= Reduced electron magnetohydrodynamics
{title2=$\partial_t\Psi=v_A^2d_i\nabla_\parallel b,\quad\partial_tb=-d_i\nabla_\parallel\nabla_\perp^2\Psi$}
= Reduced electron MHD
{synonym}
= Electron RMHD
{synonym}
With $k_\parallel/k_\perp\sim\delta B/B_0\ll1$, define $b=\delta B_\parallel/B_0$ and $\delta\mathbf B_\perp/B_0=\hat{\mathbf z}\times\nabla_\perp\Psi/v_A$ to leading order. Let $\nabla_\parallel=\partial_z+\{\Psi,\cdot\}/v_A$, where $\{f,g\}=f_xg_y-f_yg_x$. The leading <Electron> <velocities> are $\mathbf u_{e\perp}=v_Ad_i\hat{\mathbf z}\times\nabla_\perp b$ and $u_{e\parallel}=-d_i\nabla_\perp^2\Psi$. Substitution into frozen-in induction yields
$$
\partial_t\Psi=v_A^2d_i\nabla_\parallel b,\qquad \partial_tb=-d_i\nabla_\parallel\nabla_\perp^2\Psi.
$$
<Linearization> gives $\omega^2=v_A^2d_i^2k_\parallel^2k_\perp^2$, agreeing with the full <EMHD> relation to leading anisotropic order. If $b$ varies along the guide field, an order-smaller perpendicular gradient correction restores exact solenoidality; the two-field representation is not an exact decomposition at arbitrary anisotropy.
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