# Mathematical Modeling of Earth's Dynamical Systems: A Primer

Rudy Slingerland
Lee Kump
Edition: STU - Student edition
Pages: 240
https://www.jstor.org/stable/j.ctt7s5j9

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Preface
(pp. xi-xiv)
4. CHAPTER 1 Modeling and Mathematical Concepts
(pp. 1-22)

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...

5. CHAPTER 2 Basics of Numerical Solutions by Finite Difference
(pp. 23-38)

Some models give rise to relatively simple analytic solutions for a wide range of initial and boundary conditions. But this is generally not true for more complex partial differential equations of increased dimension. As the dimensions and complexity of the coefficients and boundary conditions increase, finding analytic solutions becomes prohibitively difficult, and, in fact, some nonlinear PDEs have no known analytic solutions. To circumvent this problem, numerical solution schemes have been developed that involve finding discrete solutions at specific points in time and space. Of these schemes, the simplest are of the finite difference type, and we restrict our discussion...

6. CHAPTER 3 Box Modeling: Unsteady, Uniform Conservation of Mass
(pp. 39-73)

We start our discussion of model derivations with systems that are best considered in terms of macroscopic control volumes, or “boxes”; that is, large reservoirs of mass or energy that are effectively homogeneous (well mixed) and evolve in time in response to imbalances between input and output. A familiar example is the global carbon cycle, which one typically envisions as a set of carbon reservoirs, ocean, atmosphere, living organisms, sediments, soils, and sedimentary rocks, among which carbon is transferred by a host of physical and biological processes. In such problems we are generally uninterested in spatial distributions within the reservoirs,...

7. CHAPTER 4 One-Dimensional Diffusion Problems
(pp. 74-88)

There is a very large class of problems in the earth sciences in which a conservative property moves through space at a rate proportional to some gradient (i.e., it follows a first-order rate law). Here, “first order” refers to an equation that contains the first derivative but no higher derivatives. This is different from the use in chemistry where it denotes a reaction rate proportional to the first power of a concentration. The conservative property that flows can be the moles of ions in a solution, or the thermal energy of atoms in a material, or the mass of regolith...

8. CHAPTER 5 Multidimensional Diffusion Problems
(pp. 89-110)

As noted in the past chapter, there is a large class of geoscience problems in which a quantity flows down a gradient according to a first-order rate law. Because that quantity is conserved, and assuming no other transport processes operate, the resulting mathematical descriptions all take the form of the diffusion equation. Gradients, of course, exist in all dimensions, and geoscientists are often faced with problems that demand a two-dimensional (2-D) or three-dimensional (3-D) approach. Here we extend the treatment to two dimensions, with examples that include the equations describing evolution of the landscape, flow in a pumped aquifer, and...

(pp. 111-129)

In this chapter, we consider another common process that transports mass and momentum into and out of geological reservoirs—passive transport by the motion of a medium such as water or air. The transport of a conserved property by a fluid in motion is calledadvectionorconvection. Although the terms are often used synonymously, convection is understood in some disciplines to mean the total transport of a substance by both diffusive and advective processes, whereas in others it is the transport of a substance by combined molecular and eddy diffusion as opposed to macroscopic fluid flow. Yet other disciplines...

10. CHAPTER 7 Advection and Diffusion (Transport) Problems
(pp. 130-150)

Consider a propertyPthat is conserved. It could be mass such as the amount of a dissolved species in a river or the mass of particulate load suspended in a flow, or it could be a vector property of a mudflow like momentum. Further assume thatPis passively carried along by the medium in which it exists at the flow speedu; that is, it is advected along. Also assume that within the medium,Pmoves from one point to another in proportion to its gradient; that is, it diffuses according to a first-order rate law. Finally, assume...

11. CHAPTER 8 Transport Problems with a Twist: The Transport of Momentum
(pp. 151-168)

In chapter 7, we explored the transport of a property by advection and diffusion. Whether the property was a mass of suspended sediment, dissolved ions in a stream, or heat in a lava flow, the resulting PDEs possessed the same form, and consequently the solutions behaved similarly. A signal entering the domain of interest from a boundary, or a function describing an initial condition, was translated across the domain while diffusing away. The amount of translation relative to diffusion could be qualitatively predicted if one knew the ratio of the advection speed to diffusivity (say in the form of a...

12. CHAPTER 9 Systems of One-Dimensional Nonlinear Partial Differential Equations
(pp. 169-186)

In earlier chapters, all of the problems involved one dependent variable as a function of one, two, or three independent variables. Here we introduce problems in which two dependent variables must be solved simultaneously to predict the evolution of a system. Examples include open channel flows in which the water velocity and depth are interdependent, lava flows in which the velocity depends upon a temperature-dependent lava rheology, and chemical systems in which two or more species are undergoing transport and reaction. Although systems of this sort are more complicated, their derivations require no new concepts; one need only ensure that...

13. CHAPTER 10 Two-Dimensional Nonlinear Hyperbolic Systems
(pp. 187-208)

In chapter 8, the transport equation was derived for the transport of momentum in the absence of sources and sinks. The result was a one-dimensional nonlinear hyperbolic equation for velocity. In chapter 9, this derivation was extended to include sources of momentum such as net pressure forces and sinks such as bed friction. Now we extend these ideas to two dimensions in the horizontal, and include sources and sinks of momentum as appropriate to describe a broad class of geophysical fluid flows. The concepts and approaches developed here can readily be extended to the third dimension for those using or...

14. Closing Remarks
(pp. 209-210)
15. References
(pp. 211-216)
16. Index
(pp. 217-231)