First-Passage-Time Prototypes for Precipitation Statistics

Prototype models are presented for time series statistics of precipitation and column water vapor. In these models, precipitation events begin when the water vapor reaches a threshold value and end when it reaches a slightly lower threshold value, as motivated by recent observational and modeling studies. Using a stochastic forcing to parameterize moisture sources and sinks, this dynamics of reaching a threshold is a first-passage-time problem that can be solved analytically. Exact statistics are presented for precipitation event sizes and durations, for which the model predicts a probability density function (pdf) with a power law with exponent -3/2. The range of power-law scaling extends from a characteristic small-event size to a characteristic large-event size, both of which are given explicitly in terms of the precipitation rate and water vapor variability. Outside this range, exponential scaling of event-size probability is shown. Furthermore, other statistics can be computed analytically, including cloud fraction, the pdf of water vapor, and the conditional mean and variance of precipitation (conditioned on the water vapor value). These statistics are compared with observational data for the transition to strong convection; the stochastic prototype captures a set of properties originally analyzed by analogy to critical phenomena. In a second prototype model, precipitation is further partitioned into deep convective and stratiform episodes. Additional exact statistics are presented, including stratiform rain fraction and cloud fractions, that suggest that even very simple temporal transition rules (for stratiform rain continuing after convective rain) can capture aspects of the role of stratiform precipitation in observed precipitation statistics.