Weak-to-strong transition of an Alfvénic cascade ($l_{\perp ,\ast }\sim L_{\perp }{\chi }_0^2$)

In a weak Alfvénic cascade at fixed $l_{\parallel }$, $\chi ={\tau }_A/{\tau }_{\mathrm{nl}}\propto l_{\perp }^{-1/2}$ grows as the perpendicular scale shrinks. If its outer value is ${\chi }_0\ll 1$ at $L_{\perp }$, it reaches order one at the displayed scale. Beyond it the weak-interaction ordering fails and a critical balance model may become appropriate.