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An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing $b_l=\delta B_l/B_0$, its perpendicular nonlinear interaction time is ${\tau }_{\mathrm{nl}}\sim l_{\perp }^2/(v_A{d}_i{b}_l)$. The anisotropic whistler wave time is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

For a uniform guide field, write the transverse perturbation in velocity units as $\mathbf{b}$. Linear momentum gives $\omega \mathbf{u}=-k_{\parallel }{v}_A\mathbf{b}$. Linear induction gives $\omega \mathbf{b}=-k_{\parallel }{v}_A\mathbf{u}+i{k}_{\parallel }{v}_A{d}_i\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode then yields the displayed dispersion relation. Its two positive frequency magnitudes are $|k_{\parallel }|v_A[\sqrt{1+(k{d}_i{)}^2/4}\pm k{d}_i/2]$, approaching the Alfvén wave frequency for $k{d}_i\ll 1$. For $k{d}_i\gg 1$, the fast branch is a whistler wave and the slow branch tends to $|k_{\parallel }|v_A/(k{d}_i)$.