Electron magnetohydrodynamics (${\partial }_t\mathbf{B}=\nabla \times ({\mathbf{u}}_e\times \mathbf{B})$)

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 givesFor a uniform guide field, the solenoidal linear equation implies ${\omega }^2=(c{B}_0/4\pi en{)}^2{k}_{\parallel }^2{k}^2$. With $v_A=B_0/\sqrt{4\pi n{m}_i}$ and $d_i=c\sqrt{m_i/(4\pi e^{2}n)}$, the coefficient is $c{B}_0/(4\pi en)=v_A{d}_i$.