A Stochastic Model for the Transition to Strong Convection

A simple stochastic model is designed and analyzed in order to further understand the transition to strong convection. The transition has been characterized recently in observational data by an array of statistical measures, including (i) a sharp transition in mean precipitation, and a peak in precipitation variance, at a critical value of column water vapor (CWV), (ii) an approximate power law in the probability density of precipitation event size, (iii) exponential tails in the probability density of CWV values, when conditioned on either precipitating or nonprecipitating locations, and (iv) long and short autocorrelation times of CWV and precipitation, respectively, with approximately exponential and power-law decays in their autocorrelation functions, respectively. The stochastic model presented here captures these four statistical features in time series of CWV and precipitation at a single location. In addition, analytic solutions are given for the exponential tails, which directly relates the tails to model parameters. The model parameterization includes three stochastic components: a stochastic trigger turns the convection on and off (a two-state Markov jump process), and stochastic closures represent variability in precipitation and in external forcing (Gaussian white noise). This stochastic external forcing is seen to be crucial for obtaining extreme precipitation events with high CWV and long lifetimes, because it can occasionally compensate for the heavy precipitation and encourage more of it. This stochastic model can also be seen as a simplified stochastic convective parameterization, and it demonstrates simple ways to turn a deterministic parameterization-the trigger and/or closure—into a stochastic one.