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Magnetic Reconnection

Source lecture(s): pc368_lec16_reconnection

Intuition

Magnetic field lines are not rigid. When field lines of opposite polarity are pushed together by plasma motions, they can break and reconnect, forming new topologies. This process converts magnetic energy into particle acceleration, heating, and turbulence. It powers solar flares, magnetospheric substorms, and spheromak formation.

Formal Definition

Magnetic reconnection is a topological rearrangement of magnetic flux in which non-ideal terms (\(\eta\), electron inertia, or kinetic effects) allow \(\mathbf{E}_{\parallel} \neq 0\) in a localized diffusion region, changing the connectivity of field lines.

Mathematical Formulation

Faraday’s law in resistive MHD:

\[\frac{\partial \mathbf{B}}{\partial t} = -\nabla\times\mathbf{E}\]

Ohm’s law:

\[\mathbf{E} + \mathbf{U}\times\mathbf{B} = \eta\mathbf{J}\]

Reconnection rate \(V_{\text{rec}}\) is the speed at which magnetic flux is transferred across the separator:

\[V_{\text{rec}} \sim v_A M_A\]

where \(M_A = U_{\text{in}}/v_A\) is the inflow Alfvén Mach number.

Derivation

  1. Consider two antiparallel field regions pushed together at speed \(U_{\text{in}}\).
  2. In the inflow region (\(\mathbf{U}\times\mathbf{B}\) dominates), plasma moves toward the current sheet.
  3. Inside the sheet (resistive), \(\eta\mathbf{J}\) permits \(E_\parallel\).
  4. Ampère’s law links current to field gradient; balance gives sheet thickness.
  5. Mass conservation links inflow to outflow velocity.

Worked Example

Sweet–Parker sheet: For \(S = \mu_0 v_A L/\eta = 10^8\) and \(L = 10^6\) m, the sheet half-thickness is \(\delta \sim L S^{-1/2} = 100\) m. The reconnection rate \(M_A = S^{-1/2} = 10^{-4}\). This is extremely slow; observed reconnection in the corona is orders of magnitude faster, motivating Petschek and collisionless mechanisms.

Common Mistakes

  • Reconnection is caused by resistivity alone. Astrophysics needs fast reconnection, which requires localized anomalous resistivity or collisionless Hall effects.
  • Field lines “break” physically. They are not physical ropes; their connectivity changes, not their substance.
  • Reconnection rate is universal. It depends sensitively on geometry, boundary conditions, and kinetic physics.

Quiz Questions

  1. Conceptual: Why can reconnection occur in ideal MHD at a single line of zero width but not in a finite volume?
  2. Computational: Derive the Sweet–Parker reconnection rate \(M_A = S^{-1/2}\) from force balance and mass continuity.
  3. MCQ: The most important non-ideal term in Sweet–Parker reconnection is:
  4. A) Electron inertia
  5. B) Resistivity \(\eta\)
  6. C) Hall term
  7. D) Electron pressure gradient

Further Reading

  • E. Priest & T. Forbes, Magnetic Reconnection.