Reduced electron magnetohydrodynamics (${\partial }_t\Psi =v_A^2{d}_i{\nabla }_{\parallel }b,{\partial }_{t}b=-d_i{\nabla }_{\parallel }{\nabla }_{\perp }^2\Psi$)

With $k_{\parallel }/k_{\perp }\sim \delta B/B_0\ll 1$, 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 $\{f,g\}=f_x{g}_y-f_y{g}_x$. The leading Electron velocities are ${\mathbf{u}}_{e\perp }=v_A{d}_i\hat{\mathbf{z}}\times {\nabla }_{\perp }b$ and $u_{e\parallel }=-d_i{\nabla }_{\perp }^2\Psi$. Substitution into frozen-in induction yieldsLinearization gives ${\omega }^2=v_A^2{d}_i^2{k}_{\parallel }^2{k}_{\perp }^2$, 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.