Mathematics and Climate: A New Partnership

Transcription

1 Mathematics and Climate: A New Partnership Hans G. Kaper kaper@mathclimate.org! Mathematics of Planet Earth 2013 ( invites you to enter a competition to design virtual or Department of Mathematics, Statistics, and Physics physical museum-quality exhibits (modules) Wichita State University October 2, 2013 The competition will provide Open Source material on the web that can be used by museums and schools around the world. The submitted modules will form the basis of a permanent virtual exhibition. The deadline for submissions is December 20, The first, second and third winning modules will receive respective prizes of US$ 5000, US$ 3000 and US$ 2000.

2 Summary 30-Second Elevator Speech Climate System Climate Modeling 2-Minute Station Break Conceptual Models Examples Energy Balance Model Overturning Circulation Milankovitch Cycles El Niño Southern Oscillation

3 30-Second Elevator Speech

4 30-Second Elevator Speech I am a mathematician.

5 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook.

6 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research.

7 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research. I thought all mathematics was done by 1900.

8 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research. I thought all mathematics was done by Mathematicians develop infrastructure. We create tools to solve problems of science and engineering more efficiently. At the same time, we push the boundaries of our own discipline.

9 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research. I thought all mathematics was done by Mathematicians develop infrastructure. We create tools to solve problems of science and engineering more efficiently. At the same time, we push the boundaries of our own discipline. That sounds interesting. Can you give me an example?

10 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research. I thought all mathematics was done by Mathematicians develop infrastructure. We create tools to solve problems of science and engineering more efficiently. At the same time, we push the boundaries of our own discipline. That sounds interesting. Can you give me an example? I ll give you three examples:

11 30-Second Elevator Speech I am a mathematician. You must be good at balancing your checkbook. I do research. I thought all mathematics was done by Mathematicians develop infrastructure. We create tools to solve problems of science and engineering more efficiently. At the same time, we push the boundaries of our own discipline. That sounds interesting. Can you give me an example? I ll give you three examples: 1. Mathematical algorithms for fast Google searches 2. Encryption techniques for safe internet banking 3. Uncertainty quantification for better weather forecasting

12 Take-Home Lesson Mathematics is everywhere.

13 Earth s Climate System

14 Earth s Climate System A Complex System Multicomponent system Atmosphere, troposphere, stratosphere Oceans, lakes, rivers (hydrosphere) Glaciers, ice shelves, sea ice, tundra, snow (cryosphere) Vegetation, living organisms, humans (biosphere) Soil, rocks, deserts (lithosphere) Fully coupled system Ocean atmosphere (evaporation, precipitation) Ice ocean (salinity, temperature) Atmosphere biosphere (CO2, water vapor)... Multiscale system Space (cloud formation to ocean currents) Time (atmospheric chemistry to ice ages)

15 Hierarchy of Models Comprehensive models General Circulation Models (GCMs) Atmosphere, ocean, cryosphere,... Parameterization of sub-scale phenomena Large-scale numerical simulations Intermediate complexity models Single component, feedback mechanism, etc. Computational experiments Conceptual models Energy balance, box models, etc. Ordinary differential equations Back-of-the-envelope computations

16 2-Minute Station Break Mathematics and Climate is a timely textbook aimed at students and researchers in mathematics and statistics who are interested in current issues of climate science, as well as at climate scientists who wish to become familiar with qualitative and quantitative methods of mathematics and statistics. The authors emphasize conceptual models that capture important aspects of Earth s climate system and present the mathematical and statistical techniques that can be applied to their analysis. Topics from climate science include the Earth s energy balance, temperature distribution, ocean circulation patterns such as El Niño Southern Oscillation, ice caps and glaciation periods, the carbon cycle, and the biological pump. Among the mathematical and statistical techniques presented in the text are dynamical systems and bifurcation theory, Fourier analysis, conservation laws, regression analysis, and extreme value theory. The following features make Mathematics and Climate a valuable teaching resource: Issues of current interest in climate science and sustainability are used to introduce the student to the methods of mathematics and statistics. The mathematical sophistication increases as the book progresses; topics can thus be selected according to interest and level of knowledge. Each chapter ends with a set of exercises that reinforce or enhance the material presented in the chapter and stimulate critical thinking and communication skills. Mathematics and Climate is intended for mathematicians, statisticians, data scientists, and geoscientists in academia, national laboratories, and public service organizations interested in current issues of climate and sustainability. It is written at the level of advanced undergraduate and beginning graduate students and assumes only basic familiarity with linear algebra, calculus, elementary differential equations, and statistics. Hans Kaper is affiliated with Georgetown University and is Co-Director of the Mathematics and Climate Research Network ( He spent most of his professional career at Argonne National Laboratory and served as Program Director for Applied Mathematics at the National Science Foundation. He is a corresponding member of the Royal Netherlands Academy of Sciences and was named a SIAM Fellow in Dr. Kaper is Editor-in-Chief of SIAM News, a member of the SIAM Committee on Science Policy, and Chair of the SIAM Activity Group on Dynamical Systems. Hans Engler is Professor of Mathematics at Georgetown University, where he has been since 1984 and served as Department Chair in the 1990s. He has also served as Program Director for Applied Mathematics at the National Science Foundation. He was the Founding Director of Georgetown University s MS program in mathematics and statistics. For more information about SIAM books, journals, conferences, memberships, or activities, contact: Society for Industrial and Applied Mathematics 3600 Market Street, 6th Floor Philadelphia, PA USA siam@siam.org H. Kaper H. Engler MATHEMATICS & Climate Kaper Engler MATHEMATICS &Climate ISBN OT131 Hans Kaper Hans Engler OT131 OT131_Kaper-Engler_coverC.indd 1 7/25/2013 1:31:04 PM

17 Mathematics and Climate Table of Contents Ch. 1: Climate and Mathematics Ch. 2: Earths Energy Budget Ch. 3: Oceans and Climate Ch. 4: Dynamical Systems Ch. 5: Bifurcation Theory Ch. 6: Stommel s Box Model Ch. 7: Lorenz Equations Ch. 8: Climate and Statistics Ch. 9: Regression Analysis Ch. 10: Mauna Loa CO 2 Data Ch. 11: Fourier Transforms Ch. 12: Zonal Energy Budget Ch. 13: Atmosphere and Climate Ch. 14: Hydrodynamics Ch. 15: Climate Models Ch. 16: El Niño Southern Oscillation Ch. 17: Cryosphere and Climate Ch. 18: Biogeochemistry Ch. 19: Extreme Events Ch. 20: Data Assimilation Appendix A: Units and Symbols Appendix B: Glossary Appendix C: MATLAB Codes Bibliography Index

18 Mathematics and Climate A Textbook Advanced undergraduates and beginning graduate students in the mathematical sciences Applied mathematics researchers in academia and national laboratories Mathematicians, statisticians and data scientists in public service organizations Climate scientists interested in mathematical and statistical techniques Anyone interested in modeling and qualitative analysis of Earth s climate system Prerequisites Familiarity with linear algebra, calculus, and elementary differential equations Basic knowledge of statistics

19 Mathematics and Climate Publication Details Society for Industrial and Applied Mathematics (SIAM) To appear, October 28, 2013 xx pages Printed edition, list price $59.00 Individuals SIAM members $41.30 Non-SIAM members 20% discount ebook edition, Google Play Institutions (libraries), download individual chapters

20 Energy Balance A Very Simple Model Energy received (UV radiation, from the Sun) E in = (1 α)πr 2 S 0 Energy emitted (IR radiation, from the Earth) E out = 4πR 2 σt 4 Energy balance, E in = E out = T = K

21 Energy Balance A Very Simple Model Energy received (UV radiation, from the Sun) E in = (1 α)πr 2 S 0 Energy emitted (IR radiation, from the Earth) E out = 4πR 2 σt 4 Energy balance, E in = E out = T = K Greenhouse effect, E in = εe out = T = 288 K

22 Energy Balance A More Realistic Model Snow and ice reflect more radiation than water and land Albedo (α) depends on temperature (T ) { 0.7 if T < 250 K α(t ) = 0.3 if T > 280 K

23 Energy Balance Multiple Solutions Three equilibria, T1 > T 2 > T 3 T1 = 288 K (stable) T 2 (unstable) T 3 = 234 K (stable) Current climate, T1 = 288 K Stable on time scale of a million years Includes all recent ice ages Snowball Earth, T3 = 234 K Complete glaciation, ice cover several kilometers thick Geological evidence Up to four occurrences during the Neoproterozoic age (between 750 and 580 million years ago) Take-home lesson Earth s climate system can have multiple equilibrium states

24 Great Ocean Conveyor Belt I I Driven by density differences Density depends on temperature and salinity I Thermohaline Circulation (THC)

25 North-Atlantic Overturning Circulation Two-Box Model Temperature T, Salinity S, Density ρ = ρ 0 (1 αt + βs) Flow, q = k ρ 1 ρ 2 ρ 0 = k (α(t 2 T 1 ) β(s 2 S 1 )) Virtual salt flux, H (evaporation, precipitation, runoff)

26 Governing Equations Introduce average temperature T and salinity S Use temperature and salinity anomalies dt 1 = c(t + T 1 ) + q (T 2 T 1 ) dt dt 2 = c(t T 2 ) q (T 2 T 1 ) dt ds 1 = H d(s + S 1 ) + q (S 2 S 1 ) dt ds 2 = H + d(s S 2 ) q (S 2 S 1 ) dt q = k (α(t 2 T 1 ) β(s 2 S 1 )) External forcing: evaporation, precipitation, runoff (H) Internal dynamics: temperature vs. salinity (T, S)

27 External Forcing Evaporation, Precipitation, Runoff Ignore temperature equations, fix T 2 > T 1, focus on salinity Ignore salt exchange with neighboring oceans, d = 0 ds 1 = H + q (S 2 S 1 ) dt ds 2 = H q (S 2 S 1 ) dt q = k (α(t 2 T 1 ) β(s 2 S 1 )) Note: S 1 + S 2 constant (conservation of salt) Remaining variable, S 2 S 1 d(s 2 S 1 ) dt = 2H 2k q (S 2 S 1 )

28 Two-Box Model 1-D Dynamical System Dimensionless variable, x = β(s 2 S 1 ) α(t 2 T 1 ) q/k Dimensionless flow, f = α(t 2 T 1 ) f (x) = 1 x Dimensionless surface salinity flux λ = βh/k 2 (α(t 2 T 1 )) 2 (forcing parameter) Rescale time, τ = 2α(T 2 T 1 )k 2 t Dynamical system ẋ = λ 1 x x = d dτ

29 1-D Dynamical System Bifurcation Equilibrium solutions x1 = 1 2 (1 1 4λ) if 0 < λ < 1 4 linearly stable, T-mode x2 = 1 2 ( λ) if 0 < λ < 1 4 unstable x3 = 1 2 ( λ) for any λ > 0 linearly stable, S-mode Bifurcation diagram

30 Two-Box Model Take-Home Lessons - 1 Two stable equilibrium states Current state, q > 0, temperature-driven (T-mode) Alternate state, q < 0, salinity-driven (S-mode) Model predicts reversal of overturning circulation Increasing salinity differential ( virtual salt flux )

31 Internal Dynamics Temperature vs. Salinity dt 1 = c(t + T 1 ) + q (T 2 T 1 ) dt dt 2 = c(t T 2 ) q (T 2 T 1 ) dt ds 1 = H d(s + S 1 ) + q (S 2 S 1 ) dt ds 2 = H + d(s S 2 ) q (S 2 S 1 ) dt q = k (α(t 2 T 1 ) β(s 2 S 1 )) Ignore virtual salt flux, H = 0 Note: T 1 + T 2 constant, S 1 + S 2 constant Remaining variables, T 2 T 1 and S 2 S 1

32 Stommel s Box Model d(s 2 S 1 ) = d(2s (S 2 S 1 )) 2 q (S 2 S 1 ) dt d(t 2 T 1 ) = c(2t (T 2 T 1 )) 2 q (T 2 T 1 ) dt q = k (β(s 2 S 1 ) α(t 2 T 1 )) Dimensionless variables, x = S 2 S 1 2S, y = T 2 T 1 2T Rescale time, τ = ct; δ = d/c Dimensionless flow, f = 2q Rx y f (x, y) = c λ Parameters, R = (βs )/(αt ), λ = c/(4αk) Dynamical system ẋ = δ(1 x) f x ẏ = 1 y f y

33 Two-Box Model 2-D Dynamical System Equilibrium solutions R = 2, δ = 1 λf (x, y) = φ(f ), φ(f ) = δr δ + f f

34 2-D Dynamical System Bifurcation Equilibrium solutions (x1, y 1 ), f (x1, y 1 ) > 0 q < 0 ( c, d, e, g ) (x2, y 2 ), f (x2, y 2 ) < 0 q > 0 ( b ) (x3, y 3 ), f (x3, y 3 ) < 0 q > 0 ( a ) (x2, y2 ) and (x3, y3 ) only for certain combinations of R and δ and if λ is sufficiently small Bifurcation diagram (R = 3 2, δ = 4 5 )

35 Stommel s Box Model Phase portrait c : Stable spiral point b : Saddle point (unstable) a : Stable node

36 Two-Box Model Take-Home Lessons - 2 Two stable equilibrium states Current state, q > 0, temperature-driven (T-mode) Alternate state, q < 0, salinity-driven (S-mode) Model predicts reversal of overturning circulation Decreasing the strength of the thermohaline circulation

37 Paleoclimate Box TS.6 Figure 1 - AR4 WGI Technical Summary Cyclic variation of insolation (incident Languages solar IPCC web pages radiation) Search Home Eccentricity, E Elliptical orbit Organization IPCC Fourth Assessment Report: Climate Change 2007 Working Groups / Task Obliquity, Force Activities T Tilt of equatorial plane Climate Change 2007: Working Group I: The Physical Science Basis Calendar of Meetings Meeting Documentation Precession, P Rotation of spin axis Back to report News and Events Publications and Data Reports Technical Papers Supporting Material Figures and Tables Glossary Presentations and Speeches Press Information Links Contact The Nobel Foundation IPCC honoured with the 2007 Nobel Peace Prize IPCC Phone: /84/54 Box TS.6, Figure 1. Schematic of the Earth s orbital changes (Milankovitch cycles) that drive the ice age cycles. T denotes changes in the tilt (or obliquity) of the Earth s axis, E denotes changes in the eccentricity of the orbit and P denotes precession, that is, changes in the direction of the axis tilt at a given point of the orbit. {FAQ 6.1, Figure 1} Integrate equations of celestial mechanics 4.5 million years (Laskar, 2004)

38 Milankovitch Cycles Eccentricity 400 Kyr, 100 Kyr period

39 Milankovitch Cycles Obliquity 41 Kyr period

40 Milankovitch Cycles Precession 23 Kyr period

41 Milankovitch Cycles Q 65 vs. Climate Record Obliquity (41) Eccentricity (100?) Precession (23)

42 Milankovitch Cycles Conclusions Observations Correlation between ice ages and variations in Earth s orbit Disagreement about relative importance of orbital parameters Open questions Climate record reconstructed from proxy data Feedback effects Transition from 41 Kyr cycle to 100 Kyr cycle around 1 Myr BP Take-home lesson Milankovitch theory is not the whole story

43 El Niño Southern Oscillation (ENSO) Normal conditions Trade winds blow across the tropical Pacific toward the West Sea surface near Indonesia higher than off South America Upwelling of cold water off South American coast, lower SST Cold water is nutrient-rich, high level of primary productivity El Niño conditions Weakened trade winds across the tropical Pacific Depression of thermocline off coast of South America Decreased upwelling of cold water off South America Supply of nutrient-rich water cut off Southern Oscillation Atmospheric counterpart to El Niño Weakened trade winds

44 ENSO Conceptual Model 1. Weakening trade winds in central Pacific 2. Warm equatorial Kelvin wave flowing East, 3. Cold off-equatorial Rossby waves flowing West 4. Rossby wave reflected, flows East along equator 5. Kelvin wave reaches South American coast in 1-2 months 6. Rossby wave reaches South American coast in 6 months 7. El Niño terminates

45 ENSO Mathematical Model Prognostic variable, T SST at point where equator meets eastern boundary of Pacific Kelvin wave increases T, Rossby wave decreases T Time for Kelvin wave to cross Pacific, τ K Time for Rossby wave to cross Pacific, τ R Nonlinear damping term to stabilize the system Mathematical model Ṫ (t) = at (t 1 2 τ K ) bt (t ( 1 2 τ R + τ K )) c(t (t)) 3

46 ENSO Case τ K = 0 Special case, τ K = 0 Ṫ (t) = at (t) bt (t 1 2 τ R) c(t (t)) 3 Periodic solutions, τ R = 1, ω 7 yrs a = 1.6, b = { 2.6 (solid curve) 2.7 (dashed curve), c = 0.1

47 Challenges for the Mathematical Sciences Existence of a global attractor (Dynamical systems) Weather vs. climate Climate sensitivity to radiative forcing (Dynamical systems) Assessment (IPCC, Intergovernmental Panel on Climate Change) Convergence of computational models (Numerical analysis) Grid refinement affects parameterization Averaging over ensemble of (in)dependent models (Statistics) Metrics of model reliability (IPCC: 23 GCMs)

48 Thank You!