[1] The analysis of univariate or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical field of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory. In this review we describe the connections between time series analysis and nonlinear dynamics, discuss signal-to-noise enhancement, and present some of the novel methods for spectral analysis. The various steps, as well as the advantages and disadvantages of these methods, are illustrated by their application to an important climatic time series, the Southern Oscillation Index. This index captures major features of interannual climate variability and is used extensively in its prediction. Regional and global sea surface temperature data sets are used to illustrate multivariate spectral methods. Open questions and further prospects conclude the review.

In Ensemble Kalman Filter data assimilation, localization modifies the error covariance matrices to suppress the influence of distant observations, removing spurious long distance correlations. In addition to allowing efficient parallel implementation, this takes advantage of the atmosphere's lower dimensionality in local regions. There are two primary methods for localization. In B-localization, the background error covariance matrix elements are reduced by a Schur product so that correlations between grid points that are far apart are removed. In R-localization, the observation error covariance matrix is multiplied by a distance-dependent function, so that far away observations are considered to have infinite error. Successful numerical weather prediction depends upon well-balanced initial conditions to avoid spurious propagation of inertial-gravity waves.Previous studies note that B-localization can disrupt the relationship between the height gradient and the wind speed of the analysis increments, resulting in an analysis that can be significantly ageostrophic.This study begins with a comparison of the accuracy and geostrophic balance of EnKF analyses using no localization, B-localization, and R-localization with simple onedimensional balanced waves derived from the shallow water equations, indicating that the optimal length scale for R-localization is shorter than for B-localization, and that for the same length scale R-localization is more balanced. The comparison of localization techniques is then expanded to the SPEEDY global atmospheric model. Here, natural imbalance of the slow manifold must be contrasted with undesired imbalance introduced by data assimilation. Performance of the two techniques is comparable, also with a shorter optimal localization distance for R-localization than for B-localization.

Abstract. In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field Correspondence to: S. Wiggins (s.wiggins@bristol.ac.uk) given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.

A mixture model is a flexible probability density estimation technique, consisting of a linear combination of k component densities. Such a model is applied to estimate clustering in Northern Hemisphere (NH) 700-mb geopotential height anomalies. A key feature of this approach is its ability to estimate a posterior probability distribution for k, the number of clusters, given the data and the model. The number of clusters that is most likely to fit the data is thus determined objectively. A dataset of 44 winters of NH 700-mb fields is projected onto its two leading empirical orthogonal functions (EOFs) and analyzed using mixtures of Gaussian components. Cross-validated likelihood is used to determine the best value of k, the number of clusters. The posterior probability so determined peaks at k ϭ 3 and thus yields clear evidence for three clusters in the NH 700-mb data. The three-cluster result is found to be robust with respect to variations in data preprocessing and data analysis parameters. The spatial patterns of the three clusters' centroids bear a high degree of qualitative similarity to the three clusters obtained independently by Cheng and Wallace, using hierarchical clustering on 500-mb NH winter data: the Gulf of Alaska ridge, the high over southern Greenland, and the enhanced climatological ridge over the Rockies. Separating the 700-mb data into Pacific (PAC) and Atlantic (ATL) sector maps reveals that the optimal k value is 2 for both the PAC and ATL sectors. The respective clusters consist of Kimoto and Ghil's Pacific-North American (PNA) and reverse PNA regimes, as well as the zonal and blocked phases of the North Atlantic oscillation. The connections between our sectorial and hemispheric results are discussed from the perspective of large-scale atmospheric dynamics.

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity;(3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. 7 Overall dynamics for n > 1 and m > 2 with α > 0: |y 1 (t → t c )| < +∞ (except for isolated initial conditions) and |y 2 (t → t c )| = +∞ Recall from Section 5.1 for the sub-dynamical system with only the oscillatory element that the case n > 1 corresponds to highly nonlinear oscillations with a monotonically decreasing period as the amplitude of the oscillations increases ( Figure 2). From Section 5.2 for the sub-dynamical system with only the singular element, the case m > 2 corresponds to finite-time singularity with finite increment in y 1 and infinite increment in y 2 (Figure 4). Furthermore, Section 6.3 on the phase space of the full dynamical system showed the following results:1. any trajectory y(t; y 0 , t 0 ) starting away from the origin connects the origin in backward time and |y 2 | → ∞ in forward time;