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Alfvén Speed

\[v_A = \frac{B}{\sqrt{\mu_0 \rho}}\]

Source lecture(s): pc368_lec05_mhd, pc368_lec17_sweetparker

Physical Meaning

The Alfvén speed is the characteristic speed at which transverse MHD disturbances propagate along magnetic field lines. It is the “speed of sound” for magnetic tension.

Variable Definitions

Symbol Definition SI Units
\(B\) Magnetic field strength T
\(\rho\) Mass density kg m\(^{-3}\)
\(\mu_0\) Vacuum permeability N A\(^{-2}\)

Assumptions

  • Uniform, straight magnetic field.
  • Single-fluid MHD.
  • Low-frequency perturbation (\(\omega \ll \omega_{ci}\)).

Derivation

From the linearized ideal MHD momentum and induction equations for a perturbation \(\delta \mathbf{B}\) along a uniform field \(\mathbf{B}_0 = B_0 \hat{z}\):

\[\rho_0 \frac{\partial^2 \boldsymbol{\xi}}{\partial t^2} = \frac{(\mathbf{B}_0\cdot\nabla)(\mathbf{B}_0\cdot\nabla\boldsymbol{\xi}) - B_0^2 \nabla^2\boldsymbol{\xi}}{\mu_0}\]

For a transverse wave \(\propto e^{i(kz - \omega t)}\): \(\omega^2 = k^2 v_A^2\) with \(v_A = B/\sqrt{\mu_0\rho}\).

Applications

  • Tokamak stability: \(v_A\) sets the Alfvén transit time and Alfvén eigenmodes.
  • Reconnection rate: Inflow speed \(U_{in} \sim v_A M_A\).
  • Solar wind: Observations match \(v_A\) at 1 AU.

Connections to Other Equations