Stochastic and deterministic multicloud parameterizations for tropical convection

Abstract

Despite recent advances in supercomputing, current general circulation models poorly represent the variability associated with organized tropical convection. In a recent study, the authors have shown, in the context of a paradigm two baroclinic mode system, that a stochastic multicloud convective parameterization based on three cloud types (congestus, deep and stratiform) can be used to improve the variability and the dynamical structure of tropical convection. Here, the stochastic multicloud model is modified with a lag type stratiform closure and augmented with an explicit mechanism for congestus detrainment moistening. These modifications improve the representation of intermittent coherent structures such as synoptic and mesoscale convective systems. Moreover, the new stratiform-lag closure allows for increased robustness of the coherent features of the model with respect to the amount of stochastic noise and leading to a multi-scale organization of slowly moving waves envelopes in which short-lived and chaotic convective events persist. Congestus cloud decks dominate the suppressed-dry phase of the wave envelopes. The simulations with the new closure have a higher amount of stochastic noise and result in a Walker type circulation with realistic mean and coherent variability which surpasses results of previous deterministic and stochastic multicloud models in the same parameter regime. Further, deterministic mean field limit equations (DMFLE) for the stochastic multicloud model are considered. Aside from providing a link to the deterministic multicloud parameterization, the DMFLE allow a judicious way of determining the amount of deterministic and stochastic “chaos” in the system. It is shown that with the old stratiform heating closure, the stochastic process accounts for most of the chaotic behavior. The simulations with the new stratiform heating closure exhibit a mixture of stochastic and deterministic chaos. The highly chaotic dynamics in the simulations with congestus detrainment mechanism is due to the strongly nonlinear and numerically stiff deterministic dynamics. In the latter two cases, the DMFLE can be viewed as a “standalone” parameterization, which is capable of capturing some dynamical features of the stochastic parameterization. Furthermore, it is shown that, in spatially extended simulations, the stochastic multicloud model can capture qualitatively two local statistical features of the observations: long and short auto-correlation times of moisture and precipitation, respectively and the approximate power-law in the probability density of precipitation event size for large precipitation events. The latter feature is not reproduced in the column simulations. This fact underscores the importance of gravity waves and large scale moisture convergence.

Appendix: Derivation of approximations for congestus and deep heating

Here derive approximations for congestus and deep heating. We begin by considering a regime where the dependency of the cloud fractions on CAPE (and low level CAPE) is suppressed.

In this scenario, the cloud fraction transition rates are determined by fluctuations in dryness alone. This is accomplished by replacing CAPE and low level CAPE dependency in the transition rates by the RCE value of CAPE.

$$ \hat{R}_{01}=\frac{1}{\tau_{01}}\Upgamma(\bar{C})\Upgamma(D) $$

(36)

$$ \hat{R}_{12}=\frac{1}{\tau_{12}}\Upgamma(\bar{C})(1-\Upgamma(D)) $$

(37)

$$ \hat{R}_{02}=\frac{1}{\tau_{02}}\Upgamma(\bar{C})(1-\Upgamma(D)) $$

(38)

$$ \hat{R}_{20}=\frac{1}{\tau_{20}}(1-\Upgamma(\bar{C})) $$

(39)

The hats are used to distinguish the modified rates used here for the purpose of comparison from the actual transition rates listed in Table 2. Here, $\bar{C}$ is the RCE value of scaled CAPE, and scaled low level CAPE, in (19) and (18), respectively. The two quantities are equal at RCE. As before, D is the scaled dryness of the troposphere (20).

First, we consider the congestus heating equation for, the stochastic multicloud model.

$$ H_c=\sigma_c \frac{\alpha_c \bar{\alpha} }{H_m} \sqrt{CAPE_l^+}. $$

(40)

At this point, we can take a derivative of the above expression to obtain

$$ \dot{H_c}=\dot{\sigma}_c \frac{\alpha_c \bar{\alpha} }{H_m} \sqrt{CAPE_l^+}+ \frac{\bar{\alpha} \alpha_c}{2 H_m}{\sigma_c}({CAPE_l}^+)^{-1/2} \dot {CAPE_l}. $$

(41)

We assume that variations in CAPE _l are small relative to the amount of CAPE _l , as it would be the case near equilibrium. This allows us to concentrate on the first term of the expression above. Using (30), we write

$$ \dot{H_c}\approx \frac{\alpha_c \bar{\alpha} }{H_m} (\sigma_{cs} \hat{R}_{01} -\sigma_c(\hat{R}_{10}+\hat{R}_{12})) \sqrt{CAPE_l^+}. $$

(42)

With the simplified transition rates (36) and (37) and using the definitions of the rate R ₁₀ from Table 2, we can rewrite Eq. 42 as

$$ \dot{H_c}\approx \frac{\alpha_c \bar{\alpha} }{H_m} \bigg[\bigg( \sigma_{cs} \frac{\Upgamma(\bar{C})}{\tau_{01}}+\sigma_c\frac{\tau_{01}\Upgamma(\bar{C})-\tau_{12}}{\tau_{10}\tau_{12}}\bigg)\Upgamma(D) -\sigma_c\frac{\Upgamma(\bar{C})}{\tau_{12}}\bigg]\sqrt{CAPE_l^+} $$

(43)

and from (40), we get

$$ \dot{H_c}\approx\frac{\Upgamma (\bar{C})} {\tau_{12}}\bigg(\frac{\alpha_c \bar{\alpha} \tau_{12} }{H_m \Upgamma (\bar{C})} \big( \sigma_{cs} \frac{\Upgamma(\bar{C})}{\tau_{01}}+\sigma_c\frac{\tau_{01}\Upgamma(\bar{C})-\tau_{12}}{\tau_{10}\tau_{12}}\big)\Upgamma(D) \sqrt{CAPE_l^+}- H_c\bigg). $$

(44)

Furthermore, since σ _c  ≪ σ _cs  ≈ 1, we arrive at

$$ \dot{H_c}\approx\frac{\Upgamma (\bar{C})}{\tau_{12}} \bigg( \frac{\tau_{12}\alpha_c \bar{\alpha}}{\tau_{01} H_m} \Upgamma(D) \sqrt{CAPE_l^+}- H_c\bigg). $$

(45)

In fact, we can rewrite the equation for the evolution of congestus heating in a simpler and more familiar form,

$$ \dot{H}_c\approx\frac{1}{\tau^{MFL}_c} (\alpha^{MFL}_c \Upgamma(D) \sqrt{CAPE_l^+} - H_c) $$

(46)

where

$$ \tau^{MFL}_c=\frac{\tau_{12}}{\Upgamma (\bar{C})} {\alpha^{MFL}_c}= \alpha_c \frac{\tau_{12} \bar{\alpha}}{\tau_{01} H_m}. $$

(47)

In order to derive equation for approximate deep convective heating. We consider Eq. 31 and make an assumption that the adjustment of deep convective cloud fraction is instantaneous. This yields approximation equation for the deep convective cloud fraction

$$ {\sigma_d}\approx\frac{\sigma_{cs} \hat{R}_{02}+\sigma_{c} \hat{R}_{12}}{(\hat{R}_{20}+\hat{R}_{23})}. $$

(48)

Using the simplified transition rates (37–39) and constant transition rate R ₂₃ from Table 2, this approximation takes form:

$$ {\sigma_d}\approx \bigg[\frac{(\frac{\sigma_{cs}} {\tau_{02}}+\frac{\sigma_{c}} {\tau_{12}} )\Upgamma(\bar{C})}{(\frac{1}{\tau_{23}}+\frac{1-\Upgamma(\bar{C})}{\tau_{20}})}\bigg] \big(1-\Upgamma(D) \big) $$

(49)

Again, since σ _c  ≪ σ _cs  ≈ 1, we write the last expression as

$$ {\sigma_d}\approx \bigg[ \frac{{\tau_{23} \tau_{20} \Upgamma(\bar{C}) } } {\tau_{02}( {\tau_{20}}+\tau_{23}(1-\Upgamma(\bar{C}))} \bigg] \bigg(1-\Upgamma(D) \bigg). $$

(50)

Under the above simplifying assumptions, using (23), deep convective heating is given approximately by

$$ H_d\approx\bigg[ \frac{{\tau_{23} \tau_{20} \Upgamma(\bar{C}) } } {\tau_{02}( {\tau_{20}}+\tau_{23}(1-\Upgamma(\bar{C}))} \bigg]\bigg(1-\Upgamma(D) \bigg) \left[\bar{Q}+ \frac{1}{\tau_{conv}^0 \bar{\sigma_d}} (a_1 \theta_{eb}+a_2 q - a_0 (\theta_1 +\gamma_2 \theta_2))\right]^+. $$

(51)