Josh Korotky SOOWFO PBZ

Josh Korotky

NOAA/WFO Pittsburgh

NROW Nov 1, 2005

Edward Lorenz

Workshop Agenda

Chaos and Predictability

Sources of Forecast Error

Optimizing Predictability with Ensemble Methods

Predictability and the Extended Range Forecast

The Roots of Chaos Theory

Newton and Determinism

Determinism: the philosophical belief of absolute cause and effect Every event or action is the

predictable result of preceding events and actions

Newton's laws are dynamical laws They connect the numerical

values of measurements at a given time to their values at a later or earlier time.

The measurements in Newton's laws typically include the position, speed, and direction of motion of all the objects in the system, and the strength and direction of any forces on these objects, at any given time in the history of the system.

French mathematician and physicist…many elemental contributions to mathematics, physics, and celestial mechanics.

In his research on the three body problem, Poincaré became the first person to discover a chaotic deterministic system and laid the foundations of modern chaos theory.

The 3-body problem: given the initial positions, masses, and velocities of 3 bodies, find their subsequent motions using classical (deterministic) mechanics, i.e. Newton’s laws of motion and Newton’s laws of gravity.

Poincaré’s findings: The evolution of a 3-body system is often chaotic; a slight change in one body's initial position might lead to a radically different later state. If the slight change isn't detectable by our measuring instruments, then we won't be able to predict which final state will occur

Henri Poincaré 1854 - 1912

Edward Lorenz (1917 – )

Small changes in the initial state of a system can cause major changes in the final state of the system due to non-linear feedback

“… one flap of a sea-gull’s wing may forever change the future course of the weather” (Lorenz, 1963)

The Lorenz Experiment

Lorenz was doing experiments using a simple system of equations to model convection in the atmosphere

He reran a previous experiment with 3 (.506) vs. 6 (.506127) digit accuracy (computer printout vs. internal memory), expecting to find exactly the same results

He found instead that the new forecast diverged from the previous forecast…and eventually showed a completely different solution

From nearly the same starting point, weather patterns grew farther and farther apart until all resemblance disappeared

Lorenz found the mechanism of deterministic chaos: simply-formulated systems with only a few variables can display highly complex and unpredictable behavior.

He found that slight differences in initial conditions had profound effects on the outcome of the whole system. This was one of the first clear demonstrations of sensitive dependence on initial conditions. Equally important… Lorenz showed that this occurred in a simple, but physically relevant model.

Chaos

Forecasts diverge

Forecast Time

Linear Regime Nonlinear Regime

Single Forecast Range

Forecasts Diverge

Illustration of Chaos (population growth)

Choose a number between -2 and 2 (say 0.4)

This is the “initial condition” X1

Square X1 and subtract 2 = X2

Continue to apply same rule and generate a sequence of numbers

(X2)2 - 2 …..

Blue line is sequence X1, X2, X3 …

Generate another sequence starting with 0.4001

Red dotted line is new sequence

We used deterministic rules to generate the two sequences…but they become completely uncorrelated after about the 20th iteration

This illustrates chaos…a small initial difference causes completely different solutions

The Lorenz Attractor

The Lorenz Attractor is a solution to a set of differential equations which describe the 2D flow of fluid in a simple rectangular box which is heated along the bottom.

This simple model was intended to simulate medium-scale atmospheric convection

The Lorenz attractor is a graphical representation of the time variation of three variables X(t), Y(t) and Z(t), coupled by non-linear evolution equations. In the figure, a single solution is shown evolving from an initial condition (X0,Y0,Z0)

Start two solutions running simultaneously from initial conditions separated by very small differences (e.g., dX0, dY0, dZ0 ~ 0.01)

This seemingly insignificant difference in the initial conditions will become amplified over time, until the two trajectories evolve in an uncorrelated fashion

Chaos and NWP

Weather forecasts lose skill because: Chaos…small errors in the initial conditions of a forecast grow

rapidly (initial condition uncertainty and sensitive dependence on initial conditions)

Numerical models only approximate the laws of physics (model uncertainty)

Single Model vs. Ensemble Approach to NWP

60h Eta Forecast valid 00 UTC 27 Dec 2004

“Truth” 00 UTC 27 Dec 2004

• Ignores forecast uncertainty• Potentially misleading• Oversells forecast capability

Single Model NWPSingle Model NWP

Nonlinear Regime

Forecast Time

Linear Regime

NWP: Linear and Non-linear Regimes

Linear Regime …small initial errors grow slowly (linear error growth)

Predictability maintained

Non-linear Regime …small initial errors amplify rapidly, resulting in very different forecasts over time

Predictability problematic

Weather prediction can best be understood as the time evolution of an appropriate probability density function (PDF). Ensembles offer the only reasonable way to predict the PDF beyond linear error growth

The Figure: The deterministic approach to NWP provides one single forecast (blue line) for the “true” time evolution of the system (red line). The ensemble approach estimates the PDF of forecast states (magenta shapes). Ideally, the ensemble PDF includes the true state of the atmosphere as a possible forecast outcome

Ensemble Prediction and the Lorenz Attractor

Two wings can be seen as two different weather regimes

Mild and wet (left) Cold and dry (right)

Initial circle = initial state estimate

Panel 1: small initial state errors (from the truth) do not have major effect on predictability.

High confidence in cold and dry extended forecast through 10 time steps

Panel 2: Confidence for limited time (5 or 6 time steps)…confidence in short term (mild and wet)…equal chances long term (need probability)

Panel 3: Very unpredictable…little confidence after 3 or 4 time steps

1 2

3

Predictable Less Predictable

Unpredictable

Very Predictable Somewhat Predictable

• A single model solution becomes unskillful in non-linear regime

• Ensembles extend predictability from the point where forecasts diverge until chaos dominates

• Ensembles can provide information on forecast uncertainty

• Ensembles offer the only reasonable way to predict the PDF beyond linear error growth

Ensembles Extend PredictabilityEnsembles Extend Predictability

Chaos

Forecasts diverge

Forecast Time

Linear Regime Nonlinear Regime

Single Forecast Range

Forecasts Diverge

• When predictability is limited, probability forecasting frequently extends the utility of forecasts.

• Ensemble prediction allows us to assess the relative probabilities of different outcomes. While detailed predictions of daily weather more than 3-5 days ahead are not generally practical, ensemble prediction allows us to issue some useful probabilities up out to 7-10 days.

Predictability and the Extended Range Forecast

Most of the time the atmosphere behaves like panel 2

We can predict with confidence for a few days but need to use probabilities afterward

Sometimes we have situations like panel 1

We can forecast with confidence for several days

There are times when the atmosphere shows sensitivity to initial conditions within a day or two (panel 3)

Especially when mesoscale processes (e.g., convection) dominate

Predictable Less Predictable

1 2

3

Unpredictable

Predictability and the Extended Range Forecast

Chaos imposes a limit to predictability… depends on what we are trying to predict

We can generally forecast synoptic scale systems reasonably well up to around 3 days in advance… but lots of variability around this average figure

Predictability varies according to the situation (flow dependence).

Predictability and the Extended Range Forecast

At times we can predict the general weather pattern with confidence up to a week in advance

Example: there is a large slow-moving high pressure system over the region

At other times significant errors can occur only one or two days into the forecast Advances in NWP are extending forecast utility generally, but some flows are

inherently unpredictable Some of the most difficult and unpredictable situations are associated with the

rapid development of major storms, so it is important to be able to assess the uncertainty in such situations

We can usually predict general weather patterns up to 3 days ahead, but predictability for detailed local weather (rainfall or fog formation) is limited

We may be able to predict the meteorological conditions favoring the formation of showers a few days ahead, but we may only be able to predict whether a particular location will get a shower a few hours (or less) in advance

NWP Skill as a Function of Scale

Feature/Variable < Day1 Days 1-2 Days 3-5 Days 6-7

Hemispheric flow transitions Excellent Excellent Very Good Good

Cyclone life cycle Excellent Very Good Fair-Good Low skill

Fronts Excellent Good Fair ----

Mesoscale banded structuresConvective clusters

Good Fair ---- ----

Temp / wind Excellent Very Good Skill with max/min

QPF/ mean clouds Very Good Good Some skill in 5-10 day QPF

Predictability falls off as a function of scale

Large scale features (planetary waves) may be predictable up to a week in advance

Small baroclinic systems (fronts) are well forecasted up to day 2, cyclonic systems up to day 4

Summary

Chaos imposes a limit on predictability

Predictability falls off (sometimes rapidly) as a function of scale over time

Ensemble NWP optimizes predictability for all scales, and extends the utility of forecasts…especially at extended ranges (days 4-7)

WFOs can get their greatest bang (skill) for the buck (effort) by using the HPC extended grids. The NDFD should indicate the evolution and movement of weather systems to be relevant

Grid editing for extended range forecasts should be minimal (e.g., mesoscale responses to large scale flow pattern)…avoid collaboration based on a forecasters single model “hunch”. We need to maintain a most likely large scale pattern

WFOs need to concentrate their efforts on the short term (days 1-3), where predictability is highest and customer needs are greatest

The future…HPC needs to supply WFOs with probabilistic information for local AFD input (deterministic forecast based on ensemble methods, with local forecasters explaining PDF characteristics, confidence level, etc.)

The role of WFOs in the extended range forecast?