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Estimating the Fractal Dimension and the Predictability of the Atmosphere

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  • 1 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
  • | 2 Department of Physics, Colorado State University, Fort Collins, Colorado
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Abstract

The fractal dimension, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily surface temperature and pressure over several regions of the United States and the North Atlantic Ocean with different climatic signal-to-noise ratios. Though the total number of data points (from about 13 800 to about 36 500) is larger than those used in previous studies, it is still too small to obtain a reliable estimate of the Grassberger–Procaccia correlation dimension because of the limitations discussed by Ruelle. However, it can be shown that this dimension is greater than 8. Also, it is pointed out that most, if not all, of the previous estimates of low fractal dimensions in the atmosphere are spurious. These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia are sensitive to the selection of the time delay and are thus unreliable. Geographic variability of the fractal dimension is suggested, but further verification is needed.

A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. Using this method, it is found that the error-doubling time is about 2–3 days in Fort Collins, Colorado, about 4–5 days in Los Angeles, California, and about 5–8 days in the North Atlantic Ocean. The predictability time is longer over regions with a higher climatic signal-to-noise ratio (e.g., Los Angeles), and the predictability time of summer and/or winter data is longer than for the entire year. The difference between these estimates of error-doubling time and estimates based on general circulation models (GCMs) is discussed. It is also mentioned that the computation of the Lyapunov exponents is slightly sensitive to the selection of the time delay, possibly because the fractal dimension is very high in the atmosphere. Such sensitivity has not been mentioned in previous similar studies.

Abstract

The fractal dimension, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily surface temperature and pressure over several regions of the United States and the North Atlantic Ocean with different climatic signal-to-noise ratios. Though the total number of data points (from about 13 800 to about 36 500) is larger than those used in previous studies, it is still too small to obtain a reliable estimate of the Grassberger–Procaccia correlation dimension because of the limitations discussed by Ruelle. However, it can be shown that this dimension is greater than 8. Also, it is pointed out that most, if not all, of the previous estimates of low fractal dimensions in the atmosphere are spurious. These results lead us to claim that there probably exist no low-dimensional strange attractors in the atmosphere. Because the fractal dimension has not yet been saturated, the Kolmogorov entropy and the error-doubling time obtained by the method of Grassberger and Procaccia are sensitive to the selection of the time delay and are thus unreliable. Geographic variability of the fractal dimension is suggested, but further verification is needed.

A practical and more reliable method for estimating the Kolmogorov entropy and error-doubling time involves the computation of the Lyapunov-exponent spectrum using the algorithm of Zeng et al. Using this method, it is found that the error-doubling time is about 2–3 days in Fort Collins, Colorado, about 4–5 days in Los Angeles, California, and about 5–8 days in the North Atlantic Ocean. The predictability time is longer over regions with a higher climatic signal-to-noise ratio (e.g., Los Angeles), and the predictability time of summer and/or winter data is longer than for the entire year. The difference between these estimates of error-doubling time and estimates based on general circulation models (GCMs) is discussed. It is also mentioned that the computation of the Lyapunov exponents is slightly sensitive to the selection of the time delay, possibly because the fractal dimension is very high in the atmosphere. Such sensitivity has not been mentioned in previous similar studies.

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