Transcript Climate change: can mathematics help clear the air?

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
How do we know climate change is
happening and accelerating?
FACTS
• Carbon in the atmosphere
• Human induced
From: IPCC Report WG1, 2007
PHYSICS
• Greenhouse effect
• Longer wavelength of reflected
radiation
Joseph Fourier, 1824
EVIDENCE OF A CHANGING CLIMATE
PREDICTION OF FUTURE CHANGE
Model Earth System
Real Earth System
Mathematical Replica of the Earth
3-dimensional grid: ocean/atmosphere
( xi , y j , zk )
at time
tn
Model will govern physical
properties at each grid point:
Model advances measures of
physical properties at grid points
tn  tn1
•Temperature
•Pressure
•Density
•Velocity (wind speed, current)
•Salinity (ocean)
•Water vapor (atmosphere)
Du
1
 2 u  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 Projections
IPCC WG1, 2007
Observations:
FACTS
carbon in atmosphere
EVIDENCE
rising temperatures
Theory:
PHYSICS
greenhouse effect
PREDICTION
mathematical models
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”
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.
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.
1984 TV production: Gielgud as Tiresias and Michael Pennington as Oedipus
Overriding atmosphere of dire predictions
Focus on human interaction between Tiresias
and Oedipus
Tiresias
scientist/environmentalist
Oedipus
Oedipus pushes
Doesn’t like answer
Makes accusations
Conjures up conspiracy
ccdenier/government official
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.
…
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.
Issues with Prediction
Initialization: with what do we start the computations?
Need: values of physical properties at
initial time (and at boundaries)
for example:
T0  T ( x, y, z, t  0); u0  u ( x, y, z, t  0)
0   ( x, y, z, t  0); p0  p( x, y, z, t  0)
z  zs ( x, y)
z  zs ( x, y)
above surface of land or ocean
Below surface (for ocean)
Possibilities:
1. Take all available data and interpolate, or
2. (viable method) spin-up using model while assimilating past data
Issues with Prediction
SEA ICE
Earth is a highly complex and detailed system:
many processes are unresolved in climate models
“SMALL” SCALE PROCESSES
CLOUDS
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 intervals
3. Full probability distribution function
4. 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
1. Multi-model ensembles
2. Multi-parameter ensembles
3. Multi, or stochastic parametrizations
IPCC: Ensembles of opportunity
“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
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 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?
Purpose of models and their predictions
DECISION SUPPORT:
Lenny Smith
London School of Economics
30
•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
20
10
0
1983
1993
2003
Dave Stainforth,
University of Exeter
Multi-scale dynamical systems
weather
disasters (hurricanes, volcanoes, …)
climate
Climate: slow variation (mean)
Weather: fast (noise)
Disasters: homoclinic orbit
Abrupt transitions: heteroclinic
orbits (catastrophes)
abrupt
transitions
(ice break-up,
Greenland
glacier melt,
change in
thermohaline
circulation of
ocean, tipping
point)
Extreme Weather
Climate 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!”
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
For accurate predictions, forecasters
had to wait to measure actual flood
depths at particular points and
project them downstream to Grand
Forks.
Data Assimilation
x1t (t1 ), x2t (t1 )
t  t1
State estimate:
truth
x1a (t1 ), x2a (t1 ), P a (t1 )
estimate
Model forecast:
x1f (t1 ), x2f (t1 ), P f (t1 )
x1t (t0 ), x2t (t0 )
Initial conditions:
x1f (t0 ), x2f (t0 ), P f (t0 )
t  t0
Measurement:
y1o (t1 )  x1t (t1 )  
Gain Matrix
t  t1
x1t (t1 ), x2t (t1 )
Data Assimilation
State estimate:
truth
x1a (t1 ), x2a (t1 ), P a (t1 )
estimate
Model forecast:
x1f (t1 ), x2f (t1 ), P f (t1 )
Bayes
x1t (t0 ), x2t (t0 )
Measurement:
y1o (t1 )  x1t (t1 )  
Initial conditions:
x1f (t0 ), x2f (t0 ), P f (t0 )
t  t0
P
posterior
( x y )  P ( y x) P
obs
prior
 x
Forecast step:
p(x, t0 )  p(x, t1 )
2
p  ( M i p) 1  (Qij p)


t
xi
2 xi x j
Bayes step (update/analysis):
p (x, t1 )  p (x, t1 | y o )
p (x, t1 | y ) 
o
p (y o | x) p (x, t1 )
o
p
(
y
| z ) p ( z, t1 )dz

106
But: computationally prohibitive, state ~
Techniques of Data Assimilation
Deterministic techniques
• Variational methods (3DVAR, 4DVAR)
• Kalman filter
• Ensemble Kalman filter
Requirements:
1. Gaussian
2. Close to linear
Climate:
Statistical techniques
• Particle filtering
• Dynamic Monte-Carlo
• Sampling strategies
Requirement:
Low dimension
•DA in process models
•Understanding historical climate
•Getting the ocean right!
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
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