Threshold modulation

Upper and lower limits

In terms of mass balances or budgets, geomorphic systems have three fundamental states, whereby losses or removals are either greater than, less than, or roughly equal to inputs or gains (i.e., steady-state). Thus a regolith cover, for instance, is either thinning, thickening, or maintaining steady-state relative to the rates of mass losses and inputs, and weathering and regolith formation.

The principle of threshold-mediated modulation holds that thresholds limit development on both ends (negative or positive mass balance), and that exceeding the thresholds may initiate development in the opposite direction. For instance, vertical accretion on alluvial floodplains is limited by an elevation at which regular flooding no longer occurs, thus limiting further accretion. In addition, confinement of flows within the channel may increase stream power and shear stress, thus ultimately resulting in stripping of the alluvium.

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 My most recent development (unpublished, for reasons that will become clear in a moment) is some methods for measuring the complexity of historical sequences in Earth surface systems.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.