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WIREs Clim Change
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Stochastic climate theory and modeling

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Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid‐scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large‐scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non‐Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models. WIREs Clim Change 2015, 6:63–78. doi: 10.1002/wcc.318

Regime transitions in a rotating two‐layer annulus laboratory experiment, viewed from above. Different colors correspond to different internal interface heights, through the use of a sophisticated visualization technique. In the upper row, small‐scale inertia–gravity waves are absent and large‐scale regime transitions do not occur. In the lower row, small‐scale inertia–gravity waves are present locally in the troughs of the large‐scale wave and a large‐scale regime transition does occur (from the laboratory experiments of Williams et al.).
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Mean systematic error of 500 hPa geopotential height fields (shading) for extended boreal winters (December–March) of the period 1962–2005. Errors are defined with regard to the observed mean field (contours), consisting of a combination of ERA‐40 (1962–2001), and operational ECMWF analyses (2002–2005). (a) Systematic error in a numerical simulation with the ECMWF model IFS, version CY32R1, run at a horizontal resolution of TL95 (about 210 km) and 91 vertical levels. (b) Systematic error in a simulation with a stochastic kinetic‐energy backscatter scheme. Significant differences at the 95% confidence level based on a Student's t‐test are hatched (after Berner et al.).
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