Magnetohydrodynamics: When Fluid Meets Magnetic Field

Magnetohydrodynamics (MHD) describes electrically conducting fluids — from Earth's liquid iron core to the plasma jets of black holes. Its central insight is perhaps the strangest in physics: magnetic field lines are frozen into the conducting fluid, and the fluid drags them along like rubber bands.

1. MHD Equations

MHD merges the Navier-Stokes fluid equations with Maxwell's electromagnetic equations, coupled through the Lorentz force density J×B:

Continuity: \partialρ/\partialt + \nabla\cdot(ρv) = 0 (mass conservation) Momentum: ρ(\partialv/\partialt + v\cdot\nablav) = -\nablap + J\timesB + η\nabla²v Energy: \partial(p/ρ^γ)/\partialt = ... (adiabatic, resistive, or radiative closure) Induction: \partialB/\partialt = \nabla\times(v\timesB) + η_m \nabla²B J = (1/μ₀) \nabla\timesB (Ampere's law, no displacement current in MHD) \nabla\cdotB = 0 (no magnetic monopoles) η_m = 1/(μ₀σ) (magnetic diffusivity, σ = electrical conductivity) Two dimensionless numbers control behaviour: Magnetic Reynolds number: Rm = v\cdotL / η_m Rm ≫ 1: advection dominates, field is frozen to fluid Rm ≪ 1: diffusion dominates, field decouples from fluid Alfvén Mach number: M_A = v / v_A (v_A = Alfvén speed = B/\sqrt(μ₀ρ))

2. The Frozen-In Theorem

In the ideal MHD limit (η_m → 0, perfect conductor), the induction equation becomes ∂B/∂t = ∇×(v×B). Alfvén proved in 1943 that this means:

If a closed fluid loop moves with the flow, the magnetic flux through that loop remains constant
Equivalently, magnetic field lines are "frozen" into the fluid — they move wherever the fluid moves
Field lines cannot pass through each other; they are topologically constrained

Magnetic helicity: H = \int A\cdotB dV A = magnetic vector potential (B = \nabla\timesA) H measures the "knotting" and "linkage" of field lines. In ideal MHD, H is conserved globally — you cannot change the topology of field lines. In resistive MHD, H diffuses on the timescale τ_R = L²/η_m. Solar example: L \approx 100 Mm = 10⁸ m (active region) η_m \approx 1 m²/s (coronal resistivity) τ_R \approx 10¹⁶ s \approx 10⁸ years So field lines are essentially frozen in the corona on astrophysical timescales. Reconnection (faster topology change) requires different physics (see section 5).

3. Magnetic Pressure & Tension

The magnetic force density J×B = (1/μ₀)(∇×B)×B can be decomposed into two parts:

J\timesB = -\nabla(B²/2μ₀) + (1/μ₀)(B\cdot\nabla)B = "magnetic pressure" + "magnetic tension" Magnetic pressure p_B = B²/(2μ₀) Acts like a normal gas pressure, pushing perpendicular to B A region of strong B is like a high-pressure gas Magnetic tension T = B²/(μ₀ R_c) where R_c = curvature radius Acts like elastic tension in a stretched rubber band Curved field lines want to straighten Plasma β parameter: β = p_gas / p_B = p_gas\cdot2μ₀/B² β ≫ 1: gas pressure dominates (inertial confinement, galaxy clusters) β ≪ 1: magnetic forces dominate (solar corona, magnetospheres) β \approx 1: balance (tokamak edge, accretion discs)

4. Alfvén Waves

If you perturb a magnetised plasma perpendicular to the field, the magnetic tension acts as a restoring force. The perturbation propagates along field lines as an Alfvén wave:

Alfvén speed: v_A = B / \sqrt(μ₀ ρ) B = 5 nT (solar wind at 1 AU), ρ = 7\times10⁻²¹ kg/m³ v_A \approx 5\times10⁻⁹ / \sqrt(4π\times10⁻⁷ \times 7\times10⁻²¹) \approx 42 km/s Tokamak (B = 5 T, n = 10²⁰ m⁻³, deuterium): v_A \approx 5 / \sqrt(4π\times10⁻⁷ \times 3.3\times10⁻⁷) \approx 7,700 km/s \approx 0.026c Alfvén waves are transverse (like light waves, shear waves): Fluid oscillates perpendicular to B B oscillates perpendicular to both v and original B Not compressional (unlike sound waves)

Solar corona heating mystery: The solar corona is 2 million K — 200× hotter than the photosphere (5,800 K). As you move away from a heat source, temperature should decrease. One leading explanation: Alfvén waves generated by photospheric convection propagate up into the corona and dissipate through turbulent cascade. The energy flux is sufficient (≈200 W/m²) to maintain coronal temperatures if ~10% is efficiently dissipated.

5. Magnetic Reconnection

Despite the frozen-in constraint, magnetic topology can change rapidly in thin current sheets where resistivity becomes important — a process called magnetic reconnection. Oppositely-directed field lines are brought together, neutral point forms, field lines break and rejoin in a new topology, releasing enormous magnetic energy as kinetic energy and heat.

Solar flares: Reconnection of coronal field lines releases 10²⁵–10²⁶ J in minutes — equivalent to billions of nuclear weapons. Accelerates particles to near-light speed.
Geomagnetic substorms: Reconnection in Earth's magnetotail releases stored energy as energetic particle precipitation (triggering auroras) and plasma flows.
Coronal mass ejections: Reconnection below a magnetic flux rope provides the energy release that propels billions of tonnes of plasma into space at 250–3,000 km/s.

The paradox: reconnection in ideal MHD should take millions of years (Sweet-Parker rate). It actually takes minutes in the corona. Fast reconnection models (Petschek, plasmoid instability) explain the discrepancy through localised current sheet thinning.

6. Planetary & Stellar Dynamos

Earth's geomagnetic field is maintained by the geodynamo: convective motion of liquid iron in the outer core (1,500–3,500 km radius, T ≈ 4,000–5,000 K) generates electric currents that sustain the magnetic field — a self-amplifying feedback loop. The convection is driven by:

Secular cooling of the core
Compositional buoyancy as light elements (oxygen, sulfur, silicon) rise as the inner core solidifies
Latent heat release at the inner core boundary

The geodynamo reverses polarity irregularly (on average every 300,000 years; last reversal was 780,000 years ago). Transitions take 1,000–10,000 years. Field intensity drops by ~75% during reversals. Evidence from paleomagnetism (sea-floor spreading and rock samples) records hundreds of reversals in Earth's history.

The Sun's dynamo is driven by differential rotation (equator rotates in 25 days, poles in 35 days) and helical convection. It produces the 22-year magnetic cycle (11-year sunspot cycle × 2 for full polarity reversal).

7. Engineering Applications

MHD power generation: A conducting working fluid (hot plasma or ionised gas) is pushed through a magnetic field. By Faraday's law, an EMF is induced perpendicular to both velocity and field, generating electrical current without moving parts. High-temperature MHD generators (T > 2,500 K) can reach efficiencies of 60–70% in combined cycles.
Electromagnetic pumps: Liquid metal (sodium, NaK, lead-bismuth) is pumped by the Lorentz force from crossed electric and magnetic fields — no impellers, no seals, no moving parts. Used in sodium-cooled fast nuclear reactors (BN-800, Astrid).
MHD ship propulsion: The Yamato-1 (1992) was the first ship to use seawater MHD propulsion — superconducting magnets and electrodes drive seawater backward via Lorentz force. Quiet (no propellers) but low efficiency limits practical deployment.
Mass spectrometers: Accelerated ions following circular paths in a known magnetic field — the radius uniquely identifies the mass-to-charge ratio. Basis for all modern mass spectrometry.