 Weak electron-magnetohydrodynamic cascade

For $b_l{l}_{\parallel }/l_{\perp }\ll 1$, the weak wave cascade estimate gives ${\tau }_{\mathrm{cas}}\sim l_{\perp }^3/(v_A{d}_i{b}_l^2{l}_{\parallel })$. Constant energy flux $ϵ\sim v_A^2{b}_l^2/{\tau }_{\mathrm{cas}}$ yields the displayed amplitude. An additionally isotropic local cascade has $b_l\propto l^{1/2}$ and shell-integrated magnetic energy spectrum $E_B(k)\propto k^{-2}$. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.