Climate Mathematics: Theory and Applications

Front matter
Copyright
Dedications
Contents
Preface
Acknowledgements
Main Symbols and Acronyms
1 Dimensional Analysis for Climate Science
	1.1 Dimension and Units
	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 IDim 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.l 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.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