 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 isThe wave period from part (a), using $k\sim k_{\perp }$, is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. A weak wave cascade requires that one interaction change a packet only slightly:For interactions with decorrelated phases, successive fractional changes accumulate as a random walk. An order-one change needs $N{\chi }^2\sim 1$, givingConstant energy flux per unit mass then gives $ϵ\sim v_A^2{b}_l^2/{\tau }_l\sim v_A^3{d}_i{b}_l^4{l}_{\parallel }/l_{\perp }^3$, henceThe 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.