= Electron magnetohydrodynamics
{title2=$\partial_t\mathbf B=\nabla\times(\mathbf u_e\times\mathbf B)$}
= Electron MHD
{synonym}
= EMHD
{c}
{synonym}
<Electron magnetohydrodynamics> neglects <ion> motion and <Electron> inertia while retaining transport of the <magnetic field> by the <Electron> current. At constant <Electron> <number density>, $\mathbf u_e=-\mathbf J/(en)=-c\nabla\times\mathbf B/(4\pi en)$. Flux freezing in this <velocity> gives
$$
\partial_t\mathbf B=-\frac c{4\pi en}\nabla\times[(\nabla\times\mathbf B)\times\mathbf B].
$$
For a uniform guide field, the <solenoidal> linear equation implies $\omega^2=(cB_0/4\pi en)^2k_\parallel^2k^2$. With $v_A=B_0/\sqrt{4\pi nm_i}$ and $d_i=c\sqrt{m_i/(4\pi e^2n)}$, the coefficient is $cB_0/(4\pi en)=v_Ad_i$.
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