Threshold Models for Rainfall and Convection: Deterministic versus Stochastic Triggers

Abstract

This paper investigates stochastic models whose dynamics switch depending on the state/regime of the system. Such models have been called "hybrid switching diffusions" and exhibit "sliding dynamics" with noise. Here the aim is an application to models of rainfall, convection, and water vapor, where two states/regimes are considered: precipitation and non-precipitation. Regime changes are modeled with a "trigger function," and four trigger models are considered: deterministic triggers (i.e. Heaviside function) or stochastic triggers (finite-state Markov jump process), with either a single threshold for regime transitions or two distinct thresholds (allowing for hysteresis). These triggers are idealizations of those used in convective parameterizations of global climate models, and they are investigated here in a model for a single atmospheric column. Two types of results are presented here. First, exact statistics are presented for all four models, and a comparison indicates how the trigger choice influences rainfall statistics. For example, it is shown that the average rainfall is identical for all four triggers, whereas extreme rainfall events are more likely with the stochastic trigger. Second, the stochastic triggers are shown to converge to the deterministic triggers in the limit of fast transition rates. The convergence is shown using formal asymptotics on the Master-Fokker-Planck equations, where the limit is an interesting Fokker-Planck system with Dirac delta coupling terms. Furthermore, the convergence is proved in the mean-square sense for pathwise solutions.

Cloud regimes as phase transitions

“…Nevertheless, the large susceptibility near the critical point has important implications for climate applications in either case. It would be interesting to consider the effect of adding some nonlinear features [ Stechmann and Neelin , , ; Hottovy and Stechmann , , ] to equation and to study their effect on the phase transition.…”

Deep Convective Adjustment of Temperature and Moisture

“…The slope of the line is a and the x intercept is the critical buoyancy value, B c -which is close to zero. The curvature in the ''foot'' region near B c signals a departure from the ramp function; small stochastic fluctuations in the value of B c can produce this curvature (Stechmann and Neelin 2011;Hottovy and Stechmann 2015). In Fig.…”

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

“…; ; D'Andrea et al . ; Hottovy and Stechmann, ). Indeed, one of the main applications of parcel and plume models is in convective parametrizations.…”

“…In the strong-friction limit, in a stochastic version of the model, an approximation of the mean convective initiation time can be found in analytic form. Such an analytic formula could be applied in convective parametrizations as a simple stochastic trigger (Kain and Fritsch, 1992;Lin and Neelin, 2000;Majda and Khouider, 2002;Jakob and Siebesma, 2003;Stechmann and Neelin, 2011;Gentine et al, 2013a;2013b;D'Andrea et al, 2014;Hottovy and Stechmann, 2015). Indeed, one of the main applications of parcel and plume models is in convective parametrizations.…”