## Contents

### Brief Contents

• Chapter 1 Dimensional Analysis for Climate Science
• Chapter 2 Basics of R Programming
• Chapter 3 Basic Statistical Methods for Climate Data Analysis
• Chapter 4 Climate Data Matrices and Linear Algebra
• Chapter 5 Energy Balance Models for Climate
• Chapter 6 Calculus Applications to Climate Science I: Derivatives
• Chapter 7 Calculus Applications to Climate Science II: Integrals
• Chapter 8 Conservation Laws in Climate Dynamics
• Chapter 9 R Graphics for Climate Science
• Chapter 10 Advanced R Analysis and Plotting: EOFs, Trends, and Global Data
• Chapter 11 R Analysis of Incomplete Climate Data
• Appendix A Dot Product of Two Vectors
• Appendix B Cross Product of Two Vectors
• Appendix C Spherical Coordinates
• Appendix D Calculus Concepts and Methods for Climate Science
• Appendix E Sample Solutions to the Climate Mathematics Exercises

• Preface page xv
• Acknowledgements xxii
• Main Symbols and Acronyms xxiii
• 1 Dimensional Analysis for Climate Science 1
• 1.1 Dimension and Units 1
• 1.2 Fundamental Dimensions: LMT θI-class 2
• 1.3 Dimensional Analysis for a Simple Pendulum 6
• 1.4 Dimensional Analysis for the State Equation of Air 8
• 1.5 Dimensional Analysis of Heat Diffusion 8
• 1.6 Dimensional Analysis of Rossby Waves and Kelvin Waves 10
• 1.6.1 Parameters for Rossby Waves 10
• 1.6.2 Non-Dispersive Properties of Kelvin Waves 13
• 1.7 Estimating the Shock Wave Radius of a Nuclear Explosion by Dimensional Analysis 13
• 1.8 Chapter Summary 15
• References and Further Readings 16
• Exercises 17

• 2 Basics of R Programming 21
• 2.2 R Tutorial 23
• 2.2.1 R As a Smart Calculator 23
• 2.2.2 Define a Sequence in R 24
• 2.2.3 Define a Function in R 25
• 2.2.4 Plot with R 25
• 2.2.5 Symbolic Calculations by R 26
• 2.2.6 Vectors and Matrices 27
• 2.2.7 Simple Statistics by R 29
• 2.3 Online Tutorials 31
• 2.3.1 YouTube Tutorial: For True Beginners 31
• 2.3.2 YouTube Tutorial: For Some Basic Statistical Summaries 31
• 2.3.3 YouTube Tutorial: Input Data by Reading a csv File into R 31
• 2.4 Chapter Summary 32
• References and Further Readings 33
• Exercises 34

• 3 Basic Statistical Methods for Climate Data Analysis 37
• 3.1 Statistical Indices from the Global Temperature Data from 1880 to 2015 37
• 3.1.1 Mean, Variance, Standard Deviation, Skewness, Kurtosis, and Quantiles 38
• 3.1.2 Correlation, Covariance, and Linear Trend 40
• 3.2 Commonly Used Statistical Plots 43
• 3.2.1 Histogram of a Set of Data 43
• 3.2.2 Box Plot 43
• 3.2.3 Scatter Plot 45
• 3.2.4 Q–Q Plot 47
• 3.3 Probability Distributions 48
• 3.3.1 What Is a Probability Distribution? 48
• 3.3.2 Normal Distribution 52
• 3.3.3 Student’s t-distribution 53
• 3.4 Estimate and Its Error 55
• 3.4.1 Probability of a Sample inside a Confidence Interval 55
• 3.4.2 Mean of a Large Sample Size: Approximately Normal Distribution 57
• 3.4.3 Mean of a Small Sample Size t-Test 64
• 3.5 Statistical Inference of a Linear Trend 67
• 3.6 Free Online Statistics Tutorials 69
• 3.7 Chapter Summary 70
• References and Further Readings 71
• Exercises 72

• 4 Climate Data Matrices and Linear Algebra 75
• 4.1 Matrix as a Data Array 75
• 4.2 Matrix Algebra 76
• 4.2.1 Matrix Equality, Addition, and Subtraction 77
• 4.2.2 Matrix Multiplication 78
• 4.3 A Set of Linear Equations 82
• 4.4 Eigenvalues and Eigenvectors of a Square Space Matrix 83
• 4.4.1 Matrices of Data Anomalies, Standardized Anomalies, Covariance, and Correlation 84
• 4.4.2 Eigenvectors and Their Corresponding Eigenvalues 85
• 4.5 An SVD Representation Model for Space–Time Data 87
• 4.6 SVD Analysis of Southern Oscillation Index 91
• 4.6.1 Standardized SLP Data and SOI 91
• 4.6.2 Weighted SOI Computed by the SVD Method 94
• 4.6.3 Visualization of the ENSO Mode Computed from the SVD Method 98
• 4.7 Mass Balance for Chemical Equations in Marine Chemistry 100
• 4.8 Multivariate Linear Regression Using Matrix Notations 101
• 4.9 Chapter Summary 103
• References and Further Readings 105
• Exercises 106

• 5 Energy Balance Models for Climate 108
• 5.1 EBM for Modeling the Moon’s Surface Temperature 108
• 5.1.1 Moon–Earth–Sun Orbit and Lunar Surface 109
• 5.1.2 Moon’s Surface Temperature 110
• 5.1.3 EBM Prediction for the Moon Surface Temperature 113
• 5.2 EBM for the Global Average Surface Temperature of the Earth: A Zero-Dimensional Climate Model 116
• 5.2.1 The Incoming Power from the Solar Radiation to the Earth 117
• 5.2.2 The Outgoing Power from Long-Wave Radiation Emitted by the Earth 118
• 5.2.3 EBM as a Power Balance 119
• 5.3 EBM for the Global Average Surface Temperature of an Earth with a Nonlinear Albedo Feedback 120
• 5.4 Time-Dependent Zero-Dimensional EBM for the Earth’s Global Average Surface Temperature 123
• 5.4.1 An EBM Including Time Dependence 124
• 5.4.2 Stability Analysis of the Multiple Solutions of the EBM with a Nonlinear Albedo Feedback 125
• 5.4.3 Energy Flow Budget and Greenhouse Effect for the Earth’s Climate 126
• 5.5 Increasing the Complexity of Climate Models 129
• 5.6 Chapter Summary 131
• References and Further Readings 132
• Exercises 136

• 6 Calculus Applications to Climate Science I: Derivatives 138
• 6.1 Stefan–Boltzmann Law and Budyko’s Approximation 138
• 6.2 Linear Approximation 141
• 6.3 Bisection Method for Solving Nonlinear Equations 143
• 6.4 Newton’s Method 145
• 6.5 Examples of Higher-Order Derivatives 148
• 6.6 Pressure Gradient Force and Coriolis Force 151
• 6.7 Spatiotemporal Variations of the Atmospheric and Oceanic Temperature Fields 153
• 6.8 Taylor Polynomial as a High-Order Approximation 155
• 6.8.1 Taylor’s Theorem 156
• 6.8.2 Taylor Series Example: Exponential Function 161
• 6.8.3 Numerical Integration Using Taylor Expansion 162
• Chapter Summary 163
• References and Further Readings 165
• Exercises 166

• 7 Calculus Applications to Climate Science II: Integrals 169
• 7.1 Geopotential and Atmospheric Pressure 169
• 7.1.1 Vertical Forces on a Small Parcel of Atmosphere 169
• 7.1.2 Geopotential 173
• 7.2 Hypsometric Equation: Exponential Decrease of Pressure with Respect to Elevation 175
• 7.2.1 The General Hypsometric Equation 175
• 7.2.2 An Application of the Hypsometric Equation: Calculate the Elevation of Mount Mitchell 181
• 7.2.3 Hypsometric Equation for an Isothermal Layer 182
• 7.2.4 Error Estimate of the Linear Approximation to the Hypsometric Equation 184
• 7.2.5 Applications of Geopotential Height in Radiosonde Measurements 185
• 7.3 Work Done by an Air Mass in Expansion 187
• 7.4 Internal Energy, Enthalpy, and Entropy 188
• 7.4.1 Internal Energy and Enthalpy 188
• 7.4.2 Entropy 191
• 7.5 Use of Integrals to Derive Stefan–Boltzmann’s Blackbody Radiation Formula from Planck’s Law of Radiation 191
• 7.6 Chapter Summary 196
• References and Further Readings 197
• Exercises 199

• 8 Conservation Laws in Climate Dynamics 203
• 8.1 Conservation of Mass 203
• 8.1.1 Basic Elements of the Continuum Mechanics Method for Climate Modeling 204
• 8.1.2 Lagrangian and Eulerian Observers, and Mass Conservation in the Lagrangian Framework 205
• 8.1.3 Total Derivative 207
• 8.1.4 Mass Conservation in the Eulerian Framework 208
• 8.2 Conservation of Momentum Over a Grid Box: F = ma 212
• 8.3 The Equations of Momentum Conservation in x, y, z,t Coordinates 215
• 8.4 Geostrophic Approximation of the Momentum Equations 217
• 8.4.1 Mathematical Description of the Geostrophic Approximation 218
• 8.4.2 Flow Direction Perpendicular to the PGF under the Geostrophic Approximation 218
• 8.5 The Potential Vorticity Conservation Equation 220
• 8.5.1 Absolute Vorticity and Relative Vorticity 220
• 8.5.2 Potential Vorticity and Its Conservation 220
• 8.5.3 Mathematical Derivations of the Conservation of Potential Vorticity 221
• 8.6 Chapter Summary 224
• References and Further Readings 226
• Exercises 227

• 9 R Graphics for Climate Science 229
• 9.1 Two-Dimensional Line Plots and Setups of Margins and Labels 229
• 9.1.1 Plot Two Different Time Series on the Same Plot 229
• 9.1.2 Figure Setups: Margins, Fonts, Mathematical Symbols, and More 231
• 9.1.3 Plot Two or More Panels on the Same Figure 234
• 9.2 Color Contour Maps 235
• 9.2.1 Basic Principles for an R Contour Plot 235
• 9.2.2 Plot Contour Color Maps for Random Values on a Map 235
• 9.2.3 Plot Contour Maps from Climate Model Data in NetCDF Files 237
• 9.3 Plot Wind Velocity Field on a Map 243
• 9.3.1 Plot a Wind Field Using arrow.plot 243
• 9.3.2 Plot a Surface Wind Field from netCDF Data 244
• 9.4 ggplot for Data 246
• 9.5 Animation 247
• 9.6 Chapter Summary 249
• References and Further Readings 250
• Exercises 251

• 10 Advanced R Analysis and Plotting: EOFs, Trends, and Global Data 253
• 10.1 Ideas of EOF, PC, and Variances Computed from SVD 253
• 10.2 2Dim Spatial Domain EOFs and 1Dim Temporal PCs 254
• 10.2.1 Generate Synthetic Data by R 254
• 10.2.2 SVD for the Synthetic Data EOFs, Variances, and PCs 256
• 10.3 From Climate Data Download to EOF and PC Visualization: An NCEP/NCAR Reanalysis Example 260
• 10.3.2 Space–Time Data Matrix and SVD 262
• 10.4 Area-Weighted Average and Spatial Distribution of Trend 270
• 10.4.1 Global Average and PC1 270
• 10.4.2 Spatial Pattern of Linear Trends 271
• 10.5 GPCP Precipitation Data: Analysis and Visualization by R 272
• 10.5.1 Read and Write GPCP Data 273
• 10.5.2 GPCP Climatology and Standard Deviation 275
• 10.6 Chapter Summary 277
• References and Further Readings 278
• Exercises 278

• 11 R Analysis of Incomplete Climate Data 281
• 11.1 The missing data problem
• 11.2 Read NOAAGlobalTemp and Form the Space–Time Data Matrix 282
• 11.2.2 Plot the Temperature Data Map of a Given Month 285
• 11.2.3 Extract the Data for a Specified Region 285
• 11.2.4 Extract Data from Only One Grid Box 287
• 11.3 Spatial Averages and Their Trends 287
• 11.3.1 Compute and Plot the Global Area-Weighted Average of Monthly Data 287
• 11.3.2 Percent Coverage of the NOAAGlobalTemp 288
• 11.3.3 Compare Trends and Variances at Two Different Locations 289
• 11.3.4 Which Month Has the Strongest Trend? 291
• 11.3.5 Spatial Average of Annual Data 293
• 11.3.6 Nonlinear Trend of the Global Average Annual Mean Data 293
• 11.4 Spatial Characteristics of the Temperature Change Trends 295
• 11.4.1 The Twentieth-Century Temperature Trend 295
• 11.4.2 Twentieth-Century Temperature Trend Computed under a Relaxed Condition 297
• 11.4.3 Trend Pattern for the Four Decades of Consecutive Warming: 1976–2016 298
• 11.5 Chapter Summary 299
• References and Further Readings 300
• Exercises 301

• Appendix A Dot Product of Two Vectors 303
• A.1 Two Definitions for the Dot Product 303
• A.2 Solar Power Flux to the Earth’s Surface and Seasonality 305
• A.3 Divergence Theorem for the Mass Continuity Equation in Climate Models 305

• Appendix B Cross Product of Two Vectors 307
• B.1 Definition of the Cross Product of Two Vectors 307
• B.2 Coriolis Force 309
• B.3 Vorticity 309
• B.4 Stokes’ Theorem 310

• Appendix C Spherical Coordinates 314
• C.1 Transform between the Spherical Coordinates and Cartesian Coordinates 314
• C.2 Area and Volume Differentials in Spherical Coordinates 315

• Appendix D Calculus Concepts and Methods for Climate Science 318
• D.1 Descartes’ Direct Calculus for Functions of a Single Variable 318
• D.2 Calculus from a Statistics Perspective 338
• D.3 Differentiation Methods and Higher Derivatives 348
• D.4 Calculus for Functions of Two and More Variables 351
• References and Further Readings 364
• Exercises 367

• Appendix E Sample Solutions to the Climate Mathematics Exercises 371

• Glossary 380

• Index 386