Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 8

Magnetohydrodynamics

Advanced MHD: reconnection, conservation laws, turbulence, shocks, and force-free fields

8.1 Foundations of MHD

Magnetohydrodynamics describes the macroscopic dynamics of electrically conducting fluids permeated by magnetic fields. Building on the ideal and resistive MHD equations introduced earlier, this chapter develops the deeper structure of MHD theory: conservation laws, topological constraints, reconnection physics, turbulent cascades, and shock dynamics.

Complete MHD System (Summary)

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \quad \text{(mass)}$$
$$\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \mathbf{J} \times \mathbf{B} + \rho \nu \nabla^2 \mathbf{u} \quad \text{(momentum)}$$
$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta_m \nabla^2 \mathbf{B} \quad \text{(induction)}$$
$$\frac{\partial}{\partial t}\left(\frac{p}{\rho^\gamma}\right) + \mathbf{u} \cdot \nabla \left(\frac{p}{\rho^\gamma}\right) = 0 \quad \text{(energy/entropy)}$$
$$\nabla \cdot \mathbf{B} = 0, \quad \mathbf{J} = \frac{1}{\mu_0}\nabla \times \mathbf{B} \quad \text{(constraints)}$$

Dimensionless Numbers

  • • Rm = UL/ηm (magnetic Reynolds)
  • • S = τR/τA (Lundquist)
  • • β = 2μ₀p/B² (plasma beta)
  • • MA = u/vA (Alfvén Mach)

Characteristic Speeds

  • • vA = B/√(μ₀ρ) (Alfvén)
  • • cs = √(γp/ρ) (sound)
  • • vf,s (fast/slow magnetosonic)

Timescales

  • • τA = L/vA (Alfvén transit)
  • • τR = μ₀L²/η (resistive diffusion)
  • • τrec ~ √(τAτR) (reconnection)

8.2 MHD Conservation Laws

Ideal MHD possesses a rich set of conservation laws that constrain the dynamics and govern the long-time behavior of magnetized plasmas. These arise from the symmetries of the MHD equations and the topological properties of the magnetic field.

Total Energy Conservation

The total energy density in ideal MHD combines kinetic, thermal, and magnetic contributions:

$$\mathcal{E} = \frac{1}{2}\rho u^2 + \frac{p}{\gamma - 1} + \frac{B^2}{2\mu_0}$$
$$\frac{\partial \mathcal{E}}{\partial t} + \nabla \cdot \left[\left(\mathcal{E} + p + \frac{B^2}{2\mu_0}\right)\mathbf{u} - \frac{(\mathbf{u} \cdot \mathbf{B})\mathbf{B}}{\mu_0}\right] = 0$$

The energy flux includes the enthalpy flux, Poynting flux (E×B/μ₀), and kinetic energy flux. In resistive MHD, Ohmic dissipation ηJ² converts magnetic energy to heat.

Momentum Conservation (Stress Tensor)

The MHD momentum equation can be written in conservation form using the total stress tensor:

$$\frac{\partial (\rho u_i)}{\partial t} + \frac{\partial \Pi_{ij}}{\partial x_j} = 0$$
$$\Pi_{ij} = \rho u_i u_j + \left(p + \frac{B^2}{2\mu_0}\right)\delta_{ij} - \frac{B_i B_j}{\mu_0}$$

The Maxwell stress tensor Tij = BiBj/μ₀ − (B²/2μ₀)δij captures both magnetic pressure (isotropic, compressive) and magnetic tension (anisotropic, along field lines).

Magnetic Flux Conservation

In ideal MHD, the magnetic flux through any surface S moving with the fluid is conserved (Alfvén's theorem):

$$\frac{d\Phi_B}{dt} = \frac{d}{dt}\int_S \mathbf{B} \cdot d\mathbf{S} = 0$$

This is equivalent to the frozen-in flux condition: magnetic field lines are "frozen" into the fluid and move with it. Two fluid elements connected by a field line remain connected for all time. This topological constraint is the most fundamental property of ideal MHD.

Proof sketch: From Faraday's law with E = −u × B, the rate of change of flux through a moving surface involves both the local ∂B/∂t and the convective u × B boundary terms. These cancel exactly, yielding dΦ/dt = 0. The proof relies on ∇ · B = 0 and is valid only for ideal MHD (η = 0).

Cross-Helicity Conservation

The cross-helicity measures the correlation between velocity and magnetic field:

$$H_C = \int \mathbf{u} \cdot \mathbf{B} \, d^3r$$

In ideal MHD, HC is conserved. It quantifies the degree of alignment between u and B. When |HC| is large, Alfvénic fluctuations propagate preferentially in one direction along B, and the turbulent cascade is inhibited (dynamic alignment).

8.3 Magnetic Helicity

Magnetic helicity is a topological invariant that measures the linkage, twist, and writhe of magnetic field lines. Its near-conservation even in resistive MHD imposes powerful constraints on plasma relaxation and reconnection.

Definition

For a magnetic field B = ∇ × A (where A is the vector potential):

$$H_M = \int_V \mathbf{A} \cdot \mathbf{B} \, d^3r = \int_V \mathbf{A} \cdot (\nabla \times \mathbf{A}) \, d^3r$$

HM is gauge-invariant when B · n̂ = 0 on the boundary ∂V. For two linked flux tubes with fluxes Φ₁ and Φ₂, HM = 2nΦ₁Φ₂ where n is the linking number.

Evolution Equation

In resistive MHD:

$$\frac{dH_M}{dt} = -2\int_V \mathbf{E} \cdot \mathbf{B} \, d^3r = -2\eta \int_V \mathbf{J} \cdot \mathbf{B} \, d^3r$$

While magnetic energy dissipates as ∫ηJ² d³r, helicity dissipates as ∫ηJ·B d³r. Since J·B involves the parallel current only, helicity dissipates much more slowly than energy in high-Lundquist-number plasmas:

$$\frac{\dot{H}_M}{H_M} \sim \frac{1}{\tau_R}, \quad \frac{\dot{W}_B}{W_B} \sim \frac{1}{\tau_A} \quad \Rightarrow \quad \frac{\dot{H}/H}{\dot{W}/W} \sim \frac{\tau_A}{\tau_R} = S^{-1} \ll 1$$

Significance: In high-S plasmas (S ~ 10⁶–10¹⁴ in fusion and astrophysics), helicity is effectively conserved during reconnection events, even as energy is rapidly released. This selective decay underpins Taylor relaxation theory.

Taylor Relaxation

J.B. Taylor (1974) proposed that a turbulent plasma relaxes to a minimum-energy state subject to the constraint of conserved total magnetic helicity. The variational problem:

$$\delta\left[W_B - \frac{\lambda}{2} H_M\right] = 0 \quad \Rightarrow \quad \nabla \times \mathbf{B} = \lambda \mathbf{B}$$

This yields the linear force-free (Beltrami) field with constant λ throughout the volume. The relaxed state is the Bessel function model describing the reversed-field pinch (RFP):

$$B_\theta(r) = B_0 J_1(\lambda r), \quad B_z(r) = B_0 J_0(\lambda r)$$

The toroidal field reverses direction near the wall when λa ≈ 3.11 (first zero of J₀), explaining the reversed-field pinch configuration observed experimentally.

Relative Magnetic Helicity

When B · n̂ ≠ 0 on the boundary (open systems like the solar corona), the gauge-invariant relative helicity is used:

$$H_R = \int_V (\mathbf{A} + \mathbf{A}_p) \cdot (\mathbf{B} - \mathbf{B}_p) \, d^3r$$

Bp is the reference potential field (∇ × Bp = 0) with the same normal component on ∂V. HR measures how much the actual field is twisted/sheared relative to the untwisted potential state.

8.4 Force-Free Magnetic Fields

In magnetically dominated plasmas (β ≪ 1), such as the solar corona and pulsar magnetospheres, the Lorentz force must approximately vanish: J × B ≈ 0. This requires J ∥ B everywhere.

General Force-Free Condition

$$\nabla \times \mathbf{B} = \alpha(\mathbf{r}) \mathbf{B}$$

where α(r) is a scalar function. Taking the divergence and using ∇ · B = 0:

$$\mathbf{B} \cdot \nabla \alpha = 0$$

α is constant along field lines but can vary between field lines.

Potential Field (α = 0)

Current-free: ∇ × B = 0

Minimum energy state for given boundary conditions. B = −∇φ with ∇²φ = 0.

Linear Force-Free (α = const)

Beltrami field: ∇ × B = αB

Reduces to Helmholtz equation ∇²B + α²B = 0. Taylor relaxation end-state.

Nonlinear Force-Free (α = α(r))

Variable α along the volume

Most general; requires numerical solution. Used for coronal field extrapolation.

Woltjer's Theorem

The minimum energy state of an ideal MHD system with conserved magnetic helicity HMis a linear force-free field:

$$\delta\left[\int \frac{B^2}{2\mu_0} d^3r - \frac{\lambda}{2}\int \mathbf{A} \cdot \mathbf{B} \, d^3r\right] = 0 \quad \Rightarrow \quad \nabla \times \mathbf{B} = \lambda \mathbf{B}$$

where λ = μ₀WB/HM is determined by the helicity-to-energy ratio. This provides the theoretical foundation for Taylor relaxation.

8.5 Magnetic Reconnection

Magnetic reconnection is the process by which magnetic field lines break and rejoin, converting magnetic energy into kinetic energy, thermal energy, and particle acceleration. It is the fundamental mechanism behind solar flares, magnetospheric substorms, sawtooth crashes in tokamaks, and astrophysical jets.

The reconnection problem: In ideal MHD, field lines cannot break (frozen-in flux). Resistivity allows reconnection, but classical resistive timescales (τR ~ 10⁶ years for the corona) are far too slow to explain observed phenomena (flares ~ minutes). Resolving this timescale discrepancy is the central problem of reconnection physics.

Sweet-Parker Reconnection (1957-58)

The first quantitative reconnection model considers steady-state reconnection in a thin current sheet of length L and thickness δ:

Geometry:

Antiparallel fields B0 are driven toward a diffusion region of length L (system scale) and width δ. Plasma enters slowly (vin) from above/below and is ejected rapidly (vout) from the sides.

From mass conservation, pressure balance, and Ohm's law in the diffusion region:

$$v_{\text{in}} L = v_{\text{out}} \delta \quad \text{(mass conservation)}$$
$$v_{\text{out}} = v_A = \frac{B_0}{\sqrt{\mu_0 \rho}} \quad \text{(pressure balance: outflow at Alfvén speed)}$$
$$v_{\text{in}} B_0 = \frac{\eta_m B_0}{\delta} \quad \text{(resistive Ohm's law at X-point)}$$

Combining these gives the Sweet-Parker scaling:

$$\frac{\delta}{L} = S^{-1/2}, \quad v_{\text{in}} = \frac{v_A}{\sqrt{S}}, \quad M_A^{\text{in}} = S^{-1/2}$$
$$\tau_{\text{SP}} = \sqrt{\tau_A \tau_R} = \tau_A S^{1/2}$$

where S = τR/τA = μ₀LvA/η is the Lundquist number. For the solar corona (S ~ 10¹²), this gives τSP ~ 10⁶ τA ~ months, far too slow for observed flare timescales of minutes.

Petschek Reconnection (1964)

Petschek proposed that reconnection can be much faster if the diffusion region is localized to a small region (length ℓ ≪ L) and most of the energy conversion occurs at slow-mode MHD shocks emanating from the X-point:

$$M_A^{\text{in}} \sim \frac{\pi}{8 \ln S}, \quad v_{\text{in}} \sim \frac{\pi v_A}{8 \ln S}$$

Key features of the Petschek model:

  • Diffusion region is only ℓ ~ L/√S in size (microscopic)
  • Two pairs of slow-mode shocks extend from X-point to boundaries
  • Most energy conversion occurs at the shocks, not in the diffusion region
  • Reconnection rate is nearly independent of resistivity (logarithmic dependence)
  • Outflow speed remains vA

Status: Petschek reconnection gives rates ~0.01–0.1 vA, consistent with observations. However, MHD simulations with uniform resistivity tend to collapse back to Sweet-Parker. Petschek-type reconnection requires localized resistivity enhancement or kinetic (collisionless) effects at the X-point.

Plasmoid-Mediated Reconnection

For S > Scrit ≈ 10⁴, Sweet-Parker current sheets are unstable to the plasmoid (tearing) instability:

$$\gamma_{\text{plasmoid}} \sim S^{1/4} \tau_A^{-1} \quad \text{(super-Alfvénic growth for large S)}$$

The current sheet fragments into a chain of magnetic islands (plasmoids) separated by secondary X-points. This process:

  • Breaks the single current sheet into many shorter sheets
  • Each shorter sheet reconnects faster (smaller effective L)
  • Plasmoids grow, coalesce, and are ejected along the outflow
  • Produces a fractal hierarchy of current sheets and islands
  • Yields a reconnection rate that becomes independent of S: MA ~ 0.01
$$M_A^{\text{plasmoid}} \approx 0.01 \quad \text{(approximately universal for } S > 10^4\text{)}$$

Collisionless (Hall) Reconnection

When the current sheet thickness δ approaches the ion inertial length di = c/ωpi, ions decouple from the magnetic field while electrons remain magnetized. This creates the Hall current system:

$$d_i = \frac{c}{\omega_{pi}} = \frac{c}{\sqrt{n_i e^2/(\epsilon_0 m_i)}} \sim 230 \text{ km} \cdot \left(\frac{n}{10^6 \text{ m}^{-3}}\right)^{-1/2}$$

Two-Scale Structure

  • • Ion diffusion region: ~ di
  • • Electron diffusion region: ~ de
  • • Quadrupolar out-of-plane By
  • • Hall electric field EHall = J×B/(ne)

Key Results

  • • Fast rate: MA ~ 0.1 (universal)
  • • Independent of dissipation mechanism
  • • Open outflow geometry (not bottlenecked)
  • • Confirmed by MMS spacecraft observations

Reconnection Rate Comparison

ModelRate MATimescaleMechanism
Sweet-ParkerS−1/2√(τAτR)Resistive diffusion in long sheet
Petschekπ/(8 ln S)~10 τASlow shocks + localized diffusion
Plasmoid~0.01~100 τATearing instability of current sheet
Hall/Collisionless~0.1~10 τAIon-electron decoupling

8.6 MHD Shocks and Discontinuities

MHD supports a richer variety of shocks and discontinuities than ordinary hydrodynamics, due to the anisotropy introduced by the magnetic field. The Rankine-Hugoniot jump conditions must be generalized to include electromagnetic terms.

MHD Rankine-Hugoniot Relations

In the shock frame, the jump conditions across a planar discontinuity (denoted by [[·]]) are:

$$[[\rho v_n]] = 0 \quad \text{(mass)}$$
$$\left[\left[\rho v_n^2 + p + \frac{B_t^2}{2\mu_0}\right]\right] = 0 \quad \text{(normal momentum)}$$
$$\left[\left[\rho v_n \mathbf{v}_t - \frac{B_n \mathbf{B}_t}{\mu_0}\right]\right] = 0 \quad \text{(tangential momentum)}$$
$$[[B_n]] = 0 \quad \text{(∇·B = 0)}$$
$$[[v_n \mathbf{B}_t - B_n \mathbf{v}_t]] = 0 \quad \text{(tangential electric field)}$$

Subscripts n and t denote normal and tangential components. These reduce to the usual Rankine-Hugoniot conditions when B = 0.

Classification of MHD Discontinuities

Contact Discontinuity

  • • No flow across surface (vn = 0)
  • • Density jump, continuous p and B
  • • Advected with the flow

Tangential Discontinuity

  • • No flow across, Bn = 0
  • • Arbitrary jumps in p, ρ, Bt, vt
  • • Total pressure p + B²/(2μ₀) continuous
  • • Current sheet (magnetopause example)

Rotational Discontinuity

  • • Flow crosses surface at vn = ±Bn/√(μ₀ρ)
  • • Bt rotates, |B| constant
  • • No compression (ρ, p continuous)
  • • Finite-amplitude Alfvén wave

MHD Shocks (Fast, Slow, Intermediate)

  • • Compressive (entropy increase)
  • • Fast: Bt increases across shock
  • • Slow: Bt decreases across shock
  • • Switch-on/switch-off: special cases

Perpendicular Shock (B ⊥ n̂)

When the magnetic field is perpendicular to the shock normal, the compression ratio is modified by magnetic pressure:

$$\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_f^2}{(\gamma - 1)M_f^2 + 2} \quad \text{where } M_f = \frac{v_1}{\sqrt{v_A^2 + c_s^2}}$$

The magnetic field is compressed along with the plasma: B₂/B₁ = ρ₂/ρ₁. Perpendicular shocks are common in the solar wind interaction with planetary magnetospheres (bow shocks).

Parallel Shock (B ∥ n̂)

When B is parallel to the shock normal, the tangential field is zero on both sides and the shock reduces to an ordinary hydrodynamic shock. However, the "switch-on" shock is unique to MHD:

$$B_{t1} = 0, \quad B_{t2} \neq 0 \quad \text{(switch-on: tangential field created)}$$

This can occur when MA > 1 upstream and MA < 1 downstream. The downstream flow "switches on" a tangential magnetic field component.

8.7 MHD Turbulence

Turbulence in magnetized plasmas is ubiquitous — from the solar wind and interstellar medium to tokamak edge layers and accretion disks. MHD turbulence differs fundamentally from hydrodynamic turbulence due to the presence of the magnetic field, which introduces anisotropy and wave-like dynamics.

Elsässer Variables

MHD turbulence is most naturally described using Elsässer variables, which separate the dynamics into counter-propagating Alfvén wave packets:

$$\mathbf{z}^\pm = \mathbf{u} \pm \frac{\mathbf{B}}{\sqrt{\mu_0 \rho}} = \mathbf{u} \pm \mathbf{v}_A$$

The incompressible MHD equations become:

$$\frac{\partial \mathbf{z}^\pm}{\partial t} \mp (\mathbf{V}_A \cdot \nabla)\mathbf{z}^\pm + (\mathbf{z}^\mp \cdot \nabla)\mathbf{z}^\pm = -\nabla P^* + \text{dissipation}$$

where VA = B₀/√(μ₀ρ₀) is the mean-field Alfvén velocity and P* = (p/ρ + B²/(2μ₀ρ)). The crucial feature: z⁺ and z⁻ propagate in opposite directions along B₀, and nonlinear interactions occur only between counter-propagating packets.

Iroshnikov-Kraichnan Theory (1964-65)

The first MHD turbulence theory assumed isotropic turbulence weakened by Alfvén wave propagation. Nonlinear interactions occur only during the Alfvén crossing time τA = ℓ/VA, leading to:

$$E(k) \propto (\epsilon V_A)^{1/2} k^{-3/2}$$

The −3/2 spectrum is shallower than the Kolmogorov −5/3, reflecting the Alfvén effect: wave propagation decorrelates eddies before they can fully cascade, slowing the energy transfer.

Goldreich-Sridhar Theory (1995)

The modern theory recognizes that MHD turbulence is fundamentally anisotropic. The critical balance hypothesis states that the nonlinear cascade time equals the Alfvén propagation time at each scale:

Critical Balance:

$$\tau_{\text{nl}} \sim \tau_A \quad \Rightarrow \quad \frac{\ell_\perp}{\delta u_\perp} \sim \frac{\ell_\parallel}{V_A}$$

This yields the anisotropic energy spectrum:

$$E(k_\perp) \propto \epsilon^{2/3} k_\perp^{-5/3} \quad \text{(Kolmogorov-like in } k_\perp\text{)}$$

and the scale-dependent anisotropy:

$$k_\parallel \propto k_\perp^{2/3} \quad \Rightarrow \quad \ell_\parallel \propto \ell_\perp^{2/3}$$

At smaller scales, eddies become increasingly elongated along the mean field. The cascade is predominantly perpendicular (in k⊥), with minimal transfer in k∥. This has been broadly confirmed by solar wind measurements and numerical simulations.

Dynamic Alignment & Boldyrev's Theory

Boldyrev (2005-06) proposed that velocity and magnetic field fluctuations tend to align at smaller scales (dynamic alignment), modifying the GS theory:

$$\sin\theta_\ell \propto \ell_\perp^{1/4} \quad \Rightarrow \quad E(k_\perp) \propto k_\perp^{-3/2}$$

where θℓ is the alignment angle between δu and δB at scale ℓ. This predicts a −3/2 spectrum rather than −5/3, and the anisotropy k∥ ∝ k⊥1/2. The question of whether the inertial range follows −5/3 or −3/2 remains actively debated.

Imbalanced Turbulence

In many astrophysical systems (e.g., solar wind), the Elsässer energies are unequal (z⁺ ≫ z⁻ or vice versa). This "imbalanced" or "cross-helical" turbulence exhibits:

  • Different cascade rates for dominant and subdominant components
  • Pinning of the subdominant spectrum at the dissipation scale
  • Modified scaling: dominant spectrum steepens, subdominant flattens
  • Reduced overall dissipation rate compared to balanced turbulence

Solar wind observations typically show z⁺/z⁻ ~ 2−5, with the dominant component propagating away from the Sun (outward Alfvénic fluctuations).

8.8 The Magnetic Dynamo

The dynamo problem asks: how do cosmic bodies (planets, stars, galaxies) generate and sustain their magnetic fields against resistive decay? The answer lies in the ability of conducting fluid motions to amplify magnetic fields faster than diffusion can destroy them.

Anti-Dynamo Theorems

Not all flows can sustain dynamo action. Important constraints include:

Cowling's Theorem (1933)

An axisymmetric magnetic field cannot be maintained by dynamo action. This implies that all observed cosmic dynamos must involve 3D flows breaking axisymmetry.

Zel'dovich Theorem

A purely 2D flow (uz = 0 with ∂/∂z = 0) cannot maintain a dynamo. At least some variation or flow component in the third dimension is required.

Mean-Field Dynamo Theory (α-ω Dynamo)

Decompose fields into mean and fluctuating parts: B = ⟨B⟩ + b, u = ⟨U⟩ + u'. The mean induction equation becomes:

$$\frac{\partial \langle\mathbf{B}\rangle}{\partial t} = \nabla \times (\langle\mathbf{U}\rangle \times \langle\mathbf{B}\rangle) + \nabla \times \boldsymbol{\mathcal{E}} + \eta_m \nabla^2 \langle\mathbf{B}\rangle$$

where the turbulent electromotive force (EMF) is:

$$\boldsymbol{\mathcal{E}} = \langle \mathbf{u}' \times \mathbf{b} \rangle \approx \alpha \langle\mathbf{B}\rangle - \beta \nabla \times \langle\mathbf{B}\rangle$$

α-Effect

$$\alpha \sim -\frac{\tau_c}{3}\langle \mathbf{u}' \cdot (\nabla \times \mathbf{u}') \rangle$$

Proportional to kinetic helicity of turbulence. Generates poloidal B from toroidal B (and vice versa). Requires broken mirror symmetry (rotation + stratification).

ω-Effect (Differential Rotation)

$$\nabla \times (\langle\mathbf{U}\rangle \times \langle\mathbf{B}\rangle) \supset B_r \frac{\partial \Omega}{\partial r}$$

Differential rotation stretches poloidal field into toroidal field. Combined with α-effect, gives oscillatory dynamo solutions (Parker's dynamo wave, solar cycle).

Dynamo Number and Onset

The dimensionless dynamo number determines whether the αω-dynamo is supercritical:

$$D = \frac{\alpha_0 \Omega' L^3}{\eta_T^2}$$

For |D| > Dcrit ~ 10, the dynamo grows exponentially until quenched by the back-reaction of the Lorentz force on the flow (α-quenching). The saturated state typically has the magnetic energy in near-equipartition with the turbulent kinetic energy.

8.9 Astrophysical and Laboratory Applications

Solar Flares and Coronal Mass Ejections

The solar corona stores free magnetic energy in non-potential (force-free) field configurations above active regions. When the field configuration reaches a critical point:

  • Energy storage: ~10²⁵–10³² J in stressed coronal fields (typical flare: ~10²⁵ J)
  • Trigger: Ideal MHD instabilities (kink, torus) or loss of equilibrium
  • Energy release: Reconnection converts magnetic → kinetic + thermal + particles
  • Timescale: Alfvén time ~ 10 s, flare duration ~ 10² – 10³ s
  • CMEs: Eruptive flux rope with mass ~10¹² kg ejected at 100–3000 km/s

Standard flare model (CSHKP): A rising flux rope stretches overlying field lines, forming a vertical current sheet below. Reconnection in this sheet produces the flare arcade, separating ribbons on the chromosphere, and high-energy particles accelerated in the reconnection outflow jets.

Magnetospheric Dynamics

Earth's magnetosphere is shaped by the interaction between the solar wind and the geomagnetic dipole:

Dayside Reconnection

When IMF Bz < 0 (southward), reconnection at the magnetopause opens field lines, allowing solar wind plasma entry. Rate ~ 0.1 vABn.

Magnetotail Substorms

Opened flux accumulates in the tail. When the current sheet thins to ~di, explosive reconnection releases stored energy → aurora, plasmoid ejection tailward.

Accretion Disks: The MRI

The magnetorotational instability (MRI, Balbus & Hawley 1991) is the primary mechanism for angular momentum transport in accretion disks. A weak magnetic field threading a differentially rotating disk is unstable:

$$\text{Instability condition: } \frac{d\Omega^2}{d\ln r} < 0 \quad \text{(Keplerian: always satisfied)}$$
$$\gamma_{\max} = \frac{3}{4}\Omega, \quad k_{\max} v_A = \frac{\sqrt{15}}{4}\Omega$$

The MRI operates even for arbitrarily weak fields (as long as the wavelength fits within the disk). It drives MHD turbulence that transports angular momentum outward via correlated Maxwell and Reynolds stresses, enabling accretion. The effective viscosity parameter αSS ~ 0.01–0.1.

Fusion Plasmas: Tokamak MHD

MHD governs the macroscopic stability and equilibrium of fusion devices:

  • Equilibrium: Grad-Shafranov equation determines flux surfaces, pressure profiles, safety factor q(ψ)
  • Stability limits: Internal kink (q₀ < 1 → sawtooth crashes), external kink (qa < 2), ballooning (β-limit)
  • Neoclassical tearing modes (NTMs): Bootstrap current-driven islands that degrade confinement
  • Edge-localized modes (ELMs): Peeling-ballooning instabilities at the pedestal, cause periodic energy bursts
  • Disruptions: Global MHD instabilities → rapid loss of confinement, large forces on vessel
  • Resistive wall modes: Slowly growing external kinks when wall has finite conductivity

Key Takeaways

  • ✓ MHD conserves total energy, momentum, magnetic flux (ideal), cross-helicity, and magnetic helicity
  • ✓ Magnetic helicity HM = ∫A·B d³r is nearly conserved even in resistive MHD (selective decay)
  • ✓ Taylor relaxation: minimum energy at constant helicity → force-free state ∇×B = λB
  • ✓ Sweet-Parker reconnection is too slow (τ ~ S1/2τA); Petschek and plasmoid instabilities provide fast reconnection
  • ✓ Collisionless reconnection (Hall MHD) yields universal rate MA ~ 0.1
  • ✓ MHD supports contact, tangential, rotational discontinuities, and fast/slow shocks
  • ✓ MHD turbulence is anisotropic: Goldreich-Sridhar critical balance gives k∥ ∝ k⊥2/3
  • ✓ The α-ω dynamo explains solar/stellar magnetic field generation through helical turbulence + differential rotation
  • ✓ The MRI drives accretion disk turbulence and angular momentum transport
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