E-dimension

The complexity of a nonlinear dynamical system is typically characterized by estimating the dimensionality of its dynamics. In dynamical system theory, the most frequently used estimate is the Lyapunov or Kaplan-Yorke dimension. However, for a high-dimensional, spatiotemporal dynamical system, such as a realistic dynamical model of the atmosphere, computation of the Lyapunov dimension is not feasible. Thus, quantifying the dimensionality of atmospheric dynamics requires a different approach.

A dimension definition that we specifically designed for large degree-of-freedom, spatiotemporally chaotic dynamical system, such as a weather prediction model, was the Bred Vector Dimension (Patil et al. 2001). This measure estimates the dimensionality based on a principal component analysis of an ensemble of model forecasts over localized geographical regions. We note that this local measure is called BV-dimension only for historical reasons, as it was first applied to an ensemble of bred vectors. Since it can be used for any ensemble of model solutions, regardless of the ensemble generation technique, we prefered to call it the ensemble dimension (E-dimension) in our later publications (e.g. Oczkowski et al. 2005, Szunyogh et al. 2005, Kuhl et al. 2006). The most detailed description and illustration of the E-dimension is provided in Ozkowski et al. (2005)

A critical distinction between the E-dimension and traditional definitions of dimensions, that are independent of space and time, is that the E-dimension is a local quantity that varies in both space and time. We note that the potential applications of a statistics, formally identical to the E-dimension, were discussed by Bretherton et al. (1999, J. Climate, 12, 1990-2009.) They considered its use as a measure "of the effective number of spatial degrees of freedom, or number of independently varying spatial patterns, of a time-varying field of data". We use the formula as a spatio-temporally varying measure of the effective number of degrees of freedom in a set of ensemble perturbations. Our strategy is motivated by the belief that high-dimensionality is a fundamental property of the atmospheric flow; thus searching for a limited number of independently varying global spatial pattern is not practical when synoptic- and smaller-scale motions are considered.