Climate Change: Can Mathematics Help Clear the Air?

1. Climate Change: Can Mathematics Help Clear the Air? Christopher Jones University of North Carolina at Chapel Hill and University of Warwick Center for Applied Mathematics, Cornell University, February 2009

2. How do we know climate change is happening and accelerating? PHYSICS FACTS • Carbon in the atmosphere • Human induced • Greenhouse effect • Longer wavelength of reflected radiation Joseph Fourier, 1824 From: IPCC Report WG1, 2007

3. EVIDENCE OF A CHANGING CLIMATE

4. PREDICTION OF FUTURE CHANGE Model Earth System Real Earth System

5. Mathematical Replica of the Earth 3-dimensional grid: ocean/atmosphere at time Model will govern physical properties at each grid point: • Temperature • Pressure • Density • Velocity (wind speed, current) • Salinity (ocean) • Water vapor (atmosphere) Model advances measures of physical properties at grid points

6. conservation of mass • water vapour (atmosphere) • salinity (ocean) • conservation of energy • brings in all other processes Discretize (put on grid)  connect pieces of model (boundary conditions)  initialize  solve computationally

7. IPCC Projections IPCC WG1, 2007

8. Observations: Theory: FACTS PHYSICS carbon in atmosphere greenhouse effect EVIDENCE PREDICTION rising temperatures mathematical models

10. Carbon Chastity The First Commandment of the Church of the Environment By Charles Krauthammer Friday, May 30, 2008; Page A13 I'm not a global warming believer. I'm not a global warming denier. I'm a global warming agnostic who believes instinctively that it can't be very good to pump lots of CO2 into the atmosphere but is equally convinced that those who presume to know exactly where that leads are talking through their hats. Predictions of catastrophe depend on models. Models depend on assumptions about complex planetary systems -- from ocean currents to cloud formation -- that no one fully understands. Which is why the models are inherently flawed and forever changing. The doomsday scenarios posit a cascade of events, each with a certain probability. The multiple improbability of their simultaneous occurrence renders all such predictions entirely speculative. … Krauthammer as “Climate change denier denier”

11. Carbon Chastity … Environmentalists are Gaia's priests, instructing us in her proper service and casting out those who refuse to genuflect. (…) And having proclaimed the ultimate commandment -- carbon chastity -- they are preparing the supporting canonical legislation that will tell you how much you can travel, what kind of light you will read by, and at what temperature you may set your bedroom thermostat.

12. An Allegory for the Climate Change Debate The Theban Plays by Sophocles • Oedipus Rex: • Oracle of Delphi has prophesied that Oedipus will kill his father and marry his mother. • Unbeknownst to Oedipus, it is his father whom he kills in self-defense while he leaves Corinth. • He is hailed as a hero in Thebes when he defeats the Sphinx by solving a riddle. • He becomes king and takes the late king’s wife to be his own bride. • The oracle has proclaimed that the murderer of the king must be revealed and banished from Thebes in order to cure a new plague • Oedipus confronts the blind seer Tiresias who knows the truth.

13. 1984 TV production: Gielgud as Tiresias and Michael Pennington as Oedipus

14. Overriding atmosphere of dire predictions • Focus on human interaction between Tiresias and Oedipus Tiresias scientist/environmentalist Oedipus ccdenier/government official • Oedipus pushes • Doesn’t like answer • Makes accusations • Conjures up conspiracy

15. Climate of suspicion Global warming is a fact whatever its deniers - encouraged by a cool year - have to say Fred Pearce The Guardian, Saturday June 7, 2008 … Recently I attended a conference in Reading where some of the world's top experts discussed their failings. How their much-vaunted models of the world's climate system can't reproduce El Niños, or the "blocking highs" that bring heatwaves to Europe - or even the ice ages. How their statistical mimics of tropical climate are "laughable", in the words of the official report. This sudden humility was not unconnected with their end-of-conference call for the world to spend a billion dollars on a global centre for climate modelling. A "Manhattan project for the 21st century", as someone put it. …

16. Issues with Prediction Chaos: sensitivity to initial conditions Even in 3-dimensional systems, nearby initial conditions in a dynamical system can have VERY different destinies. Can we expect to forecast in a system of size 10,000,000? Lorenz Attractor This is perhaps the least of our problems! Maybe, it even helps.

17. Issues with Prediction Initialization: with what do we start the computations? Need: values of physical properties at initial time (and at boundaries) for example: above surface of land or ocean Below surface (for ocean) Possibilities: Take all available data and interpolate, or (viable method) spin-up using model while assimilating past data

18. Issues with Prediction SEA ICE Earth is a highly complex and detailed system: many processes are unresolved in climate models “SMALL” SCALE PROCESSES CLOUDS

19. Climate Science • Developing ever-more accurate models • Aim is to progressively improve approximation to “real” Earth system • Resolve more processes by increasing complexity of model • Predict averages by averaging predictions

20. Climate is a fast/slow system weather ensembles climate

21. Debate beyond the climate change debate • How do we quantify uncertainty in climate prediction? • Can we quantify uncertainty in climate prediction? Possible answers: Mean (average) Confidence intervals Full probability distribution function Likelihood estimates Underlying issue: How do we know that the “ensembles” will render a span of the possible predictions? If modelling groups, either consciously or by “natural selection”, are tuning their flagship models to fit the same observations, spread of predictions becomes meaningless: eventually they will all converge to a delta-function. Myles Allen, Oxford Multi-model ensembles Multi-parameter ensembles Multi, or stochastic parametrizations IPCC: Ensembles of opportunity

22. Purpose of models and their predictions UNDERSTANDING: Carl Wunsch, MIT • ECCO project: Estimating the Circulation and Climate of the Ocean • Uses ocean general circulation models to obtain optimal picture of ocean circulation. • Not forecasting, but “hindcasting” • Reveals current behavior at depth which is unobservable

23. Purpose of models and their predictions TESTING HYPOTHESES: Tom Knutson Climate Dynamics and Prediction Group, Geophysical Fluid Dynamics Laboratory • Will warming of ocean lead to greater hurricane activity? • Will Increased SST make hurricanes more intense?

24. Purpose of models and their predictions DECISION SUPPORT: Lenny Smith London School of Economics • Climate predictions judged by their usefulness (information content) for making decisions. • Example: Does the Thames Flood Barrier need to be rebuilt? Will it be adequate for 500 year floods or 100? Dave Stainforth, University of Exeter

25. Multi-scale dynamical systems abrupt transitions (ice break-up, Greenland glacier melt, change in thermohaline circulation of ocean, tipping point) weather disasters (hurricanes, volcanoes, …) climate Climate: slow variation (mean) Weather: fast (noise) Disasters: homoclinic orbit Abrupt transitions: heteroclinic orbits (catastrophes)

26. Extreme Weather Climate change is expected to increase the probability of extreme weather events

28. Flood of criticism from 1997 floods: Did faulty forecasts add to disaster? For six weeks, the National Weather Service had predicted a crest of 49 feet at Grand Forks. Then, over the five days before the river burst through its restraints, forecasters methodically revised it higher, eventually to 54 feet - a difference that spelled disaster in this pancake-flat region. From evacuation centers to city offices, the same anguished question now arises: How could forecasters have been so far off? 9th April, 2007: Forecasters are still stung by the spray-painted words, many of them obscene, on what was left of flood-ruined homes after the Red River swamped this city a decade ago. Mayor of East Grand Forks: “They blew it big!”

29. Importance of Data Computer models use data collected over years, translating stream flows into depth predictions for points along the river. But when stream flows are off the chart, as they were along the Red, the models go out the window. For accurate predictions, forecasters had to wait to measure actual flood depths at particular points and project them downstream to Grand Forks. Dean Braatz, then head of the weather service's river-forecasting effort for North Dakota and Minnesota

30. Data Assimilation truth estimate Gain Matrix

31. Bayes Data Assimilation truth estimate

32. But: computationally prohibitive, state ~

33. Techniques of Data Assimilation Deterministic techniques Statistical techniques • Variational methods (3DVAR, 4DVAR) • Kalman filter • Ensemble Kalman filter • Particle filtering • Dynamic Monte-Carlo • Sampling strategies Requirements: Gaussian Close to linear Requirement: Low dimension Climate: • DA in process models • Understanding historical climate • Getting the ocean right!

34. Global climate models Process models Impact models carbon cycle Clouds and hydrologic cycle Sea ice hurricanes flooding droughts sea level rise Socio-economic models carbon trading tax structure economic incentives

35. Role of Mathematics Community • multi-scale • multi-factoral • high-dimensional • nonlinear • data-driven Features of models: MATH CLIMATE Formulatingproblems and developing ideas for systems with above features in combinations that reflect those occurring in the climate