= Solution
<Critical balance> makes the nonlinear interaction time comparable to the <wave period>:
$$
\frac{l_\perp^2}{v_Ad_i b_l}\sim\frac{l_\parallel l_\perp}{v_Ad_i},\qquad l_\parallel\sim\frac{l_\perp}{b_l}.
$$
An order-one interaction transfers energy in $\tau_l\sim\tau_{\mathrm{nl}}$, so constant <energy flux> gives $\epsilon\sim v_A^3d_i b_l^3/l_\perp^2$. Thus the <critically balanced electron-magnetohydrodynamic cascade> has
$$
\boxed{\frac{\delta B_l}{B_0}\sim\left(\frac{\epsilon}{v_A^3d_i}\right)^{1/3}l_\perp^{2/3},\qquad l_\parallel\sim\left(\frac{v_A^3d_i}{\epsilon}\right)^{1/3}l_\perp^{1/3}.}
$$
For the corresponding perpendicular <magnetic energy spectrum>, $\delta B_l^2\sim k_\perp E_B(k_\perp)$ gives $E_B(k_\perp)\propto k_\perp^{-7/3}$. The increasing ratio $l_\parallel/l_\perp\propto l_\perp^{-2/3}$ describes progressively more anisotropic fluctuations at smaller perpendicular scales, within the range in which the model and <critical balance> assumptions hold.
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