 Solution

Let $B_0=\overline{w^{\prime }{b}^{\prime }}$ be the vertical turbulent buoyancy flux, positive for heating from below. Its dimensions are length squared per time cubed. The stress velocity and height therefore admit a new dimensionless group $B_{0}z/u_{\ast }^3$. Define the signed Obukhov lengthStable cooling has $B_0<0,{\ell }_O>0$; unstable heating has $B_0>0,{\ell }_O<0$. Monin-Obukhov similarity theory expresses the mean shear asThe similarity hypothesis assumes approximately steady, horizontally homogeneous, constant-stress and constant-flux conditions, with the observation level outside the inner sublayer but below the outer-layer scales. Dimensional analysis fixes the arguments and prefactor, not the entire function ${\varphi }_m$. A measured or modeled turbulence closure supplies its form.