Preface Fractals in geophysics

Geophysical phenomena of interest to geoscientists include both atmospheric and terrestrial related processes, which can be either static or dynamic. Characterization of such phenomena requires advanced models which often include scale invariance concepts of fractal or multi-fractal type. Recently, there has been much interdisciplinary interest in the application of geophysical processes. The application of fractals and multi-fractal concepts has given rise to a better understanding of the spatio-temporal organization of geophysical phenomena from micro to macro levels [1–6]. Wider application of these concepts to analyze and characterize space–time data requires a broad cross-disciplinary exchange among mathematicians, physicists and geoscientists. This special issue on ‘‘Fractals in Geophysics’’ contributes to this exchange of ideas. We hope that it will stimulate fruitful discussions among experts in geophysics and in core-geosciences. The papers in this special issue have been classified based on the application of fractals and chaos to topics such as ocean, climatic and geophysical flow dynamics, porous media characterization and earth surface processes (e.g. channel networks, desertification, floods, earthquakes etc). These papers address various application areas of geophysical importance like climatic dynamics, earthquakes, porous media, river networks, vortex-flow, mineralogy, ocean dynamics, vegetation patterns in deserts, and soils. In particular, the following aspects are emphasized: (a) Theory and applications of scaling, fractals, and multi-fractals to quantify and characterize geophysical phenomena; (b) Techniques, algorithms, for estimation of (multi-)fractal exponents and dimensions, the characterization of attractors and time series, and the simulation of geophysical processes using multi-fractal models; and (c) Modeling of space–time geophysical phenomena. In the first article of this issue, construction of patterns over one, two and three-dimensional spaces is explained by Carlos Puente. He provides certain examples found in geophysical applications that include irregular and crystal-like patterns. Donald Turcotte describes the advantages of application of cellular automata theory [7] to deal with certain geological and geophysical problems, which cannot be solved by continuous mathematics effectively. This paper describes several successes of this approach in simulations of channel networks through diffusion limited aggregation (DLA) and in understanding avalanches and earthquakes through self-organized criticality. Understanding chaotic advection phenomenon is important in geophysical flows. In their paper, Xavier Leoncini, Leonid Kuznetsov, and George M. Zaslavsky provide qualitative insights on general transport properties of twodimensional flows, specifically geophysical flows. Fractal aspects of transport are developed to link their anomalous features to the fractal nature of the topology of the flows. Further, using a nice fractional kinetic analysis, the transport exponent is linked with the trapping time exponent. Peter Chu carries out a multi-fractal analysis of a high-resolution temperature data set to determine the nonstationarity and intermittency of the upper layer (300 m depth) in the Southwestern part of Greenland Sea, Iceland Sea, and Norwegian Sea (GIN Sea). Such a study is of great use in better understanding oceanic convection process or the secondary circulation across oceanfronts. Govindan Rangarajan and Dhananjay Sant apply fractal dimensional analysis to Indian climatic dynamics. In this paper, the time-series data of the major components of the climate––temperature, pressure, and precipitation are analyzed. Changes in climate variability changes from month to month and also from season to season are obtained. An interesting inference is made on how temperature and pressure in the February/March period influence the southwest monsoon rainfall. Understanding the flow through fractal and multi-fractal porous media is an interesting topic in geophysics. Characterization of such flows effectively is a challenging task. In the subsequent two papers, such flows have been effectively characterized by applying novel nonlinear methods, and continuum percolation theory. Veneziano and Essiam in their paper analyze the flow in a D-dimensional porous media under the condition that the hydraulic conductivity is an isotropic lognormal field with multi-fractal scale invariance. They have shown how