Physical oceanography

C830X

Physical

Oceanography

Physical

Oceanography

A Mathematical Introduction

with MATLAB®

Reza Malek-Madani

Malek-Madani

Mathematics

Physical Oceanography: A Mathematical Introduction with MATLAB®

demonstrates how to use the basic tenets of multivariate calculus to

derive the governing equations of fluid dynamics in a rotating frame.

It also explains how to use linear algebra and partial differential

equations to solve basic initial-boundary value problems that have

become the hallmark of physical oceanography. The book makes the

most of MATLAB’s matrix algebraic functions, differential equation

solvers, and visualization capabilities.

Focusing on the interplay between applied mathematics and

geophysical fluid dynamics, the text presents fundamental analytical

and computational tools necessary for modeling ocean currents. In

physical oceanography, the fluid flows of interest occur on a planet

that rotates; this rotation can balance the forces acting on the fluid

particles in such a delicate fashion to produce exquisite phenomena,

such as the Gulf Stream, the Jet Stream, and internal waves. It is

precisely because of the role that rotation plays in oceanography

that the field is fundamentally different from the rectilinear fluid flows

typically observed and measured in laboratories. Much of this text

discusses how the existence of the Gulf Stream can be explained

by the proper balance among the Coriolis force, wind stress, and

molecular frictional forces.

Through the use of MATLAB, the author takes a fresh look at advanced

topics and fundamental problems that define physical oceanography

today. He presents research from pioneers in the mathematical

modeling of physical oceanography and covers the current work

of applied mathematicians who are making significant impacts on

developing tools for modern physical oceanography.

C830X_Cover.indd 1 3/19/12 9:43 AM

Physical

Oceanography

A Mathematical Introduction

with MATLAB®

C830X_FM.indd 1 3/26/12 1:37 PM

This page intentionally left blank

Physical

Oceanography

A Mathematical Introduction

with MATLAB®

Reza Malek-Madani

C830X_FM.indd 3 3/26/12 1:37 PM

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does

not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MAT￾LAB® software or related products does not constitute endorsement or sponsorship by The MathWorks

of a particular pedagogical approach or particular use of the MATLAB® software.

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2012 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Version Date: 20120229

International Standard Book Number-13: 978-1-4398-9829-1 (eBook - PDF)

This book contains information obtained from authentic and highly regarded sources. Reasonable

efforts have been made to publish reliable data and information, but the author and publisher cannot

assume responsibility for the validity of all materials or the consequences of their use. The authors and

publishers have attempted to trace the copyright holders of all material reproduced in this publication

and apologize to copyright holders if permission to publish in this form has not been obtained. If any

copyright material has not been acknowledged please write and let us know so we may rectify in any

future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,

transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or

hereafter invented, including photocopying, microfilming, and recording, or in any information stor￾age or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copy￾right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222

Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro￾vides licenses and registration for a variety of users. For organizations that have been granted a pho￾tocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

to Jo, Behzad, and Darob

and

to all my students

This page intentionally left blank

Contents

Preface xiii

1 An Introduction to MATLAB R 1

1.1 A Session in MATLAB . . . . . . . . . . . . . . . . . . 1

1.2 Operations .*, ./ , and .^ . . . . . . . . . . . . . . . 4

1.3 Defining and Plotting Functions in MATLAB . . . . . . 9

1.4 3-Dimensional Plotting . . . . . . . . . . . . . . . . . . 16

1.5 M-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Loops and Iterations in MATLAB . . . . . . . . . . . . 20

1.7 Conditional Statements in MATLAB . . . . . . . . . . 24

1.8 Fourier Series in MATLAB . . . . . . . . . . . . . . . . 28

1.9 Solving Differential Equations . . . . . . . . . . . . . . 36

1.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . 38

1.11 References . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Matrix Algebra 41

2.1 Vectors and Matrices . . . . . . . . . . . . . . . . . . . 41

2.2 Vector Operations . . . . . . . . . . . . . . . . . . . . . 43

2.3 Matrix Operations . . . . . . . . . . . . . . . . . . . . . 45

2.4 Linear Spaces and Subspaces . . . . . . . . . . . . . . . 53

2.5 Determinant and Inverse of Matrices . . . . . . . . . . 57

2.6 Computing A−1 Using Co-Factors . . . . . . . . . . . . 67

2.7 Linear Independence, Span, Basis, and Dimension . . . 70

2.8 Linear Transformations . . . . . . . . . . . . . . . . . . 75

2.9 Row Reduction and Gaussian Elimination . . . . . . . . 77

2.10 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . 82

2.11 Project A: Taylor Polynomials and Series . . . . . . . . 88

2.12 Project B: A Differentiation Matrix . . . . . . . . . . . 90

2.13 Project C: Spectral Method and Matrices . . . . . . . . 93

2.14 Concluding Remarks . . . . . . . . . . . . . . . . . . . 94

2.15 References and Further Reading . . . . . . . . . . . . . 95

vii

viii

3 Differential and Integral Calculus 97

3.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.2 Taylor Polynomial and Series . . . . . . . . . . . . . . . 100

3.3 Functions of Several Variables and Vector Fields . . . . 104

3.4 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5 Curl and Vector Fields . . . . . . . . . . . . . . . . . . 115

3.6 Integral Theorems . . . . . . . . . . . . . . . . . . . . . 117

3.7 References and Further Reading . . . . . . . . . . . . . 122

4 Ordinary Differential Equations 123

4.1 Linear Independence and Space of Functions . . . . . . 123

4.2 Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3 General Systems of ODEs . . . . . . . . . . . . . . . . . 132

4.4 MATLAB’s ode45 . . . . . . . . . . . . . . . . . . . . . 135

4.5 Asymptotic Behavior and Linearization . . . . . . . . . 139

4.6 Motion of Parcels of Fluid in MATLAB . . . . . . . . . 145

4.7 Project A: Ekman Layer . . . . . . . . . . . . . . . . . 149

4.8 Project B: Lorenz 96 Model . . . . . . . . . . . . . . . 150

4.9 References . . . . . . . . . . . . . . . . . . . . . . . . . 152

5 Numerical Methods for ODEs 153

5.1 Finite Difference Methods . . . . . . . . . . . . . . . . . 153

5.2 Backward Euler Method (BEM) . . . . . . . . . . . . . 163

5.3 Stability of Numerical Methods . . . . . . . . . . . . . 168

5.4 Stability Analysis of Numerical Schemes . . . . . . . . 170

5.5 MATLAB Programs for the Forward Finite Difference

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.6 Stability Analysis of Numerical Schemes (continued) . . 177

5.7 Truncation Error . . . . . . . . . . . . . . . . . . . . . . 180

5.8 Boundary Value Problems and the Shooting Method . . 182

5.9 Project A: Modified Euler Method . . . . . . . . . . . . 186

5.10 Project B: Runge–Kutta Methods . . . . . . . . . . . . 189

5.11 Project C: Finite Difference Methods and BVPs . . . . 193

5.12 Project D: Method of Lines . . . . . . . . . . . . . . . . 197

5.13 Project E: Burgers Equation (Method of Characteristics) 200

5.14 Project F: Burgers Equation (Method of Characteristics

– Nonlinear Case) . . . . . . . . . . . . . . . . . . . . . 204

5.15 Project G: Burgers Equation (Formation of Singularities) 206

5.16 Project H: Burgers Equation and the Method of Lines . 208

5.17 References . . . . . . . . . . . . . . . . . . . . . . . . . 211

ix

6 Equations of Fluid Dynamics 213

6.1 Flow Representations — Eulerian and Lagrangian . . . 214

6.2 Deformation Gradient and Conservation of Mass . . . . 219

6.3 Derivation of Equation of Conservation of Mass—A

Heuristic Approach . . . . . . . . . . . . . . . . . . . . 226

6.4 Stream Function and Vector Fields A, B, C, and ABC 230

6.5 Acceleration in Rectangular Coordinates . . . . . . . . 240

6.6 Strain-Rate Matrix and Vorticity . . . . . . . . . . . . . 245

6.7 Internal Forces and Cauchy Stress . . . . . . . . . . . . 251

6.8 Euler and Navier–Stokes Equations . . . . . . . . . . . 254

6.9 Bernoulli’s Equation and Irrotational Flows . . . . . . . 257

6.10 Acceleration in Spherical Coordinates . . . . . . . . . . 260

6.10.1 Coordinate Curves . . . . . . . . . . . . . . . . . 260

6.10.2 Spherical Basis . . . . . . . . . . . . . . . . . . . 262

6.10.3 The Eulerian Formulation of Velocity and Accel￾eration Revisited . . . . . . . . . . . . . . . . . . 264

6.10.4 Velocity in Spherical Basis . . . . . . . . . . . . . 265

6.10.5 Dynamics of Basis Vectors . . . . . . . . . . . . . 267

6.10.6 Formula for Acceleration in Spherical Coordinates 267

6.11 Project A: Inviscid Linear Fluid Motions and Surface

Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . 268

6.12 Project B: Internal Gravity Waves . . . . . . . . . . . . 272

6.13 Project C: Equation for Bubble Dynamics . . . . . . . 274

6.14 Project D: Chaotic Transport . . . . . . . . . . . . . . 276

6.15 References . . . . . . . . . . . . . . . . . . . . . . . . . 279

7 Equations of Geophysical Fluid Dynamics 281

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 281

7.2 Coriolis . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

7.3 Coriolis Acceleration: 2Ω × vr . . . . . . . . . . . . . . 284

7.4 Gradient Operator in Spherical Coordinates . . . . . . 285

7.5 Navier–Stokes Equation in a Rotating Frame . . . . . . 287

7.6 β-Plane Approximation . . . . . . . . . . . . . . . . . . 287

7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . 289

8 Shallow Water Equations (SWE) 291

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 291

8.2 Derivation of Equations . . . . . . . . . . . . . . . . . . 291

8.3 Rotating Shallow Water Equations (RSWE) . . . . . . 298

8.4 Some Exact Solutions of the RSWE . . . . . . . . . . . 302

8.5 Linearization of SWE . . . . . . . . . . . . . . . . . . . 303

8.6 Linear Wave Equation . . . . . . . . . . . . . . . . . . . 305

8.7 Separation of Variables and the Fourier Method . . . . 306

x

8.8 Fourier Method in MATLAB . . . . . . . . . . . . . . . 312

8.9 Method of Characteristics . . . . . . . . . . . . . . . . . 316

8.10 D’Alembert’s Solution in MATLAB . . . . . . . . . . . 320

8.11 Method of Lines and Wave Equation . . . . . . . . . . 323

8.12 Project A: Method of Characteristics for General PDEs 326

8.13 Project B: Variations on the Method of Lines . . . . . . 329

8.14 Project C: An Inverse Problem . . . . . . . . . . . . . . 330

8.15 Project D: Exact Solutions of the Rotating Shallow Water

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 333

8.16 Project E: Courant–Friedrichs–Lewy Condition . . . . . 336

8.17 References . . . . . . . . . . . . . . . . . . . . . . . . . 339

9 Wind-Driven Ocean Circulation: Stommel and Munk

Models 341

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 341

9.2 Flow in a Rectangular Bay — Normal Modes . . . . . . 342

9.3 Eigenfunctions of the Laplace Operator . . . . . . . . . 350

9.4 Poisson Equation . . . . . . . . . . . . . . . . . . . . . 355

9.4.1 Poisson Equation with Localized Vorticity . . . . 361

9.5 Stommel Model . . . . . . . . . . . . . . . . . . . . . . 364

9.5.1 Governing PDE . . . . . . . . . . . . . . . . . . . 365

9.5.2 Non-Dimensionalization . . . . . . . . . . . . . . 370

9.5.3 Solution to the BVP . . . . . . . . . . . . . . . . 372

9.5.3.1 Determining the Particular Solution ψp 373

9.5.3.2 Determining the Homogeneous Solution

ψh

. . . . . . . . . . . . . . . . . . . . . 373

9.5.3.3 Applying the Boundary Conditions . . 374

9.6 MATLAB Programs . . . . . . . . . . . . . . . . . . . . 376

9.7 Stommel Model—A Numerical Approach . . . . . . . . 381

9.7.1 Constructing the System AΨ = B . . . . . . . . 385

9.8 MATLAB Program for the Stommel Model . . . . . . . 389

9.9 Munk Model of Wind-Driven Circulation . . . . . . . . 394

9.10 Project A: Stommel Model with a Nonuniform Mesh . . 402

9.11 Project B: Munk Model and the Finite Difference Method 403

9.12 Project C: Galerkin Method and the B. Saltzman and E.

Lorenz Equations . . . . . . . . . . . . . . . . . . . . . 405

9.13 References . . . . . . . . . . . . . . . . . . . . . . . . . 408

10 Some Special Topics 409

10.1 Finite-Time Dynamical Systems . . . . . . . . . . . . . 410

10.2 Data Assimilation and Filtering . . . . . . . . . . . . . 413

10.3 Normal Modes and Data . . . . . . . . . . . . . . . . . 416

10.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 417

xi

10.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 418

Appendix: Answers to Selected Problems 421

Index 437

This page intentionally left blank

Preface

This book is about the interplay between applied mathematics and the

field of geophysical fluid dynamics. Its primary goals are to demonstrate

how one uses the basic tenets of multivariate calculus to derive the gov￾erning equations of fluid dynamics in a rotating frame, and how one

uses methods from linear algebra and partial differential equations to

solve some of the basic initial-boundary value problems that have be￾come the hallmark of physical oceanography. MATLAB R

is the key tool

used throughout the book. Special care has been taken to take advan￾tage of this software’s matrix algebraic functions, its differential equation

solvers, and its visualization capabilities in almost every section of every

chapter. In fact, it is the use of MATLAB that allows us to consider this

highly interdisciplinary material at a level that I hope is accessible to an

undergraduate student.

The book is intended for advanced undergraduates, those who have

already completed courses in calculus, differential equations and linear

algebra. Despite requiring these prerequisites, several of the early chap￾ters are dedicated to reviewing materials from these three topics, with

varying degrees of depth and completion. While the material in these

chapters may be familiar to the reader, basic use of MATLAB as well

as simple examples that introduce features of fluid flows populate the

illustrations and exercises.

Physical oceanography, at least the part of it that we are concerned

with in this book, is characterized by the fact that the fluid flows of

interest are occurring on a planet that rotates, and that this rotation

can balance the forces acting on the fluid particles in such a delicate

fashion to produce exquisite phenomena such as the Gulf Stream, the

Jet Stream, internal waves, and the Madden–Julian Oscillation, to name

a few. Much of the development in this book is motivated by the desire

to explain how the existence of the Gulf Stream can be explained by the

proper balance between the Coriolis force, wind stress, and molecular

frictional forces. It is precisely because of the role that rotation plays in

oceanography that this field is fundamentally different from rectilinear

fluid flows, flows that we typically observe and measure in laboratories.

Although the Coriolis effect is part of our daily experience, it is dif￾xiii

xiv

ficult for most of us to develop an intuitive sense for its impact on the

behavior of motion of particles, fluid or solid. After all, the measure￾ments we make are most often carried out on the planet itself and are

therefore relative to a rotating frame. By contrast, laboratory observa￾tions are made in an inertial frame, at least at time scales that are much

faster than the time scale associated with the rotation of our planet. It

is because of this lack of familiarity that it is difficult to appreciate the

“apparent” forces that the planet is exerting on us, unless we float un￾tethered for several days (as icebergs do), an experience that most of us

have not had. It was therefore a particularly noteworthy moment when

the early practitioners of physical oceanography finally sorted out how

a current such as the Gulf Stream comes about and remains relatively

stable for centuries. The ten years between 1945 and 1955 form a period

when some of the most exciting applications of mathematics appeared in

physical oceanography; the seminal papers of H. Stommel in 1948 and

W. Munk in 1950, on the “western intensification” of ocean currents,

ushered in a new era of applications of mathematics, which is the focus

of this book.

The material in this book is not exhaustive, neither in mathematical

methods nor in oceanographic topics. Our goal has been instead to con￾centrate on introducing a set of applications that are motivated by some

of the questions we would like to investigate about our environment,

and the type of questions where mathematics could play a critical role

in their investigation. The choice of topics in many of the chapters was

motivated by the desire to direct students to topics that have appeared

in research manuscripts, most as journal articles published in the past

few decades, and, by providing the basic mathematical tools, to invite

students to begin to consult and read some of these arcticles. The new

twist for us is the availability of MATLAB, which enables us to take

a fresh look at many of the fundamental problems that define physical

oceanography today.

Most chapters in the book contain a few projects. All projects have

a significant component of MATLAB programming in them. Our hope

is that these projects may suggest templates for capstone projects or

honors theses for those students who are inclined to pursue a special

project in applied mathematics. Most of these projects, in one way or

another, are influenced by some aspect of research presented by the

founders of mathematical modeling in physical oceanography, starting

with the aforementioned Stommel amd Munk, but also works by G.

Veronis, E. Lorenz, J. G. Charney, J. Pedlosky, A. Robinson, and A. Gill,

to name a few. I believe their writing styles are accessible and inviting.

Students of mathematics can benefit enormously from spending valuable