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Sweet–Parker Reconnection Rate

\[M_A = \frac{U_{\text{in}}}{v_A} \sim \frac{1}{\sqrt{S}}\]

Source lecture(s): pc368_lec17_sweetparker

Physical Meaning

The Sweet–Parker rate gives the fraction of the Alfvén speed at which magnetic flux is reconnected in a long, thin resistive current sheet. It quantifies how efficiently magnetic energy is converted to plasma energy in the MHD paradigm.

Variable Definitions

Symbol Definition SI Units
\(U_{\text{in}}\) Inflow velocity m s\(^{-1}\)
\(v_A\) Alfvén speed m s\(^{-1}\)
\(S\) Lundquist number dimless

Assumptions

  • Steady-state 2D resistive MHD.
  • Long, thin sheet (\(L \gg \delta\)).
  • Uniform resistivity \(\eta\).
  • Incompressible flow.

Derivation

Combine:

  1. Transition balance: \(U_{\text{in}} B = \eta B / (\mu_0 \delta) \Rightarrow U_{\text{in}} \sim \eta/(\mu_0 \delta)\).
  2. Mass conservation: \(U_{\text{in}} L = U_{\text{out}} \delta\).
  3. Energy / outflow continuity: \(U_{\text{out}} \sim v_A\).
  4. Eliminate \(\delta\): \(\delta = \eta L/(\mu_0 U_{\text{in}} v_A)\).
  5. From \(U_{\text{in}} = U_{\text{out}} \delta/L\), substitute and solve for \(U_{\text{in}}\).
  6. Result: \(U_{\text{in}} \sim v_A S^{-1/2}\).

Applications

  • Solar corona: Motivated the search for faster reconnection mechanisms.
  • Laboratory plasma: Scaling of flux consumption in RFP and tokamak sawteeth.
  • Geospace: Magnetospheric substorm current sheets.

Connections to Other Equations