Climate Change and the Trillion-

Dollar Millenium Maths Problem

Tim PalmerECMWF

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?

Atmospheric Wavenumber Spectra Are Consistent With Those Of A Chaotic Turbulent

Fluid. No spectral gaps.

5/3( )E k k

ECMWF

Edward Lorenz (1917 – 2008 )

bZXYZ

YrXXZY

YXX

Is climate change predictable in a chaotic climate?

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?

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

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 June-August in 2100

Probability of >95th percentile wet June-August in 2100

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...

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 (20th C) PDF

Multi-model (20th C) ensemble PDF

Observed terciles

Observed terciles

33.3%

Lower tercile temperature DJF

<10

10-20

20-45

45-70

>70

%

From IPCC AR4 multi-model ensemble

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...

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...

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 Predictability of a Flow Which Possesses Many Scales of Motion. E.N.Lorenz (1969). Tellus.

The “Real” Butterfly Effect

Increasing scale

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)

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...

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”

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

Wilks, 2004

Redness of noise

Amplitude of noise

“Forecast” Error

Locus of minimum forecast error with non-zero noise

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

Ising Model as a Stochastic Parametrisation of Deep Convection (Khouider et al, 2003)

Above Curie Point

Below Curie Point

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

No StochasticBackscatterNo StochasticBackscatter Stochastic BackscatterStochastic Backscatter

Reduction of systematic error of z500 over North Pacific and

North Atlantic

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

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.

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 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.

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.

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)

Precip error. No stochastic backscatter

Precip error. With stochastic backscatter

El-Niño

rms error

rms spread

Red: no casbs

Blue: with casbs