Complexity, Nonlinear Dynamics, Emergence and the Origin of Stochastic Fractals in Hydrological, Biogeochemical and Sedimentation Processes (Invited)

Abstract:

According to Warren Weaver (1948), scientific problems can be divided into 3 areas: a) those involving simplicity, b) those involving disorganized complexity and c) those involving organized complexity. The modern understanding of organized complexity grew with the computer revolution, and such problems are represented by nonlinear coupled systems of 3 or more differential equations. Solutions can yield an unexpected behavior called deterministic chaotic dynamics (DCD). Although discovered near 1890 by the mathematician Henri Poincaré, the modern study of DCD was initiated by Lorenz (1963), and applications are being made in virtually all fields of science (Deutsch and Marletto, 2014) including hydrology (Faybishenko, 2002, 2004; Faybishenko and Molz, 2013; Molz and Faybishenko, 2013). The term “organized” stems from the nonlinear interactions of the dependent variables, which can result in the emergence of unexpected patterns and the loss of detailed future predictability. Unfortunately, the word “chaotic” purveys a sense of wildness and disorder. However, at the system level there is order in the form of a “strange attractor”, the locus of points resulting from a plot of the dependent variables against each other, and it has been suggested that such a locus could represent a “sustainable state” of the system (Molz and Faybishenko, 2013). A strange attractor is a type of stochastic fractal, and there is both theoretical and experimental evidence that stochastic fractal structure is related to the distribution of hydraulic conductivity (K) in sediments (Lu et al., 2002; Molz et al., 2004; Bohling et al., 2012; Meerschaert et al., 2013; Dogan et al., 2014). In real sediments K depends on the interactions of many variables (Faybishenko, 1995). The population balance (PB) equations, a coupled, nonlinear set of PDEs, have typically been used to simulate the suspended particle size distribution in a flow-field subject to fracture, flocculation and erosion/sedimentation (Lee et al., 2012; Lee and Molz, 2014). In order to simulate sediment structure and the resulting K distribution, it is necessary to elaborate on the bottom erosion/sedimentation process. This problem will be formulated and results presented.