Well-mixed ventilation temperature balance ($\mathcal{V}\dot{\theta }=\mathcal{H}-|q|\theta$)

For a distributed heat source, a well-mixed interior with temperature excess $\theta$ gains temperature-volume flux $\mathcal{H}$ and loses $|q|\theta$ by ventilation in either direction. With signed upward throughflow, $q|q|=K_A(G_T\theta \pm W)$, where $G_T=g{\alpha }_T{H}_b$ and $W$ is wind pressure divided by mass density. The plus sign is assisting wind. Steady temperatures satisfy ${\theta }^2(G_T\theta +W)={\mathcal{H}}^2/K_A$ for assistance, or ${\theta }^2|G_T\theta -W|={\mathcal{H}}^2/K_A$ for opposition. Direction restrictions must be applied to roots.

For $0<\epsilon <2/(3\sqrt{3})$, the well-mixed ventilation temperature balance has one stable upward-flow temperature and two downward-flow temperatures. The heat-removal curve on the latter interval is $\sqrt{K_A}\theta \sqrt{W-G_T\theta }$, maximal at $2W/(3G_T)$. Its lower-temperature root is stable and its higher-temperature root unstable. The two meet at a saddle-node bifurcation. The unstable root separates the basins of the stable wind-dominated and buoyancy-dominated states.

Increasing wind or reducing heating both reduce the control parameter of opposing-wind ventilation bistability. However, wind moves the reversal threshold ${\theta }_w=W/G_T$, whereas heating does not. A sufficiently abrupt wind increase can place a hot upward-flow state in the cooler state's basin. Reducing positive heating at fixed wind cannot make an upward-flow state cross ${\theta }_w$, because its temperature time derivative there is positive. Slow variation follows a persistent stable branch; a fold destroys the cooler branch when wind is weakened or heat input increased.