Weak‐ and strong‐friction limits of parcel models: Comparisons and stochastic convective initiation time

Abstract

For moist convection, models of individual parcel dynamics are valuable for their simple formulation and predictions of cloud properties. Here, two limiting idealized cases of parcel theory are investigated: the weak‐ and strong‐friction limits. The weak‐friction limit is a traditional limit with no momentum drag and the dynamics are a Hamiltonian system for the parcel's height and vertical velocity. A strong‐friction limit is derived and studied here and its limiting form involves a balance between frictional drag and buoyancy, which provides a differential equation for parcel height as a function of time. In the two limiting regimes, analytical formulas are presented and compared for quantities such as maximum vertical velocity and cloud‐top height. For example, in the strong‐friction limit, the cloud‐top height coincides with the level of neutral buoyancy (LNB), whereas in the weak‐friction limit the cloud‐top height is far above the LNB. This comparison suggests that the strong‐friction limit may provide more realistic predictions of some averaged cloud properties. In general, since frictional effects and individual parcel properties can vary even within a single cloud, the predictions of the weak‐ and strong‐friction limits can be viewed as upper and lower bounds for the behavior of more realistic finite‐friction scenarios. Finally, in a stochastic version of the parcel model, in the strong‐friction limit, analytical formulas are derived for convective initiation time. Applications to convective parametrizations are discussed; for example, the formulas for convective initiation time could be applied as stochastic convective triggers.

Energy Decompositions for Moist Boussinesq and Anelastic Equations with Phase Changes

“…Second, the variable M is associated with an additional eigenmode that arises for moist dynamics but is not present for dry dynamics. It essentially emerges from the linearization of (8) in the purely unsaturated or purely saturated case (see, e.g., Hernandez-Duenas et al 2015;Smith and Stechmann 2017, and references therein). As such, it is natural to suppose that a moist energy should have an additional term related to M 2 and associated with this additional moist eigenmode.…”

“…Another version of the potential energy, one that is not piecewise quadratic, has also been considered in the past (e.g., Pauluis 2008;Hernandez-Duenas et al 2013). Here we will write this traditional form in terms of the variables…”

“…An alternative definition of an anelastic M variable was proposed by Wetzel et al (2019). An advantageous property of M aneastic here is its physical interpretation: it is related to convective available potential energy (CAPE) (e.g., Moncrieff and Miller 1976;Emanuel 1994;Hernandez-Duenas et al 2019). In particular, notice that (45) involves the difference between an unsaturated CAPE and a saturated CAPE, since z r is similar to a lifted condensation level [see its definition in (38)] and a 1 and a 2 are levels of neutral buoyancy.…”

“…For references, see Pauluis (2008), Young (2010), Hernandez-Duenas et al (2013, 2015, and Tailleux (2013). Here, the buoyancy b is a function of the equivalent potential temperature u tot e , the total water mixing ratio q tot t , and height z, and again the integral is taken with u tot e and q tot t held fixed.…”

“…In section 3, we consider the anelastic equations, and while we cannot formulate a quadratic energy, we can still decompose a total energy into buoyant energy and latent energy released at the interface between different phases. In section 4, we consider a Kessler warm-rain parameterization of cloud microphysics, and the Fast Autoconversion and Rain Evaporation (FARE) model of Hernandez-Duenas et al (2013). In the former case, while we cannot obtain a conserved energy, we can still derive an energy principle that involves the source terms representing cloud microphysical processes.…”