Climate Change and the Trillion-Dollar Millenium Maths Problem

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Climate Change and the
Trillion-Dollar Millenium Maths Problem

• Tim Palmer
• ECMWF
• tim.palmer_at_ecmwf.int

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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.
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Edward Lorenz (1917 2008 )
Is climate change predictable in a chaotic
climate?
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Edward Lorenz (1917 2008 )
Is climate change predictable in a chaotic
climate?
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X
f2
f0
f4
f3
In the chaotic Lorenz system, forced changes in
the probability distribution of states are
predictable
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Probability of gt95th percentile warm June-August
in 2100
From an ensemble of climate change integrations.
Weisheimer and Palmer, 2005
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Probability of gt95th percentile dry June-August
in 2100
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Probability of gt95th percentile wet June-August
in 2100
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Standard Paradigm for a Weather/Climate
Prediction Model
Increasing scale
Local bulk-formula parametrisation

to represent unresolved
processes
Eg Cloud systems, flow over small-scale
topography, boundary layer turbulence..
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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 terciles
33.3
Observed (20th C) PDF
Observed terciles
Multi-model (20th C) ensemble PDF
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Lower tercile temperature DJF
From IPCC AR4 multi-model ensemble
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Standard Paradigm for a Climate Model (100km res)
Increasing scale
Bulk-formula parametrisation of cloud systems
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Standard Paradigm for Increasing Resolution (1km
res)
Increasing scale
Bulk-formula parametrisation sub-cloud physics
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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 Real Butterfly Effect
Increasing scale
The Predictability of a Flow Which Possesses Many
Scales of Motion. E.N.Lorenz (1969). Tellus.
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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)
Increasing scale
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
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Lorenz, 96
Ed Lorenz Predictability a problem partly
solved
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Model L96 in the form
Deterministic parametrisation
Stochastic parametrisation
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Forecast Error
Locus of minimum forecast error with non-zero
noise
Amplitude of noise
Redness of noise
Wilks, 2004
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Stochastic-Dynamic Cellular Automata
Eg for convection
EG Probability of an oncell proportional to
CAPE and number of adjacent on cells on
cells feedback to the resolved flow
(Palmer 1997, 2001)
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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 Backscatter Scheme
(CASBS)
smooth
scale
streamfunction forcing shape function
Cellular Automaton state

• D sub-grid energy dissipation due to numerical
  diffusion, mountain drag and convection
• r backscatter parameter

G.Shutts, 2005
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Reduction of systematic error of z500 over North
Pacific and North Atlantic
No StochasticBackscatter
Stochastic Backscatter
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Impact of stochastic backscatter is similar to an
increase in horizontal resolution
200km
40km
T95L91 CTRL
T511L91 High Resolution
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Better simulation of large-scale weather regimes
with stochastic parametrisations.
Eg ball bearing in potential well.
?
Without small-scale noise, this blocked
anticyclone regime occurs too infrequently
Without small-scale noise, this westerly-flow
regime is too dominant
?
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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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Weather Regimes Impact of Stochastic Physics
(Jung et al, 2006)
Deterministic model
Stochastic model
37.5 33.7
27.9
31.0
27.9 29.8
34.6
33.8
34.6 36.5
37.5
35.2
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Precip error. No stochastic backscatter
Precip error. With stochastic backscatter
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El-Niño
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Red no casbs Blue with casbs
rms error
rms spread