ABSTRACT

We consider a number of applications of the correlation dimension concept in the atmospheric sciences. Our emphasis is on the correlation dimension as a nonlinear signal-processing tool for characterizing the complexity of real and simulated atmospheric data, rather than as a means for justifying low-dimensional approximations to the underlying dynamics. Following an introductory exposition of the basic mathematics, we apply the analysis to a three-equation nonlinear system having some interesting points of affinity with the general subject of nonlinear energy transfers in two-dimensional fluids. Then we turn to the analysis of a 40-year data set of observed Northern Hemisphere flow patterns. Our approach deviates from most previous studies in that we employ time series of flow fields as our basic unit of analysis, rather than single-point time series. The main evidence for low-dimensional behavior is found in the patterns of interseasonal variability. However, even when this is removed, the streamfunction field shows clear indication of dimensionality in the range 20–200, rather than in the thousands. Due to problems connected with the nonequivalence of various norms in functions spaces, we do not claim that this represents the “true” dimensionality of the underlying system. Nevertheless, it does show the existence of a considerable amount of order in the system, a fact begging for an explanation.

From this, we turn to the matter of spatial complexity, examining the geometry of clouds of passive tracer mixed by spatially structured (atmospherically motivated) two-dimensional flow fields. Evidence is presented that the cloud ultimately mixes over a region characterized by dimension two. We also demonstrate that the correlation dimension of the tracer cloud is directly related to the algebraic power spectrum of the concentration distribution in Fourier space. From this we are led to some speculations on the role of chaotic mixing in the enstrophy cascade of two-dimensional turbulence.

Finally, we consider the spatial patterns of rate of predictability loss in the tracer problem. This information is obtained by computing finite-time estimates of the Lyapunov exponents for trajectories starting from various initial conditions. The rate of predictability loss is itself unpredictable, in the sense that it exhibits sensitive dependence on initial conditions. It is suggested that multifractal analysis could be used to characterize the spatial pattern of predictability.