Physical Oceanography

Physical Oceanography : A Mathematical Introduction with MATLAB

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Description

Accessible to advanced undergraduate students, Physical Oceanography: A Mathematical Introduction with MATLAB (R) 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.show more

Product details

  • Electronic book text | 456 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 85 Illustrations, black and white
  • 1584888318
  • 9781584888314

Table of contents

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 Aâ 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 DivergenceCurl 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 DynamicsFlow 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 VorticityInternal Forces and the Cauchy Stress Euler and Navier-Stokes Equations Bernoulli's Equation and Irrotational Flows Acceleration in Spherical CoordinatesProject 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 x vr Gradient Operator in Spherical Coordinates Navier-Stokes Equation in a Rotating Frame ss-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 ProblemProject 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 EquationThe Stommel ModelMATLAB Programs The Stommel Model-A Numerical ApproachThe 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.show more

About Reza Malek-Madani

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.show more