If Mother Nature has plans, those plans are flexible. She keeps her options open, allows for more than one route to a given location, and we cannot assume that the same circumstances will always produce the same outcome. To digress for a moment: accepting this need not challenge religious or philosophical beliefs about a creator. Nothing in the bible, for instance, specifies exactly how the Judeo-Christian God goes about his/her business, or specifies any single pathways or mechanisms. As a protestant minister I knew well used to say: “Religion is concerned with the ‘why’ questions, and science addresses the ‘how.’”

Indeed. 35 years in the geoscience research business has shown me that that there is no single “right” or “natural” way for the world to be. Any human notions of singular, immanent norms or optima are tied to needs, goals, or perceptions, not scientific laws or relationships. And—again—there is nothing wrong with having such goals, desires, or expectations for nature, any more than there is anything wrong with a farm or a garden. The key is to realize that there is not much point in expecting Earth surface systems to evolve toward and maintain a single specific condition, any more than we would expect a garden to maintain itself without some guidance and intervention.

For the past five years or so, I have been working on adaptations of algebraic or spectral graph theory to study geomorphic, pedological, and ecological systems. My most recent development (unpublished, for reasons that will become clear in a moment) is some methods for measuring the complexity of historical sequences in Earth surface systems.

The idea is that a historical sequence represents a series of different states or stages—for example, vegetation communities along a successional trajectory; river channel morphological states; different soils in a paleosol sequence; depositional environments in a stratigraphic sequence, nodes of phylogenetic trees in biological evolution, etc.  These are treated as directed graphs. The states or stages are the graph nodes or vertices, and the historical transitions are the edges or links between the nodes.

The standard conceptual model for pedology, soil geomorphology, and soil geography is often called the “clorpt” model, for the way it was portrayed in Hans Jenny’s famous 1941 book The Factors of Soil Formation:

S = f(cl, o, r, p, t) . . . .

This equation states that soil types or soil properties (S) are a function of climate (cl), biotic effects (o for organisms), topography (r for relief), parent material (p), and t for time, conceived as the age of the surface the soils are formed on, or the time period the soil has been developing under a given broad set of environmental controls. This factorial approach, considering soils as a function of the combined, interacting influences of environmental factors such as geology, climate, and biota, was originated by V.V. Dokuchaev in Russia in the 1880s, popularized in English by C.F. Marbut in the 1920s and 1930s, and developed by Jenny into the familiar clorpt form.

Nearly 10 years ago, while pondering complex nonlinear dynamics in geomorphic systems, I was struck by how often we reduce problems to the interplay of opposing forces (e.g. uplift vs. denudation; soil formation vs. soil erosion, etc). I began to wonder how the concept of dialectics might be applicable in Earth sciences, or maybe I just wanted to increase my pseudo-intellectual street cred by using "dialectics" in an article. Anyway, I started work on a manuscript with the working title shown above, and then dropped it. I rediscovered it on the hard drive recently, and while I still can't convince myself it is journal article material, I do think there's some potentially interesting ideas there. 

What you see below is what I wrote in early 2006 (thus the absence of reference to work since then), unmodified except for putting in a few graphics to relieve the visual tedium.

1. Introduction

The title begs at least three questions: what do I mean by dialectics, how am I defining complexity, and how do I propose to link them?

1.1.  Geomorphic Complexity

Everyone knows the classic normal distribution—the “bell curve,” where most observations cluster around the mean, and the frequency falls off toward either end, with well known statistical properties. Lots of things in nature are more-or-less normally distributed, but lots of things are not. In some cases distributions are “heavy-tailed,” such that for example there are many of the small ones, and increasingly fewer as size increases. Famous examples are the distribution of earthquake magnitudes, rank-size distributions of cities, and the distribution of wealth in societies.

Models of avalanche size distributions in (mathematically-simulated) sand piles were seminal in developing ideas about self-organized criticality and power laws, both in geomorphology and in general. Unfortunately even real sandpiles, much less more complex systems, are not necessarily well described by the models.

Scientists, including geographers and geoscientists, are easily seduced by repeated forms and patterns in nature. This is not surprising, as our mission is to detect and explain patterns in nature, ideally arising from some unifying underlying law or principle. Further, in the case of geography and Earth sciences, spatial patterns and form-process relationships are paramount.

Unfortunately, the recurrence of similar shapes, forms, or patterns may not tell us much. Over the years we have made much of, e.g. logarithmic spirals, Fibonacci sequences, fractal geometry, and power-law distributions—all of which recur in numerous phenomena—only to learn that they don’t necessarily tell us anything, other than that several different phenomena or causes can lead to the same form or pattern. The phenomenon whereby different processes, causes, or histories can lead to similar outcomes is called equifinality.

Center pivot irrigation in Kansas, USA (USGS photo).

Some comments from a reviewer on a recent manuscript of mine dealing with responses to disturbance in geomorphology got me to thinking about the concept of disturbance in the environmental sciences. Though the paper is a geomorphology paper (hopefully to be) in a geomorphology journal, the referee insisted that I should be citing some of the “foundational” ecological papers on disturbance. These, according to the referee, turned out to be papers from the 1980s and 1990s that are widely cited in the aquatic ecology and stream restoration literature, but are hardly foundational in general.

Consideration of the role of disturbance goes back to the earliest days of ecology, and is a major theme in the classic papers of, e.g., Warming, Cowles, and Clements in the late 19^th and early 20^th centuries. A general reconsideration (“reimagining” is the term many would use, but I’ve grown to hate that overused word) of the role of disturbance in ecological systems was well underway by the 1970s, and the last five years or so have seem some very interesting syntheses of these emerging ideas (two I especially like are Mori, 2011 and Pulsford et al., 2014).

Fluvial geomorphologists, along with hydrologists and river engineers, have long been concerned with the flows or discharges that are primarily responsible for forming and shaping river channels. In the mid-20^th century it was suggested that this flow is associated with bankfull stage—the stage right at the threshold of overflowing the channel—and that this occurs, on average, about every year or two in humid-climate perennial streams. If you have to choose just one flow to fixate on—and sometimes you do, for various management, design, and assessment purposes—and have no other a priori information about the river, bankfull is indeed the best choice. But, of course, nature is not that simple.

One of the classic principles/relationships in biogeography is called the species-area curve, relating the number of different species found (usually of some particular taxonomic group; e.g., birds or plants) to the area sampled. These curves are usually well fit by an exponential relationship:

S = c A ^b

where S is the number of species, A is area, c is a constant representing the number of species in the smallest area sampled, and b represents the rate of increase of species with area. While b could be greater than 1 if major biogeographical boundaries are transgressed (so that whole new sets of species are encountered), otherwise b < 1, and usually much less; 0.25 is a fairly common value.

Juanjo Ibanez and I (in separate studies) found that similar trends apply to soil diversity, with S in this case indicating number of different soil types (e.g., soil series). In his very broad scale analyses, Juanjo also found b » 0.25, while in my landscape-scale studies b was in the range of 0.6.  Syntheses of this work are found in the book Pedodiversity (CRC Press, 2013) edited by Ibanez and James Bockheim.