= Conductivity-gradient term in magnetic induction
{title2=$\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\eta\nabla^2\mathbf B-\nabla\eta\times(\nabla\times\mathbf B)$}
For scalar <electrical conductivity> $\sigma(\mathbf x)$ and constant permeability $\mu_0$, the moving-fluid Ohm law and the quasistatic <Ampere-Maxwell law> give $\mathbf E=-\mathbf u\times\mathbf B+\eta\nabla\times\mathbf B$, where $\eta=1/(\mu_0\sigma)$ is the <magnetic diffusivity>. Taking minus the <curl> and using <Faraday's law of induction> gives $\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)-\nabla\times(\eta\nabla\times\mathbf B)$. Expanding this expression and imposing $\nabla\cdot\mathbf B=0$ yields the title equation. The familiar constant-diffusivity <resistive induction equation> omits the last term because $\nabla\eta=0$; nonrelativistic motion alone does not establish that simplification.
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