An Introduction to Global Spectral Modeling

This introductory book on numerical weather prediction utilizes thespectral transform method. It is an indispensable guide to the methods beingused by nearly all major weather forecast centers in the United States, England,Japan, India, France, and Australia.

1 Introduction

1

(3)

2 An Introduction to Finite Differencing

4

(33)

2.1 Introduction

4

(1)

2.2 Application of Taylor's Series to Finite Differencing

5

(1)

2.3 Forward and Backward Differencing

6

(1)

2.4 Centered Finite Differencing

7

(1)

2.5 Fourth-Order Accurate Formulas

8

(2)

2.6 Second-Order Accurate Laplacian

10

(5)

2.7 Fourth-Order Accurate Laplacian

15

(2)

2.8 Elliptic Partial Differential Equations in Meteorology

17

(1)

2.9 Direct Method

18

(3)

2.10 Relaxation Method

21

(5)

2.11 Sequential Relaxation Versus Simultaneous Relaxation

26

(2)

2.12 Barotropic Vorticity Equation

28

(2)

2.13 The 5-Point Jacobian

30

(1)

2.14 Arakawa Jacobian

30

(5)

2.15 Exercises

35

(2)

3 Time-Differencing Schemes

37

(17)

3.1 Introduction

37

(1)

3.2 Amplification Factor

38

(2)

3.3 Stability

40

(9)

3.4 Shallow-Water Model

49

(5)

4 What Is a Spectral Model?

54

(6)

4.1 Introduction

54

(1)

4.2 The Galerkin Method

55

(3)

4.3 A Meteorological Application

58

(1)

4.4 Exercises

59

(1)

5 Lower-Order Spectral Model

60

(11)

5.1 Introduction

60

(1)

5.2 Maximum Simplification

61

(3)

5.3 Conservation of Mean-Square Vorticity and Mean Kinetic Energy

64

(2)

5.4 Energy Transformations

66

(2)

5.5 Mapping the Solution

68

(1)

5.6 An Example of Chaos

68

(2)

5.7 Exercises

70

(1)

6 Mathematical Aspects of Spectral Models

71

(33)

6.1 Introduction

71

(2)

6.2 Legendre Equation and Associated Legendre Equation

73

(3)

6.3 Laplace's Equation

76

(1)

6.4 Orthogonality Properties

77

(4)

6.5 Recurrence Relations

81

(1)

6.6 Gaussian Quadrature

82

(7)

6.7 Spectral Representation of Physical Fields

89

(4)

6.8 Barotropic Spectral Model on a Sphere

93

(3)

6.9 Shallow-Water Spectral Model

96

(4)

6.10 Semi-implicit Shallow-Water Spectral Model

100

(2)

6.11 Inclusion of Bottom Topography

102

(1)

6.12 Exercises

103

(1)

7 Multilevel Global Spectral Model

104

(34)

7.1 Introduction

104

(1)

7.2 Truncation in a Spectral Model

105

(2)

7.3 Aliasing

107

(1)

7.4 Transform Method

108

(4)

7.5 The x-y-XXX Coordinate System

112

(6)

7.6 A Closed System of Equations in XXX Coordinates on a Sphere

118

(9)

7.7 Spectral Form of the Primitive Equations

127

(7)

7.8 Examples

134

(4)

8 Physical Processes

138

(45)

8.1 Introduction

138

(1)

8.2 The Planetary Boundary Layer

138

(8)

8.3 Cumulus Parameterization

146

(13)

8.4 Large-Scale Condensation

159

(4)

8.5 Parameterization of Radiative Processes

163

(20)

9 Initialization Procedures

183

(21)

9.1 Introduction

183

(1)

9.2 Normal Mode Initialization

183

(10)

9.3 Physical Initialization

193

(8)

9.4 Initialization of the Earth's Radiation Budget

201

(3)

10 Spectral Energetics

204

(40)

10.1 Introduction

204

(1)

10.2 Energy Equations on a Sphere

204

(15)

10.3 Energy Equations in Wavenumber Domain

219

(13)

10.4 Energy Equations in Two-Dimensional Wavenumber Domain

232

(12)

References

244

(5)

Index

249

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An Introduction to Global Spectral Modeling

Summary

Table of Contents

Supplemental Materials