![Preface
Many excellent textbooks describe the physics of climate and give an introduction
to the processes that interact and feedback to produce the earth’s weather and climate
[178, 79, 133, 94, 83, 134]. In this book we approach the subject from another direction,
admitting from the outset that the definition of climate is nebulous and, in fact, still evolv-
ing. The climate will be viewed as multifaceted but always as the solution of a particular
mathematical model. That all climate models are incomplete is a consequence of our lack
of understanding of the physics as well as the incompleteness of mathematical knowledge.
The understanding of the nature of the earth’s climate has improved and changed as
the mathematical notions of what constitutes a solution to a set of partial differential
equations has changed.1
The development of theories of deterministic chaos and the ex-
istence of global attractors as invariant subsets of the time evolution associated with the
Navier–Stokes equations have had a profound influence on climate research [165]. The
invariants of flows, what is conserved, and what coherent structures persist inform the
theory of climate. How oscillations, bifurcations, and singularities arise in the solutions
of partial differential equations is fundamental to drawing a line between natural climate
variability and human induced climate change. A SIAM focus for 2013 on the “Mathe-
matics of Planet Earth” was marked with the notable publication of Kaper and Engler’s
text [100] treating many of the current conceputal models of climate.
The role of general circulation models and high-end computer simulation in the under-
standing of climate should not be underestimated. Describing the principles and practice
of this type of modeling is the primary focus of this book. The first chapter is devoted to
the observations of weather phenomena and the special programs to collect climate data
that provide a wealth of information for calibration, validation, and verification. The data
themselves, however, do not provide the interpretation of climatic events or give a means
of projecting future climate responses. Only high-end models show how the processes
interact and feedback upon one another culminating in weather phenomena and climate.
Chapter 2 introduces the governing equations of geophysical flow that model the circula-
tions of the atmosphere and ocean. Chapter 3 introduces numerical methods for solving
these equations starting from a simplified subset known as the shallow water equations.
High performance computing is a uniquely powerful tool to probe the solutions of the
equations that constitute the model, and this is introduced in Section 3.9. Numerical
methods and algorithms are the backbone of simulations based on general circulation
models. Attention is given to parallel algorithms and promising numerical methods that
will likely figure in the next generation of climate models.
Chapter 4 describes what has been learned from climate simulations, and a few case
studies are presented with the hope that interested readers and students will pursue other
1
The mathematical theory for atmospheric and ocean flows is not complete [26, 164], so there is still room
for growth.
vii](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-11-320.jpg)
![viii Preface
simulation studies following their own interests. Finally, a brief chapter introduces some
of the methods and mathematical basis for the analysis of climate. These methods must
be used to summarize simulation results and, of course, are useful in analyzing weather
data to extract climate statistics and trends.
Exercises are scattered throughout the text as well as references to MATLAB codes
that are part of these exercises and are described in supplemental material available online
at www.siam.org/books/MM19. Since methods and simulation are a thread through-
out the material, students wishing to master the material should gain experience with
computer simulation and programming through these exercises. Full-fledged simulations
using parallel computers requires more sophisticated programming than the MATLAB
environment offers. Usually simulation codes are written in FORTRAN and C++. But
access to the full code and simulation system of the Community Climate System Model
is available to the ambitious reader. For analysis, Python or the NCAR Command Lan-
guage (NCL) are the languages of choice.
Reference is also made to Supplemental Lectures [49], provided in a separate online
volume at www.siam.org/books/MM19. These lectures each start with something we
know fairly well, usually some piece of mathematics, but then branch out to things we
do not understand well. The supplemental lectures serve as a somewhat light-hearted
introduction to research areas where open questions still exist and important perspectives
are emerging. Students seem to appreciate the change of pace offered in these lectures.
The book is the result of a course sequence taught at the University of Tennessee–
Knoxville, in the Civil and Environmental Engineering graduate studies department. I am
grateful to all the graduate students who have asked questions and provided input on my
lectures. In particular, thanks to Dr. Nicki Lam, Dr. Yang Gao, Dr. Abby Gaddis, Ms.
Melissa Allen, Dr. Evan Kodra, Mr. Scott DeNeale, Dr. Abdoul Oubeidillah, Mr. Jian
Sun, and my colleagues Dr. Joshua Fu and Dr. Kathrine Evans. I am indebted to the many
exceptional researchers that I have worked with, learned from, and been inspired by at the
Oak Ridge National Laboratory Climate Change Science Institute and at the National
Center for Atmospheric Research. I owe a particular debt to Dr. David Williamson and
Dr. Warren Washington of the National Center for Atmospheric Research in Boulder.
My admiration for Dr. Washington’s book [178] will be evident throughout this text. In
the DOE National Laboratories, my colleagues Dr. Patrick Worley, Dr. David Bader,
and Dr. Jim Hack have been pioneers in the development of this computational science
discipline. Finally, I wish to thank my wife, Frances, for supporting me in this project
and providing editorial suggestions.](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-12-320.jpg)
![Chapter 1
Earth Observations
Few moments in history stand in such sharp contrast as the moment captured in Fig-
ure 1.1: the natural world in plane view from the most unnatural environment of space.
The questioning of science’s ability to provide answers for our future is part of the post-
modern social fabric in which the climate change discussion is taking place and climate
modelers are questioned intensely. It has been said that if you believe models, you will
believe anything. Yet in the quiet eye of the storm the discussion hinges on things we
know and believe, on physical, chemical, and biological processes, on cause and effect. If
we take an engineering and scientific perspective, we begin with the principles behind the
dynamics and physics of the climate system. The implications of climate change, as they
are currently understood, are strong motivation for the study that leads ultimately to the
question, “So, what are we going to do about it.”
Geologic time scales indicate large variations in earth’s climate. The recent (geologic)
past is known from ice core data obtained by drilling deep into the Antarctic ice sheet.
The research sites at Vostok Station and the European Project for Ice Coring in Antarc-
tica (EPICA) Dome C, have produced a record extending back for the past 800,000 years.
What the cores reveal is a series of eight ice ages and interglacials (the warm periods be-
tween the ice expansions) (Figure 1.2). During these variations the temperature ranges
from −10o
C to +4o
C from the modern baseline. The concentration of atmospheric car-
bon dioxide, CO2, varies from 180 parts per million (ppm) to 300 ppm. Our present
warm period began to develop 30,000 years ago and, looking at the frequency of past vari-
ation, a signal operating on the 23,000 year period emerges. According to this signal, we
are due for a cooling trend and should soon be heading into another ice age.
The timing of the observed variations are consistent with Milankovich cycles [89] and
the main theory of climate change—that the climate is forced by variations in the earth’s
orbit and the intensity and orientation of the solar input. If summer and winter are caused
by changes in the solar angle and nearness of the earth to the sun, then orbital precession,
which has a 23,000 year cycle, and orbital eccentricity, with 100,000 year cycles, are likely
causes of the longer term variations in Figure 1.3.2
Calculations of the amount of change
in the solar insolation given the variations of the orbital parameters suggest that the cli-
mate is quite sensitive to these changes [100, 131].
Looking more deeply into the past, we know that other forces have also been at work.
Some 600 million years ago, a single super continent, Pannotia, accounted for the earth’s
2
See [178, Figure 2.3] for the tilt angle (23.45o
) of the earth and the picture of the wobbling top that points
the axis of rotation at the North Star. The axis will not point at the North Star in the future.
1](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-13-320.jpg)
![2 Chapter 1. Earth Observations
Figure 1.1. NASA Apollo 8, the first manned mission to the moon, entered lunar orbit on
Christmas Eve, December 24, 1968. Said Lovell, “The vast loneliness is awe-inspiring and it makes you
realize just what you have back there on Earth.” At the height of the technical achievement of space travel,
a question is asked. Reprinted courtesy of NASA.
land mass. In what appears to be an oscillation between single and dispersed continents,
this super continent broke up and reassembled 250 million years ago to form Pangea with
the Appalachian mountains in its geographic center. The present locations of the conti-
nents are a result of plate tectonics from the breakup of Pangea [35]. Possibly caused by
climate stresses in the Permian–Triassic period 251 million years ago, a mass extinction
occurred. The high latitude temperatures were 10–30 o
C above present temperatures and
recovery took 30 million years. The Cretaceous–Tertiary extinction of 65 million years
ago, possibly the result of a large asteroid impact or increased volcanism, is responsible
for the disappearance of dinosaurs [177].
Climate changes that have occurred as earth warmed from the last ice age are also
responsible for some familiar extinctions. The Younger Dryas event 12,900 years ago
saw the extinction of mammoths and the North American camel and the disappearance
of the Clovis culture from North America. Warm periods are often called optimums,
and the Holocene climatic optimum occurred from 9,000–5,000 years before the present.
During this time the Sahara was green. With a temperature increase of +4 o
C in the high
latitudes, the United States Midwest was a desert. The Medieval Warm period occurred
from 800–1300 CE and what has been called the Little Ice Age (-1 o
C cooling) occurred
soon after. Without systematic records or temperature proxy data it is hard to tell the](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-14-320.jpg)
![6 Chapter 1. Earth Observations
• the ERA-40 archive17
from the European Centre for Medium Range Weather Fore-
casting (ECMWF) spans forty years starting in the mid-1950s at 2.5o
horizontal
resolution. Data at different atmospheric heights are part of this record and the
derived variables offer more comprehensive coverage than the available measure-
ments.
Climatologies and summaries of weather and satellite observations for the US are available
from the Earth System Research Laboratory (ESRL).18
Climate science relies heavily on
the available observations. Familiarity with available data holdings provides a rich source
of insight and fertile ground for research ideas.
1.1.3 Interpolation Basics
How does randomly distributed data (in space, time, and instrumental method) get ma-
nipulated to a usable format for analysis? The answer, in one form or another, is approx-
imation. We want to approximate a function f by another function g that is “close” to
f but more usable. Two functions are close as measured by a functional norm · , so
approximation is summarized by the following statement: given f find g so that
f − g ε, (1.1)
where ε 0 is a relatively small error in the approximation. If we think of the function f
as the “actual” surface temperature of the earth, then observations sample this function in
space and time. Of course, the measurement has some error in it but we will ignore this.
Over a day’s time, the temperature at any given point will vary quite a bit (the diurnal
variation). A more usable function is the daily or monthly average temperature. The
function g used as an approximation, in this case, is piecewise constant over a day or a
month although f is presumed to be continuous in time.
Spatial approximations have some of the same characteristics. For example, the values
of the temperature on a regular grid of lat-lon coordinates are convienent for analysis, but
since the observation network is not regular, the values of f at the grid points must be
approximated, for example, as an average of observed values near the grid point. A simple
linear interpolation among grid points could fill in all the places where no observations
exist. This is the process used in coloring contour maps. What should be noted is that
for some approximate functions g, no observation data coincide exactly in time or space.
An error or tolerance in the approximation, ε, is thus unavoidable.
One-dimensional linear interpolation fits a set of points (xi , f (xi )) with a piecewise
linear interpolant g such that g(xi ) = f (xi ). Then g may be evaluated at any point
g(x). Interpolation is a specialized form of approximation where the norm is the discrete
maximum norm, f − g = maxi |f (xi )− g(xi )| and ε = 0. This says nothing about how
good the approximation is at points not included in the discrete set.
A more accurate interpolant is obtained by choosing the interpolant from a basis set
of functions that span the space that the function f lives in. For example, we may in-
terpolate single variable functions using a polynomial basis [1, x, x2
, x3
, x4
,..., xn
]. The
function g(x) =
n
k=0
ck xk
shows the form of a higher order interpolant with polyno-
mial coefficients ck. The interpolation condition requires that g match f at given points.
The following is a set of simultaneous equations results that determine the values of the
ck :
c0 + c1xi + c2 x2
i + c3x3
i + ··· + cn xn
i = f (xi ) for each i = 0,n. (1.2)
17www.ecmwf.int/research/era/do/get/era-40
18
www.esrl.noaa.gov/psd/data/usclimate](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-18-320.jpg)
![[84
]
Fig.
2. Mesoplodon stejnegeri. Skull. Adult. Cat. No. 143132, U.S.N.M.
Ventral aspect. About ¼ nat. size.
Pterygoids and left malar defective.
Plate 7 SKULLS OF MESOPLODON
1. Mesoplodon bidens. Skull. Nantucket, Massachusetts. Female,
adult. Mus. Comp. Zoöl. No. 1727. Lateral aspect. ¼ nat. size.
Tip of beak, left pterygoid and malar defective.
2. Mesoplodon densirostris? Skull. Annisquam, Massachusetts.
Female, young. Boston Soc. Nat. Hist. Lateral aspect. ¼ nat. size.](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-23-320.jpg)
![[85
]
2. The same. Left mandibular tooth. Outer surface.
3. The same. Right mandibular tooth. Inner surface.
All figures a little more than ⅗ nat. size.
Plate 13 SKELETON AND LUNGS OF
MESOPLODON EUROPÆUS](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-32-320.jpg)
![Fig.
[86
]
1. Skull. (Type of Ziphius grebnitzkii.) Bering Island. Cat. No. 20993,
U.S.N.M. Ventral aspect. ⅙ nat. size.
2. Skull. Newport, Rhode Island. Ventral aspect. ⅙ nat. size.
Plate 20 SKULLS OF ZIPHIUS CAVIROSTRIS](https://image.slidesharecdn.com/2626024-250609213106-a75bd5db/85/Climate-Modeling-For-Engineers-And-Scientists-John-B-Drake-46-320.jpg)
Climate Modeling For Engineers And Scientists John B Drake
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- 6. About the Series The SIAM series on Mathematical Modeling and Computation draws attention to the wide range of important problems in the physical and life sciences and engineering that are addressed by mathematical modeling and computation; promotes the interdisciplinary culture required to meet these large-scale challenges; and encourages the education of the next generation of applied and computational mathematicians, physical and life scientists, and engineers. The books cover analytical and computational techniques, describe significant mathematical developments, and introduce modern scientific and engineering applications. The series will publish lecture notes and texts for advanced undergraduate- or graduate-level courses in physical applied mathematics, biomathematics, and mathematical modeling, and volumes of interest to a wide segment of the community of applied mathematicians, computational scientists, and engineers. Appropriate subject areas for future books in the series include fluids, dynamical systems and chaos, mathematical biology, neuroscience, mathematical physiology, epidemiology, morphogenesis, biomedical engineering, reaction-diffusion in chemistry, nonlinear science, interfacial problems, solidification, combustion, transport theory, solid mechanics, nonlinear vibrations, electromagnetic theory, nonlinear optics, wave propagation, coherent structures, scattering theory, earth science, solid-state physics, and plasma physics. John B. Drake, Climate Modeling for Scientists and Engineers Erik M. Bollt and Naratip Santitissadeekorn, Applied and Computational Measurable Dynamics Daniela Calvetti and Erkki Somersalo, Computational Mathematical Modeling: An Integrated Approach Across Scales Jianke Yang, Nonlinear Waves in Integrable and Nonintegrable Systems A. J. Roberts, Elementary Calculus of Financial Mathematics James D. Meiss, Differential Dynamical Systems E. van Groesen and Jaap Molenaar, Continuum Modeling in the Physical Sciences Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Müller, and Birgitt Schönfisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods Ivan Markovsky, Jan C. Willems, Sabine Van Huffel, and Bart De Moor, Exact and Approximate Modeling of Linear Systems: A Behavioral Approach R. M. M. Mattheij, S. W. Rienstra, and J. H. M. ten Thije Boonkkamp, Partial Differential Equations: Modeling, Analysis, Computation Johnny T. Ottesen, Mette S. Olufsen, and Jesper K. Larsen, Applied Mathematical Models in Human Physiology Ingemar Kaj, Stochastic Modeling in Broadband Communications Systems Peter Salamon, Paolo Sibani, and Richard Frost, Facts, Conjectures, and Improvements for Simulated Annealing Lyn C. Thomas, David B. Edelman, and Jonathan N. Crook, Credit Scoring and Its Applications Frank Natterer and Frank Wübbeling, Mathematical Methods in Image Reconstruction Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion Michael Griebel, Thomas Dornseifer, and Tilman Neunhoeffer, Numerical Simulation in Fluid Dynamics: A Practical Introduction Khosrow Chadan, David Colton, Lassi Päivärinta, and William Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems Charles K. Chui, Wavelets: A Mathematical Tool for Signal Analysis Editorial Board Alejandro Aceves Southern Methodist University Andrea Bertozzi University of California, Los Angeles Bard Ermentrout University of Pittsburgh Thomas Erneux Université Libre de Bruxelles Bernie Matkowsky Northwestern University Robert M. Miura New Jersey Institute of Technology Michael Tabor University of Arizona Mathematical Modeling and Computation Editor-in-Chief Richard Haberman Southern Methodist University MM19_Drake_FM05-06-14.indd 2 7/9/2014 11:02:15 AM
- 7. Society for Industrial and Applied Mathematics Philadelphia Climate Modeling for Scientists and Engineers John B. Drake University of Tennessee Knoxville, Tennessee MM19_Drake_FM05-06-14.indd 3 7/9/2014 11:02:15 AM
- 8. Copyright © 2014 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, info@mathworks.com, www.mathworks.com. Cover art: Visualization of time dependent fields of the community climate system model. Courtesy of Jamison Daniel. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC05-00OR22725. Figures 1.1, 1.5, and 1.10 used courtesy of NASA. Figures 1.2/5.3a, 1.4, and 1.11 used courtesy of NOAA. Figure 1.3 used with permission from Global Warming Art. Figure 1.7 © American Meteorological Society. Used with permission. Figures 1.8 and 5.4 used with permission from Elsevier. Figure 1.9 used with permission from John Wiley and Sons. Figure 1.12 used with permission from Tim Osborn, Climatic Research Unit, University of East Anglia. Figures 1.13a/5.1, 1.13b/5.2b, and 5.2a used with permission from Nate Mantua, Climate Impacts Group, University of Washington. Figure 3.5 used with permission from Sandia National Laboratories. Figure 5.5 used with permission from IOP Publishing Ltd. Library of Congress Cataloging-in-Publication Data Drake, John B. (John Bryant), author. Climate modeling for scientists and engineers / John B. Drake, University of Tennessee, Knoxville, Tennessee. pages cm Includes bibliographical references and index. ISBN 978-1-611973-53-2 1. Climatology--Data processing. 2. Climatology--Mathematical models. I. Title. QC874.D73 2014 551.501’1--dc23 2014016721 is a registered trademark. MM19_Drake_FM05-06-14.indd 4 7/9/2014 11:02:15 AM
- 9. Contents Preface vii 1 Earth Observations 1 1.1 Weather and Climate Records . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Satellite Observations Since 1979 . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Circulation Patterns of the Atmosphere . . . . . . . . . . . . . . . . . . . 11 1.4 Circulation Patterns of the Ocean . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 The Coupled Climate System . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Geophysical Flow 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Governing Equations for Mass and Momentum . . . . . . . . . . . . . . 28 2.3 Primitive Equation Formulations for Stratified, Rotating Flow . . . . 31 2.4 The Geostrophic Wind Approximation . . . . . . . . . . . . . . . . . . . 34 2.5 The Hydrostatic Approximation for a Perfect Fluid Atmosphere . . . 35 2.6 Shallow Water Equations and Barotropic Vorticity . . . . . . . . . . . . 38 2.7 Geophysical Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.8 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.9 The Model Description of the Community Climate System Model . 53 2.10 The Butterfly Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Numerical Methods of Climate Modeling 57 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Basic Equations with the Control Volume Method . . . . . . . . . . . . 58 3.3 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 The Semi-Lagrangian Transport Method . . . . . . . . . . . . . . . . . . 73 3.5 Galerkin Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.6 Continuous Galerkin Spectral Element Method . . . . . . . . . . . . . . 97 3.7 Vorticity Preserving Method on an Unstructured Grid . . . . . . . . . 101 3.8 Baroclinic Models and the Discrete Vertical Coordinate . . . . . . . . 106 3.9 Algorithms for Parallel Computation . . . . . . . . . . . . . . . . . . . . 110 4 Climate Simulation 119 4.1 What We Have Learned from Climate Simulations . . . . . . . . . . . . 119 4.2 Case Study: Paleoclimate Simulations . . . . . . . . . . . . . . . . . . . . 119 4.3 Other Paleoclimate Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4 Increased Greenhouse Gas Concentrations . . . . . . . . . . . . . . . . . 121 4.5 Case Study: Peter Lawrence Answers Pielke on Land Use . . . . . . . 122 4.6 How to Define a Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 123 v
- 10. vi Contents 4.7 What Climate Models Are and Are Not . . . . . . . . . . . . . . . . . . . 124 5 Climate Analysis 127 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Approximation of Functions on the Sphere . . . . . . . . . . . . . . . . 127 5.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.4 EOF Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.5 Canonical Correlation Analysis of Climate Time Series . . . . . . . . 136 5.6 Stochastic Dynamical System Approximation . . . . . . . . . . . . . . . 138 5.7 Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.8 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.9 Downscaling and Impact analysis . . . . . . . . . . . . . . . . . . . . . . . 145 Conclusions 149 Bibliography 151 Index 163
- 11. Preface Many excellent textbooks describe the physics of climate and give an introduction to the processes that interact and feedback to produce the earth’s weather and climate [178, 79, 133, 94, 83, 134]. In this book we approach the subject from another direction, admitting from the outset that the definition of climate is nebulous and, in fact, still evolv- ing. The climate will be viewed as multifaceted but always as the solution of a particular mathematical model. That all climate models are incomplete is a consequence of our lack of understanding of the physics as well as the incompleteness of mathematical knowledge. The understanding of the nature of the earth’s climate has improved and changed as the mathematical notions of what constitutes a solution to a set of partial differential equations has changed.1 The development of theories of deterministic chaos and the ex- istence of global attractors as invariant subsets of the time evolution associated with the Navier–Stokes equations have had a profound influence on climate research [165]. The invariants of flows, what is conserved, and what coherent structures persist inform the theory of climate. How oscillations, bifurcations, and singularities arise in the solutions of partial differential equations is fundamental to drawing a line between natural climate variability and human induced climate change. A SIAM focus for 2013 on the “Mathe- matics of Planet Earth” was marked with the notable publication of Kaper and Engler’s text [100] treating many of the current conceputal models of climate. The role of general circulation models and high-end computer simulation in the under- standing of climate should not be underestimated. Describing the principles and practice of this type of modeling is the primary focus of this book. The first chapter is devoted to the observations of weather phenomena and the special programs to collect climate data that provide a wealth of information for calibration, validation, and verification. The data themselves, however, do not provide the interpretation of climatic events or give a means of projecting future climate responses. Only high-end models show how the processes interact and feedback upon one another culminating in weather phenomena and climate. Chapter 2 introduces the governing equations of geophysical flow that model the circula- tions of the atmosphere and ocean. Chapter 3 introduces numerical methods for solving these equations starting from a simplified subset known as the shallow water equations. High performance computing is a uniquely powerful tool to probe the solutions of the equations that constitute the model, and this is introduced in Section 3.9. Numerical methods and algorithms are the backbone of simulations based on general circulation models. Attention is given to parallel algorithms and promising numerical methods that will likely figure in the next generation of climate models. Chapter 4 describes what has been learned from climate simulations, and a few case studies are presented with the hope that interested readers and students will pursue other 1 The mathematical theory for atmospheric and ocean flows is not complete [26, 164], so there is still room for growth. vii
- 12. viii Preface simulation studies following their own interests. Finally, a brief chapter introduces some of the methods and mathematical basis for the analysis of climate. These methods must be used to summarize simulation results and, of course, are useful in analyzing weather data to extract climate statistics and trends. Exercises are scattered throughout the text as well as references to MATLAB codes that are part of these exercises and are described in supplemental material available online at www.siam.org/books/MM19. Since methods and simulation are a thread through- out the material, students wishing to master the material should gain experience with computer simulation and programming through these exercises. Full-fledged simulations using parallel computers requires more sophisticated programming than the MATLAB environment offers. Usually simulation codes are written in FORTRAN and C++. But access to the full code and simulation system of the Community Climate System Model is available to the ambitious reader. For analysis, Python or the NCAR Command Lan- guage (NCL) are the languages of choice. Reference is also made to Supplemental Lectures [49], provided in a separate online volume at www.siam.org/books/MM19. These lectures each start with something we know fairly well, usually some piece of mathematics, but then branch out to things we do not understand well. The supplemental lectures serve as a somewhat light-hearted introduction to research areas where open questions still exist and important perspectives are emerging. Students seem to appreciate the change of pace offered in these lectures. The book is the result of a course sequence taught at the University of Tennessee– Knoxville, in the Civil and Environmental Engineering graduate studies department. I am grateful to all the graduate students who have asked questions and provided input on my lectures. In particular, thanks to Dr. Nicki Lam, Dr. Yang Gao, Dr. Abby Gaddis, Ms. Melissa Allen, Dr. Evan Kodra, Mr. Scott DeNeale, Dr. Abdoul Oubeidillah, Mr. Jian Sun, and my colleagues Dr. Joshua Fu and Dr. Kathrine Evans. I am indebted to the many exceptional researchers that I have worked with, learned from, and been inspired by at the Oak Ridge National Laboratory Climate Change Science Institute and at the National Center for Atmospheric Research. I owe a particular debt to Dr. David Williamson and Dr. Warren Washington of the National Center for Atmospheric Research in Boulder. My admiration for Dr. Washington’s book [178] will be evident throughout this text. In the DOE National Laboratories, my colleagues Dr. Patrick Worley, Dr. David Bader, and Dr. Jim Hack have been pioneers in the development of this computational science discipline. Finally, I wish to thank my wife, Frances, for supporting me in this project and providing editorial suggestions.
- 13. Chapter 1 Earth Observations Few moments in history stand in such sharp contrast as the moment captured in Fig- ure 1.1: the natural world in plane view from the most unnatural environment of space. The questioning of science’s ability to provide answers for our future is part of the post- modern social fabric in which the climate change discussion is taking place and climate modelers are questioned intensely. It has been said that if you believe models, you will believe anything. Yet in the quiet eye of the storm the discussion hinges on things we know and believe, on physical, chemical, and biological processes, on cause and effect. If we take an engineering and scientific perspective, we begin with the principles behind the dynamics and physics of the climate system. The implications of climate change, as they are currently understood, are strong motivation for the study that leads ultimately to the question, “So, what are we going to do about it.” Geologic time scales indicate large variations in earth’s climate. The recent (geologic) past is known from ice core data obtained by drilling deep into the Antarctic ice sheet. The research sites at Vostok Station and the European Project for Ice Coring in Antarc- tica (EPICA) Dome C, have produced a record extending back for the past 800,000 years. What the cores reveal is a series of eight ice ages and interglacials (the warm periods be- tween the ice expansions) (Figure 1.2). During these variations the temperature ranges from −10o C to +4o C from the modern baseline. The concentration of atmospheric car- bon dioxide, CO2, varies from 180 parts per million (ppm) to 300 ppm. Our present warm period began to develop 30,000 years ago and, looking at the frequency of past vari- ation, a signal operating on the 23,000 year period emerges. According to this signal, we are due for a cooling trend and should soon be heading into another ice age. The timing of the observed variations are consistent with Milankovich cycles [89] and the main theory of climate change—that the climate is forced by variations in the earth’s orbit and the intensity and orientation of the solar input. If summer and winter are caused by changes in the solar angle and nearness of the earth to the sun, then orbital precession, which has a 23,000 year cycle, and orbital eccentricity, with 100,000 year cycles, are likely causes of the longer term variations in Figure 1.3.2 Calculations of the amount of change in the solar insolation given the variations of the orbital parameters suggest that the cli- mate is quite sensitive to these changes [100, 131]. Looking more deeply into the past, we know that other forces have also been at work. Some 600 million years ago, a single super continent, Pannotia, accounted for the earth’s 2 See [178, Figure 2.3] for the tilt angle (23.45o ) of the earth and the picture of the wobbling top that points the axis of rotation at the North Star. The axis will not point at the North Star in the future. 1
- 14. 2 Chapter 1. Earth Observations Figure 1.1. NASA Apollo 8, the first manned mission to the moon, entered lunar orbit on Christmas Eve, December 24, 1968. Said Lovell, “The vast loneliness is awe-inspiring and it makes you realize just what you have back there on Earth.” At the height of the technical achievement of space travel, a question is asked. Reprinted courtesy of NASA. land mass. In what appears to be an oscillation between single and dispersed continents, this super continent broke up and reassembled 250 million years ago to form Pangea with the Appalachian mountains in its geographic center. The present locations of the conti- nents are a result of plate tectonics from the breakup of Pangea [35]. Possibly caused by climate stresses in the Permian–Triassic period 251 million years ago, a mass extinction occurred. The high latitude temperatures were 10–30 o C above present temperatures and recovery took 30 million years. The Cretaceous–Tertiary extinction of 65 million years ago, possibly the result of a large asteroid impact or increased volcanism, is responsible for the disappearance of dinosaurs [177]. Climate changes that have occurred as earth warmed from the last ice age are also responsible for some familiar extinctions. The Younger Dryas event 12,900 years ago saw the extinction of mammoths and the North American camel and the disappearance of the Clovis culture from North America. Warm periods are often called optimums, and the Holocene climatic optimum occurred from 9,000–5,000 years before the present. During this time the Sahara was green. With a temperature increase of +4 o C in the high latitudes, the United States Midwest was a desert. The Medieval Warm period occurred from 800–1300 CE and what has been called the Little Ice Age (-1 o C cooling) occurred soon after. Without systematic records or temperature proxy data it is hard to tell the
- 15. 3 extent of regional climate change and the Little Ice Age may have been a localized cooling of the European region not reflecting global conditions. Figure 1.2. Vostok Temperature, CO2 and dust from ice cores. Reprinted courtesy of NOAA (www.ngdc.noaa.gov/paleo/icecore/antarctica/vostok/). Figure 1.3. Five million years from climate record constructed by combining measurements from 57 globally distributed deep sea sediment cores. Reprinted with permission, Robert A. Rhode, Global Warming Art. Exercise 1.0.1 (Younger Dryas). What was the Younger Dryas event? When did it start and how long did it last? How was the event characterized? What are the theories about its cause? Exercise 1.0.2 (Vostok ice cores). The Vostok ice core data shows what period? How can you get the data? After obtaining the data plot CO2 versus ΔT and dust versus ΔT . According to
- 16. 4 Chapter 1. Earth Observations the data, what should the present ΔT be based on a current CO2 concentration of 400 ppm? What about for 450 ppm? How would you characterize the “error bound” on the estimate? 1.1 Weather and Climate Records The historical period of direct observations begins around 1850. After World War II, observation networks started to develop from twice daily rawinsondes3 launched at each major airport. The World Meteorological Organization (WMO)4 was formed in 1950. A variety of measurement campaigns have been launched to advance our understanding of weather and climate phenomena, for example, the Global Atmospheric Research Program (GARP), the First GARP Global Experiment (FGGE), the Tropical Ocean and Global Atmosphere Program (TOGA), the International Satellite Cloud Climatology Project (IS- CCP), the World Ocean Climate Experiment (WOCE), the Global Energy and Water Cycle Experiment (GEWEX), the International Geosphere-Biosphere Program (IGBP), and the International Polar Year (IPY). 1.1.1 Ground-based Weather Data The National Climatic Data Center in Asheville, North Carolina5 , is responsible for col- lecting and storing all weather data for the United States. A typical weather station reports the following: • air temperature including daily maximum and minimum (o C or o F ), • barometric pressure (inches or mm of mercury or hPa or atmospheres), • surface wind speed and direction (m ph, knots or m/sec), • dew point and relative humidity6 , • precipitation, and • snowfall and depth. The data from over 40,000 sites worldwide is available through the Global Historical Cli- mate Network (GHCN).7 The GHCN-Daily dataset serves as the official archive for daily data from the Global Climate Observing System (GCOS) Surface Network (GSN). It is particularly useful for monitoring the frequency and magnitude of extremes. The dataset is the world’s largest collection of daily weather data. The highest concentration of observations are from continental land masses with rel- atively fewer historical measurements for the oceans. Ocean going vessels log meteoro- logical observations including8 3A weather balloon that measures T, p, and humidity from 0 to 30km height. Wind velocity is deduced from the drift path of the balloon. 4 www.wmo.int 5www.ncdc.noaa.gov/oa/climate/stationlocator.html 6RH = pH2O p∗ H2O , where the pressures are partial pressure and saturation pressure of water. This may be approx- imated using the temperature and dew point temperature. Dew point is the temperature at which the water vapor condenses under constant temperature and is measured with a wet bulb thermometer. 7www.ncdc.noaa.gov/oa/climate/ghcn-daily 8 www.sailwx.info/index.html
- 17. 1.1. Weather and Climate Records 5 • sea surface temperature, • air temperature, • barometric pressure, • surface wind speed and direction, and • wave height. Since shipping routes do not offer a uniform coverage of the ocean and do not monitor any single point continuously through time, the data must be processed to provide maps of temperature at a given time. The weather maps and other products derived from the surface network observing system may be obtained from a number of sources. The following is a list of a few relevant links from the United States: • Unisys weather9 , • Weather Underground10 , • National Center for Atmospheric Research11 , • Geophysical Fluid Dynamics Laboratory12 , • Earth System Research Laboratory13 , and • University of Wisconsin Space Science and Engineering Center14 . Comprehensive datasets are also collected by countries other than the United States. For example, the Climate Research Unit at East Anglia University15 maintains an indepen- dent, comprehensive collection of weather and climate observations. Multiple sets of observations and processing techniques provide a scientific check on the reliability of any single source. 1.1.2 Climate Data Creating a consistent set of climate data from weather observations is not a simple matter. Several sources provide their best estimates of conditions on a latitude-longitude (lat-lon) grid for daily and monthly averaged periods. These “reanalysis” datasets are particularly useful in verification of climate models. The following two datasets are widely used: • the National Centers for Environmental Prediction (NCEP) Reanalysis Data.16 9weather.unisys.com/ 10 see www.wunderground.com/. Also see Ricky Rood’s climate change blog www.wunderground.com/blog/RickyRood/. 11 www.ncar.ucar.edu 12www.gfdl.noaa.gov 13 www.esrl.noaa.gov 14 www.ssec.wisc.edu/data/ 15www.cru.uea.ac.uk/data/ 16 www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.html
- 18. 6 Chapter 1. Earth Observations • the ERA-40 archive17 from the European Centre for Medium Range Weather Fore- casting (ECMWF) spans forty years starting in the mid-1950s at 2.5o horizontal resolution. Data at different atmospheric heights are part of this record and the derived variables offer more comprehensive coverage than the available measure- ments. Climatologies and summaries of weather and satellite observations for the US are available from the Earth System Research Laboratory (ESRL).18 Climate science relies heavily on the available observations. Familiarity with available data holdings provides a rich source of insight and fertile ground for research ideas. 1.1.3 Interpolation Basics How does randomly distributed data (in space, time, and instrumental method) get ma- nipulated to a usable format for analysis? The answer, in one form or another, is approx- imation. We want to approximate a function f by another function g that is “close” to f but more usable. Two functions are close as measured by a functional norm · , so approximation is summarized by the following statement: given f find g so that f − g ε, (1.1) where ε 0 is a relatively small error in the approximation. If we think of the function f as the “actual” surface temperature of the earth, then observations sample this function in space and time. Of course, the measurement has some error in it but we will ignore this. Over a day’s time, the temperature at any given point will vary quite a bit (the diurnal variation). A more usable function is the daily or monthly average temperature. The function g used as an approximation, in this case, is piecewise constant over a day or a month although f is presumed to be continuous in time. Spatial approximations have some of the same characteristics. For example, the values of the temperature on a regular grid of lat-lon coordinates are convienent for analysis, but since the observation network is not regular, the values of f at the grid points must be approximated, for example, as an average of observed values near the grid point. A simple linear interpolation among grid points could fill in all the places where no observations exist. This is the process used in coloring contour maps. What should be noted is that for some approximate functions g, no observation data coincide exactly in time or space. An error or tolerance in the approximation, ε, is thus unavoidable. One-dimensional linear interpolation fits a set of points (xi , f (xi )) with a piecewise linear interpolant g such that g(xi ) = f (xi ). Then g may be evaluated at any point g(x). Interpolation is a specialized form of approximation where the norm is the discrete maximum norm, f − g = maxi |f (xi )− g(xi )| and ε = 0. This says nothing about how good the approximation is at points not included in the discrete set. A more accurate interpolant is obtained by choosing the interpolant from a basis set of functions that span the space that the function f lives in. For example, we may in- terpolate single variable functions using a polynomial basis [1, x, x2 , x3 , x4 ,..., xn ]. The function g(x) = n k=0 ck xk shows the form of a higher order interpolant with polyno- mial coefficients ck. The interpolation condition requires that g match f at given points. The following is a set of simultaneous equations results that determine the values of the ck : c0 + c1xi + c2 x2 i + c3x3 i + ··· + cn xn i = f (xi ) for each i = 0,n. (1.2) 17www.ecmwf.int/research/era/do/get/era-40 18 www.esrl.noaa.gov/psd/data/usclimate
- 19. 1.2. Satellite Observations Since 1979 7 Once the ck’s are determined, g may be evaluated at any point of interest. The choice of powers of x as the basis set is far from optimal and the solution of the linear system (1.2), called the Vandermonde system, will run into numerical difficulties. This system is particularly ill-conditioned. A better conditioned choice of basis functions for the polynomials are the Lagrange polynomials, lk (x) = n i=0(i=k) (x − xi ) (xk − xi ) . (1.3) The function g(x) = n k=0 ck lk (x) and the solution to the Vandermonde system is simply ck = f (xk ). The resulting Lagrange interpolant is g(x) = n k=0 f (xk ) n i=0(i=k) (x − xi ) (xk − xi ) . (1.4) Combining this approach with piecewise interpolation, where an intepolant is constructed for each sub-interval of the domain, gives efficient and accurate interpolants. But this is for one-dimensional data and we are interested in functions on the sphere. Interpolation on a lat-lon grid can use two-dimensional methods, but care must be taken at the poles to avoid unnecessary oscillations, since the pole is a single point rather than a line, as it appears on most maps. Interpolating velocities is particularly tricky since the velocity (or any vector quantity on a lat-lon grid) has a singularity at the poles. This leads to the question, what are the natural or optimal basis for approximating functions on the sphere? This question will be answered in section 3.5.2. Exercise 1.1.1 (NCEP observational data). Interpolate the NCEP monthly mean data to produce a time series for Knoxville, TN or a location of interest to you. Compare with station data. What are the reasons for the differences? 1.2 Satellite Observations Since 1979 The first weather satellite, the Vanguard 2, was launched on February 17, 1959, but it primarily provided pictures of cloud patterns. As future missions added other instru- ments and sensors, the data became useful for weather and climate purposes. The NASA Nimbus satellites began launching in 1964 and were the first to collect data on the earth’s radiation budget, that is the basic forcing that drives climate. The data since 1979 includes sea surface temperatures and ice extent. The next few subsections are devoted to some par- ticular datasets derived from satellite data; some are supplemented with ground-based ob- servations. One of the important tasks of the last few decades has been to reconcile these two records where they are ambiguous. Since the satellite sees things from the top of the atmosphere and ground-based observations are from the bottom of the atmosphere, the accounting for differences is at the intersection of atmospheric and space science. Much of the NASA satellite data are available online19 along with an excellent image library. 1.2.1 The Sea Surface Temperature For many years the Levitus dataset20 has been the gold standard for ocean temperature and salinity data. Figure 1.4 shows the surface temperature climatology. 19http://earthobservatory.nasa.gov 20 http://gcmd.nasa.gov/records/GCMD_LEVITUS_1982.html
- 20. 8 Chapter 1. Earth Observations Figure 1.4. Annual average ocean surface temperature based on Levitus data. Note the warm pools in the western Pacific and the Indian ocean. Reprinted courtesy of NOAA. Many products have been derived from this dataset and it is used to initialize and provide SST boundary conditions for climate simulations and weather forecasts. For ex- ample, the Program for Climate Model Diagnosis and Intercomparison (PCMDI)21 com- pares atmospheric models (atmospheric model intercomparison projects (AMIPs)) and coupled models (coupled model intercomparison projects (CMIPs)) for the international community. The standard comparisons require that simulations use the same boundary conditions and forcing. A data archive of SSTs used to drive the models for a standard reality check experiment is provided. The satellite-based records have been supplemented by measurements at depth from Argo floats since 2000.22 The float data provide the instantaneous state of the ocean for the first time. This new observational record is the basis for a better understanding of ocean circulation patterns and of the southern oceans in particular. Coupled ocean and atmosphere climate models will use this new information to produce better forecasts on a seasonal to decadal timescale. 1.2.2 Biosphere and Land-Use Data Travelers to a distant planet would no doubt be curious about the temperature, the com- position of the atmosphere, and the geography of the continents and oceans, but they would be most curious about the vegetation covering the land and the life in the seas. In the 1990s, NASA started a program called Mission to Planet Earth that began to survey the globe from space, building an Earth Observing System (EOS). The observing satel- lite Terra successfully launched in December 1999 and the Aqua spacecraft launched in May 2002. The Moderate Resolution Imaging Spectroradiometer (MODIS) is one of the instruments aboard the Terra (EOS AM) and Aqua (EOS PM) satellites. Terra and 21www-pcmdi.llnl.gov/projects/amip/ 22 www-argo.ucsd.edu
- 21. 1.2. Satellite Observations Since 1979 9 Aqua’s orbits are timed so that one passes from north to south across the equator in the morning while the other passes south to north over the equator in the afternoon. The MODIS instruments gather observations for the entire earth’s surface every 1 to 2 days, acquiring data in 36 spectral bands ranging in wavelength from 0.4μm to 14.4μm. The extent and intensity of life on earth are documented in a new way due to these satellites. The near real-time images and data from MODIS have become a treasure trove for sci- entists studying the environment in the large. With spatial resolution of 250m to a few kilometers, the data have produced observations of localities that are nearly inaccessible, such as the Amazon and Indonesian rainforests. The biological activity of the earth is now being monitored and the changes in land-use documented with fine detail. MODIS data are available23 from NASA Goddard Space Flight Center. Since the albedo of the earth is dependent on the color of the surface, climate mod- els have been forced to incorporate a land surface model that includes vegetation. The seasonal change from full foliage to snow cover is an example of the essential reflective properties that go into the radiation and energy balance at the surface. Chemical uptake or release of carbon by the biosphere is another example of a critical process involving veg- etation in the climate system. MODIS vegetation indices, produced on 16-day intervals, provide spatial and temporal snapshots of vegetation greenness, a composite property of leaf area, chlorophyll, and canopy structure. Two vegetation indices are derived: the Nor- malized Difference Vegetation Index (NDVI) and the Enhanced Vegetation Index (EVI), which minimizes canopy-soil variations and improves sensitivity over dense vegetation conditions. The MODIS images also document wildfires and haze from biomass burning and dust storms. Ocean ecosystems affect the color of the ocean surface. The chlorophyll concentra- tions produced by photosynthesis in cyanobacteria and algae can be sensed remotely by processing the spectrum of reflected light. Images and measurements since 1997 are avail- able from the sea-viewing wide field-of-view sensor (SeaWIFS) instrument on the SeaStar satellite.24 The carbon content of the ocean involves the biota as well as physical and chemical processes. Since this pathway is important for the long term disposition of car- bon in the climate system, the state of the ocean and its fluxes with the atmosphere are important to track.25 1.2.3 Atmosphere and Clouds Clouds are perhaps the most visible aspect of the atmosphere and weather watchers have made an extensive catalog of types of clouds. The World Meteorlogical Organization (WMO) recommends nomenclature for use by observing stations: 23 modis.gsfc.nasa.gov 24http://oceancolor.gsfc.nasa.gov/SeaWiFS/ 25 See www.oco.noaa.gov.
- 22. Random documents with unrelated content Scribd suggests to you:
- 23. [84 ] Fig. 2. Mesoplodon stejnegeri. Skull. Adult. Cat. No. 143132, U.S.N.M. Ventral aspect. About ¼ nat. size. Pterygoids and left malar defective. Plate 7 SKULLS OF MESOPLODON 1. Mesoplodon bidens. Skull. Nantucket, Massachusetts. Female, adult. Mus. Comp. Zoöl. No. 1727. Lateral aspect. ¼ nat. size. Tip of beak, left pterygoid and malar defective. 2. Mesoplodon densirostris? Skull. Annisquam, Massachusetts. Female, young. Boston Soc. Nat. Hist. Lateral aspect. ¼ nat. size.
- 24. Fig. Distal portion of beak defective and warped. Plate 8 SKULLS OF MESOPLODON EUROPÆUS 1. Mesoplodon europæus. Skull. Atlantic City, New Jersey. Male, young. Cat. No. 23346, U.S.N.M. Lateral aspect. About ¼ nat. size. 2. Mesoplodon europæus. Skull. North Long Branch, New Jersey. Female, adult. Mus. Comp. Zoöl. Lateral aspect. About ¼ nat. size. Distal portion of beak lacking.
- 25. Fig. Plate 9 SKULLS OF MESOPLODON STEJNEGERI 1. Mesoplodon stejnegeri. Type-skull. Bering Island. Immature. Cat. No. 21112, U.S.N.M. Lateral aspect. About ¼ nat. size. Premaxillæ, maxillæ, frontals, zygomatic process, etc., defective. On account of these defects and the immaturity of the individual the forward inclination of the supraoccipital is much greater than in the skull shown in fig. 2. 2. Mesoplodon stejnegeri. Skull. Yaquina Bay, Oregon. Adult. Cat. No. 143132, U.S.N.M. Lateral aspect. ¼ nat. size.
- 26. Proximal end of premaxillæ defective. Plate 10 SKULLS OF MESOPLODON
- 27. Fig. Skulls of Mesoplodon. 1. Mesoplodon bidens. Nantucket, Massachusetts.
- 28. 2. Mesoplodon densirostris? Annisquam, Massachusetts. 3. Mesoplodon europæus. Atlantic City, New Jersey. 4. Mesoplodon europæus. North Long Branch, New Jersey. 5. Mesoplodon stejnegeri. Type-skull. Bering Island. 6. Mesoplodon stejnegeri. Yaquina Bay, Oregon. Posterior aspect. All figures ¼ nat. size. Plate 11 MANDIBLES OF MESOPLODON
- 29. igs. Mandibles of Mesoplodon. 1, 2, and 5. Mesoplodon bidens. Nantucket, Massachusetts.
- 30. 3 and 6. Mesoplodon europæus. Atlantic City, New Jersey. 4. Mesoplodon stejnegeri. Yaquina Bay, Oregon. All figures ⅕ nat. size. Plate 12 MANDIBLE AND TEETH OF MESOPLODON STEJNEGERI
- 31. Fig. 1. Mesoplodon stejnegeri. Yaquina Bay, Oregon. Mandible and tooth. ¼ nat. size.
- 32. [85 ] 2. The same. Left mandibular tooth. Outer surface. 3. The same. Right mandibular tooth. Inner surface. All figures a little more than ⅗ nat. size. Plate 13 SKELETON AND LUNGS OF MESOPLODON EUROPÆUS
- 33. Mesoplodon europæus. Atlantic City, New Jersey. Cat. No. 23346, U.S.N.M.
- 34. Fig. 1. Vertebræ, from right to left as follows: 7th thoracic, 8th thoracic, 1st lumbar, 1st caudal. Scale, 1/3.7 nat. size. 2. Sternum. Anterior aspect. 3. Left scapula. External surface. Scale 1/3.6 nat. size. 4. Right pectoral limb. External surface. Scale 1/3.7 nat. size. 5. Lungs. Dorsal aspect. About 1/8 nat. size. Plate 14 SKULLS OF ZIPHIUS CAVIROSTRIS
- 35. Fig. Ziphius cavirostris. 1. Skull. (Type of Ziphius semijunctus (Cope).) Charleston, South Carolina. Female, young. Cat. No. 21975, U.S.N.M. Dorsal aspect. ⅙
- 36. nat. size. Tip of beak slightly defective. 2. Skull. Barnegat City, New Jersey. Female, adult. Cat. No. 20971, U.S.N.M. Dorsal aspect. ⅙ nat. size. Plate 15 SKULLS OF ZIPHIUS CAVIROSTRIS
- 38. Fig. 1. Skull. Bering Island. (Topotype of Ziphius grebnitzkii.) Female (?), adult, Cat. No. 22069, U.S.N.M. Dorsal aspect. ⅙ nat. size. 2. Skull. Bering Island. (Topotype of Ziphius grebnitzkii.) Dorsal aspect. Cat. No. 21246. ⅙ nat. size. Plate 16 SKULLS OF ZIPHIUS CAVIROSTRIS
- 40. Fig. 1. Skull. (Type of Ziphius grebnitzkii Stejneger.) Bering Island. Male (?). Cat. No. 20993, U.S.N.M. Dorsal aspect. ⅙ nat. size. 2. Skull. (Topotype of Ziphius grebnitzkii.) Bering Island. Adult. Cat. No. 21245, U.S.N.M. Dorsal aspect. ⅙ nat. size. Plate 17 SKULLS OF ZIPHIUS CAVIROSTRIS
- 42. Fig. 1. Skull. (Topotype of Ziphius grebnitzkii.) Bering Island. Male (?), adult. Cat. No. 21248, U.S.N.M. Dorsal aspect. ⅙ nat. size. 2. Skull. Newport, Rhode Island. Male, adult. Cat. No. 49599, U.S.N.M. Dorsal aspect. ⅙ nat. size. Plate 18 SKULLS OF ZIPHIUS CAVIROSTRIS
- 43. Fig. Ziphius cavirostris. 1. Skull. (Type of Ziphius semijunctus (Cope).) Charleston, South Carolina. Ventral aspect. ⅙ nat. size.
- 44. 2. Skull. Barnegat City, New Jersey. Ventral aspect. ⅙ nat. size. Plate 19 SKULLS OF ZIPHIUS CAVIROSTRIS
- 46. Fig. [86 ] 1. Skull. (Type of Ziphius grebnitzkii.) Bering Island. Cat. No. 20993, U.S.N.M. Ventral aspect. ⅙ nat. size. 2. Skull. Newport, Rhode Island. Ventral aspect. ⅙ nat. size. Plate 20 SKULLS OF ZIPHIUS CAVIROSTRIS
- 48. Fig. 1. Skull. (Type of Ziphius semijunctus (Cope).) Charleston, South Carolina. Cat. No. 21975, U.S.N.M. Lateral aspect. ⅙ nat. size. 2. Skull. Barnegat City, New Jersey. Lateral aspect. ⅙ nat. size. 3. Skull. (Type of Ziphius grebnitzkii Stejneger.) Bering Island. Cat. No. 20993, U.S.N.M. Lateral aspect. ⅙ nat. size. Plate 21 SKULLS OF ZIPHIUS CAVIROSTRIS
- 49. Fig. Ziphius cavirostris. 1. Skull. Newport, Rhode Island. Lateral aspect. 1/7 nat. size.
- 50. 2. Skull. (Type of Ziphius semijunctus (Cope).) Charleston, South Carolina. Posterior aspect. 1/7 nat. size. 3. Skull. Barnegat City, New Jersey. Posterior aspect. 1/7 nat. size. 4. Skull. (Topotype of Ziphius grebnitzkii Stejneger.) Posterior aspect. 1/7 nat. size. 5. Skull. Newport, Rhode Island. Posterior aspect. 1/7 nat. size. Plate 22 MANDIBLES OF ZIPHIUS CAVIROSTRIS
- 51. Fig. Mandibles of Ziphius cavirostris. 1. Charleston, South Carolina. (Type of Z. semijunctus (Cope).)
- 52. 2. Newport, Rhode Island. 3. Bering Island. Cat. No. 22069, U.S.N.M. 4. Bering Island. Cat. No. 21248, U.S.N.M. All figures about ⅕ nat. size. Plate 23 MANDIBLES OF ZIPHIUS CAVIROSTRIS
- 53. Mandibles of Ziphius cavirostris.
- 54. Fig. 1. Bering Island. (Type of Ziphius grebnitzkii Stejneger.) Cat. No. 20993, U.S.N.M. About ⅕ nat. size. 2. Newport, Rhode Island. Symphysis. Dorsal aspect. 3. The same. Ventral aspect. Plate 24 MANDIBLES AND VERTEBRÆ OF ZIPHIUS CAVIROSTRIS
- 55. Mandibles and vertebræ of Ziphius cavirostris.
- 56. Fig. 1. (Type of Z. semijunctus (Cope).) Charleston, South Carolina. Cat. No. 21975, U.S.N.M. ⅕ nat. size. 2. (Type of Z. grebnitzkii Stejneger.) Bering Island. Cat. No. 20993, U.S.N.M. About ⅕ nat. size. 3. Barnegat, New Jersey. About ⅕ nat. size. 4. Vertebræ. (Type of Z. semijunctus (Cope).) From right to left, as follows: 1-3 cervicals, 1st thoracic, 7th thoracic, 8th thoracic, 1st lumbar, 1st caudal. About ¼ nat. size. Plate 25 SKELETON OF ZIPHIUS CAVIROSTRIS
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