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Induction Equation

\[\frac{\partial \mathbf{B}}{\partial t} = \nabla\times(\mathbf{U}\times\mathbf{B}) + \eta \nabla^2\mathbf{B}\]

Source lecture(s): pc368_lec05_mhd

Physical Meaning

The induction equation describes how magnetic field evolves due to advection by the fluid and resistive diffusion. It is the MHD analog of the vorticity equation.

Variable Definitions

Symbol Definition SI Units
\(\mathbf{B}\) Magnetic field T
\(\mathbf{U}\) Fluid velocity m s\(^{-1}\)
\(\eta\) Magnetic diffusivity m\(^2\) s\(^{-1}\)

Assumptions

  • Ohm’s law: \(\mathbf{E} + \mathbf{U}\times\mathbf{B} = \eta\mathbf{J}\).
  • Ampère’s law: \(\nabla\times\mathbf{B} = \mu_0\mathbf{J}\) (neglecting displacement current for low-frequency MHD).
  • \(\nabla\cdot\mathbf{B} = 0\) is preserved by induction.

Derivation

Start with \(\mathbf{E} = -\mathbf{U}\times\mathbf{B} + \eta\mathbf{J}\). Use Faraday’s law:

\[\frac{\partial \mathbf{B}}{\partial t} = -\nabla\times\mathbf{E} = \nabla\times(\mathbf{U}\times\mathbf{B}) - \eta\nabla\times\mathbf{J}\]

Insert \(\mathbf{J} = (1/\mu_0)\nabla\times\mathbf{B}\) and neglect displacement current to obtain the induction equation.

Applications

  • Dynamo theory: \(\eta \nabla^2\mathbf{B}\) amplifies field in stars/planets.
  • Reconnection: Resistive term allows field-line topology change.
  • Flux conservation: When \(\eta=0\), ideal induction yields frozen-in theorem.

Connections to Other Equations