Mathematical and Physical Fundamentals of Climate Change (eBook)

John C. Moore, Zhihua Zhang (Autoren)

eBook Download: PDF | EPUB

2014 | 1. Auflage
494 Seiten
Elsevier Science (Verlag)
978-0-12-800583-5 (ISBN)

Infos

Mathematical and Physical Fundamentals of Climate Change is the first book to provide an overview of the math and physics necessary for scientists to understand and apply atmospheric and oceanic models to climate research. The book begins with basic mathematics then leads on to specific applications in atmospheric and ocean dynamics, such as fluid dynamics, atmospheric dynamics, oceanic dynamics, and glaciers and sea level rise. Mathematical and Physical Fundamentals of Climate Change provides a solid foundation in math and physics with which to understand global warming, natural climate variations, and climate models. This book informs the future users of climate models and the decision-makers of tomorrow by providing the depth they need. Developed from a course that the authors teach at Beijing Normal University, the material has been extensively class-tested and contains online resources, such as presentation files, lecture notes, solutions to problems and MATLab codes. - Includes MatLab and Fortran programs that allow readers to create their own models - Provides case studies to show how the math is applied to climate research - Online resources include presentation files, lecture notes, and solutions to problems in book for use in classroom or self-study

Zhihua Zhang is a Taishan distinguished professor and director of climate modeling laboratory in Shandong University, China. His research interests are Mechanisms of Climate Change, Big Data Mining, Carbon Emissions, Climate Policy and Sustainability. Prof. Zhang has published 4 first-authored books and about 50 first-authored papers. He is a Chief Editor, Associate Editor, or Editorial Board Member in several global and regional known journals on Climate Change, Meteorology and Environmental Data.

Inhalt

Front Cover 1
Mathematical and Physical Fundamentals of Climate Change 4
Copyright 5
Contents 6
Preface: Interdisciplinary Approaches to Climate Change Research 12
Chapter 1: Fourier Analysis 14
1.1 Fourier Series and Fourier Transform 14
1.2 Bessel's Inequality and Parseval's Identity 31
1.3 Gibbs Phenomenon 35
1.4 Poisson Summation Formulas and Shannon Sampling Theorem 39
1.5 Discrete Fourier Transform 48
1.6 Fast Fourier Transform 51
1.7 Heisenberg Uncertainty Principle 55
1.8 Case Study: Arctic Oscillation Indices 58
Problems 59
Bibliography 60
Chapter 2: Time-Frequency Analysis 62
2.1 Windowed Fourier Transform 62
2.2 Wavelet Transform 64
2.3 Multiresolution Analyses and Wavelet Bases 68
2.3.1 Multiresolution Analyses 68
2.3.2 Discrete Wavelet Transform 73
2.3.3 Biorthogonal Wavelets, Bivariate Wavelets,and Wavelet Packet 75
2.4 Hilbert Transform, Analytical Signal, and Instantaneous Frequency 78
2.5 Wigner-Ville Distribution and Cohen's Class 84
2.6 Empirical Mode Decompositions 88
Problems 89
Bibliography 90
Chapter 3: Filter Design 92
3.1 Continuous Linear Time-Invariant Systems 92
3.2 Analog Filters 95
3.3 Discrete Linear Time-Invariant Systems 98
3.3.1 Discrete Signals 98
3.3.2 Discrete Convolution 99
3.3.3 Discrete System 100
3.3.4 Ideal Digital Filters 103
3.3.5 Z-Transforms 103
3.3.6 Linear Difference Equations 105
3.4 Linear-Phase Filters 106
3.4.1 Four Types of Linear-Phase Filters 108
3.4.2 Structure of Linear-Phase Filters 109
3.5 Designs of FIR Filters 110
3.5.1 Fourier Expansions 111
3.5.2 Window Design Method 112
3.5.3 Sampling in the Frequency Domain 113
3.6 IIR Filters 114
3.6.1 Impulse Invariance Method 114
3.6.2 Matched Z-Transform Method 116
3.6.3 Bilinear Transform Method 116
3.7 Conjugate Mirror Filters 117
Problems 121
Bibliography 121
Chapter 4: Remote Sensing 124
4.1 Solar and Thermal Radiation 124
4.2 Spectral Regions and Optical Sensors 126
4.3 Spatial Filtering 128
4.4 Spatial Blurring 129
4.5 Distortion Correction 130
4.6 Image Fusion 132
4.7 Supervised and Unsupervised Classification 133
4.8 Remote Sensing of Atmospheric Carbon Dioxide 134
4.9 Moderate Resolution Imaging Spectroradiometer Data Products and Climate Change 135
Problems 136
Bibliography 136
Chapter 5: Basic Probability and Statistics 138
5.1 Probability Space, Random Variables, and Their Distributions 138
5.1.1 Discrete Random Variables 139
5.1.2 Continuous Random Variables 140
5.1.3 Properties of Expectations and Variances 141
5.1.4 Distributions of Functions of Random Variables 142
5.1.5 Characteristic Functions 143
5.2 Jointly Distributed Random Variables 145
5.3 Central Limit Theorem and Law of Large Numbers 148
5.4 Minimum Mean Square Error 151
5.5 2-Distribution, t-Distribution, and F-Distribution 153
5.6 Parameter Estimation 156
5.7 Confidence Interval 161
5.8 Tests of Statistical Hypotheses 162
5.9 Analysis of Variance 163
5.10 Linear Regression 167
5.11 Mann-Kendall Trend Test 168
Problems 171
Bibliography 172
Chapter 6: Empirical Orthogonal Functions 174
6.1 Random Vector Fields 174
6.2 Classical EOFs 176
6.3 Estimation of EOFs 184
6.4 Rotation of EOFs 186
6.5 Complex EOFs and Hilbert EOFs 191
6.6 Singular Value Decomposition 195
6.7 Canonical Correlation Analysis 198
6.8 Singular Spectrum Analysis 202
6.9 Principal Oscillation Patterns 204
6.9.1 Normal Modes 204
6.9.2 Estimates of Principal Oscillation Patterns 207
Problems 208
Bibliography 209
Chapter 7: Random Processes and Power Spectra 212
7.1 Stationary and Non-stationary Random Processes 212
7.2 Markov Process and Brownian Motion 216
7.3 Calculus of Random Processes 220
7.4 Spectral Analysis 227
7.4.1 Linear Time-Invariant System for WSS Processes 227
7.4.2 Power Spectral Density 229
7.4.3 Shannon Sampling Theorem for Random Processes 232
7.5 Wiener Filtering 234
7.6 Spectrum Estimation 237
7.7 Significance Tests of Climatic Time Series 242
7.7.1 Fourier Power Spectra 242
7.7.2 Wavelet Power Spectra 245
Problems 249
Bibliography 249
Chapter 8: Autoregressive Moving Average Models 252
8.1 ARMA Processes 252
8.1.1 AR(p) Processes 253
8.1.2 MA(q) Processes 254
8.1.3 Shift Operator 257
8.1.4 ARMA(p,q) Processes 258
8.2 Yule-Walker Equation andSpectral Density 261
8.3 Prediction Algorithms 264
8.3.1 Innovation Algorithm 265
8.3.2 Durbin-Lovinson Algorithm 270
8.3.3 Kolmogorov's Formula 273
8.4 Asymptotic Theory 274
8.4.1 Gramer-Wold Device 274
8.4.2 Asymptotic Normality 278
8.5 Estimates of Means and CovarianceFunctions 280
8.6 Estimation for ARMA Models 286
8.6.1 General Linear Model 286
8.6.2 Estimation for AR(p) Processes 288
8.6.3 Estimation for ARMA(p,q) Processes 295
8.7 ARIMA Models 296
8.8 Multivariate ARMA Processes 298
8.9 Application in Climatic and Hydrological Research 300
Problems 301
Bibliography 302
Chapter 9: Data Assimilation 304
9.1 Concept of Data Assimilation 304
9.2 Cressman Method 307
9.3 Optimal Interpolation Analysis 308
9.4 Cost Function and Three-Dimensional Variational Analysis 312
9.5 Dual of the Optimal Interpolation 317
9.6 Four-Dimensional Variational Analysis 318
9.7 Kalman Filter 321
Problems 322
Bibliography 323
Chapter 10: Fluid Dynamics 326
10.1 Gradient, Divergence, and Curl 326
10.2 Circulation and Flux 332
10.3 Green's Theorem, Divergence Theorem, and Stokes's Theorem 334
10.4 Equations of Motion 335
10.4.1 Continuity Equation 335
10.4.2 Euler's Equation 337
10.4.3 Bernoulli's Equation 341
10.5 Energy Flux and Momentum Flux 344
10.6 Kelvin Law 350
10.7 Potential Function and Potential Flow 352
10.8 Incompressible Fluids 354
Problems 358
Bibliography 358
Chapter 11: Atmospheric Dynamics 360
11.1 Two Simple Atmospheric Models 360
11.1.1 The Single-Layer Model 362
11.1.2 The Two-Layer Model 363
11.2 Atmospheric Composition 365
11.3 Hydrostatic Balance Equation 367
11.4 Potential Temperature 369
11.5 Lapse Rate 371
11.5.1 Adiabatic Lapse Rate 372
11.5.2 Buoyancy Frequency 373
11.6 Clausius-Clapeyron Equation 375
11.6.1 Saturation Mass Mixing Radio 376
11.6.2 Saturation Adiabatic Lapse Rate 376
11.6.3 Equivalent Potential Temperature 378
11.7 Material Derivatives 379
11.8 Vorticity and Potential Vorticity 383
11.9 Navier-Stokes Equation 385
11.9.1 Navier-Stokes Equation in an Inertial Frame 385
11.9.2 Navier-Stokes Equation in a Rotating Frame 387
11.9.3 Component Form of the Navier-Stokes Equation 389
11.10 Geostrophic Balance Equations 391
11.11 Boussinesq Approximation and Energy Equation 393
11.12 Quasi-Geostrophic Potential Vorticity 396
11.13 Gravity Waves 399
11.13.1 Internal Gravity Waves 400
11.13.2 Inertia Gravity Waves 404
11.14 Rossby Waves 406
11.15 Atmospheric Boundary Layer 411
Problems 417
Bibliography 418
Chapter 12: Oceanic Dynamics 420
12.1 Salinity and Mass 420
12.2 Inertial Motion 421
12.3 Oceanic Ekman Layer 422
12.3.1 Ekman Currents 423
12.3.2 Ekman Mass Transport 425
12.3.3 Ekman Pumping 427
12.4 Geostrophic Currents 428
12.4.1 Surface Geostrophic Currents 428
12.4.2 Geostrophic Currents from Hydrography 431
12.5 Sverdrup's Theorem 433
12.6 Munk's Theorem 437
12.7 Taylor-Proudman Theorem 441
12.8 Ocean-Wave Spectrum 444
12.8.1 Spectrum 444
12.8.2 Digital Spectrum 445
12.8.3 Pierson-Moskowitz Spectrum 446
12.9 Oceanic Tidal Forces 448
Problems 450
Bibliography 451
Chapter 13: Glaciers and Sea Level Rise 454
13.1 Stress and Strain 454
13.2 Glen's Law and Generalized Glen's Law 456
13.3 Density of Glacier Ice 457
13.4 Glacier Mass Balance 458
13.5 Glacier Momentum Balance 459
13.6 Glacier Energy Balance 462
13.7 Shallow-Ice and Shallow-Shelf Approximations 463
13.8 Dynamic Ice Sheet Models 465
13.9 Sea Level Rise 465
13.10 Semiempirical Sea Level Models 466
Problems 467
Bibliography 467
Chapter 14: Climate and Earth System Models 470
14.1 Energy Balance Models 470
14.1.1 Zero-Dimensional EBM 470
14.1.2 One-Dimensional EBM 471
14.2 Radiative Convective Models 473
14.3 Statistical Dynamical Models 473
14.4 Earth System Models 475
14.4.1 Atmospheric Models 475
14.4.2 Oceanic Models 476
14.4.3 Land Surface Models 478
14.4.4 Sea Ice Models 478
14.5 Coupled Model Intercomparison Project 479
14.6 Geoengineering Model Intercomparison Project 480
Problems 483
Bibliography 483
Index 486

Textauszug

Chapter 1

Fourier Analysis

Abstract

Motivated by the study of heat diffusion, Joseph Fourier claimed that any periodic signals can be represented as a series of harmonically related sinusoids. Fourier’s idea has a profound impact in geoscience. It took one and a half centuries to complete the theory of Fourier analysis. The richness of the theory makes it suitable for a wide range of applications such as climatic time series analysis, numerical atmospheric and ocean modeling, and climatic data mining.

keywords

Fourier series, Fourier transform

Parseval identity

Poisson summation formulas

Shannon sampling theorem

Heisenberg uncertainty principle

Arctic Oscillation index

Motivated by the study of heat diffusion, Joseph Fourier claimed that any periodic signals can be represented as a series of harmonically related sinusoids. Fourier's idea has a profound impact in geoscience. It took one and a half centuries to complete the theory of Fourier analysis. The richness of the theory makes it suitable for a wide range of applications such as climatic time series analysis, numerical atmospheric and ocean modeling, and climatic data mining.

1.1 Fourier series and fourier transform

Assume that a system of functions {ϕn (t)}n ∈+ in a closed interval [a, b] satisfes ab|φn(t)|2dt<∞. if

abφn(t)φ¯m(t)?dt={0(n≠m),1(n=m),

and there does not exist a nonzero function f such that

ab|f(t)|2dt<∞,∫abf(t)φ¯n(t)?dt=0(n∈ℤ+),

then this system is said to be an orthonormal basis in the interval [a, b].

For example, the trigonometric system 2π,1πcosnt,1πsinntn∈ℤ+ and the exponential system 12πeint}n∈ℤ are both orthonormal bases in [−π, π].

Let f(t) be a periodic signal with period 2π and be integrable over [−π, π], write f ∈ L2π. In terms of the above orthogonal basis, let 0(f)=1π∫−ππf(t)dt and

n(f)=1π∫−ππf(t)cos(nt)?dt(n∈ℤ+),bn(f)=1π∫−ππf(t)sin(nt)?dt(n∈ℤ+).

Then a0(f), an(f), bn(f)(n ∈ +) are said to be Fourier coefficients of f. The series

0(f)2+∑1∞(an(f)cos(nt)+bn(f)sin(nt))

is said to be the Fourier series of f. The sum

n(f;t):=a0(f)2+∑1n(ak(f)cos(kt)+bk(f)sin(kt))

is said to be the partial sum of the Fourier series of f. It can be rewritten in the form

n(f;t)=∑−nnck(f)eikt,

where

k(f)=12π∫−ππf(t)e-iktdt(k∈ℤ)

are also called the Fourier coefficients of f.

It is clear that these Fourier coefficients satisfy

0(f)=2c0(f),an(f)=c−n(f)+cn(f),bn(f)=i(c−n(f)−cn(f)).

Let f ∈L2π. If f is a real signal, then its Fourier coefficients an (f) and bn (f) must be real. The identity

n(f)?cos(nt)?+?bn(f)?sin(nt)=An(f)?sin(nt+θn(f))

shows that the general term in the Fourier series of f is a sine wave with circle frequency n, amplitude An, and initial phase θn. Therefore, the Fourier series of a real periodic signal is composed of sine waves with different frequencies and different phases.

Fourier coefficients have the following well-known properties.

Property

Let f, g ∈ L2π and α, β be complex numbers.

(i) (Linearity). cn(αf + βg) = αcn(f) + βcn(g).

(ii) (Translation). Let F(t) =f(t + α). Then cn(F) = einαcn(f).

(iii) (Integration). Let (t)=∫0tf(u)du. If −ππf(t)dt=0, then n(F)=cn(f)in(n≠0)

(iv) (Derivative). If f(t) is continuously differentiable, then cn(f′) = incn(f) (n ≠ 0).

(v) (Convolution). Let the convolution f∗g)(t)=∫−ππf(t−x)g(x)dx. Then cn(f * g) = 2πcn(f)cn(g).

Proof

Here we prove only (v). It is clear that f * g ∈ L2π and

n(f∗g)=12π∫−ππ(f∗g)(t)e-intdt=12π∫−ππ(∫−ππf(t−u)g(u)du)e-intdt.

Interchanging the order of integrals, we get

n(f∗g)=12π∫−ππ(∫−ππf(t−u)e-intdt)g(u)?du.

Let v = t − u. Since f(v)e −inv is a periodic function with period 2π, the integral in brackets is

−ππf(t−u)e−intdt=e−inu∫−π−uπ−uf(v)e−invdv     ???=e−inu∫−ππf(v)e−invdv=2πcn(f)e−inu.

Therefore,

n(f∗g)=cn(f)∫−ππg(u)e−inudu=2πcn(f)cn(g).

Throughout this book, the notation f ∈ L() means that f is integrable over and the notation f ∈ L[a, b] means that f (t) is integrable over a closed interval [a, b], and the integral ℝ=∫−∞∞.

Riemann-LebesgueLemma. If f ∈ L(), then ∫f(t)e−i ω t dt → 0 as ǀωǀ →∞. Especially,

(i) if f ∈ L[a, b], then abf(t)e−iωtdt→0(|ω|→∞);

(ii) if f ∈ L2π, then cn(f) → 0(ǀnǀ → ∞) and an(f) → 0, bn(f) → 0(n → ∞).

The Riemann-Lebesgue lemma (ii) states that Fourier coeffcients off ∈ L2π tend to zero as n →∞.

Proof

If f is a simple step function and

(t)={c,a≤t≤b,0,otherwise,

where c is a constant, then

∫ℝf(t)e−iωtdt|=|∫abce−iωtdt|=|ciω(e−ibω−e−iaω)|≤2|cω|(ω≠0),

and so∫f(t)e− i ωt dt → 0(ǀωǀ → ∞). Similarly, it is easy to prove that for any step function s(t),

ℝs(t)e−iωtdt→0(|ω|→∞).

If f is integrable over , then, for ∈ > 0, there exists a step function s(t) such...

Produkt-Details

• Erscheint lt. Verlag: 6.12.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Biologie ► Ökologie / Naturschutz
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Meteorologie / Klimatologie
• Themenwelt: Technik
• ISBN-10: 0-12-800583-1 / 0128005831
• ISBN-13: 978-0-12-800583-5 / 9780128005835