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.

In Part I of this study, the impact of Ural blocking (UB) on the warm Arctic-cold Eurasian (WACE) pattern associated with the winter (DJF) arctic sea ice loss during 1979-2013 is examined by dividing the arctic sea ice reduction region into two dominant subregions: the Barents and Kara Seas (BKS) and the North American high-latitude (NAH) region (Baffin and Hudson Bay, Davis Strait, and Labrador Sea). It is found that atmospheric response to arctic sea ice loss resembles a negative Arctic response oscillation with a dominant positive height anomaly over the Eurasian subarctic region. Regression analyses of the two subregions further show that the sea ice loss over the BKS corresponds to the UB pattern together with a positive North Atlantic Oscillation (NAO 1 ) and is followed by a WACE anomaly, while the sea ice reduction in the NAH region corresponds to a negative NAO (NAO 2 ) pattern with a cold anomaly over northern Eurasia.Further analyses reveal that the UB pattern is more persistent during the period 2000-13 (P2) than 1979-99 (P1) because of the reduced middle-to-high-latitude mean westerly winds over Eurasia associated with the intense BKS warming. During P2 the establishment of the UB becomes a slow process because of the role of the BKS warming, while its decay is slightly rapid. In the presence of the long-lived UB that often occurs with the NAO 1 , the BKS-warming-induced DJF-mean anticyclonic anomaly is intensified and widened and then expands southward during P2 to amplify the WACE pattern and induce the southward displacement of its cold anomaly and the further loss of the BKS sea ice. Thus, midlatitude Eurasian cold events should be more frequent as the sea ice loss continues over the BKS.

Given the recent observational evidence that the positive (negative) phase of the North Atlantic Oscillation (NAO) is the remnant of anticyclonic (cyclonic) wave breaking, this study uses a multilevel primitive equation model to investigate important dynamical attributes of the above wave breaking behavior. For this purpose, a hierarchy of different basic states (two-and three-dimensional) and initial perturbations are used. With the three-dimensional climatological flow as the basic state, it is found that initial perturbations located equatorward (poleward) and upstream of the climatological Atlantic jet lead to wave breaking similar to that of the positive (negative) NAO phase. Consistently, analysis of observational data indeed shows that the Pacific storm track is displaced equatorward (poleward) prior the onset of the positive (negative) NAO phase. This result suggests that the latitudinal position of the Pacific storm track plays an important role for determining the phase of the NAO. Sensitivity experiments show that individual life cycles resemble each other only within the NAO region, but have large case-to-case variability outside of the NAO region. Calculations with zonally symmetric basic states fail to produce wave breaking of the correct spatial and temporal scale, underscoring the dynamical significance of the three-dimensional climatological flow.

This study applies a systematic strategy for stochastic modeling of atmospheric low-frequency variability to a realistic barotropic model climate. This barotropic model climate has reasonable approximations of the Arctic Oscillation (AO) and Pacific/North America (PNA) teleconnections as its two leading principal patterns of low-frequency variability. The systematic strategy consists first of the identification of slowly evolving climate modes and faster evolving nonclimate modes by use of an empirical orthogonal function (EOF) decomposition. The low-order stochastic climate model predicts the evolution of these climate modes a priori without any regression fitting of the resolved modes. The systematic stochastic mode reduction strategy determines all correction terms and noises with minimal regression fitting of the variances and correlation times of the unresolved modes. These correction terms and noises account for the neglected interactions between the resolved climate modes and the unresolved nonclimate modes. Loworder stochastic models with only four resolved modes capture the statistics of the original barotropic model modes quite well. A budget analysis establishes that the low-order stochastic models are dominated by linear dynamics and additive noise. The linear correction terms and the additive noise stem from the linear coupling between resolved and unresolved modes, and not from nonlinear interactions between resolved and unresolved modes as assumed in previous studies.

This study investigates the significance of trends of four temperature time series-Central England Temperature (CET), Stockholm, Faraday-Vernadsky, and Alert. First the robustness and accuracy of various trend detection methods are examined: ordinary least squares, robust and generalized linear model regression, Ensemble Empirical Mode Decomposition (EEMD), and wavelets. It is found in tests with surrogate data that these trend detection methods are robust for nonlinear trends, superposed autocorrelated fluctuations, and non-Gaussian fluctuations. An analysis of the four temperature time series reveals evidence of long-range dependence (LRD) and nonlinear warming trends. The significance of these trends is tested against climate noise. Three different methods are used to generate climate noise: (i) a short-range-dependent autoregressive process of first order [AR(1)], (ii) an LRD model, and (iii) phase scrambling. It is found that the ability to distinguish the observed warming trend from stochastic trends depends on the model representing the background climate variability. Strong evidence is found of a significant warming trend at Faraday-Vernadsky that cannot be explained by any of the three null models. The authors find moderate evidence of warming trends for the Stockholm and CET time series that are significant against AR(1) and phase scrambling but not the LRD model. This suggests that the degree of significance of climate trends depends on the null model used to represent intrinsic climate variability. This study highlights that in statistical trend tests, more than just one simple null model of intrinsic climate variability should be used. This allows one to better gauge the degree of confidence to have in the significance of trends.

Decadal and longer timescale variability in the winter North Atlantic Oscillation (NAO) has considerable impact on regional climate, yet it remains unclear what fraction of this variability is potentially predictable. This study takes a new approach to this question by demonstrating clear physical differences between NAO variability on interannual-decadal (<30 year) and multidecadal (>30 year) timescales. It is shown that on the shorter timescale the NAO is dominated by variations in the latitude of the North Atlantic jet and storm track, whereas on the longer timescale it represents changes in their strength instead. NAO variability on the two timescales is associated with different dynamical behaviour in terms of eddy-mean flow interaction, Rossby wave breaking and blocking. The two timescales also exhibit different regional impacts on temperature and precipitation and different relationships to sea surface temperatures. These results are derived from linear regression analysis of the Twentieth Century and NCEP-NCAR reanalyses and of a high-resolution HiGEM General Circulation Model control simulation, with additional analysis of a long sea level pressure reconstruction. Evidence is presented for an influence of the ocean circulation on the longer timescale variability of the NAO, which is particularly clear in the model data. As well as providing new evidence of potential predictability, these findings are shown to have implications for the reconstruction and interpretation of long climate records

Systematic strategies from applied mathematics for stochastic modelling in climate are reviewed here. One of the topics discussed is the stochastic modelling of mid-latitude low-frequency variability through a few teleconnection patterns, including the central role and physical mechanisms responsible for multiplicative noise. A new lowdimensional stochastic model is developed here, which mimics key features of atmospheric general circulation models, to test the fidelity of stochastic mode reduction procedures. The second topic discussed here is the systematic design of stochastic lattice models to capture irregular and highly intermittent features that are not resolved by a deterministic parametrization. A recent applied mathematics design principle for stochastic column modelling with intermittency is illustrated in an idealized setting for deep tropical convection; the practical effect of this stochastic model in both slowing down convectively coupled waves and increasing their fluctuations is presented here.

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...