In this Chapter

The elastic deformation of cold rock plays an essential role in geodynamic processes. In this chapter we introduce the concepts of stress and strain. Body forces and surface stresses generate a distribution of pressure, normal stress, and shear stress in an elastic medium. The concept of isostasy is essential to the understanding of the geodynamics of topography. Isostasy provides a simple explanation for the formation of mountains and sedimentary basins. Under compressional forces the continental crust thickens, forming mountains and their crustal roots. Under tensional forces the continental crust thins, leading to surface subsidence and sedimentary basins.

Pressure and stresses cause elastic solids to deform. Dilatation, normal strain, and shear strain are measures of displacement in analogy to pressure, normal stress, and shear stress. Measurements of surface displacements (strain) are an important constraint on tectonic processes. An example is the surface strain caused by rupture on a fault. Global Positioning System (GPS) observations have revolutionized the accuracy of surface displacement measurements. Absolute positions can now be determined with an accuracy of a few millimeters.

Introduction

Plate tectonics is a consequence of the gravitational body forces acting on the solid mantle and crust. Gravitational forces result in an increase of pressure with depth in the Earth; rocks must support the weight of the overburden that increases with depth. A static equilibrium with pressure increasing with depth is not possible, however, because there are horizontal variations in the gravitational body forces in the Earth's interior. These are caused by horizontal variations in density associated with horizontal differences in temperature. The horizontal thermal contrasts are in turn the inevitable consequence of the heat release by radioactivity in the rocks of the mantle and crust. The horizontal variations of the gravitational body force produce the differential stresses that drive the relative motions associated with plate tectonics.

One of the main purposes of this chapter is to introduce the fundamental concepts needed for a quantitative understanding of stresses in the solid Earth. Stresses are forces per unit area that are transmitted through a material by interatomic force fields. Stresses that are transmitted perpendicular to a surface are normal stresses; those that are transmitted parallel to a surface are shear stresses. The mean value of the normal stresses is the pressure. We will describe the techniques presently used to measure the state of stress in the Earth's crust and discuss the results of those measurements.

In this Chapter

The concept of geochemical reservoirs in the Earth provides a basis for understanding fundamental geodynamic processes. Important reservoirs in the solid Earth include the core, mantle, and continental crust. The emphasis in chemical geodynamics is on radiogenic elements. A typical example is the decay of radiogenic rubidium 87 to strontium 87. The reference isotope is strontium 86. The ratio of strontium isotopic compositions 87/86 in a rock can be used to determine its age. One example considered is the extraction of the enriched continental crust from the depleted mantle. Volcanic processes preferentially concentrate rubidium into the continental crust. Over time the production of strontium 87 relative to the reference strontium 86 can be used to determine the mean age of the continental crust.

Introduction

Radioactive heating of the mantle and crust plays a key role in geodynamics as discussed in Section 4.5. The heat generated by the decay of the uranium isotopes 238U and 235U, the thorium isotope 232Th, and the potassium isotope 40K is the primary source of the energy that drives mantle convection and generates earthquakes and volcanic eruptions. Radiogenic isotopes play other key roles in the Earth sciences. Isotope ratios can be used to date the “ages” of rocks.

The science of dating rocks by radioisotopic techniques is known as geochronology. In many cases a rock that solidifies from a melt becomes a closed isotopic system. Measurements of isotope ratios and parent–daughter ratios can be used to determine how long ago the rock solidified from a magma and this defines the age of the rock. These techniques provide the only basis for absolute dating of geological processes. Age dating of meteorites has provided an age of the solar system of 4.57 Ga. The oldest rocks on the Earth were found in West Greenland and have an age of 3.65 Ga. Lunar samples returned by the Apollo missions have ages of over 4 Ga.

Quantitative measurements of the concentrations of radioactive isotopes and their daughter products in rocks form the basis for chemical geodynamics. Essentially all rocks found on the surface of the Earth have been through one or more melting episodes and many have experienced high temperature metamorphism.

In this Chapter

The purpose of this chapter is to introduce the student to the use of the computer programming language MATLAB in the context of geodynamic problems. We first present some fundamentals of using MATLAB and we then introduce some numerical methods for solving both linear and nonlinear differential equations. Integrations of both the steady and time-dependent heat conduction equations are used to illustrate the numerical approaches and their MATLAB implementations.

Introduction

Many problems in geodynamics cannot be solved analytically. Even those that can often involve complicated functions that must be evaluated numerically. Therefore, in Chapters 11 and 12 we provide the student with the tools to evaluate complex functions and mathematical expressions and produce direct numerical solutions to problems. The material can be read at any stage in going through this book, although it would be of most benefit to the student to study this chapter early on and use the tools discussed in it throughout the book. Students with a background in numerical techniques and basic computing might already be familiar with much of this material, but we have provided it with the beginning student in mind. It is not our goal in this chapter to provide a rigorous or complete introduction to numerical analysis nor is it our purpose to train students to write sophisticated numerical codes that enable the solving of geodynamical problems. Instead we offer the student a toolbox of codes to use in problem solving with some introductory discussion of the methods employed in the codes.

We have opted to use MATLAB, a computer programming language used so widely that most students will have access to it through their college or university. There are numerous books explaining the use of MATLAB and its applications in engineering and science. We will attempt to be mostly self-contained, explaining what one needs to know to employ MATLAB as a geodynamics problem-solving package.

In this Chapter

We introduce the fundamental concepts of fluid mechanics. Our focus will be on mantle convection, but we will consider a variety of other geodynamic applications. These applications utilize Newtonian fluids in which the stress is proportional to the spatial gradient of velocity. The constant of proportionality is the viscosity. Solutions for isothermal problems require an equation for conservation of mass and a force balance equation. In our applications the force balance includes the pressure forces, viscous terms, and the gravitational body force. Temperature variations require addition of a buoyancy force and an energy equation. In addition to the terms included in the heat equation in Chapter 4, terms are required to account for the advection of heat (energy). Unlike the heat equation, the equations for fluid flow are usually nonlinear, for example, the product of velocity and temperature gradient in the energy equation. This nonlinearity greatly increases the difficulty of obtaining analytical solutions.

One of the important problems we will consider in this chapter is postglacial rebound. Under the load of ice during the last ice age, the continental crust was depressed in order to achieve isostatic compensation. The surface of Greenland is currently depressed below sea level due to the load of the Greenland Ice Cap. At the end of the last ice age, about 8000 years ago, large quantities of ice melted. The removal of this ice load results in a “rebound” of the Earth’s surface in order to re-establish the isostatic balance. This rebound demonstrated beyond doubt the fluid behavior of the Earth’s mantle. The rate of rebound quantified the viscosity of the mantle.

In this Chapter

A spherical body has a surface gravitational field that is proportional to its mass and inversely proportional to its radius squared. To a first approximation this result explains the Earth’s gravitational field. However, the Earth is rotating and this rotation results in the equatorial radius being larger than the polar radius (polar flattening and an equatorial bulge). The combination of mass and rotation gives the reference gravitational field for the Earth.

Deviations from the values given by the reference field are known as gravity anomalies. These anomalies are usually due to density variations in the Earth's interior. Surface gravity anomalies are used to search for mineral deposits and oil accumulations. We will also show that gravity anomalies can be used to quantify fundamental geodynamic processes.

Introduction

The force exerted on an element of mass at the surface of the Earth has two principal components. One is due to the gravitational attraction of the mass in the Earth, and the other is due to the rotation of the Earth. Gravity refers to the combined effects of both gravitation and rotation. If the Earth were a nonrotating spherically symmetric body, the gravitational acceleration on its surface would be constant. However, because of the Earth's rotation, topography, and internal lateral density variations, the acceleration of gravity g varies with location on the surface. The Earth's rotation leads mainly to a latitude dependence of the surface acceleration of gravity. Because rotation distorts the surface by producing an equatorial bulge and a polar flattening, gravity at the equator is about 5 parts in 1000 less than gravity at the poles. The Earth takes the shape of an oblate spheroid. The gravitational field of this spheroid is the reference gravitational field of the Earth. Topography and density inhomogeneities in the Earth lead to local variations in the surface gravity, which are referred to as gravity anomalies.

In this Chapter

In this chapter we introduce the fundamentals of elasticity. Elasticity is the principal deformation mechanism applicable to the lithosphere. In linear elasticity strain is proportional to stress. Elastic deformation is reversible; when the applied stress is removed, the strain goes to zero. Deformation of the lithosphere, in a number of applications, can be approximated as the bending (flexure) of a thin elastic plate. Examples include bending under volcanic loads, bending at subduction zones, and bending that creates sedimentary basins.

Introduction

In the previous chapter we introduced the concepts of stress and strain. For many solids it is appropriate to relate stress to strain through the laws of elasticity. Elastic materials deform when a force is applied and return to their original shape when the force is removed. Almost all solid materials, including essentially all rocks at relatively low temperatures and pressures, behave elastically when the applied forces are not too large. In addition, the elastic strain of many rocks is linearly proportional to the applied stress. The equations of linear elasticity are greatly simplified if the material is isotropic, that is, if its elastic properties are independent of direction. Although some metamorphic rocks with strong foliations are not strictly isotropic, the isotropic approximation is usually satisfactory for the Earth's crust and mantle.

At high stress levels, or at temperatures that are a significant fraction of the rock solidus, deviations from elastic behavior occur. At low temperatures and confining pressures, rocks are brittle solids, and large deviatoric stresses cause fracture. As rocks are buried more deeply in the Earth, they are subjected to increasingly large confining pressures due to the increasing weight of the overburden. When the confining pressure on the rock approaches its brittle failure strength, it deforms plastically. Plastic deformation is a continuous, irreversible deformation without fracture. If the applied force causing plastic deformation is removed, some fraction of the deformation remains. We consider plastic deformation in Section 7.11.

In this Chapter

Earthquakes are associated with brittle rupture in the Earth’s crust and mantle. Irreversible deformation occurs on preexisting faults. The behavior of faults is controlled by friction, which leads to stick–slip behavior. A fault locks after an earthquake and remains locked until the stress builds up to a level that will cause slip to occur, resulting in the next earthquake. During the build-up of stress elastic energy is stored. When slip occurs a large fraction of this stored energy is converted to elastic energy in seismic waves, this is elastic rebound.

The magnitude of an earthquake is determined from the amplitudes of surface displacements as measured by seismographs. The standard measure of magnitude is the Richter scale. We will consider the behavior of the San Andreas fault in some detail. This fault is the primary boundary between the Pacific plate and the North American plate in California. The fault is well instrumented, and surface displacements can be measured directly. The 1906 earthquake on this fault largely destroyed San Francisco.

Introduction

At low temperatures and pressures rock is a brittle material that will fail by fracture if the stresses become sufficiently large. Fractures are widely observed in surface rocks of all types. When a lateral displacement takes place on a fracture, the break is referred to as a fault. Surface faults occur on all scales. On the smallest scale the offset on a clean fracture may be only millimeters. On the largest scale the surface expression of a major fault is a broad zone of broken up rock known as a fault gouge; the width may be a kilometer or more, and the lateral displacementmay be hundreds of kilometers.

Earthquakes are associated with displacements on many faults. Faults lock, and a displacement occurs when the stress across the fault builds up to a sufficient level to cause rupture of the fault. This is known as stick–slip behavior. When a fault sticks, elastic energy accumulates in the rocks around the fault because of displacements at a distance. When the stress on the fault reaches a critical value, the fault slips and an earthquake occurs. The elastic energy stored in the adjacent rock is partially dissipated as heat by friction on the fault and is partially radiated away as seismic energy. This is known as elastic rebound. Fault displacements associated with the largest earthquakes are of the order of 30 m.

This textbook deals with the fundamental physical processes necessary for an understanding of plate tectonics and a variety of geological phenomena. We believe that the appropriate title for this material is geodynamics. The contents of this textbook evolved from a series of courses given at Cornell University and UCLA to students with a wide range of backgrounds in geology, geophysics, physics, mathematics, chemistry, and engineering. The level of the students ranged from advanced undergraduate to graduate.

Approach

We present most of the material with a minimum of mathematical complexity. In general, we do not introduce mathematical concepts unless they are essential to the understanding of physical principles. For example, our treatment of elasticity and fluid mechanics avoids the introduction or use of tensors. We do not believe that tensor notation is necessary for the understanding of these subjects or for most applications to geological problems. However, solving partial differential equations is an essential part of this textbook. Many geological problems involving heat conduction and solid and fluid mechanics require solutions of such classic partial differential equations as Laplace's equation, Poisson's equation, the biharmonic equation, and the diffusion equation. All these equations are derived from first principles in the geological contexts in which they are used. We provide elementary explanations for such important physical properties of matter as solid-state viscosity, thermal coefficient of expansion, specific heat, and permeability. Basic concepts involved in the studies of heat transfer, Newtonian and non-Newtonian fluid behavior, the bending of thin elastic plates, the mechanical behavior of faults, and the interpretation of gravity anomalies are emphasized. Thus it is expected that the student will develop a thorough understanding of such fundamental physical laws as Hooke’s law of elasticity, Fourier’s law of heat conduction, and Darcy’s law for fluid flow in porous media.

In this Chapter

Numerical solutions to a variety of problems in geodynamics are given in this chapter utilizing MATLAB. In some cases, a specific boundary value problem given in a previous chapter is generalized to arbitrary boundary conditions. One example is the bending of the lithosphere under a load. The solution for a point load given in Chapter 3 is generalized to arbitrary load distributions. Solutions for lithospheric bending under axisymmetric loads are presented. The gravity anomaly over rectangular prisms is calculated, and a new formalism based on Fourier transforms is presented to solve for the gravity over arbitrary topography. Axisymmetric solutions for postglacial rebound and crater relaxation are developed. Finite amplitude thermal convection requires the solution of nonlinear partial differential equations. These solutions must be obtained using numerical methods. A MATLAB code for a two-dimensional steady solution is given. Finally, the discussion of faulting in Chapter 8 is extended to more complex geometries and faulting scenarios.

Bending of the Lithosphere under a Triangular Load

In Section 3.16 we solved for the bending of the elastic lithosphere under the load of a volcanic island chain by representing the island chain as a line load on the plate. With a numerical solution it is possible to represent the load on the plate more realistically, e. g., by a triangular load. We first develop the numerical approach by going back to the line load problem whose analytic solution provides a benchmark against which to evaluate the numerical solution.

In this Chapter

The plate tectonic model provides a framework for understanding many geodynamic processes. Earthquakes, volcanism, and mountain building are examples. The plate velocities, 10–100 mm yr−1, imply a fluid-like behavior of the solid Earth. Hot mantle rock can flow (behave as a fluid) on geological time scales due to solid-state creep and thermal convection. The hot mantle rock is cooled by heat loss to the Earth’s surface resulting in a cold thermal “boundary layer.” This boundary layer is rigid and is referred to as the lithosphere. The surface lithosphere is broken into a series of plates that are in relative motion with respect to each other. This motion results in “plate tectonics.”

Plates are created at mid-ocean ridges, where hot mantle rock ascends. Partial melting in the ascending rock produces the magmas that form the basaltic ocean crust. The surface plates reenter the mantle at ocean trenches (subduction). The cold rock in the plate (lithosphere) is denser than the adjacent hot mantle rock. This results in a downward gravitational body force that drives the motion of the surface plate. Complex volcanic processes at subduction zones generate the continental crust. This crust is thick and light and does not participate in the platetectonic cycle. Thus the continental crust is about a factor of 10 older, on average, than the oceanic crust (1 Ga versus 100 Ma).

Interactions between plates at plate boundaries are responsible for a large fraction of the earthquakes that occur. Earthquakes are caused by episodic ruptures and displacements on preexisting faults. These displacements provide the relative motions between surface plates. Plate boundary processes are also responsible for a large fraction of the surface volcanism.

In this Chapter

One of the important aspects of geodynamics is the fluid behavior of the solid mantle. This behavior should not be a surprise since crystalline solid ice in glaciers flows freely. The similarity to ice in a glacier is that the ice is close to its melting point and rock in the mantle is close to its melting point. In both cases strongly temperature-dependent movement of crystal vacancies and/or dislocations leads to solid-state creep in a stress field. This strong temperature dependence controls the cooling of the Earth. As the amount of radioactive heat production decreases with time, the vigor of the mantle convection required to extract the heat decreases. This allows the mantle temperature to decrease and the Earth to cool, contributing significantly to the observed surface heat flow.

In this chapter we consider other rock rheologies. Viscoelasticity combines the elastic and viscous behavior of solid rock. Viscoelasticity explains why rock can transmit shear waves (requiring elastic behavior) at short time scales and behave viscously (mantle convection) at long times. Viscoelasticity will also quantify the thickness of the elastic lithosphere as a fraction of the thickness of the thermal lithosphere. When large stresses are applied to the elastic lithosphere irreversible plastic deformation occurs. The result is the development of plastic hinges in the lithosphere at some subduction zones.

Introduction

At atmospheric pressure and room temperature most rocks are brittle; that is, they behave nearly elastically until they fail by fracture. Cracks or fractures in rock along which there has been little or no relative displacement are known as joints. They occur on all scales in both sedimentary and igneous rocks. Joints are commonly found in sets defining parallel or intersecting patterns of failure related to local stress orientations. The breakdown of surface rocks by erosion and weathering is often controlled by systems of joints along which the rocks are particularly weak and susceptible to disintegration and removal. These processes in turn enhance the visibility of the jointing. Igneous rocks often develop joints as a result of the thermal stresses associated with cooling and contraction. Columnar jointing in basaltic lava flows (Figure 7.1) and parallel jointing in granitic rocks (Figure 7.2) are examples.

In this Chapter

In this chapter we consider heat flow and temperature distributions. We give a summary of surface heat flow measurements that constrain the loss of heat from the interior of the Earth. An important source of this heat is radiogenic isotopes in the Earth’s interior. Fourier’s law of heat conduction gives the linear dependence of heat flow on the spatial gradients of temperature. The heat equation is one of the most important equations in geophysics. It gives the balance between heat storage, heat generation, and heat conduction. Examples of solutions to the heat equation are given. The solution for the annual variability of the temperature at the surface of the Earth is obtained. The solution for the instantaneous cooling of a half-space is given. This solution is used to determine the cooling and thickening of the oceanic lithosphere as a function of the distance from the oceanic ridge where it was created. This solution predicts the measured values of surface heat flow and, using isostasy, explains the elevation distribution of oceanic lithosphere. A solution is given for the surface solidification of a liquid. The result predicts the rate of crustal thickening on a lava lake. Solutions of the heat equation are also applied to surface erosion.

Additional material relevant to this chapter can be found in Chapters 11 and 12 (Sections 12.3, 12.4, and 12.5). These appendices use MATLAB to numerically solve a number of the heat flow problems considered in this chapter, including temperature, surface heat flux, and depth for the plate model of the cooling oceanic lithosphere, and the post-solidification cooling of a dike.