Stability and Instability criteria for Idealized Precipitating Hydrodynamics

A linear stability analysis is presented for fluid dynamics with water vapor and precipitation, where the precipitation falls relative to the fluid at speed $V_T$. The aim is to bridge two extreme cases by considering the full range of $V_T$ values: (i) $V_T=0$, (ii) finite $V_T$, and (iii) infinitely fast $V_T$. In each case, a saturated precipitating atmosphere is considered, and the sufficient conditions for stability and instability are identified. Furthermore, each condition is linked to a thermodynamic variable: either a variable $\theta_s$ that we call the saturated potential temperature, or the equivalent potential temperature $\theta_e$. When $V_T$ is finite, separate sufficient conditions are identified for stability versus instability: $d\theta_e/dz>0$ versus $d\theta_s/dz<0$, respectively. When $V_T=0$, the criterion $d\theta_s/dz=0$ is the single boundary that separates the stable and unstable conditions; and when $V_T$ is infinitely fast, the criterion $d\theta_e/dz=0$ is the single boundary. Asymptotics are used to analytically characterize the infinitely fast $V_T$ case, in addition to numerical results. Also, the small $V_T$ limit is identified as a singular limit; i.e., the cases of $V_T=0$ and small $V_T$ are fundamentally different. An energy principle is also presented for each case of $V_T$, and the form of the energy identifies the stability parameter, either $d\theta_s/dz$ or $d\theta_e/dz$. Results for finite $V_T$ have some resemblance to the notion of conditional instability: separate sufficient conditions exist for stability versus instability, with an intermediate range of environmental states where stability or instability is not definitive.