E^{'} = E + u \times B

. Applying Ohm's law in the local fluid frame gives

E + u \times B = η J

. Neglecting the displacement current in Ampère's law,

J = \nabla \times B / μ_{0}

. Substitution in Faraday's law gives

\partial_{t} B = \nabla \times (u \times B) - \frac{η}{μ _{0}} \nabla \times (\nabla \times B) = \nabla \times (u \times B) + η_{m} \nabla^{2} B, η_{m} = \frac{η}{μ _{0}} .

(1)

Here

η_{m}

is the magnetic diffusivity; the paper's

η

is electrical resistivity, rather than magnetic diffusivity. Expanding the curl with the solenoidal magnetic-field constraint gives the resistive induction equation in its compressible form:

\partial_{t} B + (u \cdot \nabla) B = (B \cdot \nabla) u - B \nabla \cdot u + \frac{η}{μ _{0}} \nabla^{2} B .

(2)

The left side is the material derivative of the magnetic field. The three right-hand terms respectively describe field-line stretching and reorientation, amplification or dilution by compression or expansion, and magnetic diffusion. In particular, compression has

\nabla \cdot u < 0

and amplifies a field component perpendicular to the compression.

The assumptions are nonrelativistic bulk motion, the low-frequency approximation permitting omission of the displacement current, a conducting-fluid continuum, isotropic local Ohm's law with spatially constant scalar electrical resistivity, constant magnetic permeability

μ_{0}

, and

\nabla \cdot B = 0

. Hall, electron-pressure and other non-Ohmic corrections to the generalized Ohm's law are excluded. Incompressible flow is not required: imposing it would remove the compression term.