Klaus Fraedrich

Estimating the correlation dimension of an attractor from noisy and small datasets based on re-embedding

Abstract A modified version of the Grassberger-Procaccia algorithm is proposed to estimate the correlation dimension of an attractor. Firstly, a measured time series is embedded into an M dimensional phase space spanned by time delay coordinates. This, in turn, is linearly transformed into an equivalent space spanned by an orthogonal basis derived from singular value decomposition. Secondly, a subspace composed of the directions of the (first few) principal eigenvectors, is again embedded into a higher dimensional space, which is called re-embedding. Finally, the Grassberger-Procaccia algorithm is applied on a re-embedding space instead of the Takens embedding and thereby the correlation dimension ( D 2 ) is calculated. This leads to a modified version of the Grassberger-Procaccia algorithm, which is aimed at dealing with the estimation of the D 2 from noisy and relatively small data sets. In order to make full use of the available data, the delay time for the first embedding is always set to the sampling time. In order to reduce the noise level, only the principal components which are clearly above the “noise level” are used for the re-embedding. This modified algorithm is tested using low dimensional dynamical models with random noise of different levels. Here we have used the Lorenz model with D 2 about 2.0 and the Mackey-Glass equation with D 2 about 5.0. The results show that the present procedure gives a clearer scaling region in the D 2 ( M , r )-ln( r ) diagram and thus a better estimate of D 2 , especially when the data set is noisy and relatively small. This modified algorithm is applied to meteorological data and some of the problems associated with estimating the dimension of the weather and climate attractors are discussed based on the results.

Boundary-layer diffusion modelling: The Gaussian plume approach versus the spectral solution

The equation of turbulent diffusion is solved for a vertical area source within the planetary boundary layer. The traditional Gaussian-plume approach is compared with the spectral solution of the diffusion equation used together with the barotropic boundary-layer model of Lettau and Dabberdt (1970). The results of the numerical computations are presented and the differences between the solutions are discussed.

Intraseasonal Tropical—Extra-Tropical Interactions Observed in the Stratosphere

Using 34 years (July 1957–June 1991) of 50 hPa northern hemisphere geopotential height data, Φ 50 and observed tropical winds, u QBO , an attempt is made to isolate spatial patterns of interannual variability using Empirical Orthogonal Function (EOF) analysis. The dominant variability of Φ 50 in all months is zonally symmetric, representing changes in the latitudinal gradient of Φ 50 and hence in the 50 hPa northern hemisphere geostrophic wind, u G50 . EOF analysis of combined fields of u G50 and u QBO is performed, in an attempt to isolate relationships between the quasi-biennial oscillation (QBO) and the extratropics. A two-way relationship is found. There is a QBO signal in the extratropics, with enhanced mid-latitude westerlies in winters when the QBO has its west phase. A component of annual variability is evident in the tropics, with additional easterly forcing there during the west phase of the QBO (which we attribute to the enhanced equatorward propagation of planetary waves), which tends to cancel a restoration of westerly winds by mean meridional advection, leading to the observed `delay in onset’ of the easterly phase of the QBO.