The last decade has seen the success of stochastic parameterizations in short-term, medium-range, and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to represent model inadequacy better and to improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing long-standing climate biases and is relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups that show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface, and cryosphere of comprehensive weather and climate models 1) gives rise to more reliable probabilistic forecasts of weather and climate and 2) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics, and turbulence is reviewed; its relevance for the climate problem is demonstrated; and future research directions are outlined.

A new approach is proposed for stochastic parameterization of subgrid-scale processes in models of atmospheric or oceanic circulation. The new approach relies on two key ingredients: first, the unresolved processes are represented by a Markov chain whose properties depend on the state of the resolved model variables; second, the properties of this conditional Markov chain are inferred from data. The parameterization approach is tested by implementing it in the framework of the Lorenz '96 model. Performance of the parameterization scheme is assessed by inspecting probability distributions, correlation functions, and wave properties, and by carrying out ensemble forecasts. For the Lorenz '96 model, the parameterization algorithm is shown to give good results with a Markov chain with a few states only and to outperform several other parameterization schemes.

The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive highdimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Here techniques from applied mathematics are utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. The use of a few Empirical Orthogonal Functions (EOFs) (also known as Principal Component Analysis, Karhunen-Loéve and Proper Orthogonal Decomposition) depending on observational data to span the low-frequency subspace requires the assessment of dyad interactions besides the more familiar triads in the interaction between the low-and high-frequency subspaces of the dynamics. It is shown below that the dyad and multiplicative triad interactions combine with the climatological linear operator interactions to simultaneously produce both strong nonlinear dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. For a single low-frequency variable the dyad interactions and climatological linear operator alone produce a normal form with CAM noise from advection of the large scales by the small scales and simultaneously strong cubic damping. These normal forms should prove useful for developing systematic strategies for the estimation of stochastic models from climate data. As an illustrative example the one-dimensional normal form is applied below to lowfrequency patterns such as the North Atlantic Oscillation (NAO) in a climate model. The results here also illustrate the short comings of a recent linear scalar CAM noise model proposed elsewhere for low-frequency variability.low-frequency teleconnection patterns | nonlinearity | correlated noise T here is recent interest in deriving reduced stochastic models for climate and extended-range weather prediction. An attractive property of atmospheric low-frequency variability is that it can be efficiently described by just a few large-scale teleconnection patterns. These patterns exert a huge impact on surface climate and seasonal predictability. Thus, such reduced stochastic models are an attractive alternative for extended range prediction and climate sensitivity studies because they are computationally much more efficient than state-of-the-art climate models and have been shown to have comparable prediction skill (1). Because such reduced models capture the essence of low-frequency processes they allow for a better understanding of the climate system. Thus, systematic strategies are essential in successfully developing reduced models. The recently developed stochastic mode reduction strategy provides a systematic procedure for the derivation of reduced stochastic models (2-6). In these studies the functional form of reduced models is systematically derived for geophysical flow models. In refs. 7 and 8 the procedure is refined to allow a seamless way of deriving stochastic low-order models from data of complex geophysical flow...

Several different ways of constructing optimal bases for efficient dynamical modeling are compared: empirical orthogonal functions (EOFs), optimal persistence patterns (OPPs), and principal interaction patterns (PIPs). Past studies on fluid-dynamical topics have pointed out that EOF-based models can have difficulties reproducing behavior dominated by irregular transitions between different dynamical states. This issue is addressed in a geophysical context, by assessing the ability of these strategies for efficient dynamical modeling to reproduce the chaotic regime transitions in a simple atmosphere model. The atmosphere model is the well-known Charney-DeVore model, a six-dimensional truncation of the equations describing barotropic flow over topography in a ␤-plane channel geometry. This model is able to generate regime transitions for well-chosen parameter settings. The models based on PIPs are found to be superior to the EOF-and OPP-based models, in spite of some undesirable sensitivities inherent to the PIP method.