Shen & Somerville (2019) Climate Mathematics : contents

Samuel P. Shen & Richard C. Somerville, 2019: Climate Mathematics: Theory and Applications. Cambridge UK: Cambridge University Press. ISBN 978-1-108-47687-4 (hardback). DOI: 10.1017/9781108693882. [Masuda's memo (in Japanese)]

Detailed table of contents

Contents
Preface
Acknowledgements
Main Symbols and Acronyms
1. Dimensional Analysis for Climate Science
- 1.1 Dimensional analysis
- 1.2 Fundamental Dimensions: LMTθI-class
- 1.3 Dimensional Analysis for a Simple Pendulum
- 1.4 Dimensional Analysis for the State Equation of Air
- 1.5 Dimensional Analysis of Heat Diffusion
- 1.6 Dimensional Analysis of Rossby Waves and Kelvin Waves
  - 1.6.1 Parameters for Rossby Waves
  - 1.6.2 Non-Dispersive Properties of Kelvin Waves
- 1.7 Estimating the Shock Wave Radius of a Nuclear Explosion by Dimensional Analysis
- 1.8 Chapter Summary
- References and Further Readings
- Exercises
2. Basics of R Programming
- 2.1 Download and Install R and RStudio
- 2.2 R Tutorial
  - 2.2.1 R As a Smart Calculator
  - 2.2.2 Define a Sequence in R
  - 2.2.3 Define a Function in R
  - 2.2.4 Plot with R
  - 2.2.5 Symbolic Calculations by R
  - 2.2.6 Vectors and Matrices
  - 2.2.7 Simple Statistics by R
- 2.3 Online Tutorials
  - 2.3.1 YouTube Tutorial: For True Beginners
  - 2.3.2 YouTube Tutorial: For Some Basic Statistical Summaries
  - 2.3.3 YouTube Tutorial: Input Data by Reading a csv File into R
- 2.4 Chapter Summary
- References and Further Readings
- Exercises
3. Basic Statistical Methods for Climate Data Analysis
- 3.1 Statistical Indices from the Global Temperature Data from 1880 to 2015
  - 3.1.1 Mean, Variance, Standard Deviation, Skewness, Kurtosis, and Quantiles
  - 3.1.2 Correlation, Covariance, and Linear Trend
- 3.2 Commonly Used Statistical Plots
  - 3.2.1 Histogram of a Set of Data
  - 3.2.2 Box Plot
  - 3.2.3 Scatter Plot
  - 3.2.4 Q-Q Plot
- 3.3 Probability Distributions
  - 3.3.1 What Is a Probability Distribution?
  - 3.3.2 Normal Distribution
  - 3.3.3 Student's t-distribution
- 3.4 Estimate and Its Error
  - 3.4.1 Probability of a Sample inside a Confidence Interval
  - 3.4.2 Mean of a Large Sample Size: Approximately Normal Distribution
  - 3.4.3 Mean of a Small Sample Size t-Test
- 3.5 Statistical Inference of a Linear Trend
- 3.6 Free Online Statistics Tutorials
- 3.7 Chapter Summary
- References and Further Readings
- Exercises
4. Climate Data Matrices and Linear Algebra
- 4.1 Matrix as a Data Array
- 4.2 Matrix Algebra
  - 4.2.1 Matrix Equality, Addition, and Subtraction
  - 4.2.2 Matrix Multiplication
- 4.3 A Set of Linear Equations
- 4.4 Eigenvalues and Eigenvectors of a Square Space Matrix
  - 4.4.1 Matrices of Data Anomalies, Standardized Anomalies, Covariance, and Correlation
  - 4.4.2 Eigenvectors and Their Corresponding Eigenvalues
- 4.5 An SVD Representation Model for Space–Time Data
- 4.6 SVD Analysis of Southern Oscillation Index
  - 4.6.1 Standardized SLP Data and SOI
  - 4.6.2 Weighted SOI Computed by the SVD Method
  - 4.6.3 Visualization of the ENSO Mode Computed from the SVD Method
- 4.7 Mass Balance for Chemical Equations in Marine Chemistry
- 4.8 Multivariate Linear Regression Using Matrix Notations
- 4.9 Chapter Summary
- References and Further Readings
- Exercises
5. Energy Balance Models for Climate
- 5.1 EBM for Modeling the Moon's Surface Temperature
  - 5.1.1 Moon–Earth–Sun Orbit and Lunar Surface
  - 5.1.2 Moon's Surface Temperature
  - 5.1.3 EBM Prediction for the Moon Surface Temperature
- 5.2 EBM for the Global Average Surface Temperature of the Earth: A Zero-Dimensional Climate Model
  - 5.2.1 The Incoming Power from the Solar Radiation to the Earth
  - 5.2.2 The Outgoing Power from Long-Wave Radiation Emitted by the Earth
  - 5.2.3 EBM as a Power Balance
- 5.3 EBM for the Global Average Surface Temperature of an Earth with a Nonlinear Albedo Feedback
- 5.4 Time-Dependent Zero-Dimensional EBM for the Earth's Global Average Surface Temperature
  - 5.4.1 An EBM Including Time Dependence
  - 5.4.2 Stability Analysis of the Multiple Solutions of the EBM with a Nonlinear Albedo Feedback
  - 5.4.3 Energy Flow Budget and Greenhouse Effect for the Earth's Climate
- 5.5 Increasing the Complexity of Climate Models
- 5.6 Chapter Summary
- References and Further Readings
- Exercises
6. Calculus Applications to Climate Science I: Derivatives
- 6.1 Stefan–Boltzmann Law and Budyko’s Approximation
- 6.2 Linear Approximation
- 6.3 Bisection Method for Solving Nonlinear Equations
- 6.4 Newton's Method
- 6.5 Examples of Higher-Order Derivatives
- 6.6 Pressure Gradient Force and Coriolis Force
- 6.7 Spatiotemporal Variations of the Atmospheric and Oceanic Temperature Fields
- 6.8 Taylor Polynomial as a High-Order Approximation
  - 6.8.1 Taylor's Theorem
  - 6.8.2 Taylor Series Example: Exponential Function
  - 6.8.3 Numerical Integration Using Taylor Expansion
- 6.9 Chapter Summary
- References and Further Readings
- Exercises
7. Calculus Applications to Climate Science II: Integrals
- 7.1 Geopotential and Atmospheric Pressure
  - 7.1.1 Vertical Forces on a Small Parcel of Atmosphere
  - 7.1.2 Geopotential
- 7.2 Hypsometric Equation: Exponential Decrease of Pressure with Respect to Elevation
  - 7.2.1 The General Hypsometric Equation
  - 7.2.2 An Application of the Hypsometric Equation: Calculate the Elevation of Mount Mitchell
  - 7.2.3 Hypsometric Equation for an Isothermal Layer
  - 7.2.4 Error Estimate of the Linear Approximation to the Hypsometric Equation
  - 7.2.5 Applications of Geopotential Height in Radiosonde Measurements
- 7.3 Work Done by an Air Mass in Expansion
- 7.4 Internal Energy, Enthalpy, and Entropy
  - 7.4.1 Internal Energy and Enthalpy
  - 7.4.2 Entropy
- 7.5 Use of Integrals to Derive Stefan–Boltzmann’s Blackbody Radiation Formula from Planck’s Law of Radiation
- 7.6 Chapter Summary
- References and Further Readings
- Exercises
8. Conservation Laws in Climate Dynamics
- 8.1 Conservation of Mass
  - 8.1.1 Basic Elements of the Continuum Mechanics Method for Climate Modeling
  - 8.1.2 Lagrangian and Eulerian Observers, and Mass Conservation in the Lagrangian Framework
  - 8.1.3 Total Derivative
  - 8.1.4 Mass Conservation in the Eulerian Framework
- 8.2 Conservation of Momentum Over a Grid Box: F = ma
- 8.3 The Equations of Momentum Conservation in x, y, z, t Coordinates
- 8.4 Geostrophic Approximation of the Momentum Equations
  - 8.4.1 Mathematical Description of the Geostrophic Approximation
  - 8.4.2 Flow Direction Perpendicular to the PGF under the Geostrophic Approximation
- 8.5 The Potential Vorticity Conservation Equation
  - 8.5.1 Absolute Vorticity and Relative Vorticity
  - 8.5.2 Potential Vorticity and Its Conservation
  - 8.5.3 Mathematical Derivations of the Conservation of Potential Vorticity
- 8.6 Chapter Summary
- References and Further Readings
- Exercises
9. R Graphics for Climate Science
- 9.1 Two-Dimensional Line Plots and Setups of Margins and Labels
  - 9.1.1 Plot Two Different Time Series on the Same Plot
  - 9.1.2 Figure Setups: Margins, Fonts, Mathematical Symbols, and More
  - 9.1.3 Plot Two or More Panels on the Same Figure
- 9.2 Color Contour Maps
  - 9.2.1 Basic Principles for an R Contour Plot
  - 9.2.2 Plot Contour Color Maps for Random Values on a Map
  - 9.2.3 Plot Contour Maps from Climate Model Data in NetCDF Files
- 9.3 Plot Wind Velocity Field on a Map
  - 9.3.1 Plot a Wind Field Using arrow.plot
  - 9.3.2 Plot a Surface Wind Field from netCDF Data
- 9.4 ggplot for Data
- 9.5 Animation
- 9.6 Chapter Summary
- References and Further Readings
- Exercises
10. Advanced R Analysis and Plotting: EOFs, Trends, and Global Data
- 10.1 Ideas of EOF, PC, and Variances Computed from SVD
- 10.2 2Dim Spatial Domain EOFs and 1Dim Temporal PCs
  - 10.2.1 Generate Synthetic Data by R
  - 10.2.2 SVD for the Synthetic Data EOFs, Variances, and PCs
- 10.3 From Climate Data Download to EOF and PC Visualization: An NCEP/NCAR Reanalysis Example
  - 10.3.1 Download and Visualize the NCEP Temperature Data
  - 10.3.2 Space–Time Data Matrix and SVD
- 10.4 Area-Weighted Average and Spatial Distribution of Trend
  - 10.4.1 Global Average and PC1
  - 10.4.2 Spatial Pattern of Linear Trends
- 10.5 GPCP Precipitation Data: Analysis and Visualization by R
  - 10.5.1 Read and Write GPCP Data
  - 10.5.2 GPCP Climatology and Standard Deviation
- 10.6 Chapter Summary
- References and Further Readings
- Exercises
11. R Analysis of Incomplete Climate Data
- 11.1 The Missing Data Problem
- 11.2 Read NOAAGlobalTemp and Form the Space–Time Data Matrix
  - 11.2.1 Read the Downloaded Data
  - 11.2.2 Plot the Temperature Data Map of a Given Month
  - 11.2.3 Extract the Data for a Specified Region
  - 11.2.4 Extract Data from Only One Grid Box
- 11.3 Spatial Averages and Their Trends
  - 11.3.1 Compute and Plot the Global Area-Weighted Average of Monthly Data
  - 11.3.2 Percent Coverage of the NOAAGlobalTemp
  - 11.3.3 Compare Trends and Variances at Two Different Locations
  - 11.3.4 Which Month Has the Strongest Trend?
  - 11.3.5 Spatial Average of Annual Data
  - 11.3.6 Nonlinear Trend of the Global Average Annual Mean Data
- 11.4 Spatial Characteristics of the Temperature Change Trends
  - 11.4.1 The Twentieth-Century Temperature Trend
  - 11.4.2 Twentieth-Century Temperature Trend Computed under a Relaxed Condition
  - 11.4.3 Trend Pattern for the Four Decades of Consecutive Warming: 1976–2016
- 11.5 Chapter Summary
- References and Further Readings
- Exercises
Appendix A. Dot Product of Two Vectors
- A.1 Two Definitions for the Dot Product
- A.2 Solar Power Flux to the Earth’s Surface and Seasonality
- A.3 Divergence Theorem for the Mass Continuity Equation in Climate Models
Appendix B. Cross Product of Two Vectors
- B.1 Definition of the Cross Product of Two Vectors
- B.2 Coriolis Force
- B.3 Vorticity
- B.4 Stokes' Theorem
Appendix C. Spherical Coordinates
- C.1 Transform between the Spherical Coordinates and Cartesian Coordinates
- C.2 Area and Volume Differentials in Spherical Coordinates
Appendix D. Calculus Concepts and Methods for Climate Science
- D.1 Descartes’ Direct Calculus for Functions of a Single Variable
- D.2 Calculus from a Statistics Perspective
- D.3 Differentiation Methods and Higher Derivatives
- D.4 Calculus for Functions of Two and More Variables
- References and Further Readings Exercises
Appendix E. Sample Solutions to the Climate Mathematics Exercises
Glossary
Index

2020-05-13
MASUDA Kooiti

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