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PROBLEM 4–20 A mountain range can be represented as a periodic topography with a wavelength of 100 km and an amplitude of 1.2 km. Heat flow in a valley is measured to be 46 mW m−2. If the atmospheric gradi- ent is 6.5 K km−1 and k = 2.5 Wm−1 K−1, determine what the heat flow would have been without topog- raphy; that is, make a topographic correction.

PROBLEM 4–29 Estimate the effects of variations in bottom water temperature on measurements of oceanic heat flow by using the model of a semi-infinite half-space subjected to periodic surface temperature fluctuations. Such water temperature variations at a specific location on the ocean floor can be due to, for example, the transport of water with variable tem- perature past the site by deep ocean currents. Find the amplitude of water temperature variations that cause surface heat flux variations of 40 mW m−2 above and below the mean on a time scale of 1 day. As- sume that the thermal conductivity of sediments is 0.8 W m−1 K−1 and the sediment thermal diffusivity is 0.2 mm2 s−1.

PROBLEM 4–33 One way of determining the effects of erosion on subsurface temperatures is to consider the instantaneous removal of a thickness l of ground. Prior to the removal T = T0 + βy, where y is the depth, β is the geothermal gradient, and T0 is the surface temperature. After removal, the new surface is main- tained at temperature T0. Show that the subsurface temperature after the removal of the surface layer is given by

How is the surface heat flow affected by the removal of surface material?

PROBLEM 4–39 One of the estimates for the age of the Earth given by Lord Kelvin in the 1860s assumed that Earth was initially molten at a constant tem- perature Tm and that it subsequently cooled by con- duction with a constant surface temperature T0. The age of the Earth could then be determined from the present surface thermal gradient (dT/dy)0. Re- produce Kelvin’s result assuming Tm T0 = 1700 K, c=1 kJ kg−1 K−1, L =400 kJ kg−1, κ =1 mm2 s−1, and (dT/dy)0 = 25 K km−1 . In addition, determine the thickness of the solidified lithosphere. Note: Since the solidified layer is thin compared with the Earth’s radius, the curvature of the surface may be neglected.

PROBLEM4–43 Themantlerocksoftheasthenosphere from which the lithosphere forms are expected to contain a small amount of magma. If the mass frac- tion of magma is 0.05, determine the depth of the lithosphere–asthenosphere boundary for oceanic li- thosphere with an age of 60 Ma. Assume L = 400 kJ kg−1,c=1kJkg−1 K−1,Tm =1600K,T0 =275K,and κ = 1 mm2 s−1.

PROBLEM 4–52 The ocean ridges are made up of a series of parallel segments connected by transform faults, as shown in Figure 1–13. Because of the dif- ference of age there is a vertical offset on the fracture zones. Assuming the theory just derived is applica- ble, what is the vertical offset (a) at the ridge crest and (b) 100 km from the ridge crest in Figure 4–46 (ρm = 3300kgm−3,κ=1mm2s−1,αv =3×10−5 K−1,T1− T0=1300K,u=50mmyr−1).

PROBLEM 5–7 What is the value of the acceleration of gravity at a distance b above the geoid at the equator (b a)?

PROBLEM 5–18 A volcanic plug of diameter 10 km has a gravity anomaly of 0.3 mm s−2. Estimate the depth of the plug assuming that it can be modeled by a verti- cal cylinder whose top is at the surface. Assume that the plug has density of 3000 kg m−3 and the rock it intrudes has a density of 2800 kg m−3.

PROBLEM 4–20 A mountain range can be represented as a periodic topography with a wavelength of 100 km and an amplitude of 1.2 km. Heat flow in a valley is measured to be 46 mW m−2. If the atmospheric gradi- ent is 6.5 K km−1 and k = 2.5 Wm−1 K−1, determine what the heat flow would have been without topog- raphy; that is, make a topographic correction.

PROBLEM 4–29 Estimate the effects of variations in bottom water temperature on measurements of oceanic heat flow by using the model of a semi-infinite half-space subjected to periodic surface temperature fluctuations. Such water temperature variations at a specific location on the ocean floor can be due to, for example, the transport of water with variable tem- perature past the site by deep ocean currents. Find the amplitude of water temperature variations that cause surface heat flux variations of 40 mW m−2 above and below the mean on a time scale of 1 day. As- sume that the thermal conductivity of sediments is 0.8 W m−1 K−1 and the sediment thermal diffusivity is 0.2 mm2 s−1.

PROBLEM 4–33 One way of determining the effects of erosion on subsurface temperatures is to consider the instantaneous removal of a thickness l of ground. Prior to the removal T = T0 + β y, where y is the depth, β is the geothermal gradient, and T0 is the surface temperature. After removal, the new surface is main- tained at temperature T0. Show that the subsurface temperature after the removal of the surface layer is given by

How is the surface heat flow affected by the removal of surface material?

PROBLEM 4–39 One of the estimates for the age of the Earth given by Lord Kelvin in the 1860s assumed that Earth was initially molten at a constant tem- perature Tm and that it subsequently cooled by con- duction with a constant surface temperature T0. The age of the Earth could then be determined from the present surface thermal gradient ( dT/ dy)0. Re- produce Kelvin’s result assuming Tm T0 = 1700 K, c=1 kJ kg−1 K−1, L =400 kJ kg−1, κ =1 mm2 s−1, and ( dT/ dy)0 = 25 K km−1 . In addition, determine the thickness of the solidified lithosphere. Note: Since the solidified layer is thin compared with the Earth’s radius, the curvature of the surface may be neglected.

PROBLEM4–43 Themantlerocksoftheasthenosphere from which the lithosphere forms are expected to contain a small amount of magma. If the mass frac- tion of magma is 0.05, determine the depth of the lithosphere–asthenosphere boundary for oceanic li- thosphere with an age of 60 Ma. Assume L = 400 kJ kg−1, c=1kJkg−1 K−1, Tm =1600K, T0 =275K,and κ = 1 mm2 s−1.

PROBLEM 4–52 The ocean ridges are made up of a series of parallel segments connected by transform faults, as shown in Figure 1–13. Because of the dif- ference of age there is a vertical offset on the fracture zones. Assuming the theory just derived is applica- ble, what is the vertical offset ( a) at the ridge crest and ( b) 100 km from the ridge crest in Figure 4–46 (ρ m = 3300kgm−3,κ=1mm2s−1,α v =3×10−5 K−1, T1− T0=1300K, u=50mmyr−1).

PROBLEM 5–7 What is the value of the acceleration of gravity at a distance b above the geoid at the equator ( b a)?

PROBLEM 5–18 A volcanic plug of diameter 10 km has a gravity anomaly of 0.3 mm s−2. Estimate the depth of the plug assuming that it can be modeled by a verti- cal cylinder whose top is at the surface. Assume that the plug has density of 3000 kg m−3 and the rock it intrudes has a density of 2800 kg m−3.

,

Geodynamics

First published in 1982, Don Turcotte and Jerry Schubert’s Geodynamics be-

came a classic textbook for several generations of students of geophysics and

geology. In this second edition, the authors bring this classic text completely

up-to-date. Important additions include a chapter on chemical geodynamics,

an updated coverage of comparative planetology based on recent planetary

missions, and a variety of other new topics.

Geodynamics provides the fundamentals necessary for an understanding of

the workings of the solid Earth. The Earth is a heat engine, with the source

of the heat the decay of radioactive elements and the cooling of the Earth

from its initial accretion. The work output includes earthquakes, volcanic

eruptions, and mountain building. Geodynamics comprehensively explains

these concepts in the context of the role of mantle convection and plate

tectonics. Observations such as the Earth’s gravity field, surface heat flow,

distribution of earthquakes, surface stresses and strains, and distribution of

elements are discussed. The rheological behavior of the solid Earth, from an

elastic solid to fracture to plastic deformation to fluid flow, is considered.

Important inputs come from a comparison of the similarities and differences

between the Earth, Venus, Mars, Mercury, and the Moon. An extensive set

of student exercises is included.

This new edition of Geodynamics will once again prove to be a classic

dents in geology, geophysics, and Earth science.

Donald L. Turcotte is Maxwell Upson Professor of Engineering, Depart-

ment of Geological Sciences, Cornell University. In addition to this book, he

is author or co-author of 3 books and 276 research papers, including Fractals

and Chaos in Geology and Geophysics (Cambridge University Press, 1992

and 1997) and Mantle Convection in the Earth and Planets (with Gerald

Schubert and Peter Olson; Cambridge University Press, 2001). Professor

Turcotte is a Fellow of the American Geophysical Union, Honorary Fellow

of the European Union of Geosciences, and Fellow of the Geological So-

ciety of America. He is the recipient of several medals, including the Day

Medal of the Geological Society of America, the Wegener Medal of the Euro-

pean Union of Geosciences, the Whitten Medal of the American Geophysical

Union, the Regents (New York State) Medal of Excellence, and Caltech’s

Distinguished Alumnus Award. Professor Turcotte is a member of the Na-

tional Academy of Sciences and the American Academy of Arts and Sciences.

Gerald Schubert is a Professor in the Department of Earth and Space

Sciences and the Institute of Geophysics and Planetary Physics at the Uni-

iv

versity of California, Los Angeles. He is co-author with Donald Turcotte

and Peter Olson of Mantle Convection in the Earth and Planets (Cambridge

University Press, 2001), and author of over 400 research papers. He has par-

ticipated in a number of NASA’s planetary missions and has been on the

editorial boards of many journals, including Icarus, Journal of Geophysical

Research, Geophysical Research Letters, and Annual Reviews of Earth and

Planetary Sciences. Professor Schubert is a Fellow of the American Geo-

physical Union and a recipient of the Union’s James B. MacElwane medal.

He is a member of the American Academy of Arts and Sciences.

Contents

Preface page x

Preface to the Second Edition xiii

1 Plate Tectonics 1

1.1 Introduction 1

1.2 The Lithosphere 9

1.3 Accreting Plate Boundaries 10

1.4 Subduction 15

1.5 Transform Faults 23

1.6 Hotspots and Mantle Plumes 25

1.7 Continents 30

1.8 Paleomagnetism and the Motion of the Plates 36

1.9 Triple Junctions 59

1.10 The Wilson Cycle 65

1.11 Continental Collisions 70

1.12 Volcanism and Heat Flow 76

1.13 Seismicity and the State of Stress in the Lithosphere 85

1.14 The Driving Mechanism 90

1.15 Comparative Planetology 91

1.16 The Moon 92

1.17 Mercury 97

1.18 Mars 99

1.19 Phobos and Deimos 105

1.20 Venus 105

1.21 The Galilean Satellites 107

2 Stress and Strain in Solids 127

2.1 Introduction 127

2.2 Body Forces and Surface Forces 128

vi Contents

2.3 Stress in Two Dimensions 140

2.4 Stress in Three Dimensions 146

2.5 Pressures in the Deep Interiors of Planets 148

2.6 Stress Measurement 151

2.7 Basic Ideas about Strain 154

2.8 Strain Measurements 167

3 Elasticity and Flexure 185

3.1 Introduction 185

3.2 Linear Elasticity 187

3.3 Uniaxial Stress 189

3.4 Uniaxial Strain 191

3.5 Plane Stress 193

3.6 Plane Strain 196

3.7 Pure Shear and Simple Shear 197

3.8 Isotropic Stress 198

3.9 Two-Dimensional Bending or Flexure of Plates 199

3.10 Bending of Plates under Applied Moments and Vertical

3.11 Buckling of a Plate under a Horizontal Load 210

3.12 Deformation of Strata Overlying an Igneous Intrusion 212

3.13 Application to the Earth’s Lithosphere 216

3.15 Stability of the Earth’s Lithosphere Under an End Load 220

3.16 Bending of the Elastic Lithosphere under the Loads of

Island Chains 222

3.17 Bending of the Elastic Lithosphere at an Ocean Trench 227

3.18 Flexure and the Structure of Sedimentary Basins 230

4 Heat Transfer 237

4.1 Introduction 237

4.2 Fourier’s Law of Heat Conduction 238

4.3 Measuring the Earth’s Surface Heat Flux 240

4.4 The Earth’s Surface Heat Flow 242

4.5 Heat Generation by the Decay of Radioactive Elements 244

4.6 One-Dimensional Steady Heat Conduction 249

4.7 A Conduction Temperature Profile for the Mantle 253

4.8 Continental Geotherms 254

4.9 Radial Heat Conduction in a Sphere or Spherical Shell 260

4.10 Temperatures in the Moon 263

4.11 Steady Two- and Three-Dimensional Heat Conduction 264

Contents vii

4.12 Subsurface Temperature 266

4.13 One-Dimensional, Time-Dependent Heat Conduction 269

4.14 Periodic Heating of a Semi-Infinite Half-Space 271

4.15 Instantaneous Heating or Cooling of a Semi-Infinite

Half-Space 276

4.16 Cooling of the Oceanic Lithosphere 285

4.17 Plate Cooling Model of the Lithosphere 290

4.18 The Stefan Problem 294

4.19 Solidification of a Dike or Sill 300

4.20 The Heat Conduction Equation in a Moving Medium 304

4.21 One-Dimensional, Unsteady Heat Conduction in an

Infinite Region 307

4.22 Thermal Stresses 310

4.23 Ocean Floor Topography 317

4.24 Changes in Sea Level 323

4.25 Thermal and Subsidence History of Sedimentary Basins 325

4.26 Heating or Cooling a Semi-Infinite Half-Space 333

4.27 Frictional Heating on Faults 335

4.28 Mantle Geotherms and Adiabats 337

4.29 Thermal Structure of the Subducted Lithosphere 345

4.30 Culling Model for the Erosion and Deposition of Sediments 348

5 Gravity 354

5.1 Introduction 354

5.2 Gravitational Acceleration 355

5.3 Centrifugal Acceleration and the Acceleration of Gravity 365

5.4 The Gravitational Potential and the Geoid 366

5.5 Moments of Inertia 373

5.6 Surface Gravity Anomalies 378

5.7 Bouguer Gravity Formula 383

5.8 Reductions of Gravity Data 385

5.9 Compensation 387

5.10 The Gravity Field of a Periodic Mass Distribution on a

Surface 389

5.11 Compensation Due to Lithospheric Flexure 391

5.12 Isostatic Geoid Anomalies 394

5.13 Compensation Models and Observed Geoid Anomalies 397

5.14 Forces Required to Maintain Topography and the Geoid 405

6 Fluid Mechanics 411

6.1 Introduction 411

viii Contents

6.2 One-Dimensional Channel Flows 412

6.3 Asthenospheric Counterflow 418

6.4 Pipe Flow 421

6.5 Artesian Aquifer Flows 425

6.6 Flow Through Volcanic Pipes 426

6.7 Conservation of Fluid in Two Dimensions 427

6.8 Elemental Force Balance in Two Dimensions 428

6.9 The Stream Function 432

6.10 Postglacial Rebound 434

6.11 Angle of Subduction 442

6.12 Diapirism 447

6.13 Folding 456

6.14 Stokes Flow 467

6.15 Plume Heads and Tails 476

6.16 Pipe Flow with Heat Addition 481

6.17 Aquifer Model for Hot Springs 485

6.18 Thermal Convection 488

6.19 Linear Stability Analysis for the Onset of Thermal

Convection 492

6.20 A Transient Boundary-Layer Theory 500

6.21 A Steady-State Boundary-Layer Theory 505

6.22 The Forces that Drive Plate Tectonics 516

6.23 Heating by Viscous Dissipation 521

6.24 Mantle Recycling and Mixing 525

7 Rock rheology 538

7.1 Introduction 538

7.2 Elasticity 540

7.3 Diffusion Creep 553

7.4 Dislocation Creep 568

7.5 Shear Flows of Fluids 574

7.6 Mantle Rheology 588

7.7 Rheological Effects on Mantle Convection 597

7.8 Mantle Convection and the Cooling of the Earth 599

7.9 Crustal Rheology 605

7.10 Viscoelasticity 609

7.11 Elastic–Perfectly Plastic Behavior 615

8 Faulting 627

8.1 Introduction 627

8.2 Classification of Faults 628

Contents ix

8.3 Friction on Faults 632

8.4 Anderson Theory of Faulting 637

8.5 Strength Envelope 642

8.6 Thrust Sheets and Gravity Sliding 643

8.7 Earthquakes 647

8.8 San Andreas Fault 659

8.9 North Anatolian Fault 664

8.10 Some Elastic Solutions for Strike–Slip Faulting 667

8.11 Stress Diffusion 679

8.12 Thermally Activated Creep on Faults 682

9 Flows in Porous Media 692

9.1 Introduction 692

9.2 Darcy’s Law 693

9.3 Permeability Models 695

9.4 Flow in Confined Aquifers 697

9.5 Flow in Unconfined Aquifers 700

9.6 Geometrical Form of Volcanoes 717

9.7 Equations of Conservation of Mass, Momentum, and

Energy for Flow in Porous Media 722

9.8 One-Dimensional Advection of Heat in a Porous Medium 725

9.9 Thermal Convection in a Porous Layer 729

9.10 Thermal Plumes in Fluid-Saturated Porous Media 735

9.11 Porous Flow Model for Magma Migration 746

9.12 Two-Phase Convection 752

10 Chemical Geodynamics 761

10.1 Introduction 761

10.3 Geochemical Reservoirs 771

10.4 A Two-Reservoir Model with Instantaneous Crustal

Differentiation 776

10.5 Noble Gas Systems 786

10.6 Isotope Systematics of OIB 788

Appendix A Symbols and Units 795

Appendix B Physical Constants and Properties 806

Appendix C Answers to Selected Problems 815

Index 828

Preface

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

In all cases we present the material with a minimum of mathematical

complexity. We have not introduced mathematical concepts unless they are

essential to the understanding of physical principles. For example, our treat-

ment of elasticity and fluid mechanics avoids the introduction or use of

tensors. We do not believe that tensor notation is necessary for the under-

standing 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 explana-

tions for such important physical properties of matter as solid-state viscosity,

thermal coefficient of expansion, specific heat, and permeability. Basic con-

cepts 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.

The problems are an integral part of this textbook. It is only through

Preface xi

solving a substantial number of exercises that an adequate understanding

of the underlying physical principles can be developed. Answers to selected

problems are provided.

The first chapter reviews plate tectonics; its main purpose is to provide

physics, chemistry, and engineering students with the geological background

necessary to understand the applications considered throughout the rest of

the textbook. We hope that the geology student can also benefit from this

summary of numerous geological, seismological, and paleomagnetic observa-

tions. Since plate tectonics is a continuously evolving subject, this material

may be subject to revision. Chapter 1 also briefly summarizes the geologi-

cal and geophysical characteristics of the other planets and satellites of the

solar system. Chapter 2 introduces the concepts of stress and strain and dis-

cusses the measurements of these quantities in the Earth’s crust. Chapter 3

presents the basic principles of linear elasticity. The bending of thin elastic

plates is emphasized and is applied to problems involving the bending of the

Earth’s lithosphere. Chapter 4 deals mainly with heat conduction and the

application of this theory to temperatures in the continental crust and the

continental and oceanic lithospheres. Heat transfer by convection is briefly

discussed and applied to a determination of temperature in the Earth’s man-

tle. Surface heat flow measurements are reviewed and interpreted in terms

of the theory. The sources of the Earth’s surface heat flow are discussed.

Problems involving the solidification of magmas and extrusive lava flows are

also treated. The basic principles involved in the interpretation of gravity

measurements are given in Chapter 5. Fluid mechanics is studied in Chapter

6; problems involving mantle convection and postglacial rebound are empha-

sized. Chapter 7 deals with the rheology of rock or the manner in which it

deforms or flows under applied forces. Fundamental processes are discussed

from a microscopic point of view. The mechanical behavior of faults is dis-

cussed in Chapter 8 with particular attention being paid to observations of

displacements along the San Andreas fault. Finally, Chapter 9 discusses the

principles of fluid flow in porous media, a subject that finds application to

hydrothermal circulations in the oceanic crust and in continental geothermal

areas.

The contents of this textbook are intended to provide the material for a

coherent one-year course. In order to accomplish this goal, some important

aspects of geodynamics have had to be omitted. In particular, the fundamen-

tals of seismology are not included. Thus the wave equation and its solutions

are not discussed. Many seismic studies have provided important data rele-

vant to geodynamic processes. Examples include (1) the radial distribution

of density in the Earth as inferred from the radial profiles of seismic veloci-

xii Preface

ties, (2) important information on the locations of plate boundaries and the

locations of descending plates at ocean trenches provided by accurate deter-

minations of the epicenters of earthquakes, and (3) details of the structure

of the continental crust obtained by seismic reflection profiling using arti-

ficially generated waves. An adequate treatment of seismology would have

required a very considerable expansion of this textbook. Fortunately, there

are a number of excellent textbooks on this subject.

A comprehensive study of the spatial and temporal variations of the

Earth’s magnetic field is also considered to be outside the scope of this

textbook. A short discussion of the Earth’s magnetic field relevant to pale-

omagnetic observations is given in Chapter 1. However, mechanisms for the

generation of the Earth’s magnetic field are not considered.

In writing this textbook, several difficult decisions had to be made. One

was the choice of units; we use SI units throughout. This system of units

is defined in Appendix 1. We feel there is a strong trend toward the use of

SI units in both geology and geophysics. We recognize, however, that many

cgs units are widely used. Examples include µcal cm−2 s−1 for heat flow,

kilobar for stress, and milligal for gravity anomalies. For this reason we have

often included the equivalent cgs unit in parentheses after the SI unit, for

example, MPa (kbar). Another decision involved the referencing of original

work. We do not believe that it is appropriate to include a large number of

references in a basic textbook. We have credited those individuals making

major contributions to the development of the theory of plate tectonics and

continental drift in our brief discussion of the history of this subject in

Chapter 1. We also provide references to data. At the end of each chapter a

list of recommended reading is given. In many instances these are textbooks

and reference books, but in some cases review papers are included. In each

case the objective is to provide background material for the chapter or to

extend its content.

Many of our colleagues have read all or parts of various drafts of this

textbook. We acknowledge the contributions made by Jack Bird, Peter Bird,

Muawia Barazangi, Allan Cox, Walter Elsasser, Robert Kay, Suzanne Kay,

Mark Langseth, Bruce Marsh, Jay Melosh, John Rundle, Sean Solomon,

David Stevenson, Ken Torrance, and David Yuen. We particularly wish to

acknowledge the many contributions to our work made by Ron Oxburgh

and the excellent manuscript preparation by Tanya Harter.

Preface to the Second Edition

As we prepared our revisions for this second edition of Geodynamics we were

struck by the relatively few changes and additions that were required. The

reason is clear: this textbook deals with fundamental physical processes that

do not change. However, a number of new ideas and concepts have evolved

and have been included where appropriate.

In revising the first chapter on plate tectonics we placed added emphasis

on the concept of mantle plumes. In particular we discussed the association

of plume heads with continental flood basalts. We extensively revised the

sections on comparative planetology. We have learned new things about the

Moon, and the giant impact hypothesis for its origin has won wide accep-

tance. For Venus, the Magellan mission has revolutionized our information

ity data have provided new insights that emphasize the tremendous differ-

ences in structure and evolution between Venus and the Earth. Similarly,

the Galileo mission has greatly enhanced our understanding of the Galilean

satellites of Jupiter.

In Chapter 2 we introduce the crustal stretching model for the isostatic

subsidence of sedimentary basins. This model provides a simple explanation

for the formation of sedimentary basins. Space-based geodetic observations

have revolutionized our understanding of surface strain fields associated with

tectonics. We introduce the reader to satellite data obtained from the global

positioning system (GPS) and synthetic aperture radar interferometry (IN-

SAR). In Chapter 4 we introduce the plate cooling model for the thermal

structure of the oceanic lithosphere as a complement to the half-space cool-

ing model. We also present in this chapter the Culling model for the diffu-

sive erosion and deposition

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