative model of reconnection was developed independently by Peter Sweet (1958) and Eugene Parker (1957). It considers a steady-state, two-dimensional reconnection layer of length \(L\) and width \(\delta\).
and the dimensionless reconnection rate is
\[ M_A = \frac{v_i}{v_A} \sim S^{-1/2}. \]For the solar corona with \(S \sim 10^{12}\), the Sweet-Parker rate gives \(M_A \sim 10^{-6}\), corresponding to a reconnection time of order \(10^6\) Alfvén times — far too slow to explain the observed flare time scales of minutes to hours. This discrepancy has been called the “reconnection problem” and has motivated the search for faster reconnection mechanisms. The Sweet-Parker model, despite its shortcomings in predicting fast reconnection, remains important as the baseline against which all other reconnection models are compared. Its prediction that the current sheet width scales as \(\delta \sim L/\sqrt{S}\) has been confirmed numerically and provides the starting point for the plasmoid instability analysis (Section 6.4).
6.3 Petschek Reconnection
Harry Petschek (1964) proposed that the reconnection layer need not extend over the full macroscopic length \(L\). Instead, he argued that standing slow-mode shocks emanating from a compact diffusion region could mediate the conversion of magnetic energy to kinetic energy.
which is only weakly dependent on \(S\). For \(S \sim 10^{12}\), this gives \(M_A \sim 0.01\text{--}0.02\), orders of magnitude faster than Sweet-Parker.
The key insight is that the dissipation region can be much smaller than \(L\): most of the energy conversion happens at the slow shocks, not in the resistive layer. However, numerical simulations with uniform resistivity tend to produce Sweet-Parker rather than Petschek configurations. Petschek reconnection requires either localised resistivity (enhanced only near the X-point) or kinetic effects beyond fluid MHD.
6.4 The Tearing Mode Instability
Even when reconnection is not being driven externally, a current sheet can be spontaneously unstable to the tearing mode, which breaks the sheet into a chain of magnetic islands.
where \(\tau_A = a/v_A\) is the Alfvén time and \(\tau_\eta = a^2/\eta\) is the resistive diffusion time for the sheet. The growth rate is intermediate between these two time scales.
The tearing mode is important because it provides a spontaneous pathway to reconnection. In the nonlinear stage, magnetic islands grow, coalesce, and can trigger explosive reconnection when the Lundquist number exceeds a critical value \(S_c \sim 10^4\), leading to the plasmoid instability — a violent fragmentation of the current sheet into a chain of secondary islands. This plasmoid-mediated reconnection may be the mechanism responsible for fast reconnection in high-\(S\) astrophysical plasmas.
6.5 Collisionless Reconnection
In weakly collisional plasmas (the solar wind, magnetosphere, solar corona), the dissipation region is governed by kinetic physics rather than resistivity. The key insight is that at scales below the ion skin depth \(d_i = c/\omega_{pi}\), ions decouple from the magnetic field while electrons remain frozen in. This creates a two-scale structure: an outer ion diffusion region of width \(\sim d_i\) and an inner electron diffusion region of width \(\sim d_e = c/\omega_{pe}\).
The Hall term \(\mathbf{J} \times \mathbf{B}/(ne)\) breaks the symmetry of the resistive MHD reconnection layer and produces a characteristic quadrupolar out-of-plane magnetic field pattern that has been observed in spacecraft measurements at the magnetopause and magnetotail.
The Hall term introduces a characteristic quadrupolar structure in the out-of-plane magnetic field component, which is a distinctive signature of Hall-mediated reconnection. The reconnection rate in Hall MHD is \(M_A \sim 0.1\), fast and essentially independent of the dissipation mechanism. This universality was demonstrated numerically in the GEM (Geospace Environment Modeling) challenge, where fluid, hybrid, and fully kinetic simulations all converged on the same reconnection rate once the Hall term was included. The NASA Magnetospheric Multiscale (MMS) mission, launched in 2015, has provided direct observations of the electron diffusion region, confirming the predicted structure.
6.6 Astrophysical Applications
Magnetic reconnection is central to many astrophysical phenomena:
Chapter 7: Dynamo Theory
How do planets, stars, and galaxies maintain their magnetic fields against ohmic decay? The answer lies in dynamo theory: the generation and sustenance of magnetic fields by the motion of conducting fluids. The dynamo problem is one of the great challenges of mathematical physics, combining elements of MHD, turbulence theory, bifurcation theory, and differential geometry. This chapter develops the theory from the kinematic dynamo problem through the fundamental anti-dynamo theorems to the mean-field theory that provides our best understanding of the solar and geodynamo.
7.1 The Kinematic Dynamo Problem
In the kinematic dynamo problem, we study the evolution of the magnetic field under a prescribed velocity field, ignoring the back-reaction of the field on the flow (the Lorentz force). This is appropriate in the early stages of field growth, when the magnetic field is weak.
with divergence-free initial data \(\mathbf{B}(\mathbf{x}, 0)\) grows without bound as \(t \to \infty\) (for at least some initial condition), despite the diffusive term.
The kinematic dynamo is an eigenvalue problem: the induction equation is linear in \(\mathbf{B}\), so we seek solutions \(\mathbf{B} \propto e^{pt}\) with \(\text{Re}(p) > 0\). The question is: for which velocity fields \(\mathbf{u}\) does such a growing mode exist?
7.2 Anti-Dynamo Theorems
Before seeking successful dynamos, it is instructive to understand what does not work. Several fundamental results, known as anti-dynamo theorems, place restrictions on the class of velocity and magnetic fields that can sustain a dynamo.
Cowling’s theorem is deeply significant: it tells us that all real dynamos must be inherently three-dimensional. The Earth’s roughly axisymmetric dipole field cannot be maintained by an axisymmetric flow; the convection in the core must have essential three-dimensional structure.
More precisely, Zeldovich showed that the two-dimensional component of the magnetic field always decays, even though the \(z\)-component of the field can be amplified by stretching. This result, together with Cowling’s theorem, demonstrates that dynamo action requires fully three-dimensional flows. The proof relies on showing that the magnetic flux function (the two-dimensional analogue of the vector potential) satisfies a maximum principle that forces it to decay.
Additional anti-dynamo results include the theorem of Bullard and Gellman (1954) that purely toroidal flows cannot sustain a dynamo, and the result that flows confined to a plane (planar flows) cannot act as dynamos. These negative results collectively establish that successful dynamos require flows with three-dimensional structure, nonzero helicity, and sufficient complexity — conditions that are naturally met in turbulent convection in rotating bodies.
7.3 Mean-Field Electrodynamics and the Alpha Effect
Real astrophysical flows are turbulent, and we cannot hope to track the magnetic field in full spatial detail. Mean-field electrodynamics, developed by Steenbeck, Krause, and Rädler (1966), provides a systematic framework for averaging the induction equation over the turbulent fluctuations.
where the overbar denotes an ensemble (or spatial) average and \(\overline{\mathbf{u}'} = \overline{\mathbf{B}'} = \mathbf{0}\).
Averaging the induction equation gives the mean-field induction equation:
\[ \frac{\partial \overline{\mathbf{B}}}{\partial t} = \nabla \times (\overline{\mathbf{u}} \times \overline{\mathbf{B}}) + \nabla \times \boldsymbol{\mathcal{E}} + \eta \nabla^2 \overline{\mathbf{B}}, \]where \(\boldsymbol{\mathcal{E}} = \overline{\mathbf{u}' \times \mathbf{B}'}\) is the mean electromotive force (emf) arising from the correlations between the fluctuating velocity and magnetic fields.
where \(\alpha = -\frac{1}{3}\tau_c \overline{\mathbf{u}' \cdot (\nabla \times \mathbf{u}')}\) is proportional to the kinetic helicity of the turbulence and \(\beta = \frac{1}{3}\tau_c \overline{|\mathbf{u}'|^2}\) is the turbulent diffusivity. Here \(\tau_c\) is the correlation time of the turbulence.
The alpha effect is remarkable: it shows that helical turbulence can generate a mean emf parallel to the mean magnetic field. This is precisely what is needed to close the dynamo loop. In the absence of the alpha effect, the mean-field induction equation is purely diffusive and no dynamo is possible (consistent with the anti-dynamo theorems applied to the mean field).
7.4 The Alpha-Omega Dynamo
The most successful model of the solar magnetic cycle combines the alpha effect with differential rotation (the omega effect).
The \(\alpha\Omega\)-dynamo model, when applied to the Sun, naturally produces oscillatory solutions (dynamo waves) that propagate equatorward, reproducing the observed butterfly diagram of sunspot emergence over the 22-year magnetic cycle. The dimensionless dynamo number \(D = \alpha \Delta\Omega L^3/\eta_T^2\), where \(\Delta\Omega\) is the differential rotation and \(\eta_T = \eta + \beta\) the total diffusivity, must exceed a critical value for dynamo action.
Dynamo action (\(\text{Im}(\omega) > 0\)) occurs when the dynamo number \(D = \alpha_0 G L^3/\eta_T^2\) exceeds a critical value \(D_c\) of order unity.
The direction of propagation of the dynamo wave depends on the sign of \(\alpha_0 G\): the wave travels in the \(-\hat{\mathbf{x}}\) direction if \(\alpha_0 G > 0\) and in the \(+\hat{\mathbf{x}}\) direction if \(\alpha_0 G < 0\). For the Sun, equatorward propagation (matching the butterfly diagram) requires \(\alpha_0 G < 0\) in the northern hemisphere, which is consistent with the observed sign of differential rotation at the tachocline and the expected sign of the \(\alpha\)-effect from rotating convection.
7.5 Fast Dynamos
A dynamo is called “fast” if its growth rate remains bounded away from zero as \(\mathrm{Rm} \to \infty\), and “slow” if the growth rate vanishes in this limit.
Fast dynamos are of fundamental importance in astrophysics, where \(\mathrm{Rm}\) is so large that a slow dynamo (whose growth rate scales as a negative power of \(\mathrm{Rm}\)) would be ineffective. The stretch-twist-fold (STF) mechanism, introduced by Vainshtein and Zeldovich, provides the canonical example: a flow that stretches a flux tube, twists it into a figure-eight, and folds it back on itself doubles the magnetic flux in each cycle, producing exponential growth independent of \(\mathrm{Rm}\).
7.6 The Geodynamo and Solar Dynamo
The two most studied natural dynamos are the geodynamo (in Earth’s liquid iron core) and the solar dynamo (in the Sun’s convection zone).
Chapter 8: Kinetic MHD and Beyond
The MHD description, powerful as it is, has intrinsic limitations. It treats the plasma as a single, electrically conducting fluid and assumes that the distribution function of each particle species is close to a Maxwellian (local thermodynamic equilibrium). When the mean free path becomes comparable to the macroscopic scale — as in the solar wind, the Earth’s magnetosphere, and the outer regions of accretion discs — this fluid description breaks down and kinetic effects become important. This chapter surveys the extensions of MHD into the kinetic regime.
8.1 Limitations of Fluid MHD
The MHD equations are derived from moments of the Boltzmann equation under the assumption that the particle distribution functions are nearly isotropic (Maxwellian). This requires that the collision mean free path \(\lambda_{\text{mfp}}\) be much smaller than the macroscopic length scale \(L\): \(\lambda_{\text{mfp}} \ll L\). When this ordering breaks down, several new phenomena arise that are absent from fluid MHD:
(i) Pressure anisotropy (\(p_\parallel \neq p_\perp\)), giving rise to firehose and mirror instabilities. (ii) Heat flux along field lines, which cannot be captured by an isotropic thermal conductivity. (iii) Landau damping and cyclotron resonances — collisionless wave-particle interactions that damp waves and heat particles. (iv) Finite Larmor radius (FLR) effects that introduce dispersion and modify the wave spectrum at scales \(k_\perp \rho_i \sim 1\).
8.2 The Vlasov-Maxwell System
The fundamental kinetic description of a collisionless plasma is the Vlasov-Maxwell system.
where \(f_s(\mathbf{x}, \mathbf{v}, t)\) is the distribution function in phase space. The electromagnetic fields \(\mathbf{E}\) and \(\mathbf{B}\) are determined self-consistently from Maxwell’s equations with charge and current densities computed as moments of \(f_s\).
The Vlasov-Maxwell system is a nonlinear integro-differential system in six-dimensional phase space and is far more complex than the MHD equations. However, its structure admits powerful conservation laws (energy, momentum, Casimir invariants) and supports a rich spectrum of wave modes that includes all the MHD waves as low-frequency limits.
8.3 The CGL Double-Adiabatic Theory
When collisions are rare but the Larmor radius is small compared to the macroscopic scale, an intermediate description between MHD and full kinetics is possible. Chew, Goldberger, and Low (1956) derived a set of fluid equations for a collisionless, magnetised plasma by taking moments of the Vlasov equation while preserving the anisotropy of the pressure tensor.
where \(\hat{\mathbf{b}} = \mathbf{B}/B\) is the unit vector along the magnetic field, and \(p_\parallel\), \(p_\perp\) are the pressures parallel and perpendicular to \(\mathbf{B}\). In the absence of heat fluxes, the two adiabatic invariants are:
\[ \frac{d}{dt}\left(\frac{p_\perp}{\rho B}\right) = 0 \quad \text{(conservation of magnetic moment)}, \]\[ \frac{d}{dt}\left(\frac{p_\parallel B^2}{\rho^3}\right) = 0 \quad \text{(conservation of the parallel adiabatic invariant)}. \]The CGL equations reduce to isotropic MHD when \(p_\parallel = p_\perp = p\), which holds when collisions are frequent enough to isotropise the distribution. The first CGL invariant is simply the conservation of the magnetic moment \(\mu = m v_\perp^2/(2B)\) of individual particles, elevated to a fluid statement. The second reflects the conservation of the longitudinal action \(J = \oint v_\parallel dl\) for particles bouncing between magnetic mirrors.
The pressure anisotropy in CGL theory introduces new instabilities that have no counterpart in isotropic MHD.
These instabilities act as regulators of pressure anisotropy in collisionless plasmas. When the anisotropy exceeds the instability threshold, rapid growth of fluctuations scatters particles and drives the distribution back toward the marginally stable state. This self-regulation is observed in the solar wind, where the measured distribution of \(p_\perp/p_\parallel\) clusters near the firehose and mirror instability boundaries.
8.4 Gyrokinetics
For many plasma physics applications, including turbulence in fusion devices and the solar wind, the relevant fluctuations have frequencies much lower than the ion cyclotron frequency and perpendicular wavelengths much longer than the ion Larmor radius. Gyrokinetics exploits this scale separation by averaging the Vlasov equation over the fast gyromotion, reducing the dimensionality of phase space from six to five.
where \(\Omega_i = q_i B/(m_i)\) is the ion cyclotron frequency, and \(k_\parallel\), \(k_\perp\) are the parallel and perpendicular wavenumbers. Under this ordering, the distribution function is expanded as \(f_s = F_{0s} + \delta f_s\) with \(\delta f_s / F_{0s} \sim \epsilon\).
The gyrokinetic equation for the non-adiabatic part of the distribution function \(h_s = \delta f_s + (q_s \varphi / T_s) F_{0s}\), where \(\varphi\) is the electrostatic potential, takes the form of an advection equation in the reduced (five-dimensional) phase space, coupled to the field equations through quasineutrality and Ampère’s law. The gyroaveraging operation \(\langle \cdot \rangle_\alpha\), which averages over the gyrophase angle \(\alpha\), smooths out the short-scale cyclotron physics and retains only the low-frequency dynamics relevant for turbulence and transport.
8.5 Landau Damping
One of the most important kinetic effects absent from fluid MHD is Landau damping: the collisionless damping of waves due to resonant interaction between the wave and particles whose velocity matches the wave phase speed.
where the damping rate for weak damping (\(|\gamma| \ll \omega_r\)) is
\[ \gamma = -\frac{\pi \omega_r^3}{2 k^2} \frac{q^2}{m \varepsilon_0} \left.\frac{\partial f_0}{\partial v}\right|_{v = \omega_r/k}. \]For a Maxwellian distribution (which has \(\partial f_0/\partial v < 0\) for \(v > 0\)), \(\gamma < 0\) and the wave is damped.
Landau damping is a purely kinetic effect with no counterpart in fluid theory. It arises because particles travelling slightly slower than the wave gain energy from the wave’s electric field, while those slightly faster lose energy; for a Maxwellian, there are more slow particles than fast ones near the phase velocity, so the net effect is energy transfer from wave to particles. In MHD, the analogous process — the damping of Alfvén waves by resonant ions — is called cyclotron damping and requires a kinetic treatment.
8.6 Collisionless Shocks
In ordinary gas dynamics, a shock wave is a thin transition layer in which kinetic energy is irreversibly converted to thermal energy by viscous dissipation. In a collisionless plasma, particle-particle collisions are too rare to provide the necessary dissipation, yet shock waves are ubiquitous (bow shocks, supernova remnants, interplanetary shocks). The dissipation mechanism must therefore be fundamentally different.
The structure of a collisionless shock depends critically on the Mach number \(M_A = v_{\text{shock}}/v_A\) and the angle \(\theta_{Bn}\) between the upstream magnetic field and the shock normal. For quasi-perpendicular shocks (\(\theta_{Bn} \gtrsim 45°\)), the shock structure is relatively laminar and mediated by the magnetic foot — a region where reflected ions gyrate back upstream. For quasi-parallel shocks (\(\theta_{Bn} \lesssim 45°\)), the structure is turbulent and extended, with a foreshock region of backstreaming ions and ULF waves.
8.7 The Magnetorotational Instability
As a final example of MHD physics beyond the classical topics, we briefly discuss the magnetorotational instability (MRI), which has revolutionised our understanding of accretion disc physics.
The maximum growth rate is \(\gamma_{\max} = \frac{3}{4}\Omega\) (for a Keplerian disc with \(\Omega \propto R^{-3/2}\)), achieved at a wavelength of order the Alfvén wavelength \(\lambda \sim v_A/\Omega\). Crucially, the instability requires only an arbitrarily weak magnetic field.
The MRI is remarkable because the Rayleigh criterion for centrifugal stability of an unmagnetised disc requires \(d(R^2 \Omega)/dR > 0\), which is satisfied for Keplerian rotation. Thus, a Keplerian disc is stable by the Rayleigh criterion but unstable by the MRI — the weak magnetic field fundamentally changes the stability properties. The physical mechanism is elegantly simple: the magnetic field acts as a spring connecting two fluid elements at slightly different radii. The inner element orbits faster and pulls the outer one forward, giving it angular momentum and causing it to move outward; the outer element drags the inner one back, removing angular momentum and causing it to fall inward. This positive feedback loop drives the instability and provides the angular momentum transport needed to explain accretion rates in astrophysical discs.
These notes have developed the mathematical theory of magnetohydrodynamics from its electromagnetic foundations through the ideal and resistive MHD equations, wave theory, equilibria, stability, reconnection, dynamo theory, and kinetic extensions. The subject is vast and still actively evolving: recent advances in kinetic reconnection, turbulence theory, gyrokinetics, and computational MHD continue to deepen our understanding of conducting fluids across an enormous range of physical settings, from laboratory plasmas to the large-scale structure of the universe.
