Mathematical Modeling of Earth's Dynamical Systems: A Primer - Softcover

Slingerland, Rudy; Kump, Lee

Softcover

ISBN 10: 0691145148 ISBN 13: 9780691145143

Publisher: Princeton University Press, 2011

Synopsis

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

About this title

About the Author

Rudy Slingerland and Lee Kump are professors of geosciences at Pennsylvania State University. Slingerland is the coauthor of "Simulating Clastic Sedimentary Basins". Kump is the coauthor of "The Earth System".

From the Back Cover

"Written by two of the leading researchers in the field, Mathematical Modeling of Dynamical Systems is a must-read for all geoscientists, and not just students. This excellent primer offers bite-size gems of insight into the world of quantitative geosciences applications, covers both mathematical and modeling concepts, and offers practical exercises to build expertise. Course notes and methodologies will be improving across our academies."--James P. M. Syvitski, executive director, Community Surface Dynamics Modeling System

"This wonderful, timely, and necessary book is a real winner. I appreciated the amazing range of geoscience topics as well as the book's structure--each of the chapters begins with an abstract-like summary preview, followed by examples of translations, before delving more deeply into topics. The authors should be congratulated for a brilliant book and pedagogical milestone."--Gidon Eshel, Bard College

"I am impressed with the overall philosophy of the book. The authors' definition of modeling is quite lucid and there is a useful breadth to the problems presented. The book's approach is pedagogically valuable for geoscience students, and fills a niche that exists between the more traditional geophysics math methods and Earth system dynamics."--Stephen Griffies, physical scientist, NOAA Geophysical Fluid Dynamics Lab

Excerpt. © Reprinted by permission. All rights reserved.

MATHEMATICAL MODELING of Earth's Dynamical Systems

A PrimerBy Rudy Slingerland Lee Kump

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-14514-3

Contents

Preface....................................................................................xi1 Modeling and Mathematical Concepts......................................................12 Basics of Numerical Solutions by Finite Difference......................................233 Box Modeling: Unsteady, Uniform Conservation of Mass....................................394 One-Dimensional Diffusion Problems......................................................745 Multidimensional Diffusion Problems.....................................................896 Advection-Dominated Problems............................................................1117 Advection and Diffusion (Transport) Problems............................................1308 Transport Problems with a Twist: The Transport of Momentum..............................1519 Systems of One-Dimensional Nonlinear Partial Differential Equations.....................16910 Two-Dimensional Nonlinear Hyperbolic Systems...........................................187Closing Remarks............................................................................209References.................................................................................211Index......................................................................................217

Chapter One

Modeling and Mathematical Concepts

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; pronounced "systems"). CSDMS (http:// csdms.colorado.edu) is a National Science Foundation (NSF)-sponsored community effort providing cyberinfrastructure aiding the development and dissemination of models that predict the flux of water, sediment, and solutes across the earth's surface. There one can find hundreds of models that incorporate the conservation and geomorphic transport laws and that can be used to solve particular problems. A companion organization, Computational Infrastructure for Geodynamics (http://www.geodynamics .org/), provides similar support for computational geophysics and related fields.

Example I: Simulation of Chicxulub Impact and Its Consequences

Probably the most famous event in historical geology, at least from the public's perspective, is the extraterrestrial impact event at the end of the Mesozoic Era that killed off the dinosaurs. Most schoolchildren know the standard story: A large asteroid that struck the surface of the earth in Mexico's Yucatán Peninsula created the Chicxulub Crater along with a rain of molten rock, toxic chemicals, and sun-obscuring debris that eliminated roughly three-quarters of the species living at the time. To work through the specific details of what happened and to predict the consequences of such an uncommon event is not easy because the physical and chemical processes are operating in a pressure–temperature state all but impossible to obtain experimentally. It is precisely these cases that benefit most from numerical simulation.

But is an asteroid impact computable? That is, given as many conservation equations and rate laws as there are state variables, and given initial and boundary conditions, can future states of the system be predicted with an acceptable degree of accuracy? Gisler et al. (2004) thought so. They derived a model simulating a 10-km-diameter iron asteroid plunging into 5 km of water that overlays 3 km of calcite, 7 km of basalt crust, and 6 km of mantle material. The set of equations was solved using the SAGE code from Los Alamos National Laboratory and the Science Applications International Corporation, which was developed under the U.S. Department of Energy's program in Accelerated Strategic Computing. Their model contained 333 million computational cells and used 1,024 processors for a total computational time of 1,000,000 CPU hours on a cluster of HP/Compaq PCs.

The results (fig. 1.1) document the dissipation of the asteroid's kinetic energy (which amounts to about 300 teratons TNT equivalent, or ~4 x 10²¹ J). The impact produces a tremendous explosion that melts, vaporizes, and ejects a substantial volume of calcite, granite, and water. Predictions from the model aid in understanding how, why, and where the resulting environmental changes caused the extinction.

Example II: Storm Surge of Hurricane Ivan in Escambia Bay

On September 16, 2004, Hurricane Ivan made landfall about 35 mi (56 km) west of Pensacola, Florida (fig. 1.2). At the time of landfall, peak winds exceeded 125 mi h^-1 (200 km h^-1), severely damaging many buildings in the Pensacola area. Probably equally damaging, however, was the surge of water along the coast and up Pensacola Bay. Homeowners along the bay experienced significant flooding (fig. 1.3) even though some were more than 25 mi by water from the open ocean.

Was this event an unpredictable act of God or could we have predicted the flooding? As you might suspect, the answer is that not only could it have been predicted, it was (fig. 1.4).

In chapter 10, we describe how surge models of the sort used by the U.S. Army Corps of Engineers are derived.

Steps in Model Building

So how does one construct a model of a geological phenomenon? Throughout this book, we will try to follow some logical steps in model development. First, get the physical picture clearly in mind. As an example, say one wanted to model the number of flies in a room as a function of time. The physical picture includes defining the dependent variable(s) (in this case the number of flies), the independent variables (time), and the size of the room. Second, one must define the physical processes to be treated and the boundaries of the model. The processes in the case of flies are flying, crawling, hatching, and dying. The boundaries of the model are those that do not pass flies such as walls, floor, and ceiling, and open boundaries such as doors and windows. Third, write down the physical laws to be used. Generally, these will be laws such as conservation of mass, Fick's law, and so on. In the case of flies, the laws are rate laws governing the flux of flies into and out of the room and laws defining the rates at which flies are created and die within the room. Fourth, put down very clearly the restrictive assumptions made. If one assumes that the flies will enter the room in proportion to the gradient in their number between inside and outside, write that assumption down. Fifth, perform a balance, first in words and then in symbols. Usually, one balances properties such as force, mass, or number. In the case of flies, we would say

The time rate of change of flies in the room = the rate at which they enter through doors and windows - the rate at which they leave + the rate at which they are born - the rate at which they die.

We would then substitute symbols for number of flies, time, and so forth. Sixth, check units. All the terms in the balance equation must be of the same units; if they are not, we have made a mistake in our definitions, and now is the time to catch it. Seventh, write down initial and boundary conditions. By initial conditions are meant the values of the dependent variables at the start of the calculations. For example, we would specify the number of flies in the room at t = 0 as zero or some finite number. Boundary conditions are the values of the dependent variables at the edges of the spatial domain of interest. For example, we must specify the number of flies outside as a function of time and specific door or window. Lastly, solve the mathematical model. If you are lucky you can find an equation of similar form that has already been analytically solved. There is value in pursuing an analytic solution even if you need to reduce variable coefficients to constants or even drop terms, because the simplified equation will provide insight into your system's behavior. But often no analytic solutions will be available, and this step will require converting the equation set into a numerical form amenable for solution on a computer. Finally, you should verify and validate your model. According to Oberkampf and Trucano (2002), verification is the process of determining that a model implementation accurately represents your conceptual description of the model and the solution to the model. Thus, verification checks that the coding correctly implements the equations and models, whereas validation determines the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. In other words, does the model agree with reality as observed in experiments and in the field. To formalize your thinking as you approach a problem, follow all of these steps in table 1.1.

Basic Definitions and Concepts

Why Models Are often Sets of Differential Equations

We naturally find it easier to think about how an entity changes than about the entity itself. For example, my car speedometer measures my velocity, not the distance I've traveled from my garage since I started my trip. It is easier to state that the time rate of change of water in my boat equals the rate at which water enters through the open seams minus the rate at which I am bailing it out than it is to state how the volume of water actually varies with time. Changing entities of this sort are called variables, of which there are two kinds: independent (space and time) and dependent, by which we mean the state variables in question (velocity, mass of water, and so forth). The rate of change of one variable with respect to another is called a derivative, written, for example, as the ordinary derivative dV/dt if the dependent variable V is only a function of the independent variable t, or the partial derivative [partial derivative]V/[partial derivative]t if V also depends upon other independent variables. Equations that express a relationship among these variables and their derivatives are differential equations.

However, often we want to know how the variables are related among themselves, not how they are related to their derivatives. So the general procedure is to derive the differential equations from first principles and then solve them for the values of the dependent variables as functions of the independent variables and other parameters.

To solve the differential equations requires more than the differential equation itself, however. The problem must be well posed. A well-posed problem contains as many governing equations as there are dependent variables. Also, the time and space interval over which the solution is to be obtained should be specified, and additional information concerning the dependent variables must be supplied at the start time (called initial conditions, or ICs) and the boundaries of the intervals (called boundary conditions, or BCs). This information is necessary because integration of the differential equations creates constants of integration in the case of ordinary differential equations (ODEs) and functions of integration in the case of partial differential equations (PDEs). The number of constants or functions needed is equal to the order of the differential equation. Thus, for a partial differential equation that is second order in both time and space, one must supply two functions derived from the ICs specifying the dependent variable as a function of time and two functions derived from the BCs specifying the dependent variable as a function of space. There are three possible types of BC information that can be supplied.

Dirichlet Conditions

In this type of BC, the solution itself is prescribed along the boundary, as, for example, if we were to set dependent variable ITLITL(0,t) = P, where P is some temporally constant value of the dependent variable.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2011
• Language: English
• ISBN 10: 0691145148
• ISBN 13: 9780691145143
• Binding: Paperback
• Number of pages: 248