Mathematical Modeling of Earth's Dynamical Systems

A Primer

Rudy Slingerland, Lee Kump

• Publisher: Princeton University Press
• Year: 2011
• ISBN:
  9781400839117
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

A concise guide to representing complex Earth systems using simple dynamic models Mathematical Modeling of Earth's Dynamical Systems gives earth scientists the essential skills for translating chemical and physical systems into mathematical and computational models that provide enhanced insight into Earth's processes. Using a step-by-step method, the book identifies the important geological variables of physical-chemical geoscience problems and describes the mechanisms that control these variables.This book is directed toward upper-level undergraduate students, graduate students, researchers, and professionals who want to learn how to abstract complex systems into sets of dynamic equations. It shows students how to recognize domains of interest and key factors, and how to explain assumptions in formal terms. The book reveals what data best tests ideas of how nature works, and cautions against inadequate transport laws, unconstrained coefficients, and unfalsifiable models. Various examples of processes and systems, and ample illustrations, are provided. Students using this text should be familiar with the principles of physics, chemistry, and geology, and have taken a year of differential and integral calculus. Mathematical Modeling of Earth's Dynamical Systems helps earth scientists develop a philosophical framework and strong foundations for conceptualizing complex geologic systems.

• Step-by-step lessons for representing complex Earth systems as dynamical models
• Explains geologic processes in terms of fundamental laws of physics and chemistry
• Numerical solutions to differential equations through the finite difference technique
• A philosophical approach to quantitative problem-solving
• Various examples of processes and systems, including the evolution of sandy coastlines, the global carbon cycle, and much more
• Professors: A supplementary Instructor's Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://press.princeton.edu/class_use/solutions.html

CHAPTER
1

Modeling and Mathematical Concepts

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A system is a big black box
Of which we can’t unlock the locks,
And all we can find out about
Is what goes in and what comes out.

—Kenneth Boulding

Kenneth Boulding—presumably somewhat tongue-in-cheek—expresses the cynic’s view of systems. But this description will only be true if we fail as modelers, because the whole point of models is to provide illumination; that is, to give insight into the connections and processes of a system that otherwise seems like a big black box. So we turn this view around and say that Earth’s systems may each be a black box, but a well-formulated model is the key that lets you unlock the locks and peer inside.

There are many different types of models. Some are purely conceptual, some are physical models such as in flumes and chemical experiments in the lab, some are stochastic or structure-imitating, and some are deterministic or process-imitating. The distinction also can be made between forward models, which project the final state of a system, and inverse models, which take a solution and attempt to determine the initial and boundary conditions that gave rise to it. All of the models described in this book are deterministic, forward models using variables that are continuous in time and space. One should think of the models as physical–mathematical descriptions of temporal and/or spatial changes in important geological variables, as derived from accepted laws, theories, and empirical relationships. They are “devices that mirror nature by embodying empirical knowledge in forms that permit (quantitative) inferences to be derived from them” (Dutton, 1987). The model descriptors are the conservation laws, laws of hydraulics, and first-order rate laws for material fluxes that predict future states of a system from initial conditions (ICs), boundary conditions (BCs), and a set of rules. For a given set of BCs and ICs, the model will always “determine” the same final state. Furthermore, these models are mathematical (numerical). We emphasize this type of model over other types because it represents a large proportion of extant models in the earth sciences. Dynamical models also provide a good vehicle for teaching the art of modeling. We call modeling an art because one must know what one wants out of a model and how to get it. Properly constructed, a model will rationalize the information coming to our senses, tell us what the most important data are, and tell us what data will best test our notion of how nature works as it is embodied in the model. Bad models are too complex and too uneconomical or, in other cases, too simple.

Pros and Cons of Dynamical Models

The advantage of a deterministic dynamical model is that it states formal assertions in logical terms and uses the logic of mathematics to get beyond intuition. The logic is as follows: If my premises are true, and the math is true, then the solutions must be true. Suddenly, you have gotten to a position that your intuition doesn’t believe, and if upon further inspection, your intuition is taught something, then science has happened. Models also permit formulation of hypotheses for testing and help make evident complex outcomes, nonlinear couplings, and distant feedbacks. This has been one of the more significant outcomes of climate modeling, for example. If there are leads and lags in the system, it’s tough for empiricists because they look for correlation in time to determine causation. But if it takes a couple of hundred years for the effect to be realized, then the empiricist is often thwarted.

Particularly relevant for geoscientists and astrophysicists, dynamical models also permit controlled experimentation by compressing geologic time. Consider the problem of understanding the collision of galaxies—how does one study that process? Astrophysicists substitute space for time by taking photographs of different galaxies at different stages of collision and then assume they can assemble these into a single sequence representing one collision. That sequence acts as a data set against which a model of collision processes can be tested where the many millions of years are compressed. The idea of a snowball Earth provides an example even closer to home, or one could ask the question: What did rivers in the earthscape look like prior to vegetation? Questions of this sort naturally lend themselves to idea-testing through dynamical models.

But dynamical models not properly constructed or interpreted can cause great trouble. Recently, Pilkey and Pilkey-Jarvis (2007) passionately argued that many environmental models are not only useless but also dangerous because they have made bad predictions that have led to bad decisions. They argue that there are many causes, including inadequate transport laws, poorly constrained coefficients (“fudge factors”), and feedbacks so complex that not even the model developers understand their behavior. Although we think the authors have painted with too broad a brush, we agree with them on one point. A simple falsifiable model that has been properly validated [even if in a more limited sense than that of Oreskes et al. (1994)] is better than an ill-conceived complex model with scores of poorly constrained proportionality constants [also see Murray (2007) for a discussion of this point]. Finally, we should never lose sight of the fact that in a model “it is not possible simultaneously to maximize generality, realism, and precision” (atmospheric scientist John Dutton, personal communication, 1982).

An Important Modeling Assumption

We assume in this book that a fruitful way to describe the earth is a series of mathematical equations. But is this mathematical abstraction an adequate description of reality? Does reality exist in our minds as mathematical formulas or is it outside of us somewhere? For example, the current understanding of the fundamental physical laws that govern the universe—string theory—is entirely a mathematical theory without experimental confirmation. To some it unites the general theory of relativity and quantum mechanics into a final unified theory. To others it is unfalsifiable and infertile (see, e.g., Smolin, 2006).

We avoid these philosophical problems by simply asserting that mathematical descriptions of the earth both past and present have proved to be a useful way of knowing. As the Nobel Laureate Eugene Wigner noted, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve” (Wigner, 1960). An alternative view is that they are inherently quite limited in their predictive power. This view is summarized cogently by Chris Paola in a review of sedimentary models: “[A]ttempting to extract the dynamics at higher levels from comprehensive modelling of everything going on at lower levels is . . . like analyzing the creation of La Boheme as a neurochemistry problem” (Paola, 2000). Whereas we accept this point of view in the limit, we reject it for a wide range of complex systems that are amenable to reduction.

Some Examples

To set the stage for the chapters that follow, we present two problems for which modeling can provide insight. Other examples abound in the literature. Of special note for those studying Earth surface processes is the Web site of the Community Earth Surface Dynamics Modeling Initiative (CSDMS; p...

Table of contents

• Cover
• Title
• Copyright
• Contents
• Preface
• 1 Modeling and Mathematical Concepts
• 2 Basics of Numerical Solutions by Finite Difference
• 3 Box Modeling: Unsteady, Uniform Conservation of Mass
• 4 One-Dimensional Diffusion Problems
• 5 Multidimensional Diffusion Problems
• 6 Advection-Dominated Problems
• 7 Advection and Diffusion (Transport) Problems
• 8 Transport Problems with a Twist: The Transport of Momentum
• 9 Systems of One-Dimensional Nonlinear Partial Differential Equations
• 10 Two-Dimensional Nonlinear Hyperbolic Systems
• Closing Remarks
• References
• Index