1st Edition

# Physical Oceanography A Mathematical Introduction with MATLAB

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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 × v

_{r}

Gradient Operator in Spherical Coordinates

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.*

### 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.