Physical oceanography: a mathematical introduction with MATLAB® / Reza Malek-Madani

Incluye bibliografía e índice: páginas 437-440

Preface.. 1 An Introduction to MATLAB.. 1.1 A Session on MATLAB.. 1.2 The Operations *, / , and .. 1.3 Defining and Plotting Functions in MATLAB.. 1.4 3-Dimensional Plotting.. 1.5 M-files.. 1.6 Loops and Iterations in MATLAB.. 1.7 Conditional Statements in MATLAB.. 1.8 Fourier Series in MATLAB.. 1.9 Solving Differential Equations.. 1.10 Concluding Remarks.. 1.11 References.. 2 Matrix Algebra.. 2.1 Vectors and Matrices.. 2.2 Vector Operations.. 2.3 Matrix Operations.. 2.4 Linear Spaces and Subspaces.. 2.5 Determinant and Inverse of Matrices.. 2.6 Computing A−1 Using Co-Factors.. 2.7 Linear Independence, Span, Basis and Dimension.. 2.8 Linear Transformations.. 2.9 Row Reduction and Gaussian Elimination.. 2.10 Eigenvalues and Eigenvectors.. 2.11 Project A: Taylor Polynomials and Series.. 2.12 Project B: A Differentiation Matrix.. 2.13 Project C: Spectral Method and Matrices.. 2.14 Concluding Remarks.. 2.15 References and Further Reading.. 3 Differential and Integral Calculus.. 3. 1 Derivative.. 3.2 Taylor Polynomial and Series.. 3.3 Functions of Several Variables and Vector Fields.. 3.4 Divergence.. 3.5 Curl and Vector Fields.. 3.6 Integral Theorems.. 3.7 References and Further Reading.. 4 Ordinary Differential Equations.. 4.1 Linear Independence and Space of Functions.. 4.2 Linear ODEs.. 4.3 General Systems of ODEs.. 4.4 MATLAB's ode45.. 4.5 Asymptotic Behavior and Linearization.. 4.6 Motion of Parcels of Fluid in MATLAB.. 4.7 Project A: Ekman Layer.. 4.8 Project B: Lorenz 96 Model.. 4.9 References.. 5 Numerical Methods for ODEs.. 5.1 Finite Difference Methods.. 5.2 The Backward Euler Method (BEM.. 5.3 Stability of Numerical Methods.. 5.4 Stability Analysis of Numerical Schemes.. 5.5 MATLAB Programs for the Forward Finite Difference Method.. 5.6 Stability Analysis of Numerical Schemes (continued.. 5.7 Truncation Error.. 5.8 Boundary Value Problems and the Shooting Method.. 5.9 Project A: Modified Euler Method

5.10 Project B: Runge-Kutta Methods.. 5.11 Project C: Finite Difference Methods and BVPs.. 5.12 Project D: The Method of Lines.. 5.13 Project E: Burgers Equation (Method of Characteristics.. 5.14 Project F: Burgers Equation (Method of Characteristics-Nonlinear Case.. 5.15 Project G: Burgers Equation (Formation of Singularities.. 5.16 Project H: Burgers Equation and the Method of Lines.. 5.17 References.. 6 Equations of Fluid Dynamics.. 6.1 Flow Representations-Eulerian and Lagrangian.. 6.2 Deformation Gradient and Conservation of Mass.. 6.3 Derivation of Equation of Conservation of Mass-A Heuristic Approach.. 6.4 Stream Function and Vector Fields A, B, C, and ABC.. 6.5 Acceleration in Rectangular Coordinates.. 6.6 Strain-Rate Matrix and Vorticity.. 6.7 Internal Forces and the Cauchy Stress.. 6.8 Euler and Navier-Stokes Equations.. 6.9 Bernoulli's Equation and Irrotational Flows.. 6.10 Acceleration in Spherical Coordinates.. 6.10.1 Coordinate Curves.. 6.10.2 Spherical Basis.. 6.10.3 The Eulerian Formulation of Velocity and Acceleration Revisited.. 6.10.4 Velocity in Spherical Basis.. 6.10.5 Dynamics of Basis Vectors.. 6.10.6 Formula for Acceleration in Spherical Coordinates.. 6.11 Project A: Inviscid Linear Fluid Motions and Surface Gravity Waves.. 6.12 Project B: Internal Gravity Waves.. 6.13 Project C: Equations for Bubbles Dynamics.. 6.14 Project D: Chaotic Transport.. 6.15 References.. 7 Equations of Geophysical Fluid Dynamics.. 7.1 Introduction.. 7.2 Coriolis.. 7.3 Coriolis Acceleration: 2Ω × vr.. 7.4 Gradient Operator in Spherical Coordinates.. 7.5 Navier-Stokes Equation in a Rotating Frame.. 7.6 β-Plane Approximation.. 7.7 References.. 8 Shallow Water Equations (SWE.. 8.1 Introduction.. 8.2 Derivation of Equations.. 8.3 Rotating Shallow Water Equations (RSWE.. 8.4 Some Exact Solutions of the RSWE.. 8.5 Linearization of the SWE.. 8.6 Linear Wave Equation

8.7 Separation of Variables and the Fourier Method.. 8.8 Fourier Method in MATLAB.. 8.9 Method of Characteristics.. 8.10 D'Alembert's Solution in MATLAB.. 8.11 Method of Line and Wave Equation.. 8.12 Project A: Method of Characteristics for General PDEs.. 8.13 Project B: Variations on the Method of Line.. 8.14 Project C: An Inverse Problem.. 8.15 Project D: Exact Solutions of the Rotating Shallow Water Equations.. 8.16 Project E: Courant-Friedrichs-Lewy Condition.. 8.17 References.. 9 Wind-Driven Ocean Circulation: The Stommel and Munk Models.. 9.1 Introduction.. 9.2 Flow in a Rectangular Bay-Normal Modes.. 9.3 Eigenfunctions of the Laplace Operator.. 9.4 Poisson Equation.. 9.4.1 Poisson Equation with Localized Vorticity.. 9.5 Stommel Model.. 9.5.1 Governing PDE.. 9.5.2 Non-Dimensionalization.. 9.5.3 Solution to the BVP.. 9.5.3.1 Determining the Particular Solution Ψp.. 9.5.3.2 Determining the Homogeneous Solution Ψh.. 9.5.3.3 Applying the Boundary Condition.. 9.6 MATLAB Programs.. 9.7 Stommel Model-A Numerical Approach.. 9.7.1 Constructing the System AΨ =B.. 9.8 MATLAB Program for the Stommel Model.. 9.9 Munk Model of Wind-Driven Circulation.. 9.10 Project A: Stommel Model with a Nonuniform Mesh.. 9.11 Project B: Munk Model and the Finite Difference Method.. 9.12 Project C: Galerkin Method and the B. Saltzman and E. Lorenz Equations.. 9.13 References.. 10 Some Special Topics.. 10.1 Finite-Time Dynamical Systems.. 10.2 Data Assimilation and Filtering.. 10.3 Normal Modes and Data.. 10.4 Concluding Remarks.. 10.5 References.. Appendix A: Solutions to Selected Problems.. Index

"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." eng