Adam H. Monahan

The Probability Distribution of Sea Surface Wind Speeds. Part I: Theory and SeaWinds Observations

The probability distribution of sea surface wind speeds, w, is considered. Daily SeaWinds scatterometer observations are used for the characterization of the moments of sea surface winds on a global scale. These observations confirm the results of earlier studies, which found that the two-parameter Weibull distribution provides a good (but not perfect) approximation to the probability density function of w. In particular, the observed and Weibull probability distributions share the feature that the skewness of w is a concave upward function of the ratio of the mean of w to its standard deviation. The skewness of w is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). An analytic expression for the probability density function of w, derived from a simple stochastic model of the atmospheric boundary layer, is shown to be in good qualitative agreement with the observed relationships between the moments of w. Empirical expressions for the probability distribution of w in terms of the mean and standard deviation of the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surface wind vector components. The significance of these distributions for improvements to calculations of averaged air–sea fluxes in diagnostic and modeling studies is discussed.

Empirical Orthogonal Functions: The Medium is the Message

Empirical orthogonal function (EOF) analysis is a powerful tool for data compression and dimensionality reduction used broadly in meteorology and oceanography. Often in the literature, EOF modes are interpreted individually, independent of other modes. In fact, it can be shown that no such attribution can generally be made. This review demonstrates that in general individual EOF modes (i) will not correspond to individual dynamical modes, (ii) will not correspond to individual kinematic degrees of freedom, (iii) will not be statistically independent of other EOF modes, and (iv) will be strongly influenced by the nonlocal requirement that modes maximize variance over the entire domain. The goal of this review is not to argue against the use of EOF analysis in meteorology and oceanography; rather, it is to demonstrate the care that must be taken in the interpretation of individual modes in order to distinguish the medium from the message.

Nonlinear Principal Component Analysis by Neural Networks: Theory and Application to the Lorenz System

A nonlinear generalization of principal component analysis (PCA), denoted nonlinear principal component analysis (NLPCA), is implemented in a variational framework using a five-layer autoassociative feed-forward neural network. The method is tested on a dataset sampled from the Lorenz attractor, and it is shown that the NLPCA approximations to the attractor in one and two dimensions, explaining 76% and 99.5% of the variance, respectively, are superior to the corresponding PCA approximations, which respectively explain 60% (mode 1) and 95% (modes 1 and 2) of the variance. It is found that as noise is added to the Lorenz attractor, the NLPCA approximations remain superior to the PCA approximations until the noise level is so great that the lowerdimensional nonlinear structure of the data is no longer manifest to the eye. Finally, directions for future work are presented, and a cinematographic technique to visualize the results of NLPCA is discussed.

A regime view of northern hemisphere atmospheric variability and change under global warming

The leading mode of wintertime variability in Northern Hemisphere sea level pressure (SLP) is the Arctic Oscillation (AO). It is usually obtained using linear principal component analysis, which produces the optimal, although somewhat restrictive, linear approximation to the SLP data. Here we use a recently introduced nonlinear principal component analysis to find the optimal nonlinear approximation to SLP data produced by a 1001 year integration of the CCCma coupled general circulation model (CGCM1). This approximations associated time series is strongly bimodal and partitions the data into two distinct regimes. The first and more persistent regime describes a standing oscillation whose signature in the mid-troposphere is alternating amplification and attenuation of the climatological ridge over Northern Europe, with associated decreasing and increasing daily variance over Northern Eurasia. The second and more episodic regime describes a split-flow south of Greenland with much enhanced daily variance in the Arctic. In a 500 year integration with atmospheric CO2 stabilized at concentrations projected for year 2100, the occupation statistics of these preferred modes of variability change, such that the episodic split-flow regime occurs less frequently while the standing oscillation regime occurs more frequently.

Transport in time-dependent dynamical systems: Finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.

Nonlinear Principal Component Analysis: Tropical Indo–Pacific Sea Surface Temperature and Sea Level Pressure

Abstract Nonlinear principal component analysis (NLPCA) is a generalization of traditional principal component analysis (PCA) that allows for the detection and characterization of low-dimensional nonlinear structure in multivariate datasets. The authors consider the application of NLPCA to two datasets: tropical Pacific sea surface temperature (SST) and tropical Indo–Pacific sea level pressure (SLP). It is found that for the SST data, the low-dimensional NLPCA approximations characterize the data better than do PCA approximations of the same dimensionality. In particular, the one-dimensional NLPCA approximation characterizes the asymmetry between spatial patterns characteristic of average El Nino and La Nina events, which the 1D PCA approximation cannot. The differences between NLPCA and PCA results are more modest for the SLP data, indicating that the lower-dimensional structures of this dataset are nearly linear.

The preferred structure of variability of the northern hemisphere atmospheric circulation

A nonlinear generalisation of Principal Component Analysis (PCA) is applied to the 500mb geopotential height field of the Northern Hemisphere extratropical atmosphere. It is found that the low-frequency variability of the mid-troposphere is characterised by three distinct quasi-stationary states. The states are described and compared to those obtained from applications of cluster analyses and linear PCA to the height field. Evidence is provided that modes obtained through PCA (notably the Arctic Oscillation (AO)) are not independent dynamical modes of variability of the Northern Hemisphere extratropics. Rather they arise as the optimal linear compromise between the preferred quasi-stationary states of the circulation.

The Probability Distribution of Sea Surface Wind Speeds. Part II: Dataset Intercomparison and Seasonal Variability

The statistical structure of sea surface wind speeds is considered, both in terms of the leading-order moments (mean, standard deviation, and skewness) and in terms of the parameters of a best-fit Weibull distribution. An intercomparison is made of the statistical structure of sea surface wind speed data from four different datasets: SeaWinds scatterometer observations, a blend of Special Sensor Microwave Imager (SSM/I) satellite observations with ECMWF analyses, and two reanalysis products [NCEP–NCAR and 40-yr ECMWF Re-Analysis (ERA-40)]. It is found that while the details of the statistical structure of sea surface wind speeds differs between the datasets, the leading-order features of the distributions are consistent. In particular, it is found in all datasets that the skewness of the wind speed is a concave upward function of the ratio of the mean wind speed to its standard deviation, such that the skewness is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). This relationship between moments is also found in buoy observations of sea surface winds. In addition, the seasonal evolution of the probability distribution of sea surface wind speeds is characterized. It is found that the statistical structure on seasonal time scales shares the relationships between moments characteristic of the year-round data. Furthermore, the seasonal data are shown to depart from Weibull behavior in the same fashion as the year-round data, indicating that non-Weibull structure in the year-round data does not arise due to seasonal nonstationarity in the parameters of a strictly Weibull time series.

Skill Comparisons between Neural Networks and Canonical Correlation Analysis in Predicting the Equatorial Pacific Sea Surface Temperatures.

Abstract Among the statistical methods used for seasonal climate prediction, canonical correlation analysis (CCA), a more sophisticated version of the linear regression (LR) method, is well established. Recently, neural networks (NN) have been applied to seasonal climate prediction. Unlike CCA and LR, NN is a nonlinear method, which leads to the question whether the nonlinearity of NN brings any extra prediction skill. In this study, an objective comparison between the three methods (CCA, LR, and NN) in predicting the equatorial Pacific sea surface temperatures (in regions Nino1+2, Nino3, Nino3.4, and Nino4) was made. The skill of NN was found to be comparable to that of LR and CCA. A cross-validated t test showed that the difference between NN and LR and the difference between NN and CCA were not significant at the 5% level. The lack of significant skill difference between the nonlinear NN method and the linear methods suggests that at the seasonal timescale the equatorial Pacific dynamics is basically l...

The Vertical Structure of Wintertime Climate Regimes of the Northern Hemisphere Extratropical Atmosphere

A nonlinear generalization of principal component analysis (PCA), denoted nonlinear principal component analysis (NLPCA), is applied to Northern Hemisphere wintertime geopotential heights at 1000, 700, 500, 300, and 20 hPa. It is found that the optimal nonlinear approximation to the data at the four tropospheric levels is characterized by three equivalent-barotropic regimes of circulation. The NLPCA time series provides a kinematic description of variability within the regimes and transitions between them. The occupation frequencies of the regimes demonstrate substantial interannual and interdecadal variability, some of which can be associated with