Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 330 / 2 / c / Solution

Take logarithmic height $\xi =\ln (z/z_{\ast })$ with an arbitrary fixed length $z_{\ast }$, so $d/d\xi =zd/dz$. This is dimensionally meaningful and has the same derivatives as the abbreviated $\ln z$. For example,The exponent vanishes and the coefficient is $q$, by the plume similarity in power-law stratification. Likewise the momentum coefficient is $m$ and the buoyancy coefficient is $f$. The logarithmic-height plume equations are thereforeThe reference similarity solution becomes the equilibrium $(1,1,1)$ of an autonomous differential equation. Normalizing each flux by its own height-dependent similarity value is essential; simply differentiating the unnormalized fluxes would miss the three subtraction terms.