Climate Change: Can Mathematics Help Clear the Air?

Christopher JonesUniversity of North Carolina at Chapel Hilland University of Warwick

Center for Applied Mathematics, Cornell University, February 2009

How do we know climate change is happening and accelerating?

FACTS PHYSICS

• Carbon in the atmosphere• Human induced

From: IPCC Report WG1, 2007

• Greenhouse effect• Longer wavelength of reflected

radiation

Joseph Fourier, 1824

EVIDENCE OF A CHANGING CLIMATE

Real Earth System

Model Earth System

PREDICTION OF FUTURE CHANGE

Mathematical Replica of the Earth

( , , )i j kx y z at timent

3-dimensional grid: ocean/atmosphere

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

1n nt t

12

Duu p gk

Dt

• 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

IPCC WG1, 2007

IPCC Projections

FACTS PHYSICS

EVIDENCE PREDICTION

Observations: Theory:

carbon in atmosphere greenhouse effect

rising temperatures mathematical models

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.

…

Carbon Chastity

The First Commandment of the Church of the EnvironmentBy Charles KrauthammerFriday, May 30, 2008; Page A13

Krauthammer as “Climate change denier denier”

…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.

Carbon Chastity

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.

The Theban Plays bySophocles

An Allegory for the Climate Change Debate

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

Overriding atmosphere of dire predictionsFocus on human interaction between Tiresias and Oedipus

Oedipus pushes Doesn’t like answer

Makes accusationsConjures up conspiracy

Tiresias scientist/environmentalist Oedipus ccdenier/government official

…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.

…

Climate of suspicionGlobal warming is a fact whatever its deniers - encouraged by a cool year - have to say

Fred Pearce The Guardian, Saturday June 7, 2008

Issues with PredictionChaos: 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?

This is perhaps the least of our problems! Maybe, it even helps.

Lorenz Attractor

Issues with PredictionInitialization: with what do we start the computations?

Need: values of physical properties at initial time (and at boundaries)

0 0

0 0

( , , , 0); ( , , , 0)

( , , , 0); ( , , , 0)

T T x y z t u u x y z t

x y z t p p x y z t

( , )sz z x y above surface of land or ocean

( , )sz z x y

for example:

Below surface (for ocean)

Possibilities:1.Take all available data and interpolate, or2.(viable method) spin-up using model while assimilating past data

Issues with PredictionEarth is a highly complex and detailed system: many processes are unresolved in climate models

CLOUDS

SEA ICE

“SMALL” SCALE PROCESSES

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

Climate is a fast/slow system

weather

climate

ensembles

Debate beyond the climate change debate

How do we quantify uncertainty in climate prediction?Can we quantify uncertainty in climate prediction?

Possible answers:1.Mean (average)2.Confidence intervals3.Full probability distribution function4.Likelihood estimates

Underlying issue: How do we know that the “ensembles” will render a span of the possible predictions?

1. Multi-model ensembles2. Multi-parameter ensembles3. Multi, or stochastic parametrizations

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, OxfordIPCC: Ensembles of opportunity

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

Purpose of models and their predictionsTESTING HYPOTHESES:

Tom KnutsonClimate Dynamics and Prediction Group, Geophysical Fluid Dynamics Laboratory

• Will warming of ocean lead to greater hurricane activity?

• Will Increased SST make hurricanes more intense?

Lenny SmithLondon School of Economics

Purpose of models and their predictionsDECISION SUPPORT:

Dave Stainforth,University of Exeter

•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?

Barrier Closures

0

10

20

30

1983 1993 2003

Multi-scale dynamical systems

weather

climate

disasters (hurricanes, volcanoes, …)abrupt transitions (ice break-up, Greenland glacier melt, change in thermohaline circulation of ocean, tipping point)

Climate: slow variation (mean)Weather: fast (noise)Disasters: homoclinic orbitAbrupt transitions: heteroclinic

orbits (catastrophes)

Extreme WeatherClimate change is expected to increase the probability of extreme weather events

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?

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!”

For accurate predictions, forecasters had to wait to measure actual flood depths at particular points and project them downstream to Grand Forks.

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. Dean Braatz, then head of the weather service's river-forecasting effort for North Dakota and Minnesota

f f f1 1 2 1 1

Model forecast:

( ), ( ), ( )x t x t tP

t t1 1 2 1( ), ( )x t x t

t t1 0 2 0( ), ( )x t x t

f f f1 0 2 0 0

Initial conditions:

( ), ( ), ( )x t x t tP

1t t

o t1 1 1 1

Measurement:

( ) ( )y t x t

a a a1 1 2 1 1

State estimate:

( ), ( ), ( )x t x t tP

Gain Matrix

Data Assimilation

truth

estimate

0t t

f f f1 1 2 1 1

Model forecast:

( ), ( ), ( )x t x t tP

t t1 1 2 1( ), ( )x t x t

t t1 0 2 0( ), ( )x t x t

f f f1 0 2 0 0

Initial conditions:

( ), ( ), ( )x t x t tP

1t t

o t1 1 1 1

Measurement:

( ) ( )y t x t

a a a1 1 2 1 1

State estimate:

( ), ( ), ( )x t x t tP

posterior obs prior( ) ( )P x y P y x P x

Bayes

Data Assimilation

truth

estimate

0t t

0 1

2

Forecast step:

( , ) ( , )

( )( ) 1

2iji

i i j

p t p t

Q pM pp

t x x x

x x

o1 1

oo 1

1 o1

Bayes step (update/analysis):

( , ) ( , | )

( | ) ( , )( , | )

( | ) ( , )

p t p t

p p tp t

p p t d

x x y

y x xx y

y z z z

But: computationally prohibitive, state ~ 610

Techniques of Data Assimilation

Deterministic techniques Statistical techniques

• Variational methods (3DVAR, 4DVAR)• Kalman filter• Ensemble Kalman filter

Requirements:1.Gaussian2.Close to linear

• Particle filtering• Dynamic Monte-Carlo• Sampling strategies

Requirement: Low dimension

Climate: •DA in process models•Understanding historical climate•Getting the ocean right!

Global climate models

Process models Impact models

Socio-economic models

carbon cycleClouds and hydrologic cycle

Sea ice

hurricanesfloodingdroughts

sea level rise

carbon tradingtax structure

economic incentives

Role of Mathematics Community

Features of models:

• multi-scale• multi-factoral• high-dimensional• nonlinear • data-driven

Formulating problems and developing ideas for systems with above features in combinations

that reflect those occurring in the climate