Samuel P. Shen & Richard C. Somerville (2019) Climate Mathematics

  • 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 . ウェブサイトwww.cambridge.org/9781108476874

2019年の新刊。わたしはその年の年末に東京の洋書屋の店頭でみつけた。大学院生の指導をしていたらすぐ買って読ませようと思っただろうが、自分だけが読むのには かさばらないほうがよいので、ディジタル版 (わたしの習慣で Amazon Kindle)を買った。

気候の定量的研究、とくに簡単な気候モデルによる計算と、複雑な気候モデルによるシミュレーション結果の解析などをしたい人のための、数学的手法の教科書だ。数学の分野としては、線形代数、微分・積分、3次元ベクトルの内積・外積・勾配・発散・回転など、初歩的統計学などがある。統計学およびデータの図化の道具としては、プログラム言語 R をつかっている。第1章に出てくるのは、ふつう数学の範囲にはふくめない、物理量の次元と単位のあつかい、そして「次元解析」だ。このような、数学的手法や、物理科学共通の手法は、専門にすすむまえの大学初級の科目として学ぶようになっていることが多いだろう。しかし、(わたし自身、大学初級のころ思ったことだが) 専門としてやりたい課題に関連づけられると手法の勉強をする元気がでるが、手法だけを提示されても元気がでない人が多いと思う。また、手法は理解していても、専門の課題への適用例があるのはたすかる。本書のような本は必要とされていると思う。

気候特有の話題としては、第5章で、エネルギー収支にもとづく(力学をふくまない) 簡単な気候モデルがあつかわれている。第8章では、力学をふくむ大循環モデルの基本となる物理法則である、質量保存と、運動方程式(「運動量の保存」というあつかいをしている。球面を接平面で近似したあつかいなのだ)、そして地球流体特有の概念であるポテンシャル渦度の保存をあつかう。力学がふくまれるばあいのエネルギー保存の議論はみあたらないので、これで大循環モデルの力学部分の要素がそろったわけではないのだが、初級の教科書としてはこのくらいまでがよいと判断したようだ。第10-11章の統計的データ処理の材料としては、格子点化処理ずみの観測データと、観測と数値モデルを組み合わせた「データ同化」の成果である「再解析」プロダクトがとりあげられている。第11章ではとくにデータ値の欠損があるばあいをあつかっている。

Somerville さんは、数値モデルをつかった気候の研究では知られた人で、名誉教授になっているが、UCSDでの教育にはかかわっていて、近くのSan Diego State Universityで教えているShenさんに非常勤で授業をやってもらい、いっしょに本をまとめることにしたらしい。Shenさんは中国で学士、アメリカで博士をとり、カナダなどの大学で数学・統計学を教える職につきながら、研究対象としては気候変化をとりあげてきた。東大の(当時の名まえで)気候システム研究センター(CCSR)に客員としていたこともある(ことが、アーカイブとして残されているCCSRのページを見たらわかった)。

日本語版もあるとよいと思うが、わたしが率先して翻訳しようというほどではない。(いつもよい方法ではないのだが、この本のばあいは)学生がおおぜいいる研究室で分担して翻訳するとよいかもしれない。

折りかえしのあとに くわしい目次をつける。Kindle版をもとにした。MS Windows用のKindleソフトウェアで、本のなかみからのコピー(そしてWindows上のエディタに はりつけ)ができるようになったので、それを利用したうえで編集した。ただし、章・節の番号と題名がくっついてしまって(tabコードがぬけたのだと思う)、あとで空白をいれる修正をした。

== 目次 ==
Cover page
Half Title page
Title page
Copyright page
Dedication
Contents
Preface
Acknowledgements
Main Symbols and Acronyms
1 ​Dimensional Analysis for Climate Science
– 1.1 ​Dimension and Units
– 1.2​ Fundamental Dimensions: LMT θI-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

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