Journal of Chemistry

Volume 2015 (2015), Article ID 138202, 5 pages

http://dx.doi.org/10.1155/2015/138202

## Intermittent Plurisink Model and the Emergence of Complex Heterogeneity Patterns: A Simple Paradigm for Explaining Complexity in Soil Chemical Distributions

^{1}Department of Applied Mathematics, ETSI Agrónomos, Universidad Politécnica de Madrid, Avenida Complutense s/n, 28040 Madrid, Spain^{2}Department of Applied Mathematics, ETSI Informáticos, Universidad Politécnica de Madrid, Campus de Montegancedo s/n, Boadilla del Monte, 28660 Madrid, Spain

Received 31 March 2015; Accepted 16 April 2015

Academic Editor: Jianchao Cai

Copyright © 2015 Miguel Ángel Martín et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The spatial complexity of the distribution of organic matter, chemicals, nutrients, and pollutants has been demonstrated to have multifractal nature. This fact supports the possibility of existence of some emergent heterogeneity structure built under the evolution of the system. The aim of this paper is providing a consistent explanation of the mentioned results via an extremely simple model.

#### 1. Introduction: Searching Explanations for Soil Heterogeneity

Heterogeneity and complexity are ubiquitous at all scales in soil and hydrologic systems. Nowadays, new technologies are of an invaluable help for providing a great number of highly calibrated field measurements. One can get a huge amount of data from computer tomography of soil samples at microscopic scales, digital terrain catchments of landscapes, and river basins among many other technological tools. Then mathematical tools are needed to analyse and interpret those data as well to construct models to predict. However, along the way needed to get such a final purpose, scientists also need to understand why the heterogeneity is produced and what the organizing principles that might underlie the heterogeneity and complexity are (McDonnell et al. [1]). Also it is encouraged to explore the scaling behaviour of heterogeneity and the emergent properties in soil and hydrologic systems. In this paper we are mainly interested in some aspects concerning the heterogeneity in the soil scenario.

Certainly we believe that the above understanding, besides providing coherence to science, also may be useful to get the practical purpose itself. In the case of soil, an illustrating example supporting this and the issues mentioned above is the study of soil texture heterogeneity. On one hand, Multifractal Analysis of fine granulometry soil data obtained by laser diffraction techniques provides information about the scaling behaviour of particle size distribution (PSD) heterogeneity (Montero [2]). In a second step, models able to replicate the heterogeneity formerly shown may be useful for prediction purposes (Martín and García-Gutiérrez [3]). The answer to why such heterogeneity exists, however, is not an easy issue since different sources of heterogeneity should be expected. In Frisch and Sornette [4] and Sornette [5], it is suggested that the fractal behaviour might be the result of a natural mixing of simple multiplicative process that takes place along the fragmentation of different particles, also pointing out that there is no accepted theoretical explanation. Recently fragmentation algorithms were proposed to replicate the multifractal nature of soil PSD (Martín et al. [6]). In this respect it is needed to say that any partial, but coherent, explanation should help to understand the (possibly several ones) organizing principles involved.

On the other hand, the spatial complexity of the distribution of organic matter, chemicals, nutrients, and pollutants has been studied by different authors (Kravchenko et al. [7], Lehmann et al. [8]). Multifractal Analysis has been successfully used to study the spatial variability of chemicals and organic matter contents, which is characterized by the generalized fractal dimensions (Kravchenko et al. [7]). Searching why such structured heterogeneity exists, a reducionist approach based in the description of transport equations in soil, seems an unlikely choice to describe the emergence of such complex pattern across the spatial scales. On the contrary, an explanation based on the fact that many complex systems in nature evolve in an intermittent burst-like way rather than in a smooth gradual manner (Rodríguez-Iturbe and Rinaldo [9]) would be more adequate. Further, such a kind of structured heterogeneity is commonly interpreted as the result of chaos or self-organization which leads to the emergent structure built under the evolution of the system (Sornette [5]). The aim of this paper is to provide a small contribution via an extremely simple model, which gives a consistent explanation to the mentioned results on spatial variability of chemicals or pollutants in soil.

The paper is organized as follows. In Section 2.1 the model is presented and in Section 2.2 the entropy scaling analysis method is described. Section 3 is devoted to analysing the results obtained in different simulations and their discussion.

#### 2. Material and Methods

##### 2.1. The Model

Let us suppose that is a soil area square-shaped. Suppose further that at any of the four corners there is a sink () randomly acting in an intermittent manner. Suppose each sink acting with relative frecuency be the relative frequency of the appearance of such action. A pollutant deposit (“pollutant seed”) is supposedly located in an arbitrary point of the square. When a given sink acts, its suction action is able to attract the pollutant matter to another point reducing the distance to the sink in a factor , where the pollutant rests until a new (or the same) sink acts. This factor reflects the mean value of the suction power of the respective sinks. However, the “flying” pollutant matter leaves a unit of pollutant at any point where the pollutant “rests” along its travelling.

Although a much more sophisticated model might be constructed for a more realistic performance under the same essential idea, we rather prefer to emphasize how complexity may appear under quite simple and natural actions evolving in time.

##### 2.2. Measuring Heterogeneity

When the model is implemented a first goal is applying mathematical tools in order to parameterize heterogeneity in a reliable manner.

For simplicity let us assume that the unit square in the plane is the support of a distribution with highly heterogeneous features. In order to scrutinize its heterogeneity, let us consider a collection (mesh) of -boxes, , of side length , representing a partition of for each value , (see Figure 1).