دانلود کتاب Numerical Methods for Atmospheric and Oceanic Sciences

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Cover
Copyright
Contents
Figures
Foreword
Preface
1 Partial Differential Equations
1.1 Introduction
1.2 Diffusion Equation
1.3 First-order Equations
1.4 First-order Equations: Method of Characteristics
1.5 Second-order Quasilinear PDEs: Classification UsingMethod of Characteristics
1.6 Wave Equation
1.7 Linear Advection Equation
1.8 Laplace Equation
1.9 Method of Separation of Variables for theOne-dimensional Heat Equation
1.10 Method of Separation of Variables for theOne-dimensionalWave Equation
Exercises 1a (Question and answer)
Exercises 1b (Questions only)
2 Equations of Fluid Motion
2.1 Introduction
2.2 Lagrangian and Eulerian Description of Fluid Motion
2.2.1 Substantive or total derivative
2.2.2 Conservation of mass principle: Continuity equation
2.2.3 Conservation of momentum principle: Momentum equation
2.2.4 Euler’s equation of motion for an ideal fluid
2.2.5 Conservation of energy principle: Thermodynamic energyequation
2.3 Equations Governing Atmospheric Motion
2.3.1 Rotating frame of reference
2.3.2 Conservation of energy: Thermodynamic energy equation foratmosphere
2.3.3 Geostrophic balance equations
2.3.4 Hydrostatic balance equation
2.3.5 Governing equations of motion of atmosphere with pressure as avertical coordinate
2.3.6 Quasi-geostrophic equations of motion of atmosphere withpressure as a vertical coordinate
2.3.7 Shallow water equations
2.3.8 Vorticity equation for incompressible fluid: Curl of the Navier–Stokes equation
2.3.9 Vorticity equation for atmospheric and oceanic flows
2.3.10 Non-divergent vorticity equation for atmospheric and oceanicflows
2.3.11 Boussinesq approximation
2.3.12 Anelestic approximation
2.3.13 Conservation of water vapour mixing ratio equation
2.3.14 Mean equations of turbulent flow in the atmosphere
2.3.15 RANS, LES, and DNS approaches
2.3.16 Parameterization of physical processes in the atmospheric models
2.3.17 Parallel computing
Exercises 2 (Questions and answers)
3 Finite Difference Method
3.1 Introduction
3.2 Method of Finite Difference
3.2.1 Forward difference scheme
3.2.2 Backward difference scheme
3.2.3 Central difference scheme
3.2.4 Centered fourth-order difference scheme
3.2.5 Finite difference scheme for second derivatives and Laplacian
3.3 Time Integration Schemes
3.3.1 Two-time level schemes
Forward Euler scheme
Backward Euler scheme
Crank–Nicolson scheme
Matsuno forward–backward scheme
Heun’s scheme
3.3.2 Three-time level schemes
Leapfrog scheme
Adams–Bashforth scheme
Milne–Simpson scheme
Exercises 3a (Questions and answers)
Exercises 3b (Questions only)
Python examples
4 Consistency and Stability Analysis
4.1 Consistency and Stability Analysis
4.2 Basic Aspects of Finite Differences
4.2.1 Consistency
4.2.2 Convergence
4.2.3 Lax Equivalence Theorem
4.3 Errors and Stability Analysis
4.3.1 Introduction
4.3.2 Discretization error
4.3.3 Representation of real numbers in a computer: Round-off error
4.3.4 Stability analysis of FTCS scheme as applied to one-dimensionalheat conduction equation
4.3.5 Richardson central in time and central in space (CTCS) finitedifference scheme and its stability
4.3.6 DuFort–Frankel finite difference scheme and its stability
4.3.7 Backward in time and central in space (BTCS) scheme and itsstability
4.3.8 Crank–Nicolson Scheme and its stability
4.4 Two-dimensional Heat Conduction Equation
4.4.1 FTCS scheme and its stability
4.4.2 BTCS scheme and its stability
4.4.3 Alternating Direction Implicit (ADI) method
4.5 Stability Analysis of One-dimensional LinearAdvection Equation
4.5.1 Forward in time and central in space (FTCS) scheme
4.5.2 Central in time and central in space (CTCS) scheme and itsstability
4.5.3 Upwind methods
4.5.4 Lax finite difference scheme and its stability
4.5.5 Lax–Wendroff scheme and its stability
4.5.6 Backward in time and central in space (BTCS) scheme and itsstability
4.5.7 Crank–Nicolson scheme and its stability
4.6 Matrix Method of Stability Analysis
4.6.1 Matrix method for the one-dimensional heat equation
4.7 Energy Method of Stability Analysis
4.8 Aliasing and Nonlinear Computational Instability
4.9 Aliasing Error and Instability
4.10 Ways to Prevent Nonlinear Computational Instability
4.11 Arakawa’s Scheme to Prevent NonlinearComputational Instability
Exercises 4a (Questions only)
Ecercises 4b (Questions and answers)
Python examples
5 Oscillation and Decay Equations
5.1 Introduction
5.2 Properties of Time-differencing Schemes as Applied tothe Oscillation Equation
5.3 Properties of Various Two-time Level DifferencingSchemes
5.3.1 Forward Euler scheme
5.3.2 Backward Euler scheme
5.3.3 Trapezoidal scheme
5.3.4 Iterative two-time level scheme
5.3.5 Matsuno scheme
5.3.6 Heun scheme
5.3.7 Phase change of the various two-level schemes
5.4 Properties of Various Three-time Level DifferencingSchemes
5.4.1 Leapfrog scheme
5.4.2 Adams–Bashforth scheme
5.5 Properties of Various Schemes as Applied to the FrictionEquation
5.5.1 Application of various two-time level schemes to the frictionequation
Forward Euler scheme
Backward Euler scheme
Trapezoidal scheme
Iterative two-time level scheme
Three-time level schemes
Exercises 5a (Questions only)
Exercises 5b (Questions and answers)
6 Linear Advection Equation
6.1 Introduction
6.2 Centered Time and Space Differencing Schemes forLinear Advection Equation
6.3 Conservative Finite Difference Methods
6.3.1 Leapfrog scheme
6.3.2 Matsuno scheme
6.3.3 Lax–Wendroff scheme
6.4 Computational Dispersion: Phase Speed Dependenceon Wavelength
6.4.1 Group velocity
6.5 Upstream Schemes
6.5.1 Transportive property
6.6 Fourth-order Space Differencing Schemes forAdvection Equation
6.7 Higher Order Sign Preserving Advection Schemes
6.8 Two-dimensional Linear Advection Equation
6.8.1 Computational dispersion: Phase speed dependence on frequency
Exercises 6a (Questions only)
Exercises 6b (Questions and answers)
Python examples
7 Numerical Solution of Elliptic PartialDifferential Equations
7.1 Introduction
7.1.1 Commonly occurring elliptic problems
7.2 Direct Methods of Solution
7.3 Iterative Methods of Solution
7.3.1 Gauss–Seidel method
7.3.2 Successive over relaxation (SOR) method
7.3.3 Relaxation, sequential relaxation, and successive relaxationmethods
Relaxation method of solving an elliptic partial differential equation
Sequential relaxation method of solving an elliptic partial differential equation
Successive overrelaxation (SOR) method for solving an elliptic partial differentialequation
Neumann boundary condition for relaxation (sequential and SOR) methods
7.4 Multigrid Methods
7.4.1 Understanding the two-grid method
7.4.2 Full multigrid (FMG) method
7.5 Fast Fourier Transform Methods
7.6 Cyclic Reduction and Factorization Methods
Exercises 7a (Questions only)
Exercises 7b (Questions and answers)
Python examples
8 Shallow Water Equations
8.1 Introduction
8.2 One-dimensional Linear GravityWave without Rotation
8.3 Staggered Grid Arrangement for LinearOne-dimensional GravityWave
8.4 Linear Inertia–gravityWaves in One-dimension
8.4.1 Non-staggered grid arrangement – Grid ‘A’
8.4.2 Staggered grid arrangements – Grid ‘B’
8.4.3 Staggered grid arrangements – Grid ‘C’
8.4.4 Staggered grid arrangements - Grid ‘D’
8.5 Two-dimensional Linear GravityWave withoutRotation
8.5.1 Non-staggered grid arrangement (Grid ‘A’)
8.5.2 Staggered grid arrangement (Grid ‘B’)
8.5.3 Staggered grid arrangement (Grid ‘C’)
8.5.4 Staggered grid arrangement (Grid ‘D’)
8.5.5 Staggered grid arrangement (Grid ‘E’)
8.6 Two-dimensional Linear GravityWave with Rotation
8.6.1 Non-staggered grid arrangement (Grid ‘A’)
8.6.2 Staggered grid arrangement (Grid ‘B’)
8.6.3 Staggered grid arrangement (Grid ‘C’)
8.6.4 Staggered grid arrangement (Grid ‘D’)
8.6.5 Staggered grid arrangement (Grid ‘E’)
Exercises 8a (Questions and answers)
Exercises 8b (Questions only)
9 Numerical Methods for SolvingShallow Water Equations
9.1 Introduction
9.2 Linear One-dimensional Shallow Water Equationswithout Rotation
9.3 Solution of Linear One-dimensional Shallow WaterEquations without Rotation
9.3.1 Explicit schemes: Leapfrog scheme (non-staggered)
9.3.2 Explicit schemes: FTCS scheme (non-staggered)
9.3.3 Fully implicit schemes (non-staggered)
9.3.4 Forward–backward scheme (non-staggered)
9.3.5 Pressure averaging scheme (non-staggered)
9.3.6 Implicit scheme (non-staggered)
9.3.7 Staggered explicit scheme
9.3.8 Splitting method
9.3.9 Semi-implicit method
9.3.10 Stability of the semi-implicit method
9.4 Two-dimensional Linear ShallowWater Equationswithout Rotation
9.4.1 Leapfrog scheme
9.4.2 Elliassen grid
9.4.3 Forward backward scheme
9.4.4 Implicit scheme (trapezoidal method)
9.5 Semi-implicit Scheme of Kwizak and Robert
Exercises 9a (Questions and answers)
Exercises 9b (Questions only)
Python examples
10 Numerical Methods for SolvingBarotropic Equations
10.1 Introduction
10.2 Numerical Solution of a Non-divergent BarotropicVorticity Equation on a b Plane – Linear Case
10.3 Numerical Solution of a Non-divergent BarotropicVorticity Equation on a f Plane – Nonlinear Case
10.4 Numerical Solution of a Non-divergent BarotropicVorticity Equation on a b Plane – Nonlinear Case
10.5 Solving One-dimensional Linear ShallowWaterEquations without Rotation
10.6 Solving One-dimensional Linear ShallowWaterEquations with Rotation
10.7 Solving One-dimensional Nonlinear Shallow WaterEquations without Rotation
10.8 Solving Two-dimensional Linear ShallowWaterEquations without Rotation
10.9 Solving Two-dimensional Linear ShallowWaterEquations with Rotation on a b Plane
10.10 Solving Two-dimensional Nonlinear Shallow WaterEquations without Rotation
10.11 Solving Two-dimensional Nonlinear Shallow WaterEquations with Rotation on a b Plane
10.12 Equivalent Barotropic Model
Exercises 10 (Questions and answers)
Python examples
11 Numerical Methods for SolvingBaroclinic Equations
11.1 Introduction
11.2 Atmospheric Vertical Coordinates
11.3 Pressure as a Vertical Coordinate
11.4 Sigma as a Vertical Coordinate
11.5 Eta as a Vertical Coordinate
11.6 Isentropic Vertical Coordinate
11.7 Vertical Staggering
11.8 Two-layer Quasi-geostrophic Equation
11.9 Multi-level Models
11.10 Limited Area Primitive Equation Atmospheric Model
11.10.1 Finite difference equations for the limited area primitiveequation atmospheric model
11.10.2 Solution procedure
Exercises 11 (Questions and answers)
12 Boundary Conditions
12.1 Introduction
12.2 Upper Boundary Conditions
12.3 Lower Boundary Conditions
12.4 Lateral Boundary Conditions
12.5 One-way and Two-way Interactive Nesting
Exercises 12 (Questions and answers)
13 Lagrangian and SemiLagrangianSchemes
13.1 Introduction
13.2 Fully Lagrangian Scheme
13.3 Semi-Lagrangian Scheme
13.3.1 Linear one-dimensional advection equation with constantvelocity
13.3.2 Semi-Lagrangian scheme to solve the linear one-dimensionaladvection equation with constant velocity
13.3.3 Semi-Lagrangian scheme to solve the linear one-dimensionaladvection equation with non-constant velocity – three time levels cheme
13.3.4 Semi-Lagrangian scheme to solve the linear one-dimensionaladvection equation with non-constant velocity – two time level scheme
13.3.5 Semi-Lagrangian scheme to solve the linear one-dimensional advection equation with non-constant velocity in the presence ofa source term using two time level scheme
13.3.6 Stability of the semi-Lagrangian scheme to solve the linear one dimensional advection equation
13.3.7 Semi-Lagrangian scheme to solve the forced advection equation with non-constant velocity – three time level scheme
13.4 Numerical Domain of Dependence
13.5 Semi-Lagrangian Scheme to Solve Shallow WaterEquations
13.5.1 Advantages of semi-Lagrangian scheme as compared to Eulerianscheme
13.6 Interpolation
Exercises 13 (Questions and answers)
14 Spectral Methods
14.1 Introduction
14.2 Series Expansion Method
14.3 Spectral Methods and Finite Difference Method
14.3.1 Spectral methods as applied to a linear one-dimension advectionequation
14.3.2 Spectral methods as applied to a linear second-order ordinarydifferential equation
14.3.3 Spectral methods as applied to a partial differential equationinvolving time
14.3.4 Spectral methods and energy conservation
14.3.5 Spectral methods applied to nonlinear one-dimensional advection equation
14.3.6 Spectral methods applied to nonlinear one-dimensional advection equation – handling the nonlinear term
14.3.7 Spectral methods applied to barotropic vorticity equation on a bplane
14.3.8 Types of truncation
14.3.9 Advantages of spectral methods over the method of finitedifferences
14.3.10 Transform method
14.4 Spectral Methods for ShallowWater Equations
14.5 Pseudo-spectral Methods
Exercises 14 (Questions and answers)
Python examples
15 Finite Volume and Finite ElementMethods
15.1 Introduction
15.2 Integral Form of Conservation Law
15.2.1 Integral form of conservation law of mass
15.2.2 Convection equation
15.2.3 Integral form of linear momentum conservation equation
15.3 Finite Volume Method
15.3.1 Finite volume method applied to one-dimensional scalarconservation equation
15.3.2 Godunov scheme for a scalar equation
15.3.3 Rankine–Hugoniot jump condition
15.3.4 Finite volume method for one-dimensional linear heat equation
15.4 Finite Element Method
15.4.1 Finite element method as applied to an ordinary differentialequation
15.4.2 Finite element method as applied to one-dimensional advectionequation
15.4.3 Finite element method as applied to one-dimensional linearHelmholtz equation
Exercises 15 (Questions and answers)
16 Ocean Models
16.1 Introduction
16.2 Sverdrup Model for Ocean Circulation
16.2.1 Sverdrup model for ocean circulation having a zonal wind stresswith meridional variation
16.3 Stommel Model for Ocean Circulation
16.3.1 Stommel model for ocean circulation having a zonal wind stresswith meridional variation
16.4 Munk Model for Ocean Circulation
16.4.1 Munk model for ocean circulation having a wind stress with bothzonal and meridional variations
16.5 Nonlinear Model for Ocean Circulation
16.6 Vertical Coordinate for Ocean Models
16.6.1 Height (z) coordinate
16.6.2 Isopycnic coordinate
16.6.3 Sigma coordinate
16.6.4 Hybrid coordinate
16.7 Barotropic–Baroclinic Splitting
16.7.1 External and internal waves
16.7.2 Barotropic–baroclinic subsystems
16.8 Time Discretization
16.8.1 Leapfrog scheme
16.8.2 Two-time level finite difference scheme
16.9 Spatial Discretization and Horizontal Grids in OceanModels
16.9.1 Spatial arrangement of the dependent variables in the horizontal
16.10 Various Approaches to Solving the Ocean MomentumEquations
Exercises 16 (Questions and answers)
Appendix Tridiagonal Matrix Algorithm
Bibliography
Index