climate change and the trillion-dollar millenium maths problem tim palmer ecmwf [email protected]
TRANSCRIPT
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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
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These conclusions assume our
predictions of future climate are reliable.
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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?
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Atmospheric Wavenumber Spectra Are Consistent With Those Of A Chaotic Turbulent
Fluid. No spectral gaps.
5/3( )E k k
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ECMWF
Edward Lorenz (1917 – 2008 )
bZXYZ
YrXXZY
YXX
Is climate change predictable in a chaotic climate?
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ECMWF
Edward Lorenz (1917 – 2008 )
X X Y
Y XZ rX Y
Z XY b
f
Z
f
Is climate change predictable in a chaotic climate?
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ECMWF
f=0 f=2
f=3 f=4
In the chaotic Lorenz system, forced changes in the probability distribution of states are
predictable
X
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Probability of >95th percentile warm June-August in 2100
From an ensemble of climate change integrations. Weisheimer and Palmer, 2005
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Probability of >95th percentile dry June-August in 2100
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Probability of >95th percentile wet June-August in 2100
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Standard Paradigm for a Weather/Climate Prediction Model
Local bulk-formula parametrisation
to represent unresolved processes
Increasing scale
;nP X Eg Cloud systems, flow over small-scale topography, boundary layer turbulence..
1X ...
2. , ...pt
u u g u
2X 3X nX...
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Schematic of a Convective Cloud System
50km
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….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.
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Observed (20th C) PDF
Multi-model (20th C) ensemble PDF
Observed terciles
Observed terciles
33.3%
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Lower tercile temperature DJF
<10
10-20
20-45
45-70
>70
%
From IPCC AR4 multi-model ensemble
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Standard Paradigm for a Climate Model (100km res)
Bulk-formula parametrisation of cloud systems
Increasing scale
1X ...
2. , ...pt
u u g u
2X 3X nX...
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Standard Paradigm for Increasing Resolution (1km res)
Bulk-formula parametrisation sub-cloud physics
Increasing scale
1X ...
2. , ...pt
u u g u
2X 3X nX...mX...
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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?
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The Predictability of a Flow Which Possesses Many Scales of Motion. E.N.Lorenz (1969). Tellus.
The “Real” Butterfly Effect
Increasing scale
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Clay Mathematics Millenium Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory
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Clay Mathematics Millenium Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory
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Navier-Stokes Equations
For smooth initial conditions
and suitably regular boundary conditions
do there exist smooth, bounded solutions at all future times?
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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!
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Are there alternative methodologies to the “brute force” method of increasing resolution?
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An stochastic-dynamic paradigm for climate models (Palmer, 2001)
Computationally-cheap nonlinear stochastic-dynamic model, providing specific realisations of sub-grid motions rather than ensemble-mean sub-grid effects
Coupled over a range of scales
Increasing scale
1X ...2X 3X nX...
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2 1 1 1 ,1
, 1, 2, 1, 1, ,
Nc
i i i i i i j ibj
cj i j i j i j i j i j i ib
X X X X X X F x
x cbx x cbx x cx X
Lorenz, 96
Ed Lorenz: “Predictability – a problem partly solved”
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Model L96 in the form
2 1 1 1
2 3 40 1 2 3 4
i i i i i i i
i i i i i
X X X X X X F P
P a a X a X a X a X e
Deterministic parametrisation
Stochastic parametrisation
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Wilks, 2004
Redness of noise
Amplitude of noise
“Forecast” Error
Locus of minimum forecast error with non-zero noise
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Stochastic-Dynamic Cellular Automata
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)
Eg for convection
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Ising Model as a Stochastic Parametrisation of Deep Convection (Khouider et al, 2003)
Above Curie Point
Below Curie Point
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Cellular Automaton Stochastic Cellular Automaton Stochastic Backscatter Scheme (CASBS)Backscatter Scheme (CASBS)
D = sub-grid energy dissipation due to numerical diffusion, mountain drag and convection
r = backscatter parameter
Cellular Automaton state streamfunction forcing shape function
( , )x y rDt
smooth
scale
G.Shutts, 2005
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No StochasticBackscatterNo StochasticBackscatter Stochastic BackscatterStochastic Backscatter
Reduction of systematic error of z500 over North Pacific and
North Atlantic
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T95L91 CTRLT95L91 CTRL T511L91 High ResolutionT511L91 High Resolution
Impact of stochastic backscatter is similar to an increase in
horizontal resolution
2
6
Z500 Difference eto4-er40 (12-3 1990-2005)
-16
-12
-10
-8
-6
-4
-22
4
6
8
10
12
162
Z500 Difference eut3-er40 (12-3 1990-2005)
-16
-12
-10
-8
-6
-4
-22
4
6
8
10
12
16
200km 40km
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Without small-scale “noise”, this “westerly-flow” regime is too dominant
Without small-scale “noise”, this blocked anticyclone regime occurs too infrequently
Eg ball bearing in potential well.
Better simulation of large-scale weather regimes with
stochastic parametrisations.
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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
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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 computationally-demanding 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.
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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 Earth-System models should be inherently probabilistic.
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27.9%
37.5%
34.6%
31.0%
33.8%
35.2%
37.5% 33.7%
27.9% 29.8%
34.6% 36.5%
Deterministic model
Stochastic model
Weather Regimes: Impact of Stochastic Physics (Jung et al, 2006)
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Precip error. No stochastic backscatter
Precip error. With stochastic backscatter
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El-Niño
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rms error
rms spread
Red: no casbs
Blue: with casbs