1st Edition

Physical Oceanography A Mathematical Introduction with MATLAB

By Reza Malek-Madani Copyright 2012
    456 Pages 85 B/W Illustrations
    by Chapman & Hall

    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
    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.