Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 79 / 3 / ii / c / Solution

At fixed friction velocity, unstable heating gives $B_0>0$ and $\zeta <0$. The usual similarity closure has ${\varphi }_m<1$, so it reduces the mean velocity gradient relative to the neutral law of the wall. Rising warm and descending cool parcels mix momentum efficiently; less mean shear is needed to transport the same stress. Relative to a fixed near-wall reference, the speed difference aloft is reduced.

Stable cooling gives $B_0<0,\zeta >0$ and ${\varphi }_m>1$. The restoring buoyancy suppresses vertical parcel excursions and reduces the effective momentum diffusivity, requiring a steeper shear for the same stress. The speed difference from the wall reference is correspondingly larger. These are comparisons at prescribed stress; fixing an outer wind instead would allow $u_{\ast }$ and the stress to change, so it would not be the same comparison.