Spatial Flows of Information Entropy as Indicators of Climate Variability and Extremes

The objective of this study is to analyze spatial entropy flows that reveal the directional dynamics of climate change—patterns that remain obscured in traditional statistical analyses. This approach enables the identification of pathways for "climate information transport," highlights associations with atmospheric circulation types, and allows for the localization of both sources and "informational voids"—regions where entropy is dissipated. The analytical framework is grounded in a quantitative assessment of long-term climate variability across Europe over the period 1901–2010, utilizing Shannon entropy as a measure of atmospheric system uncertainty and complexity. The underlying assumption is that the variability of temperature and precipitation reflects the inherently dynamic character of climate as a nonlinear system prone to fluctuations. The analysis focuses on monthly distributions of temperature and precipitation, modeled using bivariate copula functions, with marginal distributions selected based on the Akaike Information Criterion. To enhance estimation accuracy, the study employs bootstrap techniques and numerical integration to compute Shannon entropy values at each of the 4,165 grid points, with a spatial resolution of 0.25° × 0.25°. The results indicate that entropy and its derivative are complementary indicators of atmospheric system instability—entropy proving effective in long-term diagnostics, while its derivative provides insight into short-term forecasting of abrupt changes. A lag analysis and Spearman rank correlation between entropy values and their potential support the investigation of how circulation variability influences the occurrence of extreme precipitation events. Particularly noteworthy is the temporal derivative of entropy, which reveals strong nonlinear relationships between local dynamic conditions and climatic extremes. A spatial analysis of the informational entropy field further uncovers well-defined structures of varying climatic complexity at the continental scale. This field appears to be clearly structured, reflecting not only the directional patterns of change but also the potential sources of meteorological fluctuations. A field-theory-based spatial classification allows for the identification of transitional regions—areas with heightened susceptibility to shifts in local dynamics—as well as entropy source and sink regions. The study is embedded within the Fokker–Planck formalism, wherein the change in the stochastic distribution characterizes the rate of entropy production. In this context, regions of positive divergence are interpreted as active generators of variability, while sink regions function as stabilizing zones that dampen fluctuations.