Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 73 / 4 / iv

For $q>0$, divide the two sink terms by $\mathcal{P}=C_{u}S{q}^2$. PutThe requested ratio isHere $b>0$ and $a\ge 0$. With stable stratification $a>0$, $R$ diverges at both $q\downarrow 0$ and $q\to \infty$, and has a single minimum at $q_m=\sqrt{a/b}$, with $R_{\min }=2\sqrt{ab}$. Increasing the stable density-gradient magnitude increases $a$, lifting the minimum and moving it rightward. When the mass density gradient vanishes, $a=0$ and the curve reduces to the straight line $R=bq$, with zero only as a limiting value at $q=0$ where the original ratio is undefined.

The plot uses $\tilde{q}=bq=q/(L_u{C}_{u}S)$ and $\mathcal{A}=ab=[L_{\rho }{C}_{\rho }^2/(L_u{C}_u^2)]\mathrm{Ri}$, so $R=\tilde{q}+\mathcal{A}/\tilde{q}$. The horizontal line $R=1$ selects equilibria: two positive intersections for $0<\mathcal{A}<1/4$, one double intersection at $\mathcal{A}=1/4$, and none above it. The zero-gradient line has one positive equilibrium; its formal zero root before division by $q$ is the non-turbulent state.