Climate Mathematics Theory and Applications 1st Edition by Samuel SP Shen, Richard CJ Somerville ISBN 9781108476874 1108476872

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Product details:

ISBN 10: 1108476872
ISBN 13: 9781108476874
Author: Samuel SP Shen, Richard CJ Somerville

Climate Mathematics Theory and Applications 1st Edition Table of contents:

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 La

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

B.4.1 Stokes’ Theorem

B.4.2 Green’s Theorem

B.4.3 GPS-Planimeter as a Smartphone App

B.4.4 Boundary Data and Integration inside the Boundary

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.1.1 Slope and DA Pairs

D.1.2 Height Increment and Integrals

D.1.3 Discussion and Mathematical Rigor of Direct Calculus

D.1.4 Descartes’ Method of Tangents and Brief Historical Note

D.1.5 Summary and Discussion of DD Calculus

D.2 Calculus from a Statistics Perspective

D.2.1 Arithmetic Mean, Sampling, and Average of a Function

D.2.2 Definition of Integral, Antiderivative, and DA Pair

D.2.3 Calculation of an Integral [int[sub(a)]sub(b)]] f(x) dx

D.2.4 Using Average Speed and Graphic Mean to Interpret the Meaning of a Derivative

D.2.5 Summary of Calculus from a Statistics Perspective

D.3 Differentiation Methods and Higher Derivatives

D.3.1 Rules of Differentiation

D.3.2 Higher Derivatives

D.4 Calculus for Functions of Two and More Variables

D.4.1 Introduction of Functions of Multiple Variables

D.4.2 Calculate Partial Derivatives

D.4.3 Traveling Wave Solution to the Wave Equation

D.4.4 Diffusive Solution to a Heat Equation

D.4.5 Differentials and Exact Differentials

D.4.6 Integral of a Multivariate Function

D.4.7 Line Integral and Work

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Tags: Samuel SP Shen, Richard CJ Somerville, Climate Mathematics, Applications