Mathematics of Planet Earth: Mathematicians Reflect on How to Discover, Organize, and Protect Our Planet 9781611973709

Citation preview

Mathematics

of Planet Earth

Edited by Hans Kaper Christiane Rousseau Georgetown University Washington, DC

Université de Montréal Montréal, Quebec Canada

Mathematics

of Planet Earth

Mathematicians Reflect on How to Discover, Organize, and Protect Our Planet

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2015 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. Publisher Acquisitions Editor Developmental Editor Managing Editor Production Editor Copy Editor Production Manager Production Coordinator Compositor Graphic Designer

David Marshall Elizabeth Greenspan Gina Rinelli Kelly Thomas David Riegelhaupt Matthew Bernard Donna Witzleben Cally Shrader Bytheway Publishing Services Lois Sellers

Cover image of sea ice to the west of Adelaide Island photographed from RRS James Clark Ross on passage to Rothera, December 1997. Reprinted with permission from Peter Bucktrout/British Antarctic Survey. Library of Congress Cataloging-in-Publication Data Mathematics of planet Earth : mathematicians reflect on how to discover, organize, and protect our planet / edited by Hans G. Kaper, Georgetown University, Washington, DC, Christiane Rousseau, Université de Montréal, Montréal, Quebec, Canada. pages cm. -- (Other titles in applied mathematics ; 140) Includes bibliographical references and index. ISBN 978-1-611973-70-9 1. Ecology--Mathematics. 2. Environmental sciences--Mathematics. I. Kaper, H. G., editor. II. Rousseau, Christiane, editor. QH541.15.M34M385 2015 577.01’51--dc23 2014048797 is a registered trademark.

We dedicate this volume to the members of the

SIAM Activity Group for

Mathematics of Planet Earth.

Contents Preface

xi

I

A Planet to Discover

1

1

Planet Earth 1.1 How Old Is the Earth? Christiane Rousseau . . . . . . . . . . . . . . . 1.2 The Equation of Time Christiane Rousseau . . . . . . . . . . . . . . . 1.3 How Inge Lehmann Discovered the Inner Core of the Earth Christiane Rousseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Why Do Earthquakes Change the Speed of Rotation of the Earth? Christiane Rousseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Mathematicians Listen as the Earth Rumbles Christiane Rousseau . 1.6 Flow through Heterogeneous Porous Rocks: What Average Is the Correct Average? Todd Arbogast . . . . . . . . . . . . . . . . . . . . . . 1.7 Imaging with Gaussian Beams Nick Tanushev . . . . . . . . . . . . . . 1.8 Thinking of Trees Ilya Zaliapin . . . . . . . . . . . . . . . . . . . . . . .

3 3 4

2

3

5 6 7 9 13 15

Ocean and Atmosphere 2.1 Atmosphere and Ocean Dynamics through the Lens of Model Systems Greg Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Atmospheric Waves and the Organization of Tropical Weather Joseph Biello . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nonlinear Waves and the Growth of a Tsunami Estelle Basor . . . . 2.4 Universality in Fractal Sea Coasts Christiane Rousseau . . . . . . . . 2.5 Ice Floes, Coriolis Acceleration, and the Viscosity of Air and Water Robert Miller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 There Will Always Be a Gulf Stream — An Exercise in Singular Perturbation Technique Robert Miller . . . . . . . . . . . . . . . . . . . 2.7 The Great Wave and Directional Focusing John M. Dudley and Frédéric Dias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Controlling Lightning? Christiane Rousseau . . . . . . . . . . . . . . . 2.9 Mathematician Stepping on Thin Ice Deborah Sullivan Brennan . .

19

Weather and Climate 3.1 Numerical Weather Prediction Wei Kang . . . . . . . . . . . . . . . . 3.2 Lorenz’s Discovery of Chaos Chris Danforth . . . . . . . . . . . . . . 3.3 Predicting the Atmosphere Robert Miller . . . . . . . . . . . . . . . . .

37 37 39 40

vii

19 21 23 24 25 28 31 33 35

viii

Contents

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4

II 5

6

Arctic Sea Ice and Cold Weather Hans Kaper . . . . . . . . . . . . . . Extreme Weather Event William J. Martin . . . . . . . . . . . . . . . . Wimpy Hurricane Season Brian McNoldy . . . . . . . . . . . . . . . . Extreme Events Hans Kaper . . . . . . . . . . . . . . . . . . . . . . . . . The Need for a Theory of Climate Antonello Provenzale . . . . . . Mathematics and Climate Hans Kaper . . . . . . . . . . . . . . . . . . . Climate Science without Climate Models Hans Kaper . . . . . . . . Supermodeling Climate James Crowley . . . . . . . . . . . . . . . . . . Reconstructing Past Climates Bala Rajaratnam . . . . . . . . . . . . . (Big) Data Science Meets Climate Science Jesse Berwald, Thomas Bellsky, and Lewis Mitchell . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Good Is the Milankovitch Theory? Hans Kaper . . . . . . . . Earth’s Climate at the Age of the Dinosaurs Christiane Rousseau . Two Books on Climate Modeling James Crowley . . . . . . . . . . . .

Beyond Planet Earth 4.1 Chaos in the Solar System Christiane Rousseau . . . . . . . . . . . . . 4.2 KAM Theory and Celestial Mechanics Alessandra Celletti . . . . . . 4.3 New Ways to the Moon, Origin of the Moon, and Origin of Life on Earth Edward Belbruno . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Low-Fuel Spacecraft Trajectories to the Moon Marian Gidea . . . . 4.5 Where Did the Moon Come From? Christiane Rousseau . . . . . . . 4.6 Data Assimilation and Asteroids Robert Miller . . . . . . . . . . . . . 4.7 Understanding the Big Bang Singularity Edward Belbruno . . . . .

A Planet Supporting Life

41 44 45 46 48 50 53 54 55 57 58 60 61 65 65 66 68 71 73 75 76

79

Biosphere 5.1 The Mystery of Vegetation Patterns Karna Gowda . . . . . . . . . . 5.2 How Vegetation Competes for Rainfall in Dry Regions Frank Kunkle and Karthika Muthukumaraswamy . . . . . . . . . . . . . 5.3 Biological Events in Our Water Systems Matthew J. Hoffman and Kara L. Maki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Bird Watchers and Big Data Wesley Hochachka . . . . . . . . . . . . . 5.5 It’s a Math-Eat-Math World Barry Cipra . . . . . . . . . . . . . . . . . 5.6 Ocean Acidification and Phytoplankton Arvind Gupta . . . . . . . 5.7 Ocean Plankton and Ordinary Differential Equations Hans Kaper 5.8 Prospects for a Green Mathematics John Baez and David Tanzer . .

81 81

Ecology and Evolution 6.1 Mathematics and Biological Diversity Frithjof Lutscher . . . . . . . . 6.2 Why We Need Each Other to Succeed Christiane Rousseau . . . . . 6.3 The Unreasonable Effectiveness of Collective Behavior Pietro-Luciano Buono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 From Individual-Based Models to Continuum Models Hans Kaper 6.5 Optimal Control and Marine Protected Areas Kent E. Morrison . . 6.6 Linear Programming for Tree Harvesting Peter Lynch . . . . . . . .

97 97 98

83 85 86 89 90 91 93

100 102 104 105

Contents

ix

III 7

8

9

10

IV 11

A Planet Organized by Humans

107

Communication and Representation 7.1 The Challenge of Cartography Christiane Rousseau . . . . . . . . 7.2 What Does Altitude Mean? Christiane Rousseau . . . . . . . . . . 7.3 Drawing Conformal Maps of the Earth Christiane Rousseau . . 7.4 Changing Our Clocks Hans Kaper . . . . . . . . . . . . . . . . . . . 7.5 High-Resolution Satellite Imaging Paula Craciun and Josiane Zerubia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Microlocal Analysis and Imaging Gaik Ambartsoumian, Raluca Felea, Venky Krishnan, Cliff Nolan, and Todd Quinto . . . . . . . . . 7.7 How Does the GPS work? Christiane Rousseau . . . . . . . . . . .

. . . .

. . . .

109 109 110 111 113

. . 115 . . 116 . . 120

Energy 8.1 Integrating Renewable Energy Sources into the Power Grid Wei Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Mathematical Insights Yield Better Solar Cells Arvind Gupta . . . 8.3 Mathematical Modeling of Hydrogen Fuel Cells Brian Wetton . . 8.4 Mathematical Models Help Energy-Efficient Technologies Take Hold in a Community Karthika Muthukumaraswamy . . . . . . . . . . . . 8.5 Geothermal Energy Harvesting Burt S. Tilley . . . . . . . . . . . . . . 8.6 Of Cats and Batteries Russ Caflisch . . . . . . . . . . . . . . . . . . . . .

123

Economics and Finance 9.1 Dynamic Programming for Optimal Control Problems in Economics Fausto Gozzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Solar Renewable Energy Certificates Michael Coulon . . . . . . . . . 9.3 How Much for My Ton of CO2 ? Mireille Bossy, Nadia Maïzi, and Odile Pourtallier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Social Cost of Carbon Hans Engler . . . . . . . . . . . . . . . . . 9.5 Measuring Carbon Footprints Hans Kaper . . . . . . . . . . . . . . . . 9.6 Musings on Summer Travel David Alexandre Ellwood . . . . . . . . 9.7 The Carbon Footprint of Textbooks Kent E. Morrison . . . . . . . . 9.8 Sustainable Development and Utilization of Mineral Resources Roussos Dimitrakopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Scientific Research on Sustainability and Its Impact on Policy and Management Mark Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

Human Behavior 10.1 Predicting the Unpredictable — Human Behaviors and Beyond Andrea Tosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Budget Chicken Kent E. Morrison . . . . . . . . . . . . . . . . . . . . . 10.3 Mathematics and Conflict Resolution Estelle Basor . . . . . . . . . . 10.4 Modeling and Understanding Social Segregation Laetitia Gauvin and Jean-Pierre Nadal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Modeling the Evolution of Ancient Societies Estelle Basor . . . . . . 10.6 Networks in the Study of Culture and Society Elijah Meeks . . . . .

151

A Planet at Risk Climate Change 11.1 The Discovery of Global Warming

123 124 126 130 132 133

135 137 140 141 142 143 146 147 149

151 152 154 155 157 157 161

163 Hans Kaper . . . . . . . . . . . . 163

x

Contents

Letter to My Imaginary Teenage Sister – 1 Samantha Oestreicher . Letter to My Imaginary Teenage Sister – 2 Samantha Oestreicher . Global Warming and Uncertainties Juan M. Restrepo . . . . . . . . . How to Reconcile the Growing Extent of Antarctic Sea Ice with Global Warming Hans Kaper . . . . . . . . . . . . . . . . . . . . . . . . Rising Sea Levels and the Melting of Glaciers Christiane Rousseau . Global Warming — Recommended Reading Hans Kaper . . . . . . .

164 166 167

Biological Threats 12.1 Mathematics behind Biological Invasions – 1 Mark Lewis . . . . . . 12.2 Mathematics behind Biological Invasions – 2 James Crowley . . . . 12.3 Surges in Latent Infections: Mathematical Analysis of Viral Blips Karthika Muthukumaraswamy . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Prevention of HIV Using Drug-Based Interventions Jessica M. Conway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Random Networks and the Spread of HIV Estelle Basor . . . . . . . 12.6 Talking across Fields Persi Diaconis and Susan Holmes . . . . . . . . 12.7 Using Mathematical Modeling to Eradicate Diseases Robert Smith? 12.8 Neglected Tropical Diseases — How Mathematics Can Help Robert Smith? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Contagious Behavior Estelle Basor . . . . . . . . . . . . . . . . . . . . .

173 173 174

Predicting Catastrophes and Managing Risk 13.1 Earthquake Modeling and Prediction Darko Volkov . . . . . . . . . 13.2 Seismic Risk Protection Alfio Quarteroni . . . . . . . . . . . . . . . . 13.3 Fire Season James Crowley . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Plowing Fields of Data Kent E. Morrison . . . . . . . . . . . . . . . . . 13.5 Finding a Sensible Balance for Natural Hazard Mitigation with Mathematical Models Karthika Muthukumaraswamy . . . . . . . . . . . . . 13.6 Modeling the Effects of Storm Surges on Coastal Vegetation Catherine Crawley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Königsberg’s Bridges, Holland’s Dikes, and Wall Street’s Downfall Christian Genest and Johanna G. Ne˘slehová . . . . . . . . . . . . . . . . . 13.8 Systemic Risk in Complex Systems James Crowley . . . . . . . . . .

187 187 189 190 191

11.2 11.3 11.4 11.5 11.6 11.7 12

13

169 170 170

175 176 179 180 182 183 185

191 193 194 196

Contributor Index

199

Name Index

201

Subject Index

203

Preface In 2013, the community of mathematical scientists and educators focused its collective attention on the mathematics of planet Earth. In the course of the year, a grassroots organization grew to an international partnership of more than 150 scientific societies, universities, research institutes, and organizations. The project, known as Mathematics of Planet Earth 2013 or MPE2013, received the patronage of UNESCO in December, 2012. MPE2013 was a unique event. It brought the challenges facing our planet to the attention of the mathematics research community in numerous lectures, seminars, workshops, and special sessions at conferences of the professional societies; it sponsored the development of curriculum materials for all educational levels; it organized many outreach activities, including an international juried exhibit of virtual and physical displays for use in museums and schools; and it presented a series of public lectures by renowned scientists who showed the public how mathematics contributes to our understanding of planet Earth, the nature of the challenges our planet is facing, and how mathematicians help solve them. A list of its partners and programs can be found on the MPE2013 home page, http://mpe2013.org/. From the beginning, “Mathematics of Planet Earth” was interpreted in the broadest possible terms. In addition to climate change and sustainability, it included geophysics, ecology, epidemiology, biodiversity, as well as the global organization of the planet by humans. The different topics were classified into four themes: • A PLANET TO DISCOVER: planet Earth, Earth’s climate system, weather and climate, beyond planet Earth; • A PLANET SUPPORTING LIFE: biosphere, ecology, evolution;

• A PLANET ORGANIZED BY HUMANS: communication and representation, energy, human behavior, economics and finance;

• A PLANET AT RISK: climate change, invasive species, infectious diseases, natural disasters. As part of the outreach, MPE2013 featured two blogs—an English-language blog1 and a French-language blog.2 Some individual countries featured their own blogs; for example, “Maths of Planet Earth” in Australia.3 The MPE2013 blogs were initiated in 2012. Posts appeared somewhat irregularly during the Fall of 2012, but starting on January 1, 2013, the English-language blog posted entries seven days a week until the beginning of the summer break on July 14. Blogging resumed on August 15 on a Monday through Friday 1

http://mpe2013.org/blog/

2 http://www.breves-de-maths.fr/ 3

http://mathsofplanetearth.org.au/category/blog/

xi

xii

Preface

schedule until the end of the year. Altogether, 270 posts were published, covering a wide range of topics reflecting the broad scope of MPE. The volume at hand is an anthology of the English-language blog posts published on the website of MPE2013. We have omitted posts announcing workshops, conferences, and special lectures, unless they included information of lasting value to the themes of MPE2013, and edited the posts to streamline the text where necessary. The posts are grouped in four parts, following the four MPE themes listed above. Each post is identified by its original contributor(s). Many of the entries are accessible to a general audience, some require a bit of scientific knowledge, and a few include mathematical jargon that will only appeal to the initiated. This range of entry styles reflects the breadth of MPE and the diversity of the contributors to the blog. MPE2013 was an unusually successful effort, with a legacy that will last well beyond 2013. It created exceptional opportunities for long-term collaborations within the mathematical sciences community and with other related scientific disciplines. It introduced a new generation of researchers to the scientific problems connected with climate change and sustainability, and stimulated initiatives to answer questions like “What is mathematics useful for?” Its outreach activities brought the role of the mathematical sciences in addressing some of the planet’s most pressing problems to the attention of a worldwide audience. We hope that this collection of blog posts will demonstrate that the mathematical sciences community is aware of the issues around planet Earth and stands ready to apply its expertise to the solution of the problems facing our planet.

Acknowledgments We thank our many friends and colleagues who were involved in the planning of MPE2013 and worked so hard to implement the program and make it a success. We thank, in particular, all those colleagues who contributed to the MPE2013 Daily Blog. We hope that this anthology is a tangible reminder of a worthwhile effort and time well spent. We sincerely appreciate the encouragement and advice we received from Jim Crowley, Executive Director of SIAM, and recognize the technical expertise of the staff at SIAM. It was a pleasure to work with Elizabeth Greenspan, Gina Rinelli, and David Riegelhaupt, who collectively managed the timely production of this volume while making sure that it would meet the highest standards of the publishing trade.

Hans G. Kaper Georgetown University Washington, DC January 27, 2015

Christiane Rousseau Université de Montréal Montréal, Canada

Chapter 1

Planet Earth

1.1 How Old Is the Earth? Christiane Rousseau, University of Montreal The first serious attempts to compute the age of the Earth were made in the mid-19th century. The mathematical physicist and engineer William Thomson (1824–1907) assumed that the Earth had formed as a completely molten object and estimated the age of the Earth by determining the length of time it would take for the near-surface to cool to its present temperature. Thompson used Fourier’s law of heat conduction, with the gradient of the temperature measured empirically, and two rather strong hypotheses to simplify the problem: (i) the planet is rigid and homogeneous, and (ii) the amount of thermal energy in the system is constant. Thus he arrived at an estimate between Figure 1.1. Planet Earth. Reprinted courtesy of NASA. 20 million and 400 million years for the age of the Earth. The wide range was due to the uncertainty about the melting temperature of rock, which Thomson equated with the Earth’s interior temperature. Thomson’s estimate contradicted the observations of geologists at the time and was also incompatible with the new theory of evolution of Darwin, which required a much older planet. Over the years he refined his arguments and reduced the upper bound by a factor of ten, and in 1897 Thomson, now Lord Kelvin, settled on an estimate that the Earth was 20–40 million years old. 3

4

Chapter 1. Planet Earth

It was Kelvin’s assistant John Perry who pointed out that the measured gradient of the temperature was too large to be compatible with Kelvin’s hypothesis of homogeneity. Perry proceeded to explain this larger gradient by hypothesizing the presence of a fluid core under a thin outer solid mantle, claiming that convective movements inside the core would considerably slow down the cooling in the mantle. Perry’s arguments led to estimates of the age of the Earth of two billion years or more [1]. Radioactivity, a source of heat, was discovered soon after, invalidating Thomson’s second assumption that energy was constant during the cooling process. Today we know that the age of the Earth is approximately 4.5 billion years. Perry was a visionary in his time. He was arguing that the mantle of the Earth is solid on short time scales and fluid over longer time scales, and that continents drift over time. The concept of continental drift met with strong skepticism in the scientific community, including the community of geologists, and it was only in the 1960s that it finally prevailed. Reference [1] P.C. England, P. Molnar, and F.M. Richter, Kelvin, Perry and the age of the Earth, Am. Sci. 95 (2007) 342–349.

1.2 The Equation of Time Christiane Rousseau, University of Montreal Solar noon is defined as the time of the highest position of the Sun in the sky and occurs when the Sun crosses the meridian at a given position. The length of the solar day is the time between two consecutive solar noons. The mean length of the day, 24 hours, is a little less than the period of rotation of the Earth around its axis, since the Earth makes 366 rotations around its axis during a year of 365 days. If the axis of the Earth were vertical and the orbit of the Earth around the Sun circular, the mean length of the day would correspond to the time between two consecutive solar noons and therefore to the length of the solar day. For Greenwich’s meridian, the official noon is defined as the solar noon at the spring equinox, and then for the other days of the year by applying a period of 24 hours.

Figure 1.2. The equation of time.

1.3. How Inge Lehmann Discovered the Inner Core of the Earth

5

The solar noon oscillates during the year; it coincides with the official noon only on four days during the year. The equation of time is the difference between the solar time and the official time (mean solar time). The fact that the equation of time shows oscillations with an amplitude of approximately 30 minutes can be explained by the eccentricity of Earth’s orbit around the Sun and the obliquity (tilt) of the Earth’s axis of rotation. Due to the eccentricity, when the Earth is closer to the Sun (during the winter of the northern hemisphere), it has a higher angular velocity, yielding longer solar days. This first effect has a period of one year. The obliquity of the Earth’s axis superimposes a second oscillation with a period of half a year: if the orbit of the Earth around the Sun were circular, the official noon would correspond to the solar noon at the equinoxes and at the solstices; it would be after the solar noon in fall and spring, and before the solar noon in summer and winter.

1.3 How Inge Lehmann Discovered the Inner Core of the Earth Christiane Rousseau, University of Montreal We cannot see the interior of the Earth, yet we know much about it (Figure 1.3). The inLower mantle Crust terior consists of a ferrous and solid inner core, an outer core, Upper mantle and a viscous mantle. The mantle is approximately 3200 km thick and covered by a thin solid crust. How do we know all this? I like to tell my students that we put our “mathematical glasses” on to “see” what we cannot see with our eyes. Inner core Outer core Since the interior of the Earth is not homogeneous, the speed of signals traveling inside the Earth is not uniform. When large Figure 1.3. The Earth’s interior. earthquakes occur, they generate strong seismic waves. These waves are detected and recorded by seismographs all around the world. They provide the raw data that can be further analyzed. Reconstructing the interior of the Earth by analyzing what is recorded at the surface is what is called “solving an inverse problem.” When an earthquake occurs, the first inverse problem to solve is to localize the epicenter of the earthquake. Earthquakes generate pressure waves (P-waves) and shear waves (S-waves). S-waves are strongly damped when traveling in viscous media, and hence not recorded far from the epicenter. This provides evidence for a liquid interior, as well as information on the thickness of the crust. On the contrary, P-waves travel throughout the Earth and can be recorded very far from the epicenter. Inge Lehmann (1888–1993) was a Danish mathematician. She worked at the Danish Geodetic Institute and had access to the data recorded at seismic stations around the world. In 1936 she argued, correctly, that the Earth’s core is not one single molten sphere, but that an inner core exists which has physical properties that are different from those of the outer core.

6

Chapter 1. Planet Earth

At the time it was known that the Earth’s core is surrounded by a mantle and that seismic waves travel at approximately 10 km/s in the mantle and 8 km/s in the core; hence, waves are refracted when they enter the core. This implies that there exists an annular region on the surface of the Earth, centered at the epicenter, where no seismic wave should be detected. But Lehmann discovered that signals were recorded in the forbidden region. A piece of the puzzle was missing. . . . Lehmann built a toy model (Figure 1.4) that explained the Figure 1.4. Toy model showing the effect of an inner observations. She inserted core on reflected waves (black) and refracted waves (brown). Note an inner core where the sig- that some refracted waves enter the forbidden regions. nals would travel at 8.8 km/s, slightly faster than in the outer core. According to Snell’s law of refraction, sin θ1 / sin θ2 = v1 /v2 . But this equation does not have a solution for θ2 if v1 is smaller than v2 and θ1 is sufficiently large. This means that if a seismic wave arrives on the slow side, sufficiently tangentially to the separating surface between the two media, it cannot enter the second medium, and the wave is reflected on the surface separating the inner and outer cores. The insertion of an inner core, where waves travel slightly faster than in the outer core, explained why seismic waves could be detected in the forbidden regions.

1.4 Why Do Earthquakes Change the Speed of Rotation of the Earth? Christiane Rousseau, University of Montreal MPE2013 gives us an opportunity to learn more about our planet. There are interesting features to be explored that require simple but deep principles of physics and that can become the basis of a discussion in the classroom. I frequently teach future high-school teachers and like to start by exploring questions that come from many different directions. Here is one. During an earthquake, the mass distribution in the Earth’s crust, and therefore the Earth’s moment of inertia, changes. The Earth’s moment of inertia is the sum of the moments of inertia of all the Earth’s point masses, where the moment of inertia of a point mass is the product of its mass and the square of its distance to the axis of rotation. Meanwhile, the Earth’s angular momentum is preserved. This angular momentum is the sum of the moments of inertia of all the point masses times their angular velocity. Hence, if the Earth’s moment of inertia decreases, the Earth’s angular velocity increases, and conversely, if the Earth’s moment of inertia increases, the Earth’s angular velocity decreases. The simple physical principle of conservation of angular momentum

1.5. Mathematicians Listen as the Earth Rumbles

7

thus allows us to explain disparate phenomena such as the Earth’s changing rotation rate, figure skaters spinning, spinning tops, and gyroscopic compasses. The major earthquakes in Chile (2010) and Japan (2011) increased the Earth’s spin and hence decreased the length of the day. We could imagine that the length of the day would be measured by taking, for instance, a star as some fixed point of reference. But instead, geophysicists use seismic estimates, through GPS measurements, of the movements of the fault to compute how the mass distribution and thus the length of the day has changed. According to Richard Gross, a geophysicist working at NASA’s Jet Propulsion Laboratory, the length of the day has decreased by 1.8 microseconds as a result of the 2011 Japan earthquake [1]. Instead of continuing to read Richard Gross’s interview [2], you can start playing the game and discover for yourself the answer to the next questions. For instance, earthquakes closer to the equator have a larger effect on the Earth’s spin than those close to the poles. Similarly, those with vertical motion have a larger effect than those with horizontal (transversal) motion. It is also an opportunity to start discussing the motion of a solid (here the Earth) in space. On short time intervals, the speed of the center of mass is approximately constant. Hence, if we consider a reference frame centered at the center of mass and moving uniformly, then we are left with three degrees of freedom for the movement of the solid. The derivative of this movement is a linear orthogonal transformation preserving the orientation. Hence, it is a rotation around an axis: the north-south axis. The three degrees of freedom describe the position of the axis and the angular velocity around it. But there is a second axis, which is very important: it is the Earth’s figure axis, about which the Earth’s mass is balanced. This axis is about 10 meters offset off the north-south axis. Large earthquakes abruptly move the position of this axis. For the 2011 Japan earthquake, the shift has been estimated at 17 centimeters. Earthquakes are far from being the only phenomena that change the angular speed of rotation and the position of the Earth’s figure axis. Indeed, they change with atmospheric winds and oceanic currents, but these changes are smoother than the ones observed with earthquakes. Should we care for such small changes? According to Richard Gross, we should if we work for NASA and, for instance, intend to send a spacecraft to Mars and land a rover on it. Any angular error may send us very far from our target. References [1] A. Buis, Japan Quake May Have Shortened Earth Days, Moved Axis, NASA, March 14, 2011, http://www.nasa.gov/topics/earth/features/japanquake/earth20110314.html [2] A. Moseman, How the Japan Earthquake Made the Day Shorter, Popular Mechanics, March 15, 2011.

1.5 Mathematicians Listen as the Earth Rumbles Christiane Rousseau, University of Montreal The rocks in the bottom floor of the oceans are much younger than those of the continents. On the bottom of the oceans, the most recent rocks are along the ridges where tectonic plates diverge. Indeed, there is volcanic activity along these ridges, with new rocks being formed by magma coming up from the mantle to the surface. But there are also isolated volcanic islands, like Hawaii, Tahiti, the Azores, and Cape Verde. If we look at the archipelago of Hawaii, all islands are aligned, and their age increases from the largest island at one end to the smaller islands further down. This suggested to the geophysicists that the islands were formed because of a plume—a kind of volcanic chimney

8

Chapter 1. Planet Earth

through the mantle. As we saw in Figure 1.3, the mantle goes as deep as half the radius of the Earth. Since the surface plate is moving, this could explain the successive formation of the aligned islands, the age difference of which would be calculated from the distance between the islands and the speed of the tectonic plates. But additional evidence is needed for the conjecture to be accepted by the scientific community. For instance, one would like to “see” the plume. One tool for exploring the interior structure of the Earth is remote sensing: one sends waves (signals) and analyzes the signals reflected by the boundary of some layer or refracted inside different layers. But plumes are located so deep under the Earth’s crust that the usual signals are not powerful enough to be of any help. The only waves that carry sufficient energy to analyze details at such a depth are the seismic waves generated by large earthquakes. Large databases exist which contain the recordings of these seismic waves at observing stations around the world. So the data exist. We then need an appropriate tool to analyze them. The problem is not trivial. The plumes are very thin and, moreover, the difference of the speed of a seismic wave through a plume is only of the order of 1%.

Figure 1.5. P-wave velocity perturbations (in %). Reprinted with permission from Guust Noleta.

In 2005, seismologists Tony Dahlen and Guust Noleta approached Ingrid Daubechies to see if wavelets could help in their venture. Indeed, the promising results of the student Raffaela Montelli had shown that seismic methods could be used to capture regions of perturbations of the pressure waves (P-waves) of earthquakes; see Figure 1.5. Such regions overlapped exactly the regions with isolated volcanic islands: the temperature of the ocean floor was higher in these regions. But, as mentioned above, the plumes are very thin, and the difference of the speed of seismic P-waves is very small in these regions. Hence, there is a large risk of errors in the numerical reconstruction of the inner structure of the Earth, unless we use an appropriate tool. This is where wavelets proved useful. They are the perfect tool to analyze small localized details. Moreover, one can concentrate all the energy in small regions and neglect the other regions. Recently, Ingrid Daubechies gave a short course on wavelets adapted to digital images made of pixels. Her lecture was delivered in French, but both English [1] and French [2]

1.6. Flow through Heterogeneous Porous Rocks: What Average Is the Correct Average?

9

videos of the lecture are available. She explained how wavelets allow compressing information and how we can extract very fine details in a local region while keeping the size of the data manageable. The use of wavelets to construct the images allows the removal of all errors in numerical reconstructions and making sure that the special zones identified in the image are indeed special. She showed clean images produced with wavelets in which artificial special regions had been removed, and she could announce “hot off the press” that she and her collaborators had obtained the first results with the whole Earth, and real data! References [1] I. Daubechies, Mathematicians Listen as the Earth Rumbles, http://www.videocrm.ca/MPT2013-20130410-DaubechiesEN.mov [2] I. Daubechies, Des mathématique pour faire parler la terre, UQAM.tv, https://www.youtube.com/watch?v=UohQcwo8mtU

1.6 Flow through Heterogeneous Porous Rocks: What Average Is the Correct Average? Todd Arbogast, University of Texas How fast does water flow through sand or soil? Maybe not so fast, but everyone has seen water soak into beach sand and garden soils. Most people have also noticed a concrete sidewalk soaking up a little water as rain begins to come down. But how fast does water flow through a rock? The obvious answer is incorrect: water does indeed flow through rocks. Not the rock grains themselves, but it can squeeze through the pore spaces between the rock grains. We say that the Earth’s subsurface is porous, and that it is a porous medium. Fluid flow through the planet’s subsurface is critically important to many ecological and economic activities. Subsurface flow is an important part of the entire water cycle. In fact, the United States Geologic Survey estimates that, worldwide, there is 30 times more groundwater stored in aquifers than is found in all the fresh-water lakes and rivers. Unfortunately, contaminants sometimes leach out of storage sites, either from above-ground tanks or through underground containment barriers, and travel through the Earth. Petroleum and natural gas are extracted from the subsurface by inducing these fluids to flow to production wells. To mitigate the effects of greenhouse gas accumulation in the atmosphere, technologies are being developed to sequester carbon dioxide, extracted from power plant emissions, in deep underground reservoirs. These and many other situations make it important that we have the ability to simulate the flow of fluids in the subsurface. The simulations give us a visual and quantitative prediction of the movement of the fluids, so that scientists, engineers, and regulators can design appropriate steps, for example to protect the natural ecosystem and human health, optimize the economic benefit of the world’s underground natural fluid resources, and minimize unintended impact on the natural environment. So, how fast does fluid flow in porous media? In 1856, a French civil engineer by the name of Henri Darcy determined through experimental means the speed at which fluid flows through a sand column when subjected to pressure. He gave us his now famous Darcy’s law, which states that the fluid velocity is proportional to the pressure gradient. Subsequent experiments verified that the law holds for all types of porous media. The proportionality constant in Darcy’s law, times the fluid viscosity, is called the permeability of the porous medium. The permeability is a measure of how easily fluid flows through a rock or soil.

10

Chapter 1. Planet Earth

There are many difficulties associated with simulating the flow of fluid through a natural porous medium like a groundwater aquifer or a petroleum reservoir. One of them is dealing with the extreme heterogeneity of a natural rock formation. One sees this extreme heterogeneity in outcrops, which are rock formations that are visible on the surface of the Earth, such as are often seen from a roadway which has been cut into a hill or mountain. In Figure 1.6 we show results from researchers at the University of Texas at Austin’s Bureau of Economic Geology, who measured the permeability of an outcrop. The permeability is shown on a log scale, and it varies from 10−16 (blue) to 10−11 (red) over a few meters. That is, the fluids can be moving about 100,000 times faster in one area of the rock than another less than a meter away.

Figure 1.6. Permeability of an outcrop (on a logarithmic scale).

An underground aquifer or reservoir domain can be very large, spanning a few to hundreds of kilometers. Normally a simulation of fluid flow will involve a computational grid of cells that cover the domain. The fastest supercomputers can possibly handle enough cells so that each is about ten meters square. But at that scale, we have lost all details of the permeability’s heterogeneity, and consequently many important details of the fluid flow! So the challenge is to find a way to resolve the details of the fluid flow without using a computational grid that is able to resolve the flow. It sounds like a contradiction. This is one place where mathematics and mathematicians can help. Since we cannot resolve all the details of the flow, let us take as our goal the ability to approximate the average flow within each grid cell, by simplifying the geologic structure of the porous medium. That is, we desire to replace the complex heterogeneous porous medium within a grid cell by a simple homogeneous porous medium with an average value of permeability, as depicted in Figure 1.7. But what should this average permeability be? It should be chosen so that the average amount of fluid flowing through the grid cell is the same for the true and fictitious media.

1.6. Flow through Heterogeneous Porous Rocks: What Average Is the Correct Average?

11

Figure 1.7. The heterogeneous porous medium is replaced by a homogeneous one with an ¯ chosen so that the average amount of fluid flow is the same. average permeability k,

It is fairly easy to solve the differential equations governing the flow of fluid in simple geometries. For a layered porous medium, there are two possibilities. As depicted in Figure 1.8 (left), when the flow is going along with the geologic layers, the correct way to average the permeability is to take the usual arithmetic average. In this case, it is 1 (k + k2 ). When fluid flows so that it cuts through the layers, as depicted in Figure 1.8 2 1 (right), the arithmetic average permeability does not give the correct average fluid flow. This is perhaps easy to see by considering the possibility that one of the layers becomes impermeable, say k1 = 0, so that fluid cannot flow through it. In this case, there will be no flow through the entire grid cell. But the arithmetic average is 12 k2 , which is positive, and we should expect some fluid flow through the grid cell. The correct average to take is the harmonic average, which is the reciprocal of the arithmetic average of the reciprocals—that is, 2/(k1−1 + k2−1 ).

Figure 1.8. Left: Flow going along with the geologic layers requires arithmetic averaging of the permeability k. Right: Flow that cuts through the geologic layers requires harmonic averaging of the permeability k.

The harmonic average is always less than or equal to the arithmetic average. But what about a genuinely heterogeneous porous medium like that shown in Figure 1.9? More sophisticated averaging techniques must be used, such as arise in the mathematical theory of homogenization. It can be proven that the correct average always lies between the harmonic and arithmetic averages, so our layered case actually gives the extreme cases of the problem. In fact, homogenization is also able to account for the fact that porous media may be anisotropic—that is, they do not behave the same in every direction. The example in Figure 1.9 is anisotropic, since perhaps one can see that as fluid tries to flow from left to right, it will also tend to flow upwards a bit as well.

12

Chapter 1. Planet Earth

Figure 1.9. Flow through a heterogeneous geologic region requires more complex averaging of the permeability.

Figure 1.10. A highly heterogeneous, vuggy porous medium.

These averaging techniques work well even for very complex geology. For example, consider the heterogeneous rock shown in Figure 1.10, which is a complex mixture of fossil remains, limestone, sediments, and vugs (large open spaces in the rock). Without a theoretically sound mathematical averaging technique, would you like to guess the average permeability of this rock? More complex numerical models have been developed that not only give the correct average flow but also attempt to recover some of the small-scale variability of the flow. Such multiscale numerical methods are an active area of research in mathematics today. These methods can allow scientists and engineers to simulate the flow of fluids in the subsurface in an accurate way, even on computational grids too coarse to resolve all the details. It almost seems like magic. But it is not magic; it is mathematics.

1.7. Imaging with Gaussian Beams

13

1.7 Imaging with Gaussian Beams Nick Tanushev, University of Texas Like most animals, we have passive sensors that let us observe the world around us. Our eyes let us see light scattered by objects, our ears let us hear sounds that are emitted around us, and our skin lets us feel warmth. However, we rely on something to generate light, sound, or heat for us to observe. One animal that is an exception to this is the bat. Bats use echolocation for navigation and finding prey. That is, they do not passively listen for sound; they actively generate sound pulses and listen for what comes back to figure out where they are and what’s for lunch. From an applied mathematical point of view, active sensing using waves is a complex process. Waves generated by an active source have to be modeled as they propagate to an object, reflect, and then arrive back at the receiver. Using the amplitude, phase, time lag, and possibly other properties of the waves, we have to make inferences about the location of the object that reflected the waves and its properties. It is logical to ask at this point “Well, if it is such a complex task, how can bats do it so easily?” The answer is that bats use a simplified model of wave propagation. In nearly constant media, such as air, high frequency waves propagate in straight lines, and determining the distance between the bat and the object is simply a matter of measuring the wave propagation speed and the time it takes to hear the echo back. Our brain also uses the same trick to process what our eyes are seeing. You can easily convince yourself with an old elementary school trick. Put a pencil in a clear glass of water so that half of it is submerged, and lean the pencil on the side so that it is not completely vertical. Looking from the side, it looks as if the pencil is broken at the air water interface. Of course, this is an optical illusion because our brain assumes that light has traveled in a straight line when, in fact, it takes two different paths depending on whether it is propagating in the water or in the air. A much more sophisticated version of echolocation is used by oil companies to image the subsurface of the Earth. A classical description of the equipment used to collect data is a ship that has an active source (an air gun) and a set of receivers (hydrophones) that the ship drags behind it. The air gun generates waves, and these waves propagate, reflect from structures in the Earth, and return to the hydrophones where they are recorded. The big difference is that seismic waves do not move along straight lines because the Earth is composed of many different layers with varying composition. As a result, the speed of propagation varies, and waves take a complicated path through the Earth. From the collected seismic data, a rough image of the subsurface is formed by estimating the velocity and modeling the wave fields from the source forward in time and the receiver backward in time using the wave equation. The source and receiver wave fields are then cross-correlated in time to produce an image of the Earth. The mismatch between images formed from different source-and-receiver pairs can be used to get a better estimate of the velocity. The process is iterated to improve the velocity. Thus, the key to building a good model of the subsurface is being able to solve the wave equation quickly and accurately many times. Direct discretization of the wave equation is computationally too expensive and, like the bats, we have to find a simplified wave propagation model which is sufficiently accurate and can be solved rapidly. Asymptotic methods valid for high-frequency waves are used for this purpose. Seismic waves are not considered particularly high-frequency, but “high frequency” really means that the wave length is small compared to the size of the simulation domain and the variations of the wave propagation speed, and in this case both of these assumptions are valid.

14

Chapter 1. Planet Earth

The most commonly used method is based on geometric optics, also referred to as WKB (after the physicists Wentzel, Kramers, and Brillouin, who developed the method in 1926) or ray-tracing. The key idea is to represent the wave field using an amplitude function and a phase function. Usually, these functions are slowly varying (when compared to the wave oscillations) and can be represented numerically by far fewer points than the original wave field. However, there is no free lunch, and this gain comes at a price: the partial differential equation for the phase function is nonlinear. One major problem is that the equation for the phase can only be solved classically for a short time interval, and after that we have to use more exotic solutions such as viscosity solutions. This departure from a classical solution is an observable phenomenon called “caustic regions,” and we have all observed them as the dancing bright spots at the bottom of a pool. At caustic regions, waves arrive in phase and can no longer be represented using a single phase function.

Figure 1.11. Top: Several snapshots of the real part of the wave field for a Gaussian beam. Time flows along the plotted curve, which shows the path of the packet of waves. Bottom: Several snapshots of the real part of the wave field for a Gaussian superposition beam. The plotted lines are the curves for each Gaussian beam. Note that the wave field remains regular at the caustics, where the curves cross and traditional asymptotic methods break down.

More recently, an alternative method to geometric optics, called Gaussian beams, has gained popularity. Gaussian beams are asymptotic high-frequency wave solutions to hyperbolic partial differential equations (such as the wave equation) that are concentrated on a single curve. Gaussian beams are a packet of waves that propagates coherently. A good example is a laser pointer, which forms a tight beam of light that is concentrated along a single line. Gaussian beams are essentially the same, except that the width of the beam is much smaller, and in media with a varying wave propagation speed the curves are not necessarily straight lines. The amplitude of this coherent packet of waves decays like a Gaussian distribution, whence the name. One of the most important features of Gaussian beams is that they remain regular for all time and do not break down at caustics. The trick to making Gaussian beams a useful tool for solving the wave equation is to recognize that, since the wave equation is linear, we can take many different Gaussian beams and add them together and still obtain an approximate solution to the wave equation. Such superpositions of Gaussian beams allow us to approximate solutions that are not necessarily

1.8. Thinking of Trees

15

concentrated on a single curve. Since each Gaussian beam does not break down at a caustic, neither will its superposition. The existence of Gaussian beam solutions has been known since the 1960s, first in connection with lasers. Later, they were used in pure mathematics to analyze the propagation of singularities in partial differential equations. Since the 1970s, Gaussian beams have been used sporadically in applied mathematics research, with a major industrial success in imaging subsalt regions in the Gulf of Mexico in the mid 1980s. Questions about the accuracy of solutions obtained by Gaussian beam superposition and their rate of convergence to the exact solution have only been answered in the last ten years, justifying the use of such solutions in numerical methods.

1.8 Thinking of Trees Ilya Zaliapin, University of Nevada at Reno It is October. Very soon the inspiring canvas of the fall foliage will be gone, and we will raise our eyes once in a while to enjoy an unexplained beauty of the branched architecture of the naked trees. Yet, there might be more than a sheer aesthetic pleasure in those views, and this is what this section is about.

Figure 1.12.

Nature exhibits many branching tree-like structures beyond the botanical trees. River networks, Martian drainage basins, veins of botanical leaves, lung and blood systems, and lightning can all be represented as tree graphs. Besides, a number of dynamic processes like the spread of a disease or a rumor, the evolution of a sequence of earthquake aftershocks, or the transfer of gene characteristics from parents to children can also be described by a (time-oriented) tree. This would sound like a trivial observation if not for the following fact. A majority of rigorously studied branching structures are shown to be closely approximated by a simple two-parametric statistical model, a Tokunaga self-similar (TSS) tree. In other words, apparently diverse branching phenomena (think of the Mississippi river vs. a birch tree) are statistically similar, with the observed differences being related to the value of a particular model parameter rather than qualitative structural traits. There exist two important types of self-similarity for trees. They are related to the Horton–Strahler and Tokunaga indexing schemes for tree branches. Introduced in hydrology in the mid-20th century to describe the dendritic structure of river networks, these schemes have been rediscovered and used in other applied fields since then.

16

Chapter 1. Planet Earth

Here we give some definitions (these definitions do not affect the remainder of this section and can be safely skipped): The Horton–Strahler indexing assigns orders to the tree branches according to their relative importance in the hierarchy. Namely, for a rooted tree T we consider the operation of pruning—cutting the leaves as well as the single-child chains of vertices that contain a leaf. Clearly, consecutive application of pruning eliminates any finite tree in a finite number of steps. A vertex in T is assigned the order r if it is removed from the tree during the r th application of pruning. A branch is a sequence of adjacent vertices with the same order. Quite often, observed systems show geometric decrease of the number N r of branches of Horton–Strahler order r [1]; this property is called Horton self-similarity. The common ratio R of the respective geometric series is called the Horton exponent. The stronger concept of Tokunaga self-similarity addresses so-called side branching— merging of branches of distinct orders. In a Tokunaga tree, the average number of branches of order i ≥ 1 that join a branch of order (i + k), where k ≥ 1, is given by Tk = a c k−1 . The positive numbers (a, c) are called Tokunaga parameters. Informally, the Tokunaga self-similarity implies that different levels of a hierarchical system have the same statistical structure, as is signified by the fact that Tk depends on the difference k between the child and parental branch orders but not on their absolute values. A classical model that exhibits both Horton and Tokunaga self-similarities is the critical binary Galton–Watson branching process [1], also known in hydrology as Shreve’s random topology model. This model has R = 4 and (a, c) = (1, 2). The general interest in fractals and self-similar structures in the natural sciences during the 1990s led to a quest, mainly inspired and led by Donald Turcotte, for Tokunaga selfsimilar tree graphs of diverse origin. As a result, Horton and Tokunaga self-similarities with a broad range of respective parameters have been empirically or rigorously established in numerous observed and modeled systems, well beyond river networks, including botanical trees, vein structures of botanical leaves, diffusion limited aggregation, twodimensional site percolation, nearest-neighbor clustering in Euclidean spaces, earthquake aftershock series, and dynamics of billiards; see, for example, [2, 3, 4, 5] and the references therein. The increasing empirical evidence prompts the question “What basic probability models can generate Tokunaga self-similar trees with a range of parameters?” or, more informally, “Why do we see Tokunaga trees everywhere? Is it because we are unable to reject the Tokunaga hypothesis, or is it because the Tokunaga constraint is truly important?” Burd, Waymire, and Winn [1] demonstrated that Tokunaga self-similarity is a characteristic property of the critical binary branching in the broad class of (not necessarily binary) Galton–Watson processes. Recently, Zaliapin and Kovchegov [5] studied the level-set tree representation of time series (an inverse of the Harris path) and established Horton and Tokunaga self-similarities of a symmetric random walk and a regular Brownian motion. They also demonstrated Horton self-similarity of the Kingman coalescent process and presented the respective Tokunaga self-similarity as a numerical conjecture [2]. Other recent numerical experiments suggest that multiplicative and additive coalescents, as well as fractional Brownian motions, also correspond to Tokunaga self-similar trees. These results expand the class of Horton and Tokunaga self-similar processes beyond the critical binary Galton–Watson branching, and suggest that simple models of branching, aggregation, and time series generically lead to the Tokunaga self-similarity. This makes the omnipresence of Tokunaga trees in observations less mysterious and opens an interesting avenue for further research. In particular, the equivalence of different processes via their respective tree representation (as is the case for Kingman’s coalescent and

1.8. Thinking of Trees

17

white noise [5]) may broaden the toolbox of empirical and theoretical exploration of various branching phenomena. References [1] G.A. Burd, E.C. Waymire, and R.D. Winn, A self-similar invariance of critical binary Galton-Watson trees, Bernoulli 6 (2000) 1–21. [2] Y. Kovchegov and I. Zaliapin, Horton self-similarity of Kingman’s coalescent tree, 2013, http://arxiv.org/abs/1207.7108 [3] W.I. Newman, D.L. Turcotte, and A.M. Gabrielov, Fractal trees with side branching, Fractals 5 (1997) 603–614. [4] D.L. Turcotte, J.D. Pelletier, and W.I. Newman, Networks with side branching in biology, J. Theor. Biol. 193 (1998) 577–592. [5] I. Zaliapin and Y. Kovchegov, Tokunaga and Horton self-similarity for level set trees of Markov chains, Chaos, Solitons & Fractals 45 (2012) 358–372.

Chapter 2

Ocean and Atmosphere

2.1 Atmosphere and Ocean Dynamics through the Lens of Model Systems Greg Lewis, University of Ontario Institute of Technology The atmosphere and ocean are central components of the climate system, where each of these components is affected by numerous significant factors through highly nonlinear relationships. It would be impossible to combine all of the important interactions into a single model. Therefore, determining the contribution of each factor, in both a quantitative and qualitative sense, is necessary for the development of a predictive model, not to mention a better understanding of the climate system. An approach that is appealing to mathematicians is to construct a hierarchy of models, starting from the most simple, designed to provide a “proof of concept,” and then progressively add details and complexity, as an understanding of the factors involved in the more simple model is developed and while the necessary analysis tools are developed and refined. The question arises as to how you progress from the simple to the complex, and the challenge is to justify your choice. One possibility is to study quantitatively accurate mathematical models of what I will call model systems—that is, the simplification does not come through the mathematical modeling but comes simply by considering a simpler system. For example, the mathematical models of these systems do not require the use of parameterizations of subgrid scale processes. In not so many words, these systems resemble laboratory experiments that could be, or have been, conducted. An example of a model system is the differentially heated rotating annulus, which consists of a fluid contained in a rotating cylindrical annulus while the rotation rate and the temperature difference between the inner and outer walls of the annulus are varied (Figure 2.1). Systems of this type produce an intriguing variety of flow patterns that resemble those observed in actual geophysical flows [1]. Thus, a careful study of this system is not only inherently interesting from a nonlinear-dynamics or pattern-formation perspective but can also provide insight into the dynamical properties of large-scale geophysical fluids. The study of model systems is appealing for a number of reasons. First, although still very challenging, an analysis of mathematical models of these systems may be feasible if appropriate numerical methods are considered. For example, numerical bifurcation techniques for large-scale dynamical systems as discussed in [2] can be useful in the analysis 19

20

Chapter 2. Ocean and Atmosphere

Figure 2.1. The differentially heated rotating annulus (left) and an example of a rotating wave (right), which is represented by a snapshot of a horizontal cross-section. The rotating wave rotates at constant phase speed, where the colors represent the fluid temperature (blue is cold, red is hot), and the arrows represent the fluid velocity. Left image from [8]. Right image reprinted with permission from N. Perinet.

of these systems; see, for example, [3], where bifurcation methods are applied to the differentially heated rotating annulus. Also, unlike realistic large-scale models, the results of the analysis may be quantitatively verified by comparison with observations from laboratory experiments, providing a very stringent test on the validity of any new numerical method or analysis technique that you may use. I have emphasized the mathematical perspective, but, of course, the experiments themselves are invaluable in developing an understanding of physical phenomena of interest. Such an approach can be applied even when the corresponding model system cannot be replicated in the laboratory. For example, in [4] and [5] we study a model of a fluid contained in a rotating spherical shell that is subjected to radial gravity and an equatorto-pole differential heating. The results show that, as the differential heating is increased, a transition from a one-cell pattern to a two- or three-cell pattern is observed, and we show that this transition is associated with a cusp bifurcation. This transition may be related to the expansion of the Hadley cell that has been observed in Earth’s atmosphere. Although this approach is “standard” and has fruitfully been used for many years [1], I don’t believe that it has been fully exploited in the context of atmospheric and ocean dynamics. However, recently it has received increased interest, as evidenced by the publication of a forthcoming book [6] and the proceedings of a EUROMECH workshop [7] that took place at the Freie Universität Berlin in September, 2013. References [1] P.L. Read, Dynamics and circulation regimes of terrestrial planets, Planet. Space Sci. 59 (2011) 900–914. [2] H.A. Dijkstra et al., Numerical bifurcation methods and their application to fluid dynamics, Commun. Comput. Phys. 15 (2014) 1–45. [3] G.M. Lewis, N. Périnet, and L. van Veen, The primary flow transition in the baroclinic annulus: Prandtl number effects, in Modelling Atmospheric and Oceanic Flows, Geophysical Monograph Series, edited by T. von Larcher and P.D. Williams, Geopress, Amsterdam, 2014, pp. 45–59.

2.2. Atmospheric Waves and the Organization of Tropical Weather

21

[4] W.F. Langford and G.M. Lewis, Poleward expansion of Hadley cells, Can. Appl. Math. Quart. 17 (2009) 105–119. [5] G.M. Lewis and W.F. Langford, Hysteresis in a rotating differentially heated spherical shell of Boussinesq fluid, SIAM J. Appl. Dyn. Syst. 7 (2008) 1421–1444. [6] Modelling Atmospheric and Oceanic Flows. Insights from Laboratory Experiments and Numerical Simulations, Geophysical Monograph Series, edited by T. von Larcher and P.D. Williams, Geopress, Amsterdam, 2014. [7] http://euromech552.mi.fu-berlin.de/ [8] G.M. Lewis and W. Negata, Double Hopf bifurcations in the differentially heated rotating annulus, SIAM J. Appl. Math. 63 (2003) 1029–1055.

2.2 Atmospheric Waves and the Organization of Tropical Weather Joseph Biello, University of California at Davis Though waves of one sort or another are a ubiquitous part of our daily experience (think of the light from your screen, or the sound from your kids in the other room), we have to get on with our lives and therefore tend not to think of the wave-like nature of daily phenomena.

Figure 2.2. Oceanic gravity waves. Reprinted with permission from Fotolia.com, galam. c

Those fortunate among us who can escape to the shore on a hot August day can take the time to observe the sea and the waves she sends us. We watch these waves rise as if out of nothing, break, and then crash on the beach. We see a slow, nearly periodic pattern in the swell punctuated by a burst of white water. In fact, for most of us, the word “wave” evokes this wave, the oceanic surface shallow-water gravity wave. Among all waves, it is in the oceanic gravity wave that we are able to observe (in fact experience) most of the properties of a wave. We see its wavelength, its crest, and its trough, and we can hear its period by listening to the breakers. However, what we often do not see is the origin of the wave. Far out in the ocean, storms generate strong winds which raise and depress the ocean surface, thereby generating waves. As surfers are well aware, these waves travel thousands of kilometers and ultimately release their energy on the coastlines. What we do see, but may not realize, is that the breaking of the wave ultimately releases the wave energy in a manner that cannot be re-injected into the wave. It splashes, sprays, and erodes the shoreline, thereby

22

Chapter 2. Ocean and Atmosphere

dissipating its energy. Furthermore, the wave height itself changes as the wave enters shallower water—which is why, from the shore, waves appear to grow from the ocean as if from nothing. Imagine standing at the eastern edge of a California bay and staring west at the Pacific Ocean as waves from tropical storms are rounding the point at the southern end of the bay. If we want to calculate the coastal erosion associated with these waves, we need only know some aggregate properties of the waves: wavelength, height, and average frequency of occurrence should suffice. However, if we want to determine the best surfing times, we need to know the timing of a particular tropical storm and its energy—thereby knowing the properties of the waves it produces and what time to be on the water in order to catch them. If storm systems appear in the North Pacific at the same time as tropical Pacific storms, then the waves from both systems will interact and break in a wholly different manner on the California coast than if the storm systems had not coincided.

Figure 2.3. Atmospheric internal gravity waves. Reprinted courtesy of Glen Talbot.

The atmosphere contains waves that are very similar to oceanic gravity waves. They are called “internal gravity waves” because they occur throughout the troposphere (the depth of the weather layer of the atmosphere), not just at its upper surface. We experience these waves as pressure and wind undulations with very low frequencies—wind blows from the east one day, from the west the next day, and from the east again the following day. Gravity waves exist on a wide variety of wavelengths, from meters to kilometers to thousands of kilometers. In the grossest sense, they are all generated in the same manner: something pushes the air up. The most interesting phenomena that push the air up are thunderstorms—also known as atmospheric convection. Moist air is lifted up to a height where the air is colder and the water begins to condense, releasing its latent heat and forming a cloud. This is the opposite of what occurs in a humidifier, where water is heated in order to evaporate it and moisten the air. As the latent heat is released in the atmosphere, the air warms and becomes less dense than its surroundings, thereby rising into a cooler environment and causing it to condense even more. Under the right conditions, this becomes a runaway process, whereby the condensation continues and the cloud rises until it hits the top of the troposphere. As the air rises and the thunderstorm forms, ambient air is being sucked into the bottom of the cloud while the top of the cloud is pushing air away from it. This process is highly agitating to the atmosphere and generates a gravity

2.3. Nonlinear Waves and the Growth of a Tsunami

23

wave. It is not unlike a pebble falling into a pond and generating a surface water gravity wave. However, atmospheric gravity waves differ from surface water waves in a very crucial way. As an atmospheric gravity wave travels away from the thunderstorm which generated it, it raises moist air, thereby cooling it and initiating condensation and, possibly, another thunderstorm. Thunderstorm cells cluster around one another so that, in the words of atmospheric scientist Brian Mapes (University of Miami, Florida), convection tends to be gregarious. Just as gravity waves occur on many length scales, so too does this organization. On the largest scale, thunderstorm systems over the Indian Ocean can generate gravity waves that extend over six thousand kilometers longitude. These equatorial Kelvin waves travel eastward along the equator for some twenty thousand kilometers until they hit the Andes of South America. Along the way they excite thunderstorm activity along the whole equatorial Pacific Ocean. Kelvin waves move fast, and computer simulations of climate have yet to accurately simulate them. From time to time, equatorial Kelvin waves encounter other kinds of waves (Rossby waves) coming from the North Pacific. When they do, warm moist tropical Pacific air is channeled toward the western United States. In turn, this air sends storms and rain across the North American continent. So, in order to predict weather over the U.S. we have to carefully observe what is happening over Alaska and the Indian Ocean several days before. Just as understanding coastal erosion does not require detailed knowledge of the timing of surface waves but only of their aggregate properties, so too are predictions of climate change not affected by the details of atmospheric waves, since we have a good understanding of their average properties. But surfers need to know the timing of the waves—from both the north and the south. So too, in order to improve weather forecasts and to understand how climate change will modify weather, we must understand atmospheric waves. How often do tropical Kelvin and North Pacific Rossby waves coincide? Under what conditions will they interact to cause storm systems? How do these waves interact with the ocean and thereby change the moisture of the air in which they travel? Least understood of all, what will happen to these wave interactions as our climate changes? Just as there is energy in the ocean surface waves that must be deposited on some coastline, so too there is energy in moist tropical air that must be released somewhere in the form of rain—somewhere other than where it is being released now. This means that the pattern of rains will change for the whole Pacific basin, which is all the more concerning for places like California.

2.3 Nonlinear Waves and the Growth of a Tsunami Estelle Basor, American Institute of Mathematics At the American Institute of Mathematics (AIM) in February 2013, Mark Ablowitz told me about an interesting article (with beautiful pictures) he wrote with Douglas Baldwin, entitled Nonlinear shallow ocean wave soliton interactions on flat beaches [1]. The propagation of these solitary waves may contribute to the growth of tsunamis. The article, which appeared in the journal Physical Review E, has gained a lot of media attention. It was written up as a synopsis on the American Physical Society (APS) website, with links to some nice videos. It was subsequently identified as a special focus article in the November 2012 issue of Physics Today and featured in the Bulletin of the American Meteorological Society (BAMS, January, 2013), and picked up by several science news outlets: OurAmazingPlanet, New Scientist, NRC Handelsblad (the largest

24

Chapter 2. Ocean and Atmosphere

Figure 2.4. Long stem X-type interactions with taller stem height. Taken on Venice Beach, California, by Douglas Baldwin. Reprinted with permission.

evening newspaper in the Netherlands), NBC.com, the U.S. National Tsunami Hazard Mitigation Program (NTHMP), and others. As reported in the synopsis, “Previously, the assumption was that these interactions are rare. However, the authors have observed thousands of X and Y waves shortly before and after low tide at two flat beaches, where water depths were less than about 20 centimeters. The researchers showed that the shallow waves could be accurately described by a two-dimensional nonlinear wave equation.” Reference [1] M.J. Ablowitz and D.E. Baldwin, Nonlinear shallow ocean wave soliton interactions on flat beaches, Phys. Rev. E 86 (2012) 036305, http://dx.doi.org/10.1103/PhysRevE.86.036305

2.4 Universality in Fractal Sea Coasts Christiane Rousseau, University of Montreal Sandy coasts have a smooth profile, while rocky coasts are fractal. A characteristic feature of a rocky coast is that new details appear when we zoom in, and if we were to measure the length of the coastline, we would see that this length increases significantly when we zoom in on the details. If we model the coastline as a curve, then this curve would have an infinite length. One summarizes the characteristics of the coastline through a number, its dimension, which describes the “complexity” of the curve. A smooth curve has dimension 1, and a surface has dimension 2. A dimension between 1 and 2 is typical of a self-similar object, which is thicker than a curve but has an empty interior. How does the dimension of a fractal coast depend on the coast? An article of Sapoval, Baldassarri, and Gabrielli presents a model suggesting that this dimension is independent of the coast and very close to 43 [1]. The model of Sapoval, Baldassarri, and Gabrielli describes the evolution of the coast from a straight line to a fractal coast through two processes with different time scales: a fast time and a slow time. The mechanical erosion occurs rapidly, while the chemical weakening of the rocks occurs slowly. The force of the waves acting on the coast depends on the length of the coastline. Hence, the waves have a stronger destructive power when the coastline is linear, and a damping effect takes place when the coast is fractal. The erosion model is a kind of percolation model with the resisting Earth modeled by a square lattice. The lithology of each cell—that is, the resistance of its rocks—is represented by a number between 0 and 1. The resistance to erosion of a site, also given by a number between 0 and 1, depends both on its lithology and on the number of sides exposed to

2.5. Ice Floes, Coriolis Acceleration, and the Viscosity of Air and Water

25

Figure 2.5. Rocky coasts are natural fractals. Reprinted with permission from The Fractal Foundation.

the sea. Then an iterative process starts: each site with resistance number below a threshold disappears, and the resistances of the remaining sites are updated because new sides become exposed to the sea. This leads to the creation of islands and bays, thus increasing the perimeter of the coast. When the perimeter is sufficiently large, thus weakening the strength of the waves, the rapid dynamics stops, even if the power of the waves is nonzero! During this period the dimension of the coast is very close to 43 . This is when the slow dynamics takes over, since chemical weakening of the remaining sites continues, thus reducing the resistance to erosion of sites. The slow dynamics is interrupted by short episodes of fast erosion when the resistance number of a site falls below the threshold. This dynamics of alternating short episodes of fast erosion and long episodes of chemical weakening is exactly what we observe now, since the initial fast dynamics occurred long ago. The model presented here is a model of percolation gradient, with the sea percolating in the Earth, and such models of percolation gradient exhibit universality properties. Reference [1] B. Sapoval, A. Baldassarri, and A. Gabrielli, Self-stabilized fractality of seacoasts through damped erosion, Phys. Rev. Lett. 93 (2004) 098501.

2.5 Ice Floes, Coriolis Acceleration, and the Viscosity of Air and Water Robert Miller, Oregon State University I have always wanted to run this story down since I saw the reference to a paper by G.I. Taylor [1] in Lamb’s Hydrodynamics. The paper contains a description of what oceanic and atmospheric scientists call “Ekman layers.” Physical oceanographers learn early in their careers that the Norwegian oceanographer Fridtjof Nansen (1861–1930) noted on the Fram expedition of 1893–1896 [2] that ice floes tend to drift to the right of the wind. Nansen suggested to his colleague Vilhelm Bjerknes (1862–1951) that the explanation of this phenomenon be assigned as a problem to a student. Bjerknes chose Vagn Walfrid Ekman (1874–1954), who explained the

26

Chapter 2. Ocean and Atmosphere

crosswind transport of ice floes as resulting from a balance between Coriolis acceleration and viscous drag. A description of what is now called the “Ekman layer” can be found in many places, for example the Wikipedia page [3] or just about any text on physical oceanography, e.g., [4]. Here is a bare-bones explanation. Steady flow governed by a balance of Coriolis force and viscous drag looks like this: − f v = ν uz z ,

f u = νv z z , f = 2Ω sin φ,

where (u, v) are the horizontal velocity components, Ω = 2π/86400 sec is the angular rotation rate of the Earth, φ is the latitude, and ν is the viscosity (more about this later). For flows near the ocean surface, the vector wind stress (τ (x) , τ (y) ) enters as a boundary condition, (τ (x) , τ (y) ) = ν(u, v) z . The trick is to divide both sides by ν, multiply the second equation by the imaginary unit i, and consider the system of complex conjugate scalar ODEs obtained by adding and subtracting the two equations, (u + i v) z z − i( f /ν)(u + i v) = 0, (u − i v) z z + i( f /ν)(u − i v) = 0.

These equations are readily solved, 

  z π  z π  e z/d τ (x) cos − − τ (y) sin − , df d 4 d 4   z π  2 z/d  (x)  z π  τ sin e − + τ (y) cos − , v= df d 4 d 4

u=

2

where d = (2ν/ f )1/2 . Integration over the water column from z = −∞ to z = 0 yields the result that transport is to the right of the wind. If you draw the two-component current vectors at each depth as arrows, with the tails on the z-axis, you will see that the heads trace out a nice spiral (Figure 2.6). Oceanographers talk casually about Ekman layers and Ekman transport, but, to be precise, Ekman layers do not appear in nature. The real atmosphere and ocean are turbulent. Vertical momentum transfers do not occur by simple diffusion, and characterization of penetration of surface stress into the interior fluid by a simple scalar diffusion coefficient is only the crudest approximation. Ekman knew this. He noted that if he used the measured viscosity of sea water for ν and typical wind stress magnitudes for τ (x,y) , the surface layer would be less than a meter thick. He did not refer to the work of Reynolds on turbulence [5], and much of Ekman’s dissertation is concerned with trying to model flow in the real ocean in terms of the mathematical tools available to him. G.I. Taylor was apparently unaware of Ekman’s work when he applied the same analytical machinery to form an expression for the surface velocity profile in the atmospheric surface boundary layer. Taylor was interested in quantifying the transport of physical properties by macroscopic eddies—that is, he wanted to estimate what we now recognize as eddy diffusivity and eddy viscosity. Taylor compared the solutions of his equations to observations taken from balloons and backed out estimates of kinematic viscosities varying over an order of magnitude, from about 3 · 104 to 6 · 104 cm2 /sec over land, and from 7.7 · 102 to 6.9 · 103 cm2 /sec over water. The molecular kinematic viscosity of air, by comparison, is more like 10−3 cm2 /sec. The sixth edition of Lamb’s Hydrodynamics [6]

2.5. Ice Floes, Coriolis Acceleration, and the Viscosity of Air and Water

27

Figure 2.6. The Ekman layer is the layer in a fluid where the flow is the result of a balance between pressure gradient, Coriolis, and turbulent drag forces. In this picture, the 10 m/s wind at 35◦ N creates a surface stress, and a resulting Ekman spiral is found below it in the column of water. Reprinted with permission from Texas A&M University, Department of Oceanography.

contains discussions of both Ekman’s and Taylor’s work. It’s just as well that we refer to “Ekman layers” rather than “Taylor layers.” Ekman was first, after all. Richardson, in his (1922) book Weather Prediction by Numerical Process [7], laid out specific methodology for numerical weather prediction. Richardson was a man far ahead of his time. He could not have conceived of an electronic computer—his book was published decades before the work of Turing and von Neumann. He imagined that numerical weather prediction would be done by great halls full of people working the mechanical calculators of the day. Richardson understood that he would need to specify the viscosity of air and, after examining the results available to him, including Taylor’s work, concluded that air is slightly more viscous than Lyle’s Golden Syrup [8] and slightly less viscous than shoe polish. Ninety years after the publication of Richardson’s book we have come a very long way, but we still don’t know how to deal with turbulent transports in models of the ocean and atmosphere. References [1] G.I. Taylor, Eddy motion in the atmosphere, Philos. Trans. R. Soc. London A 215 (1915) 1–26. [2] Nansen’s Fram expedition, Wikipedia, http://en.wikipedia.org/wiki/Nansen%27s_Fram_expedition [3] Ekman layer, Wikipedia, http://en.wikipedia.org/wiki/Ekman_layer [4] J.A. Knauss, Introduction to Physical Oceanography, Prentice Hall PTR, 1978. [5] D. Jackson and B. Launder, Osborne Reynolds and the Publication of His Papers on Turbulent Flow, http://www.annualreviews.org/doi/pdf/10.1146/annurev.fluid.39.050905.110241 [6] H. Lamb, Hydrodynamics, sixth edition, Dover, New York, 1945. [7] L.F. Richardson, Weather Prediction by Numerical Process, Cambridge University Press, 1922. [8] http://www.lylesgoldensyrup.com/

28

Chapter 2. Ocean and Atmosphere

2.6 There Will Always Be a Gulf Stream — An Exercise in Singular Perturbation Technique Robert Miller, Oregon State University One hears occasionally in the popular media that one possible consequence of global warming might be the disappearance of the Gulf Stream. This makes physical oceanographers cringe. The Gulf Stream and its analogues in other ocean basins exist for fundamental physical reasons. Climate change may well bring changes in the Gulf Stream; it may not be in the same place, it may not have the same strength, or it may not have the same temperature and salinity characteristics. But as long as the continents bound the great ocean basins, the sun shines, the Earth turns toward the east, and the wind blows in response, there will be a Gulf Stream. There will also be a Kuroshio, as the analogous current in the north Pacific is called, as well as the other western boundary currents, so called because, like the Gulf Stream, they form on the western boundaries of ocean basins.

Figure 2.7. Surface temperature in the western North Atlantic. North America is black and dark blue (cold); the Gulf Stream is red (warm). Reprinted courtesy of NASA.

The dynamical description of the Gulf Stream can be found in just about any text on physical oceanography or geophysical fluid dynamics. I first learned the general outlines of ocean circulation as a young postdoc, when my mentor Allan Robinson (see [1]) walked into my office and dropped a copy of The Gulf Stream by Henry Stommel on my desk and said “Read this, cover to cover.” We’ve learned a great deal about western boundary currents in the fifty years since The Gulf Stream was published, but it’s still an excellent introduction to the subject and a good read. The Gulf Stream is long out of print, but used copies can occasionally be found. I got mine at Powell’s for six bucks, with original dust jacket. Electronic versions can be found in many formats [2].

2.6. There Will Always Be a Gulf Stream — An Exercise in Singular Perturbation Technique

29

There are lots of places to learn about the wind driven ocean circulation. My purpose here is to present the fundamental picture as an exercise in perturbation technique. β-Plane. We model an ocean basin in Cartesian coordinates: x = (λ − λ0 )R cos φ0 , y = (φ − φ0 )R, z = r − R,

(2.1) (2.2) (2.3)

where (φ, λ, r ) are the coordinates of a point with latitude φ, longitude λ, and distance r from the center of the Earth, and R is the radius of the Earth; φ0 and λ0 are the latitude and longitude of a reference point in the mid-latitudes. We account for the rotation of the Earth by the Coriolis parameter f = 2Ω sin φ, where the angular speed of rotation of the Earth is Ω = 2π/86400 s. We approximate the Coriolis parameter as a linear function of latitude f = f0 + βy. This is the only effect we will consider to account for the fact that the Earth is round. Reduced-gravity model. We model the density-stratified ocean as being composed of two immiscible fluids of slightly different densities in a stable configuration, where the denser fluid lies below the less dense one. In such a configuration, there is a family of waves that is much slower than the waves on the surface of a homogeneous fluid of the same depth would be. You’ve probably seen the effect in the clear plastic boxes containing different colored fluids, usually one clear and the other blue, that are sold as toys in stationery stores or on the Internet [3]. When you tip the box you see waves propagate slowly along the interface. We simplify the two-layer model further by assuming that the deeper layer is motionless, so the thickness of the upper layer adjusts in such a way as to make the pressure gradient vanish in the lower layer. The equations of motion for the upper layer are formally identical to the shallow water equations but with the acceleration of gravity g ≈ 10 ms−2 reduced by a factor of Δρ/ρ0 , where ρ0 is the density of the upper layer. In the reduced-gravity model, the equations for steady linear flow on the β-plane in the rectangle (x, y) ∈ [0, a] × [0, b ] are − f hv + g h h x = − (τ0 /ρ0 ) cos(πy/b ) − A(u h),

(2.4)



(2.5)

(h u) x + (hv)y = 0,

(2.6)

f h u + g h hy = − A(v h),

where h is the thickness of the upper layer, (u, v) are the horizontal velocity components, and g = (Δρ/ρ0 ) g is the reduced gravity. The basin dimensions a and b are typically thousands of kilometers. We assume linear drag with constant drag coefficient A, and wind stress with amplitude τ0 in the x-direction only. The model is intended to be a schematic picture of a mid-latitude ocean basin in the northern hemisphere. The stress pattern of winds from the east in the southern half of the domain and from the west in the northern half is intended as a schematic model of the trade winds south of a relatively calm region at about 30o N, the horse latitudes [4], and westerly winds to the north. Stream function. From the continuity equation (2.6), we can define a transport stream function ψ, ψ x = hv, ψy = − h u. (2.7) The boundaries of our idealized ocean are assumed impermeable, so we choose ψ = 0 on the boundaries. Taking the curl of the momentum equations (2.5) and (2.6) leads to the

30

Chapter 2. Ocean and Atmosphere

equation A∇2 ψ + βψ x = −

τ0 π b ρ0

sin

 πy  b

.

(2.8)

Look for a separable solution of the form ψ = F (x) sin(πy/b ). Then F +

β A

F −

π2 b

2

F=−

τ0 π Ab ρ0

,

(2.9)

so F = τ0 b /(πρ0 A) + G, where G obeys the homogeneous equation G +

β A

F −

π2 b2

G = 0.

(2.10)

Hence, G = G+ e λ+ x + G− e λ− x for some constants G+ and G− , where λ± = −

β 2A





1± 1+

4π2 A2 β2 b 2

1/2 

.

This could be solved as a boundary value problem, but it is more enlightening to rescale the problem. Interior solution. With the substitutions x → x/L x , y → y/b , (2.8) becomes     L2x τ0 π Lx A sin(πy). (2.11) − ψ x x + 2 ψyy + ψ x = βL x b βb ρ0 b For basin-scale motions we may choose L x = b . Dissipation in the ocean is very weak, so A/(βb ) 1, and the first term on the left-hand side can be neglected, leaving the reduced equation  πy  τ0 π βhv = − , (2.12) sin b ρ0 b where dimensions have been restored. This is a special case of the Sverdrup relation, hv = ∇ × (τ (x) , τ (y) )/β, which says that transport in the north-south direction is proportional to the curl of the wind stress [5]. This is a good approximation in the interior, but (2.12) is a first-order equation and cannot satisfy the boundary conditions at the eastern and western boundaries, so the Sverdrup balance cannot hold uniformly. (Also, in this example, the transport in the interior of the basin is southward, and there must be a northward return flow.) Near at least one of the boundaries the momentum balance must be different. Choose ψ = 0 at x = a, so the interior solution is τ0 πa  x   πy  ψ= 1− sin . (2.13) βb ρ0 a b Boundary layer solution. We must now find an approximate solution to (2.8) in a thin strip near x = 0 with ψ(0, y) = 0 and ψ(x, y) → ((τ0 πa)/(βb ρ0 )) sin(πy/b ) as x → ∞. If we choose L x = A/β, (2.11) becomes     τ0 π A2 A sin(πy). (2.14) ψ x x + 2 2 ψyy + ψ x = − βb βb ρ0 β b

2.7. The Great Wave and Directional Focusing

31

Now, A/(βb ) 1, so to leading order near the boundary, ψ x x + ψ x = 0 and ψ = C0 + C1 exp(−x) for some constants C0 and C1 . Hence, near the boundary, in dimensional form,  τ0 π

ψ= 1 − e −xβ/A sin(πy), (2.15) βb ρ0 very close to Stommel’s solution. This boundary layer at x = 0 is the analogue, within our simple model, of the Gulf Stream. Back to the Gulf Stream. What would happen if we were to choose the interior solution as τ0 πa  x   πy  ψ= − sin (2.16) βb ρ0 a b and attempted to fit a boundary condition at x = a? It wouldn’t work. We would need to find C1 and C2 such that, in a neighborhood of the boundary, ψ = C0 + C1 e −xβ/A such that ψ(x = a, y) = 0 and ψ(x, y) → −((τ0 πa)/(βb ρ0 )) sin(πy/b ) as x → −∞, which is clearly impossible. So the fact that western boundary currents occur on the west side of ocean basins is a consequence of the fact that β is positive, and western boundary currents form in the southern hemisphere as well. References [1] F.H. Abernathy, Allan R. Robinson, obituary, Harvard Gazette, March 8, 2012, http://news.harvard.edu/gazette/story/2012/03/allan-r-robinson/ [2] H. Stommel, The Gulf Stream: A Physical and Dynamical Description, University of California Press, Berkeley and Los Angeles, 1958. [3] http://www.officeplayground.com/Liquid-Wave-Paperweight-Clown-Fish-P133.aspx [4] Horse latitudes, Wikipedia, http://en.wikipedia.org/wiki/Horse_latitudes [5] Sverdrup balance, Wikipedia, http://en.wikipedia.org/wiki/Sverdrup_balance

2.7 The Great Wave and Directional Focusing John M. Dudley, Université de Franche-Comté Frédéric Dias, University of California at Davis One of the most famous images in art is the Great Wave off Kanagawa, a woodblock print by the Japanese artist Hokusai (1760–1849). The print shows an enormous wave on the point of breaking over boats that are being sculled against the traveling wave (Figure 2.8, left). It is as famous in mathematics as it is in art: first because the structure of the breaking wave at its crest illustrates features of self-similarity, and second because the large amplitude of the wave has led it to be interpreted as a rogue wave generated from nonlinear wave effects [1]. However, we have just published a paper [2] that points out that whether the generating mechanism is linear or nonlinear does not enter into the definition of a rogue wave; the only criterion is whether the wave is statistically much larger than the other waves in its immediate vicinity. In fact, by making reference to the Great Wave’s simultaneous transverse and longitudinal localization, we show that the purely linear mechanism of directional focusing predicts characteristics consistent with those of the Great Wave. We have also been fortunate enough to collaborate with the photographer V. Sarano, who has provided us with a truly remarkable photograph of a 6 m rogue wave observed on the Southern Ocean from the French icebreaker Astrolabe, which bears a quite spectacular resemblance to the Hokusai print (Figure 2.8, right).

32

Chapter 2. Ocean and Atmosphere

Figure 2.8. Two views of a great wave. Left, Hokusai’s Great Wave. Right, a breaking wave in the southern ocean from French icebreaker Astrolabe, taken by V. Sarano [2]. Used with permission.

Figure 2.9. Computational results showing directional focusing of periodic wave trains towards an extreme wave at the focus. Reprinted with permission from Elsevier [3].

Rogue waves can arise from a variety of different mechanisms. For example, linear effects that can generate rogue waves include spatial focusing due to refraction with varying topography, wave-current interactions, and directional focusing of multiple wave trains. Nonlinear effects that have received much attention include the exponential amplification of random surface noise through modulation instability. In the case of the Great Wave, a clue as to how linear effects may play a role is seen in the localization of the Great Wave, both along its direction of travel and transversally: the wave rises from the foreground and ends in the middle ground of the print. This is in fact a characteristic linear effect of directional focusing when wave trains with different directions and phases interact at a particular point. Typical computational results of this process are shown in Figure 2.9. The modeling is based on propagation equations that include both linear and nonlinear effects, but the concentration of energy at the focus arises from linear convergence. Nonlinearity plays a role only as the wave approaches the linear focus, when it increases the steepness to the point of breaking.

2.8. Controlling Lightning?

33

The visual similarity of the numerical modeling of directional focusing with the localization properties seen in the woodcut is immediately apparent. Hence, directional focusing is clearly a mechanism that could underlie the formation of the Great Wave. In terms of the artwork of the woodcut itself, highlighting the physics of the transverse localization of the Great Wave provides room for unexpected optimism in the interpretation of the scene: the sailors may not be in as much danger as the print suggests. Is Hokusai maybe trying to highlight the Japanese crew’s skill in navigating around the wave to avoid it breaking over them? References [1] J.H.E. Cartwright and H. Nakamura, What kind of a wave is Hokusai’s “Great wave off Kanagawa”?, Notes Rec. R. Soc. London 63 (2009) 119–135. [2] J.M. Dudley, V. Sarano, and F. Dias, On Hokusai’s Great wave off Kanagawa: Localization, linearity and a rogue wave in sub-Antarctic waters, Notes Rec. R. Soc. London 67 (2013) 159–164, doi: 10.1098/rsnr.2012.0066. [3] C. Fochesato, S. Grilli, and F. Dias, Numerical modeling of extreme rogue waves generated by directional energy focusing, Wave Motion 44 (2007) 395–416.

2.8 Controlling Lightning? Christiane Rousseau, University of Montreal Halfway between chemistry and physics, the field of ultrafast laser pulses is a very promising area of research with many potential applications. These pulses range over a time scale from femtoseconds (10−15 seconds) to attoseconds (10−18 seconds, the natural time scale of the electron). Modern laser technology allows the generation of ultrafast (few cycle) laser pulses with intensities up to 3.5 × 1016 W/cm2 , exceeding the internal electric field in atoms and molecules. The interaction of ultrafast laser pulses with atoms and molecules leads to regimes where new physical phenomena can occur such as high harmonic generation (HHG), from which the shortest attosecond pulses have been created. One of the major experimental discoveries in this new regime is laser pulse filamentation (LPF), first observed by Mourou and Braun in 1995, where pulses with intense narrow cones can propagate over large distances. The discovery has led to intensive investigations in physics and applied mathematics to understand new effects such as the creation of solitons, self-transformation of these pulses into white light, intensity clamping, and multiple filamentation. Potential applications include wave-guide writing, atmospheric remote sensing, and lightning guiding. Laboratory experiments show that intense and ultrafast laser pulses propagating in the atmosphere can create successive optical solitons. These highly nonlinear nonperturbative phenomena are modeled by nonlinear Schrödinger (NLS) equations, allowing the prediction of new phenomena such as “rogue” waves, also associated with tsunamis in oceanography. (Nonperturbative means that it is impossible to apply perturbation techniques and analyze the global system by focusing on a dominant part.) Field experiments of self-guided ionized filaments for real-scale atmospheric testing are carried out, for instance by Teramobile, an international project initiated jointly by the National Center for Scientific Research (CNRS) in France and the German Research Foundation (DFG). Recently it was discovered that such intense laser pulses can create optical “rogue” waves. It is also known that these intense ultrafast pulses can generate storms or hurricanes within a distance of a few kilometers. This raises an interesting question: Is there a way to use these ultrafast laser pulses to control atmospheric perturbations? Research goes into at least two directions. The first

34

Chapter 2. Ocean and Atmosphere

Figure 2.10. Cloud of water droplets generated in a cloud chamber by laser filaments (red). The cloud is visualized through the scattering of a laser beam (green) that is collinear with the first one (red). Reprinted with permission from Jérôme Kasparian and Jean-Pierre Wolf [3].

one is to exploit the laser-filament induced condensation of water vapor, even in subsaturated conditions. A second one is to use laser filaments to control lightning, in particular to protect critical facilities. A first conference on Laser-based Weather Control (LWC2011) took place in 2011 [1] and a second (LWC2013) at the World Meteorological Organization (WMO) in Geneva in September 2013 [2]. On the home page of the conference we find the following statement: “As highlighted by the success of the first Conference on Laser-based Weather Control in 2011, ultra-short lasers launched into the atmosphere have emerged as a promising prospective tool for weather modulation and climate studies. Such prospects include lightning control and laser-assisted condensation, as well as the striking similarities between the non-linear optical propagation and natural phenomena like rogue waves or climate bifurcations. Although these new perspectives triggered an increasing interest and activity in many groups worldwide, the highly interdisciplinary nature of the subject limited its development, due to the need for enhanced contacts between laser and atmospheric physicists, chemists, electrical engineers, meteorologists, and climatologists. Further strengthening this link is precisely the aim of the second Conference on Laser, Weather and Climate (LWC2013) in Geneva, gathering the most prominent specialists on both sides for tutorial talks, free discussions as well as networking.” Where is the mathematics in all this? The phenomena induced by intense ultrafast laser pulses are nonperturbative and highly nonlinear. They are studied through nonlinear partial differential equations. References [1] First Conference on Laser-based Weather Control in 2011, Geneva, http://www.laserweatherandclimate.com/2011/index.php [2] Second Conference on Laser-based Weather Control in 2013, Geneva, http://www.laserweatherandclimate.com/index.php

2.9. Mathematician Stepping on Thin Ice

35

[3] J. Kasparian, L. Wöste, and J.-P. Wolf, Laser-Based Weather Control, Optics & Photonics News, July/August 2010, OSA.

2.9 Mathematician Stepping on Thin Ice Deborah Sullivan Brennan, San Diego Union-Tribune With a resume of scientific discoveries, and a track record of harrowing Antarctic adventures, University of Utah mathematician Ken Golden has stepped out of the ivory tower and onto thin ice. Golden, a speaker at the 2013 Joint Mathematics Meetings, held from January 9–12 at the San Diego Convention Center, gave a lecture on polar ice, a topic that has led him to the ends of the earth, and just barely back again. Over the past three decades, he’s traveled on seven voyages to Antarctica and eight to the Arctic, applying his expertise in theoretical mathematics and composite materials to questions about brine inclusions in sea ice, and the role of surface “melt ponds” on the rate of ice loss. “Our mathematical results on how fluid flows through sea ice are currently being used in climate models of sea ice,” he said. Along the way he retraced the route of the ill-fated 1914 Shackleton expedition, survived a ship fire after an engine explosion, and spent two weeks stranded on an iced-in vessel last fall. To Golden, the thrill of discovery outweighed the dangers of polar travel. “It’s like a different planet,” he said. “It’s one of the most fascinating places on earth. It’s one thing to sit in your office and prove theorems about a complicated system. It’s another thing to go down there yourself. It informs my mathematics.”

Figure 2.11. Mathematician at work: Ken Golden in Antarctica. Reprinted with permission from The San Diego Union-Tribune, LLC.

As a kind of mathematical Indiana Jones, Golden has achieved a rock star status rare among academics. The prestigious research journal Science ran a profile of Golden in 2009. Fans lined up for autographs at the Joint Mathematics Meeting and filled his lecture: one of two public events at the mostly technical conference. “Never in my wildest dreams did I imagine I’d be a math professor signing autographs,” Golden said. Golden first traveled to Antarctica during his senior year in college, along the Drake Passage, a route that Irish explorer Ernest Shackleton pioneered in 1914, after his ship

36

Chapter 2. Ocean and Atmosphere

was crushed by sea ice. Shackleton made the rescue trek in an open boat, but Golden said the journey was gut-wrenching even on a modern ship. “I have very vivid memories of crossing the Drake Passage, one of the stormiest seas in the world, and taking 50-degree rolls,” he said. Golden earned a PhD in mathematics at New York University, and landed in a professorship at the University of Utah, before returning to the realm of ice. He nearly relived Shackleton’s plight on his subsequent voyage in 1998 after the ship’s engine was destroyed in a fire. Crews sounded the emergency alarm, and then announced they were lowering the lifeboats, Golden said. “It’s not what you want to hear when you’re in the Antarctic ice pack,” he said. After five days on the ice, crews jury-rigged a backup engine and the vessel limped home, he said. Not disheartened by the mishap, Golden joined subsequent expeditions to the poles, during which he described breakthroughs in ice equations. Standing on the ice during a howling Arctic storm one night, he noticed the ground around him turning to slush, and realized “in one particular epiphany,” that the ice was reaching a percolation threshold, through which brine could flow freely. His research on the phenomenon led to his “rule of fives,” which describes the combination of temperature, salinity, and saturation at which ice becomes permeable, and helps explain how ice sheets grow. Golden’s other studies examine the role of “melt ponds” in ice, which allow ice to melt faster by reducing the amount of heat reflected. “Sea ice goes from pure white snow, to a complex, evolving mosaic of ice, snow and meltwater,” Golden said. Studying that phenomenon can quantify the overall reflectiveness of the sea ice pack, he said, helping close some gaps in current ice melt models. “Mathematics is normally rather esoteric, but Ken’s work is very applied, and it’s applied to the topic of sea ice, which is of great interest today due to climate change,” said his colleague, Ian Eisenman, a professor of climate science and physical oceanography at the Scripps Institute of Oceanography. “I think his work in general is very exciting not only for fellow scientists, but also for the general public.” This article was originally published on January 11, 2013 in the San Diego Union-Tribune.

Chapter 3

Weather and Climate

3.1 Numerical Weather Prediction Wei Kang, Naval Postgraduate School In the daily operation of weather forecasts, powerful supercomputers are used to predict the weather by solving mathematical equations that model the atmosphere and oceans. In this process of numerical weather prediction (NWP), computers manipulate vast datasets collected from observations and perform extremely complex calculations to search for optimal solutions with a dimension as high as 108 . The idea of NWP was formulated as early as 1904, long before the invention of the modern computers that are needed to complete the vast number of calculations in the problem. In the 1920s, Lewis Fry Richardson used procedures originally developed by Vilhelm Bjerknes to produce by hand a six-hour forecast for the state of the atmosphere over two points in central Europe, taking at least six weeks to do so. In the late 1940s, a team of meteorologists and mathematicians led by John von Neumann and Jules Charney made significant progress toward more practical numerical weather forecasts. By the mid-1950s, numerical forecasts were being made on a regular basis [1, 2]. Several areas of mathematics play a fundamental role in NWP. We mention mathematical modeling, the design of numerical algorithms, computational nonlinear optimization in very high-dimensional spaces, manipulation of huge datasets, and parallel computation. Even after decades of active research and increasing power of supercomputers, the forecast skill of numerical weather models extends to about only six days. Improving current models and developing new models for NWP have always been active areas of research. Operational weather and climate models are based on the Navier–Stokes equations coupled with various interactive earth components such as ocean, land terrain, and water cycles. Many models use a latitude-longitude spherical grid. Its logically rectangular structure, orthogonality, and symmetry properties make it relatively straightforward to obtain various desirable, accuracy-related properties. On the other hand, the rapid development in the technology of massively parallel computation platforms constantly renews the impetus to investigate better mathematical models using traditional or alternative spherical grids. Interested readers are referred to a recent survey paper [3]. Initial conditions must be generated before one can compute a solution for weather prediction. The process of entering observational data into the model to generate initial conditions is called data assimilation. Its goal is to find an estimate of the true state of the weather based on observations (sensor data, etc.) and prior knowledge (mathematical 37

38

Chapter 3. Weather and Climate

Figure 3.1. Weather models use systems of differential equations based on the laws of physics, fluid motion, and chemistry, and use a coordinate system which divides the planet into a 3D grid. Winds, heat transfer, solar radiation, relative humidity, and surface hydrology are calculated within each grid cell, and the interactions with neighboring cells are used to calculate atmospheric properties in the future. Reprinted courtesy of NOAA.

models, system uncertainties, sensor noise, etc.). A family of variational methods called 4D-Var is widely used in NWP for data assimilation. In this approach, a cost function based on initial and sensor error covariances is minimized to find the solution to a numerical forecast model that best fits a series of datasets from observations distributed in space over a finite time interval. Another family of data assimilation methods is ensemble Kalman filters. These are reduced-rank Kalman filters based on sample error covariance matrices, an approach that avoids the integration of a full-size covariance matrix, which is impossible even for today’s most powerful supercomputers. In contrast to interpolation methods used in the early days, 4D-Var and ensemble Kalman filters are iterative methods that can be applied to much larger problems. Yet the effort of solving problems of even larger size is far from over. Current day-to-day forecasting uses global models with grid resolutions of 16–50 km and about 2–20 km for short-term local forecasting. Developing efficient and accurate algorithms of data assimilation for high-resolution models is a longterm challenge that will confront mathematicians and meteorologists for many years to come.

3.2. Lorenz’s Discovery of Chaos

39

References [1] http://earthobservatory.nasa.gov [2] http://wikipedia.org [3] A. Staniforth and J. Thuburn, Horizontal grids for global weather and climate prediction models: A review, Quart. J. R. Meteor. Soc. 138 (2011) 1–26, doi: 10.1002/qj.958

3.2 Lorenz’s Discovery of Chaos Chris Danforth, University of Vermont This month marks the 50th anniversary of the 1963 publication of Ed Lorenz’s groundbreaking paper Deterministic nonperiodic flow in the Journal of Atmospheric Science [1]. This seminal work, now cited more than 11,000 times, inspired a generation of mathematicians and physicists to bravely relax their linear assumptions about reality and embrace the nonlinearity governing our complex world. Quoting from the abstract of his paper: “A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.” While many scientists had observed and characterized nonlinear behavior before, Lorenz was the first to simulate this remarkable phenomenon in a simple set of differential equations using a computer. He went on to demonstrate that the limit of predictability of the atmosphere is roughly two weeks, the time it takes for two virtually indistinguishable weather patterns to become completely different. No matter how accurate our satellite measurements get, no matter how fast our computers become, we will never be able to predict the likelihood of rain beyond 14 days. This phenomenon became known as the butterfly effect, popularized in James Gleick’s book Chaos [2]. Inspired by the work of Lorenz and colleagues, we are using computational fluid dynamics (CFD) simulations in my lab at the University of Vermont to understand the flow behaviors observed in a physical experiment. It’s a testbed for developing mathematical techniques to improve the predictions made by weather and climate models. A brief YouTube video describing the experiment analogous to the model developed by Lorenz can be found on the Web [3] and a CFD simulation of the dynamics observed in the experiment at [4].

Figure 3.2. Sketch of the Lorenz attractor from the original paper [1] (left, reprinted with permission from the American Meteorological Society) and a simulation of the convection loop analogous to Lorenz’s system [5] (right, reprinted with permission from the International Meteorological Institute, Stockholm).

40

Chapter 3. Weather and Climate

What is most remarkable about Lorenz’s 1963 model is its relevance to the state of the art in weather prediction today, despite the enormous advances that have been made in theoretical, observational, and computational studies of the Earth’s atmosphere. Every PhD student working in the field of weather prediction cuts his or her teeth testing data assimilation schemes on simple models proposed by Lorenz. His influence is incalculable. In 2005, while I was a PhD student in Applied Mathematics at the University of Maryland, the legendary Lorenz visited my advisor Eugenia Kalnay in her office in the Department of Atmospheric & Oceanic Science. At some point during his stay, he penned the following on a piece of paper: “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.” Even near the end of his career, Lorenz was still searching for the essence of nonlinearity, seeking to describe this incredibly complicated phenomenon in the simplest of terms. References [1] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130–141. [2] J. Gleick, Chaos: Making a New Science, Vintage, 1987. [3] Chaos in an Atmosphere Hanging on a Wall, recreationaltruth’s channel, YouTube, http://www.youtube.com/watch?v=Vbni-7veJ-c [4] Chaos in an Experimental Toy Climate, recreationaltruth’s channel, YouTube, http://www.youtube.com/watch?v=ofzPGPjPv6g [5] K.D. Harris, E.H. Ridouane, D.L. Hitt, and C.M. Danforth, Predicting flow reversals in chaotic natural convection using data assimilation, Tellus A 64 (2012) 17598, http://dx.doi.org/10.3402/tellusa.v64i0.17598

3.3 Predicting the Atmosphere Robert Miller, Oregon State University Recently, I heard a lecture by Rick Anthes, president emeritus of UCAR, former director of NCAR. His talk was entitled “Butterflies and Demons,” and the subject was predictability of weather and climate. He was a witness to and participant in the development of numerical weather prediction in the form it exists today at weather centers worldwide. It was a particularly interesting and provocative talk. Numerical weather prediction proved its worth in the forecasts of the track and severity of Hurricane Sandy. Without the forecasts, the property damage and loss of life would have been much worse than they were. One might compare the effect of Sandy to the Galveston flood of 1900, for which there was no warning and where thousands of people lost their lives. Sandy was the only hurricane in history that made landfall on the Atlantic coast from the east. Dr. Anthes showed a slide with the tracks of every Atlantic coast hurricane since 1850. Most tracked up the coast, and those that went east into the Atlantic did not return. One might reasonably question the reliability of data extending back to the age of sail, but no statistical method based on previous experience could possibly have predicted that a hurricane would go northeast from the coast and then return westward to make landfall, since it had never happened before. At this point it is important to note that the national weather centers make more than a single numerical forecast. In addition to a main central forecast, they make a collection of forecasts, numbering in the hundreds at some weather centers, each differing slightly in some respect, usually in the initial conditions. They refer to such collections of simultaneous forecasts as “ensembles.” The spread among the ensemble members is expected to reflect uncertainty in the forecast. Dr. Anthes showed the ensemble produced by the

3.4. Arctic Sea Ice and Cold Weather

41

Figure 3.3. Ensemble forecast of Hurricane Sandy. Reprinted with permission from ECMWF.

European Center for Medium-Range Weather Forecasting (ECMWF) of predicted tracks for Sandy. Nearly all of them exhibited the correct behavior. Perhaps five of the several hundred tracks predicted by the ensemble members led out into the Atlantic and did not return. Dr. Anthes said that accurate forecasts such as the ones issued by ECMWF for the track of Sandy would have been impossible 20 years ago. He emphasized the fact that advances in science, in the form of improved numerical techniques, data assimilation, and understanding of rain and clouds, along with spacecraft as well as earthbound instruments and data processing techniques, may well have saved thousands of lives and billions of dollars in property damage. It was certainly good to see benefits to society that come from my corner of the world of scientific research. It’s the received wisdom in the world of hurricane forecasting that predictions of tracks have improved considerably over the years, while improvement in prediction of intensity has been much slower. Dr. Anthes’s graph showing improvement of skill in the forecasting of hurricane tracks since 1980 didn’t strike me as being quite so impressive as other aspects of weather forecasting. If I read the graph correctly, the accuracy of present two-day storm track forecasts is about equivalent to the accuracy of one-day storm track predictions in 1980. By contrast, the graph shown by Dr. Anthes of global weather forecast skill showed that our 5-day forecasts today are as accurate as our 2-day forecasts were in 1995.

3.4 Arctic Sea Ice and Cold Weather Hans Kaper, Georgetown University Could the cold weather experienced this winter in the northern part of the Eurasian continent be related to the decrease in Arctic sea ice? This question is the subject of

42

Chapter 3. Weather and Climate

much debate in the media in Europe. This section shows some relevant weather maps and provides links to several relevant blogs and articles. Temperature distribution. First, what does the unusual temperature distribution observed this March actually look like? Figure 3.4 shows the data: freezing cold in Siberia, reaching across northwestern Europe, unusually mild temperatures over the Labrador Sea and parts of Greenland, and a cold band diagonally across North America, from Alaska to Florida. Averaged over the northern hemisphere, the anomaly disappears—the average is close to the long-term average. Of course, the distribution of hot and cold is related to atmospheric circulation, and thus the air pressure distribution. Figure 3.5 shows that there was unusually high air pressure between Scandinavia and Greenland. Since circulation around a high is clockwise (anticyclonic), this explains the influx of Arctic cold air in Europe and the warm Labrador Sea.

Figure 3.4. Mean temperature for March 2013, up to and including March 25. Reprinted courtesy of NOAA.

Arctic sea ice. Let us now discuss the Arctic sea ice. The summer minimum in September set a new record low, but also at the recent winter maximum there was unusually little ice (ranking 6th lowest—the ten years with the lowest ice extent were all in the last decade). The ice cover in the Barents sea was particularly low this winter. All in all, until March the deficit was about the size of Germany compared to the long-term average. Is there a connection with the winter weather? Does the shrinking ice cover influence the atmospheric circulation, because the open ocean strongly heats the Arctic atmosphere from below? (The water is much warmer than the overlying cold polar air.) Did the resulting evaporation of sea water moisten the air and thus lead to more snow? These are good questions that suggest new research directions.

3.4. Arctic Sea Ice and Cold Weather

43

Figure 3.5. Mean surface-level pressure, March 2013. Reprinted courtesy of NOAA.

References Here are some blogs where this problem is discussed: [1] Looking for winter weirdness 6, Arctic Sea Ice Blog: Interesting News and Data, March 30, 2013, http://neven1.typepad.com/blog/2013/03/looking-for-winter-weirdness-6.html [2] E. Rabett, Melting Ice and Cold Weather, Rabett Run, March 29, 2013, http://rabett.blogspot.de/2013/03/melting-ice-and-cold-weather.html [3] S. Rahmstorf, Eisschmelze und kaltes Wetter + Updates, SciLogs, March 28, 2013, http://www.scilogs.de/wblogs/blog/klimalounge/klimadaten/2013-03-28/eisschmelze-und-kalteswetter

Here are three references taken from Rabett Blog: [4] R. Jaiser, K. Dethloff, D. Handorf, A. Rinke, and J. Cohen, Impact of sea ice cover changes on the Northern Hemisphere winter atmospheric circulation, Tellus Ser. A Dyn. Meteor. Ocean. 64 (2012), http://www.tellusa.net/index.php/tellusa/article/view/11595/html [5] J.P. Liu, J.A. Curry, H.J. Wang, M.R. Song, and R.M. Horton, Impact of declining Arctic sea ice on winter snowfall. Proc. Natl. Acad. Sci. USA 109 (2012) 4074–4079, http://www.pnas.org/content/109/11/4074 [6] V. Petoukhov and V.A. Semenov, A link between reduced Barents-Kara sea ice and cold winter extremes over northern continents, J. Geophys. Res. Atmos. 115 (2010) D21111, http://onlinelibrary.wiley.com/doi/10.1029/2009JD013568/abstract

44

Chapter 3. Weather and Climate

3.5 Extreme Weather Event William J. Martin, Worcester Polytechnic Institute Tuesday, April 9, 2013: High 65o F—felt like 72 or so—and winds at 25 mph, gusting to 33 mph. Record high for Worcester on April 9 is 77o F. It was unusually warm and windy for early April. We piled into the toasty lecture hall with drinks and sandwich wraps in hand. Dr. Smith, with his shock of white hair and the thin frame of a marathon runner, shed his sport jacket as he recounted the 2003 European heat wave, which some claim caused up to 70,000 deaths; the 2010 Russian heatwave; the floods in Pakistan that same year; and the devastation of Hurricane Sandy last year. Trained as a probabilist and currently the Director of SAMSI and a professor of Statistics at UNC Chapel Hill, Richard Smith guided the audience through the challenges of doing reliable science in the study of climate change. Rather than address the popular question of whether recent climate anomalies are out of the statistical norm of recent millennia (and other research strongly suggests they are), Dr. Smith asked how much of the damage is attributable to human behaviors such as the emission of greenhouse gases from the burning of fossil fuels. Demonstrating a deep familiarity with the global debate on climate change and the reports of the IPCC (Intergovernmental Panel on Climate Change), Professor Smith discussed the statistical parameter “fraction of attributable risk” (FAR), which is designed to compare the likelihood of some extreme weather event (such as a repeat of the European heat wave) under a model that includes anthropogenic effects versus the same value ignoring human factors. Employing the rather flexible generalized extreme value (GEV) distribution and Bayesian hierarchical modeling, Dr. Smith walked the audience through an analysis of the sample events mentioned above, giving statistically sound estimates of how likely an event is to occur given anthropogenic effects versus without them. Smith explained how a strong training in statistics guides one to the choice of the GEV distribution as a natural model for such events; this distribution involves a parameter ξ which allows us to accurately capture the length of the tail of our observed distribution. Perhaps the most compelling graphs were plots of the estimated changes in predicted extreme weather events over time. One such plot, replicated in Figure 3.6, shows the probability of a repeat of an event in Europe similar to the 2003 heat wave, with posterior median and quartiles marked in bold and a substantial confidence interval shaded on the plot. One intriguing aspect of this cleverly designed talk was a digression about computational climate models. The NCAR Supercomputing Center in Cheyenne, Wyoming, houses the notorious “Yellowstone” with its 1.5 petaflop capabilities and 144.6 terabyte storage farm, which will cut down the time for climate calculations and provide much more detailed models (reducing the basic spacial unit from 60 square miles down to a mere 7 square miles). Dr. Smith explained the challenge of obtaining and leveraging big datasets and amassing as many reliable runs of such climate simulations as possible to improve the reliability of the corresponding risk estimates. The audience encountered a broad range of tools and issues that come into play in the science of climate modeling, and we all had a lot to chew on as a result of this talk. A lively question and answer period ensued with questions about methodology, policy, volcanoes versus vehicles, and where to go from here to make a difference. Then we all poured out into the heat of an extremely warm April afternoon, pondering whether this odd heat and wind were normal for a spring day in Massachusetts.

3.6. Wimpy Hurricane Season

45

Figure 3.6. Temperature anomaly probability vs. year, courtesy of William Sanguinet. Based on a diagram in Dr. Smith’s lecture [1].

Reference [1] Influence of climate change on extreme weather events, 2013 Math Awareness Month talk, Worcester Polytechnic Institute, April 9, 2013.

3.6 Wimpy Hurricane Season Brian McNoldy, University of Miami It was a hurricane season almost without hurricanes. There were just two, Humberto and Ingrid, and both were relatively wimpy, Category 1 storms. That made the 2013 Atlantic hurricane season the least active in more than 30 years—for reasons that remain puzzling. The season, from June through November, has an average of 12 tropical storms, of which six to seven grow to hurricane strength with sustained winds of 74 mph or greater. Typically, two storms become “major” hurricanes, Category 3 or stronger, with sustained winds of at least 111 mph. In 2013, there were 13 tropical storms, a typical number, but for the first time since 1994 there were no major tempests in the Atlantic. The last time there were only two hurricanes was 1982. The quiet year is an outlier, however, in the recent history of Atlantic cyclones. The National Oceanic and Atmospheric Administration (NOAA) notes that 2013 was only the third calmer-than-average year since 1995. The most intense storms this year had maximum sustained winds of only about 86 mph, [. . .] the weakest maximum intensity for a hurricane during a season since 1968. The first hurricane, Humberto, was just hours from matching the record for the latest first hurricane, Sept. 11. In terms of accumulated cyclone energy (ACE), the seasonal total stands at 31.1, the lowest since 1983 and just 30 percent of average. (ACE is the sum of the squares of all of the storms’ peak wind speeds at six-hour intervals, and a good measure of a storm season’s

46

Chapter 3. Weather and Climate

overall power.) Looking back to 1950, only four other years had lower ACE totals: 1972, 1977, 1982, and 1983. [. . . ] Why was this season so inactive? What did the forecasts miss? Although there are some hypotheses, it is not entirely clear. We may have to wait another couple of months, but in the meantime, there are some potential explanations. Major signals such as the El Niño–Southern Oscillation (ENSO), surface pressure, and sea-surface temperature all pointed to an average to above-average season. But there were some possible suppressing factors. • Dry air. Even over the long three-month window of August to October, the vast majority of the tropical Atlantic was dominated by drier-than-normal air, especially in the deep tropics off the coast of Africa. Dry air can quickly weaken or dissipate a tropical cyclone, or inhibit its formation. • Stable air. The average temperature profile in the region was less conducive to thunderstorm growth and development during the core months, which means that the amount of rising air in the region may have been reduced as well. • Weak African jet stream. Tropical waves, the embryos of many tropical cyclones, have their origins over continental Africa. A persistent feature called the African easterly jet stream—a fast-moving river of air in the low and middle levels of the atmosphere—extends from Ethiopia westward into the tropical Atlantic Ocean. It breaks down into discrete waves, and every few days another wave leaves the coast. Some are barely noticeable, while others become tropical storms. During the height of the hurricane season, most tropical cyclones form from disturbances off the coast of Africa. Winds in the jet stream normally cruise along at 20 to 25 mph at an altitude of 10,000 feet from August to October, but this year they were about 12 to 17 mph weaker. One would expect that to have a big impact on the amplitude of easterly waves and the hurricane season. Links to global warming? One question that inevitably is asked is how the season’s inactivity relates to climate change. It’s not possible to associate any particular season (and certainly not a specific storm) with climate change. One season’s activity does not allow any conclusions about the role of climate change. The reason is that intra- and interseasonal variability is so large that any subtle signals of influence from climate change are overwhelmed. This article was originally published on December 1, 2013 in The Washington Post.

3.7 Extreme Events Hans Kaper, Georgetown University Weather extremes capture the public’s attention and are often used as arguments in the debate about climate change. In daily life, the term extreme event can refer, for example, to an event whose intensity exceeds expectations, or an event with high impact, or an event that is rare or even unprecedented in the historical record. Some of these notions may hit the mark, but they need to be quantified if we want to make them useful for a rational discussion of climate change. The concern that extreme events may be changing in frequency and intensity as a result of human influences on climate is real, but the notion of extreme events depends to a large degree on the system under consideration, including its vulnerability, resilience, and capacity for mitigation and adaptation.

3.7. Extreme Events

47

Since extreme events play such an important role in the current discussion of climate change, it is important that we get their statistics right. The assessment of extremes must be based on long-term observational records and a statistical model of the particular weather or climate element under consideration. The proper framework for their study is probability theory—an important topic of mathematics. The recent special report on managing the risks of extreme events (SREX) prepared by the IPCC describes an extreme event as the “occurrence of a value of a weather or climate variable above (or below) a threshold value near the upper (or lower) ends of the range of observed values of the variable” [1]. Normally, the threshold is put at the 10th or 90th percentile of the observed probability distribution function, but other choices of the thresholds may be appropriate given the particular circumstances. When discussing extreme events, it is generally better to use anomalies, rather than absolutes. Anomalies more accurately describe climate variability and give a frame of reference that allows more meaningful comparisons between locations and more accurate calculations of trends. Consider, for example, the global average temperature of the Earth’s surface. We have a fairly reliable record going back to 1880, but rather than looking at the temperature itself, we use the temperature anomaly to study extremes. Figure 3.7 shows the global temperature anomaly since 1950 relative to the average of these mean temperatures for the period 1961–1990. We see that the year 2010 tied with 2005 as the warmest year on record. The year 2011 was somewhat cooler, largely because it was a La Niña year; however, it was the warmest La Niña year in recent history.

Figure 3.7. Global temperature anomaly from 1950 to 2011 relative to the base period 1961– 1990. Reprinted with permission from the World Meteorological Organization.

While the global mean temperature for the year 2010 does not appear much different from previous years, exceeding the 1961–1990 average by only about 0.5◦ C, the average June temperature exceeded the corresponding 1971–2000 average by up to 5◦ C in certain regions, and the same year brought heat waves in North America, western Europe, and Russia. Observed temperature anomalies were, in fact, much higher during certain

48

Chapter 3. Weather and Climate

months in certain regions, and it is likely that these extremes were even more pronounced at individual weather stations. Since localized extremes cause disruptions of the socioeconomic order, from crop failures to forest fires and excess deaths, they are of considerable interest to the public. To assess the likelihood of their occurrence, we need both access to data and rigorous statistical analysis. Until recently, reliable data have been scarce for many parts of the globe, but they are becoming more widely available, allowing research of extreme events on both global and regional scales. The emphasis shifts thereby from understanding climate models to assessing the likelihood of possibly catastrophic events and predicting their consequences. For what kind of extreme events do we have to prepare? How often do these extremes occur in a stationary climate? What magnitudes can they have? And how does climate change figure in all this? Adapted from the book Mathematics & Climate by H. Kaper and H. Engler, OT131, SIAM, Philadelphia, 2013. Reference [1] Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation (SREX), IPCC, Working Groups I and II, http://www.ipcc-wg2.gov/SREX

3.8 The Need for a Theory of Climate Antonello Provenzale, Consiglio Nazionale delle Richerche At the end of August 2013, Nature Climate Change published an interesting paper showing that current general circulation models (GCMs) tend to significantly overestimate the warming observed in the last two decades [1]. A few months earlier, Science had published a paper showing that four top-level global climate models, when run on a planet with no continents and entirely covered with water (an “aqua-planet”), produce cloud and precipitation patterns that are dramatically different from one model to another [2]. At the same time, most models tend to underestimate summer melting of Arctic sea ice [3] and display significant discrepancies in the reproduction of precipitation and its trends in the area affected by the Indian summer monsoon. To close the circle, precipitation data in this area also show important differences from one dataset to another, especially when solid precipitation (snow) plays a dominant role [4]. Should we use these results to conclude that climate projections cannot be trusted and that all global warming claims should be revised? Not at all. However, it would be equally wrong to ignore these findings, assume that what we know today is enough, and not invest in further research activities. Climate is a complex dynamical system, and we should be aware of the difficulties in properly understanding and predicting it. Global climate models are the most important, perhaps the only type of instrument that the scientific community has at its disposal to estimate the evolution of future climates, and they incorporate the results of several decades of passionate scientific inquiries. However, no model is perfect, and it would be a capital mistake to be content with the current state of description and think that everything will be solved if only we use bigger, more powerful, and faster computers, or organize climate science more along the lines of a big corporation. Like all sciences, the study of climate requires observations and data. Today, enormous quantities of high-resolution, precise, and reliable data about our planet are available, provided by satellites and by a dense network of ground stations. The observational datasets are now so large that we have to cope with the serious problems of storing and

3.8. The Need for a Theory of Climate

49

efficiently accessing the information provided by the many measurement systems active on Earth and making these data available to end-users such as scientists and decisionmakers. On the other hand, data must be analyzed and interpreted. They should provide the basis for conceptual understanding and for the development of theories. Here, a few problems appear: some parts of the climate system can be described by laws based on “first principles” (the dynamics of the atmosphere and the oceans, or radiative processes in the atmosphere), while others are described by semi-empirical laws. We do not know the equations of a forest, but we do need to include vegetation in our description of climate, as forests are a crucial player in the system. In addition, even for the more “mechanical” components we cannot describe all climatic processes at once: it is not feasible to describe, at the same time, the motion of an entire ocean basin or of the planetary atmosphere and take into account the little turbulent swirls at the scale of a few centimeters.

Figure 3.8. Feedback mechanisms in the Earth’s climate system.

Climate still has many aspects which are poorly understood, including the role of cross-scale interactions, the dynamics of clouds and of convection, the direct and indirect effects of aerosols, the role of the biosphere, and many aspects of ocean-atmosphere exchanges. To address these issues, the whole hierarchy of modeling tools is necessary, and new ideas and interpretations must be developed. The hydrological cycle, for example, is one of the most important components of the climate of our planet and has a crucial impact on our own life. Still, precipitation intensity and variability are poorly reproduced by climate models, and a huge effort in further investigating such themes is required. Basic research on these topics should continue to provide better descriptions and ultimately better models for coping with societal demands. Scientific activities in this field should certainly be coordinated and harmonized by large international programs, but scientific progress will ultimately come from the passion and ingenuity of the individual researchers. For all these reasons, parallel to model development and scenario runs we need to focus also on the study of the “fundamentals of climate,” analyzing available data, performing

50

Chapter 3. Weather and Climate

new measurements, and using big models and conceptual models, to explore the many fascinating and crucial processes of the climate of our planet which are still not fully understood. While continuing the necessary efforts on data collection, storage, and analysis, and the development of more sophisticated modeling tools, we also need to come up with a theory of climate. In such a construction, the role of climate dynamicists, physicists, chemists, meteorologists, oceanographers, geologists, hydrologists, and biologists is crucial, but so is the role of mathematicians. A theory of climate is needed to put together the different pieces of the climatic puzzle, addressing the most important open questions, and developing the proper mathematical descriptions, in a worldwide initiative to understand (and, eventually, predict) one of the most fascinating and important manifestations of Planet Earth. References [1] J.C. Fyfe, N.P. Gillett, and F.W. Zwiers, Overestimated global warming over the past 20 years, Nature Climate Change 3 (2013) 767–769. [2] B. Stevens and S. Bony, What are climate models missing? Science 340 (2013) 1053– 1054. [3] P. Rampal et al., IPCC climate models do not capture Arctic sea ice drift acceleration: Consequences in terms of projected sea ice thinning and decline, J. Geophy. Res. Oceans 116 (2011), doi: 10.1029/2011JC007110. [4] E. Palazzi, J. von Hardenberg, and A. Provenzale, Precipitation in the Hindu-Kush Karakoram Himalaya: Observations and future scenarios, J. Geophys. Res. Atmos. 118 (2013), 85–100, doi: 10.1029/2012JD018697.

3.9 Mathematics and Climate Hans Kaper, Georgetown University What is the role of mathematics in climate science? Climate science, like meteorology, is largely a branch of physics; as such, it certainly uses the language of mathematics. But could mathematics provide more than the language for scientific discourse? As mathematicians, we are used to setting up models for physical phenomena, usually in the form of equations. For example, we recognize the second-order differential equation L¨ x (t ) + g sin x(t ) = 0 as a model for the motion of a physical pendulum under the influence of gravity. Every symbol in the equation has its counterpart in the physical world. The quantity x(t ) stands for the angle between the arm of the pendulum and its rest position (straight down) at the time t , the constant L is the length of the pendulum arm, and g is gravitational acceleration. The mass of the bob turns out to be unimportant and therefore does not appear in the equation. The model is understood by all to be an approximation, and part of the modeling effort consists in outlining the assumptions that went into its formulation. For example, it is assumed that there is no friction in the pendulum joint, there is no air resistance, the arm of the pendulum is massless, and the pendulum bob is idealized to be a single point. Understanding these assumptions and the resulting limitations of the model is an essential part of the modeling effort. Note that the modeling assumptions can all be assessed by an expert who is not a mathematician: a clockmaker can estimate the effect of friction in the joint, the difficulty of making a slender pendulum arm, and the effort in making a bob that offers little air resistance. As mathematicians, we take the differential equation and apply the tools of the trade to extract information about the behavior of

3.9. Mathematics and Climate

51

the physical pendulum. For example, we can find its period—which is important in the design of pendulum clocks—in terms of measurable quantities. Would it be possible to develop a “mathematical model” of the Earth’s climate system in a similar fashion? Such a model should stay close to physical reality, climate scientists should be able to assess the assumptions, and mathematicians and computational scientists should be able to extract information from it.

Figure 3.9. Earth’s climate system. Reprinted with permission from IPCC [1].

Figure 3.9 gives a climate scientist’s view of the Earth’s climate system: a system with many components that interact with one another either directly or indirectly, and with many built-in feedback loops, both positive and negative. To develop a mathematical model of such a complex system, we would need to select variables that describe the state of the system (air temperature, humidity, fractions of aerosols and trace gases in the atmosphere, strength of ocean currents, rate of evaporation from vegetation cover, change in land use due to natural cycles and human activity, and many many more), take the rules that govern their evolution (laws of motion for gases and fluids, chemical reaction laws, land use and vegetation patterns, and many many more), and translate all this into the language of mathematics. It is not at all clear that this can be done equally well for all components of the system. The laws for airflow over a mountain range may be well known, but it is much harder to predict crop use and changes in vegetation. The ranges and limitations of any such model would remain subject to debate, much more so than in the case of the pendulum equation, and the resulting equations would likely cover several pages and would be far too unwieldy for a mathematical analysis. This would leave a computational approach as the only viable option. But even here we would face limitations, given the available computational resources and the scarcity of data.

52

Chapter 3. Weather and Climate

Surprisingly often, mathematics can offer perspectives that complement or provide insight into the results of observations and large-scale computational experiments. Through inspired model reduction and sometimes just clever guessing, it is often possible to come up with relativity simple models for components of the climate system that still retain some essential features observed in the physical world, that reproduce complex phenomena quite faithfully, and that lead to additional questions.

Figure 3.10. Conceptual climate model (energy balance) [2].

Figure 3.10 illustrates a simple energy balance model for the entire planet. It posits that the solar energy reaching the Earth must balance the energy that the Earth radiates back into outer space; otherwise, the planet will heat up or cool down. The model focuses on the global mean surface temperature and reproduces the current state of the climate system remarkably well with just a few physical parameters (solar output, reflectivity of the Earth’s surface, greenhouse effect). The model also shows that the Earth’s climate system can have multiple stable equilibrium states. One of these states is the “Snowball Earth” state, where the entire planet is covered with snow and ice and temperatures are well below freezing everywhere. Why is the planet at today’s climate when much colder climates are also possible? Has the planet ever been in one of the much colder climate states in the past? (The answer is yes.) Is there any danger that Earth could again revert to a much colder climate in the future? How would this happen? Mathematics can raise these questions from a very simple climate model and also support or rule out certain answers using an analysis of the model. Adapted from the book Mathematics & Climate by H. Kaper and H. Engler, OT131, SIAM, Philadelphia, 2013. References [1] S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds., IPPC, 2007: Climate Change 2007: The Physical Science Basis, Contributions of the Working Group I to the Fourth Assessment Report of the Intergovernment Panel on Climate Change, FAQ 1.2, Figure 1, Cambridge University Press. [2] H. Kaper and H. Engler, Mathematics & Climate, OT131, SIAM, Philadelphia, 2013.

3.10. Climate Science without Climate Models

53

3.10 Climate Science without Climate Models Hans Kaper, Georgetown University In June 2012, more than 3,000 daily maximum temperature records were broken or tied in the United States, according to the National Climatic Data Center (NCDC) of the U.S. National Oceanic and Atmospheric Administration (NOAA). Meteorologists commented at that time that this number was very unusual. By comparison, in June 2013, only about 1,200 such records were broken or tied. Was that number “normal”? Was it perhaps lower than expected? Was June 2012 (especially the last week of that month) perhaps just an especially warm time period, something that should be expected to happen every now and then? Also in June 2013, about 200 daily minimum temperature records were broken or tied in the United States. Shouldn’t that number be comparable to the number of record daily highs if everything was “normal”? Surprisingly, it is possible to make fairly precise mathematical statements about such temperature extremes (or for that matter, about many other record-setting events) simply by reasoning, almost without any models. Well, not quite. The mathematical framework is that individual numerical observations are random variables. One then has to make a few assumptions. The two main assumptions are that (1) the circumstances under which observations are made do not change, and (2) observations are stochastically independent—that is, knowledge of some observations does not convey any information about any of the other observations. Let’s work with these assumptions for the moment and see what can be said about records. Suppose N numerical observations of a certain phenomenon have already been made and a new observation is added. What is the probability that this new observation exceeds all the previous ones? Think about it this way: Each of these N + 1 observations has a rank, 1 for the largest value, and N + 1 for the smallest value. (For the time being, let’s assume that there are no ties.) Thus any particular sequence of N + 1 observations defines a sequence of ranks—that is, a permutation of the numbers from 1 to N + 1. Since observations are independent and have the same probability distribution (that’s what the two assumptions from above imply), all possible (N+1)! permutations are equally likely. A new record is observed during the last observation if its rank equals 1. There are N ! permutations that have this additional property. Therefore, the probability that the last observation is a new record is N !/(N + 1)! = 1/(N + 1). This reasoning makes it possible to compute the expected number of record daily high temperatures for a given set of weather stations. For example, there are currently about 5,700 weather stations in the United States at which daily high temperatures are observed. In 1963, there were about 3,000 such stations and in 1923 only about 220. Assuming for simplicity that each of the current stations has been recording daily temperatures for 50 years, one would expect that on a typical day about 2% of all daily high records are broken, resulting in about 3,000 new daily high records per month on average—provided the circumstances of temperature measurements remain the same and the observations at any particular station are independent of each other. It is fairly clear that temperature records for the same station but for the same date (day and month) are indeed independent of one another: Knowing the maximum temperature at a particular location on August 27, 2013, does not give any information about the maximum temperature on the same date a year later. However, circumstances of observations could indeed change for many different reasons. What if new equipment is used to record temperatures? What if the location of the station is changed? For example, until 1945, daily temperatures in Washington, DC, were recorded at a downtown location (24th and M St.). Since then, measurements have been made at National Airport. National Airport is adjacent to a river, which lowered

54

Chapter 3. Weather and Climate

daily temperatures measurements compared to downtown. The area around the airport, however, has become more urban over the last decades, possibly leading to higher temperature readings (the well-known urban heat island effect). And what about climate change? Perhaps it is better to use a single climate record and not thousands. Consider, for example, the global mean temperature record that is shown in Figure 3.7. It shows that the largest global mean temperature for the 50 years from 1950 to 1999 (recorded in 1998) was exceeded twice in the 11 years from 2000 to 2010. The second-highest global mean temperature for these 50 years (that of 1997) was exceeded in 10 out of 11 years between 2000 to 2010. Can this be a coincidence? There is a mathematical theory to study such questions. Given a reference value equal to the mth largest out of N observations, any observation out of n additional ones that exceeds this reference value is called an exceedance. For example, we might be interested in the probability of observing two exceedances of the largest value out of 50 during 12 additional observations. A combinatorial argument implies that the probability of seeing k exceedances of the mth largest observation out of N when n additional observations are made equals C (N + k − m, N − m)C (m + n − k − 1, m − 1) C (N + n, N )

,

(3.1)

where C (r, s) is the binomial coefficient, C (r, s) = r !/(s!(r −s)!). The crucial assumption is again that observations are independent and come from the same probability distribution. Applied to the global mean temperature record, the formula (3.1) implies that the probability of two or more exceedances of a 50-year record during an 11 year period is no more than 3%. The probability of 10 exceedances of the second-highest observation from 50 years during an 11-year period is tiny—of the order of 0.0000001%. Yet these exceedances were actually observed during the last decade. Clearly, at least one of the assumptions of stochastic independence and of identical distribution must be violated. The plot of August 20 already shows that distributions may vary from year to year, due to El Niño/La Niña effects. La Niña years in particular tend to be cooler when averaged over the entire planet. The assumption of stochastic independence is also questionable, since global weather patterns can persist over months and therefore influence more than one year. Could it be that more exceedances than plausible were observed because global mean temperatures became generally more variable during the past decades? In that case, low exceedances of the minimum temperature would also have been observed more often than predicted by the formula. That’s clearly not the case, so that particular effect is unlikely to be solely responsible for what has been observed. We see that even this fairly simple climate record leads to serious questions and even partial answers about possible climate change, without any particular climate model. Adapted from the book Mathematics & Climate by H. Kaper and H. Engler, OT131, SIAM, Philadelphia, 2013.

3.11 Supermodeling Climate James Crowley, SIAM Considering the mathematics of planet Earth, one tends to think first of direct applications of mathematics to areas like climate modeling. But MPE is a diverse subject, with respect to both applications and the mathematics itself. This was driven home to me at

3.12. Reconstructing Past Climates

55

a recent SIAM Conference on Dynamical Systems in Snowbird, Utah, when I attended a session on “Supermodeling Climate.” The application is simple enough to describe. There are about twenty global climate models, each differing slightly from the others in their handling of the subgrid physics. Typical codes discussed in the session have grid points spaced about 100 kilometers apart in the horizontal directions and about 40 vertical layers in the atmosphere. While the codes reach some general consensus on overall trends, they can differ in the specific values produced. The question is whether the models or codes could be combined in a way that would produce a more accurate result. Perhaps one approach for thinking about this is to consider something familiar to anyone who watches summer weather forecasts in the U.S., namely hurricane predictions. Weather forecasters often show a half-dozen different projected tracks for a hurricane, each based on a different model or computer code. Simply averaging the spatial locations at each time step would make little sense. While climate models or computer codes are completely different from weather models used to predict hurricanes, the problems are similar. It isn’t sufficient to average the results of the computer runs. But perhaps an intelligent way could be found to combine the computer codes so that they would produce a more accurate result with a smaller band of uncertainty. In the field of dynamical systems it is known that chaotic systems can synchronize. The session that I attended considered whether researchers could couple models by taking a “synchronization view” of data assimilation. This involves dynamically adjusting coupling coefficients between models. While the goal is to attempt this for climate models, work to date has focused on lower-order models (like the Lorenz system). If successful, this could lead to a new way of combining various models to obtain more accurate and reliable predictions. It’s one of many examples of mathematics finding useful applications in areas not originally envisioned. More generally, “interactive ensembles” of different global climate models, using intermodel data assimilation, can produce more accurate results. An active area of research is how to best do this.

3.12 Reconstructing Past Climates Bala Rajaratnam, Stanford University How sensitive is our climate to increased greenhouse gas emissions? What is the relationship between temperature and greenhouse gas concentrations? How can historical temperature measurements inform our understanding of this relationship? To what extent are temperatures during the last few decades anomalous in a millennial context? To answer these and similar questions accurately, we need data—data that are reliable, continuous, and of broad spatial coverage. The problem is that direct physical measurements of climate fields (such as temperature) are few and far between, and their quality and availability decrease sharply as one goes further back in time. Measurements of land and sea surface temperature fields cover only the post-1850 instrumental period, with large regions afflicted by missing data, measurement errors, and changes in observational practices. Temperatures during the pre-instrumental period can only be inferred from temperature-sensitive proxy data—tree rings, ice cores, corals, speleothems (cave formations), and lake sediments [1, 2].

56

Chapter 3. Weather and Climate

The reconstruction of past climates from proxy data is basically a statistical problem. It involves (1) extracting the relationship between temperature and temperature-sensitive proxy data, (2) using this relationship to backcast (or hindcast) past temperatures, and (3) quantifying the uncertainty that is implicit in such paleoclimate reconstructions—that is, making probabilistic assessments of past climates. The problem is compounded by a number of practical and methodological issues. Proxy data are not available in all areas of the globe, and their number decreases sharply back in time, while instrumental temperature data are limited in both space and time. The number of time points available to relate temperature to proxies by regression is small; hence, standard statistical methods such as ordinary least squares regression do not readily apply. Also, proxy and temperature data are correlated, both in space and in time. And finally, the traditional assumption of normality of errors is often unrealistic due to outliers in the data. The resulting nonstandard setting under which paleoclimate reconstructions have to be undertaken leads to a variety of statistical problems, with fundamental questions for mathematics, pure as well as applied. First, given the ill-posed nature of the regression problem, it is not clear which highdimensional regression methodology or type of regularization (like Tikhonov regularization) is applicable. Second, modeling a spatially random field requires specifying probability distributions in order to understand the correlation structure of temperature points and proxies. Even a coarse 5-by-5 latitude/longitude gridded field on the Earth’s surface leads to more than 2,000 spatial points. Specifying covariance matrices of this order requires estimating about 2 million parameters—a nonstarter given the fact that only 150 years of data are available. Hence, sparse covariance modeling is naturally embedded in the statistical paleoclimate reconstruction problem. Estimating covariance matrices in an accurate but sparse way leads to important questions in convex optimization. Regularization methods for inducing sparsity in covariance matrices lead to characterizing maps which leave the cone invariant. Such questions have actually been considered in a more classical setting by Rudin and Schoenberg. However, their results are not directly applicable to the paleoclimate reconstruction problem and require further generalizations and extensions [3, 4, 5, 6]. These examples show that the mathematical sciences are absolutely critical for improving our understanding of past climates. References [1] D. Guillot, B. Rajaratnam, and J. Emile-Geay, Statistical paleoclimate reconstructions via Markov random fields, Ann. Appl. Stat. 2014 (to be published), http://arxiv.org/abs/1309.6702 [2] L.B. Janson and B. Rajaratnam, A methodology for robust multiproxy paleoclimate reconstructions and modeling of temperature conditional quantiles, J. Am. Stat. Assoc. 109 (2014) 63–77, http://arxiv.org/abs/1308.5736 [3] D. Guillot and B. Rajaratnam, Functions preserving positive definiteness for sparse matrices, Trans. Am. Math. Soc. 367 (2015) 627–649, http://arxiv.org/abs/1210.3894 [4] D. Guillot and B. Rajaratnam, Retaining positive definiteness in thresholded matrices, Linear Algebra Appl. 436 (2012) 4143–4160, http://arxiv.org/abs/1108.3325 [5] D. Guillot, A. Khare, and B. Rajaratnam, Preserving positivity for rank-constrained matrices, Technical Report, Department of Mathematics and Statistics, Stanford University, 2014, http://arxiv.org/abs/1406.0042 [6] D. Guillot, A. Khare, and B. Rajaratnam, Preserving positivity for matrices with sparsity constraints, Technical Report, Department of Mathematics and Statistics, Stanford University, 2014, http://arxiv.org/abs/1406.3408

3.13. (Big) Data Science Meets Climate Science

57

3.13 (Big) Data Science Meets Climate Science Jesse Berwald, University of Minnesota Thomas Bellsky, Arizona State University Lewis Mitchell, University of Vermont Internet advertisers and the National Security Agency are not the only ones dealing with the “data deluge” lately. Scientists, too, have access to unprecedented amounts of data, both historical and real-time data from around the world. For instance, using sensors located on ocean buoys and the ocean floor, oceanographers at the National Oceanographic and Atmospheric Administration have modeled tsunamis in real time immediately after the detection of large earthquakes. This technology was used to provide important information shortly after the 2011 Tohoku earthquake off the coast of Japan.

Figure 3.11. The principle of data assimilation.

The process of incorporating data into models is nontrivial and represents a large research area within the fields of weather and climate modeling. While much of this data may be real-time observations, climate scientists also deal with a significant amount of historical data, such as oxygen isotope ratios measured from glacier ice cores. A major question facing climate modelers is how best to incorporate such data into models. As climate models increase in complexity, their results become correspondingly more intricate. Such models represent climate processes spanning multiple spatial and temporal scales and must relate disparate physical phenomena. Data assimilation (DA) is a technique used to combine observations with model forecasts in order to optimize model prediction. DA has many potential useful applications in climate modeling. Combining observational data with models is a mathematical issue which can manifest itself in many areas, including model parameterization, model initialization, and validation of the model prediction against observations. For instance, the continued improvement of weather and storm surge models can be attributed in large part to successful parameterizations and DA, in addition to greater computing power. The Hot Topics Workshop on “Predictability in Earth System Processes” at the Institute for Mathematics and Its Applications (IMA) at the University of Minnesota (November 18–21, 2013) identified challenges to assimilating data in climate processes, while highlighting mathematical tools used to approach these problems in general. Specific goals of the workshop were to identify DA problems in climate modeling and to investigate new mathematical approaches to open problems such as • improving weather forecasts by increasing accuracy to periods of several weeks to months through effective use of DA;

58

Chapter 3. Weather and Climate

• effectively applying uncertainty quantification in predictions from Earth system models arising from model errors and observational errors using DA; and • implementing topological data analysis techniques to provide insight into the state space of the model.

3.14 How Good Is the Milankovitch Theory? Hans Kaper, Georgetown University In 1941, the Serbian mathematician Milutin Milankovitch (1879–1958) suggested that past glacial cycles might be correlated to cyclical changes in the insolation (the amount of solar energy that reaches Earth from the Sun) [1]. This theory is known as the Milankovitch theory of glacial cycles and is an integral part of paleoclimatology (the study of prehistoric climates). It has been discussed in an earlier post on paleoclimate models by Christiane Rousseau [2]. The theoretical results obtained for the Milankovitch cycles can be tested against temperature data from the paleoclimate record. In the 1970s, Hayes et al. used data from ocean sediment core samples to relate the Milankovitch cycles to the climate of the last 468,000 years [3]. One of their conclusions was that “. . . climatic variance of these records is concentrated in three discrete spectral peaks at periods of 23,000, 42,000, and approximately 100,000 years.” This study was repeated recently by Zachos et al., who used much more extensive data [4]. The reconstructed temperature profile and the corresponding power spectrum show periods of 100, 41, and 23 Kyr; see Figure 3.12.

Figure 3.12. Time series and power spectrum of the Earth’s climate record for the past 4.5 Myr [6].

3.14. How Good Is the Milankovitch Theory?

59

The best data we have for temperatures during the ice age cycles come from the analysis of isotope ratios of air trapped in pockets in the polar ice. The ratio of oxygen isotopes is a good proxy for global mean temperature. In the 1990s, Petit et al. studied data from the Vostok ice core to reconstruct a temperature profile for the past 420,000 years [5]. Although the record is only half a million years long, it allows for fairly precise dating from the progressive layering process that laid down the ice. A spectral analysis shows cycles with periods of 100, 43, 24, and 19 Kyr, in reasonable agreement with previous findings and with the calculated periods of the Milankovitch cycles. At this point we might conclude that Milankovitch’s idea was correct and that ice ages are indeed correlated with orbital variations. There are, however, some serious caveats. Changes in the oxygen isotope ratio reflect the combined effect of changes in global ice volume and temperature at the time of deposition of the material, and the two effects cannot be separated easily. Furthermore, the cycles do not change the total energy received by the Earth if this is averaged over the course of a year. An increase in eccentricity, or obliquity, means the insolation is larger during part of the year and smaller during the rest of the year, with very little net effect on the total energy received at any latitude over a year. However, a change in eccentricity or obliquity could make the seasonal cycle more severe and thus change the extent of the ice caps. A possible scenario for the onset of ice ages would then be that minima in high-latitude insolation during the summer enable winter snowfall to persist throughout the year and thus accumulate to build glacial ice sheets. Similarly, times with especially intense high-latitude summer insolation could trigger a deglaciation. Clearly, additional detailed modeling would be needed to account for these effects. Even allowing for the scenario described in the previous paragraph, we would expect an asymmetric climate change, where the ice cap over one of the poles increases while the cap over the other decreases. Yet, the entire globe cooled during the ice ages and warmed during periods of deglaciation. An even more disturbing observation arises when one considers not just the periods of the cycles but also their relative strengths. In the data, the relative contributions are ordered as obliquity, followed by eccentricity, followed by precession, while for the average daily insolation at 65o North latitude at the summer solstice (denoted Q 65 , shown in Figure 3.13), the order is the reverse: precession, followed by obliquity, followed by eccentricity. The dominance of precession in the forcing term Q 65 does not even show up in the data (the “100,000-year problem”). The lesson learned here is that actual climate dynamics are very complex, involving much more than insolation and certainly much more than insolation distilled down to a single quantity, Q 65 . Feedback mechanisms are at work that are hard to model or explain. On the other hand, an analysis of the existing signals shows that astronomical factors most likely play a role in the Earth’s long-term climate. Adapted from the book Mathematics & Climate by H. Kaper and H. Engler, OT131, SIAM, Philadelphia, 2013. References [1] M. Milankovitch, Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem, University of Belgrade, 1941. [2] http://mpe2013.org/2013/02/13/paleoclimate-models/ [3] J.D. Hayes, J. Imbrie, and N.J. Shackleton, Variations in the Earth’s orbit: Pacemaker of the ice ages, Science 194 (1976) 1121–1132. [4] J. Zachos, M. Pagani, L. Sloan, E. Thomas, and K. Billups, Trends, rhythms, and aberrations in global climate 65 Ma to present, Science 292 (2001) 686–693.

60

Chapter 3. Weather and Climate

Figure 3.13. Time series and power spectrum of the average daily insolation at 65◦ North latitude at summer solstice (Q 65 ) [6].

[5] J.R. Petit et al., Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature 399 (1999) 429–436. [6] R. McGehee and C. Lehman, A paleoclimate model of ice-albedo feedback forced by variations in earth’s orbit, SIAM J. Appl. Dyn. Syst. 11 (2012) 684–707.

3.15 Earth’s Climate at the Age of the Dinosaurs Christiane Rousseau, University of Montreal Is it possible to compute the climate of the Earth at the time of dinosaurs? This question was answered by Jacques Laskar during his lecture entitled “Astronomical Calibration of the Geological Time Scales” at the workshop Mathematical models and methods for Planet Earth, held at the Istituto Nazionale di Alta Matematica (INdAM) on May 27–29, 2013. Laskar explained that Lagrange was the first to suggest a link between the past climates of the Earth and the variations of the parameters characterizing the Earth’s elliptical orbit. The latter concern the changes in the major axis, in the eccentricity, in the obliquity of the Earth’s axis, and in the precession of the Earth’s axis. These parameters undergo periodic oscillations, now called the Milankovitch cycles, with different periods between 20,000 and 40,000 years. These oscillations essentially come from the attraction exerted on the Earth by the other planets of the solar system. They have some influence on the climate, for instance when the eccentricity of the ellipse changes. The oscillations of the Earth’s axis also influence the climate: when the axis is more slanted, the poles receive more sun in summer, but the polar ice spreads more in winter. And scientists compare

3.16. Two Books on Climate Modeling

61

the measurements of ice cores and sedimentary records, showing the correlation between the past climates and the computed oscillations of the parameters of the Earth’s orbit. But how do we compute these oscillations? We use series expansions to approximate the motion of the Earth when taking into account the attraction of the other planets of the solar system. But these series are divergent, as was shown by Poincaré. Hence, the series can only provide precise information over a limited period of time. Laskar showed in 1989 that the motion of the inner planets of the solar system is chaotic, and confirmed this later in 2009 with 2,500 parallel simulations of the solar system. One characteristic of chaos is sensitivity to initial conditions, which means that errors grow exponentially in time. Hence, inevitably, the errors of any simulation will grow so much that we can no longer learn anything reliable from the simulation. The question is then “How fast do these errors grow?” This is measured by the Lyapunov time, which we will define here as the time before we lose one digit of precision—that is, the error is multiplied by 10. When modeling the planets, this Lyapunov time is 10 million years. The extinction of the dinosaurs took place 65 million years ago, and simulating the solar system over 70 million years we lose 7 digits of precision. This is a lot, but it is still tractable, and it is relatively easy to prove that the Earth is “stable” at this time horizon. But when we speak of the influence of the parameters of the Earth’s orbit on the climate, we need more precision. It does not suffice to include in the simulations all planets of the solar system as well as the Moon and the mean effect of the asteroid belt. The largest asteroids have to be considered individually, and some of them play a role. The two largest are Ceres and Vesta. Both are highly chaotic, with Lyapunov times of 28,900 years and 14,282 years, respectively. These asteroids are sufficiently large to have an influence on the orbit of the Earth, and there are other chaotic asteroids in the asteroid belt. Imagine: for each million years, the errors coming from Ceres are multiplied by 1034 and those from Vesta by 1070 ! In a paper Strong chaos induced by close encounters with Ceres and Vesta published in 2011 in Astronomy & Astrophysics, Laskar and his co-authors showed that we hit the wall and cannot obtain any reliable information past 60 million years. Hence, we may deduce the climate at the time of dinosaurs from geological observations, but there is no hope to compute it through backwards integration of the equations of motion for our solar system.

3.16 Two Books on Climate Modeling James Crowley, SIAM I am normally a great fan of book reviews, but one which covered a book on climate caught my attention. I was troubled with the review that appeared in the Philadelphia Inquirer because of the way it treated climate science in general and modeling in particular. The book review by Frank Wilson [1] concerned the book The Whole Story of Climate: What Science Reveals About the Nature of Endless Change by E. Kirsten Peters [2]. Wilson wrote: “Climate science, with its computer models, is a Johnny-come-lately to the narrative. Not so geology. ‘For almost 200 years,’ Peters writes, ‘geologists have studied the basic evidence of how climate has changed on our planet.’ They work, not with computer models, but with ‘direct physical evidence left in the muck and rocks.’ ” This seems to denigrate the role of models. It is certainly important to look to the past to better understand our climate—its trends and the mechanisms that caused those trends. However, it is also important to understand the trends on a time scale that is much smaller than the geological—that is, since the beginning of the industrial revolution—and

62

Chapter 3. Weather and Climate

Figure 3.14. Left image courtesy of Prometheus Books; www.prometheusbooks.com. Cover image 2012 c Media Bakery, Inc.; cover design by Grace M. Conti-Zilsberger. All rights reserved.

the role of increasing CO2 in the atmosphere. Modeling the physics of the atmosphere and performing simulations using high-performance computing play a crucial role in understanding the possible state of the climate in the next 100 years and beyond. Contrast the observation in Wilson’s book review to a recent textbook on climate science by Hans Kaper and Hans Engler, Mathematics & Climate [3]. This book, intended for master’s level students or advanced undergraduates, introduces students to “mathematically interesting topics from climate science.” It addresses a broad range of topics, beginning with the variability of climate over geologic history as gleaned from “proxy data” taken from deep-sea sediment cores. Certainly this variability informs our understanding of past climate history, including warming and cooling trends. The book moves from data of past climate history on to models of the ocean and atmosphere, coupled with data, covering an interesting bit of mathematics along the way. For example, students are exposed to the role of salinity in ocean circulation models, and learn something about dynamical systems that are used in these models. To give another example to show the breadth of mathematical topics covered, various statistics and analytical tools are introduced and are used to analyze the Mauna Loa CO2 data. Wilson, in his book review, states that “Using direct evidence rather than computer models, a geologist says a cold spell could be near.” That could be comforting news to some who want to ignore predictions of a warming planet, but it would be cold comfort. Mathematics, when used in the geosciences, tends to take a more balanced and calculating approach. As Kaper and Engler point out in the preface to their book, “Understanding the Earth’s climate system and predicting its behavior under a range of ‘what if’ scenarios are among the greatest challenges for science today.” Physical modeling, mathematics,

3.16. Two Books on Climate Modeling

63

numerical simulation, and statistical analysis will continue to play a major role in addressing that challenge. References [1] F. Wilson, Digging deeper on climate change, Philadelphia Inquirer, October 13, 2013. [2] E. Kirsten Peters, The Whole Story of Climate: What Science Reveals About the Nature of Endless Change, Prometheus Books, Amherst, New York, 2012. [3] H. Kaper and H. Engler, Mathematics & Climate, OT131, SIAM, Philadelphia, 2013.

Chapter 4

Beyond Planet Earth

4.1 Chaos in the Solar System Christiane Rousseau, University of Montreal

Figure 4.1. Our solar system. Reprinted courtesy of NASA.

The motion of the inner planets (Mercury, Venus, Earth, and Mars) is chaotic. Numerical evidence was given by Jacques Laskar, who showed in 1994 [1] that the orbits of the inner planets exhibit resonances in some periodic motions. Because of the sensitivity to initial conditions, numerical errors grow exponentially, so it is impossible to control the positions of the planets over long periods of time (hundreds of millions of years) using the standard equations of planetary motion. Laskar derived an averaged system of equations and showed that the orbit of Mercury could at some time cross that of Venus. Another way to study chaotic systems is to perform numerous simulations in parallel using an ensemble of initial conditions and derive probabilities of future behaviors. The shadowing lemma guarantees that a simulated trajectory for a close initial condition 65

66

Chapter 4. Beyond Planet Earth

resembles a real trajectory. In 2009, Laskar and Gastineau [2] announced in Nature the results of an ambitious program of 2,500 parallel simulations of the solar system over periods of the order of 5 billion years. The new model of the solar system was much more sophisticated and included some relativistic effects. The simulations showed a 1% chance that Mercury could be destabilized and encounter a collision with the Sun or Venus. In fact, a much smaller number of simulations showed that all the inner planets could be destabilized, with a potential collision between the Earth and either Venus or Mars likely in approximately 3.3 billion years. References [1] J. Laskar, Large-scale chaos in the solar system, Astron. Astrophys. 287 (1994) L9–L12. [2] J. Laskar and M. Gastineau, Existence of collisional trajectories of Mercury, Mars and Venus with the Earth, Nature 459 (2009) 817–819, doi: 10.1038/nature08096.

4.2 KAM Theory and Celestial Mechanics Alessandra Celletti, University of Rome Is the Earth’s orbit stable? Will the Moon always point the same face to our planet? Will some asteroid collide with the Earth? These questions have puzzled mankind since antiquity, and answers have been looked for over the centuries, even if these events might occur on time scales much longer than our lifetime. It is indeed extremely difficult to settle these questions, and despite all efforts, scientists have been unable to give definitive answers. But the advent of computers and the development of outstanding mathematical theories now enable us to obtain some results on the stability of the solar system, at least for simple model problems. The stability of the solar system is a very difficult mathematical problem, which has been investigated in the past by celebrated mathematicians, including Lagrange, Laplace, and Poincaré. Their studies led to the development of perturbation theories—theories to find approximate solutions of the equations of motion. However, such theories have an intrinsic difficulty related to the appearance of the so-called small divisors—quantities that can prevent the convergence of the series defining the solution. A breakthrough occurred in the middle of the 20th century. At the 1954 International Congress of Mathematics in Amsterdam, the Russian mathematician Andrei N. Kolmogorov (1903–1987) gave the closing lecture, entitled “The general theory of dynamical systems and classical mechanics.” The lecture concerned the stability of specific motions (for the experts: the persistence of quasi-periodic motions under small perturbations of an integrable system). A few years later, Vladimir I. Arnold (1937–2010), using a different approach, generalized Kolmogorov’s results to (Hamiltonian) systems presenting some degeneracies, and in 1962 Jürgen Moser (1928–1999) covered the case of finitely differentiable systems. The overall result is known as KAM theory from the initials of the three authors [1, 2, 3]. KAM theory can be developed under quite general assumptions. An application to the N -body problem in celestial mechanics was given by Arnold, who proved the existence of some stable solutions when the orbits are nearly circular and coplanar. Quantitative estimates for a three-body model (e.g., the Sun, Jupiter, and an asteroid) were given in 1966 by the French mathematician and astronomer Michel Hénon (1931–2013), based on the original versions of KAM theory [4]. However, his results were a long way from reality; in the best case they proved the stability of some orbits when the primary mass ratio is of the order of 10−48 —a value that is inconsistent with the astronomical Jupiter–Sun mass ratio, which is of the order of 10−3 . For this reason

4.2. KAM Theory and Celestial Mechanics

67

Hénon concluded in one of his papers, “Ainsi, ces théorèmes, bien que d’un très grand intérêt théorique, ne semblent pas pouvoir en leur état actuel être appliqués à des problèmes pratiques” [4]. This result led to the general belief that, although being an extremely powerful mathematical method, KAM theory does not have concrete applications, since the perturbing body must be unrealistically small. During one of my stays at the Observatory of Nice in France, I had the privilege to meet Michel Hénon. In the course of one of our discussions he showed me his computations on KAM theory, which were done by hand on only two pages. It was indeed a success that such a complicated theory could be applied using just two pages! Likewise, it was evident that to get better results it is necessary to perform much longer computations, as often happens in classical perturbation theory. A new challenge came when mathematicians started to develop computer-assisted proofs. With this technique, which has been widely used in several fields of mathematics, one proves mathematical theorems with the aid of a computer. Indeed, it is possible to keep track of rounding and propagation errors through a technique called interval arithmetic. The synergy between theory and computers turns out to be really effective: the machine enables us to perform a huge number of computations, and the errors are controlled through interval arithmetic. Thus, the validity of the mathematical proof is maintained. The idea was then to combine KAM theory and interval arithmetic. As we will see shortly, the new strategy yields results for simple model problems that agree with the physical measurements. Thus, computer-assisted proofs combine the rigor of the mathematical computations with the concreteness of astronomical observations. Here are three applications of KAM theory in celestial mechanics which yield realistic estimates. The extension to more significant models is often limited by the computer capabilities. • A three-body problem for the Sun, Jupiter, and the asteroid Victoria was investigated in [5]. Careful analytical estimates were combined with a FORTRAN code implementing long computations using interval arithmetic. The results show that in such a model the motion of the asteroid Victoria is stable for the realistic Jupiter–Sun mass ratio. • In the framework of planetary problems, the Sun–Jupiter–Saturn system was studied in [6]. A bound was obtained on the secular motion of the planets for the observed values of the parameters. (The proof is based on the algebraic manipulation of series, analytic estimates, and interval arithmetic.) • A third application concerns the rotational motion of the Moon in the so-called spin-orbit resonance, which is responsible for the well-known fact that the Moon always points the same face to the Earth. Here, a computer-assisted KAM proof yielded the stability of the Moon in the actual state for the true values of the parameters [7]. Although it is clear that these models provide an (often crude) approximation of reality, they were analyzed through a rigorous method to establish the stability of objects in the solar system. The incredible effort by Kolmogorov, Arnold, and Moser is starting to yield new results for concrete applications. Faster computational tools, combined with refined KAM estimates, will probably enable us to obtain good results also for more realistic models. Proving a theorem for the stability of the Earth or the motion of the Moon will definitely let us sleep more soundly!

68

Chapter 4. Beyond Planet Earth

References [1] V.I. Arnold, Proof of a theorem by A.N. Kolmogorov on the invariance of quasi– periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surveys 18 (1963) 13–40. [2] A.N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk. SSR 98 (1954) 527–530. [3] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1 (1962) 1–20. [4] M. Hénon, Explorations numériques du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique 3(1) fasc. 2 (1966) 49–66. [5] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Memoirs Am. Math. Soc. 187 (2007) 878. [6] U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Celest. Mech. Dyn. Astron. 78 (2000) 47–74. [7] A. Celletti, Analysis of Resonances in the Spin-Orbit Problem in Celestial Mechanics, PhD thesis, ETH-Zürich, 1989; see also Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance (Part I), J. Appl. Math. Phys. (ZAMP) 41 (1990) 174–204.

4.3 New Ways to the Moon, Origin of the Moon, and Origin of Life on Earth Edward Belbruno, Princeton University The field of celestial mechanics is an old one, going back to 90 AD when Claudius Ptolemy sought to describe the motions of the planets. However, the modern field of celestial mechanics goes back to the 1700s when Joseph-Louis Lagrange studied the celebrated three-body problem for the motion of three mass points under the influence of the force of gravity. Henri Poincaré in the late 19th, early 20th century made huge advances in our understanding of the three-body problem, using new methods from what today we call the theory of dynamical systems. Much of Poincaré’s work was focused on a special version of the three-body problem, namely the planar circular restricted three-body problem, where two mass points move in a plane in circular orbits about their common center of mass while a third point of zero mass moves in the same plane under the influence of the gravity of the two mass points. One can imagine these two bodies to be the Earth and the Moon, while the zero mass point can be a rock, for example a meteorite. Poincaré’s work indicated that the motion of the zero mass point could be very complicated and sensitive, so much so that it would be chaotic. If one considers the motion of a small object, say a spacecraft that is going to the Moon, under the gravitational influence of both the Earth and the Moon, then a standard route to the Moon from the Earth is called a Hohmann transfer, after the work of the engineer Walter Hohmann in the 1920s. A Hohmann transfer yields a flight time of about 3 days. The path is not chaotic since the spacecraft is going fairly fast; in fact, it looks almost linear. When the spacecraft approaches the Moon, it needs to slow down a lot, by about 1 kilometer per second, in order to be captured into lunar orbit. This requires a lot of fuel, which is very expensive—about a million dollars per pound! Is there a better way? Is there a way to find a transfer to the Moon where the spacecraft is captured into lunar orbit automatically, without the use of rocket engines? In 1987, I showed for the first time the process of ballistic capture—a transfer from the Earth that takes two years to reach the Moon and results in automatic capture [1]. The region about

4.3. New Ways to the Moon, Origin of the Moon, and Origin of Life on Earth

69

the Moon where this capture can occur is one where the spacecraft must arrive slowly enough relative to the Moon. Then the balance of the gravitational tugs of the Earth and the Moon on the moving spacecraft cause the motion of the captured spacecraft to be chaotic. Today, this region is called the weak stability boundary. A spacecraft moving in the weak stability boundary for capture is roughly analogous to a surfer catching a wave.

Figure 4.2. Hiten spacecraft. Reprinted courtesy of NASA.

This methodology was put to the test in 1991, in an effort to rescue the Japanese spacecraft Hiten (Figure 4.2) and move it to a new route to the Moon. This route was a more practical five months in duration and actually needed to use the more complex fourbody problem model between the Earth, Moon, Sun, and spacecraft. It made use not only of the weak stability boundary of the Moon, but also of the weak stability boundary of the Earth due to the gravitational interaction between the Earth and the Sun, which occurs about 1.5 million kilometers from the Earth. The spacecraft got to the Moon in ballistic capture by first flying out four times the Earth–Moon distance to 1.5 million kilometers and then falling back to the Moon for ballistic capture. This type of route, called an exterior weak stability boundary (WSB) transfer or simply a weak transfer, was used by NASA’s GRAIL mission in 2011, and there are plans to use it for more space missions in the future. In 2005, I was involved in a different application of weak transfer with J.R. Gott III to help explain the Theia Hypothesis [2]. Theia is the giant Mars-sized impactor that is hypothesized to have collided with the Earth to form the Moon from the remnants of the collision. The hypothesis is that Theia could have been formed in one of the equilateral Lagrange points in the Earth–Sun system. These are two stable locations on the Earth’s orbit about the Sun, 60 degrees ahead of and behind the Earth (Figure 4.3), where a particle of negligible mass will be trapped. The hypothesis is that Theia formed by accretion of many small rocks, called planetesimals, over millions of years. As Theia was forming, it gradually moved away from the Lagrange points where it was weakly captured. Eventually it escaped onto a weak transfer from the Lagrange region and hit the Earth with low energy. NASA investigated this theory in 2009, when it sent its two STEREO spacecrafts to these regions to look for any residual material. Since the spacecrafts were not originally designed to look for this type of material, the search was inconclusive. Together with Amaya Moro-Martin, Renu Malhotra, and Dmitry Savransky, we recently considered a new application of weak transfer to the origin of life [3]. We wanted

70

Chapter 4. Beyond Planet Earth

Figure 4.3. The five Lagrange points, L1, L2, L3, L4, L5. Reprinted courtesy of NASA.

to understand the validity of the Lithopanspermia Hypothesis: rocks containing biogenic material were ejected from a given planetary system of a star, S1, and captured by another star, for example the Sun. We considered a special type of cluster of stars, called an open star cluster, which is a loose aggregate of stars moving slowly with respect to each other with relative velocities of about one kilometer per second. It is thought by many that the Sun formed in such a cluster about 5 billion years ago. Eventually, some of the rocks would crash onto the Earth. Previous studies of this problem dealt with high-velocity ejection of rocks from S1, on the order of 6 kilometers per second. They found that the probability of capture of these rocks by another star, for example the Sun, was essentially zero. In our study, we used very low-velocity escape from S1, on the order of 50 meters per second, and examined the likelihood of weak capture by the Sun. We found that the probability increased by an order of one billion. This implies that about 3 billion rocks could have impacted the Earth over the time spans considered of about 400 million years. The time spans involved coincided nicely with the emergence of life on Earth. Last, we mention that the weak stability boundary is a very interesting region about one of the mass points—say, about the Moon—in the planar circular restricted three-body problem. It is a subset of a very complicated region consisting of invariant manifolds associated with the collinear Lagrange points L1 and L2 (Figure 4.4). References [1] E. Belbruno, Lunar capture orbits, a method of constructing Earth Moon trajectories and the lunar GAS mission, in Proceedings of the 19th International Electric Propulsion Conference, 1987, doi: 10.2514/6.1987-1054. [2] E. Belbruno and J.R. Gott III, Where did the Moon come from?, Astron. J. 129 (2005) 1724–1745. [3] E. Belbruno, A. Moro-Martin, R. Malhotra, and D. Savransky, Chaotic exchange of solid material between planetary systems: Implications for Lithopanspermia, Astrobiology 12 (2012) 754–774.

4.4. Low-Fuel Spacecraft Trajectories to the Moon

71

Figure 4.4. Slice of the weak stability boundary (red) about the Moon (at center).

4.4 Low-Fuel Spacecraft Trajectories to the Moon Marian Gidea, Northeastern Illinois University A recent blog entry discussed why celestial mechanics is part of MPE2013 [1]. Here I suggest a further argument in favor of this inclusion and call attention to some recent events and mathematical ideas in connection with explorations beyond planet Earth. There is widespread interest in finding and designing spacecraft trajectories to the Moon, Mars, other planets, and other celestial bodies (comets, asteroids) that require as little fuel as possible. This is justified mostly by the cost of these missions: each pound of payload costs roughly one million dollars. Robotic space missions typically conduct numerous observations and measurements over long periods of time, so to maximize the equipment load, it is imperative to minimize the fuel consumption of the propulsion system. One way to achieve this is to cleverly exploit, in a mathematically explicit way, the gravitational forces of the Earth, Moon, Sun, etc. Suppose we would like to design a low-energy transfer from the Earth to the Moon. Of course, the spacecraft must first be placed on some orbit around the Earth. This is expensive energy-wise but unavoidable. Surprisingly though, the second leg of the trajectory, taking the spacecraft from near the Earth to some prescribed orbit about the Moon, can be done at low energy cost (in theory, even for free). Say our goal is to insert the spacecraft at the periapsis of an elliptic orbit about the Moon of prescribed eccentricity and at some prescribed angle with respect to the Earth–Moon axis, and to do so without having to slow down the spacecraft (or maybe just a little) at arrival, thus saving the fuel necessary for such an operation. Imagine that we run the “movie” of the trajectory backwards, from

72

Chapter 4. Beyond Planet Earth

the moment when the spacecraft is on the elliptic orbit. Since the eccentricity is fixed, the semimajor axis determines the velocity of the spacecraft at the periapsis. If the semimajor axis is too short (or, equivalently, the velocity is too low), the trajectory will turn around the Moon without leaving the Moon region; such a trajectory is “stable.” By gradually increasing the semimajor axis (hence, the velocity), we find the first “unstable” trajectory that leaves the Moon region and makes a transfer to the Earth region (Figure 4.5). Exploring all angles of insertion and all values of the eccentricity of the elliptical orbit yields the “weak stability boundary.” This appears to be some sort of a fractal set (Figure 4.6). All points in the weak stability boundary correspond to arrival points of low energy transfers from the Earth to the Moon. The spacecraft trajectories designed by this method yield fuel savings of 10–15%.

Figure 4.5. Stable and unstable trajectories; P1 denotes the Earth and P2 the Moon.

The notion of a weak stability boundary was introduced by Edward Belbruno (Princeton University) in 1987; a documentary trailer on the discovery of this concept can be found on YouTube [2]. The low-energy transfer method was successfully applied for the first time in 1991 to rescue the Japanese lunar spacecraft Hiten. NASA’s GRAIL (Gravity Recovery and Interior Laboratory) mission, which took place in 2012, used the same low-energy transfer technique. The purpose of GRAIL was to obtain a high-resolution map of the gravitational field of the Moon. For this purpose, two spacecrafts had to be placed on the same orbit about the Moon. Their instruments would measure changes in their relative velocity, which would then be translated into changes of the gravitational field. This technique had been tested previously for the mapping of Earth’s gravity as part of the GRACE (Gravity Recovery and Climate Experiment) mission, a joint mission of NASA and the German Aerospace Center initiated in 2002. A key point for the GRAIL mission was to place the two spacecrafts on precisely the same lunar orbit; the weak stability boundary concept was particularly suitable for this purpose. A deeper understanding of the weak stability boundary can be obtained from the study of hyperbolic invariant manifolds. The motion of a spacecraft relative to the

4.5. Where Did the Moon Come From?

73

Figure 4.6. Weak stability boundary [3].

Earth–Moon system can be modeled through the three-body problem [3]. In this model, the intertwining gravitational fields of the Earth and the Moon determine some “invisible pathways,” called stable and unstable manifolds, on which optimal transport is possible. These manifolds, which were first introduced by Henri Poincaré, are intimately related to the weak stability boundary. Under some energy restriction, it can be proved geometrically that the weak stability boundary points lie on certain stable manifolds. References [1] C. Rousseau, Why is celestial mechanics part of MPE2013? Mathematics of Planet Earth Blog, April 20, 2013, http://mpe2013.org/2013/04/20/why-is-celestial-mechanicspart-of-mpe2013/ [2] J. Okada, Painting the Way to the Moon, YouTube, http://www.youtube.com/watch?v=zYl_3qGXuRE [3] E. Belbruno, M. Gidea, and F. Topputo, Weak stability boundary and invariant manifolds, SIAM J. Appl. Dyn. Syst. 9 (2010) 1061–1089. [4] E. Belbruno, M. Gidea, and F. Topputo, Geometry of weak stability boundaries, Qual. Theory Dyn. Syst. 12 (2013) 53–66.

4.5 Where Did the Moon Come From? Christiane Rousseau, University of Montreal You may have read Edward Belbruno’s contribution “New Ways to the Moon, Origin of the Moon, and Origin of Life on Earth” (Section 4.3). I did, and I was intrigued by his

74

Chapter 4. Beyond Planet Earth

application of weak transfer to the origin of the Moon, so I went to his 2005 joint paper with J. Richard Gott III [1]. Indeed, I was already familiar with the earlier work of Jacques Laskar on the Moon done in 1993. At the time, he had proved that it was the presence of the Moon that stabilizes the inclination of the Earth’s axis. Indeed, the axis of Mars has very large oscillations, up to 60 degrees, and Venus’s axis also had large oscillations in the past. The numerical simulations show that, without the Moon, the Earth’s axis would also have very large oscillations. Hence, the Moon is responsible for the stable system of seasons that we have on Earth, which may have favored life on our planet. Currently, the most plausible theory for the formation of the Moon is that it comes from the impact of a Mars-size planet, which we will call the impactor, with the Earth. For information, the radius of Mars is 53% of that of the Earth, its volume is 15% of that of the Earth, and its mass only 10%. Evidence supporting the impactor hypothesis comes from the geological side: the Earth and the Moon contain the same types of oxygen isotopes, which are not found elsewhere in the solar system; the Earth and Mars both have an iron core, while the Moon does not. The theory is that, at the time of the collision, the iron in the Earth and in the impactor would already have sunk into their cores, and also that the collision was relatively weak. Hence, while the two iron cores merged together, debris expelled from the mantle would later aggregate into the Moon. Indeed, the mean density of the Moon is comparable to that of the mean density of the Earth’s crust and upper mantle. Mathematics cannot prove the origin of the Moon. It can only provide models which show that the scenario of the giant impactor makes sense, and that it makes more sense than other proposed scenarios. It is believed that the planets formed by accretion of small objects, called planetesimals. Because the impactor and the Earth had similar composition, they should have formed at roughly the same distance from the Sun, namely one astronomical unit (AU). But then, why would it have taken so long before a collision occurred? It is because the Earth and the impactor were at stable positions. The Sun, the Earth, and the impactor form a three-body problem. Lagrange identified some periodic motions of three bodies where they are located at the vertices of an equilateral triangle: the corresponding points for the third bodies are called Lagrange L4 and L5 points; see Figure 4.3. These motions are stable when the mass of the impactor is much smaller than that of the Earth and the mass of the Earth is much smaller than that of the Sun. Stability is shown rigorously using KAM theory for the ideal circular planar restricted problem and numerically for the full three-dimensional three-body problem, with integration over 10 Myr. Hence, it makes sense that a giant impactor could have formed at L4 or L5: this impactor is known in the literature as Theia. Simulations show that Theia could have grown by attracting planetesimals in its neighborhood. Let’s suppose that Theia is formed at L4. Why then didn’t it stay there? Obviously, it should have been destabilized. Simulations show that some small planetesimals located near the same Lagrange point could have slowly pushed Theia away from L4. The article of Belbruno and Gott [1] studies the potential movements after destabilization. What is crucial is that, since the three bodies were at the vertices of an equilateral triangle, Theia and the Earth are at equal distances from the Sun. If the orbit of the Earth is nearly circular, Theia and the Earth share almost the same orbit! This is why there is a high danger of collision when Theia is destabilized. If the ejection speed were small, Theia would move back and forth along a trajectory resembling a circular arc centered at the Sun with additional smaller oscillations. In a frame centered at the Sun and rotating with the Earth (so the Earth is almost fixed), Theia moves back and forth in a region that looks like a horseshoe (see Figure 4.7).

4.6. Data Assimilation and Asteroids

75

In this movement it never passes close to the Earth. An asteroid with a diameter of 100 m, 2002 AA29 , discovered in 2002, has this type of orbit. This horseshoe region almost overlaps the Earth’s orbit. For a higher ejection speed, Theia would be pushed into an orbit around the Sun with radius approximately 1 AU and gradually creep towards the Earth’s orbit: it would pass regularly close to the Earth periapsis (the point of the Earth’s orbit closest to the Sun) in nearly parabolic trajectories, i.e., trajectories borderline of being captured by the Earth. Since the speed vectors of the two planets are almost parallel, the gravitational perturbation exerted by the Earth on Theia at each fly-by is Figure 4.7. Theia. small. The simulations show that these trajectories have a high probability of collision with Earth, not so long after leaving the Lagrange points (of the order of 100 years). Note that this kind of trajectory is highly chaotic, and many simulations with close initial conditions allow seeing the different potential types of trajectories. Reference [1] E. Belbruno and J.R. Gott III, Where did the Moon come from?, Astron. J. 129 (2005) 1724–1745.

4.6 Data Assimilation and Asteroids Robert Miller, Oregon State University The close approach of the asteroid that we all read about in the newspapers represented something of a coincidence for me as I prepared for the data assimilation workshop in Banff from February 17–22, 2013. Gauss invented data assimilation as we know it for the purpose of calculating asteroid orbits. The orbit of an asteroid around the Sun is determined by six parameters. Given observations of an asteroid, he chose the parameters that would minimize the sum of squared differences between the observed and predicted values. I don’t know whether he considered more complicated orbital calculations that would have taken the gravity fields of Jupiter or Mars into account. Why Gauss would have spent his time doing this, I don’t know. Two centuries later, most data assimilation systems still rely on least squares. Some may question Gauss’s title as the “inventor of data assimilation,” as Legendre did at the time. In one of his books, Eric Temple Bell describes an exchange of letters between Gauss and Legendre, in which Legendre pointed out that he had published the least-squares method before Gauss’s paper on asteroid orbits appeared. Legendre humbly asked Gauss to acknowledge his proudest achievement. Surely Gauss, for all his great accomplishments, could acknowledge what Legendre described in a charming biblical reference as “my one ewe lamb.” Gauss refused, saying that he had in fact formulated the least-squares method independently before Legendre. This turned out to be true. The least-squares method appeared in Gauss’s notebooks before Legendre’s paper, but Legendre published first.

76

Chapter 4. Beyond Planet Earth

It’s likely, though not certain, that the accounts in the media of the asteroid that recently passed within the orbits of our geostationary satellites are based on least-squares calculations. There are alternatives. Someone once told me that the trajectory of the European Space Agency’s Ariane launch vehicle is calculated by probabilistic methods based on the theory of stochastic differential equations. I asked him how they did that, and he laughed and said “Oh, they will not tell you.” However they do it, it’s a hard calculation, probably impossible without electronic computing machinery, even given Gauss’s legendary calculating abilities.

4.7 Understanding the Big Bang Singularity Edward Belbruno, Princeton University If you want to understand planet Earth, then why not go back to the beginning of the universe? The Big Bang is an event that we do not understand. It is thought to have happened about 13.75 billion years ago. What occurred, as we understand it, is mind blowing. The entire universe as we know it today seems to have come out of nowhere and very quickly. This is currently described by the theory of inflation, which estimates that the universe expanded by a factor of 1078 in volume in the time span from 1 · 10−36 seconds to 1 · 10−32 seconds after the Big Bang. Where did all this energy come from? One way to account for this energy is offered by the cyclic universe theory that says basically that prior to the Big Bang, there was another universe that contracted down in a “big crunch,” which then gave rise to the Big Bang. This process could have occurred over and over, where our universe is just one universe in the process. The cyclic universe theory has been studied by Gott and Lin [1], Figure 4.8. Edward Belbruno, Microwave Radia- Steinhard and Turok in several papers [2, 3, 4] using a stringtion of the universe, oil on canvas, 30 × 16 (2006). theory formulation, and many others.

4.7. Understanding the Big Bang Singularity

77

In the treatment of the cyclic universe theories, it is an open problem how one universe could smoothly be continued into another, since the differential equations that describe the inflation become undefined (singular) at the Big Bang itself. These differential equations are the so-called Friedmann equations, and under certain assumptions they can be reduced to a system of ordinary differential equations which are undefined at the Big Bang. In a recent paper [5], I showed how the equations can be made smooth at the Big Bang by a regularizing transformation of the position, velocity, and time variables. The regularized equations have a unique solution, not found previously, which represents a smooth transition from one universe to another. A particularly intriguing result is that the unique continuation of one universe into another is possible if and only if a key parameter in the problem, called the equation of state, can be written as a ratio of two integers which are relatively prime. Figure 4.8 features one of my paintings (details at [6]) of the universe immediately after the Big Bang. It illustrates the microwave background radiation of the universe, inspired by the Wilkinson Anisotropy Probe data. The most intense radiation is in red and the least in black-blue. References [1] J.R. Gott III and L.-X. Li, Can the universe create itself?, Phys. Rev. D 58 (1998) 023501, http://arxiv.org/abs/astro-ph/9712344 [2] P.J. Steinhardt and N. Turok, A cyclic model of the universe, Science 296 (2002) 1436–1439, http://arXiv.org/abs/hep-th/0111030 [3] P.J. Steinhardt and N. Turok, Cosmic evolution in a cyclic universe, Phys. Rev. D 65 (2002) 126003, http://arxiv.org/abs/hep-th/0111098 [4] P.J. Steinhardt and N. Turok, Is vacuum decay significant in ekpyrotic and cyclic models?, Phys. Rev. D 66 (2002) 101302, http://arxiv.org/abs/astro-ph/0112537 [5] E. Belbruno, On the regularizability of the big bang singularity, Celest. Mech. Dyn. Astr. (2013), doi: 10.1007/s10569-012-9449-4. [6] http://www.belbrunoart.com

Chapter 5

Biosphere

5.1 The Mystery of Vegetation Patterns Karna Gowda, Northwestern University

Figure 5.1. Tiger bush plateau surrounding Niamey, Niger (vertical aerial view). Vegetation is dominated by Combretum micranthum and Guiera senegalensis. Image size: 5×5 km on the ground. Reprinted courtesy of USGS.

Figure 5.1 shows a tiger bush plateau near Niamey, Niger. From above, the ground almost looks like the fur of a big cat. Vegetation and barren land array to form tiger stripes and leopard spots in the dry landscape surrounding Niamey, Niger. These types of patterns are common to semi-arid ecosystems, so-called because there is enough water 81

82

Chapter 5. Biosphere

to support some vegetation but not enough to support it uniformly. Besides being visually striking phenomena, vegetation patterns may have a lot to tell us about how ecosystems are changing. Semi-arid ecosystems are in a difficult position. They exist in dangerous limbo between vibrant vegetated ecosystems and desolate deserts. And these ecosystems are not marginal. Semi-arid ecosystems support over a third of the world’s population. If climate change pushes an ecosystem towards increased aridity, then semi-arid ecosystems can become deserts incapable of supporting people. To a great extent, these deserts would be here to stay. A simple uptick in rainfall, for instance, could not return things to the way they used to be, since desertification is an erosive process that irreversibly mars landscapes. Because of the role that vegetation patterns occupy as signals of diminishing water in an ecosystem, scientists and mathematicians are interested in what these patterns say about how close an ecosystem is to transitioning to desert. It is possible that both the characteristic width and the qualitative appearance of the patterns may function as such indicators, i.e., vegetation stripes may become spaced farther apart or turn into patchy spots as the climate gets drier? This is a compelling reason indeed to carefully understand these patterns through the lens of mathematics. It turns out that vegetation patterns in semi-arid ecosystems are probably caused by the same mechanism that causes patterns to form in lots of other systems: feedback. In this case, the feedback occurs in how plants and water interact. Plants help each other at short scales by sharing nutrients and trapping water in the soil with their root systems. This causes the feedback loop in which moderate to high densities of plants are self-sustaining. However, water is a limiting factor that stops vegetation from becoming dense, and it prevents growth in sparsely vegetated areas. These factors together result in vegetation attempting to spread outwards from areas of high density but then being restricted by the effects of limiting water so that localized structures form.

Figure 5.2. Vegetation on a hillside. Reprinted with permission from John Wiley and Sons [1].

An illustrative example comes from patterns that form on hillsides. Figure 5.2 displays a cross section of a vegetation band on a shallow slope. Water comes into the system by precipitation, which turns into runoff as it travels downhill along the soil’s surface. If this runoff travels along a bare-ground region and encounters a patch of vegetation, it becomes absorbed by the vegetation and porous soil. This helps the patch grow more robustly. Since water is limited, plants on the uphill side of the band are preferentially nourished. The density of plants away from this edge tapers off until the start of another bare-ground region. As the average water in a system diminishes, it is easy to imagine the vegetated bands becoming smaller and the bare regions becoming larger; the stripes may drift apart.

5.2. How Vegetation Competes for Rainfall in Dry Regions

83

Figure 5.3. (Top) Observed vegetation patterns for different levels of precipitation. (Bottom) Vegetation patterns predicted from mathematical models for different levels of precipitation. Reprinted with permission from APS [2].

The top panels in Figure 5.3 show three qualitatively different patterns that ecologists have observed in semi-arid regions with different amounts of mean rainfall: spots in the most arid conditions, labyrinths when there is more water, and isolated gaps in otherwise full vegetation when water becomes more abundant. Similar patterns can be obtained with mathematical models that treat vegetation and water as components that react with one another and diffuse in space, as shown in the bottom panels in Figure 5.3. Vegetation patterns are a mysterious phenomenon that we can think about in the same way as patterns that form in many other contexts. What’s more, they may have importance that transcends their beauty. If they can be used to predict whether an ecosystem is approaching collapse, they could be immensely important to the conservation of land. The mathematical study of these patterns is crucial in our understanding of the dynamics of semi-arid ecosystems, and could have an impact in how humanity responds to the danger of climate change. References [1] J.M. Thiery, J.-M. d’Herbes, and C. Valentin, A model simulating the genesis of banded vegetation patterns in Niger, J. Ecology 83 (1995) 497–507. [2] K. Gowda, H. Riecke, and M.C. Silber, Transitions between patterned states in vegetation models for semiarid ecosystems, Phys. Rev. E 89 (2014) 022701.

5.2 How Vegetation Competes for Rainfall in Dry Regions Frank Kunkle, SIAM Karthika Muthukumaraswamy, SIAM The greater the plant density in a given area, the greater the amount of rainwater that seeps into the ground. This is due to a higher presence of dense roots and organic matter in the soil. Since water is a limited resource in many dry ecosystems, such as semiarid environments and semi-deserts, there is a benefit to vegetation to adapt by forming

84

Chapter 5. Biosphere

Figure 5.4. Desert steppes in Yol Valley in Mongolia.  c l’ Christineg | Dreamstime.com, reprinted with permission.

closer networks with little space between plants. Hence, vegetation in semi-arid environments (or regions with low rainfall) self-organizes into patterns or “bands.” The pattern formation occurs where stripes of vegetation run parallel to the contours of a hill and are interlaced with stripes of bare ground. Banded vegetation is common where there is low rainfall. In a paper published in SIAM Journal on Applied Mathematics [1], author Jonathan A. Sherratt uses a mathematical model to determine the levels of precipitation within which such pattern formation occurs. “Vegetation patterns are a common feature in semi-arid environments, occurring in Africa, Australia and North America,” explains Sherratt. “Field studies of these ecosystems are extremely difficult because of their remoteness and physical harshness; moreover there are no laboratory replicates. Therefore mathematical modeling has the potential to be an extremely valuable tool, enabling prediction of how pattern vegetation will respond to changes in external conditions.” Several mathematical models have attempted to address banded vegetation in semi-arid environments, of which the oldest and most established is a system of reaction-diffusionadvection equations, called the Klausmeier model. The Klausmeier model is based on a water redistribution hypothesis, which assumes that rain falling on bare ground infiltrates only slightly; most of it runs downhill in the direction of the next vegetation band. It is here that rain water seeps into the soil and promotes growth of new foliage. This implies that moisture levels are higher on the uphill edge of the bands. Hence, as plants compete for water, bands move uphill with each generation. This uphill migration of bands occurs as new vegetation grows upslope of the bands and old vegetation dies on the downslope edge. In his paper, the author uses the Klausmeier model to determine the critical rainfall level needed for pattern formation based on a variety of ecological parameters, such as rainfall, evaporation, plant uptake, downhill flow, and plant loss. He also investigates the uphill migration speeds of the bands. “My research focuses on the way in which patterns change as annual rainfall varies. In particular, I predict an abrupt shift in pattern formation as rainfall is decreased, which dramatically affects ecosystems,” says Sherratt. “The

5.3. Biological Events in Our Water Systems

85

mathematical analysis enables me to derive a formula for the minimum level of annual rainfall for which banded vegetation is viable; below this, there is a transition to complete desert.” The model has value in making resource decisions and addressing environmental concerns. “Since many semi-arid regions with banded vegetation are used for grazing and/or timber, this prediction has significant implications for land management,” Sherratt says. “Another issue for which mathematical modeling can be of value is the resilience of patterned vegetation to environmental change. This type of conclusion raises the possibility of using mathematical models as an early warning system that catastrophic changes in the ecosystem are imminent, enabling appropriate action (such as reduced grazing).” The simplicity of the model allows the author to make detailed predictions, but more realistic models are required to further this work. “All mathematical models are a compromise between the complexity needed to adequately reflect real-world phenomena, and the simplicity that enables the application of mathematical methods. My paper concerns a relatively simple model for vegetation patterning, and I have been able to exploit this simplicity to obtain detailed mathematical predictions,” explains Sherratt. “A number of other researchers have proposed more realistic (and more complex) models, and corresponding study of these models is an important area for future work. The mathematical challenges are considerable, but the rewards would be great, with the potential to predict things such as critical levels of annual rainfall with a high degree of quantitative accuracy.” With 2013 being the year of Mathematics of Planet Earth, mathematics departments and societies across the world are highlighting the role of the mathematical sciences in the scientific effort to understand and deal with the multifaceted challenges facing our planet and our civilization. “The wider field of mathematical modeling of ecosystemlevel phenomena has the potential to make a major and quite unique contribution to our understanding of our planet,” says Sherratt. Reference [1] J.A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments V: The transition from patterns to desert,” SIAM J. Appl. Math. 73 (2013) 1347–1367 (online publish date: July 3, 2013).

5.3 Biological Events in Our Water Systems Matthew J. Hoffman, Rochester Institute of Technology Kara L. Maki, Rochester Institute of Technology There is much interest in understanding, detecting, and predicting biological events in coastal ocean systems, estuaries, and lake systems. This requires a combination of numerical modeling and observation from both the ground and the air. Mathematical modeling of the biology and chemistry, both simplified and complex, is a rich and active area of research. But the predictive accuracy of these models is often degraded by the accuracy of hydrodynamic inputs such as temperature, salinity, and currents. These input fields can come from both numerical models and observations. Models that numerically solve the (incompressible, typically hydrostatic) Navier–Stokes equations have the advantage of providing all variables at any specified time and location. The downside in the ocean is the same as it is in the atmosphere: all models have errors, and those errors can lead to significant errors in the model results. Observations, either in situ or remote, sample the real system but typically only at infrequent times and over a fraction of the domain of interest. Satellite images, for example, can only observe certain surface variables; they are limited by cloud cover and are inaccurate near the coast.

86

Chapter 5. Biosphere

Some of these issues can be addressed through data assimilation—a process that has been mentioned a few times in this volume. At a simple level, data assimilation can be thought of as an interpolation of both observational data and model predictions, where each piece is weighted by its uncertainty. Where no observations exist, the model can give reasonable estimates, while observations can be used to move the model fields closer to the true state. Data assimilation is used at all of the major operational weather centers in the world to improve initial conditions for weather forecasting. It is also something we have worked on to produce more accurate flow fields for driving biological and chemical models of the Chesapeake Bay [1]. Data assimilation can incorporate satellite observations, but the satellite observations themselves warrant some consideration. Satellite temperature observations are available, but satellites do not actually observe temperature directly. Instead, radiation given off by the ocean and passed through the atmosphere is recorded (the observation is known as a radiance). These values are turned into temperatures through what is mathematically an inverse problem. Inverse modeling requires some knowledge of the physical relationship between the desired quantity (in this case temperature) and the emitted radiation. This relationship is less clear when the desired quantity is something like an abundance of algae. As an alternative to inverse modeling, statistical models can be developed by using satellite radiances as predictors for some quantity observed in situ. One example is using this to estimate salinity values in the Chesapeake Bay [2], but this type of statistical modeling can also be used to predict other quantities, such as biology, as done experimentally for sea nettles in the Chesapeake Bay [3]. What’s next? Mathematicians are working on enhancing the utility of these types of models with the long-term goal of guiding environmental and public health officials making policy decisions. References [1] M.J. Hoffman, T. Miyoshi, T. Haine, K. Ide, R. Murtugudde, and C.W. Brown, Advanced data assimilation system for the Chesapeake Bay, J. Atmos. Oceanic Technol. 29 (2012) 1542–1557, doi: 10.1175/JTECH-D-11-00126.1. [2] E.M. Urquhart, J. Hoffman, B.F. Zaitchik, S. Guikema, and E.F. Geiger, Remotely sensed estimates of surface salinity in the Chesapeake Bay, Remote Sens. Envir. 123 (2012) 522–531, doi: 10.1016/j.rse.2012.04.008. [3] Sea Nettle Probability of Encounters, National Weather Service Ocean Prediction Center, NOAA, http://www.opc.ncep.noaa.gov/Loops/SeaNettles/prob/SeaNettles.shtml

5.4 Bird Watchers and Big Data Wesley Hochachka, Cornell University You would be forgiven for not initially recognizing some of the high-level similarities between the practice of research in sciences such as physics and research in ornithology. One basic similarity is that we are all constrained in what we can measure. Quantum physics has its uncertainty principle that describes limits on what can be measured. Ornithologists are at times limited in what they can measure by the very things that they are trying to observe: birds will sometimes actively avoid detection. Additionally, we all have to deal with imperfect measuring devices and the need to create calibrations for these devices. And we all need to do “big science” to find answers to some of our questions. In the

5.4. Bird Watchers and Big Data

87

case of ornithology, various groups are building sensor networks that span countries if not entire continents. It’s just that ornithologists call their sensors “bird watchers.” One of these ornithological sensor networks was prototyped roughly sixteen years ago across the United States and Canada, called the Great Backyard Bird Count (GBBC) [1]. It served as a test platform for engaging the general public in reporting bird observations over a large geographical area, as well as the web systems needed to ingest and manage the information that was provided. While the GBBC still happens each year, engaging tens of thousands of people over a single long weekend in February, some of the GBBC’s participants keep counts and report on the birds that they see year-round, and from the GBBC a global bird-recording project emerged, named eBird [2]. eBird collects a lot of data, thousands of lists of birds each day, at a rate fast enough that you can watch the data coming into the database [3]. So, what do we do with all of those data? That’s where mathematics comes into the picture in a big way. As I already wrote, we know that the lists of birds that people report are not perfect records of the birds that were present. Some subset of the birds, and even entire species, almost certainly went undetected. We need to account for these uncertainties in what observers detect in order to get an accurate picture of where birds are living and where they’re traveling. Sometimes we have enough background information to be able to write down the statistical equations that describe the processes that affect the detection of birds and the decisions that birds make about where to live. There are other times, however, when we do not know enough to be able to write down an accurate statistical model in advance, but instead we need to discover the appropriate model as part of our analyses of the information. In these instances, our analyses fall into the realm of data mining and machine learning. Using a novel machine-learning method, we are able to describe the distributions of bird species across the United States, accurately showing where species are found throughout the entire year. The map in Figure 5.5 shows the distribution of a bird species called Orchard Oriole that winters in Central and South America [4]. In spring, most of these orioles fly across the Gulf of Mexico to reach the United States, where the migrants divide up into two distinct populations: one living in the eastern United States and a second population in the Great Plains states. Then in fall, both populations take a more westerly route back to their wintering grounds along the east coast of Mexico. Being able to accurately describe the seasonally changing distribution of these orioles and other species of birds means that our machine-learning analyses were able to use information on characteristics of the environment, such as habitat, in order to identify the preferred habitats of birds as well as how these habitat preferences change over the course of a year [5]. So, not only do these analyses tell us where birds are living, but these analyses also provide insights into the reasons why birds choose to live where they do. Reference [6] links to an animated map (created by The Cornell Lab of Ornithology). Knowing where birds live isn’t an end in itself. Being able to create an accurate map of a species’ distribution means that we understand something about that species’ habitat requirements. Additionally, knowledge of birds’ distributions, especially fine-grained descriptions of distributions, can have very practical applications. This observation was the basis for a very practical application: determining the extent to which different parties are responsible for conservation and management of different bird species. This effort, jointly undertaken by a number of governmental and nongovernmental agencies, took the continent-wide range maps throughout the year and superimposed them on information about land ownership throughout the United States. The result was the first assessment of the extent to which many bird species were living on lands that were publicly or privately owned, and within the public lands the agencies most responsible for management were

88

Chapter 5. Biosphere

Figure 5.5. Distribution of Orchard Orioles at four dates across the nesting season, as predicted by machine-learning analyses using bird watchers’ observations. Early May sees orioles moving into the eastern U.S. after crossing the Gulf of Mexico, by late May a later-arriving population has returned to the Midwest to nest, in early August the eastern orioles are moving southward, largely along the western edge of the Gulf of Mexico, and it is only in late August that Midwestern orioles follow the same route south. See [6] for an animated version. Reprinted with permission from Cornell Lab of Ornithology.

also identified. The product, the State of the Birds report for 2011, provided the first quantitative assessment of management responsibilities for a large number of species across their U.S. ranges [7]. The computational work underlying the State of the Birds report is a final point of similarity between ornithology and other big-data sciences: all of the model building is well beyond the capacity of a desktop computer. Climatology, astronomy, and biomedical research all readily come to mind as areas of research that make heavy use of highperformance computer systems (or supercomputers) in which a larger task can be broken into many smaller pieces that are each handled by one of a large number of individual processing units. The building of hundreds of year-round, continent-wide bird distribution models lends itself to this same divide-and-conquer process, because the continent-wide distribution models are built from hundreds of submodels that each describe environmental associations in a smaller region and narrow slice of time. The collection of citizen-science data in the Great Backyard Bird Count and eBird is only the start of a long process of gaining insights from these raw data. Extracting information from the data has required collaboration between ornithologists, statisticians, and computer scientists working together in the interface between biology and mathematics. As an ornithologist by training, it has been an interesting and exciting journey for me to travel. References [1] The Great Backyard Bird Count, http://www.birdsource.org/gbbc/ [2] eBird, http://ebird.org

5.5. It’s a Math-Eat-Math World

89

[3] eBird: Real Time Checklist Submissions, http://ebird.org/ebird/livesubs [4] Orchard Oriole, All About Birds, The Cornell Lab of Ornithology, http://www.allaboutbirds.org/guide/Orchard_Oriole/id [5] eBird: Occurrence Maps, http://ebird.org/content/ebird/occurrence/ [6] Orchard Oriole Animated Map, http://mpe2013.org/wp-content/uploads/2013/02/Orchard_Oriole_animated_map.gif [7] The State of the Birds, http://www.stateofthebirds.org

5.5 It’s a Math-Eat-Math World Barry Cipra, Northfield, Minnesota A book review in the January 11, 2013, issue of Science magazine begins with a wonderful line: “It is not often that mathematical theory is tested with a machine gun.” The book under review is How Species Interact: Altering the Standard View on Trophic Ecology, by Roger Arditi and Lev R. Ginzburg (Oxford University Press, 2012). In it, according to reviewer Rolf O. Peterson at the School of Forest Resources and Environmental Science at Michigan Technological University, the authors argue in favor of a theory they developed in the 1980s, that predator-prey dynamics, which is classically viewed as principally depending on prey density, is better viewed as depending on the ratio of prey to predator. “I admit to being impressed by the immediate usefulness of viewing predation through ratio-dependent glasses,” Peterson writes. The details of the debate lie within the equations, which Peterson, quoting from the classic essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner, calls a “wonderful gift.” (Peterson ruefully adds that many of his colleagues in wildlife management “have an unfortunate phobia for all things mathematical.” Trophic ecology, by the way, is rooted in the Greek word trophe, for “nourishment.” It is basically the study of food chains, which are mathematical from top to bottom.) The two “sides” of the debate may actually lie at opposite ends of a spectrum, with prey density predominating when predators are rare and the prey-to-predator ratio taking over when predators become dense enough themselves to interfere or compete with one another. The best course mathematically may be to take a page from the fundamentalists and “teach the Figure 5.6. Reprinted with permission controversy.” from Oxford University Press. As for the machine gun, you’ll want to read Peterson’s review, which, if you subscribe to Science, is available [1]—suffice it here to say, it has something to do with wolves, Alaska, and aircraft. But the story predates Sarah Palin by a couple of decades.

90

Chapter 5. Biosphere

Reference [1] R.O. Peterson, It’s a wonderful gift, Science 339 (2013) 142–143, doi: 10.1126/science.1232024.

5.6 Ocean Acidification and Phytoplankton Arvind Gupta, Mitacs The health of the world’s oceans has been in the news a lot over the last few months. Recent reports suggest that the oceans are absorbing carbon dioxide at unprecedented rates [1]. The ocean is the dominant player in the global carbon cycle [2], and the sequestering of more carbon dioxide—a major greenhouse gas—sounds like a good thing. However, researchers have measured significant increases in ocean acidity, and they worry that this will have a negative impact on marine life, especially phytoplankton [3]. That would be a big deal, as phytoplankton, which are the foundation of the ocean food chain, are vitally important to life on Earth. They capture the radiant energy from the Sun, converting carbon dioxide into organic matter, and they produce half of the Earth’s oxygen as a by-product. The amount and distribution of phytoplankton in the world’s oceans are measured in two ways. Remote sensing satellites in space detect chlorophyll pigments by quantifying how green the oceans are. Since the amount of phytoplankton is proportional to that of chlorophyll, this technique measures the amount of near-surface chlorophyll over a very large scale.

Figure 5.7. Satlantic Radiance Camera being lowered in the water. Reprinted with permission by Marlon Lewis.

A second, more accurate and mathematically intensive approach uses special cameras lowered into the ocean to measure the radiance field in the water, both at the surface and at various depths. These measurements are then used to infer the properties of the water and its constituents. The ocean’s radiance field is determined primarily by the Sun and sky

5.7. Ocean Plankton and Ordinary Differential Equations

91

and is influenced by a host of factors, including the behavior of the air-sea interface, the inherent optical properties of the water, scattering effects, etc. Determining the makeup of the water from the measured radiance distribution has proven to be a difficult inverse problem for oceanographers to solve. The first step to solving this problem is having a precise specification of the radiance field. This provides oceanographers with the means to calculate quantities that can be used to assess phytoplankton populations. This is the subject of work by researchers Marlon Lewis and Jianwei Wei of Dalhousie University in Nova Scotia, supported by the Mitacs internship program [4]. Lewis and Wei helped to develop a new camera that can be used as an oceanographic radiometer. The high-resolution device makes it possible to resolve the spherical radiance distribution at high frequency at surface and at depth. Their work established the precision and reliability of the radiance camera and has provided scientists with a new tool to monitor phytoplankton populations as well as the health of the world’s oceans. References [1] NSF Press Release, http://www.nsf.gov/news/news_summ.jsp?cntn_id=123324 [2] Carbon Cycle, NASA, http://science.nasa.gov/earth-science/oceanography/ ocean-earth-system/ocean-carbon-cycle/ [3] B. Plumer, The oceans are acidifying at the fastest rate in 300 million years. How worried should we be? The Washington Post, August 31, 2013, http://www.washingtonpost.com/blogs/wonkblog/wp/2013/08/31/the-oceans-are-acidifyingat-the-fastest-rate-in-300-million-years-how-worried-should-we-be/

[4] MITACS Accelerate, https://www.mitacs.ca/accelerate

5.7 Ocean Plankton and Ordinary Differential Equations Hans Kaper, Georgetown University As applied mathematicians we love differential equations. So, if you are looking for an interesting set of ordinary differential equations (ODEs) with relevance for planet Earth that is a bit more complicated than the predator-prey models, you might take a look at the so-called NPZ model of biogeochemistry. The N, P, and Z stand for nutrients, phytoplankton, and zooplankton, respectively (or, rather, for the nitrogen concentrations in these species), and the NPZ model describes the evolution of the plankton population in the ocean, dP dt dZ dt dN dt

= f (I ) g (N )P − h(P )Z − i(P )P, = γ h(P )Z − j (Z)Z, = − f (I ) g (N )P + (1 − γ )h(P )Z + i(P )P + j (Z)Z.

The first equation gives the rate of change of the phytoplankton; phytoplankton increases (first term) due to nutrient uptake (g (N )), which is done by photosynthesis in response to the amount of light available ( f (I ), I is the light intensity), and decreases due to zooplankton grazing (second term, h(P )) and death and predation by organisms not included in the model (third term, i(P )). The second equation gives the rate of change of the zooplankton; zooplankton increases (first term) due to grazing (h(P )), but only a fraction γ of the

92

Chapter 5. Biosphere

Figure 5.8. Reprinted with permission from the EPA.

harvest is taken up, and decreases due to death and predation by organisms not included in the model (second term, j (Z)). The third equation gives the rate of change of the nutrients; nutrients are lost due to grazing by phytoplankton (first term, f (I ) and g (N )) and increased by the left-over fraction 1−γ of harvested zooplankton (second term, h(P )) and the remains of phytoplankton (third term, i(P )) and zooplankton (fourth term, j (Z)). Note that the right-hand sides of the equations sum to zero, so the NPZ model conserves the total amount of nitrogen in the system, (N + P + Z)(t ) = (N + P + Z)(0),

t ∈ .

The most common use of NPZ models is for theoretical investigations, to see how the model behaves if different transfer functions are used. A survey of the merits and limitations of the NPZ model is given in the review article [1], with an interesting follow-up article [2] by the same author. See also the recent textbook [3]. References [1] P.J. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application, J. Oceanography 58 (2002) 379–387. [2] P.J.S. Franks, Planktonic ecosystem models: Perplexing parameterizations and a failure to fail, J. Plankton Res. 31 (2009) 1299–1306. [3] H. Kaper and H. Engler, Mathematics & Climate, OT131, SIAM, Philadelphia, 2013, Chapter 18.

5.8. Prospects for a Green Mathematics

93

5.8 Prospects for a Green Mathematics John Baez, University of California at Riverside David Tanzer, New York University It is increasingly clear that we are initiating a sequence of dramatic events across our planet. They include habitat loss, an increased rate of extinction, global warming, the melting of ice caps and permafrost, an increase in extreme weather events, gradually rising sea levels, ocean acidification, the spread of oceanic “dead zones,” a depletion of natural resources, and ensuing social strife. These events are all connected. They come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as a permanent condition. Deep changes will occur as these idealizations bring us crashing into the brick wall of reality. If we do not muster the will to act before things get significantly worse, we will need to do so later. While we may plead that it is “too difficult” or “too late,” this doesn’t matter: a transformation is inevitable. All we can do is start where we find ourselves and begin adapting to life on a finite-sized planet. Where does mathematics fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions in mathematics. Starting near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals and geometry. Centuries later, the Industrial Revolution brought us calculus, and eventually a flowering of mathematics unlike any before. Now, as the 21st century unfolds, mathematics will become increasingly driven by our need to understand the biosphere and our role within it. We refer to mathematics suitable for understanding the biosphere as green mathematics. Although it is just being born, we can already see some of its outlines. Since the biosphere is a massive network of interconnected elements, we expect network theory will play an important role in green mathematics. Network theory is a sprawling field, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology, and more. Computation plays an important role here, both because it has a network structure—think of networks of logic gates—and because it provides the means for simulating networks. One application of network theory is to tipping points, where a system abruptly passes from one regime to another [1]. Scientists need to identify nearby tipping points in the biosphere to help policy makers head off catastrophic changes. Mathematicians, in turn, are challenged to develop techniques for detecting incipient tipping points. Another application of network theory is the study of shocks and resilience. When can a network recover from a major blow to one of its subsystems? We claim that network theory is not just another name for biology, ecology, or any other existing science, because in it we can see new mathematical terrains. Here are two examples. First, consider a leaf. In The formation of a tree leaf by Qinglan Xia, we see a possible key to nature’s algorithm for the growth of leaf veins [2]. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it. The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a

94

Chapter 5. Biosphere

thicker pipe to transport materials together. Another parameter β expresses the turning cost. Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.

Figure 5.9. A growing network. Reprinted with permission from EDP sciences [2].

Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish. A second example comes from stochastic Petri nets, which are a model for networks of reactions. In a stochastic Petri net, entities are designated by “tokens” and entity types by “places” which hold the tokens [3]. “Reactions” remove tokens from their input places and deposit tokens at their output places. The reactions fire probabilistically, in a Markov chain where each reaction rate depends on the number of its input tokens. Perhaps surprisingly, many techniques from quantum field theory are transferable to stochastic Petri nets. The key is to represent stochastic states by power series. Monomials represent pure states, which have a definite number of tokens at each place. Each variable in the monomial stands for a place, and its exponent indicates the token count. In a linear combination of monomials, each coefficient represents the probability of being in the associated state. In quantum field theory, states are representable by power series with complex coefficients. The annihilation and creation of particles are cast as operators on power series. These same operators, when applied to the stochastic states of a Petri net, describe the annihilation and creation of tokens. Remarkably, the commutation relations between

5.8. Prospects for a Green Mathematics

95

Figure 5.10. A stochastic Petri net.

annihilation and creation operators, which are often viewed as a hallmark of quantum theory, make perfect sense in this classical probabilistic context. Each stochastic Petri net has a “Hamiltonian” which gives its probabilistic law of motion. It is built from the annihilation and creation operators. Using this, one can prove many theorems about reaction networks, already known to chemists, in a compact and elegant way. See the Azimuth network theory series for details [3]. Conclusion: The life of a network, and the networks of life, are brimming with mathematical content. We are pursuing these subjects in the Azimuth Project [4], an open collaboration between mathematicians, scientists, engineers, and programmers trying to help save the planet. On the Azimuth Wiki [4] and Azimuth Blog [5] we are trying to explain the main environmental and energy problems the world faces today. We are also studying plans of action, network theory, climate cycles, the programming of climate models, and more. If you would like to help, we need you and your special expertise. You can write articles, contribute information, pose questions, fill in details, write software, help with research, help with writing, and more. Just drop us a line, either here or on the Azimuth Blog. References [1] The Azimuth Project: Tipping Point, http://www.azimuthproject.org/azimuth/show/Tipping+point [2] Q. Xia, The formation of a tree leaf, ESAIM Control Optim. Calc. Var. 13 (2007) 359–377, https://www.math.ucdavis.edu/∼qlxia/Research/leaf.pdf [3] J. Baez, Network Theory, October 13, 2014, http://math.ucr.edu/home/baez/networks/ [4] The Azimuth Project: Azimuth Library, http://www.azimuthproject.org/azimuth/show/Azimuth+Library [5] Azimuth: The Official Blog of the Azimuth Project, http://johncarlosbaez.wordpress.com/about/

Chapter 6

Ecology and Evolution

6.1 Mathematics and Biological Diversity Frithjof Lutscher, University of Ottawa Picture a meadow in spring: grasses and flowers abound; different species are competing for our attention and appreciation. But these various species also compete for other things. They compete for water and essential resources to grow, for space, and for light. A similar picture emerges when we look at animals: insects, birds, mammals, fish. Usually, several species Figure 6.1. Reprinted courtesy of WorldIslandInfo. coexist in the same location, albeit not always peacefully, as we shall see. This diversity of species is beautiful, but it can be puzzling as well. Mathematical models and early experiments by G.F. Gause indicate that if two or more species compete for the same single limiting resource (for example water or nitrogen in the case of plants), then only the one species that can tolerate the lowest resource level will persist and all others will go extinct at that site. Granted, there may be more than one limiting resource. But Simon Levin extended the mathematical result: no more species can stably coexist than there are limiting factors. In less mathematical terms: if one wants a certain number of biological species to coexist, then one needs to have at least that same number of ecological opportunities or niches. In the words of Dr. Seuss (from On Beyond Zebra), And NUH is the letter I use to spell Nutches Who live in small caves, known as Nitches, for hutches. These Nutches have troubles, the biggest of which is The fact there are many more Nutches than Nitches. 97

98

Chapter 6. Ecology and Evolution

Each Nutch in a Nitch knows that some other Nutch Would like to move into his Nitch very much. So each Nutch in a Nitch has to watch that small Nitch Or Nutches who haven’t got Nitches will snitch. These insights were powerful incentives to develop new mathematical models and devise new ecological experiments to explain why there is so much biological diversity almost everywhere we look. Most of the answers come in the form of some kind of “trade-off.” There is the spatial trade-off. Some species are better at exploiting a resource; some are better at moving toward new opportunities. After a patch of land opens up in the Canadian foothills—maybe from fire or logging—aspen trees move in quickly and establish little stands there. Given enough time, the slower moving spruce will come and replace the aspen, because it can utilize the resources more efficiently. Another trade-off is temporal when the resource fluctuates. For example, water is fairly plentiful in the Canadian prairies in the spring, but later in the summer and the fall there can be serious droughts. Some plants grow very fast early in the year when water is abundant but are out-competed later in the year by others who can tolerate low water levels. When grazing or predatory species enter the picture, new opportunities for biodiversity open up. The trade-off now lies in being good at competing for resources or being good at fending off predators. Sometimes, predation can induce cycles in an ecological system. A particularly well known cyclic system are the snowshoe hares and lynx in Western Canada. Evidence for cycles can already be found in the trading books of the Hudson’s Bay company. When the internal dynamics of predation induce cycling, the temporal trade-off can give additional opportunities for coexistence of competing species. Recently, researchers in ecology and mathematics have recognized that not all competitive relationships between species are truly competitive. Sometimes they are mutualistic. For example, owls and hawks hunt similar rodent species, but one hunts at night and the other during the day. Combined they don’t give their prey a safe time to forage. In some cases, researchers have found that the predation rate of two competing predators combined is higher than the sum of the predation rates in isolation. Mathematical models then demonstrate that even moderate amounts of mutualistic behaviors between competing species can lead to stable coexistence—one more mechanism to promote the coexistence of a large diversity of biological populations.

6.2 Why We Need Each Other to Succeed Christiane Rousseau, University of Montreal It is a great scientific problem to understand why biodiversity has become so large on Earth. Mutations allow for the appearance of new species. Selection is an important principle of evolution. But selection works against biodiversity. The biodiversity we observe on Earth is too large to be explained only through mutations and selection. Another force is needed. Martin Nowak identified this other force in 2003: cooperation! Nowak gave a public lecture dealing with this theme at CRM on November 6, 2013; the title of his lecture was “The evolution of cooperation: Why we need each other to succeed.” Nowak explained how cooperation is widespread in the living world. Bacteria cooperate for the survival of the species. Eusociality describes the very sophisticated behavior of social insects like ants and bees, where each individual works for the good of the community. Human society is organized around cooperation, from the good Samaritan to

6.2. Why We Need Each Other to Succeed

99

Figure 6.2. Reprinted courtesy of Tenan.

the Japanese worker who accepts to work on the cleaning of Fukushima’s nuclear plant: “There are only some of us who can do this job. I’m single and young, and I feel it’s my duty to help settle this problem.” Cells in an organism cooperate and replicate only when it is timely, and when cells stop cooperating cancer occurs. Nowak gave a mathematical definition of cooperation. A donor pays a cost, c, for a recipient to get a benefit, b , greater than c. This brings us to the prisoner’s dilemma. Each prisoner has the choice of cooperating and paying the cost c, or defecting. From each prisoner’s point of view, whatever the other does, his better choice is to defect. This means that the rational player will choose to defect. But the other prisoner will make the same reasoning. Then they will both defect . . . and get nothing. They could each have got b − c if they had behaved irrationally and had chosen to cooperate. Natural selection chooses defection, and help is needed to favor cooperators over defectors. In evolution, we can identify five mechanisms for cooperation: direct reciprocity, indirect reciprocity, spatial selection, group selection, and kin selection. Direct reciprocity (I help you, you help me). Tit-for-tat is a corresponding good strategy for repeated rounds of the prisoner’s dilemma: I start with cooperation; if you cooperate, I will cooperate; if you defect, I will defect. This strategy leads to communities of cooperators, but it is unforgiving when there is an error. This leads to a search for better strategies: the generous tit-for-tat strategy incorporates the following difference: If you defect, I will defect with probability q = 1 − b /c. This leads to an evolution of forgiveness. Another good strategy is win-stay, lose-shift. Novak explained for each of these strategies under which mathematical conditions it performs well and leads to a community of cooperators. Indirect reciprocity (I help you, somebody helps me). The experimental confirmation is that, by helping others, one builds one’s reputation: people help those who help others, and helpful people have a higher payoff at the end. Games of indirect reciprocity lead to the evolution of social intelligence. Since individuals need to be able to talk to each other, there must be some form of communication in the population. Spatial selection. Each individual interacts with his or her neighbors. Cooperators incur a cost if neighbors receive a benefit. The strategy favors cooperation if b /c > k, where k is the average number of neighbors. The mechanism is studied through spatial

100

Chapter 6. Ecology and Evolution

games, games on graphs (the graph describing a social network), and evolutionary set theory. Group selection. “There can be no doubt that a tribe including many members who [. . .] are always ready to give aid to each other and to sacrifice themselves for the common good, would be victorious over other tribes; and this would be natural selection.” (Charles Darwin, The Descent of Man, 1871). In group selection, you play the game with others in your group, offspring are added to the group, groups divide when reaching a certain size, and groups die. The strategy favors cooperation if b /c > 1 + n/m, where n is the group size and m the number of groups. Kin selection occurs among genetically related individuals. The strategy is related with Hamilton’s rule. Nowak explained the scientific controversy around Hamilton’s rule and inclusive fitness, which is a limited and problematic concept. Direct and indirect reciprocity are essential for understanding the evolution of any pro-social behavior in humans. Citing Novak: “But what made us human is indirect reciprocity, because it selected for social intelligence and human language.” Novak ended his beautiful lecture with an image of the Earth and the following sentence: “We must learn global cooperation . . . and cooperation for future generations.” This started a passionate period of questions, first in the lecture room and then around a glass of wine during the vin d’honneur.

6.3 The Unreasonable Effectiveness of Collective Behavior Pietro-Luciano Buono, University of Ontario Institute of Technology

Figure 6.3. Reprinted courtesy of D. Dibenski.

Observing collective phenomena such as the movement of a flock of birds, a school of fish, or a migrating population of ungulates is a source of fascination because of the mystery behind the spontaneous formation of the aggregating behavior and the apparent cohesiveness of the movements. However, collective phenomena can also be the cause of major environmental and social problems. Think, for example, of a swarm of voracious locusts ravaging crops and putting many communities under severe stress. Swarming is a collective phenomenon of particular interest: a large group of animals seems to form spontaneously and moves in an apparently coordinated fashion without an obvious “director.” Because of its ubiquity in the biological world, swarming is an intrinsically interesting topic of research. But it is also critical for the design of strategies, for example to prevent the aggregation of locusts before it happens, or to perturb or possibly stop its progression and the destruction of crops once it has started. On the lighter side,

6.3. The Unreasonable Effectiveness of Collective Behavior

101

a dance company used the scientific study of collective movement in an original initiative to bridge the gap between scientific and artistic languages. Collective movement of animals can be modeled in a Lagrangian or an Eulerian mode. The Lagrangian mode is an “individual-based” modeling strategy, where the movements of individuals are simulated under simple rules for their mutual interactions. The emerging aggregations and collective motions are observed directly [1, 2]. The interaction rules can be defined in terms of distances between group members or topologically in terms of a collection of neighbors, independently of distance [3]. In the Eulerian mode, one follows the evolution of animal densities using equations similar to those used in fluid dynamics or statistical mechanics. One makes hypotheses about the social interactions between individuals in a group in terms of repulsion, attraction, and alignment. Repulsion is strongest in a zone close to an individual, while attraction is strongest in a zone furthest from the subject. These two zones can overlap, and the alignment zone sits somewhere between the two zones. In the case of a herd of prey, individuals want to keep a comfortable distance between each other and repel the ones that are too close; on the other hand, an individual too far from the group is vulnerable to predators and will seek to get closer to the group. These social interactions are often nonlocal and can be modeled using integration of so-called interaction kernels. A typical example of an interaction kernel is a Gaussian function with its peak at the distance where the interaction is strongest. One can also define modes of communication corresponding to visual, auditive, or tactile interactions and depending on whether individuals in a group move towards or away from each other [4]. These modes of communication encode topological properties of the animal groups. In [5], Topaz et al. use an Eulerian approach to model locust aggregation. They show that the solitary state of locusts becomes unstable as the population density reaches a critical value. They also show the occurrence of hysteresis in the gregarious mass fraction of locusts, which means that to dissolve a locust swarm, population densities have to be reduced to a level well below the critical density at which the swarm forms. This result suggests that control strategies that prevent the formation of swarms by limiting the population density are probably more successful than control strategies that try to reduce the population density once the swarm has been formed. Theoretical and simulation studies with the above modeling methods help to demystify the mechanisms of collective motion. They produce new hypotheses for empirical studies and are also important in understanding and controlling insect invasions. Moreover, they have led to the development of new mathematics and improvement of numerical simulation algorithms. A better understanding of these collective phenomena does not remove any of the magic of the gracious ballet of a flock of birds; in fact, it adds to their enchantment. References [1] V. Grimm and S.F. Railsback, Individual-based Modeling and Ecology, Princeton University Press, 2005. [2] Ian Couzin’s lab, http://icouzin.princeton.edu/ [3] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA 105 (2008) 1232–1237. [4] R. Eftimie, G. de Vries, M.A. Lewis, and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol. 69 (2007) 1537–1565.

102

Chapter 6. Ecology and Evolution

[5] C.M. Topaz, M.R. D’Orsogna, L. Edelstein-Keshet, and A.J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PloS Comput. Biol. 8 (2012) e1002642.

6.4 From Individual-Based Models to Continuum Models Hans Kaper, Georgetown University At the recent workshop on “Sustainability and Complex Systems” at the Mathematical Biosciences Institute (MBI) at Ohio State University [1] the question came up of how to apply dimension-reduction techniques to derive continuum models from individualor agent-based models (IBMs). IBM seems to be the model of choice in mathematical ecology, and the point was made that continuum models might give a more system-based description. The discussion reminded me of my earlier interest in gas dynamics, where a similar issue arises: how to derive continuum models like the Navier–Stokes equations from the equations of motion of the individual molecules that make up the gas. This issue has a long history in kinetic theory, going back to Ludwig Boltzmann (1844–1906). While the initial discussions were mostly heuristic, mathematical research in the latter part of the 20th century has provided a rigorous framework for the various approximations, so today the theory is on a more or less solid foundation [2, 3, 4]. A recent paper by Gorban and Karlin [5] revisits the problem in the broader context of Hilbert’s Sixth Problem. In kinetic theory, a gas is thought of as a collection of mutually interacting molecules, possibly moving under the influence of external forces. We assume for simplicity that the molecules are all of the same kind (a “simple gas”), that they have unit mass, and that there are no external forces acting on the molecules. Each molecule moves in physical space; at any instant t , its state is described by its position vector x and its velocity vector v. Molecules interact, they may attract or repel each other, and as they interact their velocities change. The interactions are assumed to be local and instantaneous and derived from some potential (for example, the Lennard-Jones potential). If the interaction is elastic, then mass, momentum, and kinetic energy are preserved, so the velocities of two interacting molecules are determined uniquely in terms of their velocities before the interaction. At the microscopic level, the state of a gas comprising N molecules is described by an N -particle distribution function fN with values fN (x1 , . . . , xN , v1 , . . . , vN , t ). This function evolves in a 6N -dimensional space according to the Liouville equation, ∂ fN ∂t

+

N i =1

(∇xi fN ) · vi +

N i =1

(∇vi fN ) · ai = 0,

ai = −

N j =1

∇xi Φij ,

where Φij is the potential at xi due to a molecule at x j . Note that the Liouville equation is linear in fN . The Liouville equation is transformed into a chain of N differential equations by taking s = 1, 2, . . . , N and, for each s, averaging the equation over N − s molecules,

s s s ∂ fs + (∇xi f s ) · vi + (∇vi f s ) · ai = − ∇vi (∇xi Φi ,s +1 ) f s +1 d x s +1 d v s +1 . ∂t i =1 i =1 i =1 This is the so-called BBGKY hierarchy (named after its developers, Bogoliubov, Born, Green, Kirkwood, and Yvon) [6], which is a description of a gas at the microscopic level. The first equation of the chain (s = 1) connects the evolution of the one-particle distribution function f1 with the two-particle distribution function f2 , the second equation

6.4. From Individual-Based Models to Continuum Models

103

(s = 2) connects the two-particle distribution function f2 with the three-particle distribution function f3 , etc. The equations in the hierarchy define a linear operator in the space of chains of length N of density functions. From the BBGKY hierarchy one obtains a description at the mesoscopic level by taking the equation for the one-particle distribution function and employing a closure relation to express the two-particle distribution function as the product of two one-particle distribution functions. This is the (in)famous “Stosszahlansatz,” which leads to the Boltzmann equation—an integrodifferential equation for the one-particle distribution function f1 (which we denote henceforth simply by f , with values f (x, v, t )) with a quadratic nonlinearity on the right-hand side, ∂f ∂t

+ (∇x f ) · v + (∇v f ) · a = Q( f ),

where Q is the collision operator. The step from the BBGKY hierarchy to the Boltzmann equation introduces not only a nonlinearity but also introduces irreversibility: the Boltzmann equation is time-irreversible (Boltzmann’s H -theorem) [7].  The macroscopic variables of the gas are the mass density ρ = f d v, the hydrody  namic velocity u = v f d v, and the temperature T = (v − u)2 f d v (which is a measure of the internal energy). Because mass, momentum, and internal energy are preserved in a molecular interaction, we have the identities

Q( f ) d v = v Q( f ) d v = (v − u)2 Q( f ) d v = 0. Multiplying both sides of the Boltzmann equation with 1, v, and (v − u)2 and integrating over all molecular velocities, we obtain the continuity equation, ∂ρ ∂t

+ ∇ · (ρu) = 0,

and the equation of motion, ∂ (ρu) ∂t

+ ∇ · (ρuu) − ∇ ·  = 0,

where  is the Cauchy stress tensor. If the gas is in hydrostatic equilibrium, the Cauchy stress tensor is diagonal, the shear stresses are zero, and the normal stresses are all equal. The hydrostatic pressure p is the negative of the normal stresses, so  = − p . The equation of motion reduces to ∂ (ρu) ∂t

+ ∇ · (ρuu) + ∇ p = 0.

This is the Navier–Stokes equation of fluid dynamics, which describes the evolution of the gas at the macroscopic level. The procedure outlined above to derive the Navier–Stokes equation from the Boltzmann equation is known as the Chapman–Enskog procedure. It is essentially an asymptotic analysis based on a two-time scale singular perturbation expansion, where the macroscopic variables evolve on the slow time scale and the one-particle distribution function on the fast time scale. Thus, there exists a very systematic procedure to get from the microscopic level (the Liouville equation and the BBGKY hierarchy) to the mesoscopic level (the Boltzmann

104

Chapter 6. Ecology and Evolution

equation) and from there to the macroscopic level (the Navier–Stokes equation). Given that the IBMs are the analogue in ecology of the Liouville equation in gas dynamics, I suspect that there is a similar procedure to reduce the IBMs to more manageable equations at the macroscopic level. Food for thought. References [1] H. Kaper, MBI Workshop “Sustainability and Complex Systems”, Mathematics of Planet Earth Blog, October 7, 2013, http://mpe2013.org/2013/10/07/mbi-workshop-sustainability-and-complex-systems/ [2] J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, 1954. [3] J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North–Holland, Amsterdam, 1972. [4] S. Chapman, T.G. Cowling, and C. Cercignani, The Mathematical Theory of Nonuniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge Mathematical Library, 1991. [5] A.N. Gorban and I. Karlin, Hilbert’s 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Am. Math. Soc. 51 (2014) 187–246. [6] BBGKY hierarchy, Wikipedia, http://en.wikipedia.org/wiki/BBGKY_hierarchy [7] H-theorem, Wikipedia, http://en.wikipedia.org/wiki/H-theorem

6.5 Optimal Control and Marine Protected Areas Kent E. Morrison, American Institute of Mathematics

Figure 6.4. Reprinted courtesy of Stoon | FreeDigitalPhotos.net.

There are two standard ways to restrict harvesting of fish in order to maintain or improve the population. One way is to establish marine protected areas (MPAs), where fishing is prohibited; the other is to allow fishing everywhere but at something less than maximal capacity. I recently noticed an interesting preprint in arXiv [1] by Patrick De Leenheer, who is an applied mathematician at the University of Florida. It sets up a mathematical framework for deciding whether to use protected areas and, if so, where they should be established.

6.6. Linear Programming for Tree Harvesting

105

By formulating the problem as a problem of optimal control theory and allowing a harvesting rate that varies from point to point along the one-dimensional coastline, the author creates a much broader spectrum of possible harvesting strategies. An interval over which the harvesting rate is zero corresponds to a protected area, so there could conceivably be several protected areas separated by regions that allow some harvesting. De Leenheer proposes maximizing a weighted sum of the total yield and the average fish density: The motivation for choosing this measure is that it incorporates two of the main measures that have been used in the past, namely yield and density, coupled with the fact mentioned earlier, that MPA’s are believed to have opposite effects on these measures. To me it is not intuitively clear what to expect for the optimal strategy, but De Leenheer’s analysis shows that there are three possible optimal strategies. Just which one occurs depends on two parameters: (1) the weight given to the average density in the objective function, and (2) the length of the coastline. When the weight of the average density is below a threshold, then it is optimal to allow fishing at maximal capacity everywhere. But when the weight exceeds the threshold value, then the length of the coastline comes into play. Below a certain critical value for the length parameter it remains optimal to allow fishing everywhere, but above the critical length it is optimal to install a single marine reserve in the middle of the coast line. Reference [1] P. De Leenheer, Optimal Placement of Marine Protected Areas, http://front.math.ucdavis.edu/1307.1581

6.6 Linear Programming for Tree Harvesting Peter Lynch, University College Dublin A Berkeley graduate student, George Dantzig, was late for class. He scribbled down two problems from the blackboard and handed in solutions a few days later. But the problems on the board were not homework assignments; they were two famous unsolved problems in statistics. The solutions earned Dantzig his PhD. With his doctorate in his pocket, Dantzig went to work with the U.S. Air Force, designing schedules for training, stock distribution, and troop deployment—activities known as programming. He was so efficient that, after the second World War, he was given a well-paid job at the Pentagon, with the task of mechanizing the program planning of the military. There he devised a dramatically successful technique, or algorithm, which he named linear programming (LP). LP is a method for decision making in a broad range of economic areas. Industrial activities are frequently limited by constraints. For example, there are normally constraints on raw materials and on the number of staff available. Dantzig assumed these constraints to be linear, with the variables, or unknown quantities, occurring in a simple form. This makes sense: if it requires four tons of raw material to make 1,000 widgets, then eight tons are needed to make 2,000 widgets. Double the output requires double the resources. LP finds the maximum value of a quantity, such as output volume or total profit, subject to the constraints. This quantity, called the objective, is also linear in the variables. A real-life problem may have hundreds of thousands of variables and constraints, so a systematic method is needed to find an optimal solution. Dantzig devised a method ideally suited to LP, called the simplex method.

106

Chapter 6. Ecology and Evolution

At a conference in Wisconsin in 1948, when Dantzig presented his algorithm, a senior academic objected, saying: “But we all know the world is nonlinear.” Dantzig was nonplussed by this put-down, but an audience member rose to his defense, saying: “The speaker titled his talk ‘Linear Programming’ and carefully stated his axioms. If you have an application that satisfies the axioms, then use it. If it does not, then don’t.” This respondent was none other than John von Neumann, the leading applied mathematician of the 20th century. LP is used in a number of Irish industries. One interesting application, used by Coillte, is harvest scheduling. This enables decisions to be made about when and where to cut trees in order to maximize the long-term financial benefits. A more advanced system, which incorporates environmental and social constraints in addition to economic factors, is being developed by Coillte and UCD Forestry.

Figure 6.5. Coillte uses linear programming to make decisions about when and where to cut trees to maximize long-term benefits. Reprinted with permission from The Irish Times.

The acid test of an algorithm is its capacity to solve the problems for which it was designed. LP is an amazing way of combining a large number of simple rules and obtaining an optimal result. It is used in manufacturing, mining, airline scheduling, power generation, and food production, maximizing efficiency and saving enormous amounts of natural resources every day. It is one of the great success stories of applied mathematics. This article was originally published on October 8, 2013 in The Irish Times.

Chapter 7

Communication and Representation

7.1 The Challenge of Cartography Christiane Rousseau, University of Montreal The shape of the Earth can be approximated by a sphere. If we want to be more precise, we use a geoid, which is an ellipsoid of revolution, since the radius of the Earth is larger at the equator than at the poles. It was Carl Friedrich Gauss who first proved that there exists no isometry between a surface of nonzero curvature and a surface of zero curvature like a plane, a cone, or a cylinder (Theorema Egregium). All map projections distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections. A fundamental projection classification is based on the type of projection surface onto which the globe is conceptually projected. The surface can be cylindrical (e.g., Mercator), conic (e.g., Albers), or azimuthal or plane (e.g., stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Another way to classify projections is according to properties of the model they preserve. Some of the more common categories preserve directions (azimuthal), local shapes (conformal or orthomorphic), area (equal-area or equivalent), or distance along certain curves (equidistant). Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal. The horizontal projection of a sphere onto a cylinder tangent to the sphere at the equator is equivalent; this was already known to Archimedes of Syracuse (c. 287 BC– c. 212 BC). The stereographic projection of the sphere from one pole to a plane tangent to the other pole is conformal; this was already known to Hipparchus of Nicaea (c. 190 BC– c. 120 BC). Composing with biholomorphic transformations on the plane yields other conformal projections. When the biholomorphic transformation is the function log z, one obtains the Mercator projection on a cylinder. Other natural functions are z a , where a is real and positive; they yield conformal projections on a cone. The geoid is a Riemann surface, and there are conformal transformations from it to the plane, but the formulas are more involved. Most of what we have presented so far is several centuries old. The mathematician John Milnor, 1962 Fields medalist, got interested in determining which projection minimizes the distortion of distances in a neighborhood of a given point. But what is meant 109

110

Chapter 7. Communication and Representation

Figure 7.1. An azimuthal equidistant projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere. Reprinted courtesy of USGS.

by distortion? Consider any two points P and Q in the neighborhood, and let P and Q be their images on the map. Let A and B be the sharp lower and upper bounds, respectively, of the ratio dist(P, Q)/dist(P , Q ). Then the distortion is defined as the logarithm of the ratio B/A. Milnor showed that, for any given region, there always exists a projection that minimizes the distortion. Furthermore, for the particular case where the region is bounded by a circle on the sphere, this optimal projection is the azimuthal equidistant projection.

7.2 What Does Altitude Mean? Christiane Rousseau, University of Montreal If we model the Earth as a sphere of radius R, then the altitude of a point is its distance to the center of the Earth minus R. But we know that the surface of the Earth is not exactly a sphere and is, in fact, better approximated by an ellipsoid. Again, it is possible to generalize the definition of altitude for an ellipsoid of revolution. For a given point A, we consider the segment joining it to the center O of the ellipsoid. The half-line OA cuts the ellipsoid in a point B, and the altitude is the difference between the length of OA and that of OB. So far, no problem. However, the Earth is not a perfect ellipsoid of revolution. Then what is the center of the Earth, and what is meant by altitude? When we represent the Earth by a solid sphere or ellipsoid, we implicitly assume that we have a surface that approximately fits the surface of the Earth’s oceans. (Of course, the surface of the oceans varies with the tides, and we must consider the mean surface of the Earth’s oceans.) This surface is called a geoid. We then add the topographical details to the geoid. The point of view taken in geodesy is to consider the gravitational field generated by the Earth. On the surface of the Earth, the gravitational field is directed toward the Earth’s interior, and the center of gravity of the Earth, which is a natural candidate for the Earth’s

7.3. Drawing Conformal Maps of the Earth

111

Figure 7.2. Geoids. Reprinted courtesy of NASA.

center, is a singular point of this gravitational field. The gravitational field comes from a potential, and it is natural to consider the level surfaces of this potential. Thus, a geoid will be an equipotential of the gravitational field, chosen to give the best fit of the surface of the oceans with the geoid corresponding to the mean sea level (MSL). The differences between the geoid and an ellipsoid come not only from the presence of mountains but also from the density variations inside the Earth. The geoid is then taken as the surface of altitude zero, and the altitude of a point A is defined as its distance to the geoid measured along the normal through A to the geoid. This normal is easily determined in practice, since it corresponds to the vertical as indicated by a carpenter’s level or a surveyor’s plumb bob. Let O be the center of the geoid and B the intersection point of the half-line OA with the geoid. In general, the altitude of A is not exactly equal to the difference between the length of OA and that of OB because the normal to the geoid through A may not pass through O. The difference between the geoid and an ellipsoid of revolution approximating the Earth can be up to 100 meters; hence, it is quite significant. The first GPS would calculate the altitude as the distance to an ellipsoidal model of the Earth. Modern receivers are now able to correct this measurement and give the real altitude over the geoid.

7.3 Drawing Conformal Maps of the Earth Christiane Rousseau, University of Montreal This contribution can be seen as a follow-up to Section 7.2, where I discussed the Earth as a deformed sphere that geodesists choose to approximate by the geoid. The geoid is the level surface of the gravitational field corresponding to the mean sea level (MSL). It has been known since Gauss that it is not possible to draw maps of the Earth that preserve ratios of distances. But it is possible to find projections of the sphere that preserve ratios of areas: these projections are called equivalent. A typical example is the horizontal projection of Lambert, which was in fact already known from Archimedes. It is also possible to find projections of the sphere that preserve angles: these projections are called conformal. One of them is the stereographic projection, which was already known to

112

Chapter 7. Communication and Representation

Figure 7.3.

the Greek Hipparchus of Nicaea (190 BC–120 BC). A second one is the Mercator projection. It is remarkable that Hipparchus and Mercator (1512–1594) could prove that their respective projections are conformal without the use of differential calculus. But is it possible to draw conformal maps of the geoid? The answer is positive, but the proof is more subtle. For instance, we find elements of a proof in the book Differential Geometry of Curves and Surfaces by Do Carmo [1], but he refers to Riemann Surfaces by Lipman Bers for a full proof [2]. The proof amounts to showing that a regular differentiable surface can be given a conformal structure, i.e., is a Riemann surface, and the conformal structure is obtained by solving a Beltrami equation. Let us discuss the particular case where the geoid is rotationally symmetric around the Earth’s axis, by generalizing Mercator’s strategy. We consider two angles—the longitude L and the latitude —and make the hypothesis that any half-line from the center of the Earth along the direction corresponding to longitude L and latitude  cuts the Earth in a point at a distance R() from the center of the Earth. The intersection curves of the geoid with the half-planes where L is constant are called the meridians of the geoid, and we wish them to be represented by vertical lines on the map. Similarly, the intersection curves of the geoid with the cones where  is constant are called the parallels of the geoid, and we represent them on the map by horizontal segments of length 2π, parameterized by L. Let us consider a small region corresponding to a width d L and a height d , and with a corner at (L, ) on the geoid. On the geoid, the length of this small region in the direction

of the meridians is approximately R2 + (R )2 d , and its width (in the direction of the parallels) is approximately R cos  d L. Hence, the diagonal makes an angle θ with the parallels such that  tan θ 

R2 + (R )2 d  R cos 

dL

.

7.4. Changing Our Clocks

113

Now we must compute the projection on the map of a point of longitude L and latitude . Its coordinates will be (L, F ()). On the map, the small region is represented by a rectangle of width d L and height F () d . Hence, the diagonal makes an angle θ with the horizontal direction, such that tan θ = F ()

d dL

.

The mapping is conformal if θ = θ, which means that  R2 + (R )2 , F () = R cos  from which F can be obtained by integration. Note that we could have used the same technique to find an equivalent mapping. If we let r () = R()/R(0), then the projection preserves ratios of areas if  F () = r cos  r 2 + (r )2 , from which F can again be obtained by integration. References [1] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Pearson, 1976. [2] L. Bers, Riemann Surfaces, New York University, 1957.

7.4 Changing Our Clocks Hans Kaper, Georgetown University On the first Sunday of November, most of the United States and Canada changes from Daylight Saving Time (DST) to Standard Time: at 2:00 a.m. local time, clocks fall back to 1:00 a.m. This event is the reverse of what happens in the spring: on the second Sunday in March at 2:00 a.m., clocks spring forward to 3:00 a.m. Effectively, DST moves an hour of daylight from the morning to the evening. The modern idea of daylight saving was first proposed by the New Figure 7.4. Reprinted with Zealand entomologist George Vernon Hudson in a paper presented to the Wellington Philosophical permission from Thinkstock. Society in 1895 [1]. It was first implemented on April 30, 1916, by Germany and its war-time ally Austria–Hungary as a way to conserve coal. This annual ritual does not happen everywhere and does not happen everywhere at the same time. In the U.S. and Canada, each time zone switches at a different time. DST is not observed in Hawaii, American Samoa, Guam, Puerto Rico, the Virgin Islands, the Commonwealth of Northern Mariana Islands, and Arizona. The Navajo Nation participates in the DST policy, even in Arizona, due to its large size and location in three states. However, the Hopi Reservation, which is entirely surrounded by the Navajo Nation, doesn’t observe DST. In effect, there is a donut-shaped area of Arizona that does observe DST, but the “hole” in the center does not.

114

Chapter 7. Communication and Representation

The timing of the changeover, 2:00 a.m., was chosen because it was practical and minimized disruption. Most people are at home at that hour, and shifts usually change at midnight. It is the time when the fewest trains are running. It is late enough to minimally affect bars and restaurants, and it prevents the day from switching to yesterday, which would be confusing. It is early enough that the entire continental U.S. switches by daybreak, and the changeover occurs before most early shift workers and early churchgoers are affected. In the U.S., the dates of the changeover were set in 2007. Widespread confusion was created during the 1950s and ’60s when each U.S. locality could start and end DST as it desired. One year, 23 different pairs of DST start and end dates were used in Iowa alone. For exactly five weeks each year, Boston, New York, and Philadelphia were not on the same time as Washington D.C., Cleveland, or Baltimore—but Chicago was. And, on one Ohio to West Virginia bus route, passengers had to change their watches seven times in 35 miles! The Minnesota cities of Minneapolis and St. Paul once didn’t have twin perspectives with regard to the clock. These two large cities are adjacent at some points and separated only by the Mississippi River at others, and are considered a single metropolitan area. In 1965, St. Paul decided to begin its DST period early to conform to most of the nation, while Minneapolis felt it should follow Minnesota’s state law, which stipulated a later start date. After intense intercity negotiations and quarreling, the cities could not agree, and so the one-hour time difference went into effect, bringing a period of great time turmoil to the cities and surrounding areas. Indiana has long been a hotbed of DST controversy. Historically, the state’s two western corners, which fall in the Central Time zone, observed DST, while the remainder of the state, in the Eastern Time zone, followed year-round Standard Time. An additional complication was that five southeastern counties near Cincinnati and Louisville unofficially observed DST to keep in sync with those cities. Because of the longstanding feuds over DST, Indiana politicians often treated the subject gingerly. In 1996, gubernatorial candidate Rex Early firmly declared, “Some of my friends are for putting all of Indiana on Daylight Saving Time. Some are against it. And I always try to support my friends.” In April 2005, Indiana legislators passed a law that implemented DST statewide beginning on April 2, 2006. The North American system is not universal. The countries of the European Union use Summer Time, which begins the last Sunday in March (one or two weeks later than in North America) and ends the last Sunday in October (one week earlier than in North America). All time zones change at the same moment, at 1:00 a.m. Universal Time (UT, the successor of Greenwich Mean Time). The only African countries and regions which use DST are the Canary Islands, Ceuta and Melilla (Spain), Madeira (Portugal), Morocco, Libya, and Namibia. In Antarctica, there is no daylight in the winter and months of 24-hour daylight in the summer. But many of the research stations there still observe DST anyway, to synchronize with their supply stations in Chile or New Zealand. Proponents of DST generally argue that it saves energy, while opponents argue that actual energy savings are inconclusive. DST’s potential to save energy comes primarily from its effects on residential lighting, which consumes about 3.5% of electricity in the United States and Canada [2]. Delaying the nominal time of sunset and sunrise reduces the use of artificial light in the evening and increases it in the morning. As Franklin’s 1784 satire pointed out, lighting costs are reduced if the evening reduction outweighs the morning increase, as in high-latitude summer when most people wake up well after sunrise. An early goal of DST was to reduce evening usage of incandescent lighting, formerly

7.5. High-Resolution Satellite Imaging

115

a primary use of electricity. Although energy conservation remains an important goal, energy usage patterns have greatly changed since then, and recent research is limited and reports contradictory results. Electricity use is greatly affected by geography, climate, and economics, making it hard to generalize from single studies [2]. References [1] G.V. Hudson, On seasonal time-adjustment in countries south of lat. 30o . Trans. Proc. New Zealand Inst. 28 (1895) 734. [2] M.B.C. Aries and G.R. Newsham, Effect of daylight saving time on lighting energy use: A literature review, Energy Policy 36 (2008) 1858–1866, doi: 10.1016/j.enpol.2007.05.021.

7.5 High-Resolution Satellite Imaging Paula Craciun, INRIA Josiane Zerubia, INRIA Do you know that over 50 satellites are launched every year to orbit the Earth? Have you ever wondered what the purpose of these satellites is? Here is one of them. With the launch of the first satellite, a new way of gathering information about the Earth’s surface emerged. Highly sophisticated cameras are built on the satellites to obtain very high resolution images. Satellites nowadays provide images at a resolution of 0.3 meters, which means that you can even identify your own scooter! Huge amounts of data are collected every day using these cameras. Still, all this data is meaningless, unless the images are further analyzed and understood. A first step in understanding what is represented in an image is to identify the objects which it contains. We will focus here on identifying boats in a harbor. Boat extraction in harbors is a preliminary step in obtaining more complex information from images such as traffic flow within the harbor, unusual events, etc. When you look at a satellite image of a harbor, you can visually detect the boats based on their characteristics, such as the fact that they are usually in water, their white color, or their elliptical shape. All these characteristics make it easy for us humans to correctly identify the boats and discriminate them from other objects such as cars, buildings, or trees. Humans know the concept of a boat, but computers don’t. Tell a computer to identify a boat, and it won’t know what you’re talking about.

Figure 7.5. Reprinted with permission from CNES/INRIA.

116

Chapter 7. Communication and Representation

In order to use a computer to detect boats, one must first identify all the characteristics that make a boat unique. Some of them were mentioned before; can you think of others? Once you write down a list of all such characteristics, you then have to define them in a mathematical manner. Put all these mathematical characteristics together and you have developed a mathematical model for boats in harbors. Keep in mind that you must model the boat itself, as well as the relationships between the boats. One example of a relationship between two boats is the fact that they are usually not allowed to overlap. Note that if the final result is not satisfying, it probably means that the model is poor, and you should try to improve it. The last step is to integrate this model into a framework that allows you to extract only those objects that fit the model and neglect all others. Probabilities play an important role in this step. The computer will search for a configuration of objects until it finds the one that best describes the real data in the image. In the best case scenario you’ll end up with a configuration that incorporates all the boats in the harbor. At that point, you can move on to do more interesting stuff with this information.

7.6 Microlocal Analysis and Imaging Gaik Ambartsoumian, The University of Texas at Arlington Raluca Felea, Rochester Institute of Technology Venky Krishnan, TIFR Centre for Applicable Mathematics Cliff Nolan, University of Limerick Todd Quinto, Tufts University Modern society is increasingly dependent on imaging technologies. Medical imaging has become a vital part of healthcare, with X-ray tomography, MRI, and ultrasound being used daily for diagnostics and monitoring the treatment of various diseases. Meteorological radar is essential for weather prediction, sonar scanners produce sea-floor maps, and seismometers aid in geophysical exploration. In all these techniques, the imaged medium is probed by certain physical signals (X-rays, electromagnetic or sound waves, etc.) and the response recorded by a set of receivers. For example, in computerized tomography (CT), X-rays are sent at various angles through the human body and the intensity of outgoing rays is measured. In ultrasound tomography, sound waves are sent through the body and the transducers located on the surface of the body collect the resulting echoes. Imaging modalities differ in the physical nature of input and output signals, their interaction with the medium, as well as geometric setups of data acquisition. As a result, the mathematical description of the underlying processes and collected data are different, too. However, many of them fall into a common mathematical framework based on integral geometry and the wave equation. In particular, one can model scattered waves (the recorded data) as integrals along certain trajectories of a function that describes physical or biological properties of the medium. To create an image of the medium, one would like to recover the latter function from the data, i.e., to invert the integral transform. Integral geometry is a branch of mathematics that studies properties of such transforms and their inversion. For example, in X-ray tomography the data are essentially integrals of the density of the object over lines. In many imaging applications, regaining the unknown function modeling the medium is not possible, either because the data are complicated or because not enough data are taken to obtain exact reconstruction formulas. In fact, full knowledge of the function is

7.6. Microlocal Analysis and Imaging

117

not always necessary. For example, if one is looking for a tumor in a part of the human body, then the location and shape of the tumor are already useful information even if the exact values of the tumor density are not recovered. The location of the tumor can be easily determined from the singular support of the density function of the body—that is, the set of points where the function changes values abruptly. For example, the electromagnetic absorption coefficient of a cancerous tissue is far greater than that of a healthy tissue. A better understanding of the tumor regions can be obtained if we can recover the shape of the tumor as well. In other words, more precise information can be obtained if we can attach certain directions to the singular support at a point. In mathematical terms, such information can be obtained by looking at the Fourier transform of the function.

Figure 7.6. This picture represents the function that takes the values 1 inside and 0 outside the circle. The wavefront set is the set of normals to the boundary of the disk.

A smooth function that is zero in the complement of any ball has the property that its Fourier transform decays rapidly at infinity; in other words, the decay at any point is faster than any negative power of the distance of that point from the origin. One could then study the local behavior as well as the directional behavior of a function near a singular point by localizing the function near that point and by looking at the directions where its (localized) Fourier transform is not rapidly decaying. Such directions are in the wavefront set of the function. For example, if f is the function that takes the values 1 inside and 0 outside the disk in Figure 7.6, then the function is not smooth at the boundary circle. The wavefront directions are those normal to the boundary. Intuitively, these are the directions at which the jump in values of f at the boundary is most dramatic. Microlocal analysis is the study of such singularities and what operators (such as those in tomography) do to them. In cases where there are no exact reconstruction formulas, approximate backprojection reconstruction can be used. Microlocal analysis of such reconstruction operators gives very useful information. Let f be a function, and let x be a point one wants to image (i.e., find f (x)). The data are integrals of f over lines the X-rays traverse. Figure 7.7 shows what happens when f is the function that takes the values 1 inside and 0 outside the disk. For

118

Chapter 7. Communication and Representation

Figure 7.7. A disk and the backprojection reconstruction from X-ray data. The four lines in the data set are horizontal, vertical, and lines at 45 and 135 degrees. Note how the reconstruction “backprojects” the values of the line integrals over all points in the line. These are then added up to get the reconstruction. With lines at more angles, the reconstruction will look much better.

each line L in the plane, the data R f (L) is the length of the intersection of L with the disk. So, R f (L) is 0 if L does not meet the disk and R f (L) is the value of the diameter of the circle if L goes through the center of the disk. For such functions g (L) defined on a set of lines one can define a backprojection operator R∗ , which maps g (L) to a function h(x) as follows. For every fixed x, the value h(x) is equal to the “average” of g (L) over all L passing through x. Now, applying R∗ to R f one obtains the so-called normal operator of f , which is often used as an “approximate reconstruction operator” of f ; i.e., h(x) = R∗ R f (x) is an approximation to f . The study of normal operators and how well h(x) approximates f (x) in a given setup is one of the important problems in integral geometry. Ideally, one would like to have a situation when the wavefront set of h is the same as that of f . In this case, the singularities of the reconstruction, h, would be in the same locations as f . However, in many cases h may have some additional singularities (artifacts) or lack some of the singularities of f . One of the goals in such cases is to describe these artifacts, find their strengths, and diminish them as much as possible. Similar problems arise for transforms integrating along other types of curves, for example the transform R that integrates over ellipses. This elliptical transform is related to the model of bistatic radar [1]. In this case, the reconstruction operator includes a backprojection plus a sharpening algorithm. Consider the reconstruction shown in Figure 7.8. The function to be reconstructed is the characteristic function of a unit disk above the x-axis. The ellipses have foci in the interval [−3, 3] along the y-axis. Two important limitations of backprojection reconstruction methods are visible in this reconstruction. First, the top and bottom of the disk are visible but the sides are not. Second, there is a copy of the disk below the x-axis, although the object is above the axis. This is to be expected because the ellipses are all symmetric with respect to the x-axis—an object above the axis would give the same data as its mirror image below the axis. This same left-right ambiguity happens in synthetic aperture radar [2, 3], and it is important to understand the nature of the artifacts and why they appear. As can be seen in Figure 7.8, there is an artifact below the flight path. The artifact is as pronounced as the original disk, and microlocal analysis shows that such artifacts will always be as strong as the original object (e.g., [4]). However, if the flight path is not straight, microlocal analysis shows that the artifacts change position, and in certain cases, some artifacts can be eliminated [5, 6]. This problem comes up in other areas, such as electron microscopy and SPECT [7].

7.6. Microlocal Analysis and Imaging

119

Figure 7.8. Reconstruction of a disk on the y-axis from integrals over ellipses centered on the x-axis and with foci in [−3, 3]. Notice that some boundaries of the disk are missing. There is a copy of the disk below the axis. Reprinted with permission from Springer [8].

References [1] V. Krishnan and E.T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging, Inverse Probl. Imaging 5 (2011) 659–674. [2] C.J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, J. Fourier Anal. Appl. 10 (2004) 133–148. [3] L. Wang, C.E. Yarman, and B. Yazici, Theory of passive synthetic aperture imaging, in Excursions in Harmonic Analysis, Volume 1, edited by T.D. Andrews, R. Balan, J.J. Benedetto, W. Czaja, and K.A. Okoudjou, Springer-Birkhäuser, 2013, ISBN 978-0-81768375-7. [4] G. Ambartsoumian, R. Felea, V. Krishnan, C. Nolan, and E.T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, J. Funct. Anal. 264 (2013), 246–269. [5] R. Felea, Displacement of artifacts in inverse scattering, Inverse Problems 23 (2007) 1519–1531. [6] P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math. 73 (2013) 1596–1612. [7] R. Felea and E.T. Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math Anal. 43 (2011) 1145–1157. [8] V. Krishnan and H. Levinson, Microlocal analysis of elliptical radon transforms with foci on a line, in The Mathematical Legacy of Leon Ehrenpreis, 1930–2010, edited by I. Sabadini and D. Struppa, Springer Proceedings in Mathematics, Vol. 16, Springer, Berlin, 2012, pp. 163–182.

120

Chapter 7. Communication and Representation

7.7 How Does the GPS work? Christiane Rousseau, University of Montreal A global positioning system (GPS) is composed of at least 24 satellites orbiting around the Earth in six planes making an angle of 55 degrees with the equatorial plane. The position of the satellites is known at all times, and the clocks of the satellites are perfectly synchronized. The satellites send signals which are repeated periodically. The receiver of these signals, which is equipped with a clock and at least four channels, receives the signals of at least four satellites and measures (on its clock!) the transit times of at least four signals. However, since its clock is not necessarily synchronized with the clocks of the satellites, these times are fictitious transit times. To find the actual location of the receiver one must solve a system of four equations in four unknowns: the three coordinates of the receiver and the time shift between the clock of the receiver and that of the satellites. The system has two solutions, one of which corresponds to the location of the receiver. In practice, the computation is more complicated. Indeed, the speed of the satellites is sufficiently large that all of the calculations must be adapted to account for the effects of special relativity. Since the clocks on the satellites are traveling faster than those on Earth, they run slower. Furthermore, the satellites are in relatively close proximity to the Earth, which has significant mass. General relativity predicts a small increase in the speed of the clocks on board of the satellites. If the Earth is modeled as a large nonrotating spherical mass without any electrical charge, the effect is relatively easy to compute using the Schwarzschild metric, which describes the effects of general relativity under these simplified conditions. Fortunately, this simplification is sufficient to capture the actual effect to high precision. The two effects must both be considered because even though they are in opposite directions, they only partially cancel each other out. One source of errors in the computations is the estimation of the speed of the signal. To improve the estimate one uses differential GPS, which has a much higher precision. GPSs are used in geography, for example to measure the height of mountains and measure their growth. They have been used to establish the official height of Mount Everest

Figure 7.9. A GPS system consisting of 24 satellites orbiting the Earth.

7.7. How Does the GPS work?

121

and confirm that it is indeed the highest mountain on Earth. In another application, GPSs have been used to measure the displacement of tectonic plates. In the process, smaller plates were discovered, so we now know that, in addition to the 12 large ones, there are at least 52 smaller tectonic plates. There are very interesting mathematics in the signal of the GPS; see, for instance, [1, Chapter 1]. Reference [1] C. Rousseau and Y. Saint-Aubin, Mathematics and Technology, Springer Undergraduate Series in Mathematics and Technology, Springer, Berlin, 2008.

Chapter 8

Energy

8.1 Integrating Renewable Energy Sources into the Power Grid Wei Kang, Naval Postgraduate School For many years there has been a global push to increase our use of clean renewable electric energy. State and local governments of many countries have adopted renewable portfolio standards, which require a certain percentage of electric energy production to come from renewable resources. Reliable power system operation requires a balance of supply and deFigure 8.1. Wind power at work at Kinderdijk, The Nethermand at every moment lands. Reprinted courtesy of Lidia Fourdraine. in time. However, largescale integration of variable generators like wind turbines and solar panels can significantly alter the dynamics in a grid because wind and sunlight are intermittent resources. The power output can have fast fluctuation for various reasons, such as weather change and system reliability of a large number of turbines. Generators that use renewable energy to produce electricity often must be sited in locations where wind and solar resources are abundant and sufficient space exists for harnessing them. However, these locations are likely far away from population centers that ultimately consume the energy. The required transmission grids present additional challenges in various aspects including operational control, economic concerns, and policy making. Mathematical models that adequately represent the dynamic behavior of the entire wind or solar plant at the point of interconnection are a critical component for daily analysis and for computer model simulations. The analysis and simulations are used by 123

124

Chapter 8. Energy

Figure 8.2. Maintaining the integrity of the nation’s power grid is one of the major challenges for mathematics. Reprinted with permission from Metrics.

system planners and operators to assess the potential impact of power fluctuations, to perform proper assessment of reliability, and to develop operating strategies that retain system stability and minimize operational cost and capital investment. Traditional models used by the power industry cannot meet this goal for power grids with a large-scale integration of intermittent generators, but active research on better models is being carried out by several organizations and institutions. IEEE Power and Energy Magazine had two issues (Vols. 9(9) and 11(6)) that focused on several aspects of wind power integration. Technically, storage is an ideal flexible resource that is quick to respond to the fluctuation of generation and demand. Its functions include provision of energy arbitrage, peak shifting, and storing of otherwise-curtailed wind. In the case of battery storage it can be deployed close to the load in a modular fashion. However, efficiency issues coupled with the high capital costs make the justification of new storage difficult. A 2012 workshop report from the American Institute of Mathematics (AIM) dealt with some technical problems related to storage, such as the linear programming model that optimizes the required battery storage size and a nonlinear optimal control problem for batteries of predetermined size [1]. Review articles can also be found in the IEEE magazine issues mentioned above. Power systems are reliability-constrained; i.e., they must perform their intended functions under system and environmental conditions. Intuitive or rule-of-thumb approaches currently used in the industry will be inadequate for future power systems. More sophisticated quantitative techniques and indices have been developed for many years, and they are still an active focus of research. The work involves many areas of mathematics, including the mathematical concepts and models of reliability, nonlinear optimization, and large-scale simulations. References can easily be found in many journals such IEEE Transactions on Power Systems. Reference [1] S. Chen et al., Battery Storage Control for Steadying Renewable Power Generation, IMA Preprint Series No. 2373, July 2011.

8.2 Mathematical Insights Yield Better Solar Cells Arvind Gupta, Mitacs On November 19, 2013, I had the pleasure of attending the “Third Annual Mitacs Awards” ceremony in Ottawa. These awards recognize the outstanding R&D innovation

8.2. Mathematical Insights Yield Better Solar Cells

125

achievements of the interns supported by the various Mitacs programs—Accelerate, Elevate, and Globalink. This year, I was particularly inspired by the story of the winner of the undergraduate award category, a Globalink intern from Nanjing University in China named Liang Feng. The Globalink program invites top-ranked undergraduate students from around the world to engage in four-month research internships at universities across Canada. Liang Feng spent this summer in the lab of Professor Jacob Krich of the University of Ottawa Physics Department studying intermediate band (IB) photovoltaics, a technology that is being used to design the next generation of solar cells. Modern solar cells are based on silicon and other semiconductor materials and have been around for nearly 60 years. The first practical device, the “solar battery,” was invented in Bell Labs in 1954 and achieved 6% efficiency in converting incident sunlight into electricity. By 1961, it was determined that the “theoretical limit” for solar cell efficiency based on p-n semiconductors is 33.7%. As with many theoretical limits, creative scientists have found ways to break the rules, and the best solar cells today use multilayer structures and exotic materials to achieve more than 44% efficiency in converting sunlight into electricity.

Figure 8.3. Photovoltaic band gap diagram. Reprinted with permission from AIP [1].

In IB solar cells, additional semiconducting materials such as quantum dots are added to make it easier for electrons to be liberated by sunlight. Instead of requiring a single higher energy photon to knock an electron from the valence band (VB) to the conduction band (CB), the job can now be done by two or more low-energy photons. Thus more of the sunlight’s spectrum is harnessed by the cell by providing electrons with several possible steps on their staircase to freedom. The challenge for physicists to design such cells is how electrons will behave at the interfaces between the materials. Physicists use computational device models to design and model the behavior of multilayer cells. The best device models are both accurate and also computationally inexpensive, though in practice as approximations are made, the models become simpler to evaluate but less accurate. Professor Krich assigned Liang Feng the task of improving the model he had developed over the previous years, which allowed

126

Chapter 8. Energy

him to bring his mathematical and physical intuition to bear on the problem. According to Krich The most sophisticated previously existing IBSC device models all made an approximation that the boundary condition at the interface between a standard semiconductor and the intermediate-band semiconductor should be Ohmic, meaning that electrical current flows freely through it. This boundary condition was motivated by an analogous structure, the p-i-n diode, in which it is quite successful. Liang immediately disliked the Ohmic boundary condition for the case. While I gave him the standard explanations as to why it was an appropriate approximation, he came back to me time and again with different arguments as to why the Ohmic condition simply could not be accurate. His own persistence, intuition, and mathematical and computational abilities led him to his somewhat radical hypothesis (i.e., all previously published models for IBSC’s fail in a large range of cases), which he then convincingly proved. Liang Feng has made a significant and original contribution to improving device modeling for intermediate band solar cells. The achievement is truly his alone, because I actively discouraged him from pursuing it for several weeks. It is no exaggeration to say that this change may significantly aid the development of highly efficient and affordable solar cells. The outstanding achievement by Liang Feng during his Globalink internship is a great example of how surprising advances in the mathematical sciences are often driven by individual creativity and persistence in the face of skepticism. Through such thinking we consistently discover that theoretical limits are only temporary obstacles on the road of innovation. Reference [1] J.J. Krich, B.I. Halperin, and A. Aspuru-Guzik, Nonradiative lifetimes in intermediate band photovoltaics–Absence of lifetime recovery, J. Appl. Phys. 112 (2012) 013707.

8.3 Mathematical Modeling of Hydrogen Fuel Cells Brian Wetton, University of British Columbia I was a participant in the MPE workshop Batteries and Fuel Cells, running from November 4–8, 2013, in Los Angeles. This was part of a term-long thematic program “Materials for a Sustainable Energy Future,” organized by the Institute for Pure and Applied Mathematics (IPAM) at UCLA [1]. I was invited because I was involved in a decadelong project (1998–2008) modeling hydrogen fuel cells. This was a very applied project in collaboration with scientists at Ballard Power Systems, a Vancouver company that is a world leader in the development of these devices [2]. It was a group project, with several other faculty members participating, notably Keith Promislow, who became a close personal friend. In this section I’ll give a description of what we did on that project. The activity serves as an example of how academic mathematicians can become involved in work that has a direct impact on the world. Hydrogen fuel cells are of interest as an alternative energy technology. They are electrochemical systems that combine hydrogen and oxygen (from air) to produce electrical energy. They have potential for use in many applications, including automotive, stationary power, and small-scale power for mobile electronics. Unlike batteries, the energy source (hydrogen gas) flows through the device, so they are not intrinsically limited in capacity. The devices fit into a possible new energy economy where energy from different

8.3. Mathematical Modeling of Hydrogen Fuel Cells

127

sources is stored as hydrogen gas. Fuel cells have two main benefits over existing technology. They are very efficient when fueled by hydrogen, and the only end product is water, so they are nonpolluting. It should be said that currently hydrogen is produced mainly by refining fossil fuels. Fuel cells are now proven technology; however, they are more costly than current technologies. Current research focuses on the development of new materials to lower the cost and increase the lifetime of these devices. Modeling these new materials is the subject of my current research with Keith. The project I was involved with began under the umbrella of MITACS, a Canadian network that supported industrial mathematics activity from 1999 to a few years ago. Actually, MITACS still exists but has broadened its scope to cover all disciplines (so the “M” no longer stands for Mathematics). There were a number of connections that led to the collaboration, but the one that makes the most interesting story starts with John Kenna, who was an engineer at Ballard at that time. He had been trained as a fuel cell engineer but had worked at Hughes as an aeronautical engineer before coming to Ballard. Most of the design work for aircraft now is done with computational tools. That is, new design ideas are not made as physical models and tested experimentally. Rather, the physics of airflow and the behavior of the aircraft structure in response to stresses while in flight are described (approximately) by mathematical equations. This is the process of “mathematical modeling” in the title of the section. These equations can’t be solved exactly. For example, there is no way to get a written formula for the air speed at every point around an airplane wing. Thus, the next step is to approximate the solutions to the equations using numerical methods. This field is known as “scientific computation” and is my original research area. The resulting computational tools can be used to quickly and cheaply test many new airplane designs and optimize performance and safety. So John Kenna came to work at Ballard and with his background expected to have some simulation tools to help with design. However, he discovered that such tools had not yet been developed for the fuel cell industry. He was a guy with vision and thought that these would be a real help to the company. He started by taking a graduate level mathematics course at Simon Fraser University taught by Keith, who was working there at the time. Quite soon, he realized that it would be easier to get Keith involved in the activity than learn the math himself and pushed for the collaborative project with us from within the company. Keith and I had funding from MITACS and Ballard, and formed a group to develop models and computational simulation tools for hydrogen fuel cells. They are electrochemical systems that combine hydrogen and oxygen (from air) to produce energy. Rather than generating thermal energy through combustion, they generate electrical power with two electrochemical steps (hydrogen separating into protons and then combining with oxygen to form water) separated by a membrane that only conducts protons. This is shown schematically in Figure 8.4; the red arrows depict hydrogen movement from channels to catalyst sites, green oxygen movement from channels, and blue the movement of product water. Electrons travel through an external circuit doing useful work. The membrane is a key element to these devices. There are several types of fuel cells. The ones we looked at were polymer electrolyte membrane fuel cells (PEMFC)—low temperature devices (80o C) in which the membrane is a polymer material with acidic side chains. The electrochemical reactions on either side of the membrane have to be catalyzed to run at appreciable rates. Currently, platinum is used as a catalyst. This is one of the limitations to widespread use since platinum is expensive and rare. Some more details of the processes in the membrane electrode assembly (MEA) between the fuel cell channels are shown in Figure 8.5. Much of what we did was modeling, that is, writing equations that describe processes within a fuel cell and then thinking of ways to compute approximations to these models

128

Chapter 8. Energy

Figure 8.4. Schematic of a fuel cell [3].

Figure 8.5. Membrane electrode assembly [3].

efficiently. The models were what is known as “multiscale,” since details of processes from channel to channel (about 1 mm) affected performance along the length of the cell (up to 1 m long), and a number of cells (up to 100) are combined in a fuel cell stack to make appreciable power. Much of what we did is summarized in the review article PEM fuel cell: A mathematical overview [3].

8.3. Mathematical Modeling of Hydrogen Fuel Cells

129

I found some pictures from our group from the early years (late 1990s). Shown in Figure 8.6 from left to right are me (looking young and using an overhead projector!), Keith Promislow, and Radu Bradean, who worked with us as a post-doctoral fellow and then went to a position at Ballard. You can see we had fun with this project. As mathematicians, we really brought something to this project and this industry. Standard engineering computational tools such as computational fluid dynamics packages are not a good fit to models from this industry due to their multiscale nature, the stiff electrochemical reaction rates, and the capillary-dominated two-phase flow in the electrodes.

Figure 8.6. Brian Wetton (left), Keith Promislow (middle), and Radu Bradean (right). Reprinted with permission from Brian Wetton.

However, I have to say that I was initially reluctant to be involved in the project and viewed it as a distraction from my research work at the time on more abstract questions in scientific computation. In hindsight, I am happy I did get involved, but my initial reservation is common to many mathematicians. In my department (Mathematics at the University of British Columbia) I would say that only 10 of 60 faculty members would be open to an interdisciplinary project like the one I described above, and this is a higher ratio than in most departments. Concentrating on research in a single technical abstract area is seen as the best path to professional success. In some departments (not that uncommon) most of the work I did on this project would not count towards professional advancement (tenure and promotion), since it was not mathematics research but rather the use of “known” mathematics in a new application (known to us but not to the application scientists). I am not advocating that all mathematicians should work on such projects: it was the high-level mathematical training I received in a mathematics-focused environment that gave me the skills to contribute to this project. However, I believe such projects should be encouraged and rewarded. Events like MPE2013 highlight the contributions that mathematicians can make to our world, and I am very happy to be a part of it. References [1] International Conference on Microelectronic Systems Education, http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?reload=true&punumber=6560461

[2] Ballard, http://www.ballard.com/ [3] K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math. 70 (2009) 369–409.

130

Chapter 8. Energy

8.4 Mathematical Models Help Energy-Efficient Technologies Take Hold in a Community Karthika Muthukumaraswamy, SIAM Mathematical models can be used to study the spread of technological innovations among individuals connected to each other by a network of peer-to-peer influences, such as in a physical community or neighborhood. One such model was introduced in a paper published recently in SIAM Journal on Applied Dynamical Systems [1]. Authors N.J. McCullen, A.M. Rucklidge, Catherine Bale, T.J. Foxon, and W.F. Gale focus on one main application: the adoption of energy-efficient technologies in a population and, consequently, a means to control energy consumption. By using a network model for adoption of energy technologies and behaviors, the model helps evaluate the potential for using networks in a physical community to shape energy policy.

Figure 8.7. Compact fluorescent light bulbs beat traditional light bulbs at energy efficiency. Reprinted with permission from amasterphotographer/Shutterstock.

The decision or motivation to adopt an energy-efficient technology is based on several factors, such as individual preferences, adoption by the individual’s social circle, and current societal trends. Since innovation is often not directly visible to peers in a network, social interaction—which communicates the benefits of an innovation—plays an important role. Even though the properties of interpersonal networks are not accurately known and tend to change, mathematical models can provide insights into how certain triggers can affect a population’s likelihood of embracing new technologies. The influence of social networks on behavior is well recognized in the literature outside of the energy policy domain: network intervention can be seen to accelerate behavior change. “Our model builds on previous threshold diffusion models by incorporating sociologically realistic factors, yet remains simple enough for mathematical insights to be developed,” says author Alastair Rucklidge. “For some classes of networks, we are able to quantify what strength of social network influence is necessary for a technology to be adopted across the network.” The model consists of a system of individuals (or households) who are represented as nodes in a network. The interactions that link these individuals—represented by the

8.4. Mathematical Models Help Energy-Efficient Technologies Take Hold in a Community

131

edges of the network—can determine probability or strength of social connections. In the paper, all influences are taken to be symmetric and of equal weight. Each node is assigned a current state, indicating whether or not the individual has adopted the innovation. The model equations describe the evolution of these states over time. Households or individuals are modeled as decision makers connected by the network, for whom the uptake of technologies is influenced by two factors: the perceived usefulness (or utility) of the innovation to the individual, including subjective judgments, as well as barriers to adoption, such as cost. The total perceived utility is derived from a combination of personal and social benefits. Personal benefit is the perceived intrinsic benefit for the individual from the product. Social benefit depends on both the influence from an individual’s peer group and influence from society, which could be triggered by the need to fit in. The individual adopts the innovation when the total perceived utility outweighs the barriers to adoption. When the effect of each individual node is analyzed along with its influence over the entire network, the expected level of adoption is seen to depend on the number of initial adopters and the structure and properties of the network. Two factors in particular emerge as important to successful spread of the innovation: the number of connections of nodes with their neighbors, and the presence of a high degree of common connections in the network. This study makes it possible to assess the variables that can increase the chances for success of an innovation in the real world. From a marketing standpoint, strategies could be designed to enhance the perceived utility of a product or item to consumers by modifying one or more of these factors. By varying different parameters, a government could help figure out the effect of different intervention strategies to expedite uptake of energyefficient products, thus helping shape energy policy. “We can use this model to explore interventions that a local authority could take to increase adoption of energy-efficiency technologies in the domestic sector, for example by running recommend-a-friend schemes, or giving money-off vouchers,” author Catherine Bale explains. “The model enables us to assess the likely success of various schemes that harness both the householders’ trust in local authorities and peer influence in the adoption process. At a time when local authorities are extremely resource-constrained, tools to identify the interventions that will provide the biggest impact in terms of reducing household energy bills and carbon emissions could be of immense value to cities, councils and communities.” One of the motivations behind the study—modeling the effect of social networks in the adoption of energy technologies—was to help reduce energy consumption by cities, which utilize over two-thirds of the world’s energy, releasing more than 70% of global CO2 emissions. Local authorities can indirectly influence the provision and use of energy in urban areas, and hence help residents and businesses reduce energy demand through the services they deliver. “Decision-making tools are needed to support local authorities in achieving their potential contribution to national and international energy and climate change targets,” says author William Gale. Higher quantities of social data can help in making more accurate observations through such models. As author Nick McCullen notes, “To further refine these types of models, and make the results reliable enough to be used to guide the decisions of policymakers, we need high quality data. Particularly, data on the social interactions between individuals communicating about energy innovations is needed, as well as the balance of factors affecting their decision to adopt.”

132

Chapter 8. Energy

Reference [1] N.J. McCullen, A. M. Rucklidge, C.S.E. Bale, T.J. Foxon, and W.F. Gale, Multiparameter models of innovation diffusion on complex networks, SIAM J. Appl. Dyn. Syst. 12 (2013) 515–532 (online publication date March 26, 2013), http://epubs.siam.org/doi/abs/10.1137/120885371

8.5 Geothermal Energy Harvesting Burt S. Tilley, Worcester Polytechnic Institute As energy needs are expected to exceed the energy content of the available fossil-fuel resources before the end of the 21st century, interest in renewable energy sources has increased dramatically during the past decade. One source of interest is geothermal energy harvesting, where an energy (heat) is retrieved from the interior to be used on the surface of the Earth—either directly, for example to provide heat to a community, or indirectly by converting it to electrical energy. As the energy is brought from the deepest depth to the surface, some of this energy is transferred to the surrounding soil. In conventional deep wells (depths of 4 km or more), this transfer is a true loss, while in more shallow residential geothermal heat-pump systems (depths of 100 m), this transfer is the main mechanism for harnessing energy. Recently, my collaborator T. Baumann (Technical University Munich, TUM) and I applied some classical mathematical techniques to model the temperature attenuation in a fluid from a deep aquifer at a geothermal facility in the Bavarian Molasse Basin [1]. Energy losses (potentially up to 30%) depend on the production rate of the facility. Our approach takes advantage of the small aspect ratio of the radius of the well to its length and the balance between the axial energy transport in the fluid and the radial transport in the soil. We find that the dominant eigenfunction for the radial problem in the fluid captures this balance, and that the corresponding eigenvalue provides the appropriate constant relating the effective axial energy flux with the temperature drop over the length of the well. In the design of these wells, this constant is traditionally prescribed phenomenologically. This approach may be quite useful in the construction of the shallow residential geothermal heat-pump systems. Although the operation of these systems costs about onethird of the operation of conventional heating and cooling systems, they are currently not economically viable, since the installation cost of the wells depends significantly on the well depth required for the power needs of the residence. These systems are used year round, with energy deposited into the soil from the residence in the summer months and then retrieved for heating in the winter months. Recently, a group of undergraduate students participated at the NSF-funded Research Experiences for Undergraduates (REU) program at Worcester Polytechnic Institute [2] to work on this problem, which was brought to us from the New England Geothermal Professionals Association [3]. With our modeling approach, the eigenvalue and the axial behavior give a characteristic length for the well over which an energy attenuation of 1/e is achieved. Hence, three of these characteristic lengths are needed to attain over 90% of the possible energy available. We are currently extending these approaches to horizontal piping systems. References [1] B.S. Tilley and T. Baumann, On temperature attenuation in staged open-loop wells, Renewable Energy 48 (2012) 416–423. [2] Research Experience for Undergraduates (REU) in Industrial Mathematics and Statistics, Worcester Polytechnic Institute, http://www.wpi.edu/academics/math/CIMS/REU [3] New England Geothermal Professional Association, http://www.negpa.org

8.6. Of Cats and Batteries

133

8.6 Of Cats and Batteries Russ Caflisch, University of California at Los Angeles What do cats and batteries have in common? Not much, you might think. After all, cats are cuddly and purr. Batteries? They power your flashlights and cellphones, but no one wants a battery sitting on their lap while they watch TV. Cats were the subject of a recent, surprising news item [1]. A group of computer scientists at Google and Stanford University fed YouTube videos to a computer that was running a “machine learning” program. This program “trains” on the input to find clusters of similar images and, once it’s trained, the computer can classify new images as belonging to one of the clusters. After training on images from ten million YouTube videos, the computer learned to reliably identify images of cats. Like a newborn baby, the computer started with no knowledge but learned to identify objects—in this case cats—based on what it had already seen. This exercise illustrates the ability of machine learning to enable recognition tasks such as speech recognition, as well as classification tasks such as identifying cat faces as a distinct category of images. Batteries deserve attention on this website because of their essential role in any strategy for sustainable energy. Batteries are a primary means for storing, transporting, and accessing electrical energy. For example, they provide storage of excess energy from wind and solar sources and enable electrical power for cars and satellites. Today’s hybrid and electric vehicles depend on lithium-ion batteries, but the performance of these vehicles is limited by the energy density and lifetime of these batteries. To match the performance of internal combustion vehicles, researchers estimate that the energy density of current batteries would need to increase by a factor of 2 to 5. Strategies for achieving these gains depend on identifying new materials with higher energy densities. The traditional method for finding new materials is proposing a material based on previous experience, fabricating the new material, and measuring its properties, all of which can be expensive and time consuming. More recently, computational methods, such as density functional theory, have been used to accurately predict the properties

Figure 8.8. Where is Tigger? Reprinted with permission from Hein Nouwens/Shutterstock.

134

Chapter 8. Energy

of hypothetical materials. This removes the fabrication step but can involve large-scale computing. Although both of these methods have produced many successful new materials, the time and expense of the methods limit their applicability. Cats—more precisely, the machine learning program that recognized cats—could come to the rescue. Instead of watching YouTube videos, a machine learning method could train on existing databases (from both experiment and computation) of properties for known materials and learn to predict the properties for new materials. Once the machine learning method is trained (which can be a lengthy process), its prediction of material properties should be very fast. This would enable a thorough search through chemical space for candidate materials. Machine learning methods have not yet been used for finding materials for batteries, but they have been used for prediction of structural properties, atomization energies, and chemical reaction pathways. Their use in materials science is growing rapidly, and we expect that they will soon be applied to materials for batteries and other energy applications. Reference [1] J. Markoff, How Many Computers to Identify a Cat? 16,000, New York Times, June 25, 2012, http://www.nytimes.com/2012/06/26/technology/in-a-big-network-ofcomputers-evidence-of-machine-learning.html?_r=0

Chapter 9

Economics and Finance

9.1 Dynamic Programming for Optimal Control Problems in Economics Fausto Gozzi, Luiss University Some history. Since its beginnings in the 1950s, optimal control theory has found many applications in various areas of the natural and social sciences, and as more difficult applied problems were attacked, further advances were made in the theory. Here we consider the optimal control theory of infinite-dimensional systems, which has recently found interesting applications in the world of economics and finance. Infinite-dimensional systems are usually dynamical systems whose evolution is described by a partial differential equation (PDE) or a delay differential equation (DDE). They are infinite-dimensional in the sense that they can be rephrased as standard ordinary differential equations (ODEs) in abstract infinite-dimensional spaces such as Hilbert or Banach spaces. The study of optimal control problems for infinite-dimensional systems began in the 1970s with the two main tools of optimal control theory: Bellman’s Dynamic Programming and Pontryagin’s Maximum Principle. Here we discuss the use of the dynamic programming method with the associated Hamilton–Jacobi–Bellman (HJB) equations for the optimal control of heterogeneous systems [1]. Why model heterogeneity in economics? Economic models have traditionally been built under several simplifying assumptions for a number of reasons, including tractability. We give three examples. Considering a single agent to represent the average behavior of a large number of consumers greatly simplifies the analysis of an economic system and has enabled the development of a large and coherent body of economic research. As an example, neoclassical growth theory, which has been tremendously influential, considers a representative consumer and a representative firm instead of thousands (or millions) of separate consumers and firms. Capital homogeneity—the lumping together of all the forms of capital investment, including human and physical capital—is another simplifying assumption often made in economics. Again, the neoclassical growth theory makes this assumption and treats capital investments at different times (vintages) as identical. This, of course, is hardly realistic, since new vintages typically embody the latest technical improvements and are likely to be significantly more productive. This was clearly stated by Solow in 1960 when he wrote 135

136

Chapter 9. Economics and Finance

“This conflicts with the casual observation that many if not most innovations need to be embodied in new kinds of durable equipment before they can be made effective. . . .” Finally, while space has been recognized as a key dimension in several economic decision-making problems for quite some time, it has seldom been explicitly incorporated, even in models of growth, trade, and development where this dimension seems natural. This trend lasted until the early 1990s, as mentioned by Krugman (2010) in a retrospective essay: “What you have to understand is that in the late 1980s mainstream economists were almost literally oblivious to the fact that economies aren’t dimensionless points in space and to what the spatial dimension of the economy had to say about the nature of economic forces. . . .” Beyond analytical simplicity and internal consistency, the prevalence of such simplifying assumptions is due to the widely shared belief that departing from these assumptions would NOT improve our understanding of the main mechanisms behind the observed economic facts and would, at the same time, make economic models analytically intractable. But since the late 1990s, accounting for heterogeneity has become an essential aspect of research. The representative agent assumptions and other homogeneity assumptions have been heavily questioned, and new analytical frameworks explicitly incorporating heterogeneous agents and/or goods have been put forward and studied. Basically, this evolution is due to two important factors: • The emerging view that heterogeneity is needed to explain key economic facts. For example, the resurgence of the vintage capital work in the late 1990s is fundamentally due to new statistical evidence on the price of durable goods in the U.S., showing a negative trend in the evolution of the relative price of equipment, only compatible with embodied technical progress, thus making legitimate the explicit vintage modeling of capital. • The rapid development of computational economics. Advances in computational techniques and the advent of more powerful computational resources make it feasible to deal with models having heterogeneous agents. Special issues of the reference journal in the field, Journal of Economic Dynamics and Control, have been devoted to this specific area (Issue 1 in 2010, Issue 2 in 2011), suggesting that it is one of the hottest areas in the field of computational economics. An illustrative example. Consider, for example, the vintage capital model. Beginning with the easiest neoclassical growth model (the so-called AK model), one generalizes it to the case when capital is heterogeneous—that is, differentiated by age (vintage capital). The basic equation—the state equation, in the language of optimal control—becomes a DDE. Using Bellman’s dynamic programming method, it becomes possible to characterize the optimal trajectories, which “should” describe the behavior of economic systems. The introduction of heterogeneity allows a more faithful description of the features of the economic system. Indeed, the graphs in Figure 9.1 show the output y(t ) (the production) of the model (after a detrending, which is done for the sake of clarity) before and after the introduction of heterogeneity. Fluctuations of the output is a well-known feature that is captured with infinite-dimensional optimal control models. Further directions. Further work needs to be done with heterogeneity resulting from the spatial and population distribution of economic activity. This heterogeneity is a key feature of contemporary economic systems, and a deep study of such models incorporating heterogeneity should provide both more insight into the behavior of such systems and more help for decision makers. The particular issues under study are • environmentally sustainable growth regimes;

9.2. Solar Renewable Energy Certificates

137

Figure 9.1. Production according to the classical AK model before (horizontal line) and after (oscillating line) the introduction of heterogeneity. Reprinted with permission from Elsevier [2].

• land use; • the socioeconomic and public finance problems related to aging on one hand and to epidemiological threads on the other; and • incorporation of the age structure of human populations in the analysis of key economic decisions like investment in health and/or investment in pension funds both from the private and social optimality points of view. References [1] G. Fabbri and F. Gozzi, Solving optimal growth models with vintage capital: The dynamic programming approach, J. Econ. Theory 143 (2008) 331–373. [2] R. Boucekkine, O. Licandro, L.A. Puch, and F. del Rio, Vintage capital and the dynamics of the AK model, J. Econ. Theory 120 (2005) 39–72.

9.2 Solar Renewable Energy Certificates Michael Coulon, Sussex University In recent years, governments around the world have experimented with many different policy tools to encourage the growth of renewable energy. In particular, it is clear that subsidies are needed to stimulate investment in clean technologies like wind and solar that are not yet able to compete effectively on cost alone (especially in the U.S. today, where cheap natural gas is showing the potential to dominate!). Economists, politicians, and journalists actively debate the merits and limitations of various subsidies, tax incentives, or of feed-in tariffs in electricity markets, popular in many European countries. However, an interesting alternative is also growing rapidly at the state level in the U.S.: markets for tradable renewable energy certificates (RECs), or as a subcategory, solar renewable energy certificates (SRECs). Here we discuss the vital role that mathematics can play in helping to better understand these important new markets.

138

Chapter 9. Economics and Finance

Figure 9.2. Solar power station. Reprinted with permission from Kajano/Shutterstock.

Over the last decade or so, about 30 states have implemented specific targets for renewable energy growth as part of a so-called renewable portfolio standard (RPS). Among these, many have a specific “solar carve-out,” a target for the solar sector in particular, in addition to renewables overall. To achieve these goals, about 10 states have launched SREC markets, with New Jersey (NJ) being the largest and most ambitious so far (targeting 4.1% solar electricity by 2028). It is worth noting that similar markets for “green certificates” also exist in various countries around the world. The basic idea is that the government sets specific requirement levels for solar energy in the state in each future year as a percentage of total electricity generation. Throughout each year, certificates (SRECs) are issued to solar generators for each MWh of solar power that they produce. These can then be sold in the market to utility companies, who must submit the required number of SRECs at each compliance date (once per year). Anyone not meeting the requirement must instead pay a penalty (known as the SACP), which is typically chosen to decrease from year to year, but has been as high as $700 per MWh in the NJ market. While the concept is straightforward and intuitive (and parallels that of a cap-andtrade market for CO2 emissions), the implementation is far from simple, with different states already trying many variations for setting future requirement and penalty levels. Another important policy consideration is the number of “banking” years permitted, meaning how long SRECs remain valid for compliance after they are first issued (e.g., currently a 5-year lifetime in NJ). A fundamental challenge for regulators is trying to choose appropriate requirement levels many years in advance, such that the market does not suddenly run into a large over- or undersupply of certificates, causing prices to swing wildly. In NJ, for example, SREC market prices dropped from over $600 throughout most of 2011 to under $100 by late 2012 in the wake of a huge oversupply, and this despite a major rule change passed in 2012 (more than doubling the 2014 requirement) to help support price levels. On the one hand, the large oversupply was good news, signaling the success of the SREC market in enabling solar in NJ to grow very rapidly between 2007 and 2012 (from under 20MW to nearly 1,000 MW of installed capacity). On the other hand, this initial success of the market brings with it some risk for its future. At only $100 an SREC and with the possibility of further price drops, will investors now shy away from new solar projects? Like all financial markets, SREC markets can provide very rewarding opportunities for investing (in new solar farms in this case). But they also come with significant risk due

9.2. Solar Renewable Energy Certificates

139

to volatile price behavior. Financial mathematics, a field that has grown rapidly over several decades now, is well versed in analyzing and modeling such risks and returns. However, most financial mathematicians work on classical markets for stocks or bonds, instead of venturing into the peculiarities of commodity prices, and even more so those of RECs. Nonetheless, commodities, energy, and environmental finance is a rapidly growing subfield and popular research area these days (see, for example, [1]). So how can mathematical modeling help us to better understand SREC markets? And why is it important to do so? In recent and ongoing work at Princeton University [2], we propose an original approach to modeling SREC prices, which is able to reproduce NJ’s historical price dynamics to an encouraging degree. Drawing on some ideas from existing literature in carbon allowance price modeling, we create a flexible framework that can adapt to the many rule changes that have occurred. In particular, we treat SREC prices as combinations of “digital options” on an underlying process for total solar power generation, since SRECs essentially derive their value from the probability of the market being short of certificates and paying a penalty at one or more future compliance dates. However, a key additional challenge comes in capturing an important feedback effect from prices onto the stochastic process for generation. As today’s prices increase, future generation growth rates should also increase (as more solar projects are built), which in turn reduces the probability of future penalty payments, feeding back into today’s price. An equilibrium price emerges, which can be solved for via dynamic programming techniques. This is an example of a “structural” model, which combines economic fundamentals of supply and demand with tractable stochastic processes and convenient mathematical relationships. Academic literature on energy-price modeling covers a wide range of different approaches and makes use of a diverse set of mathematical tools, from partial differential equations (PDEs) to stochastic processes, optimization, and statistical estimation procedures. The feedback discussed above has even been shown to produce interesting applications of complicated “forward-backward SDEs” in the case of carbon markets. Nonetheless, the specific application to SREC markets is extremely new, and we hope to encourage more research in this young and exciting field. Understanding the behavior of SREC prices is crucial both for investors contemplating a new solar project and for regulators determining how best to design the market or set the rules. How does price volatility vary with regulatory policy? For example, can we effectively implement a requirement growth rule which dynamically adapts to the shortage or surplus of SRECs in the previous year? (as has in fact been attempted in Massachusetts). Can this avoid the need for frequent legislation to rewrite the rules at great uncertainty to all market participants? How can we best avoid sudden price swings, while preserving the attractive features of these markets and their abilities to stimulate growth of solar? While our model allows us to begin to address such important market design issues, many interesting and relevant questions remain to be investigated, and we look forward to continuing to explore this promising new area of applied mathematics! The reference for our first paper on this topic is given below [2]. For further details on the NJ SREC market, the websites of NJ Clean Energy [3], SREC trade [4], and Flett Exchange [5] all provide useful and up-to-date information. References [1] M. Ludkovski, Fields Institute—Focus Program on Commodities, Energy, and Environmental Finance, Mathematics of Planet Earth Blog, May 7, 2013, http://mpe2013.org/2013/05/07/fields-institute-focus-program-on-commodities-energyand-environmental-finance/

140

Chapter 9. Economics and Finance

[2] M. Coulon, J. Khazaei, and W.B. Powell, SMART-SREC: A Stochastic Model of the New Jersey Solar Renewable Energy Certificate Market, working paper, Department of Operations Research and Financial Engineering, Princeton University. [3] New Jersey’s Clean Energy Program, http://www.njcleanenergy.com/ [4] SRECTrade, http://www.srectrade.com/ [5] FlettExchange Environmental Service, http://www.flettexchange.com/

9.3 How Much for My Ton of CO2 ? Mireille Bossy, INRIA Nadia Maïzi, MINES ParisTech Odile Pourtallier, INRIA Mathematics analyzes numerous aspects of financial markets and financial instruments. For the markets trading CO2 emissions (direct or the CO2 equivalent for other greenhouse gases), mathematics is used to decide how cap-and-trade rules will operate. The capand-trade mechanism sets future caps for pollution emissions and issues emission rights that can be bought and sold by the companies concerned. Producers that overstep their allotted cap must pay a penalty. Designing a market involves deciding in particular on a timetable for using permits, the initial mode of distribution (e.g., free attribution, bidding system, etc.), and how penalties operate.

Figure 9.3. Reprinted with permission from Geek Culture.

These trading markets are still in their early stages. Under the impetus of the Kyoto Protocol and its extension, a few attempts have been made to open up markets in some

9.4. The Social Cost of Carbon

141

countries, beginning with the European Community in 2005 and followed by more recent initiatives in Australia, China, and some American states. Emissions trading provides a financial incentive to reduce greenhouse gas emissions from some sectors of economic activity. From theory to practice, mathematics can help obtain a clearer view of design choices so that setting up emissions trading can lead to effective reduction. Emissions trading also affects the price of goods, because greenhouse gas emissions are an externality of production, as well as the price of raw materials. Game theory can be used to analyze interaction between stakeholders in these markets and to understand the connection between the design of cap and trade, the prices set for commodities such as electricity, windfall effects, and favored production technologies. In addition, industrial production models can be used by a stakeholder subject to CO2 penalties. For a fixed market design, digital simulation can be used to quantify the stakeholder’s activity balance sheet (i.e., wealth produced, CO2 discharged) and calculate its subjective price to acquire a permit. Using this information, different designs can be tested and compared in line with varied criteria, including emissions reduction. Reference [1] CarbonQuant software, http://carbonvalue.gforge.inria.fr/index.html

9.4 The Social Cost of Carbon Hans Engler, Georgetown University

Figure 9.4. Reprinted with permission from Climate Reality Project.

Recently, the United States Environmental Protection Agency (EPA) increased its estimate for the net societal cost of an additional ton of carbon dioxide (CO2 ) that is released into the atmosphere to $36 from $22. One of the immediate effects was a change of energy standards for household appliances, for example microwave ovens. Other consequences are expected to follow, for example possibly in emission standards for automobiles and power plants, and in other regulations. What exactly is the definition of the “social cost of carbon” (SCC)? Who is interested in determining this quantity? Who is interested in its value? Can this even be done and, if so, how accurately? And how is it done? Is there any mathematics in it? The SCC is generally defined as the net economic damage (overall cost minus overall benefits, accumulated over time and discounted) of a small additional amount of CO2 (a metric ton, 1,000 kg, produced by burning about 440 liters of gasoline) that has been released into the atmosphere. Mathematically, it’s a rate of change; economically, it’s a marginal cost. Economists have been trying to determine this in order to estimate the cost of mitigation of climate change: In an ideal situation, the cost of mitigating the effects of an additional ton of CO2 in current dollars should be equal to the SCC, and if a tax were assessed on releasing CO2 , it should equal the SCC.

142

Chapter 9. Economics and Finance

Concretely, suppose a new regulation is proposed with the goal of reducing greenhouse gas emissions. Implementing the regulation will cost money. If the expected cost exceeds the SCC, it is unlikely to be enacted, at least in the US and in the EU. Regulators have to include a cost-benefit analysis, and the new regulation will come up short. Therefore, the SCC furnishes an immediate connection between climate science and climate policy. It’s one way to “monetize” the results of anthropogenic climate change. Since a higher SCC is expected to make regulations easier, it will generate resistance from groups that are opposed to regulation. It is very difficult to obtain a reasonable number for the SCC. Clearly, climate science models that connect the release of greenhouse gases to climate changes must be used (and that’s where mathematics comes in, but it’s not the only place). But there are many additional input variables that influence it. Climate system variables include the overall climate sensitivity to CO2 emissions, the extent to which a climate model can predict abrupt climate changes, and the level of geographic detail in the model. Higher climate sensitivity, the inclusion of abrupt changes, and more details all tend to increase the SCC. There are also economic variables and model details that influence SCC, such as the discount rate (used to turn future costs into present day costs), the economic value placed on the quality of human life and ecosystems, the capacity of a society to adapt to changes in climate conditions, and the extent to which indirect costs of climate change are incorporated. A lower discount rate (meaning a long-term view into the future), high economic valuation of ecosystems, and detailed inclusion of indirect costs will all increase the SCC. In addition, the SCC is generally expected to increase in the future as economies become more stressed due to results of previous climate change. A ton of CO2 that is released in 2030 will be more expensive. Current models used by the United States EPA try to assess costs up to the year 2300—which may be longer than the time horizon of many climate models that are currently being used. It is perhaps no surprise that all SCC calculations end up with a range of numbers, rather than with a fixed value, and that these ranges vary widely. In fact, there are low estimates of an SCC of about $2 (that is, a small net benefit of increased CO2 emissions) and high estimates of $200 or more. Generally, research in this important area lags behind the state-of-the-art of physical climate models, mainly due to the additional economic components that have to be included. I mentioned that the mathematical connection comes from climate models which are used to make predictions. But there is a broader, more general connection. Using models that include physical, social, and economic factors, all with their own uncertainties, presents new challenges to the emerging mathematical field of uncertainty quantification. Perhaps over time mathematics can contribute to improving the methods by which the SCC is computed.

9.5 Measuring Carbon Footprints Hans Kaper, Georgetown University Releasing a ton of carbon dioxide (CO2 ) into the atmosphere has quite a different effect on the global average temperature than releasing a ton of methane (CH4 ). Have you ever wondered how the effects of different greenhouse gases are compared? Designing appropriate metrics is nontrivial but essential for setting standards and defining abatement strategies to limit anthropogenic climate change, as was done, for example, in the Kyoto Protocol. Yes, we are talking about a “carbon footprint.” Do you know how it is defined?

9.6. Musings on Summer Travel

143

The standard unit for measuring carbon footprints is the carbon dioxide equivalent (CO2 e), which is expressed as parts per million by volume, ppmv. The idea is to express the impact of each different greenhouse gas in terms of the amount of CO2 that would create the same amount of warming. That way, a carbon footprint consisting of lots of different greenhouse gases can be expressed as a single number. Standard ratios are used to convert the various gases into equivalent amounts of CO2 . These ratios are based on the so-called global warming potential (GWP) of each gas, which describes its total warming impact relative to CO2 over a set period of time (the time horizon, usually 100 years). Over this time frame, according to the standard data, methane scores 25 (meaning that one metric ton of methane will cause the same amount of warming as 25 metric tons of CO2 ), nitrous oxide comes in at 298, and some of the superpotent greenhouse gases score more than 10,000. The adequacy of the GWP has been widely debated since its introduction. The choice of a time horizon is a critical element in the definition. A gas which is quickly removed from the atmosphere Figure 9.5. may initially have a large effect but for longer time periods becomes less important as it has been removed. Thus methane has a potential of 25 over 100 years but 72 over 20 years. Conversely, sulfur hexafluoride has a GWP of 23,900 over 100 years but 16,300 over 20 years. Relatively speaking, therefore, the impact of methane—and the strategic importance of tackling its sources, such as agriculture and landfill sites—depends on whether you’re more interested in the next few decades or the next few centuries. The 100-year time horizon set by the Kyoto Protocol puts more emphasis on near-term climate fluctuations caused by emissions of short-lived species (like methane) than by emissions of long-lived greenhouse gases. Since the GWP value depends on how the gas concentration decays over time in the atmosphere, and this is often not precisely known, the values should not be considered exact. Nevertheless, the concept of the GWP is generally accepted by policy makers as a simple tool to rank emissions of different greenhouse gases.

CO2

9.6 Musings on Summer Travel David Alexandre Ellwood, Harvard University Thanks to the affordability of air travel, an increasing number of us have the opportunity to visit exotic locations around the globe. Back in my student days, I was enthralled by the idea of attending conferences in cultural centers like Paris and Edinburgh, as well as remote villages like Les Houches in the French Alps, or Cargèse in Corsica. The idea of pitching a tent next to the beach and spending a week learning about the latest developments in theoretical physics made me feel like I was the luckiest person in the world. Back then I didn’t think twice about the unintended consequences of my travels, but now that the scientific case for anthropogenic global warming (AGW) is firmly established, we scientists can no longer ignore the externalities of our summer gatherings. A comprehensive analysis of the evolution of scientific consensus for AGW was published recently in Environmental Research Letters [1]. The study identified more than

144

Chapter 9. Economics and Finance

4,000 abstracts that stated a position on the cause of global warming out of 11,944 in the peer reviewed scientific literature over the last 21 years. Out of these 97% endorsed the view that human activity is the unambiguous cause of such trends. The study found that “the number of papers rejecting the consensus on AGW is a vanishingly small proportion of the published research.” Unfortunately, the degree of consensus within the academic community is poorly represented in the popular media, and there continues to be widespread public perception that climate scientists disagree about the significance of human activity in driving these changes. Indeed a 2012 poll showed that although an increasing number of Americans believe there is now solid evidence indicating global warming, more than half still either disagree with or are otherwise unaware of the consensus among scientists that human activity is the root cause of this increase [2]. The ability to implement effective climate policy is noticeably impaired by the public’s confusion over the position of climate scientists. In the Pew Research Center’s annual policy priorities survey, just 28% said that dealing with global warming should be a top priority, ranking climate policy last amongst the 21 priorities tested [3]. Since becoming involved with MPE2013, I’ve tried to develop a better understanding of the work of climate scientists, as well as the economic and technological challenges we must face to meet future energy demands. Although many of my colleagues share similar concerns about the urgency of addressing AGW, very few have made any significant changes to their personal/professional life choices. As members of the academic community, the intellectual milieu in which we work exposes us to trends and ideas far ahead of their widespread adoption; just think of our use of information technology and the Web. But various lunchtime conversations on the topic soon made me realize how difficult it will be to bring about the kind of awareness necessary to meet the challenges of global issues like climate change, even amongst the educated elite. On this point I must say that I’m personally grateful to everyone who has worked so hard to make MPE2013 a success. Before last year I’d never estimated my carbon footprint, let alone compared it to those of my friends and colleagues from abroad. But after attending an MPE planning meeting, I started following activities and some of the topics kindled a sense of personal responsibility quite beyond the usual intellectual curiosity I might feel for other disciplines. I hope one of the legacies of MPE2013 will be an influx of new talent into the wide range of intricate mathematical problems highlighted in the lectures, workshops, and conferences currently taking place around the world. But I also hope that many more of us will take note of a science that connects us back to the world in which we live and a greater personal awareness of the energy choices we make in our lives. So if you’ve never taken the time before, I think you might enjoy playing with one of the various carbon calculators available on the Web. The U.S. Environmental Protection Agency has one on their website [4], or National Geographic has a personal “energy meter” that is both educational and easy to share with your neighbors and friends [5]. You might be surprised to see how you measure up to your nonacademic peers. Are you part of the energy avant-garde or lagging the national average? How you fare may crucially depend on whether you frequently visit international colleagues or have a penchant for traveling abroad to conferences and workshops. In an article on January 26, 2013, The New York Times suggested “your biggest carbon sin may be air travel” [6]. Have you ever purchased carbon credits to offset your flights, or would you consider declining an invitation for a professional meeting to reduce your score? Some airlines and several of the popular online travel agencies offer the opportunity to purchase offsets when you buy your air tickets. If you haven’t adopted any such scheme, you are not alone; indeed you are in good company! As reported in the New York Times article [6],

9.6. Musings on Summer Travel

145

Figure 9.6. Reprinted with permission from Leszek Kobusinski/Shutterstock.

Last fall, when Democrats and Republicans seemed unable to agree on anything, one bill glided through Congress with broad bipartisan support and won a quick signature from President Obama: the European Union Emissions Trading Scheme Prohibition Act of 2011. This odd law essentially forbids United States airlines from participating in the European Union Emissions Trading System, Europe’s somewhat lonely attempt to rein in planet-warming emissions. Under this program, the aviation sector was next in line to join other industries in Europe and start paying for emissions generated by flights into and out of EU destinations. After an uproar from both governments and airlines, as well as a slew of lawsuits from the United States, India, and China, the European Commission delayed full implementation for one year to allow an alternative global plan to emerge. But already back in 2007, the most contentious matter on the 36th assembly of the International Civil Aviation Organization’s (ICAO) agenda was the environmental impact of international aviation. Stratospheric ozone depletion and poor air quality at ground level are also effects of aircraft emissions, and although the Kyoto Protocol of 1997 assigned the ICAO the task of reducing the impact of aircraft engine emissions, so far the organization has resisted measures that would impose mandatory fuel taxes or emissions standards. This set the stage for a legal dispute of gargantuan proportions between the ICAO’s European member countries and foreign airlines and governments who do not want to comply. The ICAO’s general assembly meets once every three years, and the 38th Assembly began on September 24, 2013. The hottest topic on the agenda is sure to be the pending EU legislation and the need to find common ground on aviation emissions standards and trading, but what position will the United States, India, and China now adopt? In his 2013 Inaugural Address, President Obama promised to make dealing with climate change part of his second-term agenda. The volume of air travel is increasing much faster than gains in fuel efficiency, and emissions from many other sectors are falling. The meetings taking place at the ICAO assembly might have been some of the most significant in the fight against AGW in 2013, but will our government finally take the lead in bringing about the kind of binding legislation our planet so desperately needs?

146

Chapter 9. Economics and Finance

References [1] J. Cook, D. Nuccitelli, S.A. Green, M. Richardson, B. Winkler, R. Painting, R. Way, P. Jacobs, and A. Skuce, Quantifying the consensus on anthropogenic global warming in the scientific literature, Environ. Res. Lett. 8 (2013) 024024, doi: 10.1088/1748-9326/8/2/024024. [2] More Say There Is Solid Evidence of Global Warming, Pew Research Center, October 4–7, 2012, http://www.people-press.org/2012/10/15/more-say-there-is-solid-evidenceof-global-warming/ [3] Climate Change: Key Data Points from Pew Research, Pew Research Center, June 24, 2013, http://www.pewglobal.org/2013/06/24/climate-change-and-financial-instabilityseen-as-top-global-threats/ [4] Household Carbon Footprint Calculator, United States Environmental Protection Agency, http://www.epa.gov/climatechange/ghgemissions/ind-calculator.html [5] The Great Energy Challenge, National Geographic, http://environment.nationalgeographic.com/environment/energy/great-energy-challenge/

[6] E. Rosenthal, Your Biggest Carbon Sin May Be Air Travel, The New York Times, January 26, 2013, http://www.nytimes.com/2013/01/27/sunday-review/the-biggestcarbon-sin-air-travel.html?smid=pl-share&_r=0

9.7 The Carbon Footprint of Textbooks Kent E. Morrison, American Institute of Mathematics Compared with a conventional textbook it’s obvious that an e-text saves energy and reduces greenhouse gas emissions—or is it? When you actually look at the way students use both kinds of textbooks, the obvious turns out to be not so obvious. Looking at the behavior of college students is exactly what Thomas F. Gattiker, Scott E. Lowe, and Regis Terpend did in order to determine the relative energy efficiency of electronic and conventional hard copy textbooks. They used survey data from 200 students combined with life cycle analysis of digital and Figure 9.7. Reprinted courtesy of conventional textbooks and found that on the average the carbon footprint for digital textadamr/FreeDigitalPhotos.net. books is a bit smaller but not as much smaller as you would hope. In a short summary article for The Chronicle of Higher Education [1], Gattiker and Lowe write We discovered that when we consider all greenhouse-gas emissions over the life cycle of the textbook, from raw-material production to disposal or reuse, the differences between the two types of textbooks are actually quite small. Measured in pounds of carbon-dioxide equivalent (CO2 e), a common unit used to measure greenhouse-gas emissions, the use of a traditional textbook resulted in approximately 9.0 pounds of CO2 e per student per course, versus 7.8 pounds of CO2 e for an e-textbook.

9.8. Sustainable Development and Utilization of Mineral Resources

147

However, there is a wide variability in the energy used by individual students, and the reasons are easy to understand. Some of the factors that matter are • the device on which the e-text is read (desktop computer, laptop, dedicated e-reader);

• the number of pages printed by the student and whether the pages are two-sided or single-sided; • the source for the electric power (hydro, coal, natural gas); and • the number of times a hard-copy book is resold.

Compare a 500-page conventional text with the same text in digital format. If the student reads it on a desktop computer located where electric power is generated by burning coal, and if the student prints 200 one-sided pages, then the carbon footprint is much greater for the e-book. But if the student reads it on an e-reader, doesn’t print much, and gets hydroelectric power, then the e-book has a much smaller carbon footprint. They identify three “levers” that college faculty and students can use to reduce the carbon load associated to textbooks: • encourage multiple use of hard-copy textbooks;

• read e-texts on laptops and dedicated readers rather than desktop computers; and • print on both sides with recycled paper.

Reference [1] T.F. Gattiker, S.E. Lowe, and R.Terpend, Online texts and conventional texts: Estimating, comparing, and reducing the greenhouse gas footprint of two tools of the trade, Decision Sci. J. Innov. Ed. 10 (2012) 589–613, http://onlinelibrary.wiley.com/doi/10.1111/j.1540-4609.2012.00357.x/abstract

9.8 Sustainable Development and Utilization of Mineral Resources Roussos Dimitrakopoulos, McGill University The sustainable development and utilization of mineral resources and reserves is an area of critical importance to society, given the fast growth and demand of new emerging economies and environmental and social concerns. Uncertainty, however, impacts sustainable mineral resource development, including the ability of ore bodies to supply raw materials, operational mining uncertainties, fluctuating market demand for raw materials and metals, commodity prices, and exchange rates. Throughout the last decade, new technological advances in stochastic modeling, optimization, and forecasting of mine planning and production performance have shown to simultaneously enhance production and return on investment. These advances shifted the paradigm in the field, showed initially counterintuitive outcomes that are now well understood, and outlined new areas of research needs. The old paradigm—based on estimating mineral reserves, optimizing mine planning, and production forecasting—resulted in single, often biased, and flawed forecasts. The flaws were due largely to the nonlinear propagation of errors associated with ore bodies throughout the chain of mining.

148

Chapter 9. Economics and Finance

The new stochastic paradigm addresses these limits, and application of the stochastic framework increases net present value (NPV) of mine production schedules (20–30%). It also allows for stochastically optimal pit limits that are about 15% larger in total tonnage when compared to conventionally optimal pit limits, adding about another 10% NPV. Related technical developments also impact sustainable utilization of mineral resources; uncertainty quantification and risk management; social responsibility through improved financial performance; enhancement of production and product supply; contribution to management of mine remediation; and objective, technically defendable decision making. Ongoing research efforts focus in particular on the quantification of geological uncertainty and uncertainty in metal supply, and the development of global stochastic optimization techniques for mining complexes and mineral supply chains. New methods for quantifying geological uncertainty and uncertainty in metal supply include a high-order modeling framework for spatial data based on spatial cumulants— combinations of moments of statistical parameters that characterize non-Gaussian random fields. Advantages include the absence of distributional assumptions and pre- or postprocessing steps like data normalization and training image (TI) filtering; the use of high-order relations in the data (data-driven, as opposed to TI-driven simulation process); and the generation of complex spatial patterns reproducing any data distribution, variograms, and high-order spatial cumulants. The method offers an alternative to the multiple-point methods applied by our Stanford University colleagues, with additional advantages. Research directions include the search of new methods for high-order simulation of categorical data (e.g., geology of mineral or petroleum deposits, ground water aquifers, sites for CO2 sequestration), as well as high-order simulations for spatially correlated attributes and principal cumulant decomposition methods. The new modeling framework has significant impacts in mine production planning, scheduling, and forecasting, from single mines to mineral supply chains. Global stochastic optimization techniques for mining complexes and mineral supply chains are a core element of mine design and production scheduling, because they maximize the economic value generated by the production of ore and define a technical plan to be followed from the development of the mine to its closure. This planning optimization is a complex problem due to its large scale, the uncertainty in key geological, mining, and financial parameters, and the absence of a method for global or simultaneous optimization of the individual elements of a mining complex or mineral supply chain. Our research aims to develop a new stochastic optimization framework integrating multiple mines, multiple processing streams—including blending stockpiles and waste dumps designed to meet quality specifications and minimize environmental impact—waste management issues, and transportation methods. The ability to manage and simultaneously optimize all aspects of a mining complex leads to mine plans that not only minimize risk related to environmental impact and rehabilitation but also increase the economic value, reserves, and life-of-mine forecast, thus contributing to the sustainable development of the nonrenewable resource. Stochastic integer programming has been a core technique in our stochastic optimization efforts. However, the scale of the scheduling and material flow through a mineral supply chain is very large and requires the development of efficient solution strategies. For example, a hybrid approach integrating metaheuristics and linear programming permits the linking of long-term and short-term production schedules, where information gleaned from one can be used to improve the other, leading to a globally optimal and practical mining plan. Extensive testing and benchmarking are underway, and the more promising approaches are being field-tested at mine sites with collaborating companies from North to South America, Africa, and Australia.

9.9. Scientific Research on Sustainability and Its Impact on Policy and Management

149

For more information, see the related webinar available at the McGill website [1]. For a short video with simple explanations, see the NSERC website [2]. References [1] R. Dimitrakopoulos, Strategic Mine Planning Under Uncertainty: From Complex Orebodies and Single Mines to Mining Complexes, McGill University, https://connect.mcgill.ca/p7iglvm7if3/?launcher=false&fcsContent=true&pbMode=normal

[2] NSERC Presents 2 Minutes with Roussos Dimitrakopoulos, Department of Mining and Materials Engineering, McGill University, http://www.nserc-crsng.gc.ca/MediaMedia/2minutes-2minutes/Roussos-Roussos_eng.asp

9.9 Scientific Research on Sustainability and Its Impact on Policy and Management Mark Lewis, University of Alberta I recently had the opportunity to lecture on “Aquaculture and Sustainability of Coastal Ecosystems” at the NSF-funded Mathematical Biosciences Institute (MBI) in Columbus, Ohio. The MBI focuses on different theme programs; in the fall of 2013 the theme program was “Ecosystem Dynamics and Management.” In my lecture, I focused on work done over the last 10 years, with grad students and colleagues, on disease transfer between aquaculture and wild salmon. This turns out to be a key issue for sustainability of wild salmon, particularly pink salmon, in coastal ecosystems. Our work investigates the dynamics of parasite spillover and spillback between wild salmon and aquaculture. It employs mathematical methods, such as dynamical systems and differential equations, in order to analyze the biological processes. It also involves large amounts of data collected by field researchers on wild and domestic salmon parasites. Over the years, the work has received a great deal of scientific and public scrutiny. Our work, showing how aquaculture can impact wild salmon populations, has been enthusiastically endorsed by some and has also been criticized by others. However, it has connected with policy makers and the general public, and we believe that it can make and has made a difference in how we manage aquaculture. A reflection on how scientific research can impact policy and decision making is given in a new book, Bioeconomics of Invasive Species: Integrating Ecology, Economics, Policy and Management [1]. The lecture has been recorded and can be viewed at http://beta.mbi.ohio-state.edu/video/player/?id=2760 Reference [1] R.P. Keller, M.A. Lewis, D.M. Lodge, J.F. Shogren, and M. Krko˘sek, Putting bioeconomic research into practice, in Bioeconomics of Invasive Species: Integrating Ecology, Economics and Management, edited by R.P. Keller, D.M. Lodge, M.A. Lewis, and J.F. Shogren, Oxford University Press, 2009, Chapter 13, pp. 266–284.

Chapter 10

Human Behavior

10.1 Predicting the Unpredictable — Human Behaviors and Beyond Andrea Tosin, Istituto per le Applicazioni del Calcolo No matter how surprising, outlandish, or even impossible it may seem, one of the next challenges of modern applied mathematics is the modeling of human behaviors. This has nothing to do with the control of minds. Rather, thanks to its innate reductionism, mathematics is expected to help shed some light on those intricate decision-based mechanisms which lead people to produce, mostly unconsciously, complex collective trends out of relatively elementary individual interactions. The flow of large crowds, the formation of opinions impacting socioeconomic and voting dynamics, the migration fluxes, and the spread of criminality in urban areas are examples which, although quite different, have two basic characteristics in common. First, individuals operate almost always on the basis of a simple one-to-one relationship. For instance, they try to avoid collisions with one another in crowds, or they discuss with acquaintances or are exposed to the influence of media about some issues and can change or radicalize their opinions. Second, the result of such interactions is the spontaneous emergence of group effects visible at a larger scale. For instance, pedestrians walking in opposite directions on a crowded sidewalk tend to organize in lanes, or the population of a country changes its political inclination over time, sometimes rising suddenly against the regimes. In all these cases, a mathematical model is a great tool for schematizing, simplifying, and finally showing how such a transfer from individual to collective behaviors takes place. Also, a mathematical model raises the knowledge of these phenomena, which is generally initiated mainly through qualitative observations and descriptions, to a quantitative level. As such, it allows one to go beyond the reproduction of known facts and face situations which have not yet been empirically reported or which would be impossible to test in practice. In fact, one of the distinguishing features of human behaviors is that they are hardly reproducible at one’s beck and call, just because they pertain to living, not inert “matter.” As a matter of fact, historical applications of mathematics to more “classical” physics (think, for example, of fluid or gas dynamics) are also ultimately concerned with the quantitative description and simulation of real-world systems, so what is new here? True, but what makes the story really challenging from the point of view of the mathematical 151

152

Chapter 10. Human Behavior

research is the fact that to date we do not have a fully developed mathematical model for the description of human behaviors. The point is that the new kinds of systems mentioned above urge applied mathematicians to face some hard stuff, which classical applications have only marginally been concerned with. Just to mention a few key points: • A nonstandard multiscale question. Large-scale collective behaviors emerge spontaneously from interactions among few individuals at a small scale. This is the phenomenon known as self-organization. Each individual is normally not even aware of the group s/he belongs to and of the group behavior s/he is contributing to, because s/he acts only locally. Consequently, no individual has full access to group behaviors or can voluntarily produce and control them. Therefore, models are required to adopt nonstandard multiscale approaches, which may not simply consist in passing from individual-based to macroscopic descriptions by means of limit procedures. In fact, in many cases it is necessary to retain the proper amount of local individuality also within a collective description. Moreover, the number of individuals involved is generally not as large as that of the molecules of a fluid or gas, which can justify the aforesaid limits. • Randomness of human behaviors. Individual interaction rules can be interpreted in a deterministic way only up to a certain extent, due to the ultimate unpredictability of human reactions. It is the so-called bounded rationality, which makes two individuals react possibly not the same, even if they face the same conditions. In opinion formation problems this issue is of paramount importance, for the volatility of human behaviors can play a major role in causing extreme events with massive impact known as black swans in the socioeconomic sciences. Mathematical models should be able to incorporate, at the level of individual interactions, these stochastic effects, which in many cases may not be schematized as standard white noises. • Lack of background field theories. Unlike inert matter whose mathematical modeling can be often grounded on consolidated physical theories, living matter still lacks a precise treatment in terms of quantitative theories whence to identify the most appropriate mathematical formalizations. If, on the one hand, this is a handicap for the “industrial” production of ready-to-use models, on the other hand it offers mathematics the great opportunity to play a leading role in opening new ways of scientific investigation. Mathematical models can indeed fill the quantitative gap by acting themselves as paradigms for exploring and testing conjectures. They can also put in evidence facts not yet empirically observed, whereby scientists can be motivated to perform new specific experiments aiming at confirming or rejecting such conjectures. Finally, mathematics can also take advantage of these applications for developing new mathematical methods and theories. In fact, nonstandard applications typically generate challenging analytical problems, whereby the role of mathematical research as a preliminary necessary step for mastering new models also at an industrial level is enhanced.

10.2 Budget Chicken Kent E. Morrison, American Institute of Mathematics The political wrangling over the government shutdown (and the looming debt ceiling) is more and more described as a game of CHICKEN. As you probably know, the game of CHICKEN is the suicidal, hormonally charged confrontation of two teenage boys driving

10.2. Budget Chicken

153

Figure 10.1. One possible outcome of the game of CHICKEN. Can c Stock Photo Inc./Cla78. Reprinted with permission.

down a highway straight at each other. Whoever swerves first loses, but if neither swerves they also lose. It does seem more complicated than that to me, but it could be instructive to analyze the game of CHICKEN from the perspective of classical game theory. For this we assign numerical values to the various outcomes for the players’ choices. I will use the numbers in Philip Straffin’s book Game Theory and Strategy published by the MAA [1]. There are two players A and B (Administration and Boehner). Each has two strategies: swerve or don’t swerve. The rows of the payoff matrix represent A’s choices and the columns the choices of B. There is a pair of numbers for each of the four outcomes with the first number being the payoff to A and the second the payoff to B. 

B choice swerve   swerve (0, 0) don’t (1, −2) A choice

don’t (−2, 1) (−8, −8)

For example, the pair (−2, 1) in the upper right corner means that if A swerves and B doesn’t, then A loses two units and B gains one unit. CHICKEN is not a zero-sum game. There are two Nash equilibria in the payoff matrix. These are the upper right and lower left corners, in which one player swerves and the other doesn’t. With these scenarios neither player can do better by switching to a different option when the other player does not switch. (The definition of a Nash equilibrium is just that: it is a simultaneous choice of strategies for all the players so that no player can improve his or her lot by switching under the assumption that the other players do not change their choices.) In addition, these Nash equilibria are optimal in the sense that there is no other outcome that improves the lot of at least one of the players without making it worse for another player. (This is called Pareto optimality.) There is also a Nash equilibrium among the mixed strategies, where a mixed strategy is a probabilistic mixture of the two pure strategies. That is, for each p between 0 and 1, there is the mixed strategy of swerving with probability p and not swerving with probability 1 − p. Then one can show that the mixed strategy with p = 67 (i.e., swerve with probability 67 , don’t swerve with probability 1 ) is also a Nash equilibrium, which means that neither player can do better by using a 7 different mixed strategy, assuming that the other player sticks with this one. In this case the payoff to each player is − 27 . Now, the payoffs are equal but this outcome is not Pareto optimal because both players can do better with the strategy of swerving, in which case each receives 0.

154

Chapter 10. Human Behavior

And so it seems that there is no satisfactory solution to the game of CHICKEN and related games such as the PRISONER’S DILEMMA—at least within the confines of classical game theory. For some current commentary on game theory and the budget stalemate read the interview with Daniel Diermeier in The Washington Post [2]. References [1] P.D. Straffin, Jr., Game Theory and Strategy, New Mathematical Library Series No. 36, Mathematical Association of America (MAA), 1993. [2] D. Matthews, How a game theorist would solve the shutdown showdown, The Washington Post, October 4, 2013, http://www.washingtonpost.com/blogs/wonkblog/wp/2013/10/04/how-a-gametheorist-would-solve-the-shutdown-showdown/

10.3 Mathematics and Conflict Resolution Estelle Basor, American Institute of Mathematics

Figure 10.2. Reprinted courtesy of the United Nations.

One of the main ideas behind the MPE2013 project was to showcase how mathematics solves the problems of the planet in ways that are analytical and useful. At the heart of this initiative is the belief that when one uses mathematical models, the results are unemotional and valid, at least given that the model is a good approximation to the problem at hand. The hope then is that those in power will pay attention to the mathematics. This

10.4. Modeling and Understanding Social Segregation

155

of course assumes something about the reasonableness of those in power, but for topics like climate change, the neutrality of mathematics should be an advantage in arguing for policy change. The November 2012 issue of the AMS Notices has an intriguing article about the use of mathematics to help solve the Middle East conflict [1]. The authors, Thomas L. Saaty and H.J. Zoffer, discuss how the analytic hierarchy process (AHP) can be used to help sort out the complex issues of the Israeli–Palestinian conflict. In their words, the advantages of the AHP in dealing with conflicts is “that the process creatively decomposes complex issues into smaller and more manageable segments. It also minimizes the impact of unrestrained emotions by imposing a mathematical construct, pairwise comparisons and prioritization with a numerical ordering of the issues and concessions.” The article reports in detail (and fills in some of the mathematics at its core) on a meeting of the two sides held in Pittsburgh, Pennsylvania, in August of 2011, where important progress was made in addressing the critical issues of the conflict. The Pittsburgh Principles were the outcome of that meeting, and they are described at the end in the article. Note: The article by Saaty and Zoffer generated a spirited outburst in the mathematics community; see [2] for a statement from the Notices editor, a selection of letters received by the editor, and a response from the authors. References [1] T.L. Saaty and H.J. Zoffer, Principles for implementing a potential solution to the Middle East conflict, Notices of the AMS 60 (2013) 1300–1322, http://www.ams.org/notices/201310/fea-saaty-with-link.pdf [2] Notices of the AMS 61 (2014) 240–242.

10.4 Modeling and Understanding Social Segregation Laetitia Gauvin, École Normale Supérieure Jean-Pierre Nadal, École Normale Supérieure Since the death of Nelson Mandela, there has been a lot of talk about social segregation. Apartheid was an institutionalized segregation, which was imposed by the dominant class using force. Similarly, the southern states of the United States imposed separation of blacks and whites in public life by law. When this separation lost its legal basis and the public’s attitudes changed, would one observe more of a mixed society? A collection of individual (or individualistic) strategies does not always lead to a collectively desirable outcome. Given this perspective, we address here the issue of social segregation. In the 1970s, the American political scientist and economist Thomas Schelling (Nobel Prize in economics, 2005) observed that individual behavior, when depending on other individuals’ behavior, can lead to social phenomena that are not necessarily anticipated or even desired. An early student of collective phenomena in the social sciences, Schelling introduced simple mathematical models, the analysis of which can serve to provide information for political decision making. Social segregation in urban environments is undoubtedly the issue addressed by Schelling that has given rise to the largest number of studies, although the general framework of Schelling’s work allows for the study of other types of social segregation as well. In the context of the ethnic segregation of blacks and whites in the United States, Schelling proposed a model that in the social sciences is often considered as the paradigm of selforganization. Figure 10.3 shows the final frame of a video that can be viewed at [1]. The video shows a grid with a large number of cells. A cell can be empty (white) or occupied by

156

Chapter 10. Human Behavior

Figure 10.3. Final stage in the evolution of a random initial population distribution, with a tolerance threshold S = 13 .

a red agent or a blue agent. At every time step, an agent randomly chooses one of its eight neighboring cells. If the cell is empty, the agent “decides” to move if the ratio of its neighbors of the same color is less than a certain threshold value S. Thus, the population becomes more “tolerant” as S gets smaller. Starting from a random distribution of the two colored agents on the grid and repeating this process many times, Schelling observed that the initial distribution rapidly evolved toward a globally segregated population of red and blue agents, even when the threshold value is as small as S = 13 . Thus, despite a strong tolerance level of each agent for a neighborhood unlike them, the individuals’ independent decisions lead to a totally segregated collective state. Recent investigations have shown that this phenomenon is generic, i.e., the result is qualitatively the same for a wide range of model parameters (tolerance threshold S, fraction of empty cells, etc.) and for a variety of similar models. More recently, some of these models have been reconsidered by researchers in various disciplines—sociologists, geographers, economists, computer scientists, physicists, mathematicians—often in interdisciplinary collaborative projects. Modern techniques, particularly those developed in statistical physics, applied mathematics, and scientific computing, enable increasingly detailed and complete explorations of the properties of these agent-based models, leading to a better understanding of their general characteristics, as well as their limits. The initial models can be made more complex by including economic features such as individual income levels, so comparisons can be made with real data. Eventually, by coupling the models with data, various scenarios can be tested to study the evolution of social mixing. This post is based on joint work by Laetitia Gauvin, Jean-Pierre Nadal, Jean Vannimenus, and Annick Vignes at the laboratories LPS (ENS Paris), CAMS (EHESS), and ERMES (Université Paris 2). Reference [1] http://mpt2013.fr/modeliser-comprendre-et-combattre-la-segregation-sociale/

10.5. Modeling the Evolution of Ancient Societies

157

10.5 Modeling the Evolution of Ancient Societies Estelle Basor, American Institute of Mathematics

Figure 10.4. 13th century Mongol horsemen on the attack (reimagined).

Another mathematical modeling success is highlighted in a September 23, 2013, ScienceDaily story that describes the evolution of ancient complex societies [1]. One interesting fact reported is that intense warfare is the evolutionary driver of complex societies. The findings accurately match the historical records. The study was done by a trans-disciplinary team at the University of Connecticut, the University of Exeter, and NIMBioS and is available as an open access article in Proceedings of the National Academy of Sciences [2]. To see a simulation go to [3] and for more information see the press release [4]. References [1] National Institute for Mathematical and Biological Synthesis (NIMBioS), Math explains history: Simulation accurately captures the evolution of ancient complex societies, ScienceDaily, September 23, 2013. [2] P. Turchin, T, Currie, E. Turner, and S. Gavrilets, War, space, and the evolution of Old World complex societies, Proc. Natl. Acad. Sci. USA 110 (2013) 16384–16389, doi: 10.1073/pnas.1308825110. [3] http://www.eurekalert.org/multimedia/pub/62059.php?from=249418 [4] C. Crawley, Math explains history: Simulation accurately captures the evolution of ancient complex societies, http://www.eurekalert.org/pub_releases/2013-09/nifm-meh091813.php

10.6 Networks in the Study of Culture and Society Elijah Meeks, Stanford University The use of computational methods to explore complex social and cultural phenomena is growing ever more common. Geographic information science in the service of better understanding the shape and scale of the Holocaust [1], natural language processing techniques leveraged to detect style and genre of 19th century literature, or the use of information visualization to present and interrogate each of these subjects is already happening. Among these techniques, it is the study of networks and how they grow that may be the most interesting. Modern mathematical network analysis techniques have been around for decades, whether as developed to identify centrality in social networks or to distort topography

158

Chapter 10. Human Behavior

to reflect topology in transportation networks, such as the work of geographer Waldo Tobler. But the growing accessibility of tools and software libraries to build, curate, and analyze networks, along with the growing prominence of such networks in our everyday lives, has led to a wealth of applications in digital humanities and computational social sciences. When we use networks to study culture and society, we perform an important shift in perspective away from the demographic and biographical to a focus on relationships. The study of networks is the study of the ties that bind people and places and objects, and the exhaustive details of those places and people, which are so important to traditional scholarship, are less important when they are viewed in a network. It is the strength and character of the bonds that define an actor’s place in a network, not the list of accomplishments that actor may have, though one would expect some correlation. In changing our perspective like this, we discover the nature of the larger system, and gain the ability to identify overlooked individuals and places that may have more prominence or power from a network perspective.

Figure 10.5. This classic map by Charles Joseph Minard of Napoleon’s March shows the losses sustained by the French Army during the 1812 Russian Campaign. The map succeeds in combining five different variables: the size of the army, time, temperature, distance, and geographic location.

Many of the networks studied by researchers are social networks, with the historical kind being the most difficult to approximate and comprehend. Historical networks deal with difficult problems of modeling and representation. In the 16th century Spanish scientists shared geographic locations and subject matter of study, but some gamed the system and claimed a connection to other, more prominent scientists or activity in fields that was not true. The China Biographical Database [2] has nearly 120,000 entries for Chinese civil servants, their kinship ties, their offices and posting, and the events in their lives, but only one-half of them have known affiliations. In the case of historical networks, the unevenness of the data may not be systematic, and it might even be the result of intentional misrepresentation. Other networks are not social networks per se. In “ORBIS: The Stanford Geospatial Network Model of the Roman world” [3, 4], the goal was to build a parsimonious transportation network model of the Roman world with which to compile and better understand movement of people and goods in that period and region. To do so required

10.6. Networks in the Study of Culture and Society

159

not only the tracing of Roman roads using GIS, but the simulation of sailing to generate coastal and sea routes to fill out the network. The result of such a model is to provide the capacity to plan a trip from Constantinople to Londinium in March and see the cost according to Diocletian’s Edict and the time according to a schematic speed for the vehicle selected. But more than that, the ORBIS network model is an argument about the shape and nature of the Roman world, and embedded in it are claims such as that the distance of England from the rest of Rome was variable, and that changing the capital—moving the center of the network—would have systematic effects on the nature of political control. Networks are inherently models that involve explicit, formal representation of the connection between individual elements in a system. But the accessibility of tools to represent and analyze such models has outstripped the familiarity with the methods for doing so. You can now calculate the eigenvector centrality of your network with the push of a button, but understanding what eigenvector centrality is still takes time and effort. More complex techniques for understanding the nature of networks, like the exponential random graph models studied at the AIM workshop from June 17–21, 2013, require even more investment to understand and deploy. But the results of the use of computational methods in the exploration of history and culture are worth that investment. It may be that information or data visualization will play a role in the greater adoption and understanding of these complex techniques. This is especially true as we move away from the static representation of data points and toward the visual representation of processes, such as Xueqiao Xu’s interactive visualization of network pathfinding [5]. Such visualizations make meaningful the processes and functions to audiences that may not be familiar with mathematical notation or programming languages. Scheidel, in his paper [4], utilizes dynamic distance cartograms—made possible as a result of creating a network—to express a Roman world view with a highly connected Mediterranean coastal core and inland frontiers. While this relatively straightforward transformation of geographic space to represent network distance could have been expressed with mathematical notation, data visualization is more accessible to a broader audience. Networks are allied with notions of social power, diffusion, movement, and other behavior that have long been part of humanities and social science scholarship. The interconnected, emergent, and systematic nature of networks and network analysis is particularly exciting for the study of culture and society. Other computational methods do not so readily promote the creation of systems and models like networks do. But doing so will often require dealing with issues of uncertainty and missing evidence, especially in the case of historical networks, and require a better understanding of how networks grow and change over time. It will also require some degree of formal and explicit definition of connection that reflects fuzzy social and cultural concepts that, until now, have only been expressed in linear narrative. References [1] The Spatial History Project: Holocaust Geographies, Stanford University, http://www.stanford.edu/group/spatialhistory/cgi-bin/site/project.php?id=1015 [2] China Biographical Database Project (CBDP), Harvard University, http://isites.harvard.edu/icb/icb.do?keyword=k16229 [3] ORBIS: The Stanford Geospatial Network Model of the Roman World, http://orbis.stanford.edu/ [4] W. Scheidel, The shape of the Roman world, http://orbis.stanford.edu/assets/Scheidel_59.pdf [5] X. Xu, Path finding visual, http://qiao.github.io/PathFinding.js/visual/

Chapter 11

Climate Change

11.1 The Discovery of Global Warming Hans Kaper, Georgetown University

Figure 11.1. Joseph Fourier (1768–1830).

“As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial rays, produces a local heightening of the temperature at the Earth’s surface.” Thus in 1862 John Tyndall described the key to climate change. He had discovered in his laboratory that certain gases, including water vapor and carbon dioxide (CO2 ), are opaque to heat rays. He understood that such gases high in the air help keep our planet warm by interfering with escaping radiation. This kind of intuitive physical reasoning had already appeared in the earliest speculations on how atmospheric composition could affect climate. It was in the 1820s that the French scientist Joseph Fourier (pictured above) first realized that the Earth’s atmosphere retains heat radiation. He had asked himself a deceptively simple question, of a sort that 163

164

Chapter 11. Climate Change

physics theory was just then beginning to learn how to attack: What determines the average temperature of a planet like the Earth? When light from the Sun strikes the Earth’s surface and warms it up, why doesn’t the planet keep heating up until it is as hot as the Sun itself? Fourier’s answer was that the heated surface emits invisible infrared radiation, which carries the heat energy away into space. He lacked the theoretical tools to calculate just how the balance places the Earth at its present temperature. But with a leap of physical intuition, he realized that the planet would be significantly colder if it lacked an atmosphere. (Later in the century, when the effect could be calculated, it was found that a bare rock at Earth’s distance from the Sun would be well below freezing temperature.) How does the Earth’s blanket of air impede the outgoing heat radiation? Fourier tried to explain his insight by comparing the Earth with its covering of air to a box with a glass cover. That was a well-known experiment—the box’s interior warms up when sunlight enters while the heat cannot escape. This was an overly simple explanation, for it is quite different physics that keeps heat inside an actual glass box, or similarly in a greenhouse. (As Fourier knew, the main effect of the glass is to keep the air, heated by contact with sun-warmed surfaces, from wafting away. The glass does also keep heat radiation from escaping, but that’s less important.) Nevertheless, people took up his analogy and trapping of heat by the atmosphere eventually came to be called “the greenhouse effect.” Reference [1] Simple Models of Climate Change, American Institute of Physics, http://www.aip.org/history/climate/simple.htm#L_M085

11.2 Letter to My Imaginary Teenage Sister – 1 Samantha Oestreicher, University of Minnesota Doing your school research paper on climate change sounds like a great idea! Let me see if I can get you started. I’ll even put a few references at the end in case you want to look those up for your school report (hint hint!). First, I totally agree, popular culture is becoming inundated with the buzz words “green,” “ecofriendly,” and “global warming,” but I’m not sure society is explaining things to you very well. You have some really good questions about what global warming means. Even your pop idol, Miley Cyrus, is singing “Everything I read is global warming, going green, I don’t know what all this means. . .” [1]. And if she doesn’t get it, then why should the adults expect you to understand? The truth is, no one really knows all the answers about the problem. The climate is really complicated, and scientists don’t always get things right the first try. It takes them a little while to figure something out, just like it takes you a little while to learn something new. (Remember all those cooking failures when you were young?) But we do know a LOT about climate change. And we do know that something needs to change, or we may be in some serious trouble. Okay, let’s talk about scientist lingo. The scientists who wrote the IPCC (or Intergovernmental Panel on Climate Change) report say, “A global assessment of data since 1970 has shown it is likely that anthropogenic warming has had a discernible influence on many physical and biological systems” [2]. So, what does that mean? The report is saying that it is likely that humans are affecting the world around us. There is even a note that clarifies: “Likely” means 66–90%. So we may—or may not—be affecting the global climate. Well, if that isn’t vague, I don’t know what is! But, maybe, just maybe, the statement has to be vague. There were “more than 2500 scientific expert reviewers, more than 800 contributing authors, and more than 450 lead authors” who worked on writing the IPCC report [3]. Okay, so you know how you and I can’t always agree? We are only two

11.2. Letter to My Imaginary Teenage Sister – 1

165

people. Now, imagine trying to get 800+ people to all agree on the same thing. It would be impossible! All of a sudden, that range of 66–90% is looking a little more reasonable. No matter what, all the scientists think it’s more than 50% likely that we, humans, are changing the climate. We are affecting our planet. (Well, there are people who think climate change is not our fault. But if a person can’t believe 800 of the top scientists who all agree, then do we really want to believe them?) The real problem that we should be worrying about is that we don’t know what’s going to happen to Earth under our influence. This is what the scientists are currently arguing about. What can we expect from the climate? What should we do?

Figure 11.2. How much salt in your climate pancakes? Reprinted courtesy of HassleFreeClipArt.com.

Now, imagine you are making pancakes. (It sounds random, but just trust me for a minute, okay?) If you add too much salt, then your pancakes start to taste funny. But a couple extra grains aren’t going to make a difference. However, there is some critical mass of salt which ruins the pancakes. And you can’t just take the salt out once it’s mixed up! The pancakes are ruined, and you have to start over. This is what we are doing to our atmosphere. Only, we are adding extra carbon and other GHGs (greenhouse gases) to the mix instead of salt. In our atmosphere carbon is measured in parts per million, or ppm, instead of teaspoons. So how much salt in supposed to be in our climate pancakes? Pre-industrial levels of carbon were around 275 ppm. This is going to be our baseline recipe value. We know from this cool science which uses really old ice that Earth has had atmospheric carbon values between 180 and 280 ppm for the last 800,000 years. We also know that global temperature is closely correlated to carbon levels [4]. The scientists from the IPCC think it’s “very likely” that GHGs, including carbon, are the cause of global warming (very likely means 90–99%) [5]. We now have 400 ppm instead of the baseline of 275 ppm! Our batter is getting pretty salty. Eww! Salty enough that the scientists are starting to wish we could throw it out and start over. But we only have one planet and one atmosphere. We can’t throw it out and start over. How much more salt are we willing to dump in our mixing bowl and still eat the pancakes? Love, Samantha P.S. Let me know if you have any further questions I can help with! References [1] M. Cyrus, Wake up America, Lyrics, Breakout, Hollywood Records, 2008. [2] S. Soloman et al., IPCC, Fourth Assessment Report, Working Group 2 (2007) Summary for Policymakers, p. 9. [3] Press Flyer announcing IPPC AR4, http://www.ipcc.ch/pdf/press-ar4/ipcc-flyerlow.pdf

166

Chapter 11. Climate Change

[4] S. Soloman et al., IPCC, Fourth Assessment Report, Working Group 1 (2007) Summary for Policymakers, p. 3. [5] S. Soloman et al., IPCC, Fourth Assessment Report, Working Group 1 (2007) Technical Summary, p. 24.

11.3 Letter to My Imaginary Teenage Sister – 2 Samantha Oestreicher, University of Minnesota I was thrilled to get your last letter. I’m glad to see you are looking at some of the references I sent you last time. Figuring out who is responsible for higher atmospheric levels and how to respond to climate change can be difficult. First, let’s talk about where the carbon is coming from. Some of my mathematical research is to try to prove that we need to stop adding carbon to the atmosphere, so I have a couple specific ideas for you to think about. One of the pieces of evidence I study is the Keeling curve, which is the upward curve of measured carbon in the atmosphere. The measurements are taken in Hawaii (pretty sweet location, right?). Well, scientists and mathematicians have actually figured out that we can determine where the carbon was released despite the location of the measurements [1]. So we can conclusively know which area of the world added the carbon to the atmosphere. The kicker is that most of the carbon is from industrialized nations like the U.S. and China. In 2000, the U.S. added more carbon to the atmosphere than every other country on Earth. I’ve attached a clever map I found of the world, where each country is scaled based on that country’s carbon emissions in 2000 [2].

Figure 11.3. Carbon dioxide released into the atmosphere, by country. Worldmapper, c reprinted with permission.

See how big the U.S. is? As Americans, I think it’s our responsibility to fix some of what we caused. Sadly, my research alone will not solve the problem of global warming. But there are lots of real things that anyone can do to decrease the amount of salt they add to the batter.

11.4. Global Warming and Uncertainties

167

So, to answer your second question: Yes! There are lots of ways that you can help. The IPCC reports that lifestyle choices can contribute to climate change mitigation across all sectors by decreasing GHG emissions [3]. There are these two guys, Robert Socolow and Stephen Pacala, who present 15 ways to reduce GHGs, any seven of which would hold the carbon emissions constant [4]. They are things like decreasing the amount of energy we use in our homes by 25% or using more wind power, stuff our society already knows how to use. You might try to convince your school to recycle more, put up solar panels, or use energy efficient air conditioners the next time they remodel. You could also drive less. . .. So, I agree with your idol Miley Cyrus when she says we need to “wake up America” [5]. I think we need to start passing laws and legislation to decrease the amount of greenhouse gasses we are emitting and put money into developing new greener technologies. We are putting too much carbon in the atmosphere, and the scientists aren’t sure what’s going to happen. We don’t know how much carbon is too much, and we don’t have a good way to pull it back out of the atmosphere. (Remember the salty pancake analogy from my last letter?) Thus, we, as a society, need to put some serious thought into the problem. The good news is, there are actions you can take that we already know will help. Love, Samantha P.S. As always, if you have any more questions, please send them my way. References [1] W. Buermann, B. Lintner, C. Koven, A. Angert, C. Tucker, and I. Fung, The changing carbon cycle at Mauna Loa Observatory, Proc. Natl. Acad. Sci. USA 104 (2007) 4249– 4254, http://www.pnas.org/cgi/doi/10.1073/pnas.0611224104 [2] SASI Group and M. Newman, Map 295, University of Michigan and University of Sheffield, 2006, http://www.worldmapper.org [3] S. Soloman et al., IPCC, Fourth Assessment Report, Working Group 3 (2007) Summary for Policymakers, p. 12. [4] R. Socolow and S. Pacala, A plan to keep carbon in check, Scientific American, September 2006. [5] M. Cyrus, Wake up America, Lyrics, Breakout, Hollywood Records, 2008.

11.4 Global Warming and Uncertainties Juan M. Restrepo, University of Arizona Because my research is concerned with uncertainty quantification (UQ) in climate data and dynamics, I am often asked whether global warming is occurring and whether human activities are responsible for the situation. Indeed, the global warming issue is informed by climate science and UQ. I have worked with the data, understand how they have been processed, and am familiar with the challenges involved in making the analyses. I have not seen any data or analysis that demonstrates that warming is not occurring. The data also show that this warming is correlated with human activity and that this warming event is not the same as other warming events in the Earth’s past. But global warming and humans’ role in it are risk-analysis questions. Just as correlations are high between cigarette smoking and cancer, there are data correlations between human activities and global warming. However, to date, no one knows exactly how cigarettes cause cancer. The causal relationship between human activities (e.g., burning hydrocarbons) and global warming is not fully understood. In the cigarette

168

Chapter 11. Climate Change

case, the risk analysis made it clear that it was better to curb smoking than not. No one waited for the causation mechanism to be fully elucidated. Unlike the cigarette case, the complex, global, climate change risk-analysis problem has no clear options. It’s not clear what should or could be done to reduce global warming. It is valid to ask, as some climate change debaters do, whether it would be better to spend (presumably) huge resources on tackling climate change right now, or to spend these resources on wiping out hunger, say, or ravaging diseases. It is also valid to point out that there are inconsistencies in climate model outcomes. But the data that is used to show a global warming trend do not rely on model forecasts. To claim that global warming does not occur or does not involve human activities because climate models are wrong is an obvious non-sequitur. Moreover, it demonstrates incompetence in the person’s knowledge of climate science: a global climate model is nothing more than a compendium of dynamics that fit best with our expectations of outcomes and data; there are no theorems in this business. And, furthermore, this is how most science outside of mathematics is done: via compelling evidence, not necessarily evidence beyond a shadow of a doubt. To say that current models have flaws is something that no climate modeler will disagree with. The challenge of improving these models is precisely what he or she has agreed to take on. Will energy conservation and a shift to renewable energy do the trick? No one really knows. But it does seem to be a sensible idea for political, environmental, and social reasons, beyond climate. People who have a self-interest in “denying” global warming fail to recognize the simple fact that climate scientists would be employed, whether the Earth is warming up or not. Climate scientists working on global warming trends would readily modify their conclusions if they were presented with data that show a different picture of what’s happening. We should be able to agree that any risk analysis on climate change that sidesteps the data or distorts the data and its uncertainties is irresponsible. Even more damaging than denying global warming are the deliberate efforts to curb research on climate science with the goal of changing the risk balance on climate change. The legacy of the efforts by environmentalists to slow down research in nuclear energy, as a way to change the risk balance on energy, is that we are now further away from knowing how to safely use this stuff. A similar situation is shaping up for the risk balance associated with climate. So what’s the fuss over a little bit of warming? If you live in the southwest of the U.S., you are familiar with pre-Columbian communities that disappeared because of sudden drought, so a “little” change, up or down, can be accompanied by drastic local changes, whether they are induced by us or otherwise. Natural or man-made, some of the people who stand to lose a lot from the change are those very people who rely on cheap and plentiful energy to make it through a change that can have devastating effects on society, the economy, and their wealth. The main reason I cannot offer more than a personal opinion on the implications of global warming is that I am not qualified to venture more than a personal opinion on the risk-analysis problem, even as it relates to something I know about—climate. The more we know about climate dynamics, the more informed any risk-analysis decision is. But the little I know of probability and the little I know of climate dynamics fall short on skill set, just as someone who might know something about probability and know how to design cars will be poorly equipped to design liability insurance instruments. As a concerned citizen, I am troubled that we do not have a plan. As a scientist, I focus on understanding climate and hope that my efforts and those of others might inform the people who want to take on the risk-analysis issue. As a concerned citizen, I applaud the carbon tax as a serious attempt to use market incentives that have worked so well in the

11.5. How to Reconcile the Growing Extent of Antarctic Sea Ice with Global Warming

169

hydrocarbon business, to lead to a shift in how we use resources. It improves our resilience and ability to cope with dramatic economic consequences of a changing environment. But I am a scientist, and I am not trained to venture an answer to the question whether such a proposal could shift the risk of global warming.

11.5 How to Reconcile the Growing Extent of Antarctic Sea Ice with Global Warming Hans Kaper, Georgetown University My colleague Hans Engler (Georgetown University) alerted me to an interesting article in Le Monde of March 31, 2013, entitled “En Antarctique, le réchauffement provoque une extension de la banquise” [1]. The article was based on a technical paper co-authored by four scientists from the Royal Netherlands Meteorological Institute (KNMI) in De Bilt, The Netherlands, “Important role for ocean warming and increased ice-shelf melt in Antarctic sea-ice expansion,” published online on the same day in Nature Geoscience [2]. The problem offers a nice challenge for mathematicians.

Figure 11.4. Antarctic sea ice. Reprinted with permission from Peter Bucktrout/British Antarctic Survey.

It is well known that sea ice has a significant influence on the Earth’s climate system. Sea ice is highly reflective for incident radiation from the Sun, and at the same time it is a strong insulator for the heat stored in the upper (mixing) layer of the ocean. While global warming causes Arctic sea ice to melt at a measurable and significant rate, sea ice surrounding Antarctica has actually expanded, with record extent in 2010. How can this somewhat paradoxical behavior be reconciled with global warming? Various explanations have been put forth. Usually, the expansion of the Antarctic sea ice is attributed to dynamic changes that induce atmospheric cooling. But the authors of the paper present an alternate explanation, which is based on the presence of a negative feedback mechanism. The authors claim that accelerated basal melting of Antarctic ice shelves is likely to have contributed significantly to sea-ice expansion. Observations indicate that melt water from Antarctica’s ice shelves accumulates in a cool and fresh surface layer that shields

170

Chapter 11. Climate Change

the surface ocean from the warmer deeper waters that are melting the ice shelves. Simulating these processes in a coupled climate model they found that cool and fresh surface water from ice-shelf melt indeed leads to expanding sea ice in austral autumn and winter. This powerful negative feedback counteracts Southern Hemispheric atmospheric warming. Although changes in atmospheric dynamics most likely govern regional seaice trends, their analyses indicate that the overall sea-ice trend is dominated by increased ice-shelf melt. Cool sea surface temperatures around Antarctica could offset projected snowfall increases in Antarctica, with implications for estimates of future sea-level rise. References [1] En Antarctique, le réchauffement provoque une extension de la banquise, Le Monde, March 31, 2013, http://www.lemonde.fr/planete/article/2013/03/31/en-antarctiquele-rechauffement-provoque-une-extension-de-la-banquise_3151102_3244.html

[2] R. Bintanja, G.J. van Oldenborgh, S.S. Drijfhout, B. Wouters, and C.A. Katsman, Important role for ocean warming and increased ice-shelf melt in Antarctic sea-ice expansion, Nature Geoscience, http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo1767.html

11.6 Rising Sea Levels and the Melting of Glaciers Christiane Rousseau, University of Montreal We regularly hear warnings by scientists of the significant rise of the sea level that will occur before the end of the century. The worst scenario usually predicts a rise of less than a meter before 2100. Where does this number come from? The common answer is that the rise of the sea level comes both from the melting of glaciers and the dilation of the seawater due to the increase of its temperature. I have made the exercise of calculating the volume of the glaciers of Greenland and Antarctica. The area of glaciers in Greenland is 1,775,637 km2 and their volume is 2,850,000 km3 . The area of Antarctica is 14,000,000 km2 and the thickness of ice is up to 3 km. If we take a mean thickness of 2 km, then this gives a volume of 28,000,000 km3 . Hence, the total volume of ice of the glaciers of Greenland and Antarctica is of the order of 30,850,000 km3 . Now, the area of the oceans is 335,258,000 km2 . Hence, if all glaciers were to melt and produce the same volume of water (OK, it is a little less, but the water will dilate when its temperature increases), we would see a rise of the sea level of 92 meters! Can we explain the difference? Of course, my model is very rough. It is not clear that all the new water will stay in the oceans. Some could percolate in the soil, and some could evaporate in the atmosphere. I have asked the question recently to Hervé Le Treut, from the Institut Pierre Simon Laplace in Paris. His answer was that ice melts slowly, and hence it takes much more than 90 years for all the glaciers to melt. But his answer raises another question: “Why do we stop our predictions in 2100? Is sustainability no longer necessary past 2100?”

11.7 Global Warming — Recommended Reading Hans Kaper, Georgetown University Global warming, one of the most important science issues of the 21st century, challenges the very structure of our society. It touches on economics, sociology, geopolitics, local politics, and individuals’ choice of lifestyle. For those interested in learning more

11.7. Global Warming — Recommended Reading

171

about the complexities of both the science and the politics of climate change, I recommend a nice little book by Mark Maslin, Global Warming, A Very Short Introduction, Oxford University Press, 2009 (ISBN 978-0-19-954824-8). Mark Maslin FRGS, FRSA is a Professor of Climatology at University College London. His areas of scientific expertise include causes of past and future global climate change and its effects on the global carbon cycle, biodiversity, rain forests, and human evolution. He also works on monitoring land carbon sinks using remote sensing and ecological models and international and national climate change policies.

Chapter 12

Biological Threats

12.1 Mathematics behind Biological Invasions – 1 Mark Lewis, University of Alberta When asked to give an invited lecture at the first ever Mathematical Congress of the Americas, I jumped at the chance. This would be an opportunity to meet new colleagues from the Americas and to share my interest in mathematical ecology. My talk focused on The Mathematics behind Biological Invasions, a subject near and dear to my heart. I enjoy talking about it for three reasons: it has a rich and beautiful history, going back to the work of Fisher, Kolmogorov, Petrovskii, Piskunov, and others in the 1930s [1, 2]; the mathematics is challenging and the biological implications are significant; and, finally, it is an area that is changing and growing quickly with much recent research. The major scientific question addressed in my talk was “How quickly will an introduced population spread spatially?” Here the underlying equations are parabolic partial differential equations or related integral formulations. The simplest models are scalar, describing growth and dispersal of a single species, while the more complex models have multiple components, describing competition, predation, disease dynamics, or related processes. Through the combined effects of growth and dispersal, locally introduced populations grow and spread, giving rise to an invasive wave of population density. Thus the key quantity of interest is the so-called spreading speed: the rate at which the invasive wave sweeps across the landscape. Ideally one would like to have a formula for this speed, based on model parameters, that could be calculated without having to numerically simulate the equations on the computer. It turns out that such a formula can be derived in some situations and not in others. My talk focused on when it was possible to derive a formula. One useful method for deriving a spreading speed formula is based on linearization of the spreading population about the leading edge of the invasive wave and then associating the spreading speed of the nonlinear model with that of the linearized model. If this method works, the spreading speed is said to be linearly determined. It turns out that the conditions for a linearly determined spreading speed, while well understood for scalar models, are challenging to analyze for multicomponent models of the sort that include interactions between species. In some cases, such as competition, the results have been worked out, but in many other cases it remains an open question. I was gratified that the talk generated discussion and questions, and I hope that the subsequent followup will result in new collaborations with colleagues in the Americas who are interested in similar questions. 173

174

Chapter 12. Biological Threats

References [1] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugen. 7 (1937) 355– 369. [2] A.N. Kolmogorov, I.G. Petrovskii, and N.S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech. 1 (1937) 1–25.

12.2 Mathematics behind Biological Invasions – 2 James Crowley, SIAM Invasive species are a big deal today. One need only do a simple Google search and see all the exotic species that are hitching a ride on container cargo to find a niche on a new continent. The U.S. Environmental Protection Agency (EPA) has a website devoted to invasive species [1], as does the U.S. National Oceanographic and Atmospheric Agency (NOAA) [2]. There are a lot of discussions in the scientific literature as well, addressing topics from ecology to biological diversity. Interestingly, there is a long history of contributions in the mathematical literature to this topic as well. One such example is the work of Mark Lewis, captured in part in his invited talk at the Mathematical Congress of the Americas in Guanajuato, Mexico in July 2013 [3] and in his blog post “Mathematics behind biological invasions” for the MPE2013 Daily Blog reproduced in Section 12.1. Mathematicians construct and analyze models of biological invasions, asking questions like “Can the invader establish itself and, if so, under what conditions?” and “Will the invading population spread and, if so, how fast?” Lewis, in his talk The Mathematics behind Biological Invasion Processes, looked at the second of these questions, focusing on the spread of populations. Such models must take into account the growth rate of the population under various conditions as well as the diffusion of the population. Populations may compete with other species or cooperate. Lewis gave two examples. The first example was the invasion of the grey squirrel into the U.K., a country where the red squirrel had been prevalent prior to the introduction of the grey squirrel in the 19th century [4]. Interacting species can compete for similar resources. Grey squirrels are larger and more aggressive than their cousins. Will their population eventually replace the red squirrel? Lewis discussed various mathematical models and the conclusion. A second example was West Nile virus, introduced into the U.S. in the late 1990s [5]. The spread of the virus depends on hosts (birds and mosquitoes in this case) and has spread rapidly since its introduction. A notable feature of mathematics is that seemingly disparate phenomena can have very similar mathematical models. The mathematics lends itself to analysis that can be applied generally to many different situations. Lewis traced some of the early history of such models, going back to the work of R.A. Fisher, through to modern dynamical systems. One can learn about the mathematics behind biological invasions by reading the post by Mark Lewis, “Mathematics behind biological invasions,” in Section 12.1 and listening to the recording of the talk online [6]. References [1] Invasive Species, EPA, http://www.epa.gov/glnpo/invasive/ [2] What is an Invasive Species?, NOAA, http://oceanservice.noaa.gov/facts/invasive.html [3] Mathematical Congress of the Americas 2013, Guanajuato, Mexico, August 5–9, 2013, http://www.mca2013.org/

12.3. Surges in Latent Infections: Mathematical Analysis of Viral Blips

175

[4] FAQ: Why are red squirrel populations in decline in Scotland?, Saving Scotland’s Red Squirrels, http://www.scottishsquirrels.org.uk/squirrel-facts/squirrel-faqs/why-arered-squirrel-populations-in-decline-in-scotland/ [5] West Nile Virus in the United States, Wikipedia, http://en.wikipedia.org/wiki/West_Nile_virus_in_the_United_States [6] Video Webcasting of lectures and interviews, Mathematical Congress of the Americas 2013, Guanajuato, Mexico, August 5–9, 2013, http://www.mca2013.org/en/component/content/article/181.html

12.3 Surges in Latent Infections: Mathematical Analysis of Viral Blips Karthika Muthukumaraswamy, SIAM

Figure 12.1. Three-dimensional rendering of a colony of pathogen viruses. Reprinted with permission from Natalia Lukiyanova.

Recurrent infection is a common feature of persistent viral diseases. It includes episodes of high viral production interspersed by periods of relative quiescence. These quiescent or silent stages are hard to study with experimental models. Mathematical analysis can help fill in the gaps. A recent paper [1] has presented a model to study persistent infections. In latent infections (a type of persistent infection), no infectious cells can be observed during the silent or quiescent stages, which involve low-level viral replication. These silent periods are often interrupted by unexplained intermittent episodes of active viral production and release. “Viral blips” associated with human immunodeficiency virus (HIV) infections are a good example of such active periods. “Mathematical modeling has been critical to our understanding of HIV, particularly during the clinically latent stage of infection,” says author Pei Yu. “The extremely rapid turnover of the viral population during this quiescent stage of infection was first demonstrated through modeling (David Ho, Nature, 1995), and came as a surprise to the clinical community. This was seen as one of the major triumphs of mathematical immunology: an extremely important result through the coupling of patient data and an appropriate modeling approach.” Recurrent infections also often occur due to drug treatment. For example, active antiretroviral therapy for HIV can suppress the levels of the virus to below-detection limits

176

Chapter 12. Biological Threats

for months. Though much research has focused on these viral blips, their causes are not well understood. Previous mathematical models have analyzed the reasons behind such viral blips and have proposed various possible explanations. An early model considered the activation of T-cells, a type of immune cell, in response to antigens. Later models attributed blips to recurrent activation of latently infected lymphocytes, which are a broader class of immune cells that include T-cells. Asymmetric division of such latently infected cells, resulting in activated cells and latently infected daughter cells, were seen to elicit blips in another study. These previous models have used exogenous triggers such as stochastic or transient stimulation of the immune system in order to generate viral blips. In this paper, the authors use dynamical systems theory to reinvestigate in-host infection models that exhibit viral blips. They demonstrate that no such exogenous triggers are needed to generate viral blips and propose that blips are produced as part of the natural behavior of the dynamical system. The key factor for this behavior is an infection rate which increases but saturates with the extent of infection. The authors show that such an increasing, saturating infection rate alone is sufficient to produce long periods of quiescence interrupted by rapid replication, or viral blips. These findings are consistent with clinical observations where even patients on the best currently available HIV therapy periodically exhibit transient episodes of viremia (high viral load in the blood). A number of reasons have been proposed for this phenomenon, such as poor adherence to therapy or the activation of a hidden reservoir of HIV-infected cells. “If adherence is the underlying factor, viral blips are triggered when the patient misses a dose or several doses of the prescribed drugs,” explains Yu. “If activation is the cause, blips may be triggered by exposure to other pathogens, which activate the immune system. Our work demonstrates that viral blips might simply occur as a natural cycle of the underlying dynamical system, without the need for any special trigger.” The authors propose simple 2- and 3-dimensional models that can produce viral blips. Linear or constant infection rates do not lead to blips in 2-, 3-, or 4-dimensional models studied by the authors. However, a 5-dimensional immunological model reveals that a system with a constant infection rate can generate blips as well. The models proposed in the paper can be used to study a variety of viral diseases that exhibit recurrent infections. “We are currently extending this approach to other infections, and more broadly to other diseases that display recurrence,” says Yu. “For example, many autoimmune diseases recur and relapse over a timescale of years, and once again, the triggers for episodes of recurrence are unknown. We would like to understand more fully what factors of the underlying dynamical system might be driving these episodic patterns.” Reference [1] W. Zhang, L.M. Wahl, and P. Yu, Conditions for transient viremia in deterministic in-host models: Viral blips need no exogenous trigger, SIAM J. Appl. Math. 73 (2013) 853– 881.

12.4 Prevention of HIV Using Drug-Based Interventions Jessica M. Conway, The Pennsylvania State University Viruses replicate by infecting a host’s healthy cells and hijacking those cells’ genetic machinery to make copies of itself. Figure 12.2 shows this process schematically for HIV, which infects primarily immune system cells, in particular helper T-cells.

12.4. Prevention of HIV Using Drug-Based Interventions

177

Figure 12.3 illustrates typical disease progression in a host: after an initial spike during the acute phase of infection, the virus slowly depletes T-cells, crippling the immune system. Average lifetime is on the order of 10 years. Antiretroviral treatments (ARTs) for HIV target viral replication and have proved very effective in improving quality and length of life in HIV+ individuals. They decrease viral load and allow the T-cell population to recover somewhat, as shown in Figure 12.4. But they are not a cure: at interruption of treatment, viral load jumps right back up and T-cell depletion proceeds as before. Another interesting use of HIV treatments is for infection prevention, taken shortly after accidental exposure to the virus. The hope is that viral replication will be halted before the infection can take hold. This use has proved effective in hospital settings. Starting with analyses of [1], using differential equation models fit to data to estimate viral production and clearance rates, we have relied on mathematical models to gain insights into HIV infection. While deterministic (differential equation) models remain the standard, stochastic modeling approaches have gained momentum as well (for example, see [2]). Basic viral dynamics models are deterministic models examining the interplay between virus and host cells. One approach we use to test hypotheses with regards to immune system activation and its role in HIV infection progression is building models and Figure 12.2. Schematic of HIV replication. Reprinted courtesy of Daniel Beyer and fitting them to data. We use statistical tools to evaluate goodness of fit and whether our Raul564. hypotheses are consistent with data. However, during the earliest stages of HIV infection—and in chronically infected, treated patients—viral and infected cell populations can be very low. In these regimes a deterministic approach, focusing on average behavior, is inappropriate. We utilize instead a nontraditional, stochastic approach: multitype, continuous-time branching formulations. Loosely speaking we consider coupled birth-and-death processes on different cell types. Such systems can be described by differential Chapman–Kolmogorov equations. From these equations, we derive differential equations for the probability generating function and use a novel numerical technique to extract probability distributions over time for the different cell types. We can also use this formulation to investigate small-probability, large deviations from the mean viral load (viral blips) [3], infection clearance times [3], and risk of infection after exposure to the HIV virus [4]. These clinically important quantities are inaccessible using traditional ODE formulations.

178

Chapter 12. Biological Threats

Figure 12.3. Typical infection timecourse [5]. Reprinted courtesy of Jurema Oliveira.

ARTs taken as post-exposure prophylaxis (PEP) have been shown to effectively prevent HIV infection after accidental exposure [4]. Current WHO and Canadian guidelines recommend initiation of PEP as early as possible (