Critically balanced electron-magnetohydrodynamic cascade ($b_l\propto l_{\perp }^{2/3},l_{\parallel }\propto l_{\perp }^{1/3}$)

Critical balance sets ${\tau }_{\mathrm{nl}}\sim {\tau }_w$, hence $l_{\parallel }\sim l_{\perp }/b_l$. A local strong cascade with constant energy flux has $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$, giving the displayed exponents and perpendicular magnetic energy spectrum $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The anisotropy becomes stronger towards small perpendicular scales.