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

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.

For a rough boundary, integrate the similarity shear from the roughness length. A usual moderate-stability closure iswith positive dimensionless coefficients. These are empirical closures, not quantities determined by similarity alone. An often-used choice is $c_s=5,c_u=16$. The unstable quarter-power relation is tested over approximately $-1\lesssim z/{\ell }_O\lesssim -0.01$ in Dyer's flux-gradient study; the stable coefficient convention is recorded in the surface-flux formulation. Near neutrality one takes their smooth limits.

For moderate stable conditions, integration givesFor the unstable closure set $x(\zeta )=(1-c_u\zeta {)}^{1/4}$ and defineThen $1-\zeta {\psi }_m^{\prime }(\zeta )={\varphi }_m(\zeta )$ andFor weak unstable departure, $|c_{u}z/{\ell }_O|\ll 1$, this reduces to the stable-form expression with $c_s$ replaced by $c_u/4$ and a negative ${\ell }_O$. The full quarter-power expression is useful for moderately unstable conditions with $|z/{\ell }_O|$ of order one; it must not be extrapolated automatically to the distinct asymptotic free-convection regime. The stable linear expression similarly describes a continuously turbulent surface layer, rather than guaranteeing validity after strong stratification destroys the constant-flux state. Both profiles require $z_0\ll z\ll {\delta }_{\tau }$ and approximately uniform fluxes.

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.

Write $S=U_z$ and $N^2=d\bar{b}/dz$. The gradient Richardson number and Flux Richardson number areThe first compares the restoring buoyancy gradient with mean shear; the second is the fraction of shear production consumed by stable buoyancy work. With this convention, both are positive for stable mixing and negative for unstable stratification. In local steady equilibrium, $ϵ=P+B_0=P(1-{\mathrm{Ri}}_f)$, so a positive dissipation requires ${\mathrm{Ri}}_f<1$. The inviscid Miles–Howard theorem concerning ${\mathrm{Ri}}_g\ge 1/4$ is a linear stability criterion, not a universal equality for turbulent fluxes.

Use the gradient-diffusion closures $-\overline{u^{\prime }{w}^{\prime }}=K_{m}S$ and $B_0=-K_b{N}^2$. The turbulent Prandtl number is ${\mathrm{Pr}}_t=K_m/K_b$, givingUnder the ideal Reynolds analogy $K_m=K_b$, the two Richardson numbers coincide. It is an additional closure, not an automatic consequence of the definitions. If the buoyancy-gradient similarity function is defined by $N^2=-B_0{\varphi }_h/(\kappa z{u}_{\ast })$, then