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 VT. The aim is to bridge two extreme cases by considering the full range of VT values: (i) VT 5 0, (ii) finite VT, and (iii) infinitely fast VT. 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 us, which denotes the saturated potential temperature, or the equivalent potential temperature ue. When VT is finite, separate sufficient conditions are identified for stability versus instability: due/dz . 0 versus dus/dz , 0, respectively. WhenVT5 0, the criterion dus/dz5 0 is the single boundary that separates the stable and unstable conditions; and when VT is infinitely fast, the criterion due/dz 5 0 is the single boundary. Asymptotics are used to analytically characterize the infinitely fast VT case, in addition to numerical results. Also, the small-VT limit is identified as a singular limit; that is, the cases of VT 5 0 and small VT are fundamentally different. An energy principle is also presented for each case of VT, and the form of the energy identifies the stability parameter: either dus/dz or due/dz. Results for finite VT 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.