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Lundquist Number

\[S = \frac{\mu_0 v_A L}{\eta}\]

Source lecture(s): pc368_lec17_sweetparker

Physical Meaning

The Lundquist number is the ratio of magnetic advection time to magnetic diffusion time. It is the magnetic Reynolds number in MHD.

Variable Definitions

Symbol Definition SI Units
\(\mu_0\) Vacuum permeability N A\(^{-2}\)
\(v_A\) Alfvén speed m s\(^{-1}\)
\(L\) Characteristic length m
\(\eta\) Magnetic diffusivity m\(^2\) s\(^{-1}\)

Assumptions

  • Single-fluid resistive MHD.
  • Global scale \(L\) is much larger than resistive dissipation scale.

Derivation

Advection time: \(\tau_A = L/v_A\).

Diffusion time: \(\tau_\eta = \mu_0 L^2/\eta\).

Ratio:

\[S = \frac{\tau_\eta}{\tau_A} = \frac{\mu_0 v_A L}{\eta}\]

Applications

  • Sweet–Parker sheet: \(M_A \sim S^{-1/2}\).
  • CRM: Fast reconnection requires \(S\) locally to drop or kinetic processes.
  • Dynamo: Field sustains if \(S \gg 1\) and the flow has suitable helicity.

Connections to Other Equations