Climate Change
and the Trillion-Dollar
Millenium Maths
Problem
Tim Palmer
ECMWF
tim.palmer@ecmwf.int
Stern Review: The Economics of
Climate Change
• Unmitigated costs of climate change
equivalent to losing at least 5% of GDP
each year
• In contrast, the costs of reducing
greenhouse gas emissions to avoid the
worst impacts of climate change – can be
limited to around 1% of global GDP each
year
• Global GDP is around 60 trillion dollars
These conclusions
assume our
predictions of future
climate are reliable.
How predictable is climate?
How reliable are predictions of
climate change from the current
generation of climate models?
What are the impediments to
reducing uncertainties in climate
change prediction?
E (k )
k 5/ 3
Atmospheric Wavenumber Spectra Are Consistent With
Those Of A Chaotic Turbulent Fluid. No spectral gaps.
Edward Lorenz (1917 – 2008 )
X  X  Y
Y   XZ  rX  Y
Z  XY  bZ
Is climate change
predictable in a
chaotic climate?
ECMWF
Edward Lorenz (1917 – 2008 )
X   X   Y  f
Y   XZ  rX  Y  f
Z  XY  bZ
Is climate change
predictable in a
chaotic climate?
ECMWF
X
f=0
f=2
f=3
f=4
In the chaotic Lorenz system, forced changes in
the probability distribution of states are
predictable
ECMWF
Probability of >95th percentile warm
June-August in 2100
From an ensemble of climate change integrations.
Weisheimer and Palmer, 2005
Probability of >95th percentile dry JuneAugust in 2100
Probability of >95th percentile wet JuneAugust in 2100
Standard Paradigm for a Weather/Climate Prediction Model


2
   u.  u   g  p    u,
 t

X1 X 2 X 3 ...
...
... X n
Increasing scale
Eg Cloud systems, flow
over small-scale
topography, boundary
layer turbulence..
Local bulk-formula
parametrisation
P  X n ; 
to represent
unresolved processes
Schematic of a Convective Cloud System
50km
….and yet climate models
have substantial biases (in
terms of temperature, winds,
precipitation) when verified
against 20th Century data.
These biases are typically as
large as the climate-change
signal the models are trying
to predict.
Observed
terciles
33.3%
Observed (20th C)
PDF
Observed
terciles
Multi-model (20th C)
ensemble PDF
Lower tercile temperature DJF
%
>70
45-70
20-45
10-20
<10
From IPCC AR4 multi-model ensemble
Standard Paradigm for a Climate Model (100km res)


2
   u.  u   g  p    u,
 t

X1 X 2 X 3 ...
...
... X n
Increasing scale
Bulk-formula
parametrisation of
cloud systems
Standard Paradigm for Increasing Resolution (1km res)


2
   u.  u   g  p    u,
 t

X1 X 2 X 3 ...
... X n
...
... X m
Increasing scale
Bulk-formula
parametrisation
sub-cloud
physics
Higher resolution allows more scales of motion
to be represented by the proper laws of physics,
rather than by empirical parametrisation and
gives better representation of topography and
land/sea demarcation etc.
But running global climate models over century
timescales with 1km grid spacing will require
dedicated multi-petaflop high-performance
computing infrastructure.
How much will accuracy of simulations improve
by increasing resolution to, say, 1 km
resolution?
The “Real” Butterfly Effect
Increasing scale
The Predictability of a Flow Which Possesses Many
Scales of Motion. E.N.Lorenz (1969). Tellus.
Clay Mathematics Millenium
Problems
•
•
•
•
•
•
•
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills Theory
Clay Mathematics Millenium
Problems
•
•
•
•
•
•
•
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills Theory
Navier-Stokes Equations
For smooth initial conditions
and suitably regular
boundary conditions
do there exist smooth,
bounded solutions at all
future times?
The Millenium Navier Stokes problem concerns
the finite-time downward cascade of energy from
large scales to arbitrarily small scales.
It is closely related to the Real Butterfly Effect
which concerns the finite time upward cascade
of error to large scales, from arbitrarily small
scales.
Ie moving parametrisation error from cloud
scales to sub-cloud scales may not improve
simulation by as much as we would like!
Are there alternative methodologies to the “brute
force” method of increasing resolution?
An stochastic-dynamic paradigm for
climate models (Palmer, 2001)
X1 X 2 X 3 ...
... X n
Increasing scale
Coupled
over a
range of
scales
Computationally-cheap
nonlinear stochastic-dynamic
model, providing specific
realisations of sub-grid
motions rather than
ensemble-mean sub-grid
effects
Lorenz, 96
N
X i   X i 2 X i 1  X i 1 X i 1  X i  F  bc  x j ,i
j 1
x j ,i  cbx j 1,i x j  2,i  cbx j 1,i x j 1,i  cx j ,i  bc X i
Ed Lorenz:
“Predictability – a
problem partly
solved”
Model L96 in the form
X i   X i 2 X i 1  X i 1 X i 1  X i  F  Pi
Pi  a0  a1 X i  a2 X  a3 X  a4 X  e
2
i
3
i
4
i
Deterministic
Stochastic
parametrisation parametrisation
“Forecast” Error
Locus of
minimum
forecast
error with
non-zero
noise
Amplitude
of noise
Redness of noise
Wilks, 2004
Stochastic-Dynamic Cellular Automata
Eg for convection
EG Probability of an “on”cell proportional to CAPE and
number of adjacent “on” cells – “on” cells feedback to the
resolved flow
(Palmer; 1997, 2001)
Ising Model as a Stochastic Parametrisation of Deep Convection
(Khouider et al, 2003)
Above Curie Point
Below Curie Point
Cellular Automaton Stochastic
Backscatter Scheme (CASBS)
smooth
scale
Cellular Automaton state
streamfunction forcing shape 
function

  ( x, y )  rD
t
D = sub-grid energy dissipation due to numerical diffusion,
mountain drag and convection
G.Shutts, 2005
r = backscatter parameter
Reduction of systematic error of z500 over North Pacific and
North Atlantic
No StochasticBackscatter
Stochastic Backscatter
Impact of stochastic backscatter is similar to an increase in
horizontal resolution
200km
40km
T95L91 CTRL
T511L91 High Resolution
Z500 Difference eto4-er40 (12-3 1990-2005)
Z500 Difference eut3-er40 (12-3 1990-2005)
16
2
6
16
12
6
2
12
-2
10
6
10
2
8
2
2
-2
-2
6
-2
-6
4
2
-2
-2
-2
-2
-4
2
-2
2
-6
-2
-4
-2
-6
-2
-6
-8
-8
-2
2
-10
-12
2
-16
4
-2
-2
-2
-6
-6
8
2
6
-10
2
2
-2
2
-10
-12
2
-16
Better simulation of large-scale
weather regimes with stochastic
parametrisations.
Eg ball bearing in potential well.

Without smallscale “noise”, this
“westerly-flow”
regime is too
dominant

Without small-scale
“noise”, this blocked
anticyclone regime
occurs too infrequently
Advantages of Stochastic Weather
Climate Models
• Capable of emulating some of the impact of
increased resolution at significantly reduced
cost.
• Explicit representations of forecast
uncertainty
Conclusions
• Climate change is “the defining issue of our age” (Ban
Ki-moon). Reliable climate predictions are essential to
guide mitigation and regional adaptation strategies
• Climate prediction is amongst the most computationallydemanding problems in science. All climate models have
significant biases in simulating climate.
• Dedicated multi-petaflop computing is needed to allow
resolution to be increased from 100km to 1km grids.
However, there is no theoretical understanding of how
the accuracy of climate simulations will converge with
increased model resolution.
• Stochastic representations of unresolved processes
offers a promising new approach to improve the realism
of climate simulations without substantially increasing
computational cost. Importing ideas from other areas of
physics (eg Ising models) may be useful.
If an Earth-System model purports
to be a comprehensive tool for
predicting climate, it should be
capable of predicting the
uncertainty in its predictions.
The governing equations of EarthSystem models should be
inherently probabilistic.
Weather Regimes: Impact of Stochastic Physics
(Jung et al, 2006)
Deterministic
model
Stochastic
model
31.0%
37.5%
33.7%
27.9%
33.8%
27.9%
29.8%
34.6%
35.2%
34.6%
36.5%
37.5%
Precip error.
No
stochastic
backscatter
Precip error.
With
stochastic
backscatter
El-Niño
Red: no casbs
Blue: with casbs
rms error
rms
spread