 Solution

If $\delta$ is the cross-stream transition width and $\Delta U$ its speed change, dominant horizontal shear requiresBalancing the two terms in the reduced ODE gives the scaleThe approximation is strongest in the margin where $U^{\prime }$ is large, for a thin basal layer $h\ll H$ and parameters making this lateral scale short compared with the scale on which vertical shear changes the plug velocity. It fails sufficiently far into either uniform region because $U^{\prime }\to 0$ while the $O(ϵ)$ vertical shear needed to transmit the driving stress remains. It can also fail in narrow neighborhoods where neglected free-surface, bed-transition, or three-dimensional effects vary on the same scale.