One of the most intriguing facets of the climate system is that it exhibits variability across all temporal and spatial scales; pronounced examples are temperature and precipitation. The structure of this variability, however, is not arbitrary. Over certain spatial and temporal ranges, it can be described by scaling relationships in the form of power laws in probability density distributions and autocorrelation functions. These scaling relationships can be quantified by scaling exponents which measure how the variability changes across scales and how the intensity changes with frequency of occurrence. Scaling determines the relative magnitudes and persistence of natural climate fluctuations. Here, we review various scaling mechanisms and their relevance for the climate system. We show observational evidence of scaling and discuss the application of scaling properties and methods in trend detection, climate sensitivity analyses, and climate prediction.Plain Language Summary Climate variables are related over long times and large distances. This shows up as correlations for averages on long intervals or between distant areas. An important finding is that the majority of correlations in climate can be described by a simple mathematical relationship. We present such correlations for temperature on long times. Similarly, the intensity of precipitation events depends on their frequency in a simple manner. A useful concept is scaling where a scale denotes the width of an average. Scaling says that averages on different scales are related by a simple function-mathematically, this is a power law with the scaling exponent as a characteristic number. Scaling has impacts on predictability, temperature trends, and the assessment of future climate changes caused by anthropogenic forcing.

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Precipitation is an important meteorological variable which is critical for weather risk assessment. For instance, intense but short precipitation events can lead to flash floods and landslides. Most statistical modelling studies assume that the occurrence of precipitation events is based on a Poisson process with exponentially distributed waiting times while precipitation intensities are typically described by a gamma distribution or a mixture of two exponential distributions. Here, we show by using hourly precipitation data over the United Sates that the waiting time between precipitation events is non‐exponentially distributed and best described by a fractional Poisson process. A systematic model selection procedure reveals that the hourly precipitation intensities are best represented by a two‐distribution model for about 90% of all stations. The two‐distribution model consists of (a) a generalized Pareto distribution (GPD) model for bulk precipitation event sizes and (b) a power‐law distribution for large and extreme events. Finally, we analyse regional climate model output to evaluate how the climate models represent the high‐frequency temporal structure of U.S. precipitation. Our results reveal that these regional climate models fail to accurately reproduce the power‐law behaviour of intensities and severely underestimate the long durations between events.