Zhang / Moore Mathematical and Physical Fundamentals of Climate Change

1. Auflage 2014
ISBN: 978-0-12-800583-5
Verlag: William Andrew Publishing
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 494 Seiten, Format (B × H): 152 mm x 229 mm

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.

Zielgruppe

Upper-level UG/Grad Students, post-docs, researchers in meteorology, climatology, oceanography, earth science and environmental science

Autoren/Hrsg.

Zhang, Zhihua

Moore, John C.

Fachgebiete

• Geowissenschaften Umweltwissenschaften Klimawandel, Globale Erwärmung
• Geowissenschaften Geologie Marine Geologie, Ozeanographie (Meereskunde)
• Geowissenschaften Geologie Meteorologie, Klimatologie

Inhaltsverzeichnis

Preface: Interdisciplinary Approaches to Climate Change Research 1. Fourier Analysis 2. Time-Frequency Analysis 3. Filter Design 4. Remote Sensing 5. Basic Probability and Statistics 6. Empirical Orthogonal Functions (EOFs) 7. Random Processes and Power Spectra 8. Autoregressive Moving Average (ARMA) Models 9. Data Assimilation 10. Fluid Dynamics 11. Atmospheric Dynamics 12. Oceanic Dynamics 13. Glaciers and Sea Level Rise 14. Climate and Earth System Models

Leseproben

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 {?n (t)}n ?+ in a closed interval [a, b] satisfes ab|fn(t)|2dt<8. if

abfn(t)f¯m(t)?dt={0(n?m),1(n=m),

and there does not exist a nonzero function f such that

ab|f(t)|2dt<8,?abf(t)f¯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 2p,1pcosnt,1psinntn?Z+ and the exponential system 12peint}n?Z are both orthonormal bases in [-p, p].

Let f(t) be a periodic signal with period 2p and be integrable over [-p, p], write f ? L2p. In terms of the above orthogonal basis, let 0(f)=1p?-ppf(t)dt and

n(f)=1p?-ppf(t)cos(nt)?dt(n?Z+),bn(f)=1p?-ppf(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+?18(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)=12p?-ppf(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 ?L2p. 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+?n(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 ?n. 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 ? L2p and a, ß be complex numbers.

(i) (Linearity). cn(af + ßg) = acn(f) + ßcn(g).

(ii) (Translation). Let F(t) =f(t + a). Then cn(F) = einacn(f).

(iii) (Integration). Let (t)=?0tf(u)du. If -ppf(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)=?-ppf(t-x)g(x)dx. Then cn(f * g) = 2pcn(f)cn(g).

Proof

Here we prove only (v). It is clear that f * g ? L2p and

n(f*g)=12p?-pp(f*g)(t)e-intdt=12p?-pp(?-ppf(t-u)g(u)du)e-intdt.

Interchanging the order of integrals, we get

n(f*g)=12p?-pp(?-ppf(t-u)e-intdt)g(u)?du.

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

-ppf(t-u)e-intdt=e-inu?-p-up-uf(v)e-invdv     ???=e-inu?-ppf(v)e-invdv=2pcn(f)e-inu.

Therefore,

n(f*g)=cn(f)?-ppg(u)e-inudu=2pcn(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=?-88.

Riemann-LebesgueLemma. If f ? L(), then ?f(t)e-i ? t dt ? 0 as |?| ?8. Especially,

(i) if f ? L[a, b], then abf(t)e-i?tdt?0(|?|?8);

(ii) if f ? L2p, then cn(f) ? 0(|n| ? 8) and an(f) ? 0, bn(f) ? 0(n ? 8).

The Riemann-Lebesgue lemma (ii) states that Fourier coeffcients off ? L2p tend to zero as n ?8.

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-i?tdt|=|?abce-i?tdt|=|ci?(e-ib?-e-ia?)|=2|c?|(??0),

and so?f(t)e- i ?t dt ? 0(|?| ? 8). Similarly, it is easy to prove that for any step function s(t),

Rs(t)e-i?tdt?0(|?|?8).

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