= Solution
Let $b_l=\delta B_l/B_0$. The nonlinear term has two perpendicular spatial <derivatives>, so its rate relative to the fluctuation amplitude is
$$
\tau_{\mathrm{nl}}^{-1}\sim\frac{v_Ad_i b_l}{l_\perp^2},\qquad \tau_{\mathrm{nl}}\sim\frac{l_\perp^2}{v_Ad_i b_l}.
$$
The <wave period> from part (a), using $k\sim k_\perp$, is $\tau_w\sim l_\parallel l_\perp/(v_Ad_i)$. A <weak wave cascade> requires that one interaction change a packet only slightly:
$$
\chi=\frac{\tau_w}{\tau_{\mathrm{nl}}}\sim b_l\frac{l_\parallel}{l_\perp}\ll1.
$$
For interactions with decorrelated phases, successive fractional changes accumulate as a <random walk>. An order-one change needs $N\chi^2\sim1$, giving
$$
\boxed{\tau_l\sim N\tau_w\sim\frac{\tau_{\mathrm{nl}}^2}{\tau_w}=\frac{l_\perp^3}{v_Ad_i b_l^2l_\parallel}.}
$$
Constant <energy flux> per unit mass then gives $\epsilon\sim v_A^2b_l^2/\tau_l\sim v_A^3d_i b_l^4l_\parallel/l_\perp^3$, hence
$$
\boxed{\frac{\delta B_l}{B_0}\sim\left(\frac{\epsilon l_\perp^3}{v_A^3d_i l_\parallel}\right)^{1/4}.}
$$
The <weak electron-magnetohydrodynamic cascade> scaling uses both locality in scale and the decorrelation assumption; a small amplitude by itself does not specify the cascade time.
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