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  • Uncertainties in weather and climate prediction

    Henk Dijkstra Institute 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

    dt (0) = 0

    d2θ

    dt2 +

    g

    L θ = 0

    θ(t) = θ0 cos( √

    g/L t)

    | θ |� 1

    θ

    mg

    : mg sin θ

    mL d2θ

    dt2 = −mg sin θ

  • Creative aspect 1: Phase space

    Geometry of motion!

    Gibbs Boltzmann

    Poincare

    d2θ

    dt2 +

    g

    L θ = 0

    dx

    dt = y

    dy

    dt = − g

    L x

    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 = lim t→∞

    1

    t ln | 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 − 109

  • 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 kinetic energy

    in press (2015)

  • Creative aspect 3: Ensemble forecasting

    Forecast time

    Te m

    pe ra

    tu re

    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)

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

    UK

    Control Analysis Ensemble

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

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

    UK

    Control Analysis Ensemble

    ECMWF ensemble forecast - Air temperature Date: 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 80 100 0

    0.2

    0.4

    0.6

    0.8

    1

    Lead time [years from 2000]

    F ra

    ct io

    na l u

    nc er

    ta in

    ty

    Internal variability

    Scenario

    Model

    Total

    Global, decadal mean surface air temperature

  • Summary Chaos 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