 Logarithmic-height plume equations

Normalize volume, momentum and buoyancy by a positive plume similarity in power-law stratification and use $\xi =\ln (z/z_{\ast })$. The normalized equations are ${\hat{Q}}_{\xi }=q(\sqrt{\hat{M}}-\hat{Q})$, ${\hat{M}}_{\xi }=m(\hat{F}\hat{Q}/\hat{M}-\hat{M})$ and ${\hat{F}}_{\xi }=f(\hat{Q}-\hat{F})$. They form an autonomous differential equation.