= Solution
Take logarithmic height $\xi=\ln(z/z_*)$ with an arbitrary fixed length $z_*$, so $d/d\xi=z\,d/dz$. This is dimensionally meaningful and has the same derivatives as the abbreviated $\ln z$. For example,
$$
\frac{d\widehat Q}{d\xi}=\frac{zQ_z}{Q_s}-q\widehat Q=\frac{2\alpha\sqrt{M_o}}{Q_o}z^{1+m/2-q}\widehat M^{1/2}-q\widehat Q.
$$
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 therefore
$$
\boxed{\begin{aligned}
\widehat Q_\xi&=-q(\widehat Q-\widehat M^{1/2}),\\
\widehat M_\xi&=-m(\widehat M-\widehat F\widehat Q/\widehat M),\\
\widehat F_\xi&=-f(\widehat F-\widehat Q).
\end{aligned}}
$$
The 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.
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