VOLUME 86, NUMBER 26 PHYSICAL REVIEW LETTERS 25 JUNE 2001 Local Low Dimensionality of Atmospheric Dynamics D. J. Patil,* , † Brian R. Hunt,* Eugenia Kalnay, ‡ James A. Yorke,* and Edward Ott § University of Maryland, College Park, Maryland 20742 (Received 10 January 2001) A statistic, the BV (bred vector) dimension, is introduced to measure the effective local finite-time dimensionality of a spatiotemporally chaotic system. It is shown that the Earth’s atmosphere often has low BV dimension, and the implications for improving weather forecasting are discussed. DOI: 10.1103/PhysRevLett.86.5878 PACS numbers: 05.45.Jn, 89.75.–k, 92.60.Wc From the dynamical systems point of view, the behavior of the Earth’s atmosphere is extremely high dimensional (e.g., a realistic atmospheric model based on a modal ex- pansion would necessarily include many modes). In spite of the atmosphere’s high dimensionality, in this Letter we demonstrate that, in a suitable sense, the local finite-time atmospheric dynamics is often low dimensional. Further- more, as we discuss at the end of this Letter, we believe that this finding has important implications for weather fore- casting. More generally, this behavior may be common to other physical spatiotemporally chaotic systems, and these systems may also be amenable to the type of analysis that we introduce for the atmosphere. The study that we report in this Letter is based on data from numerical weather forecasts posted at regular inter- vals on the Internet by the United States National Weather Service (NWS). The data provide a unique opportunity to study a state-of-the-art model and a real complex physical system that are closely tied together by data assimilation. Here, the phrase data assimilation refers to the process by which the representation of the atmosphere in the com- puter model is periodically adjusted to attempt to make it consistent with current physical measurements of the at- mospheric state. On 7 December 1992 the NWS implemented operational ensemble forecasts [1]. At regular time intervals several perturbations to the model atmospheric state are created. The original atmospheric state (referred to below as the main solution) and the ensemble of perturbed states are evolved forward in time by the model to create an ensemble of forecasts. The difference between the main solution and a per- turbed solution is similar to a Lyapunov vector in the familiar calculation of the evolution of differential dis- placements from chaotic trajectories [2,3]. A difference here is that for the NWS computation, the perturbations, although small, are not infinitesimal. The individual per- turbations obtained from the ensemble forecasts are called the bred vectors (BV) [4]. If the perturbations were in- finitesimal (rather than finite), then, in the limit of infinite time evolution, the bred vectors would point in the direc- tion of dominant growth, and the corresponding exponen- tial growth rate of their magnitudes would be the largest Lyapunov exponent. For the analysis presented in this Letter, we used en- sembles consisting of five perturbed forecasts [3,5]. The ensemble forecasts are made available on the Internet every 24 h by the NWS and give the forecasts at 12-h intervals spanning 8 days [6]. In this study we focus on the wind vector field at the height where the pressure is 500 mbar (approximately 5 km in altitude). We now describe how these data can be analyzed to ob- tain a useful local measure of the dimensionality of the atmospheric dynamics. We consider square regions of roughly 1100 km 3 1100 km, choosing a grid point in the center of the region plus 24 uniformly distributed neigh- bors. The north-south and east-west wind components of a bred vector at the 500 mbar pressure level at each of the 25 points in such a region form a 50-dimensional column vector which we refer to as a local bred vector. We normal- ize each local bred vector in two stages: first we scale the north-south velocities so that their mean squared value is the same as for the east-west velocities. Then we normal- ize the full 50-dimensional vector to have magnitude one. If there are k local bred vectors (k  5 in our case), the issue we want to address is the degree of linear inde- pendence of these k local bred vectors. That is, we want to determine the effective dimensionality of the subspace spanned by the local bred vectors. To do this we use princi- pal component analysis (PCA) [7] (also known as empiri- cal orthogonal functions). The underlying idea is to find the lowest dimensional subspace that, in a least squares sense, optimally represents the majority of the data. The k local bred column vectors form a 50 3 k matrix, B. The k 3 k covariance matrix of B is C  B T B, where B T is the transpose of B. Since the covariance matrix is non-negative definite and symmetric, its k eigenvalues l i are non-negative l i $ 0, and its eigenvectors, after mul- tiplying by B and normalizing, form an orthonormal set of vectors y i which span the column space of B. We order the eigenvalues by l 1 $l 2 $ ··· $l k . The singular values of B are s i  p l i . The eigenvalues l i are a measure of the extent to which the k column unit vectors making up B point in the direction y i , and each s 2 i  l i is said to represent the amount of variance in the set of the k unit vectors that is accounted for by y i . By identifying how many of the vectors of y i represent most of the variance of B we can identify an effective dimension spanned by the 5878 0031-9007 01  86(26)  5878(4)$15.00 © 2001 The American Physical Society