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978-0-521-88916-2 — Light Scattering by Ice Crystals
Kuo-Nan Liou , Ping Yang
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LIGHT SCATTERING B Y ICE CRYSTA LS
Fundamentals and Applications
This research volume outlines the scientific foundations that are central to our current understanding of light scattering, absorption and polarization processes involving ice crystals.
It also demonstrates how data from satellite remote sensing of cirrus clouds (comprising
various ice crystal sizes and morphologies) can be combined with radiation parameterizations in climate models to estimate the role of these clouds in temperature and precipitation
responses to climate change. Providing a balanced treatment of both the fundamentals and
applications, this book synthesizes the authors’ own work, as well as that of other leading
researchers in this area, in a coherent and logical presentation. Numerous illustrations are
included, including three-dimensional schematics, in order to provide a concise discussion
of the subject and enable easy visualization of the key concepts.
This book is intended for active researchers and advanced graduate students in atmospheric science, climatology and remote sensing, as well as scholars in related fields such
as ice microphysics, electromagnetic wave propagation, geometric optics, radiative transfer
and cloud–climate interactions.
kuo-nan liou is a Distinguished Professor of Atmospheric Sciences and founding
director of the Joint Institute for Regional Earth System Science and Engineering at the
University of California, Los Angeles. Dr. Liou has received numerous awards including
the 1998 Jule G. Charney Award from the American Meteorological Society, for his pioneering work in the theory and application of radiative transport and its interaction with
clouds, and the 2013 Roger Revelle Medal from the American Geophysical Union for outstanding contributions in atmospheric sciences. He is a member of the National Academy
of Engineering and of Academia Sinica.
ping yang is Professor and Head of the Department of Atmospheric Sciences, and the
David Bullock Harris Chair in Geosciences, at Texas A&M University, where his research
interests cover the areas of remote sensing and radiative transfer. He received the 2013
Ascent Award from the Atmospheric Sciences Section of the American Geophysical Union,
of which he is an elected Fellow.
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Cambridge University Press
978-0-521-88916-2 — Light Scattering by Ice Crystals
Kuo-Nan Liou , Ping Yang
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L I G H T S C AT T E R I N G B Y
I C E CRY S TA L S
Fundamentals and Applications
K U O -NA N L I OU
University of California, Los Angeles
and
P I N G YA N G
Texas A&M University
with contributions by
YO S H I H I D E TA K A N O
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Kuo-Nan Liou , Ping Yang
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University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
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C Kuo-Nan Liou and Ping Yang 2016
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no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2016
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978-0-521-88916-2 — Light Scattering by Ice Crystals
Kuo-Nan Liou , Ping Yang
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Epigraph
To all the happy ice crystals in planetary atmospheres
Let there be light.
Let there be beautiful ice crystals in the air and mountain ranges.
And here come the reindeers and Santa Claus carrying Maxwell’s equations, and
light rays are shining in the wonderlands.
Let the glory of Geometric Optics for ice crystals, Newton’s optics, and sun’s
light rays rise again from the horizon.
Let ice crystals’ old friends – black carbon and dust – be not forgot for Auld
Lang Syne.
And ice crystals are carried by the ceaseless winds; and
After traveling thousands of miles up and down, the sky looks very blue.
Let there be space missions to tender ubiquitous light rays in the sky,
And all things considered, let light scattering by ice crystals in remote sensing
and climate change be a delight.
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978-0-521-88916-2 — Light Scattering by Ice Crystals
Kuo-Nan Liou , Ping Yang
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Contents
Preface
1 Ice in the Earth’s atmosphere
1.1 Introduction to clouds
1.2 Geographical and temporal distributions of ice in the Earth’s atmosphere
1.2.1 A global perspective on clouds
1.2.2 An example of global cloud climatology
1.2.3 View from ground-based instruments
1.2.4 View from specific satellite instruments
1.3 Formation of ice crystals
1.3.1 The structure of ice crystals
1.3.2 Homogeneous and heterogeneous nucleation
1.3.3 Atmospheric ice nuclei
1.3.4 Secondary ice particle production
1.4 Growth of ice crystals
1.4.1 Growth by diffusion
1.4.2 Growth by accretion
1.5 Illustration of ice crystal morphology
1.5.1 Some historical perspectives
1.5.2 Classification of ice crystals
1.5.3 Computer generation of three-dimensional ice crystals
1.6 Ice crystal size and shape distributions in cirrus clouds
1.6.1 Introductory remarks
1.6.2 Ice crystal size and shape spectra
1.6.3 Correlation between ice crystal maximum dimension and width
1.7 Correlation between ice water content and ice crystal size
1.7.1 Introductory note
1.7.2 Correlation analysis
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1.8
Numerical modeling of cirrus cloud formation
1.8.1 A review
1.8.2 A two-dimensional cirrus cloud model
1.8.2.1 Ice microphysics
1.8.2.2 Radiative transfer
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2 Fundamentals of light scattering by ice crystals
2.1 Introductory remarks
2.2 Maxwell’s equations, wave equations, and boundary conditions
2.2.1 Maxwell’s equations
2.2.2 Boundary conditions
2.2.2.1 Normal components
2.2.2.2 Tangential components
2.2.3 Wave equations and some analytic solutions
2.2.3.1 Spherical coordinates
2.2.3.2 Cylindrical coordinates
2.2.3.3 Spheroidal coordinates
2.3 Optical properties of ice: index of refraction
2.3.1 Complex index of refraction and dispersion of light
2.3.2 Optical properties of ice
2.3.3 Indices of refraction for ice
2.4 Definition of the single-scattering and polarization properties
of ice crystals
2.4.1 Representation of a simple wave
2.4.2 Representation of electromagnetic waves by Stokes parameters
2.4.3 Scattering phase matrix for an ice crystal
2.4.4 Extinction cross-section for a non-spherical ice crystal
2.4.5 Scattering phase matrix for an ensemble of ice crystals
2.4.6 Some examples of the single-scattering properties of ice spheres,
cylinders, and spheroids
2.5 Introduction to radiative transfer in ice crystal clouds: link to single
scattering
2.5.1 Formulation of scalar and vector radiative transfer equations
2.5.2 Radiative transfer in horizontally oriented ice particles
3 Principles of geometric optics for application to light scattering by ice crystals
3.1 Application of geometric optics to light scattering by ice crystals
3.1.1 Snell’s and Fresnel’s laws for geometric optics
3.1.2 Absorption effects in the context of geometric optics
3.1.3 Geometric ray tracing and the Monte Carlo hit-and-miss approach
to ice crystals
3.1.4 Illustration of ice optics in the atmosphere
3.1.4.1 22° and 46° halos
3.1.4.2 Sundogs and their geometry
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3.1.4.3 Horizontally oriented plate and column crystals
3.1.4.4 An observed halo and arc complex pattern
3.2 Fraunhofer diffraction for light scattering by ice crystals
3.2.1 Sphere
3.2.2 Rectangular parallelepiped
3.2.3 Hexagonal cylinder
3.2.4 Spheroid
3.2.5 Some numerical results
3.3 Generalized geometric optics for light scattering by ice crystals
3.3.1 Introduction
3.3.2 Conventional approach
3.3.3 Improved geometric-optics approach using the electromagnetic
equivalence theorem
3.3.4 Ray-by-ray approach to extinction and absorption calculations
3.3.5 Contributions of surface waves to spheres: the edge effect
3.3.6 The geometric-optics and surface-wave approach for spheres
3.3.7 Application of the geometric-optics and surface-wave approach to
ice crystals
3.3.7.1 Fundamentals
3.3.7.2 Geometric optics
3.3.7.3 Addition of surface waves
3.3.7.4 Comparison with available FDTD and DDA results
3.3.8 Geometric-optics and surface-wave approach coupled with the
Rayleigh–Gans–Debye approximation
3.4 A unified theory of light scattering by ice crystals based on generalized
geometric optics
3.4.1 The essence of a unified theory of light scattering by ice crystals
3.4.2 Laboratory scattering and ice microphysics experiments with ice
crystals
3.4.3 Applications to snow grains contaminated by black carbon (BC)
and dust
4 Other useful approaches to light scattering by ice particles
4.1 Finite-difference time domain method
4.1.1 A brief review
4.1.2 Numerical discretization of Maxwell’s curl equations
4.1.3 Absorbing boundary conditions and electromagnetic fields in the
frequency domain
4.1.4 Near-to-far-field mapping and optical property calculations
4.1.5 Validation of the method and some pertinent results
4.1.6 Pseudo-spectral time domain method
4.2 T-matrix numerical method
4.2.1 Fundamentals of the T-matrix approach
4.2.2 Invariant imbedding T-matrix method
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Contents
4.3
Discrete dipole approximation
4.3.1 Fundamentals of the DDA approach
4.3.2 Comparison among DDA, FDTD, and PSTD
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5 Application of light scattering by ice crystals to remote sensing
5.1 Introduction to remote sensing of ice clouds in the Earth’s atmosphere
5.1.1 Atmospheric composition and structure
5.1.2 Atmospheric absorption spectrum
5.1.3 Relative position of the sun–satellite system and radiative transfer
5.1.4 Bidirectional reflectance spectra of ice
5.1.5 The A-Train satellite constellation
5.2 Determination of cirrus cloud optical depth and ice crystal size
5.2.1 Definition of optical depth and mean effective size
5.2.2 Two-channel inversion technique
5.2.3 The importance of ice crystal phase function in the two-channel
approach
5.2.4 The current MODIS operation program for ice cloud studies
5.2.5 Global mapping of cirrus clouds
5.2.5.1 MODIS
5.2.5.2 AIRS
5.2.5.3 CERES-2
5.2.6 Detection of thin cirrus
5.3 Remote sensing of cirrus cloud vertical size profile
5.3.1 The importance of cirrus cloud vertical profiles
5.3.2 Development of a retrieval algorithm for vertical sizing
5.3.3 Validation of the vertical sizing algorithm
5.4 Remote sensing of cirrus clouds using reflected polarization
5.4.1 An historical review of polarization measurements
5.4.2 Reflected polarization of sunlight from ice crystal clouds
5.5 Principle of backscattering depolarization for differentiation of ice
crystals and water drops
5.5.1 Theoretical basis
5.5.2 Issues of orientation and theoretical results
5.5.3 Experimental arrangements and results
5.5.4 Application to lidar (laser radar)
5.6 Remote sensing of cirrus using reflected line spectra in
1.38 and 0.76 µm bands
5.6.1 A brief review of the use of line spectra for Earth
remote sensing
5.6.2 Reflected line spectra in the 1.38 µm band
5.6.3 Reflected line spectra in the 0.76 µm band
5.6.4 Retrieval of cirrus cloud parameters using reflected line spectra
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Contents
6 Application of light scattering by ice crystals to climate studies
6.1 Physical basis for the spectral single-scattering parameterization
of ice crystals
6.1.1 Physical foundations
6.1.2 A generalized single-scattering parameterization for ice crystals
6.1.3 Absorption line formation in the scattering of light by ice crystals
6.1.4 Single-scattering properties of combined ice crystals and
absorbing gases
6.2 Radiative transfer in cirrus cloudy atmospheres
6.2.1 The subject of radiative transfer for application to climate studies
6.2.2 Radiative transfer model results versus aircraft observations for
cirrus clouds
6.2.3 Radiative transfer model results compared with satellite
observations
6.3 Radiative forcing of cirrus clouds
6.3.1 A discussion of the radiative forcing of cirrus clouds
6.3.2 Theoretical analysis for cirrus radiative forcing
6.4 Climatic effects of cirrus clouds
6.4.1 Investigation using one-dimensional climate models
6.4.1.1 A one-dimensional cloud–precipitation–climate model
6.4.1.2 A one-dimensional climate–cirrus-cloud model
6.4.2 Ice microphysics sensitivity investigation using GCMs
6.4.2.1 The effect of interactive ice crystal size on precipitation
simulations
6.4.2.2 The effect of ice crystal shape on cloud radiative forcing
simulations
6.5 Climate issues associated with cirrus clouds
6.5.1 Contrails and contrail cirrus
6.5.1.1 Early studies
6.5.1.2 Contemporary programs and investigations
6.5.2 The role of cirrus in upper-troposphere and lower-stratosphere
exchange
6.5.2.1 Introductory notes
6.5.2.2 Cirrus clouds and the mechanism of stratosphere
dehydration
6.5.3 Thin cirrus clouds and climate
6.5.3.1 Thin cirrus detection and comparison with GCM output
6.5.3.2 Some notes on high thin clouds in global radiative and
climate forcing
References
Index
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Preface
The preparation of Light Scattering by Ice Crystals: Fundamentals and Applications began
about seven years ago. We thought that sufficient material should be available to compose
a high-level text reflective of the complex and intricate domain of ice crystals in the Earth’s
atmosphere and their interaction with “light” from the sun and that emitted from the Earth
and the atmosphere, with applications to remote sensing and climate studies. This text was
supposed to be a three- to four-year project; however, after sifting through the literature
for about two years, gaps emerged on various subjects, including both fundamentals and
applications. For this reason, we conducted additional research in an attempt to bridge
various gaps that are essential, from our perspective, to the unification of all subjects in a
coherent and logical manner with reference to light scattering by ice crystals. Accordingly,
we are pleased to present this text for active researchers and advanced graduate students
who are interested in general areas of atmospheric physics, atmospheric radiative transfer,
atmospheric optics, computational modeling, cloud–climate interactions, and remote sensing of the atmosphere and oceans within the purview of atmospheric and climate sciences.
It is intended to complement other researchers who work in the field of light scattering by
non-spherical particles, which includes ice crystals.
“Ice in the Earth’s atmosphere,” the title of Chapter 1, plays a key role in the hydrological
cycle and precipitation processes. Furthermore, ice clouds in the upper troposphere through
their solar albedo and infrared greenhouse effects are critical elements in determining
surface and atmospheric temperature patterns within the context of greenhouse warming
and climate change induced by man-made perturbations in greenhouse gases and regional air
pollution. In Chapter 1, we introduce cloud classification, a global view of clouds in general,
and cirrus clouds in particular, followed by discussion of the formation and growth of ice
crystals. We then illustrate the complex nature of ice crystals with reference to their size and
three-dimensional morphology based on findings obtained from laboratory experiments and
aircraft observations, to establish a correlation between ice water content and ice crystal
size. This correlation is important in developing radiative transfer parameterizations in
climate models and in understanding the role of ice in climate radiative forcing. Lastly,
we present a two-dimensional cirrus cloud model to illustrate interactions of winds, ice
microphysics, and radiative transfer.
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Preface
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In Chapter 2, “Fundamentals of light scattering by ice crystals,” a number of fundamental
subjects are presented in relation to light scattering by ice crystals. We discuss the scope
and boundaries of light scattering by ice crystals and present the fundamental Maxwell’s
equations, leading to vector wave equations whose solutions require the imposition of
boundary conditions. We show that exact analytic solutions of vector wave equations exist
only for spherical, circular cylindrical, and spheroidal coordinates. The optical properties
of ice are then introduced, followed by a discussion that defines the single-scattering and
polarization properties of non-spherical ice crystals, including the meaning of the scattering
phase matrix. We subsequently discuss the link between single-scattering properties of ice
crystals, deduced from the independent scattering concept, and the transfer of radiation,
including multiple scattering and emission within ice crystal clouds.
Chapter 3, entitled “Principles of geometric optics for application to light scattering
by ice crystals,” presents the geometric-optics approach to light scattering by ice crystals,
starting with an overview of the essence of geometric optics, including diffraction and
surface waves, from several historical perspectives. We then illustrate fascinating ice optics
produced by randomly and horizontally oriented ice particles by means of Monte Carlo
geometric ray tracing. Subsequently, we demonstrate that exact solutions for diffraction
involving a number of ice crystal shapes can be analytically derived, followed by discussion of conventional and improved geometric-optics approaches, and, based on a number of
postulations, introduce surface-wave contributions – the edge effect – for spheres with modification to hexagonal ice crystals. Lastly, we present a unified theory of light scattering by
ice crystals on the basis of the geometric-optics surface-wave approach. In this discussion,
theoretical phase functions are compared with those determined from controlled laboratory
light scattering and spectroscopic experiments, as well as application to light absorption
and scattering by snow grains internally contaminated by black carbon and dust particles
wherein stochastic processes are further introduced.
In Chapter 4, “Other useful approaches to light scattering by ice particles,” we confine
our presentations to three contemporary numerical approaches to light scattering by nonspherical particles within the purview of their applications to light scattering by ice crystals,
namely the finite-difference time domain method, the T-matrix numerical method, and the
discrete dipole approximation. Single-scattering and phase matrix results determined from
these methods for applicable ice crystal size and shape ranges have been used to cross check
and calibrate those computed from a number of geometric-optics approaches. Moreover, we
develop numerical techniques to improve the first two methodologies in terms of particle
size applicability, as well as application to intricate particle shapes.
The subject of “Application of light scattering by ice crystals to remote sensing” is
presented in Chapter 5, wherein we first discuss atmospheric composition and structure,
the atmospheric absorption spectrum, sun–satellite geometry, radiative transfer, and the
contemporary A-Train satellite constellation, which are important for cirrus cloud detection
and quantification. From that base, we then present the subject of retrieving the optical depth
and ice crystal size of cirrus clouds using reflected visible and near-infrared radiation and
illustrate the importance of the phase function of ice crystals. A discussion follows on
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xiv
Preface
detecting thin cirrus and vertical sizing of cirrus cloud layers. We subsequently cover the
subjects of remote sensing of ice clouds using reflected polarization and the principle of
backscattering depolarization to differentiate ice crystals and water droplets. Lastly, we
present reflected line spectra in the 1.38 µm band and the oxygen A-band for inferring the
composition and optical properties of high clouds.
Chapter 6, the last chapter of the text, comprises discussions of “Application of light
scattering by ice crystals to climate studies.” Herein, we present the physical foundations
for parameterization of the extinction and absorption coefficients and phase function, for
ice crystals imbedded in gaseous absorption line formation. We then discuss delta-twostream and delta-four-stream approximations for efficient radiative flux transfer in nonhomogeneous plane-parallel atmospheres and compare theoretical results with aircraft
and satellite observations. On this basis, we present radiative forcing of cirrus clouds
from the viewpoint of theoretical calculations and point out the prevalence of the infrared
greenhouse effect over its solar albedo counterpart. This is followed by a presentation of
climatic effects of cirrus clouds from the perspective of one-dimensional climate models,
where we point out that cloud cover, ice water path, and ice crystal size are influenced by
temperature increases in greenhouse warming scenarios. Also discussed are examples of the
impacts of microphysics on precipitation and radiative forcings using results analyzed from
global climate model simulations. We then present a number of climatic issues associated
with cirrus clouds, including contrails and induced contrail cirrus produced by high-flying
aircraft, a man-made perturbation, their role in upper troposphere and lower stratosphere
exchanges, and the usefulness of optically thin cirrus data determined from modern satellite
instrumentation for ice cloud parameterization development in climate models.
In view of the above, the subject of light scattering by ice crystals as presented in this
text has made definitive contributions to fundamental understanding of and insight into
light scattering, absorption, and polarization processes involving ice crystals. Moreover,
light scattering by ice crystals has provided a new dimension and valuable datasets to the
development of satellite remote sensing of ubiquitous cirrus clouds comprising various ice
crystal sizes and morphologies, as well as to radiation parameterizations for these clouds
in climate models to investigate uncertainties surrounding their role in temperature and
precipitation responses to global warming and climate change.
The senior author wishes to acknowledge the National Science Foundation for its support over the last 30 years or so of his basic research on light scattering by ice crystals
and related subjects in radiative transfer at the University of Utah and the University of
California, Los Angeles. In particular, he would like to thank the Atmospheric Sciences
Section for a Creativity Award based on contributions to “Light Scattering by Ice Crystals:
Theory and Experiment” (1996) through the recommendation of the late Ronald Taylor and
subsequent support from R. R. Rogers, Bradley Smull, and Chungu Lu, Program Directors
for Dynamical and Physical Meteorology. Additionally, he thanks Roberto Peccei, a particle physicist and UCLA Emeritus Vice Chancellor for Research, and Joseph Rudnick, a
condensed matter physicist and Senior Dean of the UCLA College of Letters and Science,
for unwavering support in his pursuit of academic and research excellence and the founding
of the Joint Institute for Regional Earth System Science and Engineering at UCLA. He is
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xv
also exceedingly grateful to Wilfried Brutsaert and Richard Goody for their encouragement
and advice.
The co-author is grateful to Kuo-Nan Liou, Warren Wiscombe, George Kattawar, and
Jerry North for long-term mentorship. He would also like to thank Hal Maring and Lucia
Tsaoussi, NASA Program Managers, and Chungu Lu, a National Science Foundation
Program Director, for their encouragement in his pursuit of excellence in light scattering
research over the years.
We are indebted to several individuals whose efforts have directly impacted the final
production of this volume: Tara Fickle, a former graduate student in the UCLA English
Department and a part-time editor from 2008 to 2014, for her assistance with typing,
editing, and computer graphics; Janice Amar, retired Chief Operating Officer at the UCLA
Joint Institute for Regional Earth System Science and Engineering and a recalled Senior
Editor in 2015, for her assistance with final editing, permission requests, and compliance
with Cambridge University Press format requirements; Christine Cho for general support
to the authors on various aspects of the book project; Steve Ou for gathering references
and other materials for a number of sections in Chapters 5 and 6; Yu Gu for assistance
with the discussion in Section 1.8; and Lei Bi and Zhibo Zhang for contributions to
the presentation of Chapter 4 and Subsection 5.2.4, respectively. It would not have been
possible to complete the book project without the dedicated support of Yoshihide Takano,
contributing author, who has also made significant contributions to the field, in carrying
out additional calculations, checking fundamental equations, gathering references from
the Internet, as well as making revisions to three-dimensional and color presentations for
numerous figures in the text. We thank Qiang Fu, Professor of Atmospheric Sciences at
the University of Washington, who has made pioneering contributions in areas presented in
Chapter 6, for his critical review and constructive comments. Lastly, we wish to thank Bryan
Baum, Qiming Cai, Andrew Heymsfield, Yongxiang Hu, Hironobu Iwabuchi, Brian Kahn,
George Kattawar, Michael King, Patrick Minnis, Michael Mishchenko, Steven Platnick,
and Manfred Wendisch for research collaboration during the course of our work on light
scattering by ice crystals.
Finally, we acknowledge the generosity of the American Geophysical Union and John
Wiley & Sons (Figures 1.7, 1.8, 1.23, 1.24, 3.32a, 3.32c, 3.33, 5.9, 5.10, 5.11, 5.22,
5.23, 5.24, 5.25, 5.32, 6.1, 6.9, 6.12, 6.15, 6.18, 6.19, 6.20), the American Meteorological
Society (Figures 1.4, 1.5, 1.16, 1.17, 1.22, 1.25, 3.5, 4.5, 5.14, 5.26, 5.30, 5.31, 5.33, 6.3,
6.5, 6.6, 6.7, 6.13), Elsevier (Figures 1.3b, 1.13, 2.7, 2.9, 3.4, 3.11, 3.12, 3.18, 3.19, 3.20,
3.21, 3.22, 4.10, 4.13, 4.17, 4.18, 4.19, 4.20, 4.21, 4.24, 6.16), the European Geophysical
Union (Figures 5.18, 5.27), Hokkaido University (Figures 1.14, 1.15), the Institute of
Electrical and Electronics Engineers (Figures 5.17, 5.19), Nature Publishing Group (6.17),
Taylor & Francis Ltd. (Figure 1.9), the Optical Society of America (Figures 2.4, 2.8, 2.20,
3.28, 3.29, 3.30, 3.31, 4.3, 4.4, 4.6, 4.7, 4.9, 4.11, 4.12, 4.28), Oxford University Press
(Figures 1.10, 5.34), Springer-Verlag (Figure 4.8), and colleagues who have furnished
unpublished material – Les Cowley (Figure 3.9b), Edwin W. Eloranta (Figure 1.6), Marko
Riikonen (Figure 3.9a), Ulrich Schumann (Figure 1.3a), and Stefan Bauer, Janek Zimmer,
and Uwe Reiss (Figure 1.1) – for permission to use this material in this text.
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