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# Physical Oceanography A Mathematical Introduction with MATLAB

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ISBN 9781584888307
Published April 20, 2012 by Chapman and Hall/CRC
456 Pages - 85 B/W Illustrations

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## Book Description

Accessible to advanced undergraduate students, 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 (PDEs) 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. The projects in each chapter incorporate a significant component of MATLAB programming. These projects can be used as capstone projects or honors theses for students inclined to pursue a special project in applied mathematics.

An Introduction to MATLAB
A Session on MATLAB
The Operations *, / , and ^
Defining and Plotting Functions in MATLAB
3-Dimensional Plotting
M-files
Loops and Iterations in MATLAB
Conditional Statements in MATLAB
Fourier Series in MATLAB
Solving Differential Equations
Concluding Remarks

Matrix Algebra
Vectors and Matrices
Vector Operations
Matrix Operations
Linear Spaces and Subspaces
Determinant and Inverse of Matrices
Computing A−1 Using Co-Factors
Linear Independence, Span, Basis and Dimension
Linear Transformations
Row Reduction and Gaussian Elimination
Eigenvalues and Eigenvectors
Project A: Taylor Polynomials and Series
Project B: A Differentiation Matrix
Project C: Spectral Method and Matrices
Concluding Remarks

Differential and Integral Calculus
Derivative
Taylor Polynomial and Series
Functions of Several Variables and Vector Fields
Divergence
Curl and Vector Fields
Integral Theorems

Ordinary Differential Equations (ODEs)
Linear Independence and Space of Functions
Linear ODEs
General Systems of ODEs
MATLAB’s ode45
Asymptotic Behavior and Linearization
Motion of Parcels of Fluid in MATLAB

Numerical Methods for ODEs
Finite Difference Methods
The Backward Euler Method (BEM)
Stability of Numerical Methods
Stability Analysis of Numerical Schemes
MATLAB Programs for the Forward Finite Difference Method
Stability Analysis of Numerical Schemes (continued)
Truncation Error
Boundary Value Problems and the Shooting Method
Project A: Modified Euler Method
Project B: Runge–Kutta Methods
Project C: Finite Difference Methods and BVPs
Project D: The Method of Lines
Project E: Burgers Equation (Method of Characteristics)
Project F: Burgers Equation (Method of Characteristics—Nonlinear Case)
Project G: Burgers Equation (Formation of Singularities)
Project H: Burgers Equation and the Method of Lines (MOL)

Equations of Fluid Dynamics
Flow Representations—Eulerian and Lagrangian
Deformation Gradient and Conservation of Mass
Derivation of Equation of Conservation of Mass—A Heuristic Approach
Stream Function and Vector Fields A, B, C, and ABC
Acceleration in Rectangular Coordinates
Strain-Rate Matrix and Vorticity
Internal Forces and the Cauchy Stress
Euler and Navier–Stokes Equations
Bernoulli’s Equation and Irrotational Flows
Acceleration in Spherical Coordinates
Project A: Inviscid Linear Fluid Motions and Surface Gravity Waves
Project B: Equations of Motion for Bubbles
Project C: Chaotic Transport

Equations of Geophysical Fluid Dynamics
Introduction
Coriolis
Coriolis Acceleration: 2Ω × vr
Navier-Stokes Equation in a Rotating Frame
β-Plane Approximation

Shallow Water Equations (SWE)
Introduction
Derivation of Equations
The Rotating Shallow Water Equations (RSWE)
Some Exact Solutions of the RSWE
Linearization of the SWE
Linear Wave Equation
Separation of Variables and the Fourier Method
The Fourier Method in MATLAB
The Characteristics Method
D’Alembert’s Solution in MATLAB
Method of Line and the Wave Equation
Project A: Derivation of the Characteristics Method
Project B: Variations on the Method of Line
Project C: An Inverse Problem
Project D: Exact Solutions of the RSWE

Wind-Driven Ocean Circulation: The Stommel and Munk Models
Introduction
Flow in a Rectangular Bay—Normal Modes
Eigenfunctions of the Laplace Operator
Poisson Equation
The Stommel Model
MATLAB Programs
The Stommel Model—A Numerical Approach
The MATLAB Program for the Stommel Model
The Munk Model of Wind-Driven Circulation
Project A: Stommel Model with a Nonuniform Mesh
Munk Model and the Finite Difference Method
Project C: The Galerkin Method and the B. Saltzman and E. Lorenz Equations

Some Special Topics
Finite-Time Dynamical Systems
Data Assimilation
Normal Modes and Data

Appendix A: Solutions to Selected Problems

References appear at the end of each chapter.

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## Author(s)

### Biography

Reza Malek-Madani is a professor in the Department of Mathematics at the U.S. Naval Academy. He earned a Ph.D. in applied mathematics from Brown University. His research interests include mathematical modeling and stability analysis in fluid flows and dynamical systems.