= Solution
At fixed <number density> $n$, the <electron> current gives
$$
\mathbf u_e=-\frac{\mathbf J}{en}=-\alpha\nabla\times\mathbf B,\qquad\alpha=\frac{c}{4\pi en},\qquad\alpha B_0=v_Ad_i.
$$
The second equality uses <Ampère's law> without <displacement current> and stationary <ions>. A perturbation with characteristic perpendicular <wave number> $k_\perp$ therefore produces $\delta u_e\sim\alpha k_\perp\delta B$. Its nonlinear magnetic-advection rate is
$$
\boxed{\tau_{\mathrm{nl}}^{-1}\sim k_\perp\delta u_e\sim v_Ad_i k_\perp^2\frac{\delta B}{B_0}.}
$$
For the anisotropic <electron magnetohydrodynamics> <wave>, $k\simeq k_\perp$ and $|\omega|\sim v_Ad_i|k_\parallel|k_\perp$. Equating this to the nonlinear rate by <critical balance> gives
$$
\boxed{\frac{\delta B}{B_0}\sim\frac{|k_\parallel|}{k_\perp}\sim\epsilon\ll1.}
$$
This is small-amplitude but strong <turbulence>: a small perturbation can have order-one nonlinear distortion over one <wave> period because its perpendicular gradients are much larger than its parallel gradients.
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