Uncertainties in weather and climate prediction

Henk DijkstraInstitute for Marine and Atmospheric research Utrecht

Utrecht University

Weather Forecasts Climate projections

A classical mechanical system: the pendulum (in vacuum)

Given initial position and velocity -> determine future position

θ(0) = θ0

dθ

dt(0) = 0

d2θ

dt2+

g

Lθ = 0

θ(t) = θ0 cos(√

g/L t)

| θ |� 1

θ

mg

:mg sin θ

mLd2θ

dt2= −mg sin θ

Creative aspect 1: Phase space

Geometry of motion!

GibbsBoltzmann

Poincare

d2θ

dt2+

g

Lθ = 0

dx

dt= y

dy

dt= − g

Lx

x = θ ; y =dθ

dt

Nonlinear mechanical (fluid) systems

The Lorenz system

Lorenz model

demo

dx

dt= −c(x − y)

dy

dt= ax − y − xz

dz

dt= b(xy − z)

~ vertical temperature difference

Edward Lorenz (1917-2008)

a

Lorenz attractor

Creative aspect 2: Sensitivity to initial conditions

Trajectory

Lyapunov exponent

0.9056, 0, -14.5723Lorenz system:

λ > 0 → chaotic motion

d(t) = x′(t)− x(t)

λi = limt→∞

1

tln | di(t)

di(0)|

Numerical Weather Prediction Model

Grid: N x M x L

# Observables (temperature, humidity, velocities, etc.): k

Dimension phase space: d = k x N x M x L

Typically: d = 105 − 10

9

Origin of the ‘plume’ in weather forecasts

Numerical weather prediction models: many Lyapunov exponents > 0

Daily mean temperature January 1940 in `grid box’ the Netherlands

Weather prediction

Lorenz (1969): … one flap of a sea-gull’s wing may forever change the

future course of the weather

Leith (1984): … even talking about the weather can

change the weather!

Examples of finite-time error growth on the

Lorenz attractor for three probabilistic predictions

starting from different points on the attractor.

Error growth in the Lorenz attractor

Perron-Frobenius or transfer operator

Lorenz model with ‘noise’

σ : noise amplitude X

Z

= 0.1 density of trajectories

Regime transitions in atmospheric flows

Transition through preferred pathways increases predictive skill

Atmospheric pressure anomalies (hPa)

Mean kinetic energy

Eddy kineticenergy

in press (2015)

Creative aspect 3: Ensemble forecasting

Forecast time

Tem

pera

ture

Complete description of weather prediction in terms of a Probability Density Function (PDF)

Initial condition Forecast

Numbers of observational items assimilated over a 24 hour period on 13 February 2006

Starting a forecast: The initial conditions

Flow dependence of forecast errors

If the forecasts are coherent (small spread) the atmosphere is in a more predictable state than if the forecasts diverge (large spread)

34

30

28

26

24

22

20

18

16

14

12

10

0 1 2 3 4 5 6 7 8 9 10Forecast day

UK

Control Analysis Ensemble

ECMWF ensemble forecast - Air temperatureDate: 26/06/1994 London Lat: 51.5 Long: 0

30

28

26

24

22

20

18

16

14

12

10

8

0 1 2 3 4 5 6 7 8 9 10Forecast day

UK

Control Analysis Ensemble

ECMWF ensemble forecast - Air temperatureDate: 26/06/1995 London Lat: 51.5 Long: 0

26th June 1995 26th June 1994

Example of 66 h probabilistic forecast for 15–16 October 1987.

Slingo J , and Palmer T Phil. Trans. R. Soc. A 2011;369:4751-4767

Processes limiting predictability: formation of H and L pressure systems and their interaction

Surface pressure maps UK, North Sea

Predictability limits

Statistics of ensemble mean forecast error (r.m.s.e.; solid line) and ensemble spread (dotted line) in Northern

Hemisphere systems

Predictability horizon

How about climate projections?

Climate is what you expect, weather is what you get!

Weather: state of the climate system at a given time and place

Climate: statistics of weather conditions over a decade or more

Representative concentration pathways

IPCC, Chapter 11, 2013

Scenario:

Climate models

Community Earth System Model

ocean/sea ice grid box: 10 km

atmosphere/land grid box: 50 km

d ~ 109

Tiled Panel Display Visualization (BBG 611)

Climate projection results

Back to the basics … Pendulum in air

vacuum

air

air

`weather’ of the pendulum: small scale (chaotic) motions induced by interaction with the air

`climate’ of the pendulum: longer time scale (regular) motion controlled by gravity

Also the `climate’ state can display chaotic behavior (but with a very different

Lyapunov exponent than for the weather)

The double pendulum

Projection uncertainties

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Lead time [years from 2000]

Fra

ctio

nal u

ncer

tain

ty

Internal variability

Scenario

Model

Total

Global, decadal mean surface air temperature

SummaryChaos plays a very different role in uncertainties of

weather forecasts and climate projections

Future weather forecasts:- relevant processes are instabilities of the large-scale atmospheric circulation with typical time scales of up to 5 days- limited prediction skill beyond 10 days

Future climate change:- relevant processes are several large-scale feedbacks in the climate system associated with the radiation balance- projection skill is limited by emission scenario

Creative aspect: Phase Space; connecting geometry and motion

Creative aspects: Sensitivity to initial conditions; Ensemble forecasting

Creative aspect: Predictability horizons connected to specific physical processes