What can critical processes tell us about climate extremes?

Tools from complexity theory can provide potentially powerful insights into aspects of the climate system, including extreme events that have increasingly large impacts. However, analogies to critical processes should not be applied blindly, and understanding arising from the known equations of fluid mechanics can yield modifications to approaches motivated by simple, well-controlled systems. This is illustrated here with examples drawn from understanding probability distributions of climate variations, especially hydrological cycle extremes. While parallels to self-organized criticality prove useful in anticipating related properties, stochastic models for variations across a transition can be placed in more quantitative correspondence with the underlying partial differential equations. Adaptations of first-passage processes, point-process neuronal spike train models and branching processes are reviewed for insights into, respectively: probability distributions of precipitation intensities and the related moisture distributions, heatwave and dry-spell durations, and spatial clusters of precipitation (including such phenomena as mesoscale convective systems). A scale-free range occurs in each case, but while this might be of interest to specialists in critical phenomena, in no case is it the most important part of the behavior for applications. Rather, the "cutoff regime", which terminates the approximately power law range, controls the most impactful events and the increases in probability of extreme events under climate change. This regime requires domain-specific understanding of the climate processes involved, which yields fruitful ground for interplay between fluid dynamical theory and large numerical simulations. Examples will be reviewed in which simpler systems aid in designing pragmatic diagnostics for full climate system models.