Threshold Models for Rainfall and Convection: Deterministic versus Stochastic Triggers

We study two types of convective triggers in models of single column water vapor for idealized single column dynamics. The models are idealizations of a general circulation model's convective parametrization trigger. In one model, we use a stochastic trigger that initiates strong convection at a random time after the water vapor has reached a critical value. In the other, the trigger is deterministic and occurs immediately when a fixed threshold is reached. With the stochastic trigger model it is difficult to analyze and compute statistics, while the deterministic trigger model has statistics which are easier to solve exactly. We show that the stochastic trigger model converges to the deterministic trigger model as the jump rate tends to infinity, and thus the deterministic model can be used as an approximation. We compare, numerically and analytically, the probability densities of the stationary state and the event time densities for both models.